Hyperfine Interactions 123/124 (1999) 127­274 127 Coherent -ray optics J.P. Hannon and G.T. Trammell Physics Department, Rice University, Houston, TX 77005-1892, USA With the advent of high brightness synchrotron radiation sources, an important new field has been opened up involving coherent nuclear excitations induced by synchrotron radiation pulses traversing a piece of matter. We review the theory of coherent resonant -ray optics, including some of the interesting new phenomena which occur when systems of nuclei are excited by synchrotron radiation pulses, such as the creation of nuclear exciton states, superradiant and subradiant decay, spatially coherent quantum beats, and temporal Pendell¨osung. We also discuss the relation between the nuclear exciton states and multi- photon Dicke superradiance and -ray lasers, and comment on neutron phasors and neutron superradiance in resonant neutron optics. The interesting features of coherent enhancement, superradiant decay, and dynamical beats are discussed from the fundamental perspective of the radiative normal modes of a system of nuclear resonators. AMS subject classification: PACS numbers: 73.20.Hb, 31.50.+w, 79.20.Rf, 79.20.-m 1. Introduction For most M¨ossbauer transitions, the -ray wavelength ( 0 0.1­1.0 A ) is well suited for diffraction experiments. Tuned to resonance, the scattering amplitude f is proportional to 0, and can be very large, 400r0 for the 14.4 keV 57Fe transition. The linewidths are exceedingly sharp ( 10-9­10-6 eV), allowing the easy variation of the phase and amplitude of the resonantly scattered wave by Doppler shifting, with negligible variation of the nonresonant electronically scattered wave. Correspondingly, the time response of the resonant nuclear scattering is very long, 10-9­10-6 s (com- pared to 10-15 s for a 1 eV width atomic resonance, and 10-14 s for characteristic crystal vibration times). The signal emitted by a "nuclear oscillator" over a duration of 10-7 s has a longitudinal coherence length lc c 30 m, or a "quality factor" Q 1012, the number of perfectly spaced radiation crests in the signal, comparable to the highest precision optical lasers. In magnetic crystals, and in the presence of crystal electric fields, there is hyperfine splitting of the nuclear resonances, and the amplitude and polarization of the resonantly scattered photons are strongly dependent upon the directions of the local fields. M¨ossbauer transitions also occur with well-defined multi- polarity E1, M1, E2 (and in a few cases M1­E2), in contrast to the electronic response of atoms which occurs as a hierarchy of multipole contributions E1 E2 M1, etc. These unique features ­ large scattering amplitudes, the long scattering time, the sharp variation with frequency near resonance, the sensitivity to crystalline electric J.C. Baltzer AG, Science Publishers 128 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 and magnetic fields, and the well defined multipolarities ­ give rise to important appli- cations for chemical and magnetic structure determinations, new possibilities for long coherence length X-ray interferometry, time-resolved spectroscopy on the nanosecond scale, and interesting optical effects such as strong Faraday-type effects and higher multipole Borrmann effects. Because of the low brightness of most radioactive M¨ossbauer sources, however, until recently nearly all M¨ossbauer experiments involved only resonance absorption, which, while valuable, does not exhibit to the fullest the interesting effects due to coherence in the emission or scattering of -rays from a system of nuclei. This situation has changed dramatically with the advent of high brightness synchrotron radiation sources, and an active field of coherent -ray optics utilizing nuclear excitation by synchrotron radiation pulses has emerged. We review the theory of coherent resonant -ray optics, and discuss some of the interesting new phenomena which occur when systems of nuclei are excited by syn- chrotron radiation pulses, such as the creation of nuclear exciton states, superradiant and subradiant decay, spatially coherent quantum beats, and temporal Pendell¨osung. We also discuss the relation between the nuclear exciton states and multi-photon Dicke superradiance and -ray lasers, and comment on neutron phasors and neutron super- radiance in resonant neutron optics. In the final section, we discuss one of the most unique and fundamental aspects of M¨ossbauer -ray optics ­ collective nuclear states established by the radiative interaction between the nuclei. 2. Simple features of resonant -ray scattering In this section we discuss some of the simple features of resonant nuclear scat- tering, and make comparisons with atomic X-ray scattering. 2.1. Atomic scattering The usual X-ray scattering from an atom consists of three contributions: the direct A2 interaction, which gives the "free electron" Thomson scattering f0, and two virtual excitation/deexcitation diagrams (with photon absorption preceding or following emission), which give the "anomalous scattering" f + if , Ze2 f ¯ 0 R ¯ 0 R e = - F ( ) + + mec2 (E (E n n - E0) - - i T/2 n n - E0) + = f0 + f + if . (2.1) These contributions are indicated schematically in figure 1. In the anomalous dispersion contribution, R/ is the rate for the radiative decay n 0, with R proportional to the square of the electric dipole moment, R | n|x| 0 |2, and T/ is the total decay rate of the intermediate excited state n due to all possible radiative and nonradiative decay processes (such as the Auger emission III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 129 Figure 1. X-ray scattering contributions. Figure 2. Cu K-edge. process, in which the excitation energy of n is transferred via the Coulomb interaction to an outer electron which is ejected into the continuum). Typically, R 0.01­0.1 eV, while T 1­10 eV. (For simplicity, we have omitted the polarization dependence, and a statistical/angular momentum factor 1. Also, 0 is the resonance wavelength and ¯ 0 0 2 .) The Thomson scattering f0 Zr0, where r0 = e2/mec2 is the "classical elec- tron radius", while the resonant contribution to the anomalous scattering has a peak amplitude ¯ 0( R/ T), which, as for any resonant scattering, is independent of r0, and instead is determined entirely by the resonance wavelength 0 and the branching ratio R/ T. Typically R/ T 10-3­10-2, while ¯ 0/r0 5 × 103 for 1 A radia- tion, so that the peak value of (f + if ) 1 - 10r0 near an absorption edge. Since there is a "continuum" of resonators above the absorption edge, rather than a sharp resonance response, (f + if ) exhibits a "distributed" resonance response, spread over 10­100 eV, such as the well-known Cu K-edge spectrum shown in figure 2. 2.2. Nuclear scattering The scattering of a photon by a nucleus is determined by the same three diagrams, where the A2 interaction is now for the protons of the nucleus, and the n are the 130 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 virtual excited states of the nucleus, so again Ze2 f ¯ 0 ¯ 0 n = - + + , (2.2) mpc2 (E (E n n - E0) - - i /2 n n - E0) + where the first term is the nuclear Thomson scattering, the second term is the resonant -ray scattering, and the third term is the nonresonant second order contribution. For nuclear scattering, we use the notation R and T . Now, however, the relative magnitude of the three contributions and the nature of the response are very different from the atomic scattering for two reasons: First, the nuclear Thomson scattering involves the proton mass mp rather than the electron mass me, and hence, is down by me mp 1/2000, so the nuclear Thomson scattering is typically 0.01r0 and can usually be ignored. Secondly, the nuclear resonances are dramatically sharper than the atomic resonances, typically, 10-8 eV (and 10-9 eV) in contrast to T 1­10 eV for atomic excited states. As a consequence, there is no appreciable scattering from the nucleus unless the incident photon energy is tuned to within a few widths 10-8 eV of a nuclear resonance, in which case 1 f n ¯ 0 , (x - i) where x = [(En - E0) - ]/( /2) is the deviation from resonance in units of the halfwidth. At resonance, the scattering amplitude peaks to f n i¯ 0 , which can be very large. For the 14.4 keV 57Fe transition, / 1/9, and |fn| 440r0 at resonance! Also, in contrast to the continuum distribution of oscillators above an atomic absorption edge, the nuclear excited states n are separated by 10's or 100's of keV. As a consequence, the nuclear scattering exhibits a strikingly simple Lorentzian response, with f n = ¯ 0( / )/(x2 + 1) peaking to ¯ 0( / ) 440r0 at resonance, and f n = ¯ 0( / )x/(x2 + 1) varying from +220r0 to -220r0 as is tuned from - /2 + /2 about resonance, as shown in figure 3. The narrow linewidths ( 10-9­10-6 eV) allow the easy variation of the phase and amplitude of the resonantly scattered wave by Doppler shifting, with negligible variation of the electronic scattering amplitude fe. Complementary to the exceedingly sharp frequency response, the time response of the resonant nuclear scattering is very slow, 10-9­10-6 s (compared to 10-15 s for a 1 eV width atomic resonance, and 10-14 s for characteristic crystal vibration times). The signal emitted by a "nuclear oscillator" over a duration of 10-7 s has a longitudinal coherence length lc = c 30 m, or a "quality factor" Q 1012, the number of perfectly spaced radiation crests in the signal, comparable to that of the highest precision optical lasers. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 131 Figure 3. Lorentzian response of 57Fe. The striking difference in the sharpness of a nuclear -ray resonance compared to an atomic X-ray resonance is primarily a size effect: 2 R n|x| 0 (radius)2, and thus, 2 R(nucleus) R(nucleus) 10-8. R(atom) R(atom) That is, the dipole moment for a nuclear transition is about four orders of magnitude smaller than that for an atomic transition, leading to a radiative decay rate eight orders of magnitude smaller. In magnetic crystals, and in the presence of crystal electric fields, there is hy- perfine splitting of the nuclear resonances, and the amplitude and polarization of the resonantly scattered photons are strongly dependent upon the directions of the local fields. M¨ossbauer transitions also occur with well-defined multipolarity E1, M1, E2 (and in a few cases, M1­E2), in contrast to the electronic response of atoms, which occurs as a hierarchy of multipole contributions E1 E2 M1, etc. The best known M¨ossbauer resonance is the 14.4 keV ( 0 = 0.86 A ) transi- tion of 57Fe, indicated schematically in figure 4. This is an M1 transition from a ground state j0 = 1/2 to an excited state j1 = 3/2. The lifetime of the excited state is = / = 141 ns. In a magnetic material the spin polarized 3d elec- trons induce, via exchange, a net spin polarization of the s-electron density at the nucleus, which, by the Fermi contact interaction, gives a strong magnetic field at the nucleus, Bint 30­40 T, resulting in Zeeman splitting of the ground and ex- cited levels, E(mj) = -mj jBint. In -Fe, Bint = 32.2 T and j j = 0.09µN and -0.15µN (µN = nuclear magneton), respectively for the ground and excited states, so the separation between levels is g0 = 39.7 for the ground state, and g1 = 22.4 for the excited state. Since the transition is M1, the selection rule is jz = 0, ±1, and there are six allowed transitions. In the presence of crystal fields, the coupling of the 132 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Figure 4. 57Fe M¨ossbauer transition. Figure 5. Decay from an excited nucleus. quadrupole moment of the nucleus in the state n to the electric field gradient (EFG) gives an additional shift to the nuclear levels, typically on the order of 10­20 . The coherent elastic scattering amplitude fn is then a superposition of six sharp resonance contributions, the separation between the resonances being typically on the order of 20­40 . In an excited state n, a nucleus will decay via radiative decay (at the rate / ), or more likely, by internal conversion decay (at the rate / ), which is the direct analogue of Auger decay: the excitation energy of n is transferred via the Coulomb interaction to an outer electron, which is ejected into the continuum, as indicated in figure 5. The total decay width is = + . For the 14.4 keV level of 57Fe, / 8 so that an isolated excited nucleus is eight times more likely to give up its energy via conversion. However, for a collection of nuclei, as we will discuss in section 4.1, the probability for radiative decay can be greatly enhanced by spatial coherence. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 133 2.3. Fast versus slow scattering The effect of zero point motion and temperature motion of a scatterer bound in a solid on the X-ray or -ray coherent elastic scattering amplitude f depends on whether the collision time is short or long relative to the characteristic vibration time, -1 m ( 10-14­10-13 s), of the scatterer about its mean position. For nonresonant Thomson scattering, the A2 scattering is "instantaneous", and the scattering amplitude f0 will contain the factor exp[-i(kf - k0) · r], where r is the position of the nucleus of the scatterer relative to its mean position. When averaged over a Gibbs ensemble this gives the Debye­Waller factor 1 f 2 D( ) = exp -i(kf - k0) · r = exp - (k . (2.3) 2 f - k0) · r For resonant scattering, the coherent elastic scattering amplitude can be put in the time-dependent representation [1], f res = dt ei( - 0)te-( /2 )t e-ikf·r(t)eik0·r(0) , (2.4) 2ik 0 where r(t) is the nuclear center of mass operator in the Heisenberg representation. For resonant X-ray scattering, the characteristic scattering times are / 10-16­10-15 s and, hence, fast compared to -1 m . Then effectively t 0, which gives the Debye­Waller factor exp -ikf · r(t 0) exp ik0 · r(0) fD( ). (2.5) On the other hand, for M¨ossbauer resonances, the scattering is very slow, / 10-9­10-6 s -1 m , so that effectively t and the motions r( ) and r(0) are uncorrelated, giving the Lamb­M¨ossbauer factor exp -ikf · r(t ) exp ik0 · r(0) exp -ikf · r exp[ik0 · r] = fM. (2.6) Interesting cases of intermediate scattering times -1 m can occur for neutron resonance scattering as discussed in section 5. 2.4. Polarization response of resonant scattering The nonresonant Thomson scattering has a simple e f·e0 polarization dependence, so there is no orthogonal scattering, and no sensitivity to local fields. On the other hand, resonant X-ray and -ray scattering is generally sensitive to the directions of the local fields (B and EFG axis) at the scatterer, and as a consequence, the scattering necessarily has a strong tensor component in the polarization response (giving, e.g., strong orthogonal scattering). For a pure QL multipole transition (2L ­ pole electric (Q = E) or magnetic (Q = M) multipole; thus, E1 = electric dipole, etc.), with quantization axis zJ (for 134 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 simplicity, we assume Jz is a good quantum number), the coherent elastic scattering amplitude is [2­7] (see appendices A.1 and A.2) L f n(efkf; e0k0) = 4 ¯ e f · Y(q) LM kf Y(q) LM k0 · e0 F(q) LM , (2.7) M=-L where f j0 C(j F (q) M 0Lj1; m0M )2 LM = . (2.8) (2j0 + 1) x(m m 0M ) - i 0=-j0 The Y(q) LM with q = e or m are the vector spherical harmonics [8], x(m0M ) = 2 [ E(j1; m0 + M)- E(j0; m0)- ]/ and fM is the Lamb­M¨ossbauer factor, m0 runs over the (2j0 + 1) hyperfine levels of the ground state, and M gives the Jz change (m1 - m0). The scattering amplitude is a coherent superposition of resonances, such as the six allowed transitions for 57Fe. For a Jz = M resonance line of a qL-multipole transition, the polarization response e f · Y(q) LM kf Y(q) LM k0 · e0 (2.9) is precisely the polarization response of a qL, Jz = M classical oscillator. For example, for an E1 transition, there are three types of oscillators: M = 0 which is a linear electric dipole oscillation along the quantization axis zJ, M = +1 is a right- hand circular oscillation about the +zJ axis, and M = -1 is a left-hand circular oscillation about zJ. If the resonances are well separated ( ), then there is necessarily strong orthogonal scattering and a strong dependence on the direction of the local fields (zJ). For example (see figure 6), tuning in to an M = +1, E1 resonance, regardless of the incident polarization (as long as e0 drives the oscillator), the scattered radiation will be RH circular for kf in the direction +zJ, LH circular in the -zJ direction, linear for kf zJ, and elliptical for all other directions; and the whole pattern will rotate with zJ if the direction of zJ is changed. Figure 6. M = +1 electric dipole oscillator. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 135 The overall form of the polarization response is dictated by symmetry consider- ations and can depend only on e0, k0, e f, kf and zJ since these are the only defined directions of the scattering (in the absence of crystal fields). For an E1 transition, the explicit Y(e) 1M expressions give the scattering amplitude [3] fE1 = A0e f · e0 + iA1 e f × e0 · zJ + A2 e f · zJ (e0 · zJ), (2.10) where factors A0, A1 and A2 are combinations of the F(e) 1M (see appendix A.2). But the general form can be deduced entirely by symmetry arguments: the lowest order E1 approximation is made by replacing exp(±ik · x) 1, and the resonant scattering amplitude fE1 is then proportional to 0|p·e f| n n|p·e0| 0 . Thus, fE1 is a scalar which is linear in the components of e0 and e f, and independent of k0 and kf. In the absence of crystal fields, the only defined directions which enter into fE1 are e0, e f and zJ, the direction of the quantization axis defined by the local moment of the ion, as indicated in figure 7. Thus the distinct polarization responses possible are the various scalar products which can be constructed from e0, e f and zJ, with e0 and e f occurring linearly, and with zJ allowed to any order. These combinations are simply e f · e0, i(e f × e0) · zJ and (e f · zJ)(e0 · zJ). In the linear term, the factor i is required for time-reversal invariance. For a magnetic dipole transition (M1), similar arguments show that the same form holds, but with the E-polarization vector e replaced by the B-polarization vector b = k × e. For an E2 transition, the scattering amplitude is linear in both e and k. Thus, for E2 scattering the defined vectors are e0, k0, e f, kf and zJ. The scattering ampli- tude fE2 is a scalar comprised of products of these vectors, with each term linear in e0, k0, e f and kf, and containing zJ to any order [3­5]. The explicit expressions for the electronic and nuclear contributions fe and fn to the elastic scattering amplitude are given in appendices A.1 and A.2 (for E1, M1, E2 and mixed M1­E2 nuclear transitions, assuming good Jz). The modifications of fn when the electric field gradient (EFG) is noncoaxial with Bint are also discussed in appendix A.2. Figure 7. Scattering geometry. 136 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 3. Dynamical theory of M¨ossbauer optics 3.1. Basic equations The quantum theory of crystal optics is most conveniently developed using the Green-function techniques of quantum electrodynamics [9].1 Using the simple intuitive Feynman procedures, basic equations are obtained bearing a close formal relation to those of classical optics [10]. For an incident photon [9]2 4 c2 1/2 A0µ(x) = e0 2 µ exp(-i 0tx + ik0 · x), (3.1) 0 the induced current for coherent elastic scattering is c-1Jµ(x) = d4yMµ (x, y)A0 (y), (3.2) where Mµ (x, y) is the coherent elastic scattering operator i Mµ (x, y) = Mµ (x, y, tx - ty) = - j | . (3.3) c2 0|T µ(x)j (y) 0 Here jµ(x) is the Heisenberg current operator, T is the time-ordering operator, 0 is the initial Heisenberg state vector for the system, and the outer brackets · · · indicate an ensemble average over the initial state of the system 0. The Feynman photon potential for the coherent elastic scattered wave3 is then exp[i(k|z - x|)] Asµ(z; ) = c-1 J |z - x| µ(x, ) dx, exp[i(k|z - x|)] = M |z - x| µ (x, y, )A0 (y, ) dx dy, (3.4) or in obvious matrix notation As = +MA0. (3.5) 1 We make extensive use of the Feynman techniques and nomenclature as given in his book [9]. 2 We use the four-vector conventions of [9]. x, y are four-vectors (ctx, x), (cty, y). Four-vector contrac- tions are denoted as either kx = (ck4tx - k · x) or by repeated indices AµBµ = A4B4 - A · B. The four-dimensional gradient operator used later is µ = [(1/c)( / t), - ]. 3 As we have discussed in [10], at large distances from the source the "Feynman potential" is A (x, t) = 0| A (x)| (t) , where A (x) is the usual transverse vector potential operator at the point x, | (t) represents the electromagnetic field state vector at t, and |0 represents the vacuum state. More generally, the "Feynman potential" is a photon Green's function as discussed in [10, appendix A]. Our treatment and nomenclature are designed to maintain a close correspondence with the conventional semiclassical treatment of X-ray optics. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 137 Here the index indicates the Fourier transform. For coherent elastic scattering, the final state of the system is unchanged, f = 0, and consequently, As is coherent with the primary wave A0. The total Feynman potential at z is then Aµ(z) = A0µ(z) + Asµ(z). (3.6) The Feynman potential may be used to compute the scattering cross-section in the following manner: at large distances from the scatterer the expected energy flux is given by c S = Re E × B , (3.7) 2 where E and B are computed from the potential Asµ(z) in the usual manner. The quantum flux is ( )-1S. Dividing by the incident quantum flux, the coherent elastic scattering cross-section is d R2|As = lim (z; 0)|2 . (3.8) d R |a0 |2 We have assumed an incident photon of well-defined energy 0. However, if the incident wave packet is a time varying signal, A0µ(z) = A0µ(z, )e-i tz d , (3.9) then the time-dependent coherent elastic scattered wave packet is Asµ(z) = Asµ(z, )e-i tz d . (3.10) For time-dependent scattering, the expected flux (probability/cm2/s) for a photon at (z, t) is n(z, t) = 0 A 2, (3.11) 2 c (z, t) where 0 is now the central resonance frequency of the M¨ossbauer transition, and the frequency spectrum (probability/cm2/unit frequency interval) for a photon at (z, ) is n(z, ) = 0 A 2. (3.12) 4 2 c (z, ) As discussed in [10], for scattering from a single atom, the scattering matrix M includes all internal processes involving virtual photon exchange, e.g., radiative re- action, virtual internal conversion, virtual photoelectric and Auger absorption, etc. When calculated exactly, M gives all contributions for the coherent elastic scattering amplitude f for the atom. Similarly, for scattering from a multiatom system, in addition to the complete radiative processes for each atom, M now also includes all possible virtual photon exchanges between atoms, and the resulting coherent elastic scattered wave As includes 138 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 all possible single and multiple scattering processes which leave the final state of the system unchanged, f = 0. The basic problem for coherent X-ray and -ray optics is to obtain the coherent elastic scattering operator M for the system. As discussed in the following section, the scattering operator M can be developed in terms of single atom scattering operators leading to simple multiple scattering equations which determine the coherent elastic wave. 3.2. Multiple scattering equations The scattering amplitude f for resonant X-ray or -ray optics gives strong tensor scattering (e.g., ). The appropriate dynamical theory to handle this complication has already been developed in connection with M¨ossbauer optics, and can be taken over intact for resonant X-ray optics. The multiple scattering equations for X-ray or -ray optics are developed in terms of the scattering operators Mj of the individual atoms. In terms of Mj, the coherent elastic scattering amplitude fj(kfef; k0e0) for atom j to scatter an incident |k0e0 photon into a |kfef photon is given by (see [10]) fj(kf, ef; k0, e0) = fje + fjn = e f · Mj µ µ (kf, k0; ) · (e0) , (3.13) including all atomic and nuclear scattering processes. For an arbitrary incident field Aj, M j determines the scattering response of the jth atom, i.e., the coupling to the inci- dent wave, and the amplitude and polarization of the scattered photon. As noted, this response can be quite complex, depending on the direction of incidence, the polariza- tion and frequency of the incident photon, the direction and time-dependence of the local fields, and the direction of the scattered photon. For a system of scatterers, a simple set of multiple scattering equations determine the coherent elastic wave (see [10]) exp(ik|z - R A i|) µ(z, ) = A0µ(z, ) + Mi k k Ai |z - R µ i, i; (Ri, ), (3.14) i i| where Mi is the coherent elastic atomic scattering operator for atom i, and Ai is the coherent elastic wave incident on atom i. Ai satisfies exp(ik|R Ai i - Rj|) µ(Ri, ) = A0µ(Ri, ) + Mj k k Aj |R µ j, j; (Rj, ). (3.15) j =i i - Rj| Here kj = i j and kj = -i j, where the gradient is with respect to Rj, the equilibrium position of atom j, and the gradients operate on the factors to the left or right as indicated. In obvious matrix notation, these equations become A = A0 + +z,iMiAi (3.16) i III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 139 and Ai = A0 + +ijMjAj, (3.17) j =i where +ij = exp(ikRij)/Rij. Equations (3.17) are a set of coupled equations which determine the incident fields Ai, and then (3.16) gives the coherent elastic wave A at (z, ). Equation (3.17) simply says that the total field incident on the ith atom is the primary field A0, plus the scattered waves emanating from all the other atoms j under the inluence of the total fields Aj incident at those sites. Equation (3.16) then gives the total field incident at r as the primary field A0(r, t) plus the scattered waves emanating from all the atoms j under the inluence of the total fields Aj incident at those sites. Equations (3.16) and (3.17) are the basic optical equations for coherent X-ray and -ray optics. These equations are identical in form to the semiclassical equations used to develop the dynamical theory of X-ray diffraction (the important distinction between the classical and quantum equations is that the quantum field equations are complex, while the classical equations are real). Except for a small inaccuracy in accounting for "mirror terms" [10, appendix B], eqs. (3.16) and (3.17) are the exact single-quantum coherent scattering equations from a collection of scatterers. Using these equations, the dynamical theory for X-ray or -ray optics has been de- veloped using a generalized Darwin­Prins approach, which is applicable even to mono- layer films [2,11­13]. An alternative generalized Laue development of M¨ossbauer optics has been given by Afanas'ev and Kagan [14­16].4 In the following subsection we review the Darwin­Prins approach. The resulting optics is very rich because of the exceeding sharpness of the nuclear resonances, the well defined multipolarity of the transitions, and the strong dependence of the scattering on the directions of the local fields. There are strong magneto- optical effects (Kerr, Faraday, etc.), and striking dynamical effects such as the strong coherent enhancement of the radiative width ( c + ), which occurs for Bragg excitation, and, in the Laue geometry, the nuclear Borrmann effect, which involves quite different modes of the radiation field than the X-ray Borrmann effect because of the higher multipolarity of the nuclear resonators, and which has a much deeper penetration depth because of the long scattering times (because the Lamb­M¨ossbauer factor replaces the Debye­Waller factor in the scattering amplitude) [14,17]. 3.3. Generalized Darwin­Prins dynamical theory The multiple scattering equations which describe the interaction of a -ray with a M¨ossbauer medium are given by eqs. (3.16) and (3.17). These equations are identical in form to the semiclassical equations used to develop the dynamical theory of X-ray 4 The generalized Laue development is also given in [2, appendix D], starting from the basic multiple scattering equations (3.16) and (3.17). 140 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Figure 8. (a) Schematic representation of the crystal geometry used in developing the dynamical theory (dashed lines indicate crystal planes); (b) representation of open radiation channels for a plane layer; (c) Bragg reflection; (d) Laue transmission. diffraction,5 and as a consequence, much of the X-ray theory can be taken over directly. However, it is necessary to generalize the X-ray theory to account for the very strong polarization mixing which occurs in M¨ossbauer optics. We will generally limit our discussion here to perfect single crystals (except for the cases of critical reflection and transmission off-Bragg, where the expressions are independent of crystal structure). The reflection and transmission coefficients depend somewhat on the shape of the crystal, and, as in the usual X-ray development, we take the form shown in figure 8: we assume that some set of infinite crystalline planes is parallel to the surface, which we take as the xy plane, and that the crystal is of finite thickness l in the z direction, l = Md, where M is the number of "planes", and d is the interplanar distance. The thickness of the layers is rather arbitrary, but they must be sufficiently thin so that the Born approximation is good within the layer. We generally assume a unit cell thickness (chemical or magnetic, whichever is larger). For example, if there is a magnetic spiral axis which is commensurate with the crystal structure, it is convenient to use the magnetic unit cell, which contains the full spiral, so that each "plane" of the crystal has the same internal field structure. If a photon A0µ(z, t) = a0µ exp[i(k0 · z - t)] = A0µ(z) exp(-i t) is incident on a plane layer (at z = 0), then in the Born approximation the coherent elastic photon amplitude at R is given by6 Aµ(R) = A0µ(R) + Asµ(R), (3.18) 5 For a discussion of the Darwin­Prins and Laue formulations of the dynamical theory of X-ray diffraction, see [18]. 6 An exact treatment of planar scattering, i.e., including multiple scattering within the plane, is given in [11] and discussed in section 6. For most cases the planar self-action contribution is negligible. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 141 where Asµ(R) is the scattered wave, i As 0n µ(R) = ei[gs|z|+(k0xy+ s)·R]M -i a0 sin µ R, k0 (3.19) s s with g 2 s = k20 - k0xy + s , (3.20) g sin s s = , (3.21) k0 and n is the planar density of unit cells (number/cm2). Mµ is understood to be the coherent scattering operator of the unit cell Mµ k , k = M( ) µ k , k e-i(k -k)· . (3.22) c In deriving (3.19), we used the planar relation eik|R-Rj|eik0xy·Rj = i2 n (1/g |R - R s)ei[gs|z|+(k0xy+ s)·R]. (3.23) j j| s The s in eqs. (3.19) and (3.20) are the planar reciprocal lattice vectors for the unit cells, and the sum is over all s for which gs is real (the sum over the s for which gs is imaginary gives an exponentially damped contribution on the order of e-k0|z|/k0|z|). The scattered photon amplitude is thus a superposition of a finite number of plane- wave channels (s±) having wave vectors k(s±) = (k0xy + s, ±gs). These channels are symmetric about the scattering plane, i.e., for each forward scattered wave in the k(s+) direction (the (s+) channel), there is a reflected wave in the k(s-) direction (the (s-) channel) (see figure 8(b)). Finally, we note that the operator -i R in eq. (3.19) operates on the exponential plane-wave factor to the left of Mµ , and is replaced by the wave vector k(s+) = (k0xy + s, gs) if z > 0 (corresponding to forward scattering k0 k(s+)), and by k(s-) = (k0xy + s, -gs) if z < 0 (corresponding to reflection k0 k(s-)). In a crystal of finite thickness, any wave of appreciable magnitude is built up by constructive interference of the planar radiation channels. All the (open) planar channels contribute to the general dynamical equations (eqs. (3.24), (3.34), (3.35) below), but in the usual approximation, only constructively interfering channels need be considered open for the crystal. Thus, there will generally only be one or two radiation channels open in a crystal (off-Bragg, only the (0+) channel is open, for a single Bragg reflection, the (0+) and a (1-) channel are open, and for Laue transmission, the (0+) channel and a (1+) channel are open). This approach to the problem follows the Darwin­Prins development of the dy- namical X-ray diffraction. The alternative Laue formulation for -ray optics is dis- 142 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 cussed in [14­16]. For a discussion of the Darwin­Prins and Laue formulations of the dynamical theory of X-ray diffraction, see, for example, [18]. The generalization of the Darwin­Prins dynamical equations is straightforward: denoting by A(m) µ (k(s+)) the wave incident on the mth plane (from "above") in the (s+) channel, and similarly, by A(m) µ (k(s-)) the wave incident on the mth plane (from "below") in the (s-) channel, then the waves incident from above on the mth plane are related to the waves incident on the (m - 1)th plane as A(m) µ (k(s+)) = eigsd A(m-1) µ (k(s+)) i + 0nd M(m-1) k A(m-1) k , (3.24) sin µ (s+), k(s ) (s ) s s =± where d is the interplanar distance and n is the unit-cell density (cm-3). M(m-1) µ is the unit-cell scattering operator of the (m - 1)th plane. We will generally limit our considerations to uniform M¨ossbauer crystals (i.e., crystals with a uniform distribution of M¨ossbauer atoms, and for which the unit-cell internal field structure is the same throughout the crystal) for which the plane designation of the scattering operator can be omitted. Equation (3.24) is an obvious result: the amplitude of a wave A(m) µ (k(s+)) incident on the mth plane (from above) in the (s+) channel is equal to the amplitude A(m-1) µ (k(s+)) of such a wave incident on the (m - 1)th plane, plus the forward scattering i 0nd M(m-1) k A(m-1) k (3.25) sin µ (s+), k(s+) (s+) s of this wave by the (m - 1)th plane, plus the amplitudes of all other channels incident on the (m - 1)th plane scattered into the (s+) channel by this plane. These effects are propagated to the mth plane by the phase factor eigsd. Similarly, the waves incident on the mth plane from below are related to the waves incident on the (m + 1)th plane, with A(m) µ (k(s-)) given by eq. (3.24) with the replacements (s+) (s-), (m - 1) (m + 1). Denoting the first plane as the m = 0 plane, and the last plane as the m = M plane (the total number of planes is M + 1), the general boundary conditions to be satisfied are A(m=0) µ (k(s+)) = s0a0µeik0·R0, (3.26) A(m=M) µ (k(s-)) = 0. (3.27) That is, the only wave incident from above on the first plane is A0µ, and there are no waves incident from below on the last plane. It has already been implicitly assumed in deriving eq. (3.24) that k0d 1, which is well satisfied for M¨ossbauer frequencies. Thus, the longitudinal and scalar components need not be considered and it is convenient to express eq. (3.24) in terms of orthogonal transverse amplitudes. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 143 We define by e(s ) x , e(s ) y ( = + or -) any convenient orthogonal basis per- pendicular to k(s ). In terms of this basis, the transverse part of A(m) µ (k(s )) is given by A(m) µ (k(s+)) = e(s+) x T (s) x,m + e(s+) y T (s) y,m eik(s+) xy ·Rm , (3.28) A(m) µ (k(s-)) = e(s-) x R(s) x,m + e(s-) y R(s) y,m eik(s-) xy ·Rm , (3.29) where T (s) x,m = e(s+) x · A(m)(k(s+))e-ik(s+) xy ·Rm , etc., so that T (s) x,m gives the scalar amplitude of a photon |e(s+) x , k(s+) incident on the mth layer from above, etc. We will denote the (s+) channels by the index t, and the (s-) channels by r, and define the column vectors Ttm, Rrm as Tt T (t) x,m m = T (t) , (3.30) y,m Rr R(r) x,m m = R(r) . (3.31) y,m The "planar" scattering matrix F ss (and fss ) (s, s = t or r) for scattering an (s ) channel into an (s) channel is defined by d F ss F ss fss = xx F ss xy , (3.32) sin s F ss yx F ss yy where d d F ss fss k , (3.33) sin = 0nf s, es ; ks , es s sin s , = x or y, and where f(ks, es ; ks , es ) is the unit cell coherent elastic scattering amplitude as given by eqs. (3.13) and (3.22). The planar scattering amplitudes F ss are dimensionless, while the amplitudes fss have the dimensions of a linear mass absorption coefficient, i.e., length-1. In terms of these matrix quantities, the dynamical equations for M¨ossbauer optics are given by the coupled equations Tt m = eigtd Ttm-1 + i F tt Tt m-1 + i F trRrm-1 , (3.34) t r Rr m = eigrd Rrm+1 + i F rr Rr m+1 + i F rtTtm+1 , (3.35) r t 144 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 with the boundary conditions Tt 0 = t0a0 k, , (3.36) RrM = 0. (3.37) Equations (3.34), (3.35) are the generalization of the Darwin­Prins equations of X-ray optics. These equations include not only the polarization mixing effects, but by including the contribution of all (open) planar radiation channels, the scattering from any set of Bragg planes can be computed. 3.4. General solution For a uniform M¨ossbauer crystal it is straightforward to obtain a general solu- tion from eqs. (3.34)­(3.37) for the amplitude and polarization of the reflected and transmitted waves by making the substitutions T(t) Tt x,k m = Ttk eimk d eimk d, (3.38) T (t) k k y,k R(r) Rr x,k m = Rrk eimk d eimk d. (3.39) R(r) k k y,k Substitution of (3.38), (3.39) into (3.34)­(3.37) gives e-i(gt-k )d - 1 Tt k - i F tt Tt k - i F trRrk = 0, (3.40) t r e-i(gr+k )d - 1 Rr k - i F rr Rr k - i F rtTtk = 0, (3.41) r t and the boundary conditions T (t) ,k e(t) = t0a0 k0, 0 , (3.42) k R(r) ,k eik Md=0. (3.43) k The dispersion equation to determine the allowed values of k is then det C k = 0, (3.44) III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 145 where C(k ) is the coefficient matrix of eqs. (3.40) and (3.41), [(e-i(g0-k )d - 1) I- i F00] -i F 0t -i F t0 [(e-i(gt-k )d - 1) I- i F tt] . . C(k ) = . . . . -i F r0 -i F rt .. . . .. · -i F 0r · · -i F tr · . · .. · . (3.45) · [(e-i(gr+k )d - 1) I- i F rr] · . · .. · Here I is the 2×2 unit matrix and the F ss are the dimensionless planar scattering ma- trices defined in eq. (3.33). C(k ) includes the (0+) channel, and all other transmission channels (t) and reflection channels (r) that are open. If there are n open channels, then eq. (3.44) gives 2n values of k (which are independent of the polarization basis of representation). There are then 4n2 amplitudes T (t) ,k , R(r) ,k to be determined from the 4n2 linear relationships given by eqs. (3.40)­(3.43). In particular, for a given value of k , the 2n - 1 independent equations (3.40), (3.41) give the determinate solutions T (t) ,k = Dt0 x k T (0) k T (0) x,k , R(r) ,k = Dr0 x x,k , in terms of T (0) x,k . The 2n values of T (0) x,k are then determined from the boundary conditions (3.42), (3.43). The reflected and transmitted waves are then given by Rr = e(r) R(r) ,k , (3.46) k Tt = e (t)T (t) ,k eik Md. (3.47) k Actually, eqs. (3.46) and (3.47) give Rr0, the wave incident from below on the first plane (m = 0), and TtM, the wave incident from above on the last plane (m = M). The exact reflected and transmitted waves Rr and Tt can then be calculated from equations (3.34), (3.35) with the boundary conditions (3.36), (3.37): Rr = Rr 0 + i F rr Rr 0 + i F r0T00, (3.48) r Tt = Tt M + i F tt Tt M. (3.49) t Usually, however, the scattering effects of a single plane can be neglected (unless we are interested in the scattering from very thin films, say M 1­10), and it suffices to use Rr = Rr0 and Tt = TtM as given by eqs. (3.46) and (3.47). 146 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 The reflection and transmission coefficients are then given by |Rr|2 |Tt|2 Rr = g d 0 d k , T t = g d , (3.50) 0 I 0 d k0 0 I0 where I 2 0 = d 0 d k a , 0 0 0, k0 g = sin s/ sin 0 (figure 8(c,d)) is the Jacobian | k / )| for s = r or t, respectively, 0 ks and the integration range is taken sufficiently small that the relative variation of g and 0 is negligible. The coherent elastic wave scattered into the ks-channel is linear in the incident photon potential a0 (= a0e0) and can always be expressed in the general form S ; ks, k0; e0 = S ; ks, k0 a0, (3.51) where S = Rr or Tt for reflection or transmission channels, respectively. The compo- nents of the 2 × 2 scattering matrix S (= R or T), which directly relates the coherent elastic wave scattered into the ks-channel to the incident photon a0, can be extracted directly from the solutions for Rr and Tt. The internal field in the crystal can be expressed as a superposition of radiation eigenwave contributions, A(r) = Ak (r), (3.52) k where the k eigenwave contribution is Ak (r) = eik z Ttk ei(kt)xy·r + Rrk ei(kr)xy·r . (3.53) t r In certain cases it is convenient to solve in terms of eigenwaves, e.g., off-Bragg (section 3.5), Laue (section 3.8), and multibeam Borrmann (section 3.9). 3.5. Off-Bragg Off-Bragg, to a very good approximation, we only need consider the (0+) chan- nel open. This channel only involves forward scattering amplitudes, and the optical solutions are thus valid for a noncrystalline medium as well as for a crystal. A com- plicating, but interesting feature of -ray optics is the presence of strong optical activ- ity, which gives strong polarization mixing (i.e., orthogonal scattering ex ey) and Faraday-type effects [10,13,19,20]. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 147 3.5.1. Eigenwaves Keeping only the (0+) channel in (3.40) and taking if ei(k -g0)d = 1 + d , (3.54) sin 0 the eigenpolarizations and indices of refraction are determined by the eigenvalue equa- tion fxx fxy Tx,k Tx,k f = f , (3.55) yx fyy Ty,k Ty,k where fxx, fyy, etc. are the planar forward-scattering amplitudes defined by (3.33) and again we have used ex, ey to denote any two orthogonal polarizations perpendicular to k0; fab (a, b = x or y) is related to the unit-cell coherent forward scattering amplitude by fba(k, ) = n f( )(eb, k; ea, k), (3.56) where n is the unit-cell density, the sum is over the unit cell, and f( )(eb, k; ea, k) is the coherent elastic scattering amplitude of atom for scattering an |ea, k photon into an |eb, k photon. From (3.55) we obtain (for = 1 or 2) 1 1 2 1/2 f = Tr f + (-1)( +1) Tr f - det f 2 2 1 1 1/2 = (f (f , (3.57) 2 xx + fyy) + (-1)( +1) 4 xx - fyy)2 + fxyfyx where the square root is taken to have a positive imaginary part. The corresponding eigenpolarizations e are given by7 (f e - fxx) k, = K( ,x) ex + e , (3.58) f y xy where (f 2 -1/2 K - fxx) ( ,x) = 1 + . (3.59) f xy 7 There are, of course, a number of alternative forms for e which are obtained by using the determinant equation (3.55), i.e., (fxx - f )(fyy - f ) - fxyfyx = 0, and by modifying the factor K( ,x). In particular, we can interchange all x and y subscripts in (3.58) and (3.59). In going from one basis to another, over-all phase factors are absorbed into K( ,x). 148 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 From (3.54) we see that k d = g0d+f d/ sin 0 +O(f d)2, so the e eigenwave propagates in the medium (to first order) as exp(in k0 · r), with the complex index of refraction given by n = 1 + ¯ 0f . (3.60) We note that since the scattering matrix in (3.55) is, generally, non-Hermitian [2, 19], the eigenwaves e are, generally, nonorthogonal. Only in limiting cases will the eigenwaves be strictly orthogonal: well-isolated resonances (splitting large compared to ); unsplit resonances (splitting small compared to ); or otherwise, if Jz is a good quantum number, and k0 · zJ = 0, ±1. Although it is not immediately obvious, it can be verified that e and f are invariant with respect to the choice of orthogonal basis ex, ey, as they must be. 3.5.2. Transmission through a M¨ossbauer film The polarization and intensity of the coherent elastic wave transmitted through a M¨ossbauer medium are easily obtained using the eigenwaves e . The polarization and amplitude of the ( k, ) component of the incident photon we write as a a0x 0 k, = a0x k, ex k + a0y k, ey k = a , (3.61) 0y where ex( k) and ey( k) are an orthogonal polarization basis perpendicular to the mo- mentum vector k of the incident photon. Expressing ex and ey in terms of the eigenwaves e1 and e2 by inverting (3.58), the ( k, ) component of the wave transmitted through a M¨ossbauer film of normal thickness l is then given by T k, = T k, ; l a 0 k, , (3.62) where the 2 × 2 transmission matrix is given by T k, ; l = Txx Txy Tyx Tyy 1 f f e xx - fyy e xy 2 (+) + e(-) f (-) f = 1 - f2 1 - f2 f 1 f .(3.63) e yx xx - fyy (-) e f (+) - e(-) 1 - f2 2 f1 - f2 Here e (±) k, ; l = eif1l ± eif2l eig0l, (3.64) where l = l/ sin 0 is the thickness along the direction of propagation. The intensity of the transmitted packet is then given by I = d d 2 k T k, . (3.65) III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 149 If the incident packet is unpolarized or partially polarized, one can use the density matrix formalism discussed in [13]. The transmitted wave T( k, ) is a superposition of eigenwaves e1 and e2. Since the indices of refraction of the two waves are different, i.e., 1 1/2 n1 - n2 = 2¯ (f , (3.66) 4 xx - fyy)2 + fxyfyx there are Faraday effects involved in the transmission through a M¨ossbauer medium. The real part of n1 - n2 gives the usual Faraday rotation effect, while the imaginary part gives a "selective absorption" effect. Since the electronic scattering gives essen- tially no contribution to (3.66) (assuming the M¨ossbauer resonance frequency is far from any X-ray absorption edges of the atom), n1 - n2 depends upon the nuclear scattering amplitude and is primarily imaginary at resonance, while the real part dom- inates off-resonance. Thus, at resonance the Faraday effect in M¨ossbauer optics is primarily one of selective absorption. That is, if the resonances are well separated and if the incident frequency (packet) is within a few widths of a resonance, so that I(f1) I(f2), the e1 eigenwave is damped out much more rapidly. For such a case, the thickness l of the polarizing medium can be chosen sufficiently thick that almost all of the e1 wave is absorbed, but still sufficiently thin that the e2 wave is only slightly attenuated, and the transmitted wave is nearly e2 polarized. If the incident frequency is adjusted off-resonance (within a few ), the real part of n1 - n2 dominates and the optical activity is primarily Faraday rotation. 3.5.3. Blume­Kistner approach An alternative approach to treating off-Bragg transmission is to convert the finite difference equation (3.34) to a differential equation. This is essentially the approach of Blume and Kistner [19]. With only the (0+) channel open, (3.34) is T(0) m = eig0d 1 + i F 00 T(0) m-1. (3.67) Treating the interplanar separation d as an infinitesimal (i.e., d dz, md z, T(0) m T(z), and F 00 dz f/ sin 0), and taking normal incidence for simplicity (i.e., 0 = /2 so sin 0 = 1, g0 = k0 and l = l), eq. (3.67) becomes the differential equation dT(z) = ik dz 0 nT(z), (3.68) where n = [1+ ¯ 0f ] is the 2 × 2 index of refraction operator, defined in (3.60), which, combined with the boundary condition (3.36), gives the solution for the transmitted wave amplitude T k, = T(z = l) = T k, ; l a 0 k, , (3.69) where the 2 × 2 transmission matrix is now given by T k, ; l = exp ik0 nl = exp i k0 + f l . (3.70) 150 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Expressing the 2×2 planar scattering matrix f in terms of the Pauli matrices, i.e., f = a+b· , and making use of the operator identity exp(ib· z) = cos(bz)+i( b· ) sin(bz), T( k, ; l) can be expressed as T k, ; l = ei(k0+a)l cos(bl) + i b · sin(bl) , (3.71) where 1 1 i i a = (f (f (f (f 2 xx + fyy), bx = 2 xy + fyx), by = 2 xy - fyx), bz = 2 xx - fyy), (3.72) and 1 1 b = b · b = (f bz bx - iby . 4 xx - fyy)2 + fxyfyx, b · = b bx +iby -bz Noting that b = (f1 - f2)/2 and that exp[i(k0 + a)l] cos(bl) = e(+)( k, ; l)/2 and exp[i(k0 + a)l] sin(bl) = -ie(-)( k, ; l)/2 (assuming normal incidence), it is easy to verify the equivalence of the two expressions (3.63) and (3.71) for the transmission matrix T ( k, ; l). 3.6. Bragg For a single-Bragg reflection from a M¨ossbauer crystal, two channels are open: the t0 = (0+) channel and a "reflection" channel r (Bragg case, shown in figure 8(c)), or a "transmission" channel t (Laue case, shown in figure 8(d)). In this section we consider the Bragg case. The general dynamical equations (3.40), (3.41) now contain only the two ampli- tudes T0k and Rrk , C k T0k Rr = 0, (3.73) k where the coefficient matrix C(k ) is now the 4 × 4 matrix, C k = (e-i(g0-k )d - 1) I- i F 00 -i F 0r -i F r0 [(e-i(g . (3.74) r +k )d - 1) I - i F rr The dispersion equation (3.44) is now of the fourth order, and except for a few special cases, there is no simple analytical solution. Of course, the numerical solution is easily obtained, as outlined in section 3.4. The special cases occur when the 2 × 2 forward scattering matrices F 00 and F rr, and the reflection matrices F r0 and F 0r can be simultaneously diagonalized with an appropriate choice of orthogonal polarization bases e(0) and e(r) ( and = x, y). Three important cases where this is possible are the following: 1. In the "isotropic limit" (magnetically disordered state for which the splittings are negligible compared to , e.g., 57Fe in stainless steel), the polarization response of III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 151 the resonant M¨ossbauer scattering amplitude, given by eqs. (A.47)­(A.49) of appen- dix A.2 for E1, M1, and E2 nuclear transitions, respectively, is that of a free classi- cal oscillator of the corresponding multipolarity. For example, for an E1 resonance, fE1 e f · e0, which is the response of a free electric dipole oscillator: the incident e0 polarized E field drives the oscillator with an acceleration a e0 exp(-i t), and the emitted radiation in the kf direction is proportional to the transverse projection a of the acceleration relative to the kf direction. For the "isotropic limit", we also assume that the resonance energy of the M¨ossbauer transition is far removed from any atomic absorption edges, so that the electronic contribution to the scattering amplitude has the simple limiting form of eq. (A.13), i.e., fe e f · e0. We make this same assumption for the other two special cases discussed below. For Bragg scattering from a crystal of isotropic scatterers, there is no orthogonal scattering between the and polarizations ( 0 = r is the unit vector lying parallel to kr × k0, while s (s = 0, r) lies in the (kr, k0) scattering plane and perpendicular to ks, with the convention that ( s, s, ks) forms a right-hand coordinate system). That is, in the isotropic limit, polarization scatters only into polarization (both for forward scattering 0 0, r r and for reflections 0 r), and polarization scatters only into polarization. With the x, y polarization basis perpendicular to ks taken as (e(s) x , e(s) y ) = ( s, s), there is no polarization mixing x y, and the four coupled linear equations of eq. (3.73) separate into two sets of 2 × 2 coupled linear equations, one set containing only the x amplitudes and the second only the y amplitudes. 2. A second special case occurs when Jz is conserved, and the quantization axis zJ (direction of the internal magnetic field and symmetry axis of the quadrupole field) at each M¨ossbauer nucleus is perpendicular to the scattering plane (kr, k0). Each hyperfine split resonance is now a good Jz transition. As discussed in section 2.4, for elastic scattering, the polarization response of a pure qL multipole transition with Jz = M is precisely that of a qL, Jz = M classical oscillator, with symmetry axis along the quantization axis zJ (which is perpendicular to the scattering plane (kr, k0)). In this situation, there is no polarization mixing x y if the polarization basis is again taken as (e(s) x , e(s) y ) = ( s, s). For example, for an E1 resonance, the Jz = ±1 resonances correspond to right(+) and left(-) hand-side circularly polarized electric dipole oscillations about + zJ, while Jz = 0 resonances correspond to linear oscillation along ± zJ. With zJ (kr, k0), incident 0 polarized radiation couples only to the Jz = ±1 oscillators, which emit f polarized radiation in the (kr, k0) plane, and incident radiation couples only to the Jz = 0 oscillators, which emit polarized radiation in the (kr, k0) plane.3. A third interesting special case where it is possible to simultaneously diago- nalize the scattering matrices is for a pure magnetic reflection from a simple antifer- romagnet, under the conditions that Jz is conserved, and that the quantization axis zJ at each M¨ossbauer nucleus lies in the (kr, k0) scattering plane. 152 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 This is a more complex situation because for forward scattering (0 0 or r r) polarized radiation scatters only into , and scatters only into , but for reflection (0 r), scatters into , and scatters into . To see this, consider the simple case of an E1 M¨ossbauer transition (fE1 = A0e f · e0 + iA1(e f × e0) · zJ + A2(e f · zJ)(e0 · zJ)), with two atoms (± zJ) in the magnetic unit cell. The unit cell forward scattering amplitude is then proportional to 2A0e f · e0 + 2A2(e f · zJ)(e0 · zJ) (plus an electronic contribution proportional to e f · e0, assuming the resonance energy is far removed from any atomic absorption edges), while for a pure magnetic reflection, the unit cell reflection amplitude is proportional to 2A1(e f×e0)· zJ (for a pure magnetic reflection, the electronic contribution vanishes, as do the resonant terms of fE1 which contain even powers of zJ). With zJ in the (kr, k0) scattering plane, zJ · = 0, and it is easy to verify that the polarization response for forward scattering and reflection from the unit cell is that discussed above. Because for forward scattering the polarization state ( or ) is conserved, while reflection changes the state ( or ), it follows that if we take the x, y basis vectors for the (0+) and (r) channels as (e(0) x , e(0) y ) = ( 0, ) and (e(r) x , e(r) y ) = ( , - r), there is no polarization mixing x y. For all cases where there is no polarization mixing x y, the four coupled linear equations of (3.73) separate into two sets of 2 × 2 coupled linear equations, one set containing only the x amplitudes and the other only the y amplitudes. There is then a simple analytical solution for the reflected and transmitted waves: If the incident radiation is near a Bragg angle for a reflection channel r = (1-) as shown in figure 8(c), then (g0 + g1)d = 2n + 2 . (3.75) We note that with g0 and g1 given by eq. (3.20), eq. (3.75) defines in terms of the angle of incidence 0 and, for asymmetric reflections, the x, y orientation of k0. For a symmetric reflection (r = (0-)), = k0d cos 0 , where 0 is now understood to be the exact Bragg angle, and is the deviation of the incidence angle from exact Bragg.The dispersion equation then gives two values of k ( = x,y), g k 0d - + ± = , (3.76) d where 1 = F00 , (3.77) 2 - F 11 = 2 - F01 F 10 , Im > 0, (3.78) 1 = + F 00 , (3.79) 2 + F 11 and where the F ss are the planar scattering amplitudes defined by eq. (3.33). III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 153 The reflected wave in the k1 channel is then given by R(1)( , ) = R ; k 1, k0; M a0 k0, , (3.80) where the 2 × 2 reflection matrix R is diagonal, R ; k1, k0; M = im F 10 (3.81) with m given by i m = . (3.82) + i cot(M ) Here (M + 1) is the number of layers. As noted before in eqs. (3.46) and (3.48), R ( , ) is strictly the wave incident from below on the first plane (m = 0). For thick crystals (such that Im(M ) 1), m i/( + ), and F 10 R ( , ) - e(0) , k , (3.83) · a0 + which is the usual Darwin thick-crystal expression.8 In the opposite limit of thin films (and very near Bragg), such that |M | 1, then m M/(1 - iM ), and iMF 10 R ( , ) e(0) , k . (3.84) 1 - iM · a0 The transmitted wave in the k0 channel is given by T(0)( , ) = T ; k 0, k0; M a0 k0, , (3.85) where the diagonal 2 × 2 transmission matrix T is T ; k0, k0; M = Q-1 iM( . (3.86) exp + + g0d - ) Here Q + - = 1 - ei2M + im 1 - ei2M . (3.87) 2 ( - ) + The denominator Q has simple limiting forms for thick and thin crystals: Q = 1 for thick films, and Q = [1 - iM( - )] for thin films. 3.6.1. Coherent enhancement One of the most interesting features of resonant -ray optics is the phenomenon of "coherent enhancement" near a Bragg excitation ­ under proper conditions, a strong enhancement of the coherent radiative width occurs, due to a constructive collective interaction between the nuclei interacting through the resonant radiation field [11]. The 8 In comparing with the standard results, e.g., [18, p. 428], note that the | m|, | 0| factors which appear in the X-ray results for asymmetrical reflections are the same as the sin (s) factors which are included in the F ss . 154 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 integrated intensity of the reflected wave then maximizes near resonance, in optimum cases approaching 100% reflectivity, and also the transmitted intensity, although min- imizing near resonance, is generally much greater than the off-Bragg intensity. (This occurs because the transmitted wave depends primarily on the eiM factor in (3.86), and , which depends on the square root of the difference between 14(F00 + F 11 )2 and F 01 F 10 , generally has a much smaller imaginary part than F 00 , which determines the off-Bragg transmission.) The optimum conditions are to have a highly enriched crystal in which the res- onant nuclear contribution to the coherent elastic scattering amplitude strongly dom- inates the nonresonant electronic contribution; to have a symmetric Bragg reflection (i.e., the r = (0-) channel) so that F 11 = F 00 and F 10 = F 01 ; and to have a situation where the resonant nuclear scattering is isotropic in the (k0, k1) scattering plane, so that F 10 = F 00 for the nuclear scattering. These conditions optimize the radiative coupling between the contributing planes. To illustrate coherent enhancement, we consider the particularly simple case of an "isotropic" M1 M¨ossbauer transition and a simple cubic lattice containing a fraction C of the resonant nuclei. We assume incident -polarized radiation (the incident B field, which couples to the M1 transition, is then -polarized and the scattering is isotropic in the (k0, k1) scattering plane), and a symmetric Bragg reflection. For this case we have (with e(s) x = (s) for s = 0, 1) F 00 xx = F 11 xx = Fn + Fe, (3.88) F 10 xx = F 01 xx = Fn + re cos(2 0), where 2 0 is the scattering angle ( 0 is the angle of incidence with respect to the xy planes), and nd nd 1 2J 1 F 1 + 1 n = f , sin n = 0 sin 0 2 ¯ CfM 2J0 + 1 [x(J0, J1) - i] nd nd F e e = f -Zr , (3.89) sin e(0) = 0 + i 0 sin 0 4 ¯ nd nd r e e = f -F (2 , sin e(2 0) = 0)r0 + i 0 sin 0 4 ¯ where x(J0, J1) = [E(J1) - E(J0) - ]/( /2). Fe and re in (3.89) should, of course, be the coherent average of the M¨ossbauer and non-M¨ossbauer atoms. For a numerical estimate we take a thin iron crystal with M 104 plane layers, containing 65% 57Fe. Then for ex = polarized resonant radiation incident at Bragg (with C/ sin 0 = 1), we have T 0.04, R 0.67; while off-Bragg, T 2×10-4 and R 2×10-7. Thus, at Bragg, the reflectivity is increased by six orders of magnitude, and the transmitted intensity, although still small, is increased by two orders of magnitude compared to off-Bragg. This enhancement of the coherent scattered radiation can be viewed as due to an enhancement of the partial width c for coherent scattering. For a highly enriched III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 155 crystal for which we can neglect Fe and re with respect to Fn near resonance, we have, in the thin-crystal limit of eq. (3.84), 1 i( R c/2) x( , ) , (3.90) 1 - iM [ 0 - - i( /2) - (i c/2)/(1 - iM ) where the partial width for spatially coherent scattering is c = 2 n¯ 2l (k0) coh. (3.91) Here coh = [fMC(2j1 + 1) /(4j0 + 2)] and l (k0) = Md/ sin 0 is the thickness of the crystal in the direction of k0. (Strictly, the total width = + in the denominator of (3.90) should be + , where is the partial width for spatially incoherent radiative decay and is the partial width for internal conversion decay.) Thus, the effect of the coherent interaction is to enhance the resonance width to , = c + + = 1 + (M )2 + + c at exact Bragg ( = 0), (3.92) and additionally to shift the resonance frequency about Bragg, M c 0 = 0 + c( ) = 0 + . (3.93) 1 + (M )2 2 The enhancement of c is proportional to the number of layers of the crystal, and for sufficiently thick crystals, typically M 103 - 104 layers, c will exceed ( + ) and coherent decay will dominate incoherent decay. 3.6.2. Nuclear excitons, superradiance and coherent enhancement That radiative coupling can radically alter the radiative decay rate of an excited system relative to that of an isolated atom is well known [11,21­25]. Following [11], consider two identical nuclei with nondegenerate ground states a1 and a2 and excited states b1 and b2. Because of the radiative coupling between the nuclei, the exponentially decaying states (normal modes) are not b1a2 or a1b2, but rather the symmetric and antisymmetric combinations (b1a2 ± a1b2)/ 2. If is the partial width for radiative decay from either nucleus alone, the radiative decay rates of these two modes are 1 sin kR ± = 1 ± , (3.94) 2 |1 ± exp(ik · R)|2 = kR where k is the wave vector of the emitted radiation, and R is the relative displacement between the two nuclei. For kR 1, we have + 2 and - 0. In this limit, the symmetric mode has an enhanced decay rate while the antisymmetric mode becomes nonradiative. We call the phenomenon that a collective state of a system has a radiative decay rate larger than that of a single nucleus "superradiance". If the decay rate is smaller than that of a single nucleus, the mode is "subradiant". 156 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 This behavior is easy to understand classically. The symmetric mode corresponds to two oscillators oscillating together in-phase, which, for kR 1, doubles the dipole moment. The radiation fields are doubled and the radiation power is increased to four times that of a single oscillator. Since the total mechanical energy T is also doubled, the decay rate for the system is -(d T/dt)/ T = 2 . In contrast, in the antisymmetric mode, the oscillators move 180 out of phase and the radiation fields from the two oscillators interfere destructively, giving a very weak (electric quadrupole) radiative decay.In addition to the altered decay rates, the normal modes also exhibit shifted resonance frequencies. For the two-nuclei system, the resonance shifts are (averaged over the direction R) cos(kR) ± , (3.95) kR which will give large resonance shifts for kR 1, with the symmetric mode oscillating at the lower frequency. For a crystal of resonant nuclei, the effects can be much more pronounced. As a first approximation, the wave functions for the exponentially decaying excited states (normal modes) in a crystal might be expected to be of the form of Bloch waves, N 1 e(q) = eiq·Rl|e G , (3.96) N l 0(l) l=1 where |el |G0(l) represents the state in which the lth nucleus is excited while all other nuclei remain in their ground states, Rl is the position of the lth nucleus and q is the Bloch vector. The partial width for the radiative decay of this "qth mode" is (in the Golden Rule approximation) 2 N c(q) = d (n) exp -i(k - q) · R , (3.97) 4 N l l=1 where k = ( 0/c)n is the wave vector of the outgoing radiation ( 0 is the resonance frequency of the nuclei). Most states | e(q) are subradiant, with c(q) on the order of /N. However, if for some specific emission direction n, the condition k - q = holds, where is a reciprocal vector of the crystal lattice, then the mode | e(q) will have a highly directional coherent decay about the q + direction at an enhanced decay rate c = n¯ 2l (q + ) , (3.98) where l (q + ) is the thickness of the crystal in the direction of (q + ) and n is the number of nuclei per unit volume. We will give a more detailed discussion of this result in section 4.1.4. For = 1 A , n = 0.1 A -3, we get a decay rate c = for a crystal with a thickness of only 100 A . Thus, in such a superradiant mode, if a crystal III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 157 is sufficiently thick, the coherent decay rate will dominate the incoherent radiative decay rate and the the decay rate due to internal conversion . Now, in fact, the Bloch waves | e(q) are generally not the true radiative normal modes in a crystal. As discussed in [11] and shown in section 6, the proper forms of the eigenmodes for a parallel sided crystal have a sine or cosine modulation (coming in from the crystal faces) with a complex wave vector. Generally, the Bloch state | e(q) is a superposition of these radiative eigenmodes, which have a spread of eigenfrequencies and decay rates. The important exception is the case in which |q| = 0/c and q satisfies a Bragg condition q · = (1/2) 2, where is a reciprocal lattice vector which is perpendicular to the parallel faces. In this case, | e(q) is a superradiant eigenmode, radiating at the natural resonance frequency 0, and decaying at an enhanced decay rate including the contributions from coherent decay into both the q and q + directions, c(q) = n¯ 2 l (q) + l (q + ) . (3.99) An important general result is that the eigenmodes are not Hermitian orthogonal, and, as a consequence, the decay of a general superposition state depends upon the spread of mode eigenfrequencies as well as the individual eigenmode decay rates.This issue is of considerable current interest because it is now possible to create spatially coherent single exciton nuclear states | e(k0) of the Bloch form by illu- minating crystals containing resonant M¨ossbauer nuclei (e.g., 57Fe) with synchrotron radiation pulses [26­28]. The resulting state has k0 = ( 0/c)n0, where 0 is the resonance frequency and n0 is the incidence direction of the synchrotron pulse. By proper choice of n0, k0 will satisfy a Bragg condition and the state | e(k0) becomes a superradiant eigenmode decaying into the k0 and k0 + directions. For k0 off-Bragg, | e(k0) is a superposition of various normal modes of different eigenfrequencies and decay rates. Nevertheless, as we will discuss in section 4.1.4, such a state still has an initial superradiant decay into the "forward" k0 direction, with c given by eq. (3.98) with = 0. The normal mode breakup gives a different perspective on the scattering of external resonant radiation at Bragg and the coherent enhancement effect given in eqs. (3.90)­(3.93): the intermediate excited states in the scattering process are now the collective normal mode exciton states, and at exact Bragg ( = 0) for a sym- metric Bragg reflection, the phasing is such that only the superradiant eigenmode state | e(q = k0) is excited, with resonance occurring at the natural resonance fre- quency 0, and with the coherent decay rate c given by eq. (3.99) (which agrees with eq. (3.91), noting that for the symmetric Bragg reflection l (k0 + ) = l (k0)). Rocking off-Bragg, in addition to the superradiant eigenmode, various other normal mode states are virtually excited, the weighted resonance energy is shifted from 0, and the effective width is decreased. 158 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 3.6.3. Thick crystal reflectivity curves A qualitative insight into the dependence of the integrated reflected resonance scattering on the beam collimation, the rocking angle, the frequency, and the enrich- ment C of M¨ossbauer atoms is obtained by examining the angular dependence of the reflectivity curves |R ( , )|2 for different incidence frequencies . In figures 9(a) and (b) we plot |R ( , )|2 as a function of for a symmetric Bragg reflection from a thick crystal of 57Fe in stainless steel, with enrichment concentrations C = 0.02, 0.65, respectively, and with 0 = 10 . The incident polarization is ex = . Rx( , ) is then given by eq. (3.83), with the planar scattering amplitudes given by eqs. (3.88) and (3.89). The curves labeled x = are the purely electronic scattering curves. As is well known, the reflected intensity for this case has a region of near total reflection for - (Fe ± re), where Fe and re are given by eq. (3.89). In figure 9(a), the crystal has a weak concentration of M¨ossbauer atoms, C = 0.02. For this case, Fn 1.1×10-4(x-i)-1 while Fe -6.1×10-4+i2.0×10-5 and re -3.3 × 10-4 + i2.0 × 10-5. Thus, even at resonance, the dominant contributions to the coherent scattering amplitudes are the real parts of Fe and re, and the region of maximum scattering is still contained within the region - (Fe ± re). As the frequency of the incident radiation nears the resonance frequency, Fn is primarily imaginary and acts as an absorptive contribution to Fe and re. As shown in figure 9(a), the intensity within the maximum scattering region is strongly decreased, and it is clear that the integrated intensity will exhibit a minimum in the resonance region. In figure 9(b), C is taken as 0.65, for which Fn 3.5×10-3(x-i)-1, and Fn gives the dominant contribution to the scattering amplitudes near resonance. For this case the region of strong scattering is greatly broadened near resonance, with a maximum spread on the order of |Fn| = . This broadening of the scattering region can lead to a strong 0 maximum for the integrated intensity. Of course, within the region - (Fe ± re), Figure 9. Thick crystal reflectivity curves for 57Fe stainless steel with enrichments (a) C = 0.02, and (b) C = 0.65. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 159 |Rx( , )|2 as a function of frequency again strongly decreases near resonance, so if the incident beam is collimated to this region, a minimum will occur (for the case being considered, this corresponds to an angular collimation 4 × 10-4 rad). Qualitatively then, the reflected wave will maximize near resonance if the rms spread satisfies > |Fn| = > |r 0 e|, i.e., the most favorable conditions for a maximum are for highly enriched crystals, large scattering angles, and collimation spread larger than the region - (Fe ± re). Finally, we note that polarization is important. For example, in the simple case considered above, for incident ey = polarization, Ry( , ) is given by eq. (3.83) with the planar scattering amplitudes F 00 yy = F 11 yy = Fn + Fe, (3.100) F 10 yy = F 01 yy = Fn cos(2 0) + re. An interesting point is that the reflected intensity will always minimize near resonance for a Bragg angle of 0 = 45 . This occurs because cos(2 0) = 0, so that the numerator of Ry( , ) only contains re and is frequency independent, while the denominator contains Fn and maximizes at resonance. This is analogous to the Brewster angle effect in optics, but modified by the multipolarity of the scattering: for 0 = 45 and incident ey = , the magnetic dipoles respond to the incident 0 polarized B field, and hence oscillate in the direction of the reflected wave and do not contribute to the scattered field. Thus, the scattered wave is produced entirely by the electrons, which do strongly scatter the incident -polarized E field. However, the reflected wave from a layer will interact with the magnetic dipoles in upper layers, and this interaction dissipates energy from the reflected wave. 3.7. Grazing incidence For grazing incidence reflection, there are again two channels open: the t0 = (0+) channel and the symmetric reflection channel r = (0-). Grazing incidence reflection is, in fact, a special case of Bragg reflection with Bragg angle B = 0. If the incident photon is near grazing incidence with respect to the xy planes, the four scattering matrices F s s (s , s = 0, r) in eq. (3.74) can be taken equal to the forward scattering matrix F 00 to order (where is the angle of grazing incidence). In this approximation, the matrices can all be simultaneously diagonalized, and the equations separate into two sets of Darwin­Prins equations, as in the special Bragg cases considered in section 3.6. Furthermore, since only F 00 enters, the diagonalizing bases are the off-Bragg nonorthogonal eigenwaves e ( k, ) given by eq. (3.58). Since only forward scattering amplitudes are involved, the optical solutions are valid for noncrystalline films as well as for crystals. The equivalence of this "two-wave" dy- namical theory approach and the "one-wave" index of refraction approach is discussed in [29, appendix A]. 160 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 The Bragg reflection solutions of section 3.6 can then be used, with appropriate modifications for the nonorthogonality of the bases. For an incident e0 = e ( k0, ) photon, the reflected wave is given by eq. (3.83), which after some manipulation becomes R k r, = R e kr, , (3.101) 1 - (1 + 2 R ¯ f / 2)1/2 = , (3.102) 1 + (1 + 2¯ f / 2)1/2 where f is given by eq. (3.57). For the special cases for which the eigenbases are orthogonal, the reflected wave for an incident photon of arbitrary polarization is given by R kr, = e · e0 R e . (3.103) =1,2 For the general case where the e are nonorthogonal, the development follows the off-Bragg treatment, and the reflected wave is given by R k r, = R kr, k0, a0 k0, , (3.104) where the incident wave a0 is taken as a two-component column vector as in eq. (3.61), and the 2 × 2 reflection matrix is given by R k Rxx Rxy r, k0, = Ryx Ryy 1 f f R xx - fyy xy (+) + R(-) R(-) = 2 f1 - f2 f1 - f2 f 1 f , R yx xx - fyy (-) R f (+) - R(-) 1 - f2 2 f1 - f2 (3.105) where R(±) = R1 ± R2. (3.106) Here f and R ( = 1, 2) are given by eqs. (3.57) and (3.102). In figure 10(a) we plot |R( kr, )|2 as a function of for for several different frequencies about a resonance. Here we consider only a single resonance so that fn = f0/(x - i) with x = 2(Eres - )/ , and for f0 we use the factors appropriate for grazing incidence of right-circular-polarized radiation in near resonance with the m0 = +1/2 m1 = +3/2 resonance of 57Fe with B k0. Off resonance there is a region of near total reflection for < c = [-2¯ (f )]1/2 3.8 × 10-3 rad. In the resonance region, the scattering amplitude f becomes primarily imaginary, and there III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 161 Figure 10. (a) |R( , )|2 vs. for reflection from a resonant 57Fe mirror for several different frequencies. The crosses indicate the angles c( ). The X-ray critical angle is c = 3.8 mrad. (b) c( ) vs. / = ( - Eres)/ . The scale is mrad, and the dotted line indicates the X-ray critical angle c. is no well-defined critical angle. Nevertheless, a rough characterization of the curves is given by the "frequency-dependent critical angle" c( ), c( ) = max 2¯ (f ) , 2¯ (f ) . In figure 10(a) the positions of c( ) are indicated by crosses. The region < c( ) is a region of strong scattering for a wave of frequency . In figure 10(b) we plot c( ) for the same resonance. c( ) is enhanced on the high-frequency side of the resonance, where the nuclear contribution to the index of refraction interferes constructively with the electronic contribution, and the angular region of strong reflection is much broader than far off resonance ( c). Correspondingly, there is a minimum for c( ) on the low- frequency side of the resonance, where the nuclear scattering interferes destructively with the electronic scattering, and the angular region of strong scattering is much smaller than c. From figure 10(a) it is clear that if the beam is incident below the electronic c, then as a function of frequency the reflected intensity will exhibit a sharp minimum on the low-frequency side of the resonance (where c( ) minimizes). Similarly, if > c, the reflected intensity will have a sharp maximum on the high-frequency side of the resonance. In figures 11(a) and (b) we plot |R( , )|2 vs. for grazing-incidence reflection from 57Fe at angles = 3.0 and 4.4 mrad, which lie below and above the electronic critical angle c = 3.8 mrad. Here the magnetic field at the nucleus lies in the plane of the film and perpendicular to k0, so the eigenpolarizations are the linear basis , . The solid lines give the response to incident . The two M = 0 transitions couple only to radiation, and the four M = ±1 transitions couple only to radiation. 162 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Figure 11. (a) |R( , )|2 vs. for reflection from a resonant 57Fe mirror for (a) B k0, = 3.0 mrad, and (b) = 4.4 mrad. The solid line gives the response for incident polarization, the dashed line for incident . In (c), B k0, = 4.4 mrad. The solid line is for incident e(+1), the dashed line for incident e(-1). The strongest polarization coupling is that of polarization to the linear M = 0 oscillators (| · Y(m) 10 ( k0)|2 = 1, while | · Y(m) 1±1( k0)|2 = 1/2). The Clebsch­Gordan coefficients are C2 = 2/3 for the two M = 0 transitions, C2 = 1 or 1/3 for the two M = +1 transitions (the strongest is the +1/2 +3/2 transition), and similarly C2 = 1 or 1/3 for the two M = -1 transitions. For this geometry, the strongest oscillator strengths then occur for the M = 0 transitions (C2| · Y(m) 10 |2 = 2/3, while C2| · Y(m) 1±1|2 = 1/2 or 1/6), and we see in figure 11(b) that the strongest effects (higher peak reflectivities and broader widths) occur near these resonances. In agreement with our earlier discussion, for < c, a sharp minimum occurs on the low-frequency side of each resonance, while for > c a sharp maximum occurs on the high-frequency side of each resonance. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 163 In figure 11(c) we give the corresponding results for = 4.4 mrad with B parallel to k0, for which the eigenpolarizations are the circular basis e(±1). Now the solid line gives the response for e(+1) radiation, and the dashed line for e(-1) radiation. In this geometry the M = +1 transitions couple only to e(+1) radiation, the M = -1 transitions couple only to e(-1) radiation, and the M = 0 transitions are unexcited. The polarization coupling is now maximal, |e (±1) · Y(m) 1±1|2 = 1, and the oscillator strengths are C2|e (±1) · Y(m) 1±1|2 = 1 or 1/3. We note that in this geometry the oscillator strengths for the two strong M = ±1 transitions (±1/2 ±3/2, for which C2 = 1) are stronger than those of the M = 0 transitions with B k0, and, in fact, this geometry gives the maximum possible oscillator strength in the case of hyperfine splitting. The only stronger coupling is for the unsplit case, for which the oscillator strength is increased from 1 to 4/3. An important extension of these considerations is grazing incidence reflec- tion from multilayer media of resonant and nonresonant films, discussed by R¨ohlsberger [30] and Chumakov et al. [31] (in this issue), and in [29,32,33]. Thin-film interference techniques are of fundamental importance in modern optics. In the optical region, thin dielectric films are used to produce antireflection coatings, broad and nar- row bandpass frequency filters, polarizers, dichroic mirrors for color-selective beam splitting, and multilayer high-reflectance dielectric mirrors. Many of these techniques have been extended to the UV and X-ray regions. For -ray optics, the important point is that if one or more of the films are highly enriched M¨ossbauer films, the index of refraction for near resonant radiation differs strongly from the index of refraction far off resonance, and as a consequence it is possible to use thin-film interference techniques to strongly suppress the off-resonance grazing incidence reflectivity while maintaining a very "bright" reflection for near resonance radiation, offering an "ul- tranarrow" bandpass filter for synchrotron radiation. By using different combinations of films and substrates, the response can be tailored to give narrow bandpass widths and correspondingly long delayed scattering times to optimize time filter- ing, or at the other extreme, to produce filters of very broad resonance width with 100 , which would be ideal for a high resolution X-ray source [32,33]. 3.8. Laue For a Laue-case Bragg reflection from a M¨ossbauer crystal, the two open channels are the t0 = (0+) incidence channel, and a "transmission" channel t1 = (1+), as shown in figure 8(d). In parallel to the Bragg case, the general dynamical equations (3.40), (3.41) now contain only the two amplitudes T0k and T1k , C k T0k T1 = 0, (3.107) k 164 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 where the coefficient matrix C(k ) is now the 4 × 4 matrix e-i(g0-k )d - 1 I - i F 00 -i F 01 C k = . (3.108) -i F 10 e-i(g1-k )d - 1 I- i F 11 As in the Bragg case, the dispersion equation (3.44) is again of fourth order and, except for a few special cases, there is no simple analytical solution, but the numerical solution is easily obtained, as outlined in section 3.4. The special cases again occur when the 2 × 2 forward scattering matrices F 00 and F 11, and the reflection matrices F 10 and F 01 can be simultaneously diagonalized with an appropriate choice of orthogonal polarization bases e(0) and e(1) ( and = x, y). For these special cases, there is no polarization mixing x y, the four coupled linear equations of (3.107) separate into two sets of 2 × 2 coupled linear equations, and the resulting transmission matrices are diagonal. If the incident radiation is near a Bragg angle for a transmission channel t = (1+), as shown in figure 8(d), then (g0 - g1)d = 2n + 2 , (3.109) and the dispersion equation then gives ( = x, y) k = (g0d - + ± )/d, (3.110) where, for the Laue case, 1 = F00 , (3.111) 2 + F 11 = 2 + F01 F 10 , (3.112) 1 = + F 00 . (3.113) 2 - F 11 The square root is again taken so that ( ) > 0. The transmitted waves in the k0 and k1 channels are then given by T(0)( , ) = T ; k 0, k0; l a0 k0, , (3.114) T(1)( , ) = T ; k 1, k0; l a0 k0, ), (3.115) where the 2 × 2 transmission matrices T are diagonal, sin M T ; k 0, k0; l = cos M eiM( +g0d- ), (3.116) + i sin M T ; k 1, k0; l = iF 10 eiM( +g0d- ). (3.117) From eqs. (3.116) and (3.117), we see that for a thin crystal, very near Bragg, T (1) iMF 10 and T (0) (1 + iM ). As a function of frequency, the k1 wave in this III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 165 case will exhibit a maximum in the region of constructive interference between the electronic and nuclear scattering, while the k0 wave exhibits a weak minimum. For a thick crystal the dominant contributions to T (1) and T (0) come from the exponential term depending on the difference between and , i.e., 1 F 10 T (1) ( , ) - e(0) eiM( - +g0d- ), 2 · a0 (3.118) 1 T (0) ( , ) e(0) 1 - eiM( - +g0d- ). 2 · a0 The exponent in (3.118) determines the Laue penetration depth l for the radiation field, l -1 ( ) = d ( - ) . (3.119) Since the imaginary part of - will generally maximize in the resonance region, it is clear that, as a function of frequency, both the k0 and k1 waves will exhibit a minimum in the resonance region for a thick crystal (and it is only for the thin- film limit that a maximum will occur). Nevertheless, because the penetration depth depends on the difference between and , the transmitted intensity for the Laue case is greater than for the off-Bragg case, and for certain high symmetry situations in simple crystals, it is possible to strongly suppress either the nonresonant photoelectric absorption (X-ray Borrmann effect), or the resonant internal conversion absorption ( -ray suppression effect). 3.8.1. X-ray Borrmann effect For a symmetric Laue reflection for which the scattering planes are perpendicular to the crystal surface, 1 = 0 and F11 = F 00 . If, furthermore, there is only one atom per unit cell, then F 01 = F 10 . Under these conditions, at exact Bragg ( = 0), the penetration depth for e(0) incident radiation is l -1 ( = 0) = d F 00 - F 10 . (3.120) If the frequency of the radiation is large compared to any of the atomic absorption edge frequencies, the coherent elastic X-ray scattering amplitude is given to good approximation by the isotropic limit, eq. (A.13). When this holds, the imaginary part of F 10 differs from the imaginary part of F 00 by two factors: the Debye­Waller factor fD( ) and the polarization factor e f · e0, which is 1 for polarization and cos( ) for polarization. The Laue penetration depths are then 2 cos( /2) 1 l ( = 0) = , n e 1 - fD( ) (3.121) 2 cos( /2) 1 l ( = 0) = . n e 1 - cos( )fD( ) 166 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 For polarization, the Laue penetration depth is greatly enhanced over the off-Bragg penetration depth, 2 cos( /2) lOB = , (3.122) n e the enhancement being l 1 1 d2 = 1 - f -1 ¯ 2 = B , (3.123) l D( ) OB 2 sin2( /2) x2 2m2 2 x2 where x is the displacement of an atom from its equilibrium position due to zero point and thermal motion, and where, in the last expression, we have used the Bragg condition ksin = m /dB, where dB is the interplanar distance for the particular set of Bragg planes. For polarization, the enhancement is l = 1 - cos( )f -1 2sin2( /2) -1. (3.124) l D( ) OB For an iron lattice at liquid-He temperature, and = 20 , the l enhancement is 2.2 × 102, and the l enhancement is 17. The deep penetration of the polarization occurs because for this polarization the radiation field establishes a standing wave pattern in the crystal with nodes at the lattice sites, greatly reducing the photoabsorption. When the difference between the wave vectors k0 and k1 of two plane waves in the crystal is very nearly equal to a reciprocal lattice vector , as shown in figure 12(a), the waves satisfy the Bragg condition for reflection from the set of planes associated with and will be strongly reflected one into the other. Neither the wave in k0 nor that in k1 by itself constitutes an eigenwave of the radiation field in the crystal; rather, the eigenwaves are linear combinations of the waves in these two channels. There are two linearly independent polarization vectors associated with each beam in the mode, and , and thus there are four eigenmodes associated with each two-beam Laue reflection. When the Bragg condition is satisfied exactly, the wave field of these eigenmodes within the crystal consists of traveling waves along the direction of propagation (along k0 + k1) and standing waves perpendicular to that direction. At exact Bragg ( = 0) for a symmetric Laue reflection, incident -polarized radiation is a superposition of two eigenmodes: 1 E0 0eik0·r = E + E , (3.125) 2 |2 2 |1 2 where the eigenmodes are E E m x E 0 0 |2 = = eikz cos( /2)2i y sin , (3.126) 2 2 0eik0·r + 1eik1·r 2 dB E E m x E 0 0 |1 = = eikz cos( /2)2 y cos . (3.127) 2 2 0eik0·r - 1eik1·r 2 dB III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 167 Figure 12. (a) Structure of the two-beam symmetric eigenmode |2 2, corresponding to the reciprocal lattice vector , where the relative phase of the two beams is that which occurs at an atomic site. The lower figure shows a projection of the mode onto the plane perpendicular to the symmetry axis z of the mode. (b) Standing-wave fields produced within the crystal in mode |2 2. The solid vertical lines through the lattice points denote planes of nodal E. Here we have used the geometry shown in figure 12 (and figure 14 in the next sec- tion).In figure 12(b) we plot the standing waves of E|2 and its associated magnetic 2 field B|2 ,2 E B 0 |2 = - 2 2 0eik0·r - 1eik1·r -E m x m x = 0 eikz cos( /2) 2i x cos sin - 2 zsin cos . 2 2 dB 2 dB (3.128) Note that, in this mode, the equilibrium position of each atom in the crystal lies within a nodal plane of the electric field. In contrast, for the E|1 mode, eq. (3.127), each 2 atom is in an antinode of the electric field. At typical X-ray energies, photoabsorption is predominantly electric dipole, so the photoabsorption rate is proportional to the average squared electric field at the atom, |E|2 . In the Borrmann mode (|2 2), each atom lies in a region of low electric field and the photoabsorption rate of this mode is reduced from the off-Bragg value of µOB = n e to a value µ = geµOB = gen e, (3.129) where, to first order, ge |2 2 = 2k2 sin2( /2) x2 . (3.130) 168 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 The factor k2 sin2( /2) x2 is the product of the mean-square amplitude of oscillation of the atom about its equilibrium position and the square of the x, y component of each wave vector in the mode. This occurs because the atoms oscillate about their equilibrium positions, and the electrons sample the nonzero electric field on either side of the node. Comparing eqs. (3.130) and (3.123), we see that ge(|2 2) = lOB/l ; ge(|2 2) can be quite small ( 0.1­0.01), and radiation in the Borrmann mode will penetrate 10­100 times further in a good crystal than will radiation off-Bragg in the same material. In contrast, the |1 2 mode has an enhanced E-field at each site, giving ge |1 2 = 2, (3.131) and the two modes also have nonzero E-fields at the lattice sites, giving ge |2 2 = 2 sin2( /2), (3.132) ge |1 2 = 2 cos2( /2), (3.133) and as a consequence, the penetration depths (= lOB/ge) of these modes are much less than for the two-beam Borrmann mode |2 2. The E and B standing waves of the |2 2 mode are shown in figure 13. Returning to eq. (3.125), incident polarization is a superposition of E|2 2 and E|1 . Within the crystal, the E component is rapidly absorbed out, leaving 2 |1 2 the E|2 pattern deep in the crystal. The transmitted field at the bottom surface is 2 then E|2 , which then divides into two equal amplitude channels k 2 0 and k1, each carrying 25% of the incident intensity (if l l ). This is just the solution for T (0) and Figure 13. (a) Structure of the two-beam symmetric eigenmode |2 2, corresponding to the reciprocal lattice vector , where the relative phase of the two beams is that which occurs at an atomic site. The lower figure shows a projection of the mode onto the plane perpendicular to the symmetry axis z of the mode. (b) Standing-wave fields produced within the crystal in mode |2 2. The solid vertical lines through the lattice points denote planes of nodal B. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 169 T (1) given by eq. (3.118) at exact Bragg, for the highly symmetric situation considered here. 3.8.2. -ray suppression effect For the resonant M¨ossbauer nuclei, the effect is even more striking: because the resonant nuclear transitions take place over times much longer than crystal vibration times (10-8 s or longer as compared to 10-13 s), in contrast to the electronic interac- tions, which occur over a much shorter time period ( 10-16 s), it is possible for the nuclei to effectively decouple from a two-beam Laue mode. Under these conditions, there is a complete suppression of all elastic and inelastic resonant -ray processes, and the nuclei become completely transparent to the mode. However, because the multipolarity of the M¨ossbauer transition is generally M1 or higher, the suppression effect mode will generally be different from the E1 Borrmann mode |2 2. This effect was first discussed by Afanas'ev and Kagan [14,34,35]. As for the X-ray Borrmann effect, the suppression effect is optimal in a simple crystal with one atom per unit cell (so F 01 = F 10 for the "isotropic" limits) and for a symmetric Laue reflection (so 1 = 0 and F11 = F 00 ). Then at exact Bragg ( = 0), the penetration depth l again depends inversely on (F00 - F 10 ), as given by eq. (3.120), but now the planar scattering amplitudes F contain both the nuclear and electronic scattering amplitudes, F = n d(fe +fn)/ sin 0. Assuming the simplest situation of an unsplit M¨ossbauer line, the nuclear scattering amplitudes fn are given by the isotropic limits given in eqs. (A.47)­(A.49) for E1, M1 and E2 transitions, respectively. Because the resonant scattering is slow compared to crystal vibration times, the nuclear scattering amplitude contains the M¨ossbauer phonon factor fM = exp(-k2 x2 ) for both k0 k1 scattering and for forward scattering k0 k0. Thus, in contrast to the electronic scattering, which contains the Debye phonon factor fD( ), F10 (n) differs from F 00 (n) only by the polarization factor of the nuclear scattering. As a consequence, for the nuclear scattering it is possible to have a complete equality of the 0 1 and the 0 0 scattering amplitudes, F 10 (n) = F 00 (n). When this occurs, the nuclei become completely transparent to the deeply penetrating Laue mode ­ i.e., there is a complete suppression of all resonant nuclear scattering and absorption in this mode. The penetration depth of the suppression effect mode (SE mode) is determined entirely by photoabsorption, l = d/ (F00 (e) - F 10 (e)). If the M¨ossbauer transition is E1 (e.g., the 6.25 keV transition of 181Ta or the 25.7 keV transition of 161Dy), then fn e f · e0 for an unsplit line, just as for fe in the isotropic limit. Then for incident 0 radiation, F10 (n) = F 00 (n), and the deeply penetrating SE mode is just the Borrmann |2 2 mode with nodes of the E field at the lattice sites, with penetration depth given by eq. (3.121). The other two-beam Laue modes (|1 2, |2 2, |1 2), which all have antinodes of E at the nuclei, are strongly absorbed at resonance. Most M¨ossbauer transitions, however, are M1 or E2. For an M1 M¨ossbauer transition, for an unsplit line, fn b f · b0, where b = ( k × e) is the magnetic 170 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 polarization vector of the radiation field. Thus, an M1 transition responds to the b vector of the radiation field just as an E1 transition responds to the e vector. As a consequence, it is for e = radiation (for which b = ), that the SE condition holds, i.e., F 10 (n) = F 00 (n). The penetration depth l is strongly limited by photoabsorption, however, as given by eq. (3.123). The effective decoupling of the polarization from the M1 nuclear resonance occurs because for this polarization the radiation field is able to establish a standing wave pattern in the crystal with nodes of the magnetic field at the lattice sites. At exact Bragg ( = 0) for a symmetric Laue reflection, incident -polarized radiation is a superposition of two eigenmodes, 1 E0 0eik0·r = E + E , (3.134) 2 |2 2 |1 2 in parallel to eq. (3.125). The structure of the |2 2 mode is shown in figure 13. Note that in this mode, the equilibrium position of each atom in the crystal lies within a nodal plane of the magnetic field, but in an antinode of the electric field. In the |1 2 mode, there is a 180 phase change of the k1 wave, and the resulting standing wave patterns are shifted by 90 from the |2 2 mode, so the |1 2 mode has antinodes of both E and B at the lattice sites. Thus, at resonance, the |1 2 component of the incident radiation is rapidly absorbed out by resonant nuclear absorption, leaving the |2 2 pattern deep in the crystal. Resonant nuclear transitions take place over times much longer than crystal vibration times (10-8 s or longer as compared to 10-13 s), in contrast to the electronic interactions, which occur over a much shorter time period ( 10-16 s). The coupling between the radiation mode and the nuclei is, thus, proportional to the square of the average of the appropriate component of the field at the nucleus, in this case | B |2 = 0, rather than the average of the square |B|2 as one would expect for a fast process [1]. Note that the M1 suppression effect mode |2 2 is not a Borrmann mode, since it has a nonzero electric field at the equilibrium site of the atom and thus couples strongly to the electrons. Conversely, the Borrmann mode |2 2 will couple strongly to M1 M¨ossbauer nuclei, since, in this mode, each nucleus lies in a region of strong magnetic field (see eq. (3.128) and figure 12(b)). This mode will couple to higher-multipole transitions as well since the electric field gradient (EFG) and higher derivatives of the field are also large at the oscillator. In general, it is not possible to obtain simultaneous suppression of both the nuclear and electronic coupling in the two-beam case for M¨ossbauer transitions of multipolarity M1 or higher in the absence of Zeeman splitting. As we shall see in the next section, this is possible for multibeam modes. For an unsplit E2 M¨ossbauer transition, f n kf · k0 e f · e0 + kf · e0 k0 · e f . Because fn depends on the directions of both the wave vector k and of the polar- ization e, the E2 scattering is anisotropic for both and polarizations, F 10 (n) = F 00 (n) cos( ) and F 10 (n) = F 00 (n) cos(2 ), where is the scattering angle k0 k1. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 171 The classical radiation pattern being produced by the "isotropic" E2 oscillator driven by incident |k0e0 radiation is a fourfold E2 pattern, with lobes along ±e0 and ± k0. Thus, for incident 0, which lies perpendicular to the (k0, k1) scattering plane, F 10 (n) = F 00 (n) only for 180 backscattering, k1 = -k0; while for incident 0, which lies in the (k0, k1) scattering plane, F10 (n) = F 00 (n) for 90 scattering as well as for 180 scattering. As a consequence, there is no suppression effect possible for an unsplit E2 transition, except for the very special situation of a 90 Bragg angle, in which case the SE condition holds for radiation, F 10 (n) = F 00 (n). Higher multipoles will also exhibit anisotropic scattering in any scattering plane, so in the absence of hyperfine splitting, there can be no two-beam suppression effect for transitions of multipolarity E2 or higher, except for certain accidental situations where a Bragg angle coincides with a scattering symmetry. To this point we have assumed that there is no hyperfine splitting. If there is hyperfine splitting, then it is possible to have a two-beam suppression effect for selected Jz nuclear transitions for any multipole order and it is also possible to obtain a simultaneous suppression of both the electronic and nuclear coupling in the two-beam case: The simplest situation is to have the local quantization axis zJ of each nucleus oriented parallel to the symmetry axis z of the symmetric Laue channel (i.e., the sample is ferromagnetically ordered in the z direction). For this symmetrical arrangement, k0 and k1 each make a polar angle /2 relative to zJ (see figure 12), and the axial angles 0 and 1 differ by , so that exp[iM( 1 - 0)] = (-1)M. It then follows that the couplings of the oscillator to the two channels are related by e(1) · Y(q) LM k1 = (-1)M e(0) · Y(q) LM k0 , (3.135) for e = or (see, e.g., the explicit Y(q) LM expressions for E1, M1 and E2 given in eqs. (A.22)­(A.24)). As a consequence, using the simple interference criteria discussed in the following section, for even order transitions ( Jz = 0, ±2, . . .), there is no coupling to either |1 2 or |1 2; and for odd order transitions ( Jz = ±1, ±3, . . .), there is no coupling to either the two-beam Borrmann mode |2 2 or to the mode |2 2. Thus, simultaneous SE and Borrmann effects occur for any well-isolated odd order Jz transition (symmetrically oriented as discussed above), for example, for an M1 transition with Jz = +1. From the semiclassical point of view, the B-field standing wave of the |2 2 mode is in the z direction, as shown in figure 12, which is orthogonal to the ( x, y)-oscillation plane of the M1, Jz = +1 oscillator, so the |2 2 mode does not couple to this nuclear transition. However, the mode does couple to the Jz = 0 components, so strong resonance absorption will occur if the frequency is shifted to a Jz = 0 resonance. 3.8.3. Interferometry, Borrmann and suppression The photon mode established deep in the crystal by the scattering/absorption in the upper layers (the |2 2 mode for the Borrmann case, or the |2 2 mode for the isotropic M1 suppression effect) is a coherent two-channel mode. This is exactly 172 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 the situation produced in Bonse­Hart X-ray interferometry in the final plate of the interferometer where the two beams are brought back together. The suppression of absorption (and scattering) that occurs in the Borrmann and SE effects is due to destructive interference between competing quantum processes. Consider an atom which can be moved about the unit cell (for example, on the lower surface of the crystal). Let its equilibrium position be r, with r = 0 a lattice site (or extended lattice site on the lower surface). If the photon is in the |2 2 state, then the transition matrix element for K-shell photoabsorption is f |HI| 0 k |p · ei(k0-ke)·x ei(k0-ke)·r e 0| K f 0 + k |p · ei(k1-ke)·x ei(k1-ke)·r, (3.136) e 1| K f 0 where ke is the wave vector of the ejected photoelectron, and 0 and f denote the initial and final crystal vibrational states. Since 1 = - 0, there will be destructive interference between the two processes if the atom is located at a lattice site (r = 0), while at an interstitial, the two absorption amplitudes interfere constructively. The resulting total photoabsorption cross-section is then = e 1 - fD(k1 - k0) cos(k1 - k0) · r , (3.137) where e is the usual photoabsorption cross-section, and fD(k1 - k0) = fD( ) is the Debye phonon factor for photon scattering k0 k1. Thus, at a lattice site, photoabsorption is suppressed by the factor [1 - fD( )] 2 sin2( /2) x2 /¯ 2. For an iron lattice at liquid-He temperature, with 14.4 keV radiation and = 20 , the suppression is 4.5 × 10-3. Scattering is also suppressed in the Borrmann effect. The scattering amplitude for photon scattering |2 2 |k, e from an atom located at a position r within the unit cell is 1 k, e| fe(r)|2 2 = f , (3.138) 2 e(k, e; k0, 0)ei(k0-k)·r + fe(k, e; k1, 1)ei(k1-k)·r which is the coherent superposition of the two scattering processes |k0, 0 |k, e and |k1, 1 |k, e . These two contributions interfere destructively if the atom is located at a lattice site (r = 0) and if fe is given by the isotropic limit (A.14), but the cancelation is not as complete as for photoabsorption: in addition to having different phonon factors as in eq. (3.136), the real parts also have different form factors, n(Q0) and n(Q1) . The resulting cross-section for scattering is then suppressed by a factor [1 - n(Q0) n(Q1) d k/ n(Q0) 2 d k], typically 10-1. For the nuclear scattering and absorption processes, it is possible to have com- plete suppression. This occurs for two reasons. First, the M¨ossbauer phonon fac- tor fM = exp(-k2 x2 ) replaces the Debye­Waller factor fD(Q) = exp(-12Q2 x2 ). The important point is that fM is essentially isotropic, while fD depends on the momentum transfer and, hence, varies with the scattering angle . Secondly, there is no Q-dependent form factor for nuclear scattering. Instead, the angular depen- dence is determined entirely by the multipole pattern of the particular QLM os- III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 173 cillator. Thus, for a properly oriented nuclear oscillator, it is possible to have fn(k, e; k0, e0) = fn(k, e; k1, e1), and complete suppression can be achieved. For resonant scattering or absorption, the transition amplitude has the general form f|HI| n n|HI(em)| 0 , (3.139) [E n n - E0 - - i n/2] where HI(em) = -c-1 dx jn(x) · A(x) is the electromagnetic interaction potential for the nucleus, and HI is the appropriate interaction potential for the decay. If the incident photon is in a coherent two-beam superposition state |m 2 (m = 1 or 2, = or ), then the photon absorption part of the resonant amplitude (3.139) has the form 0 H I (em) m J(q) Y(q) k · e(0) k · e(1) , (3.140) 2 LM LM 0 + (-1)mY(q) LM 1 where we have assumed that the nuclear transition is a pure QLM multipole transi- tion, and that the nucleus is located at a lattice site. Substituting (3.140) into (3.139), the resonant transition amplitude becomes a coherent superposition of two compet- ing amplitudes, arising from the incident photon being in the k0- or the k1-channel, respectively. Complete destructive interference between these two amplitudes occurs if Y(q) LM k0 · e(0) + (-1)mY(q) LM k1 · e(1) = 0. (3.141) If the local quantization axis zJ of each nucleus is oriented parallel to the symmetry axis z of the symmetric Laue channel (i.e., the sample is ferromagnetically ordered in the z-direction), then Y(q) LM ( k0) · e(0) and Y(q) LM ( k1) · e(1) are related by eq. (3.135), leading to the following conclusions: for even order transitions ( Jz = 0, ±2, . . .), there is a complete suppression of resonant scattering and absorption in both the |1 2 and the |1 2 modes; and for odd order transitions ( Jz = ±1, ±3, . . .), there is complete suppression in the two-beam Borrmann mode |2 2 and in the |2 2 mode. Since the even and odd Jz transitions couple to different sets of modes, it is generally not possible to have an SE effect if there is no hyperfine splitting, except for E1 or M1 transitions. For an unsplit E1 transition, if we take zJ parallel to z as above, the Jz = 0 transitions correspond to a classical charge oscillating along the z axis and there is no coupling to -polarization. Thus the |2 2 mode will not couple to the Jz = 0 transitions and, as discussed above, suppression holds for this mode for the remaining Jz = ±1 transitions. Similarly, for an unsplit M1 transition, the SE effect holds for the |2 2 mode because the Jz = 0 transitions will not couple to -polarization, and suppression holds for the Jz = ±1 transitions in this mode. 3.8.4. Enhancement and suppression The enhancement effect and the nuclear Borrmann effect are distinct effects, but there is an aspect linking the two effects. In the Russian literature, the anomalous transmission at Laue, and the near total reflection occurring at Bragg in a thick crystal, 174 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 are both considered aspects of a single effect, the "suppression of inelastic chan- nels" [14,15]. We use the term "suppression effect" to mean specifically the nuclear Borrmann effect. For the nuclear Borrmann effect, which occurs in the symmetric Laue geometry, there is a complete suppression of all elastic and inelastic resonant processes. This is a quantum interference effect in which the photon is first put into a coherent two- beam symmetric Laue state (the |2 2 state for E1 Borrmann or the |2 2 state for M1 Borrmann) by scattering and absorption from the atoms in the upper layers of the crystal. Once the mode is established, which requires 50% absorption of the incident |k0, e(0) radiation, there is a complete suppression of all further resonant scattering or absorption processes for a M¨ossbauer atom placed at a lattice site. This occurs because there are two coherent amplitudes leading to any final state | f ­ either the absorbed photon is incident in the |k0, e(0) channel, or it is incident in the |k1, e(1) channel ­ and there is complete destructive interference between the two amplitudes. The nuclear Borrmann effect is not a collective effect of coupled resonators. Once the coherent photon state is established, suppression occurs equally well for a single M¨ossbauer atom at a lattice site as for a collection of M¨ossbauer atoms at different lattice sites. Furthermore, the resonators need play no role in establishing the coherent two-beam photon state. For example, for E1 resonators, the |2 2 mode can be established by normal X-ray scattering and absorption in the upper layers, with the M¨ossbauer isotope atoms only being introduced at depths on the order of the Borrmann penetration depth, or at extended lattice sites on the exit surface. The enhancement effect, on the other hand, is manifestly a collective effect between the interacting resonators: when a crystal of M¨ossbauer atoms is excited resonantly by an external monochromatic plane wave at a Bragg angle (in the Bragg geometry), there is an enhancement of the partial width for emission via coherent decay ( c = 2 n¯ 2l (k0)) and a consequent suppression of the probability for incoherent decay (P = /( c + + )), and of the probability for internal conversion decay (P = /( c + + )). In the most ideal case, a single superradiant normal mode of the system of nuclei is excited. c is proportional to the thickness of the crystal, so for sufficiently thick crystals the probability for coherent decay approaches unity. From the temporal standpoint, the time scales for the various decay modes are / c, / and / , respectively, so if the crystal is sufficiently thick that c ( + ), then with high probability the excited state decays "quickly" by coherent emission, rather than by the "slow" incoherent, inelastic decay processes. This suppression of the inelastic processes which occurs at Bragg does have a "standing wave" interference aspect, analogous to the X-ray Borrmann and nuclear Borrmann effects: at exact Bragg, the excitation amplitudes of the nuclei are uniform throughout the crystal, each reduced to 1/M (relative to "unity" in the incident plane wave E0 alone). This occurs because in each layer, the wave Tm incident from above and the wave Rm incident from below almost cancel, the total field at the plane being reduced by 1/M. For example, for a system of E1 resonators, with the incident field E0 being polarized, Tm + Rm = ±(1/M)E0. (At the bottom layer, RM = 0 III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 175 and TM = ±(1/M)E0; at the top layer, T0 = E0 and R0 = -(1 - 1/M)E0.) Thus, for thick crystals, the local field which each nucleus is subjected to is very nearly a |2 2 mode, which is the E1 Borrmann mode. (The propagation direction now lies in the plane in the direction of (k0)xy, with "nodal" planes coinciding with the M layers.) The probability of incoherent scattering or absorption from each nucleus is then reduced by (1/M)2. But there is, of course, no suppression of the coherent elastic scattering by the crystal: the amplitude for elastic scattering by each nucleus is reduced by 1/M, but these amplitudes add coherently over the entire crystal, resulting in the enhanced coherent decay rate c and near total reflection. In the enhancement effect, there is a strong enhancement of coherent elastic scattering, and correspondingly, a strong suppression of the inelastic processes; in the nuclear Borrmann effect, there is a complete suppression of all processes, elastic as well as inelastic. For a system of resonators, there is a fundamental distinction between the Bragg and Laue geometries: for Bragg excitation, a superradiant normal mode is excited, while for Laue or off-Bragg excitation this is not the case. The normal modes of decay for a crystal of parallel planes are necessarily Bragg geometry modes because for a normal mode all planes must be mutually coupled ­ i.e., the upper planes must communicate to lower planes (T -channels) and the lower planes must communicate to the upper planes (R-channels). For the Laue case, and for off-Bragg, there is only forward communication (T -channels). Radiative normal modes are discussed in section 6. When a crystal is excited by a synchrotron pulse rather than by a monochro- matic plane wave, we also use the term "coherent enhancement" to describe the superradiant decay of the resulting nuclear exciton | e(k0) , because this is again a collective effect involving the same enhancement of the coherent decay rate, c(k0) = m · n¯ 2l (k0) , where k0 is the phasing vector (= ( 0/c)n0), and m is the number of open channels. But there is an important difference now: the initial decay is superradiant regardless of the orientation of k0 ­ i.e., whether k0 satisfies Bragg-, off-Bragg-, or Laue-conditions. But the origin of the superradiance, and the delayed decay (t > -1 c ), is radically different depending on whether or not k0 satis- fies the Bragg condition. In all cases, there is also a suppression of inelastic processes, but this is now accomplished by the speedup of the coherent decay mode relative to the unaltered decay rates into the inelastic modes (there is no standing wave interference involved here). These aspects are discussed in section 6. 3.9. Multibeam Borrmann In the two-beam Borrmann modes, the electric field is zero at the equilibrium position of each atom in the crystal, but the magnetic field and higher derivatives of the field (electric field gradient, magnetic field gradient, . . . ) are generally quite large in the vicinity of the atoms. Other modes exist in which the magnetic field is zero at each atomic site, but which have a large electric field near the atoms. Although there 176 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 are no two-beam modes in which the electric and magnetic fields are simultaneously zero at the atomic sites, there are multibeam modes, which occur when three or more beams simultaneously satisfy the Bragg condition, which do have both the electric and magnetic fields (or, in some cases, other higher-multipole derivatives of the field) simultaneously zero at the lattice sites. Other multibeam Borrmann modes exist in which the magnetic field (and other field derivatives) are enhanced near the vicinity of the atoms. (We use the term "Borrmann mode" to refer to any mode in which the electric field is zero at each atomic site in the crystal.) As discussed by Hutton et al. [17], these properties of multibeam modes, coupled with the well-defined multipolarity of M¨ossbauer transitions, lead to three particularly interesting effects which occur at multibeam points of good single crystals containing M¨ossbauer nuclei: (1) Certain multibeam Borrmann modes exhibit suppressed coupling both to the elec- trons and to the nuclei and will be anomalously transmitted at resonance, even when the nuclear transition is of multipolarity M1 or higher and there is no Zee- man splitting. Such simultaneous suppression is not possible in the usual two-beam case. (2) Other multibeam Borrmann modes exhibit particularly strong coupling to the nu- clei, so that nuclear absorption is greatly enhanced. (3) Excited M¨ossbauer nuclei located at lattice sites within a crystal will have en- hanced emission into certain of these weakly attenuated multibeam Borrmann modes ("anomalous emission"). While these last two effects can also occur in the two-beam case, greater enhancements are possible in multibeam modes, since the coupling is proportional to the number of beams in the mode. 3.9.1. Multibeam modes Multibeam modes of the radiation field occur when the Bragg condition is satisfied simultaneously by three or more plane waves (beams) within the crystal. Except for certain very special frequencies, all the reciprocal lattice vectors involved in such a mode will be coplanar such that their endpoints lie on a circle. The wave vectors then lie on the surface of a cone whose base is defined by this circle, as shown in figure 14. The difference between any two of these vectors is then nearly equal to a reciprocal lattice vector and the wave in each channel will, in general, be strongly reflected into each of the others. For multibeam modes, we take to be the half-apex angle of the cone, as shown in figure 14 (previously, was the scattering angle k0 k1). Again, there are two linearly independent polarization vectors associated with each beam, so there are 2m eigenmodes associated with each m-beam point. We restrict our consideration here to modes formed about high-symmetry axes. In our analysis we assume a perfect crystal which contains resonant nuclei located at lattice sites. Finally, we limit our discussion to that point in phase space where the III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 177 Figure 14. General structure of a multibeam mode, showing the polarization basis chosen for each beam. The 's lie in the plane containing the reciprocal lattice vectors (along (k) in the spherical coordinate system ( k, , ), where now is the angle from the symmetry axis z to the wave vector k), and the 's lie out of this plane, along the (k) direction. The figure on the right shows a projection of the mode onto the plane perpendicular to z. The four-beam case is shown, but similar modes exist for other numbers of beams. Bragg condition is satisfied exactly for all the reflections in the mode and we restrict our attention to those modes whose structure is fully determined by the symmetry of the crystal [36]. The symmetry of a mode is determined by the symmetry of the axis of the reciprocal lattice about which it is formed. When there is no hyperfine splitting, the symmetry group of a mode centered about an n-fold rotation axis of the reciprocal lattice is the group of an n-sided regular polygon, that is, the group of rotations and reflections which leave that polygon invariant. When there is Zeeman splitting, with zJ parallel to the mode symmetry axis z, reflections are no longer valid symmetry operations and the group reduces to the group of rotations which leaves the n-sided polygon invariant. The set of eigenmodes associated with an m-beam case whose cone is centered about an m-fold rotation axis of the reciprocal lattice will contain either two (if m is odd) or four (if m is even) eigenmodes which are fully determined by symmetry,9 and as such will be identical to the corresponding eigenmodes in the X-ray case, as determined by Joko and Fukuhara [37]. In complete analogy to the two-beam 9 The set of 2m basis vectors {| s , | s }, which spans the space of an m-beam mode, defines a regular representation of the symmetry group Cmv. In such a case, the number of times each irreducible representation appears among the eigenmodes is equal to the dimension of that representation (see, e.g., L.D. Landau and E.M. Lifshitz, Quantum Mechanics. Non-Relativistic Theory (Pergamon, Ox- ford, 1965)). One-dimensional irreducible representations appear only once and the structure of the eigenmodes to which they correspond is fully determined by symmetry. Cmv has either two (if m is odd) or four (if m is even) such one-dimensional irreducible representations. Cmv (m > 2) also has one or more two-dimensional irreducible representations. The modes to which they correspond are not fully determined by symmetry and their structure thus depends upon the multipolarity of the nuclear transition and the relative strength of the nuclear and electronic responses. 178 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Figure 15. Structure of the three- and four-beam |m m, |(m/2) m and |(m/2) m Borrmann modes. In each case, the excitation pattern shown is that which occurs at an atomic site. results, these modes have either all or all polarization channels excited with equal amplitude. |m m and |m m have neighboring beams in phase so that they are invariant under a rotation of 2 /m about the cone axis (the notation used for the modes is that of Afanas'ev and Kohn [38]10). In contrast, |(m/2) m and |(m/2) m have neighboring beams out of phase, so that they change sign under such a rotation [38]. |(m/2) m and |(m/2) m exist only when m is even and are Borrmann modes for m 4, while |m m is always a Borrmann mode; |m m is never a Borrmann mode. The structure of the three- and four-beam |m m, |(m/2) m, and |(m/2) m Borrmann modes are shown in figure 15. 3.9.2. Photoabsorption in the multibeam Borrmann modes As for the two-beam Borrmann case, in the multibeam Borrmann modes each atom lies in a region of low electric field and the photoabsorption rate is reduced to µ = gen e, where the electronic coupling factors ge for these modes are, to first order, m x2 sin2 , |m m , g 2 k2 m, 2 m e = m m (3.142) 2 k2 x2 cos2 sin2 , 2 , m where is now the half-apex angle of the cone, as shown in figure 14. These modes differ in symmetry from the two-beam Borrmann mode in that here each atom lies on 10 The subscript denotes the number of beams, the Greek letter denotes the polarization, while the first number gives the symmetry of the mode. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 179 a nodal line of the electric field, rather than in a nodal plane. This accounts for the additional factor 2 in eq. (3.130).11 3.9.3. Multipole coupling to the multibeam Borrmann modes Multibeam modes couple only to particular Jz components of multipole tran- sition currents (where Jz is the change of angular momentum about the symmetry axis of the mode, which we take to be the z axis). As a consequence of this and certain polarization properties discussed below, multibeam modes couple only to par- ticular electric multipole transitions of order L (EL) and particular magnetic multipole transitions (ML). The rules for coupling the multibeam Borrmann modes considered above to multipole currents may be summarized as follows:12 Mode Couples only to Couples to multipole |m m Jz = 0, ±m, ±2m, . . . ML: L 1; EL: L m |(m/2) m, |(m/2) m Jz = ±m/2, ±3m/2, . . . ML: L m/2; EL: L m/2 As discussed below, these results are an immediate consequence of the rotational symmetries and polarization properties of the multibeam Borrmann modes and of the multipole radiation fields emitted by the various Jz transitions. The radiation pattern of |m m is invariant under rotation by 2 /m about the symmetry axis of the mode, while the amplitude of the radiation field emitted by a Jz transition transforms as exp(i Jz ). Hence, |m m can couple only to Jz = 0, ±m, ±2m, . . . transitions. For magnetic multipole transitions (ML), |m m couples to all orders (L) through the Jz = 0 components. This is easy to see physically since the circulating radiation pattern associated with these modes is polarized entirely along and, for example in the M1 case, corresponds exactly to that of a classical magnetic dipole oscillating along the z axis. The radiation pattern associated with Jz = 0 components of electric transitions, on the other hand, is polarized entirely along and is thus everywhere orthogonal to the field in |m m. Thus, these modes do not couple to any electric transition (EL) with L < m. For example, |3 3 does not couple to E1 or E2 transitions, but will couple to E3 transitions through their Jz = ±3 components. Similarly, |4 4 will couple to E4 transitions, but not to E1, E2, or E3. Similarly, the radiation patterns of |(m/2) m and |(m/2) m change sign under a rotation by 2 /m about the symmetry axis of the mode, so these modes couple only to Jz = ±m/2, ±3m/2, . . . components of multipole currents, which have radiation 11 E1 polarization response goes as e 1 · e2, so that to this order, |4 4 and |2 4 may be regarded as two independent |2 2 modes. Thus ge for these two modes must equal that for |2 2, hence, the additional factor 2 in eq. (3.130). 12 Other modes exist (see [17,37,38]) which couple to components of multipole currents other than Jz = 0, ±m/2, ±m, . . . . For example, there exist four- and six-beam modes which couple only to Jz = ±1, ±(m + 1), . . . components, such as the |1 , 4 mode discussed in [17]. Similarly, there are six-beam modes which couple only to Jz = ±2, ±8, . . . components. 180 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 patterns that go as exp[i(±m/2) ] and exp[i(±3m/2) ], . . . , respectively. Thus, |2 4 and |2 4 couple to M2, E2, and higher multipole nuclear transitions, but do not couple to M1 or E1. Similarly, |3 6 and |3 6 couple to M3 and E3 transitions, but not to M1, E1, M2, or E2. A nuclear coupling factor gn may be defined in a manner exactly analogous to that used to define ge in eqs. (3.129) and (3.142), so that the rate at which a mode is absorbed by the nuclei is given by µn = gnn n, (3.143) The nuclear coupling factor gn is given by ( g B| fn| B ) n = , (3.144) ( OB| fn| OB ) where | B is a multibeam eigenmode of the crystal such as those considered above, consisting of a coherent superposition of single plane waves, |ki, e(i) , given by | B = i ki, e(i) , (3.145) i with | i |2 = 1. (3.146) i | OB = |k, e is a single plane wave off-Bragg. If there is no Zeeman splitting, then OB| fn| OB is independent of k and e. The gn's for coupling the Borrmann modes described above to the first few nuclear multipole transitions are given in table 1. The expression gn,M2 for |2 2 differs from that for the other |m m modes because |2 2 couples both to Jz = 0 and to Jz = ±2 components, which are allowed for M2 transitions, while the other |m m modes couple only to Jz = 0. Table 1 gn's for coupling Borrmann modes to good multipole nuclear transitions. Only the first two or three nonzero coupling factors are shown in each case. The symbols C and S used in the table refer to cos and sin , respectively. Note that since these are Borrmann modes, none of them couple to E1 nuclear transitions. Transition Mode E1 M1 E2 M2 E3 M3 |2 2 0 2S2 2S2 4S2C2 |m m 0 mS2 0 (3/2)mS2 |2 4 0 0 4S2 4S2C2 |2 4 0 0 4S2C2 4S2 |3 6 0 0 0 0 (45/8)S4 (45/8)S4C2 |3 6 0 0 0 0 (45/8)S4C2 (45/8)S4 III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 181 3.9.4. Multibeam Borrmann modes in -ray optics These properties of multibeam modes, coupled with the well-defined multipolarity of M¨ossbauer transitions, lead to three particularly interesting effects which occur at multibeam points of good single crystals containing M¨ossbauer nuclei: Simultaneous suppression of electronic and nuclear coupling. It follows immediately from the simple coupling rules given above that certain multibeam Borrmann modes do not couple to M¨ossbauer transitions of particular multipolarities (see table 1). Since both the electric field and the multipole component of the electromagnetic field driving the nuclear transition are zero at the equilibrium site of the atoms in such a mode, the coupling to both the electrons and the nuclei will be suppressed, regardless of whether or not there is Zeeman splitting of the nuclear transition. As noted before, this is not possible in the two-beam case. For example, in the |2 4 Borrmann mode, there is no coupling to an M1 M¨ossbauer transition. This mode may be regarded as two independent |2 2 modes, rotated by 90 from each other and 180 out of phase. As a consequence, the nonzero magnetic standing wave Bz of the |2 2 mode, shown in figure 12, is canceled, so the electric and magnetic fields are simultaneously zero at the lattice site in the |2 4 Borrmann mode. It is also interesting to note that for any |m m Borrmann mode with m 3, there is no coupling to an E2 M¨ossbauer transition. In contrast, in the two-beam case there is no E2 suppression effect if there is no Zeeman splitting. Enhanced nuclear absorption in multibeam Borrmann modes. While some Borrmann modes do not couple to particular nuclear transitions, others have an enhanced coupling and will be absorbed more strongly than will off-Bragg modes. While enhanced absorption occurs in the two-beam case [12,13,39,41], much greater enhancements are possible for multibeam modes. For example, for an unsplit M1 transition, the coupling to the |m m Borrmann mode is gn = m sin2 , directly proportional to the number of beams. This occurs because the M1 multipole couples to the mode through the Jz = 0 transition which is driven by the Bz component of the incident radiation, and for the |m m mode, the m separate contributions to Bz coming from the m beams of the mode all add constructively at the lattice sites, leading to the m-fold enhancement of the resonant cross-section (m rather than m2 because there is also a factor 1/ m in the normal- ization of the photon state |m m). Anomalous emission in multibeam Borrmann modes. It may be shown by applying the reciprocity theorem that the factor gn, which gives the coupling between a radiation mode and a nuclear absorber, also describes the coupling between that mode and a nuclear emitter at the same site (section 3.10.2). Thus, |m m modes are fed by M1 and higher-order magnetic multipole emitters, |3 6 and |3 6 by E3 and higher, etc. The fact that M1 and higher multipolarity excited nuclei can emit into certain of these weakly attenuated Borrmann modes is responsible for the predicted "anomalous 182 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 emission effect", in which waves emerge from a crystal from sources located at lattice sites deep within the crystal [13,17,39]. 3.9.5. Multibeam Borrmann modes and crystalline -ray lasers If a -ray laser can be achieved, it will necessarily have to rely upon the M¨ossbauer effect [42], so that the resonant nuclei must reside in a cool solid. There are important practical difficulties associated with pumping such a system while main- taining the conditions necessary for an appreciable recoilless fraction [43]. If these difficulties can be overcome, then there are significant advantages to be obtained by using a single crystal [40,41,44,45,48]. Lasing requires that the gain from stimulated emission exceed the losses from nonresonant absorption. The Schallow­Townes [46] steady-state lasing condition is K 0, where K is the net gain in intensity per unit length, which for a -ray laser is given by K = K0 - µe = ng n n - eg en0. (3.147) For simplicity, we have assumed no hyperfine splitting. According to the linearized theory [47], the wave in the lasing mode will then amplify by eKl on traversing a distance l through the crystal. In eq. (3.147) n0 is the number density of atoms in the crystal, while e is the nonresonant atomic absorption cross-section, which for M¨ossbauer energies (10­ 100 keV) will be almost entirely due to the electric dipole contribution to photoelectric absorption, n is the nuclear absorption-stimulated emission cross-section at exact resonance, given by 2 ¯ 2fM 1 n = . (3.148) (1 + )(1 + a) 1 + 2 Here is the internal conversion coefficient, while (1 + a) represents the effects of inhomogeneous broadening. 1 and 2 are the total widths of the upper and lower lasing levels, respectively. n = n1 - (g1/g2)n2 is the population-inversion density, where n1 and n2 are the population densities in the upper and lower lasing levels, while g1 = 2j1 + 1 and g2 = 2j2 + 1 are the spin degeneracy factors. g e and g n describe the coupling of the electromagnetic field of the lasing mode to the atomic electrons and the nuclear resonators, respectively, and are related to ge and gn as given by eqs. (3.142) and (3.144) by g g s s = , s = e or n. (3.149) cos Here is the angle between the beams in the eigenmode and the direction of energy propogation (i.e., the symmetry axis of the eigenmode, as noted in figure 14). The factor 1/ cos gives the increase in path length of the radiation in the mode as compared to a single plane wave traveling along the mode axis. In effect, the crystal consists of a series of weakly reflecting mirrors so that it acts as an etalon at the Bragg angles. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 183 Setting K = 0 in eq. (3.147) and solving for n yields the threshold population- inversion density nt given by g n e e t = n g 0. (3.150) n n Equation (3.147) may thus be rewritten as g K = n n ( n - n cos t). (3.151) The threshold condition (3.150) favors minimizing the photoabsorption coupling ge, hence Borrmann modes, and maximizing the coupling gn to the lasing nuclei, hence the strongest anomalous emission mode (which will be the mode with the largest number of beams). Thus, for a crystal containing M1 M¨ossbauer nuclei, the favored lasing mode will be the |m m mode with maximum m. Above threshold, the gain (3.151) is proportional to gn/ cos , which favors either the anomalous emission mode with the largest number of beams, or the mode with the largest cone angle , to maximize the etalon effect of the crystal. A number of examples are discussed in [41,45,48]. Now in fact, as discussed in [44,45,48], once the population inversion has been established to make a practical laser, the threshold for Dicke superradiance will also have been surpassed. Since the time required to superradiate is much shorter than the time required to lase, the system will predominantly de-excite by superra- diant, rather than stimulated, emission. The same factors, gn/ge, that determine the favored Borrmann lasing modes, however, will also determine the superradiant modes. Thus, the advantages of using a crystalline sample apply whether the system lases or superradiates. 3.10. Internal sources When an emitting atom is located within a crystal, the emitted photon is diffracted by the surrounding lattice and the radiation pattern outside the crystal exhibits a set of light and dark Kossel cones, whose positions and intensities are determined by the crystal structure. The theory of the Kossel effect for X-rays and its application to crystal structure determinations have been discussed extensively in the literature (see, e.g., [21, pp. 430­448]). The -ray emission problem has been discussed in detail by the authors [12,13,49­52] and by Alexandrov and Kagan [16]. Of particular interest are the anomalous emission effect, magnetic Kossel cones, and phase determination using the holographic nature of the Kossel cones. For -ray optics, the source can be a radioactive isotope, or alternatively, the wave arising from incoherent scattering, e.g., nuclear spin-flip (Raman) scattering by a M¨ossbauer atom or the isotope effect. For X-ray optics, the source is normally fluorescence radiation following photoabsorption. Historically, the internal-source problem for X-ray optics was first discussed by von Laue by using the classical reciprocity theorem. A similar approach can be taken 184 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 in M¨ossbauer optics, using a more general form of the theorem to treat the higher- order multipolarity of the sources and the magnetic nature of the medium. However, it is also straightforward to obtain a direct solution to the problem without recourse to reciprocity. The direct solution adds considerable insight to the problem and is more convenient for treating the anomalous emission effect, magneto-optical effects, and polarization effects. 3.10.1. Direct solution The spherical wave emanating from an emitting atom at Rj may be expressed as a sum of plane waves in accordance with exp(ik|R - Rj|) = S |R - R 1 + S2, (3.152) j| where ik S1 = exp ik · (R - R d 2 j) k (3.153) + and S 2 = k exp -k|z - zj| J0 k| - j 1 + 2 d . (3.154) 0 In eq. (3.154) is the component of R in the xy plane. In eq. (3.153) the integration over the solid angles of the k vector is over the upper hemisphere kz > 0 if (z-zj) > 0, and over the lower hemisphere kz < 0 if (z - zj) < 0. The contribution S1 to (3.152) is a sum of plane waves k, whereas the contribution S2 is a sum of exponentially damped waves as a function of |z - zj|. In sections 3.5­3.9 we obtained the response of a crystal to plane waves incident upon a crystal surface. These expressions can be used to immediately give the response of the crystal to S1. The contribution of S2 to the field in the crystal is, in fact, generally very small except in the near zone, kR 1, of the radiating atom. For X-ray wavelengths this means that it can be neglected relative to the radiation field, S1, except in computing the scattering and absorption processes occurring within the radiating atom itself. These intra-atomic processes determine the internal conversion width of the decaying nucleus, and give rise to small ( 10-3) multipolarity dependent phase shifts of the waves emitted by the atom, as discussed in appendix A.2.4. In the radiation zone, keeping only S1 in eq. (3.152), the emitted radiation field of a decaying atom located at Rj is ik A0(R, ; j) = exp ik · (R - R Jf0 2 j) (-k, ; j) d k. (3.155) + III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 185 Here Jf0 (-k, ; j) is the component of the transition current at Rj perpendicular to k, Jf0 f|e-ik·rj | 0 jf0 (-k) (-k, ; j) = , (3.156) + Ef + f - E0 - 0 + i /2 where the first factor gives the usual recoil factor, jf0(-k) = c-1 dx f|e-ik·xj(x)| 0 is the Fourier transform of the current density of the atom Rj, and in the resonance denominator E gives the internal state energy of the atom and gives the vibrational state energy of the crystal. The general multiple scattering equations are now given by eqs. (3.16) and (3.17), with A0 now given by eq. (3.155). The propagation of each plane wave contribution through the crystal can be obtained directly from our previous results. If a wave vector k0 is incident on the mth layer, then, as discussed in section 3.3, there will be waves ks± scattered from that layer, ks± = ±gs z + k0xy + s, (3.157) with gs = k20 - (k0xy + s)2. (3.158) As before, the s are planar reciprocal-lattice vectors, with 0 0. Within a crystal, for each component k0, there will be additional waves traveling between the crystal planes with all these wave vectors ks ( = + or -), and waves with wave vectors ks+ and ks- emerging from the top and bottom faces of the crystal, respectively; but, as discussed in section 3.3, it is only when the Bragg condition is nearly satisfied that a ks wave will be built up by constructive interference to an appreciable magnitude. The near Bragg condition is 2 n ks - k0 s + z, (3.159) d or 2 n gs - g0 (Laue case), (3.160) d 2 n gs + g0 (Bragg case). (3.161) d The right-hand side of (3.159) with s a planar reciprocal-lattice vector is, of course, a crystal reciprocal-lattice vector. If (3.159) is satisfied by a particular k0, then it is satisfied by all the k0's making the same angle with the reciprocal lattice vector as the original k0. The k0's and ks 's then form a double cone, the Kossel cone, whose axis is along the reciprocal-lattice vector. The Laue case holds for those points for which (ks )z and (k0)z are of the same sign, the Bragg case when they are of opposite sign. In the Laue case the two waves arising from each other by internal Bragg reflections will emerge from the crystal through the same surface (top or bottom), as shown in 186 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 figure 16; while in the Bragg case they will emerge through opposite surfaces, as shown in figure 17. A given Kossel cone can be purely Laue, purely Bragg, or both. Equation (3.157) gives the wave vectors ks (k0) of the various waves arising by scattering if a given k0 is incident on the crystal. This relation can be inverted to give ks 0 (k), representing the various incident waves which can give rise to a wave vector k: ks 0 (k) = k2 - (kxy - s)2 z + kxy - s. (3.162) In our case the "incident" wave is the spherical wave A0 given by (3.155). We are interested in obtaining the wave emerging from the upper face of the crystal, which is layer M using the emission geometry shown in figures 16 and 17 ( z is now "up", Figure 16. Schematic representation of a Laue emission channel. The wave fed directly into the k channel by the source is indicated by the double line, and the wave reflected into the k channel from the k1 channel which interferes with the direct wave is indicated by the single line. Figure 17. Schematic representation of a Bragg emission channel. The wave fed directly into the k channel by the source is indicated by the double line, and the wave reflected into the k channel from the k1 channel which interferes with the direct wave is indicated by the single line. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 187 the emitting atom is located in layer mj, the bottom layer is m = 0, and the top layer is m = M). If we write this emergent wave as ik AM(R) = A ik · R - z d 2 M (k) exp M z k, z > zM, (3.163) kz>0 then A M (k) = DM j k, ks 0 k) J -ks 0 (k), ; j , (3.164) s, where we have used (3.155) for the source of the incident waves and the sum is over those s's for which ks 0z is real (open channels). For large R (kR 1), eikR AM(R) AM(k) , (3.165) R where k = k R, so that the direction of observation R determines the "channel" k = k R at the top of the crystal which contributes to AM(R). The -ray flux may be obtained by inserting these expressions for AM(R) into (3.12). Equation (3.164) represents the wave AM(k) emerging from the crystal in the "k channel" as resulting from the coherent "feeding" of the channels ks 0 (k) by the source currents at Rj. The 2 × 2 matrix DM j k, ks 0 (k) gives the amplitude of the coherent wave in the k channel at layer M, which arises, due to scattering within the medium, from the wave in the ks 0 channel at layer mj. The amplitude with which the ks 0 channel is being fed at the mj layer is given by J (-ks 0 (k), ; j). The sum is over all the open channels ks 0 that reflect into the k channel. As indicated, the propagation matrix DM j is a function of the frequency , the thickness of the crystal, the position at which the photon is being emitted, which channel is being fed (i.e., the ks 0 channel), and the final direction k. Now DM j (k, ks 0 (k)) is negligible unless (3.159) is satisfied (using k as the independent variable, (3.159) takes the form k - ks 0 s + (2 n/d) z, similarly for (3.160) and (3.161)). In particular, the propagation matrix depends upon whether the k channel satisfies the conditions for Laue, Bragg, off-Bragg, or grazing emission. Off-Bragg emission. For nearly all k's (3.159) is satisfied only for s = 0, = +, n = 0 for which k0+ 0 (k) = k, and (3.164) becomes AM(k) = DM j (k, k)J (-k, ; j) (off-Bragg). (3.166) DM j (k, k) is simply the off-Bragg transmission matrix which propagates a photon from the emission position Rj to the top surface of the crystal, i.e., DM j (k, k) = T k, ; lj , (3.167) where T( k, ; lj) is given by eq. (3.63) with lj = zM - zj = (M - mj)d Mjd. 188 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Laue emission. If k is near a Bragg angle for the Laue case, then k1(k) is given by (eq. (3.162)) k1 = k2 - (kxy - 1)2 z + kxy - 1, (3.168) where 1 is some planar reciprocal lattice vector. The near-Laue condition (3.160) becomes (g - g1)d = 2 m + 2 , (3.169) where g kz = (k2 - k2xy)1/2 and is small ( = 0 is the exact Bragg condition). In terms of the deviation = ( - B) of the rocking angle ( 0) from the exact Bragg angle B, is given by = (kd sin ) /2 sin 1, where is the scattering angle (figure 16). If (3.168) and (3.169) are satisfied, we say that k and k1 are "Laue emission channels". In this case we have AM(k) = DM j (k, k)J (-k, ; j) + DM j (k, k1)J (-k1, ; j) (Laue), (3.170) and channel k at the top of the crystal is "fed" coherently by the waves emitted by the atom at Rj into both the channel k and the channel k1. This is shown schematically in figure 16. For k in the vicinity of a Laue channel, DM j (k, k) and DM j (k, k1) are rapidly varying functions of k and give rise to the appearance of the (Laue) Kossel cones in these directions. The Laue coherent wave propagation matrices DM j are given in terms of the Laue transmission matrices T by (see [13]) DM j (k, k) = T ; k, k; lj e-ik· j (3.171) and sin DM 0 j (k, k1) = T ; k, k1; lj sin e-ik1· j, (3.172) 1 where lj = zM - zj = (M - mj)d Mjd is the depth below the surface where the emitter is located, and j is the position of the emitter within the unit cell, taking the origin as the lattice point of the unit cell in which the emitter is located. The T( ; k, k; lj) and T ( ; k, k1; lj) give the amplitude in the channel k at the top of the crystal for unit amplitude waves incident at the layer mj + 1 in channels k and k1, respectively. The factors exp[-ik · j] and exp[-ik1 · j] give the relative phases of the contributions emerging at the top of the crystal in the k channel depending on the position of the emitter within the unit cell. The factor sin 0/ sin 1 in (3.172) is the Jacobian |d k /d 1 k| = |kz/k1z| = sin 0/ sin 1, which converts a k1 solid angle into a k solid angle on reflection. A general procedure for obtaining the T matrices is given in section 3.8. In the case of strong polarization mixing numerical procedures must be resorted to. For III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 189 the cases of no polarization mixing, T ( ; k, k; lj) and T( ; k, k1; lj) are given by eqs. (3.116) and (3.117), respectively, with M replaced by Mj = (M - mj). Bragg emission. For the Bragg channels, k is near a Bragg angle for some set of crystal planes and the wave field is determined by the k channel and a reflection channel k1 = -g1 z + kxy + 1, which travels in the - z direction, as indicated in figure 17. The Bragg condition (3.161) is now (g0 + g1)d = 2n + 2 , (3.173) where is again given by = (kd sin ) /2 sin 1, as in the Laue case. For Bragg emission, AM(k) = DM j (k, k)J (-k, ; j) + DM j (k, k1)J (-k1, ; j) (Bragg), (3.174) and channel k at the top of the crystal is fed coherently by the waves emitted by the atom at Rj into both the channel k and the channel k1, as shown schematically in figure 17. The D matrices are now related to the Bragg transmission and reflection matrices T and R, and there is a new complication not present in the off-Bragg and Laue cases: a wave k incident on layer mj + 1 can be transmitted directly to layer M in the k channel by the Bragg transmission matrix T ( ; k, k; M - mj). But also the k wave can be reflected back into the k1 channel by the upper M - mj layers, and then reflected back into the k channel before being transmitted to the upper surface. The combined matrix for this process is T ; k, k; M - m j R ; k, k1; mj R ; k1, k; M - mj . Or rather than a double reflection, there could be four reflections before transmission, and so on. As a consequence, DM -1 j (k, k) = T ; k, k; M - mj 1 - R ; k, k1; mj R ; k1, k; M - mj e-ik· j. (3.175) Similarly, a wave k1 incident on layer mj - 1 can be reflected into the k channel by the lower mj layers, and then transmitted through the upper M - mj layers, with the combined matrix being T ( ; k, k; M - mj) R( ; k, k1; mj). But again there can be multiples of double reflections between the parts of the crystal above and below the emitting atom at Rj, as discussed above. The resulting D matrix is DM -1 j (k, k1) = T ; k, k; M - mj 1 - R ; k, k1; mj R ; k1, k; M - mj sin × R ; k, k 0 1; mj sin e-ik1· j. (3.176) 1 The Jacobian and phase factors have the same significance as in eqs. (3.171) and (3.172). 190 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 For the special cases of no polarization mixing, the T and R matrices are given by eqs. (3.86) and (3.81), with M replaced by M - mj or mj, as appropriate. More generally, the numerical procedure must be used. 3.10.2. Reciprocity In X-ray optics, the problem of diffraction when the source lies within the crystal was solved by von Laue [53] by using the classical reciprocity theorem. For X-ray optics, the theorem can be stated as follows [18, pp. 430­448]: "If a source of radiation and a point of observation are interchanged, the intensity, measured in terms of electric displacement, is the same at the new place of observation as at the old." This theorem allows one to solve the emission problem in terms of the solutions to the external source problem. As stated, however, the theorem only applies to E1 sources and a nonmagnetic scattering medium, or more exactly, a medium which is unchanged by time reversal. The reciprocity theorem can, of course, also be applied in M¨ossbauer optics to relate the internal source problem to that of an external source; but its statement must be generalized somewhat to take into account the higher-order multipolarity of M¨ossbauer sources, and the possibly strong magnetic effects in the scattering medium. The general classical reciprocity theorem such as given by Landau [54] and utilized in [16] is valid for arbitrary multipolarity of the sources and for an absorbing medium, but it is not suitable for describing cases where magneto-optical effects occur, since then the assumed symmetry of the dielectric tensor ik breaks down. A more general theorem based on the supposed invariance of physical laws under time reversal is derived in [12]. The resulting reciprocity relation for single photon optics (eq. (46) of [12]) applies to arbitrary multipole currents, for photon propagation through a dispersive medium with arbitrary magneto-optical effects, and is valid for inelastic as well as elastic photon propagation. For the coherent elastic wave A(R, ; j) of frequency , which is propagated through the crystal to a distant point R following an initial decay at Rj (either recoilless or with recoil), the theorem gives eikR A (R, ; j) = - e A R 0J0 f µ kj, ; j µ(Rj, ; e0). (3.177) e0= x, y Here x and y are any convenient orthogonal basis vectors perpendicular to k = -k = -( /c) R; kj = -i j, where the gradient is with respect to Rj, the equilibrium position of atom j, and the gradient operates on the factors to the right as indicated; the current matrix element is the absorption process between the time reversed states, |f |0 ; and A µ(Rj, ; e0) is the total coherent field at Rj due to an external plane wave source e0 exp(ik · x) incident in the k direction on the time-reversed scattering medium, which is the external source problem already solved. To use (3.177) to determine the (k, ) coherent wave at a distant point R fol- lowing a decay |0 |f at Rj, one first calculates the coherent wave A µ(Rj, ; e0) at Rj caused by a plane wave of polarization e0 and frequency incident in the III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 191 -k direction on the medium in which the B field at each site is reversed (while the electric field gradient remains unchanged). The photon flux at R, given by inserting expression (3.177) for A (R, ; j) into the photon flux eq. (3.12), is then proportional to the excitation probability |J0 f µ A µ|2 for inducing the transition from the ground state |f = T |f to the excited state |0 = T |0 in the time reversed process. To obtain the recoilless M¨ossbauer fraction, the transition is taken as recoilless, while for the recoil fraction the transition is taken with recoil f 0. Using this procedure, we can treat the various cases discussed above. Although the reciprocity approach is more cumbersome, it gives a useful insight into the emission problem. 3.10.3. Anomalous emission, magneto-optical effects, holographic phase determination There are a number of interesting features of the internal source problem, among which are the following: "Anomalous emission effect" for M¨ossbauer optics. If a M¨ossbauer emitter of mul- tipolarity M1 or higher is located at a lattice site deep in a crystal (at a depth greater than the photoelectric absorption depth but less than the Borrmann penetration depth), and if the absorption from the remaining crystal sites is primarily the E1 photoelectric absorption, then a -ray emitted from the site into a Borrmann channel will emerge from the crystal. On the other hand, for an E1 emitter, such emission is strongly suppressed [13,39]. The effect is most easily seen using the reciprocity theorem, which states that the amplitude of the radiation field produced at a distant point in a direction k from the crystal is proportional to the coupling of the atom to the radiation field produced inside the crystal by a plane wave incident on the crystal in the direction - k. In the present case the reciprocal problem is an external plane wave incident in a Borrmann channel, as discussed in section 3.8.1, and the diffraction effects lead to nodes in the electric field at regular lattice sites, and antinodes for the magnetic field and electric field gradients at such sites (figure 12). Hence, there is no coupling to an E1 oscillator at such a site, but strong coupling to an M1 or E2 oscillator, which, by reciprocity, implies the anomalous emission effect stated above. Alternatively, the effect occurs because the wave emitted directly into the k direction by the source interferes constructively or destructively with the wave which is Bragg reflected into the k direction by the crystal (figure 16), with the nature of the interference being determined by the multipolarity and position of the source. As discussed in [13,39], the interference is constructive if the source is M1 or higher, and located at a lattice site. The polarization of the anomalous emission mode is parallel to the scattering planes, and the emitted radiation is narrowly collimated with a collimation angle typically on the order of 10-4 rad. Thus, the effect is a natural source of polarized, collimated radiation. 192 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 The radiation field set up within the crystal by these deep crystal sources will also have E-field nodes at lattice sites, and B-field and EFG antinodes at lattice sites. Hence, this field can stimulate emission from similar emitters of multipolarity M1 or higher located at lattice sites. As discussed in [92,93,95] and section 3.9.4, the effect is potentially useful for the construction of -ray lasers. Magnetic Kossel cones. The Kossel pattern exhibits the crystal symmetry and affords a sensitive measure of crystal parameters. In X-ray optics, Kossel pattern analysis has been used to make precision determinations of lattice parameters and orientation, and to determine the variation of the parameters with temperature and strain, the components of the strain tensor, and the degree of crystal perfection [55]. In M¨ossbauer optics these features are also present and a new phenomenon can occur: the M¨ossbauer scattering is sensitive to the direction of the magnetic field and electric field gradient tensor at the nucleus, and if the internal field "unit cell" is greater than the chemical unit cell, there will appear extra Kossel cones ("magnetic Kossel cones") exhibiting the structure of the internal fields [13,39]. Thus, for example, analysis of the magnetic Kossel line pattern of rare-earth M¨ossbauer metals and alloys will permit accurate determinations of the spiral structures, and of their dependence on temperature and strain. For example, if a magnetic crystal has a spiral axis z with a spiral angle , there will be Bragg or Laue channels k and k1 open when (k - k1 + n ) is equal to a reciprocal lattice vector, where = ( /ds) z and ds is the interplanar distance along the spiral axis z. These lines will, of course, only appear if the internal fields are strong enough to cause appreciable hyperfine splitting. Faraday effects. The most interesting effects in off-Bragg emission are the magneto- optical effects involved in the propagation of the photon through the surrounding medium. The matrix DM j (k, k) = T( k, ; lj) given by eq. (3.63) determines the prop- agation of the emitted photon |J (-k, ; j) through the medium. As discussed in section 3.5, for a given frequency and direction of propagation k, there are two eigenwaves e ( k, ), = 1, 2, which are generally nonorthogonal and have different complex indices of refraction, n = 1 + f . This leads to Faraday and selective ab- sorption effects, and as a consequence, for recoilless emission, the polarization state of the photon emitted from the crystal will generally differ from the polarization state at the source. The density matrix formalism for treating the polarization state of the emitted photon is discussed in [13]. Holography and phase determination. The problem of determining the structure of complex molecules by X-ray diffraction is greatly complicated by the fact that the reflected intensity only gives the magnitude |Ah| of the unit-cell scattering factor and not the phase h. The most widely used method to extract h is to use controlled variation of the structure factor, for example, by heavy atom replacement [56,57] or anomalous scattering [58]. In these procedures, the unit cell scattering amplitude is changed III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 193 from A0h = -r0Fh, where r0 = e2/mc2 and Fh = |Fh|ei h is the unit-cell structure factor, to Ah = A0h + A1h, where A1h is the added amplitude due to the extra scattering (heavy atom, anomalous dispersion). Since the integrated reflected intensity is directly proportional to |Ah|2 for small crystals, h can be deduced from knowledge of A1h and measurement of |A0h|2 and |A0h + A1h|2. In [13,59­61] it was pointed out that internal source emission is a holographic process, and offers a sensitive method to determine the structure factor phases h of the chemical unit cell (and in the M¨ossbauer case, also the phases h (B) for the structure of the internal field unit cells): the interference of the direct wave emitted in a given direction with that Bragg-reflected into this direction (figure 17) allows the phase of the latter to be obtained from the intensity distribution within the Kossel line, as in the holographic technique, where phase information is obtained by "beating" the diffracted wave with a primary reference wave. If the source is located at a lattice site Rj near the top surface of the crystal, then the total amplitude of the wave traveling in the k direction is (see [60]) e-ik·Rj + ei he-ik1·RjD( /Wh), (3.178) where Wh is the half-width of the total reflection region, and D( /Wh) is the usual Darwin reflection amplitude. The cross term in the resulting intensity depends upon the phase of the structure factor, allowing h to be determined. The holographic nature of internal source emission has more recently also been pointed out by Sz¨oke [62], and has been utilized to do surface hologra- phy using photoelectron sources [63]. Furthermore, as pointed out by Timmermans et al. [64­66], if the neighboring atoms have spins, then due to exchange scatter- ing, there is a "spin hologram" contribution to the interference between the direct and scattered electron waves which can be extracted to give the magnetic struc- ture. X-ray interferometry and phase determination. As just discussed, for the Kossel lines, the total field A(R, ; j) in the k = R direction, emitted by a source atom at the site Rj, is a superposition of the directly emitted wave k, and the wave Bragg-reflected from k1 into k by the crystal (figure 17 and eq. (3.178)). In the time reversed process, the total field at the site Rj due to a plane wave of frequency incident in the - k direction, is a superposition of the direct wave (-k) and the wave Bragg-reflected (-k) to (-k1) by the crystal. The superposition of the two waves produces an X-ray standing wave pattern. By the reciprocity theorem eq. (3.177), the intensity |A(R, ; j)|2 at R, which gives the Kossel line profile, is directly proportional to the excitation probability |J0 f µ A µ|2 for inducing the transition from the ground state |f = T|f to the excited state |0 = T |0 in the time reversed process. Thus, observing the fluorescence yield of an atom due to the field in the standing wave created by rocking through a Bragg peak is equivalent to observing the Kossel line intensity profile produced by the decay of that same atom, and consequently has the same phase information, as pointed out 194 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 by Hutton et al. [67] and Gog et al. [68]. X-ray holography using this technique has recently been demonstrated [69,70]. 4. Coherent excitation of nuclei by synchrotron radiation pulses Because of the low brightness of most radioactive M¨ossbauer sources, until re- cently nearly all M¨ossbauer experiments involved only resonance absorption, which, while valuable, does not exhibit to the fullest the interesting effects due to coherence in the emission or scattering of the -rays from the various nuclei in the sample. In 1974, however, Ruby [71] and M¨ossbauer [72] pointed out that the spectral brightness in the then existing synchrotron radiation sources at, e.g., the 14.4 keV 57Fe reso- nance line exceeded that of 100% 57Co radioactive sources. Since it was clear that with the use of wigglers or undulators the synchrotron brightness could be very much enhanced at the resonance lines, a number of groups undertook the quest of "filter- ing" out the "M¨ossbauer slice" from the white synchrotron beam. The first dramatic success was achieved by Gerdau et al. [73] in 1985, where filtering was first unam- biguously achieved, and since that time there has been rapid progress and interest in the field [74].13 It is now clear that an important new field has been opened up involving coherent nuclear excitations induced by synchrotron radiation pulses traversing a piece of matter. Several important new areas of application are: Inelastic scattering studies. Until recently, inelastic neutron scattering has been the primary method available for obtaining the low energy excitations (phonons, magnons, etc.) in condensed matter. X-ray inelastic scattering was not useful for such experi- ments because of poor energy resolution. However, perfect Si crystal backscattering monochromators have been con- structed at Brookhaven [75] and Hasylab [76], which have achieved resolutions of E 5­7 meV for X-ray energies. The use of such monochromators to filter syn- chrotron radiation and to measure energy gains or losses in the scattered beam will furnish another and perhaps more powerful means to determine condensed matter ex- citations. To exceed the limit E 1 meV obtainable by wavelength selecting mono- chromators, one must use nuclear resonance filters (frequency or energy selecting monochromators) where one can obtain submicrovolt resolution, E < 10-6 eV at E = 14.4 keV (for 57Fe filters), with a flux of the order of 105 photons/s/mm2 or higher.Resonant absorber Doppler shifting detectors capable of speeds up to 30 m/s have now been developed, making possible inelastic measurements corresponding to energy transfers of 10-5­1 meV [77]. This should furnish a good method for measuring 13 The current experimental situation using both synchrotron and conventional M¨ossbauer radiation sources is reviewed elsewhere in this issue. For a review of earlier work, see [74]. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 195 very slow relaxation and fluctuation processes ( 107 s-1) in the critical region, and faster processes up to 3 × 1011 s-1 (1 meV). There has been considerable recent progress in this field, as discussed in sec- tion 5.1 of this paper, and by Chumakov and R¨uffer [78]. Interferometry, metrology, holography. Filtered synchrotron radiation opens up quite new possibilities in X-ray interferometry by utilizing the very long coherence length of the radiation. The usual Bonse­Hart interferometer [79] is a large single crystal of Si or Ge, cut to have three parallel plates ("arms"), with a set of crystal planes running per- pendicular to the faces of the plates. Set in a Laue transmission geometry, the first plate separates the incident X-ray radiation into two components (the direct and Bragg reflected beams), and Bragg reflection of the two beams in the second arm recombines the beams at the third plate, which acts as an analyzer. It is then possible to insert a very thin "transparent" wedge of material (several microns) into one of the paths to give an index of refraction change of the optical path length. In principle, observation of the interference pattern vs. wedge thickness can give an accurate determination of the index of refraction, and can, for example, be used to make precision determinations of the anomalous scattering near a K-shell absorption edge [80]. A restriction here is that the coherence length of the radiation is lc 10-4 cm, which means path length differences are restricted to 10-4 cm (the Bragg scattering in the interferometer arms, or from a prior monochromatizing crystal, monochrom- atizes the X-ray radiation to 0.1 eV, giving a coherence length of lc = c/ 2 × 10-4 cm). Since the pathlength differences must be kept to less than lc to ob- tain interference in the recombined beams, this effectively restricts us to transmission elements (e.g., a transparent thin wedge) in one of the beams. However, if filtered synchrotron radiation is used as the source, then 10-8­ 10-6 eV and the coherence length is lc 2 cm­20 m, and it would then be possible to use a scattering element in a Bonse­Hart interferometer. For example, in place of the transparent wedge, it would now be possible to insert two parallel cut grazing in- cidence reflectors for surface studies, where now the optical path length is changed by rotation of the reflectors. Another possibility would be to insert a multiple Bragg reflec- tion etalon grating, used, for example, to make multiple passes through a monolayer. A backscattering monochromater has been demonstrated by Shvyd'ko et al. [81]. It is also possible to do long coherence length interferometry with unmonochro- matized synchrotron radiation pulses by utilizing the long coherence time response of a resonance detector [82]. For example, a synchrotron radiation pulse of duration t 0.1 ns sent through the two arms of an interferometer (e.g., Bonse­Hart, Michel- son, . . . ) will emerge as two coherent sharp pulses, separated by a time T L/c, where L is the optical path difference between the two arms. If the pulses then impinge on a film containing resonant M¨ossbauer nuclei, the nuclei are excited under the coher- ent superposition of two sharp pulses, giving an excitation probability proportional to 196 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 cos2( L/ 0). This offers the interesting metrological possibility of measuring changes of the length L 0 1 A for L's up to lc 2 cm­20 m. Structure determination. Another important application will be for the structure de- termination of complex biomolecules. The problem of making structure determinations by X-ray diffraction is greatly complicated by the fact that the reflected intensity only gives the magnitude of the unit-cell scattering factor and not the phase. For this rea- son, considerable effort has been devoted to devising methods for extracting phase information. To determine the phase, the usual method employed is impurity substitution, in which one determines the phase of the unit cell scattering amplitude relative to the phase of the wave scattered from a limited number of impurity sites. In practice this is a tedious process, involving duplicate experiments on pure samples and impurity doped samples. An alternate, and very powerful method is offered by M¨ossbauer diffraction [83,84]. In this case, one substitutes resonant 57Fe for 56Fe in the biomolecules to be studied, leaving the electronic scattering unchanged. Because of the exceedingly sharp resonance, it is then possible to vary the phase and amplitude of the resonantly scattered wave by Doppler-shifting with negligible variation in the nonresonant electronically scattered wave. Consequently, the inter- ference between the waves is easily varied in a controlled manner. As discussed by Black [83], and as demonstrated by Parak et al. [84], it is then possible to extract the phase of the chemical structure factor of the unit cell relative to the structure factor due to the resonant scattering at the M¨ossbauer sites. The problem is then reduced to determining the relative positions of the M¨ossbauer atoms within the unit. (A related technique for phase determination is to tune near a K-shell absorption edge and use the frequency variation of the anomalous scattering f K + if K over eV-regions. However, this contribution is typically only several electrons even in the immediate vicinity of a K shell, and more importantly, because of the continuum nature of the intermediate states involved, there is no sharp dispersion variation across the K edge. For example, the measurements of Materlik and Bonse [80] on Ni show that f K varies from -3 electrons at = K +150 eV, decreasing to -7.5 electrons at K. In contrast, the real part of the resonant nuclear scattering amplitude for 57Fe scatters as +220 electrons for = 0 - /2 , and as -220 electrons for = 0 + /2 , with = 5 × 10-9 eV.) However, in practice, because of the high collimation required and the limited brightness of natural sources, very long counting times are required ( several counts per hour), making the M¨ossbauer experiments exceedingly difficult. Filtered synchrotron radiation sources are ideal for such experiments, providing a pre-collimated beam with a brightness 3­4 orders of magnitude greater than that pro- vided by a natural source, making the M¨ossbauer technique applicable to the analysis of significant biological molecules. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 197 Hyperfine splitting, isomer shifts, . . . by monitoring the quantum beats in the coherent forward and Bragg scattering channels. Synchrotron radiation from a source oper- ating in the single-bunch mode consists of sharp pulses of about 10-10 s duration and about 10-6 s separation between pulses. Excited nuclear states of energy less than 100 keV commonly have lifetimes in the range -1 = 10-6­10-10 s. If the pulse, monochromatized to perhaps 1 eV bandwidth at the nuclear transition, impinges on a small crystal containing the resonant nuclei, then the electronically scattered X-rays, photoelectrons, etc. will emerge promptly during the 10-10 s pulse, while those processes involving nuclear excitation will be delayed a mean time -1. There- fore, by using a timed detector which can recover from the prompt pulse in a time short compared to -1, the resonant and nonresonant events can be separated temporally. There are "quantum beats" in the coherently scattered radiation in the forward and Bragg channels, arising from the interference between waves emitted from different sites [26]. The essential feature is that the sharp synchrotron radiation pulse excites the various nuclear excited state sublevels suddenly and coherently, which then "oscillate" at their various natural frequencies, giving beats in subsequent decays. These beats occur at frequencies B n, m, ; n , m , = nm( ) - n m (4.1) corresponding to the difference frequencies of all allowed nuclear hyperfine transitions nm( ) from all the different nuclear sites , from which the hyperfine splittings of both the excited and ground states may be found, as well as any energy shifts between nuclei located in different chemical or magnetic sites. In contrast, the quantum beats occurring in the usual time dependent perturbed angular correlation (TDPAC) measurements are spatially incoherent, with the beats arising from the interference between transitions to the same ground state. Thus, the beat frequencies give only the splittings of the excited levels, with no information on ground state splittings or on shifts between different sites. The quantum beats in the coherent scattering give a very sensitive measure of the hyperfine splittings and isomer shifts, which can be even more precise than the usual M¨ossbauer absorption measurements because the quantum beats give a direct measure of the energy differences, while the precision of the absorption measurements depends on the precision of the velocity control. An important practical point is that because of the pulsed nature of the synchrotron radiation source, the quantum beats can be measured in the forward direction scattering following the pulse excitation [85]. Since coherent forward scattering occurs for any sample (crystalline, mosaic, amorphous), all normal M¨ossbauer experiments can be carried out in the time domain. 4.1. Nuclear exciton states There are two approaches for treating the interaction of a synchrotron pulse with a system of nuclei. Because the temporal duration of the synchrotron pulse and its transit 198 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 time across the system are short compared to the lifetime of the nuclear resonance, the formation of the intermediate excited state and its subsequent decay can be treated as two independent quantum-mechanical processes. Alternatively, the interaction can be treated purely as a resonance scattering problem. The first approach brings out interesting new physics: the creation of nuclear excitons with synchrotron radiation pulses, and the nature of the subsequent exciton decay. The second approach, on the other hand, gives the simplest method for obtaining the scattered signal. The time dependent amplitude of the scattered signal is simply proportional to the Fourier trans- form of the frequency dependent reflection and transmission amplitudes R( k1, k0; ) and T ( k1, k0; ) already obtained for the various cases of off-Bragg, Bragg, Laue and grazing incidence. In this subsection we discuss the exciton approach, and in the following subsection 4.2 we outline the scattering approach. From the pure physics standpoint, the nuclear exciton produced by the syn- chrotron radiation pulse when it traverses a sample containing M¨ossbauer nuclei is a new state of matter exhibiting interesting effects of coherence, notably: * Speedup of the decay (relative to that of an isolated excited nucleus) resulting from superradiance in the coherent channels (and a slowdown is also possible, resulting from subradiance in some coherent decay channels). * Quantum beats resulting from the interference of waves of different frequencies from different nuclei, giving a time dependent spatially coherent decay rate, c(t), varying periodically between superradiant and subradiant emission into the coherent channels. 4.1.1. Nuclear exciton The important new element in the synchrotron pulse experiment is that because both the pulse duration and the transit time of the pulse across the crystal are short compared to the excited state lifetime -1, the pulse creates a collective nuclear excited state | e(k0) which is a spatially coherent superposition of the various excited state hyperfine levels of all the nuclei in the crystal, with a spatial phasing wave vector k0 [26,28]. The subsequent radiative decay is radically affected by coherence, exhibiting both a speed-up due to "coherent enhancement" [2,10­13,25,26,28,32,86,87] and a quantum beat modulation of the decay rate [26,28,32]. As discussed in [26], the resulting recoilless excitation just after the pulse tra- verses the system is N e(k0) = eik0·Rl cn |e G | lml lnl 0(l) 0 , (4.2) l=1 nl where k0 = c-1 0n0 (n0 is the direction of propagation of the incident pulse and 0 is the mean transition frequency for all the nuclei) and cnm is the amplitude for exciting the nucleus at Rl from |glml |elnl , with no change in the vibrational state | 0 , III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 199 and |G0(l) indicates that all the other nuclei (l = l) are in their initial ground states |gl ml . For a nuclear dipole transition (E1 or M1), cnm is given by i cn = f e , (4.3) lml lnl| µ|glml · F 0 where µ is the nuclear dipole moment operator (E1 or M1), F is the resonant Fourier 0 component of the incident synchrotron pulse F(t) (F = E for E1 and F = B for M1), and f is the M¨ossbauer factor. | e(k0) is a collective nuclear excited state, i.e., a spatially coherent superposi- tion of the various excited state hyperfine levels of all the nuclei in the crystal. In each contributing term, one nucleus is excited into one of its excited state hyperfine levels, and all the remaining nuclei remain in their initial ground states. The relative spatial phase is exp(ik0 · Rl). Hence, | e(k0) is a single exciton collective state, with one excitation distributed coherently over the entire system, with a spatial phasing wave vector k0. 4.1.2. Radiative decay of the nuclear exciton Because the temporal duration of the incident pulse is short compared to the lifetime of the nuclear resonance, it is legitimate to regard the formation of the inter- mediate excited state | e(k0) and its subsequent decay as two independent quantum- mechanical processes [88]. Consequently, we can use the general theory for emission discussed in section 3.10. The initial state of the crystal is now the exciton state | e(k0) with one excitation spread coherently over all the nuclei of the system. The photon potential Aµ(z) due to a transition to a final crystal state | f corresponding to all M¨ossbauer nuclei being in their ground states is given by A µ(z) = c-1 f | d4x Dµ (z, x)J (x, l) e(k0) , (4.4) l where J (x, l) is the source current of the lth nucleus and Dµ (z, x) is the photon prop- agator. If we neglect scattering, then Dµ (z, x) is given by the free photon propagator D(0) µ (z, x), e-ik(z-x) D(0) µ (z, x) = -4 gµ d4k. (4.5) k2 + i The source transition current J (x, l) is given to good approximation by Jµ(x, l) = eiH0txj(l) µ (x)e-i(H0- /2)(tx- l)1(tx - l), (4.6) where j(l) µ (x) is the current-density operator for the atom l, is the width of the excited level, H0 is the Hamiltonian for the complete crystal with electromagnetic interactions replaced by instantaneous Coulomb and magnetic interactions, and 1(t) is the Heaviside step function (1(t < 0) = 0, 1(t > 0) = 1). In (4.6) it is assumed that the excited nuclear state is produced at tx = l. For a synchrotron pulse sweeping 200 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 across a crystal in the direction n0, arriving at the origin at t = 0, the lth nucleus is "turned on" at l = c-1 n0 · Rl. In the first Born approximation, the width would simply be the excited state width of an isolated nucleus, = + . But the effects of coherent decay will radically effect as we discuss below. For our purposes, it suffices to calculate in the second Born approximation, using the Fermi Golden Rule if there is no hyperfine splitting (section 4.1.3), and the direct time dependent flux calculation if there is hyperfine splitting (section 4.1.6). The exact treatment of the excited state propagator for an initial exciton state | e(k0) is discussed in section 6. 4.1.3. Spatially coherent and incoherent decay The radiative decay of the exciton state e(k0) | f + k (4.7) can be either spatially coherent, giving highly directional emission into the coherent channels, or spatially incoherent, giving essentially isotropic emission into 4 sr. Spatially coherent decay occurs when | e(k0) makes a radiative transition to the original ground state, i.e., | f = | 0 . For the decay | e(k0) | 0 + k, the photon potential (for the elastic wave) becomes 1(t ) Ael (R, t) = F t e-i(k-k0)·Rl, (4.8) R l,ml l where k = c-1 0R/R, t = (t-R/c) is the retarded time, and the emission amplitude from site Rl is Fl,m t = f g dx j e-i( nlml-i / )t. (4.9) l lml|c-1 (x)e-ik·x|elnl cnlml nl The coherent elastic wave, obtained by ensemble averaging over | 0 , is then 1(t ) A (R, t) = e-i(k-k0)·RlF t , (4.10) R c l where the ensemble average emission amplitude Fc(t) is C Fc(t) = F (t), (4.11) (2j l,ml 0 + 1) ml where C is the abundance of the resonant nuclear isotope. If there are several different resonant sites within the unit cell, the sum in eq. (4.10) is also carried out over , and Fc(t) will also depend on . Thus for spatially coherent decay | e(k0) | 0 + k, the emitted wave is a superposition of waves exp[-i( n - i )t] emanating from all the various sites R lml l which add up to give highly directional emission, with the intensity modulated by the various difference frequencies. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 201 Viewed as a scattering process, as discussed in section 4.2, the system resonantly scatters a photon |k0 in the incident synchrotron pulse to a final state |k , with the nuclear system undergoing the transition | 0 | e(k0) | 0 , i.e., k + | | 0 0 e(k0) 0 + k. (4.12) This is coherent elastic resonant scattering and corresponds to "multi-slit interference": it is impossible to tell at which nucleus the scattering took place, so the scattered wave amplitude is then the sum of the individual scattered wave amplitudes from all the nuclear sites, giving highly directional coherent scattering (as well as a weak diffuse contribution due to the "isotope effect"). If | f = | 0 , the decay will be spatially incoherent (ignoring, for the moment, spatially coherent decay with recoil). For single photon decay, | f can differ from | 0 only by the spin state of a single nucleus, e.g., | f = glm l G0(l) m l = ml. (4.13) In this case the lth site is "tagged" by "spin flip", and the emitted wave reduces to a single contribution, 1(t ) A (R, t) = F t e-i(k-k0)·Rl. (4.14) R l,ml Thus, for spatially incoherent emission, Aµ(z) emanates from a single site Rl, giving emission into 4 sr, with beat frequencies dependent only on the hyperfine splitting of the excited state of the nucleus at Rl. Viewed as a scattering process, the scattering definitely occurred at the lth nucleus, and the scattered wave amplitude only contains this single contribution, giving essentially isotropic (diffuse) scattering. Similarly, if | e(k0) decays by internal conversion, then one of the atoms will be tagged, now by the difference in the initial and final electronic states. As a conse- quence, there can be no spatial coherence in internal conversion decay. Finally, the exciton may decay with the emission of one or more phonons in addition to the photon. Since the vibrational states are nonlocalized, this decay is spatially coherent if there is no spin flip. However, because of the continuum of phonon states, there is no coherent enhancement to these decay modes (the partial width is essentially (1 - f) ). 4.1.4. Coherent enhancement and coherent speedup For an isolated nucleus, the total width for the excited level is = + , where is the partial width for radiative decay and is the partial width for internal conversion decay. For the synchrotron produced nuclear exciton state | e(k0) , the total decay width is increased by the effect of spatially coherent radiative decay, = c + + , (4.15) where is the partial width for spatially incoherent radiative decay and c is the partial width for spatially coherent decay | e(k0) | 0 + k. If there is no hyperfine 202 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 splitting of the nuclei, then c will be time independent and can be calculated by the Fermi Golden Rule. For an E1 or M1 resonance, 2 2 c(k0) = cohN -1(4 )-1 d (n) 1 - n · 0 S(k - k0) , (4.16) where the diffraction factor S(k - k0) is given by N S(k - k0) = e-i(k-k0)·Rl. (4.17) l=1 Here k0 = k0n0, k = k0n, 0 is the "appropriate" polarization of the incident synchrotron pulse ( 0 = e0 for E1, or 0 = b0 = k0e0 for M1, respectively), and coh = fC(2j1 +1) /(4j0 +2), where f is the M¨ossbauer factor, C is the abundance of the resonant nuclear isotope, and j0 and j1 are the spins of the nuclear ground and excited states. The classical analogue is a phased array of dipole oscillators, oscillating with frequency 0 in the 0-direction, with the amplitude of the radiator at Rl phased as exp(ik0 · Rl). The differential radiation rate is then d (k0) 3 = (4 )-1 1 - n · 2 S(k - k 2. (4.18) d (n) 2 0 0) |S(k-k0)|2 gives the relative intensity, due to the interference among the various source waves, of the radiation emitted in the k direction. The factor [1 - (n · 0)2] gives the polarization dependence: each oscillator has an acceleration a 0 exp(-i t), and the emitted radiation in the k direction is proportional to the transverse projection a of the acceleration relative to the k direction. Integrating over d (n) gives the total radiation rate for the system, (k0). Comparing with eq. (4.16), the partial width c(k0) for spatially coherent decay | e(k0) | 0 + k corresponds to the classical radiation rate per oscillator, (k0)/N. The relative intensity, |S(k-k0)|2, peaks to N2 in directions n where constructive interference occurs for the wave amplitudes exp[-i(k - k0) · Rl] emitted from all the nuclei. Thus in the small solid angle about n0 for which (n - n0) · (Ri - Rj) for all interatomic distances, d (k0)/d (n) N2/4 . The contribution to (k0) of this "forward direction" coherent radiation is then N2 (k coh 0) , (4.19) 4 where is the solid angle about n0 for which (n - n0) · Ri - Rj < for all Rij. depends strongly on the dimensionality and shape of the crystal. If the radiators are uniformly distributed in a prism of dimensions l (k0), l(1) (k0) and l(2) (k0) relative to the direction n0, with the convention that l(1) (k0) l(2) (k0), and if l and l , then there are three distinct cases: III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 203 (i) (1-D crystal) (4 /l ) if l(1) l(2) < l ; (ii) (2-D crystal) (4 /l(1) )(2 /l ) if l(1) l and l(2) < l ; and (iii) (3-D crystal) ( /l )2 if l(1) l(2) > l . In the opposite limit of long wavelengths, such that l , l (0-D crystal), 4 . Our main interest is in samples satisfying (iii), which is the usual situation, and we restrict our attention here to this case. The other three cases are discussed briefly in the following section. For (iii) satisfied, we have 2 c(k0) (k N = 0) coh = coh n N 4 l ¯ 2l , (4.20) where n = N/(l l2 ) is the number of radiators per unit volume. Equation (4.20) shows that there is a coherent enhancement of the radiation rate per atom (k0)/N proportional to the number of atoms in a squared wavelength of the material in the thickness measured in the n0 direction. This is generally a very large number: if we take 1 A and n 0.1 A -3, then n¯ 2 10-2 A -1, so that a thickness of only 100 A will double the radiation rate of properly phased atoms, and even for a very thin sample with l 1 µm, condition (iii) allows l 1 cm, giving enhancements of 106. However, the actual length l over which coherence can occur will be limited by scattering and absorption processes: for the 14.4 keV resonance in 57Fe, photoabsorption limits the coherence length to l 2 × 10-3 cm and the enhancements to 103. A very nice demonstration of the linear dependence of c on the thickness has been given by van B¨urck et al. [89]. In addition to the enhancement due to coherence in the "forward" direction n n0, in a crystal, a further enhancement occurs if k0 makes a Bragg angle with a set of crystalline planes. Constructive interference then also occurs for those k ( kn) lying in the small solid angle about the (k0 + ) direction ( is a reciprocal-lattice vector) such that [k - (k0 + )]·Rij 2 for all Rij. For each open channel (k0 + ), there will be a contribution [2,11,87] ( ) 2 c = coh 1 - n · 0 n¯ 2l (k0 + ), (4.21) where l (k0 + ) is the thickness of the crystal in the direction of (k0 + ), and n is a unit vector in the direction of (k0 + ). While the forward enhancement ( = 0) is always present, each additional channel (k0 + ) only contributes for k0 within the small angular region d/[m cos( 0)l (k0 + )] around the Bragg angle 0 (d = interplanar spacing, m = order of Bragg reflection). Thus, if the incidence direction n0 is taken so that k0 satisfies the condition for a symmetric Bragg reflection from a thin crystal of resonant nuclei M-layers thick, then 204 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 there will be two coherent channels open, the reflection and transmission channels R and T , and the partial width for coherent decay will be c(k0) = (T) c + (R) c ( ), (4.22) where, for the coherent emission into the T -channel, (T) c = n¯ 2l (k0) coh (4.23) and for the coherent emission into the R-channel, (R) 2 sin2(M ) c ( ) = (T ) c 1 - n1 · 0 . (4.24) (M )2 Here 0 = 0B + is the incidence angle with respect to the crystal planes, 0B is the Bragg angle, d is the interplanar separation, and = kd cos( 0B) . We have also assumed that the -angular spread of the incident synchrotron pulse is small compared to the Bragg width, B /(Md cos 0B). If B, then (R) c ( ) is integrated over the angular distribution of the incident pulse intensity. If 0 n1, then near Bragg ( B), c doubles to 2 (T) c , which agrees with the enhanced width obtained in eq. (3.99) for excitation at Bragg by a near resonant monochromatic wave of frequency . For a sufficiently thick crystal, typically M 103­104 layers, c will exceed ( + ), and coherent decay will dominate incoherent decay. The nuclear exciton then de-excites at the speeded-up rate ( c + + )/ , primarily via coherent radiative decay into either the R- or T -channel. This speed-up of coherent decay at Bragg has been observed, e.g., in [90]. Because the forward scattering channel ( = 0) is always open, an exciton | e(k0) created by a synchrotron radiation pulse is always superradiant (for a 3-D crystal geometry), regardless of whether k0 satisfies a Bragg condition or not. However, there is an important distinction between the Bragg- and off-Bragg cases. As discussed in section 6, in simple situations the Bragg mode excitation is a true semi-stationary state radiative eigenmode with an enhanced width ( c + + ), while the off- Bragg (or Laue) excitation is in fact a superposition of radiative eigenmodes of slightly differing frequencies. In all cases, the initial decay rate of | e(k0) is ( c+ + )/ (where c depends on the number of open channels, as discussed above), but the character of the decay at delayed times /( c + + ) is radically different, depending on whether the exciton | e(k0) is an eigenmode or not. In the case of an eigenmode, the scattered signal (emission probability) has a pure exponential decay exp[-( c+ + )t/ )] at the enhanced decay rate. On the other hand, if | e(k0) is a superposition of eigenmodes, the superradiant components die out quickly, leaving a superposition of slowly damped components exp[-( + )t/ )], with a spread of eigenmode frequencies, producing a slowly decaying "ringing" signal at delayed times. This delayed time behavior is discussed in sections 4.2 and 6. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 205 4.1.5. c in 0-, 1-, and 2-dimensions As noted above, the solid angle of constructive interference depends strongly on the dimensionality and shape of the crystal (relative to the direction of the phasing vector k0). For "end-fire" emission from a 1-dimensional shaped rod (l (k0) l (k0)2/ ), the solid angle of constructive interference is ( /l ). For a linear chain of N atoms with an interatomic spacing d, Rc(k0) coh . (4.25) N 4d Thus, there is no appreciable enhancement for a one-dimensional array with d. For grazing "surface mode" emission from a 2-dimensional planar sample l l(1) (k0) (k0) l (k0), l(2) (k0) , and · 2 . l(1) l For a square array of N = M2 atoms with an interatomic spacing d, R 2 c(k0) l 2 ¯ coh , (4.26) N d where l = Md. This result also holds for a planar crystal if the orientation of k0 is such that a surface mode channel ks is just opened (rather than for k0 being a surface mode). Thus for a planar sample and short wavelength resonance radiation d, there is an enhancement of coherent decay l / of the exciton | e(k0) if the phasing vector k0 just opens a surface mode for the plane. In the opposite limit of long wavelegth resonances with the dimensions of the sample (0-dimensional crystal), there is constructive interference in all directions, so = 4 and Rc(k0) = N coh . (4.27) N Thus, for a 0-dimensional crystal c would be strongly enhanced, directly proportional to the number of nuclei N. Of course this long wavelength situation will not be realized for the M¨ossbauer transitions of interest to us, where always d. 4.1.6. Spatially coherent quantum beats and temporal Pendell¨osung If there is hyperfine splitting of the nuclear levels, or if there are shifts between the levels of nuclei located in different chemical sites, there will be periodic construc- tive/destructive interference between waves of different frequency emitted from the 206 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 different sites. This results in a striking "quantum beat" modulation of the spatially coherent decay rate, c c(t), (4.28) varying periodically between "superradiant" and "subradiant" emission into the coher- ent channels, with beat frequencies determined by the frequency differences among the various transition resonances. Furthermore, if there are several coherent channels open, the "superradiant" condition can oscillate between channels, giving a striking temporal Pendell¨osung effect for the probability for photon decay. The coherent elastic photon potential A (R, t) for the spatially coherent decay | e(k0) | 0 + k is given by eqs. (4.9)­(4.11), where the decay width is given in the first Born approximation by = = + , while in the second Born approximation, 1 t t = dt. (4.29) t c(t) + + 0 c(t), and give the various partial widths of | e(k0, t) at time t, i.e., d 1 = - , (4.30) dt e(t) e(t) c(t) + + e(t) e(t) so that within the spirit of this approximation, | e(t) decays as e(t) e(t) = e- (t)t/ e(k0) e(k0) , (4.31) with (t) as given above. The partial width c(t) for coherent emission is given by R2 d Rnc(R, t) c(t ) = , (4.32) e(t )| e(t ) where the photon flux (probability/cm2/s) for spacially coherent emission is given by n 0 2 c(R, t) = · A . (4.33) 2 c (R, t) If there is no hyperfine splitting, then c(t) = c, the enhanced coherent decay width given by eq. (4.16). In the presence of splitting, A (R, t) is a superposition of waves of all the various resonance frequencies nm( ) of all the nuclei within each unit cell, and there will be a "quantum beat" modulation of the coherent decay rate, with c(t)/ varying between a strongly enhanced peak decay rate and a strongly suppressed minimum decay rate. Because of the high directionality of |S(k - k0)|2, c(t) is the sum of the open channel contributions, c(t) = (m) c (t). (4.34) m III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 207 (m) c (t)/ gives the instantaneous rate for radiative decay into the mth channel, so the expected signal I(m) c (t) in the channel (prob./s a photon is emitted into the channel) is (m) (m) I(m) c (t) c (t) c (t) = · = · e- (t)t/ · . (4.35) e(t) e(t) e(k0) e(k0) For the spin flip and internal conversion incoherent decay modes, there is no interference between waves from different sites, and at a particular site, the decay is cnm|en |gm , so the quantum beats only exhibit the splittings of the excited levels. There is no information about the ground state splittings or of the energy shifts between different sites. Because there is no spatial coherence, there is no enhancement of the partial widths for incoherent decay, + . Furthermore, and the spin flip contribution to are time-independent, in sharp contrast to c(t). The spatially incoherent decay goes into all directions, and in any particular direction a quantum beat modulation will be observed. But since the contributing waves arise from different Jz-"oscillators", there will be no net interference when integrated over d R. (There will, however, be a time dependence in the contribution to , which arises from the diffuse elastic scattering associated with the isotope effect. But this is only a small contribution to the total width (t) so we will not consider it further.) A striking example of the quantum beat modulation of the coherent rate due to spatial coherence has been observed in the experiments on yttrium­iron­garnet (YIG) [91]. In YIG there are two inequivalent sites, d1 and d2, with a shift E 6 between the resonance frequencies at the two sites arising from the different orien- tation of the electric field gradient (EFG) relative to the internal hyperfine field Bint at the two sites. There is also hyperfine splitting at each site, resulting in additional higher frequency quantum beat modulations. But to simplify the discussion, we take an idealized case with a single unsplit resonance at each site, with two inequivalent sites d1 and d2 within each unit cell, and with a difference E between the resonance frequencies at the two sites. With k0 set to satisfy the Bragg condition for a symmetric reflection from an M-layer crystal, the partial width for coherent decay is c(t) = (R) c (t) + (T ) c (t), where for m = R, T , (m) c (t) = c · cos2 ( Et - m)/2 . (4.36) Here m gives the relative spatial phase of the d1 and d2 contributions for emission into the mth channel, m = (km - k0) · ( d1 - d2), (4.37) and c gives the peak partial width for coherent emission into a single channel, c = n¯ 2l (k0) coh. (4.38) The crystal "planes" now have unit cell thickness, with a separation d between plane centers, and n is the density of unit cells (/cm3). We have also assumed 0 (k0, k(0-)) 208 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Figure 18. The probabilities/s for photon emission into the R- and T -channels (solid and dashed lines, respectively) following a pulse excitation of an 57Fe enriched YIG crystal with k0 satisfying the condition for an (0 0 2) Bragg reflection. and an exact Bragg condition, so that [1 - (nR · 0)2] · [sin(M )/(M )]2 = 1 for the reflection channel. Thus, the decay rate (m) c (t)/ for coherent emission into each channel oscil- lates between superradiant and subradiant peaks occurring at the quantum beat fre- quency E. Furthermore, if the relative spatial phase factor m = 2n , then the peak decay rates into the R- and T -channels occur at different times, with the time separation being a direct measure of m. For the (0 0 2) Bragg reflection in YIG, (R) = and (T) = 0, so that the instantaneous partial widths for emission into the R- and T -channels are 180 out of phase, (R) c (t) = c sin2( Et/2), (4.39) (T) c (t) = c cos2( Et/2). (4.40) Thus, for the (0 0 2) reflection, the total partial width c(t) for coherent decay is independent of time, c(t) = (R) c (t) + (T ) c (t) = c, (4.41) and the decay parameter (t) also becomes time-independent, (t) = = c + + . (4.42) The probabilities/s for photon emission into the R- and T -channels, eq. (4.35), are then 1 I(R) c (t) = e- t/ sin2( Et/2) · , (4.43) c e(k0) e(k0) 1 I(T) c (t) = e- t/ cos2( Et/2) · . (4.44) c e(k0) e(k0) III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 209 In the reflection channel R(002), there is thus an initial complete suppression of the coherent decay, followed by a delayed enhanced probability for photon emission, as shown in figure 18. The decay rate (R) c (t)/ for coherent decay into R peaks at odd multiples of / E 74 ns. Because the probability of excitation e(t)| e(t) is decaying proportional to exp(- t/ ), the observed signal I(R) c (t) peaks at earlier times tn = [(2n - 1) / E - ], which are the positive time solutions of tan( Et/2) = E/ . For = 4 , as observed in the experiment in [91], 32 ns, and the first delayed peak occurs at t1 42 ns. The enhanced total width 4 is due to coherent enhancement of the radiative width c 3 30 , a 30-fold enhancement of the radiative decay rate. Equally remarkable is the initial complete suppression of coherent decay into R, followed by a delayed peak probability after 42 ns. This is in marked contrast to the radiative decay from an isolated nucleus, which follows an exp(- t/ ) dependence. For the transmission channel, the relative spatial phase is (T) = 0, giving initial constructive interference, and the coherent emission into T(002) is 180 out of phase with the emission into R(002). Thus, for the (0 0 2) Bragg reflection, the coherent decay should exhibit a striking "temporal Pendell¨osung" effect, oscillating back and forth between the T - and R-channels at frequency E. This behavior can be changed by changing the order of the Bragg reflection. For example, for the (0 0 4)-Bragg reflection, the spatial phase factors are (R) = (T) = 0 (mod 2 ), so in this case the R- and T -channels beat in phase with a cos2( Et/2) dependence, with the partial width for coherent decay being c(t) = (R) c (t) + (T ) c (t) = 2 c cos2( Et/2). (4.45) The decay parameter (t), eq. (4.29), is now also time-dependent, sin( (t) = Et) + + c 1 + , (4.46) Et and the probabilities/s for photon emission into the R- and T -channels are then 1 I(R) c (t) = I(T ) c (t) = e- (t)t/ cos2( Et/2) · . (4.47) c e(k0) e(k0) Thus, for the (0 0 4) reflection, in contrast to the (0 0 2) reflection, c(t) is time- dependent, with the probability for photon decay (into either channel) being initially strongly enhanced, followed by complete suppression of coherent decay at a time / E 74 ns. 4.1.7. Nuclear exciton states, Dicke superradiance, and -lasers Steady state grazing is very unpromising because it would involve maintain- ing crystallinity during several natural lifetimes ( 10-9­10-6 s) while each atom is discharging 104 eV energy. Therefore, to prevent excessive heating, the radiation pulse must be emitted quickly, before the crystal is destroyed. Hence, "superradiance" appears essential for gamma lasing [21,43,44]. 210 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 In the case of an initially inverted system, we find that multi-photon Dicke su- perradiance will take place in the mode for which the single exciton state | e(k0) discussed above has the maximum decay rate c(k0)/ : For a long thin needle, with all N atoms initially excited, the initial decay rate for emission into the end-fire mode [21] is N / , where = d /4 is the partial width for single atom decay into the end-fire mode, with = ( /d)2 and d = needle diameter. However, coherent emission into the mode enhances the rate for subsequent emissions into the mode. According to Dicke [21], if p photons have already been emitted into the mode, then the rate of emission of the next photon is R coh coh(p + 1) = (p + 1)(N - p), (4.48) where coh(p + 1) = (p + 1) (4.49) is the probability/s of radiating into the end-fire mode per excited atom. Thus, coh(p + 1) exhibits a "coherent enhancement" over the single atom decay rate by a factor of (p + 1). The power Rcoh(p + 1) will maximize at p = N/2, giving Rcoh(N/2) = (N/2)2 , (4.50) but coh(p + 1) continues to increase, reaching its maximum, N (= n¯ 2l = c(k0), as discussed in section 4.1.4), at p = N - 1, i.e., when only a single excitation remains. Thus, the last state in the cascade, the single exciton state | e(k0) , is the most superradiant (i.e., has the greatest decay rate per excited atom). Conversely, the single exciton state (mode) | e(k0) with the greatest decay rate c(k0)/ will be the favored superradiant decay mode for an initially inverted system. Thus by studying the coherent enhancement of single exciton modes, in crystals as well as amorphous samples, we get direct information regarding the development of superradiance in an initially inverted system. As discussed in section 3.9, coherent multibeam modes resulting from crystallinity have greater coherent enhancements than any single beam modes and would in the case of an initially inverted crystal be the superradiant decay modes. The connection between Dicke superradiance and single exciton states is very important because the coherent enhancement of single exciton modes is subject to experimental investigation now. Thus, it is of interest to investigate, both theoretically and experimentally, the optics of single exciton states induced in crystals by syn- chrotron radiation pulses, not only because of the intrinsic interest in this new field of optics, but also because the results will have a very important bearing on the possibility of -ray lasing/superradiance. Of particular interest will be thin crystal excitations in multibeam Bragg and Laue modes, as well as thick crystal excitations in multibeam Borrmann modes [92­95]. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 211 4.2. Coherent elastic scattering of synchrotron radiation pulses The exciton approach brings out the interesting new physics involved in syn- chrotron pulse excitation of M¨ossbauer nuclei. However, a much simpler method for obtaining the scattered signal is to treat the interaction as a coherent elastic scattering problem. Synchrotron radiation is a very bright, white source of radiation with fluxes in the 10 keV region ranging from 1012­1014 photons/s per eV per mrad of y (see figure 19). The radiation is highly collimated about the plane of the synchrotron, with the divergence at 10 keV being x 0.1­0.2 mrad, with about 90% of the intensity associated with the component parallel to the plane of the orbit ( y in figure 19). There is also a stable periodic time structure, with the duration of each pulse being 10-10 s, and the separation between pulses, depending on the electron bunching, being 10-6 s for operation in the single bunch mode. As discussed in [26,32,86], each synchrotron pulse consists of many photons emitted by the many electrons in the bunch incoherently, one with the other. The pulse from a given electron has a duration c/c 10-19 s, or, if this radiation is filtered to about 1 eV, the coherence time becomes Tc 10-15 s, which is still generally very short compared to the response time I(s) c (t) of the coherently scattered signal (even with strong coherent enhancement, say c 100 , the response time (100 )-1 10-15 s). The duration of the entire pulse depends on the bunching of the electrons and is typically 10-10 s. The scattered flux is then computed starting from eqs. (3.9)­(3.11), where A0(t) represents a one-photon amplitude of duration on the Figure 19. Schematic of scattering geometry. 212 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 order of the coherence time Tc. Because Tc is very short compared to the scattering response time, the time dependent amplitude of the coherent elastic scattered wave is simply proportional to the Fourier transform of the frequency dependent reflection and transmission amplitudes R( ; k1, k0) and T ( ; k1, k0) already obtained for the various cases of off-Bragg, Bragg, Laue and grazing incidence: The time-dependent coherent elastic photon flux which is scattered into the ks channel following excitation of the crystal by a synchrotron pulse of polarization e0 incident at t0 = 0 is given by I(s)(t, ) = 2 gI 2 0p( ) S t; ks, k0; e0 , (4.51) where S(t; ks, k0; e0) gives the amplitude response to a -function pulse, 1 + S t; ks, k0; e0 = d S ; k e-i t. (4.52) 2 s, k0; e0 - Here, S ; ks, k0; e0 = S ; ks, k0 e0 (4.53) is the wave coherent elastically scattered into the ks = ( /c) ks channel when a wave k0 = ( /c) k0 is incident on the crystal with polarization e0. For a reflection channel kr, S( ; kr, k0; e0) is the reflected wave Rr; and for a transmission channel kt, S = Tt. S( ) is given in general by eq. (3.51), and for the special cases of off-Bragg, Bragg, grazing incidence, and Laue, by eqs. (3.63), (3.81, 3.86), (3.105) and (3.114), (3.115), respectively. In (4.51), I0p( ) is the number of photons/eV per mrad2 per synchrotron pulse incident on the crystal at the rocking angle , with polarization e0 at frequency 0; and g = sin s/ sin 0 is the geometrical factor relating the cross-sectional areas of the incident and scattered beams. For simplicity, we also assume that the crystal is oriented so that the scattering plane (k0, ks) is perpendicular to the crystal surface. The y divergence is then maintained on scattering (see figure 19). I(t, ) dt d y then gives the expected number of photons scattered into the ks channel between t and t + dt. If the duration of the incident synchrotron pulse is appreciable compared to the response time of |S(t)|2, then the response must be folded over the initial pulse shape P0(t), I(s)(t, ) = 2 gI 2 0p( ) dt0 S(t - t0; ks, k0; e0) P0(t0). (4.54) In most cases there is no simple analytical solution for S(t; k1, k0), but instead one numerically computes the fast Fourier transform of S( ; k1, k0), which is calculated using the general procedure outlined in section 3.4. There are simple solutions in several important limiting cases, however. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 213 4.2.1. Quantum beats in the Born approximation In the first Born approximation, S ; k1, k0 = M F 10 = n l (k1) f( ). (4.55) Here F 10 is the planar scattering amplitude as given in eq. (3.32) and M is the number of layers. In the second line, l (k1) = Md/ sin 1 is the thickness of the crystal in the k1 direction, and f( ) = f( ) e + f( ) n ( )]e-i(k1-k0)· (4.56) is the 2×2 matrix for the unit cell scattering amplitude. The resonant nuclear scattering amplitude f( ) n ( ) is a sum of Lorentzian resonances corresponding to the various allowed transitions between the excited and ground state hyperfine levels at the unit cell site , f( ) f( ) 0 (nm; k1, k0) /2 n ( ) = . (4.57) [ - nm nm( ) + i /2] The transform S(t; k1, k0) is then a superposition of damped harmonic waves, S t; k1, k0 e-i(k1-k0)· f( ) 0 (nm; k1, k0)e-i[ nm( )-i /2 ]t, (4.58) nm and, hence, the intensity |S(t)|2 is modulated at the various quantum beat frequencies B n, m, ; n , m , = nm( ) - n m (4.59) corresponding to the difference frequencies of all allowed nuclear hyperfine transitions nm( ) from all the different nuclear sites , from which the hyperfine splittings of both the excited and ground states may be found, as well as any energy shifts between nuclei located in different chemical or magnetic sites. The electronic contribution to f( ) gives a prompt time response which does not interfere with the slowly decaying signal from the nuclear resonances. (The fre- quency insensitive parts of fe yield a delta-function response, and the absorption edge resonances with widths on the order eV give signals which decay out in 10-15 s.) The first Born approximation will give an accurate description of the resonant signal if the effects of coherent enhancement are small, i.e., if c . Since c n¯ 20l fM , where n is the density of resonators and fM is the M¨ossbauer factor, the crystallite dimensions must be l 4 /nfM¯ 20, so typically a µm or less. For optically thick samples, the interaction among the resonators can lead to additional "dynamical beats" which must be subtracted from the quantum beat spectrum. The dynamical beats are discussed below. 214 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 4.2.2. Superradiant normal mode excitation at Bragg For a symmetric Bragg reflection from a highly enriched thin film crystal of unsplit M1 M¨ossbauer nuclei, and for incident e0 = radiation, S( ; k1, k0; e0) = R( , ) is given by the simple Lorentzian form of eq. (3.90), with an enhanced width ( ) = + + c( ) and shifted resonance 0( ) = 0 + c( ), as given by eqs. (3.92) and (3.93). Here = k0d cos 0 , where is the rocking angle from exact Bragg at resonance. The transform time signal R(t, ) is then a damped harmonic wave, R(t, ) exp -i 0( ) - i ( )/ t . (4.60) At exact Bragg ( = 0), the signal decays at the natural frequency 0 of the nuclear resonators, with the enhanced decay rate ( + c)/ , with the Bragg enhancement being c = 2 ¯ 2l (k0) coh as given by eqs. (3.99) and (3.90). Rocking off Bragg, the frequency is shifted above or below 0, depending on the sign of , and the decay rate decreases. As discussed in section 3.6.2 and the appendix, the proper intermediate excited states in the scattering process are the collective normal mode exciton states, and at exact Bragg ( = 0) for a symmetric Bragg reflection, the phasing is such that only the superradiant eigenmode state | e(q = k0) is excited, with resonance occurring at the natural resonance frequency 0, and with the coherent decay width c given by eq. (3.99). Rocking off Bragg, in addition to the superradiant eigenmode, various other normal mode states are virtually excited, the weighted resonance energy is shifted from 0, and the effective width is decreased. At exact Bragg, the eigenmode decays symmetrically into the forward (k0+) and reflected (k1-) channels. 4.2.3. Superradiance, dynamical beats, quantum beats and conversion electrons off-Bragg For off-Bragg transmission (or transmission through a noncrystalline film), S( ; k0, k0) = T ( ; k0, k0) as given by eq. (3.63). If there is no hyperfine split- ting, T is diagonal for any basis vectors (ex, ey), and for any incident polarization the transmitted amplitude is T ( ) = exp[in fel] exp i c/( 0 - - i /2) , (4.61) where c = n¯ 2l coh, the enhanced coherent scattering width of the exciton | e(k0) , as given by eq. (4.20). Correct to terms of order l2 ( 2c), the resonance factor can be expressed as a simple Lorentzian response, with enhanced total width ( + c), i i exp c = 1 + c + O l3 . (4.62) 0 - - i /2 ( 0 - ) - i( + c)/2 This expression determines the short time behavior of the transform. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 215 The time response is then given by (see [86]) T (t) = -ie-i[ 0-i /2 ]tJ c 1 4 ct/ . (4.63) t Here we have dropped a prompt (t) contribution and we have suppressed the elec- tronic damping factor exp(in fel). For short times (t < 4 / c the first zero of J1( 4 ct/ )/ t ), i T (t) - c e-i[ 0-i( + c)/2]t, (4.64) which is just the transform of the Lorentzian response of eq. (4.62). For long times (t > 4/ c), using the asymptotic behavior J1(z) 2/ z cos(z - 3 /4), i 3 T (t) - c e-i[ 0-i /2 ]t 4 c cos 4 . (4.65) t3 ct/ - 4 Thus, the delayed forward scattered signal following the creation of a nuclear exciton | e(k0) with k0 off-Bragg is 2 ( c/ )e- t/ J1 4 ct/ /t, I(T) c (t) = ( (4.66) c/ )2e-( + c)t/ , t < 4 / c, c/ t3e- t/ cos2 4 ct/ - 3 /4 , t > 4 / c. In figure 20 we plot I(T) c (t) vs. t for an 57Fe film with c = 4 . The natural lifetime is -1 = 141 ns. The solid line shows the exact transform, and the dashed line gives the short time solution, exp[-( + c)t/ ]. Figure 20(a) gives a linear plot over two natural lifetimes, and (b) gives a log plot over ten lifetimes. For comparison, Figure 20. Off-Bragg time response. 216 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 the dotted line in each figure gives the thin film Bragg response coherently scattered into either the R- or T -channels, with l (k0) taken equal to the off-Bragg thickness l, so that c(Bragg) = 2 c, and I(R) c = I(T) c = exp[-( + 2 c)t/ ]. Superradiance and dynamical beats. Looking at the linear plot, the interesting feature is that the decay is superradiant, for the reasons discussed in section 4.1.4, with I(T) c closely following the simple superradiant decay, exp[-( + c)t/ ], over the effective lifetime of the exciton. With high probability, the off-Bragg exciton | e(k0 decays during this superradiant period. For delayed times t 4 / c, the decay of the off-Bragg exciton has a markedly different character: the fall-off is now slow, exp(- t/ )/ t3, in contrast to the initial superradiant decay exp[-( + c)t/ ], and there is the onset of dynamical beats, cos2( 4 ct/ - 3 /4). In contrast to this complex off-Bragg decay, the Bragg exciton | e(k0) has a simple exponential decay for all times, at the superradiant decay rate ( + 2 c)-1, as shown by the dotted lines in figures 20(a), (b). From the normal mode approach, discussed in section 6, this very different behav- ior of the off-Bragg and Bragg cases can be understood as follows: the exciton | e(k0) created by the synchrotron pulse is a Bloch wave of the form given by eq. (3.96). How- ever, the Bloch waves are generally not the true radiative normal modes in a crystal, but instead, the proper forms of the eigenmodes for a parallel sided crystal have a sine or cosine modulation (coming in from the crystal faces) with a complex wave vector. Generally, the Bloch state | e(k0) is a superposition of these radiative eigenmodes, which have a spread of eigenfrequencies and decay rates. The important exception is the case in which k0 satisfies a Bragg condition k0 · = (1/2) 2, where is a reciprocal lattice vector which is perpendicular to the parallel faces (i.e., a symmetric Bragg reflection). In this case, | e(k0) is a superradiant eigenmode, radiating at the natural resonance frequency 0, with a simple exponential decay at an enhanced decay width ( + c), including the contributions from coherent decay into both the k0 and k0 + directions, c(k0) = n¯ 2 l (k0) + l (k0 + ) . (4.67) When k0 is off Bragg, then | e(k0) is a superposition of radiative eigenmodes. An important general result, discussed in section 6, is that the eigenmodes are not Her- mitian orthogonal. As a consequence, there will be interference effects between the different modes in the evolution of the exciton expectation value, e(k0; t)| e(k0; t) . These interference effects lead to both the appearance of the dynamical beats, and to a "dephasing" of the expectation value, which contributes both to the initial su- perradiant decay, and to the weak 1/ t3 enhancement of the delayed decay of e(k0; t)| e(k0; t) . Correspondingly, the emitted wave packet A(z, t) is a superposition of waves of the various normal mode frequencies, a spread of frequencies ± c/ about 0, leading to initial superradiance, dynamical beats and a weakly enhanced long time III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 217 decay of the emitted signal. For a simplified example, if the normal mode wave contributions entered with equal amplitude over a frequency range 0 ± c/ , and if all modes decayed at the natural , then the emitted signal would have dynamical beats of frequency c/ and a weakly enhanced long time decay |T (t)|2 exp(- t)/t2: + c/ 2 sin( T (t) d e-i( 0+ -i /2 )t = ct/ ) e-i( 0-i /2 )t. (4.68) - t c/ This illustration is much too simple and does not give a proper description of the beats or long time decay, and there is no initial superradiance. But as shown in section 6, making the correct resolution of | e(k0) into normal mode contributions, and using the correct complex mode frequencies, the numerically calculated signal I(T) c (t) is essentially indistinguishable in form from the analytic transform eq. (4.66). A simple alternative approach, which gives a different perspective to the problem, is discussed in [96]: directly solve for the time dependent response of the dipoles in each layer, acting under the transient radiation fields from all the other layers, following the initial prompt pulse excitation. The initial phasing of the dipoles within each layer is such that the layer will radiate equally into both the "forward" k0+ = k0 direction and the "reflected" k0- direction. Initially, the phasing between different planes is such that there will be construc- tive interference in the forward direction k0 between radiation emitted from each plane layer ­ so that "behind" the film (z > Md), the field amplitude is enhanced M-fold, E(+)(z) = ME0, while at the (m + 1)th layer inside the film, the forward propagating wave is enhanced m times, E(+)(md) = mE0. If k0 is off-Bragg, then there is no constructive interference in the reflected k0- channel and the problem is an asymmetrical "one-way communication": the mth layer acts under the influence of the preceding (m - 1) "upstream" layers, but the "downstream" (M -m) layers have essentially no effect on the mth layer. In particular, there are no external forces acting on the first layer, and the dipoles in this layer decay at their natural frequency 0 and natural decay rate (neglecting the small self action of the plane on itself). In contrast, in the Mth layer, the dipoles are driven by the fields from all the preceding layers, and there are two consequences: the component of the force which is 90 out-of-phase with the oscillator motion contributes to the damping of the oscillator, while the in-phase component acts to change the oscillator's phase relative to the "upstream" oscillators, which act under weaker driving forces. The phasing set by the synchrotron pulse is such that, initially, the radiative damping forces add layer by layer, so that the initial decay rate varies from / at the first layer to ( + 2 n¯ 2l (k0) )/ at the Mth layer, with the average decay rate being ( + n¯ 2l (k0) )/ , which is the effective off-Bragg superradiant decay rate of | e(k0) . Due to the asymmetry of the forces, however, the "downstream" planes are even- tually forced out-of-phase with the "upstream" planes, which results in both "dynamical beats", and a dephased exp(- t/ )/ t3 long time decay. 218 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 If, on the other hand, k0 satisfies the symmetrical Bragg condition, then the waves are constructive in both the k0+ and k0- directions, and the problem becomes symmetrical: the driving forces on each oscillator are always equal, and normal mode motion results, with superradiant decay c = 2 n¯ 2l (k0) , and natural frequency 0. A nice demonstration of the dynamical beats has been given by Kikuta [97]. Conversion electrons. As noted above, following the creation of an off-Bragg exciton | e(k0) , there is a marked asymmetry in the subsequent decay rates across the film, with the initial decay rates varying from / at the first layer, to ( +2 n¯ 2l (k0) )/ at the Mth layer, with the average decay rate being ( + n¯ 2l (k0) )/ , which is the effective off-Bragg superradiant decay rate of | e(k0) . This asymmetry of the pla- nar decay rates can easily be detected by observing the incoherent resonant processes (conversion electrons, incoherently scattered gamma radiation, . . . ) occurring at dif- ferent depths of the film. For example, conversion electrons emitted near the top surface will emerge during the natural lifetime -1, while conversion electrons emit- ted near the exit surface will primarily emerge during the much shorter excitation time [ + 2 n¯ 2l (k0) ]-1, and then there will be subsequent weak oscillations. Quantum beats. In the case of hyperfine splitting, the quantum beats B, which arise from the wave interference between different frequency transitions, will be modulated by the dynamical beats, which arise from the interaction between oscillators of the same natural frequency, as discussed above. In the presence of hyperfine splitting, the transmission matrix T ( ; k0, k0), eq. (3.63), is generally not diagonal, and there are strong Faraday type effects due to the orthogonal scattering ex ey. There is generally no simple analytical trans- form in this case, but T (t; k0, k0) is easily computed using the fast Fourier transform programs [85]. There are a few special cases where T( ; k0, k0) can be diagonalized: if Bint k0 (e.g., a ferromagnetic or antiferromagnetic thin film with the magnetization lying in the plane of the film), then the linear vectors ex Bint, ey Bint are the basis vectors; and if Bint k0, the basis vectors are the circular polarizations e±1. For these special cases, the diagonal elements of the transmission matrix are (a = x, y) Taa ; k0, k0 = exp(in faal) i = exp(in f c(m0M ; a) el) exp , (4.69) m 0(m0M ) - - i /2 0M where c(m0M; a) = n lf0(m0M; a) , with Cf f M 2 0(m0M ; a) = 4 ¯ C2(j e k . (4.70) (2j 0Lj1; m0M ) a · Y(q) LM 0 0 + 1) If there are several atoms per unit cell, as in the antiferromagnetic case, then n is the unit cell density, and the f0 contributions must be summed over each site in the unit cell. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 219 Figure 21. Off-Bragg quantum beat spectrum for a ferromagnetic 57Fe film, with Bint lying in the plane of the film and perpendicular to the linear polarization e0 of the incident synchrotron pulse: (a) gives a linear plot of I(T) c (t), and (b) gives a log plot of the signal. Even though T ( ) is diagonal, Taa( ) is a product of exponentials, one factor for each contributing resonance, and there is no simple transform for T (t). However, if the quantum beat frequencies are large compared to the natural width, i.e., B / , then correct to terms of order / B, Taa t; k0, k0 -i e-i[ 0(m0M)-i /2 ]tJ c(m0M ; a)/ 1 4 c(m0M; a)t/ . (4.71) t m0M Here we have again dropped the prompt (t) contribution, and we have suppressed the electronic damping factor exp(in fel). As an example, in figure 21 we show the quantum beat spectrum for a ferro- magnetic 57Fe film, with magnetization lying in the plane of the film and perpendic- ular to the linear polarization of the incident synchrotron pulse. In this case, only the two M = 0 transitions are excited (the +1/2 +1/2 and -1/2 -1/2 transitions), which have equal oscillator strengths, and a quantum beat frequency B = (+1/2, 0) - (-1/2, 0) = 62.1 / between the two transitions. Equa- tion (4.71) is then a good approximation, and the observed signal is I(T) c 2 c (t) cos2( J 4 /t. (4.72) Bt/2)e- t/ 1 ct/ In figure 21, we have taken the film thickness so that c = 4 , as was done in figure 20 for the single resonance. The solid line gives I(T) c (t), as given by eq. (4.72), and the rapid oscillations give the quantum beats, with the period being = 2 / B = 14.3 ns. The dotted curve gives the dynamical beat modulation e- t/ [J1( 4 ct/ )]2/( ct/ ), produced by each of the transitions separately. The important feature to notice is that 220 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 although the amplitude of the signal is modulated by the dynamical beat envelope, the quantum beat spectrum can still easily be discerned. A nice demonstration of this effect is given in [97]. If B is rotated, a very different quantum beat spectrum will be obtained. For example, if B e0, then e0 couples to the four M = ±1 transitions (±1/2 ±3/2, each with relative oscillator strength 1, and ±1/2 1/2, each with relative oscillator strength 1/3). The resulting quantum beat pattern is shown below for grazing incidence reflection in figure 24(b). 4.2.4. Thick crystal pure nuclear Bragg reflection Another important special case that has a simple analytical solution for the time transform is for a pure nuclear Bragg reflection from a thick crystal, with an unsplit (or well isolated) M¨ossbauer transition. In a thick crystal, the initial excitation amplitude within the crystal exponentially decreases exp(-n ez/2 sin 0). The exciton produced, | e(k0) , is "asymmetrical" and no longer a simple Bloch wave as given by eqs. (3.96) or (4.2), and | e(k0) is no longer a normal mode excitation even if k0 satisfies a symmetrical Bragg condition. As a consequence, the coherently scattered signal no longer exhibits a pure exponential decay exp[-( + c)t/ ], as for the thin crystal symmetric Bragg case, but instead has a more complex behavior discussed below. By pure nuclear reflection, we mean that the electronic scattering is strongly suppressed, while the nuclear scattering contribution gives a strong reflection. One method for obtaining such reflections is to produce pure nuclear Bragg reflections by utilizing the unique features of the resonant nuclear scattering: (1) The strength of the resonant scattering can greatly exceed that of the electronic scattering, and the scattering is isotropic (except for the multipole pattern). For ex- ample, an unsplit 57Fe nucleus scatters as 440 electrons at the 14.4 keV resonance, while the electronic scattering is 26 electrons in the forward direction, dropping off to 7.6 electrons at 90 . Consequently, very pure nuclear Bragg reflections can be obtained from crystalline thin films approximately several thousand angstroms thick, which are optically thick for the resonant scattering but essentially transpar- ent for the electronic scattering [98]. (2) The resonant scattering depends on the direction of the magnetic field and elec- tric field gradient (EFG) at the nucleus, while the X-ray scattering is insensitive to internal fields, so magnetic superlattice Bragg reflections are possible in anti- ferromagnetic crystals [11,98­102]. EFG superlattice reflections have also been observed [102,103]. (3) The nuclear transitions are multipole oscillators which (excluding unsplit E1 tran- sitions) have a quite different polarization response than the electronic system, which scatters essentially as an isotropic E1 oscillator. For a 90 scattering with kf in the direction of the incident linear polarization e0, the electronic scattering vanishes, while there will generally be strong resonant scattering, so very pure III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 221 nuclear reflections can be obtained by 90 Bragg reflection of -polarized incident radiation [98]. (4) In complex systems with several atoms per unit cell, including 57Fe, pure nuclear reflections can be obtained at special Bragg reflections where the unit-cell structure factor "accidentally" vanishes due to destructive interference [104]. (5) Grazing incidence techniques can also be used to obtain pure nuclear reflections, as discussed in the next section. As a specific example, we consider a symmetric Bragg reflection from the (3 3 2) planes of an 57Fe crystal, for which the Bragg scattering angle is B = 89.46 for 14.4 keV radiation. In order to simplify the discussion, we assume no hyperfine split- ting. If the crystal is oriented so that the linear polarization of the incident synchrotron pulse corresponds to -polarization for the scattering geometry, then the electronic scat- tering is strongly suppressed, since k1 0, while the M1 nuclear resonance, which is driven by the -polarized magnetic field of the incident radiation pulse, scatters very strongly in the (k0, k1) plane perpendicular to . R ( ; k1, k0) is then given by eqs. (3.83) and (3.88), with the simplification that cos(2 0) = 0, so that the planar reflection arises entirely from the nuclear scattering, i.e., F 10 xx = Fn in eq. (3.88). For this special case there is a simple time transform14 [32,86], R(t, ) = -ie-i[ 0+ c( )]te- t/2 J1[ c( )t], (4.73) t where the complex dynamical beat frequency c( ) is given by i c( ) c/2 c( ) c( ) - = . (4.74) 2 Me + i Here c is given by c = 2 n¯ 2l coh, (4.75) where 2 l = 2l n e, (4.76) e and Me is given by l M sin 0 e = = k d 0le sin(2 0) . (4.77) is now measured from the actual Bragg (including the Thomson scattering index of refraction effect). l gives the thickness over which the initial excitation amplitude decreases by 1/e, and Me gives the effective number of cooperating plane layers in the exciton | e(k0) . 14 This result was presented by the authors at the Workshop on New Directions in M¨ossbauer Spectroscopy, Argonne (July 1977); and, independently, by Yu. Kagan at the Moscow­Munich M¨ossbauer Workshop (October 1977). 222 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Just as for the thin crystal case given by eq. (3.99), c gives the enhanced partial width for coherent decay into two open channels (at exact Bragg), but now with the cooperation length l (k0) determined by the photoabsorption length le. However, because | e(k0) is not an eigenmode, ( c + )/ only gives the initial decay rate. As we will see below, the "long time" development is characterized by an exp(- t/ )/t3 fall-off, and, rocking slightly off exact Bragg, by the onset of dynamical beats. There are two distinct -regions to the response: Very near exact Bragg such that Me < 1 ( < ¯ /le), then c is primarily imaginary: i c c( ) - . (4.78) 2 Then for short times t 2 / c, so that J1( ct) ct/4, there is a pure exponential decay at the enhanced decay rate ( c + )/ , I(R) c t < 2 / c, < ¯ /le exp -( c + )t/ . (4.79) For long times t 2 / c, 2 3 i J ct 1( ct) cos exp , ct - ct 4 ct 2 and the decay becomes nonexponential, exp(- t/ ) I(R) c t 2 / c, < ¯ /le . (4.80) t3 For Me > 1 ( > ¯ /le), c is primarily real, and now becomes independent of the photoabsorption length le 2 n c ¯ 3 coh c( ) . (4.81) 2 Me In this region, the initial decay is simply the natural decay exp(- t/ ), while for delayed times t > 1/ c, the decay again goes as exp(- t/ )/t3, but now modulated by dynamical beats of frequency c( ), exp(- t/ ) 3 I(R) c t > 1/ c, > ¯ /le cos2 . (4.82) t3 ct - 4 In figure 22 we illustrate these results for the (3 3 2) 90 pure nuclear Bragg reflection in 57Fe, with the simplifying assumption of no hyperfine splitting. For this case, le 1.9 × 10-3 cm, giving c 338 ( a 3,000-fold enhancement over ). Figure 22(a) shows the strong superradiant decay which occurs during the first few nanoseconds at exact Bragg = 0, while figure 22(b) gives a log plot over two natural lifetimes, and shows the delayed exp(- t/ )/t3 fall-off for = 0, and the dynamical beats c( ) for = 0.01 and 0.02 mrad. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 223 Figure 22. Time response I(R) c (t, ) for a pure nuclear Bragg reflection in a thick crystal of 57Fe: (a) gives a linear plot of I(R) c (t, ) at exact Bragg ( = 0), showing the superradiant decay which occurs during the first few ns; (b) gives a log plot of the signal for = 0, 0.01 and 0.02 mrad and shows the delayed exp(- t/ )/t3 fall-off for = 0, and the angle dependent dynamical beats for = 0. Superradiant decay occurs only within the very narrow central region < ¯ /le 0.7 × 10-6 rad. Outside of this central region, the most notable feature is the dynamical beats, with the beat frequency ( ) decreasing linearly with as given by eq. (4.81). For 57Fe, this gives a beat frequency of c 12 / for = 0.01 mrad, decreasing to c 1.2 / for = 0.1 mrad. Because of the rapid variation of c with , the Bragg reflection dynamical beats will normally not be resolved: the incident synchrotron pulse typically has an angular spread 0.1 mrad, and the dynamical beat modulation will "wash out", cos2( ct - 3 /4) 1/2. The observed decay will then be exp(- t/ )/t3. Just as in the off-Bragg case, it is the fact that the initial exciton | e(k0) is a superposition of radiative eigenmodes, which leads to the dynamical beats and the weak 1/t3 enhancement of the long time decay. 4.2.5. Pure nuclear reflections at grazing incidence An alternative method for producing pure nuclear reflections is to use grazing incidence reflection from mirrors coated with antireflection films, in which either the films or the substrate contain resonant M¨ossbauer nuclei [29,32,33]. The important point is that the index of refraction for near resonant radiation differs strongly from the index of refraction far off resonance, and as a consequence, it is possible to use thin-film interference techniques to strongly suppress the off-resonance reflectiv- ity while maintaining a very bright reflection for near resonance radiation. Typically, nonresonant reflectivities can be suppressed to 10-4­10-3, while maintaining reso- 224 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 nant reflectivities 70%. Furthermore, by using different combinations of films and substrates, the response can be tailored to give narrow bandpass widths / and correspondingly long delayed scattering times to optimize time filtering, or at the other extreme, to produce filters of very broad resonance width with 100 / , which would be ideal for a high resolution X-ray source. Because this is an index or refraction technique, no crystals are needed and the system is free of the restrictions imposed by a crystalline Bragg condition. As a consequence, the filter can accept the full beam divergence of the synchrotron, and the system is stable against lattice parameter changes, such as might be caused by heating or radiation damage. Furthermore, the films are relatively easy to fabricate, and the techniques can be applied to a number of low-energy M¨ossbauer transitions. The interesting features of the time response are again the quantum beats B which determine the various hyperfine splittings; and, for an isolated resonance, the coherent speedup of the decay and the dynamical beats. Also, for multiple reflection from a system of m parallel mirrors, there is a multiple-reflection delay to the scattering, which can be a useful aid for time filtering. As a specific example, we consider a resonant 57Fe mirror coated with an im- pedance matched quarter-wave film of Te, for which the optimum parameters are l1 76 A and 0 = 4.4 mrad ­ i.e., the 57Fe mirror is coated with a 76 A film of Te, and the grazing incidence angle is set for 0 = 4.4 mrad. The electronic reflec- tivity is then suppressed to |Re|2 9.4 × 10-4, while the peak resonant reflectivity is 0.7 [32]. The filter system is then taken as a system of m parallel mirrors of 57Fe, each coated with a quarter-wave film of Te. The internal magnetic fields B in the separate mirrors will be assumed to be in parallel alignment. For parallel alignment of the internal fields, the filter system has well-defined eigenpolarizations e ( k0, ), = 1, 2, which are determined by the resonant 57Fe medium as given explicitly by eq. (3.58). The eigenpolarizations are generally fre- quency dependent and nonorthogonal, but for the special cases of B k0, or B k0, or if there is no hyperfine splitting, then the bases are orthogonal and frequency in- dependent. In particular, if B k0, then the eigenpolarizations are the circularly polarized vectors e(±1); while if B k0, lying in the plane of the mirror, then the eigenpolarizations are the linear vectors and . For the particular cases of orthogonal eigenbases, I(R) c (t, ) simplifies to I(R) 2 2 c (t, ) = 2 I0p( ) R(m) (t, ) e · e0 , (4.83) =1,2 where R(m) (t, ) is now the Fourier transform of the eigenpolarization response R(m) ( , ) of the m-mirror system. As discussed in [32], the effect of the impedance- matched quarter-wave film is to effectively remove all electronic scattering, so that, to III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 225 a very good approximation, R(m) ( , ) is a pure nuclear reflection, 1 - m R(m) n( ) ( , ) = , (4.84) 1 + n( ) where 1 + n 2f 1/2 n( ) n( ) = . (4.85) ( 2 - 2c + in¯ e) Here c = [-2¯ (fe)]1/2 3.8 mrad is the X-ray grazing incidence critical angle for the 57Fe substrate. The nuclear forward scattering amplitude fn( ) is a superposition of oscillators, f f 0(m0M ; ) n( ) = , (4.86) x(m m 0M ) - i 0M where 2[E x(m 1(j1m0 + M ) - E0(j0m0) - ] 0M ) = , (4.87) and the oscillator strength of the m0M resonance for e eigenpolarization is 2 f 0fM C 2 0(m0M ; ) = C2(j e k . (4.88) 2j 0Lj1; m0M ) · Y(q) LM 0 0 + 1 Here the notation is that of appendix A.2. Isolated resonance: Dynamical effects. For a well isolated resonance with transition frequency (m0M), we take f f 0(m0M ; ) n( ) = (4.89) x(m0M) - i in eq. (4.85). There is then a simple time transform [32] R(m) (t, ) = (-i)me-i[ (m0M)+ c(m0M)]te- t/2 mJm[ c(m0M )t] , (4.90) t where Jm is the mth-order cylindrical Bessel function and the complex dynamical beat frequency is given by i c(m0M ; ; ) c(m0M ; ; ) c(m0M ; ; ) - 2 n = ¯ 2f0(m0M; ) . (4.91) 2 - 2c + in¯ e To emphasize the close connection between the grazing incidence and Bragg cases, we rewrite c, in direct analogy to eq. (4.74), as i c( ) c/2 c c( ) - = , (4.92) 2 Me + i 226 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 where now c = 2 n¯ lef0(m0M; ) (hyperfine splitting) (4.93) = 2 n¯ 2le coh (no hyperfine splitting). (4.94) The effective cooperation length is again determined by photoabsorption, le = 1/n e, and Me is now defined as Me k0le( 2 - 2c). For 57Fe coated with Te ( = 4.4 mrad, c = 3.8 mrad) with B k0, then c( ) 10 / and c( ) 2.8 for the strong M = ±1 resonances. From eqs. (4.83) and (4.90) the time spectrum is then I(R) c (t, ) e-( + c)t/ |m Jm( ct)|2 . (4.95) t2 For short times t < m| c|-1, Jm(z) zm/(2mm!), so that I(R) c (t, ) t2m-2e-( + c)t/ . (4.96) For t > m| c|-1, Jm(z) 2/ z cos(z - /4 - m /2), giving e-( + c)t/ I(R) 2 c (t, ) cos( . (4.97) t3 ct - /4 - m /2) For t m ( c/2)-1, | cos( ct)|2 exp( ct/ )/4, and the long time decay becomes e- t/ I(R) c (t, ) . (4.98) t3 For a single mirror, there is coherent enhancement of the initial decay, exp[-( + c)t/ ]. For intermediate times, the decay is additionally modulated by dynamical beats, exp[-( + c)t/ ] cos2( ct - 3 /4), while for long times, there is a weakly enhanced exp(- t/ )/t3 decay. The dynamical beats and the weakly enhanced long time decay have the same origin as for the off-Bragg and thick crystal Bragg cases: the initial exciton | e(k0) is a superposition of radiative eigenmodes, but the eigenmodes are not Hermitian orthogonal (but instead are transpose orthogo- nal), and as a consequence there is interference between the modes in the subsequent excitation probability e(k0; t)| e(k0; t) , leading to both dynamical beats and a long time "dephasing" of the decay. For multiple reflections, there is an inititial delay of the peak signal. Explicitly, the signal goes as |Jm( ct)/t|2 exp[-( + c)t/ ], which increases initially t2m-2, building up to a maximum at the delayed time tmax (m - 1) /( + c). The nature of this delayed response from the multi-mirror system is the following: since all mirrors are coated with anti-reflection films, only the first mirror sees the sharp synchrotron pulse ("white" over several eV about resonance). Following the initial pulse excitation, the first film sends out a long coherence length wave exp[-i( 0 + c - i /2 )t]J1( ct)/t. In the second film, the resonant absorption/re-emission of the nearly monochromatic incident wave gives a delay in the response. Classically, the near resonant incident wave must first build up the oscillations of the resonators in III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 227 Figure 23. Time response log[I(R) c (t, )] vs. t following excitation of an m-mirror system with a syn- chrotron pulse for m = 1, 2 and 4. The resonant 57Fe mirrors are each coated with 76 A of Te, so = 4.4 mrad, and in all mirrors the internal fields are taken B k0. Here we only include the effects of a single isolated resonance and in (a) use the parameters appropriate for the two strong M = ±1 resonances for which c (10 - i1.4) / ; while in (b) we use the parameters appropriate for the two weak M = ±1 resonances for which c (3.3 - i0.5) / . In both figures, the solid straight line at the top is - t/ , indicating the natural decay of an isolated nucleus. In (a), the dotted straight line is -( + c)t/ , indicating the initial superradiant decay from a single mirror. the second film before any appreciable radiation is sent out. Thus the initial response from the compound system is zero, as given explicitly by the t2 dependence, and since the resonators of the second film respond collectively with a resonance half- width ( + c), the maximum response occurs after a delay /( + c). The features of enhanced decay, dynamical beats, and multiple-reflection delay are illustrated in figure 23, which shows the results for a system of parallel 57Fe mirrors ( c = 3.8 mrad), coated with a /4-antireflection film of Te (lTe = 76 A , and = 4.4 mrad), and in all mirrors the internal fields are taken B k0. The oscillating lines give log[I(R) c (t, )] vs. t for m = 1, 2 and 4 parallel mirrors. Here we only include the effects of a single isolated resonance and in (a) use the parameters appropriate for the two strong M = ±1 resonances for which c 10 / and c 2.8 ; while in (b) we use the parameters appropriate for the two weak M = ±1 resonances for which c 3.3 / and c 0.9 . The solid straight line at the top is - t/ , indicating the natural decay of an isolated nucleus, and the dotted straight line is -( + c)t/ , indicating the initial superradiant decay from a single mirror. Quantum beats. The dynamical beats are a collective effect occurring for each iso- lated resonance. Whenever there is hyperfine splitting, the fundamentally more impor- tant quantum beats B(m0M), arising from the interference between the waves emitted by the various resonances, will be superimposed on the dynamical beat modulation. 228 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 For e eigenpolarization, the nuclear forward scattering amplitude fn( ) is a superposition of contributing resonances, as given by eq. (4.86). There is then no simple time transform of R(m) ( , ), but if the quantum beat frequencies are large compared to the natural width, i.e., B / , then correct to terms of order / B, R(m) (t, ) = (-i)m e-i[ 0(m0M)+ c(m0M; )]t e- t/2 mJm[ c(m0M; )t]. (4.99) t m0M Because the complex dynamical beat frequencies c(m0M; ) appear in the time- dependent exponential phase factor exp{-i[ 0(m0M) + c(m0M; )]t} for each con- tributing transition in R(m) (t, ), then in the initial development, the dynamical beats will shift the quantum beats, B = B + c, (4.100) where the dynamical beat shift is c m0M; m 0M ; = c(m0M; ) - c m 0M ; . (4.101) However, at delayed times t > m ( c/2)-1, then exp(-i c)Jm( ct) exp(- /4 - m /2)/ 2 ct, and the natural beat frequencies B are obtained. The quantum beat spectrum depends on which transitions contribute to each eigenpolarization e , and how the incident synchrotron polarization e0 couples to each eigenpolarization. As a consequence, the quantum beat spectrum is strongly dependent on the orientation of the B-field relative to k0 and e0. As an example, let the incident synchrotron radiation be linearly polarized, and take the mirror surfaces oriented parallel to e0, so that e0 = for the scattering geometry. Then, if B k0, the eigenpolarizations are the circular bases e(±1). Since = [e(+1) + e(-1)]/ 2, the time spectrum is then 1 I(R) 2 2 c (t, ) R(m) + R(m) . (4.102) 2 (+1)(t, ) (-1)(t, ) For this orientation of B, e(+1) couples only to the M = +1 transitions, and e(-1) cou- ples only to the M = -1 transitions. For 57Fe, there are two M = +1 transitions, and two M = -1 transitions, and for pure Zeeman splitting, E(M = +1) = E(M = -1) 62.1 . In this case, there is a single quantum beat frequency B 62.1 / in the spectrum. In figure 24(a) we show I(R) c (t, ) for a two-mirror system with B k0. The "smearing" of the beats in the early development is due to the dynamical beat shifts c(±1) 6.7 / , which are in the opposite directions for two eigenwave contributions to I(R) c (t, ). If the internal field is rotated so that B k0, with B lying in the plane of the film (i.e., B ), then the eigenpolarizations are the linear bases and . Since e0 = , the time spectrum then becomes I(R) 2 c (t, ) R(m) (t, ) . (4.103) III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 229 Figure 24. Time response I(R) c (t, ) vs. t following excitation by a synchrotron radiation pulse for an 57Fe mirror system: two parallel mirrors coated with Te antireflection films. The rapid oscillations give the quantum beats. In (a) the internal fields of the two films are oriented B k0, while in (b), B e0. For this orientation of B, couples to the four M = ±1 transitions of 57Fe. For pure Zeeman splitting, this results in four distinct quantum beat frequencies B: 106.9 / between the strong +1 and -1 transitions, 62.1 / between the two +1 transitions or between the two -1 transitions, 44.8 / between the strong +1 and the weak -1 (or vice-versa), and 17.3 / between the weak -1 and the weak +1 transitions. The resulting beat pattern is shown in figure 24(b). The dominant feature is the sharp B = 106.9 / quantum beats due to the interference between the strongest two transitions, ±1/2 ±3/2. The irregular modulation of the peaks is due to the other quantum beat contributions. The dramatic change between figures 24(a) and (b) illustrates the sensitivity of the quantum beat spectrum to the directions of the hyperfine fields. 5. Resonant neutron optics To this point we have been concerned with resonant X-ray and -ray optics, but the general features are universal for any type of resonant scattering. There are a number of neutron resonances with energies less than a few volts among the heavy nuclei [105,106]. With the advent of neutron spallation sources giving high intensities in the several eV region, it is now possible to use neutron resonance scattering for condensed matter investigations. We comment briefly here on two interesting aspects of resonant neutron optics. 230 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 For a neutron of incident energy E, the resonance coherent elastic scattering amplitude is fres = ¯ 0 n , (5.1) ER - E - i( n + )/2 where n is the partial width for the decay of the excited compound nucleus |(A + 1) to the ground state |A with the emission of a neutron n (|fres| 16.6×10-12 cm), is the partial width for |(A+1) to decay to a lower energy state |(A+1) with the emis- sion of a photon, ER is the resonance energy for the transition n + |A |(A + 1) , and 0 is the neutron wavelength at resonance. For simplicity, we have left out a statistical factor (2j1 + 1)/(2j0 + 1). Typically, the resonance energy ER 1 eV, so 0 0.3 A . The resonances have widths = n + 10­100 meV, and the branching ratio n/ 10-2­ 10-1, so that at resonance |fres| 10-10­10-11 cm, which is 1­2 orders of magnitude larger than the nonresonant (potential) scattering length, b 10-12 cm. For example, for 119In, ER = 1.5 eV, = 72 meV, n = 3.3 meV, so at resonance |fres| 16.6 × 10-12 cm, while b = 0.4 × 10-12 cm. Thus, just as for resonant -ray scattering, two notable features of resonant neu- tron scattering are that the scattering amplitudes near resonance are large compared to the nonresonant (potential) amplitudes, and the phase of the amplitude increases from 0 to as one increases the energy of the incident neutron from well below the resonance to well above it. One simplicity of resonant neutron scattering is that this is a scalar wave, and the scattering is primarily s-wave scattering, so there is no complex polarization or angular dependence as for resonant X-ray or -ray scattering. An important distinction is that the scattering time / 10-14­10-13 s cor- responds to medium collision times, in contrast to the fast X-ray scattering and the very slow M¨ossbauer -ray scattering. That is, speaking roughly, during the residence time ( / ) of the neutron in the compound nucleus, the nucleus (atom) executes an appreciable fraction of a vibration period. As a consequence, for inelastic res- onant neutron scattering, the differential scattering cross-section of a small sample, rather than giving the van Hove S(Q, ), gives W (k0, kf) [1,105], which only reduces to S in the extremely fast collision limit. As an example, for incoherent scattering S(Q, ) g(r, t), the self-correlation function of the scattering particles, while W gives an integral over a higher order correlation function that gives information on the path taken by the particle in moving a distance r in a time t. Two interesting aspects of resonant neutron optics, which we briefly discuss, are the determination of molecular motions with neutron phasors, and neutron superra- diance. A discussion of the suppression effect in resonant neutron optics is given in [107]. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 231 5.1. Determining molecular motions The structure determination of large biological molecules is advancing very rapidly. But to understand the functioning of these molecules their motions must be determined, ultimately at the individual atoms' level, and there has been very little progress in this very important field. One might have hoped, with the advances in high resolution microscopy, that in the near future these motions could be studied by focusing on a single molecule. However, because of the effects of the various possible illuminating radiations, this seems to be impossible at the atomic level of resolution (although X-ray microscopy at the 25 A resolution level and neutron holography at the 100 A resolution level remain open possibilities) [108]. This being the case, to determine the molecular mo- tions at the atomic level we must prepare a macroscopic sample of identical molecules (generally a crystal) and deduce the motions in the individual molecules by measuring the radiation scattered by the ensemble. In this case measurement of the inelastic differential scattering cross-section of a small sample, d2 /d d , gives S(Q, ) [109]: S(Q, ) = eiQ·re-i tg(r, t) dr dt, (5.2) where Q and are the momentum and energy transferred to the sample in the scattering process and g(r, t) = (r + x, t) (x, 0) dx, (5.3) where is the scattering density (electron density for X-ray scattering and the nuclear scattering length density in the case of neutron scattering) and the angular brackets indicate the ensemble average. g(r, t) is the information on the sample which can be obtained from the scattering experiments. The elastic scattering measurements give S(Q, 0) g(r, ), g(r, ) = (r + x) (x) dx, (5.4) the Patterson map of the molecule [110]. Except for the simplest molecules g(r, ) cannot be solved to give the molecular structure (r) (phase of the structure factor problem in crystallography). g(r, ) has "peaks" at all the N(N - 1)+1 interatomic vectors of the N atom molecule. Roughly it is the "N-ply" exposed image obtained by translating the molecule N times, bringing in turn each atom to a fixed point and exposing an image. For small molecules, with the knowledge of some chemistry, we can determine (r) to within an enantiomorph from g(r, ), but when N2 becomes large it is very difficult to determine (r) from the Patterson map. Similarly, g(r, t) is incapable of yielding the detailed structure and motion, the dynamical structure, of any but the simplest systems. For example, while the eigen- 232 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 values (frequencies) of the vibrational eigenmodes of a molecule can be determined from inelastic scattering measurements, the eigenvectors, which give the amplitudes and phases of the displacements of the individual atoms, are not determined. This can be called the phase of the "dynamic structure factor" problem which will become acute when we try to study experimentally the concerted motions in large biomolecules. The methods of isomorphous replacement, heavy atom tagging, and anomalous scattering used to solve the problem in the time averaged static structure determinations can also be used to help in the dynamics determinations. There are differences, however: isomorphous replacements may not change the static structure, but may affect the motions appreciably. Also, X-ray scattering, so useful for static structure determinations, cannot be used to measure low energy vibrational motions until X-ray sources with energy resolution three or more orders of magnitude greater than present monochromators can yield are available. Thus, it appears necessary to rely on neutron scattering measurements, and the use of resonant neutron phasors may furnish the best method for solving the problem. The resonance scattering amplitudes of the low energy neutron resonances are one to two orders of magnitude larger than nonresonance nuclear scattering amplitudes so that the resonance­nonresonance interference term in the cross-section for neutron scattering from a molecule containing a resonant atom is large. Furthermore, by varying the incident neutron energy the phase and amplitude of the resonantly scattered wave can be varied in a controlled manner. This makes it possible to determine [106,111] gR(r, t) = (r + x, t)R(x, 0) dx, (5.5) where R(x) is the probability density of the resonant atom. gR(r, t) gives the posi- tions and the motions of the N nonresonant atoms in the molecule relative to that of the resonant atom, whereas g(r, t), as discussed above, just gives the time dependent Patterson map. Put another way, the resonant atom serves as a reference wave phasor which allows determination of the dynamic structure factor. In summary, the determination of molecular motions is a very important chem- ical problem, and the use of nuclear resonance phasors can serve to determine these motions. 5.2. Neutron superradiance As we discussed earlier, the decay of a system of -ray emitters is radically affected by the spatial coherence of the emitters, and, for example, can be either superradiant or subradiant, depending on the relative phases of the emitters. Now in fact, if the initial coherent state corresponds to just a single excitation, then the nature of the emitted particle (i.e., whether it is a boson or a fermion) plays no role in determining the emission rate. Thus a spatially coherent system of neutron emitters will exhibit the same features of superradiance or subradiance. This is simply a consequence of the quantum superposition principle: to com- pute the probability of a process, we add the amplitudes (with proper phases) of the III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 233 various ways the process can take place, and take the absolute square of the sum of the amplitudes. This leads, e.g., to the characteristic interference pattern for particles traversing a two-slit apparatus, but also, as a consequence, the same coherence effects that affect the decay rates of classical systems emitting classical waves (electric os- cillators radiating electromagnetic waves, Helmholtz resonators radiating sound, etc.) apply to particle emission (electron, neutron, photon, etc.) decay rates in quantum systems [11,112,113]. For example, for two identical neutron emitters (nuclei) separated by a distance R, the semistationary single exciton states are the symmetric (|+ ) and antisymmetric (|- ) combinations, 1 |± = |e , (5.6) 2 1 |g2 ± |g1 |e2 where |el = |(A + 1) l designates the excited state of the compound nucleus l, and |gl = |(A)l designates the ground state of nucleus l. The corresponding decay widths (±) for neutron emission, |± |g1 |g2 + n, are given by sin(kR) (±) n 1 ± , (5.7) kR with eigenwidths E(±), cos(kR) E n (±) = ER . (5.8) 2 kR If kR 1, then the symmetric mode decays at twice the rate of a single nucleus ("superradiance") while the antisymmetric mode decays very slowly ("subradiance"). This corresponds classically to two electric dipoles oscillating together in phase (sym- metric mode) or 180 out of phase (antisymmetric mode), which exhibits the same superradiant or subradiant behavior when kR 1. Similarly, for a crystal of such neutron radiators, the single exciton state | e(q) , N e(q) = eiq·Rl|el G0(l) , (5.9) l=1 will be superradiant or subradiant depending on the phasing vector q: if q = k0 = (2 / 0)n, where 0 is the neutron resonance wavelength, then | e(k0) emits a neutron superradiantly at the coherently enhanced decay rate c(k0), c(k0)/ = n¯ 2L (k0) n, (5.10) with emission occurring in a narrow solid angle ( 0/L )2 about k0 (where L is the transverse dimension of the crystal k0), while if |q| = |k0| the emission is subradiant. If the phasing vector k0 should satisfy a Bragg condition for neutron diffraction from the crystal, then there will be superradiant contributions c(k0 + ) for each open Bragg channel, just as for the -ray case. 234 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 For resonant -ray optics, it is possible to create superradiant single exciton states | e(k0) using synchrotron radiation pulse excitation. There are, of course, no such sources available for neutrons. However, the state | e(k0) is excited virtually when a nearly resonant neutron E ER is incident externally at a Bragg angle, and as the incidence energy E is tuned through resonance, the width of the Bragg scattered intensity will show an enhanced width ( c + ). This coherent enhancement of the width has recently been observed by Hastings et al. [114]. For -ray emission, the superradiant single exciton state | e(k0) is closely related to multiphoton Dicke superradiance: the single exciton state | e(k0) with the greatest decay rate c(k0)/ , is the favored mode of decay for an initially inverted system. Since there is single exciton superradiance for a system of neutron emitters just as for a system of -ray emitters, is it possible to have multineutron Dicke superradiance? The answer of course is no, because the Fermi statistics of the neutron prevents a superradiant cascade of neutrons into the endfire mode. 6. Radiative normal modes In this final section, we discuss one of the most unique and fundamental as- pects of M¨ossbauer -ray optics ­ collective nuclear states established by the radiative interaction between the nuclei. M¨ossbauer -ray optics is rich and complex, with a variety of competing processes ­ sharp resonant scattering from the various nuclear hyperfine transitions, internal conversion absorption, Thomson scattering from the atomic electrons, pho- toelectric absorption, resonant and nonresonant magnetic X-ray scattering, etc. The most straightforward development of the coherent optics is in terms of the multiple scattering equations (3.5) and (3.6), with the effects of all the various processes in- corporated on an equal footing in the coherent elastic scattering amplitudes fj of each atom.Some of the most interesting features involve the collective effects of the inter- actions between the nuclear oscillators, e.g., the "enhancement effect" at Bragg, and the superradiant or subradiant decay of a nuclear exciton. Since these phenomena only involve the nuclei, considerable insight can be gained by approaching the prob- lem from a different point of view: solve directly for the radiative normal modes (semi-stationary states) of the system of nuclei. The scattering of an external plane wave k0 then involves virtual excitation of these modes as intermediate states; and the decay of a nuclear exciton is a superposition of exponentially decaying semi-stationary states.In this section, we discuss the normal modes of decay of a system of identical resonators. We give the effective Hamiltonian equation of motion which determines the complex normal mode frequencies n and give simple sum rules which govern the real and imaginary parts of n. Because of retardation, the Hamiltonian is symmetric rather than Hermitian, and as a consequence, the eigenmodes | n satisfy a transpose orthogonality condition rather III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 235 than Hermitian orthogonality. This has an important consequence in the decay of a general superposition exciton state | e = an| n , because interference between the modes now leads to "dephasing" and to dynamical beats in the decay of the excitation probability e(t)| e(t) . Of particular interest is the decay of an exciton | e(k0) created by a synchrotron pulse. We show that if k0 satisfies the symmetric Bragg condition, then a single superradiant eigenmode is excited, and the subsequent decay is a simple enhanced exponential decay. If k0 is off-Bragg, then | e(k0) is a superposition of normal modes, and the spread of frequencies and Hermitian-nonorthogonality play an important role both in the initial superradiance and subsequent dynamical beats. The development we give here follows [11,96,112,115]. 6.1. Classical and quantum equations of motion For a freely decaying system of identical classical electric dipole oscillators, each oscillating in the xo direction, di(t) = exi(t) xo, the coupled equations of motion are15 m¨xi + m xi + m 20xi = e xo Eij(t), (6.1) j =i where Eij(t) = Ej(Ri, t) is the electric field at Ri due to dipole j (Eij(t) is assumed to be uniform over the dimensions of the dipole, i.e., |xi| ). For normal mode motion, each dipole oscillates as xj(t) = xj exp(-i t), where is the complex frequency of the mode. The field at R emitted by the jth oscillator is then 1 Ej(R, t) = × × d i k|R - R , (6.2) |R - R j exp j| - t j| where dj = exj xo. Equation (6.2) is valid for the "small source" approximation, which requires that the dimension of the oscillator |xj| be much smaller than the distance |R - Rj| and the wavelength . For "M¨ossbauer radiators", 1 A and |xj| 10-4 A , so the condition |xj| is well satisfied, and (6.2) would be a good approximation even for |R - Rj| 10-3 A , well inside the atom. Assuming normal mode oscillation, substituting (6.2) into (6.1), and linearizing the resulting equations in by making the replacements 2 - 20 = ( + 0)( - 0) 2 0( - 0) and exp(ikRij) exp(ik0Rij), gives an effective Hamiltonian equation of motion hX = X, (6.3) 15 is customarily a frequency in classical electrodynamics, while is customarily an energy in quantum electrodynamics (= classical). Since the factors of are not germane to the discussion here, in this section we suppress all factors of . 236 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 where, for the classical system of oscillators, X is the N-component state vector x1 x X = 2 , (6.4) .. . xN and the Hamiltonian is i hii = 0 - , (6.5) 2 eik0Rij h ij(i = j) = - . (6.6) 2 ij k0Rij In (6.6), eik0Rij 3 exp(ik 0|Ri - Rj|) ij = x × × x , (6.7) k o o 0Rij 2k30 i i |Ri - Rj| so that 3 2 1 i 3 n - 1 - (near zone R 2 ij · xo (k k ij ), 0Rij)2 0Rij ij (6.8) 3 1 - n 2 (far zone R 2 ij · xo ij ). Equation (6.3) is an eigenvalue equation, which determines the complex nor- mal mode frequencies n (n = 1, . . . , N) and the corresponding normal mode state vectors Xn. The corresponding quantum system is a system of nuclei with nondegenerate ground states |al (l = 1, . . . , N) and excited states |bl . The normal modes of decay (semi-stationary states) are now nuclear exciton states: c1 N c 2 X = | e = cl|bl |G0(l) = . , (6.9) . l=1 . cN where |bl |G0(l) means that nucleus l is excited and all the remaining nuclei (l = l) are in their ground states |al . An initially excited nucleus i will decay due to self-action (virtual radiative and internal conversion emission/absorption, and "mirror terms" involving virtual photon exchange with the other nuclei, as discussed in [12, appendix] and [2, appendix A]), and, if initially unexcited, can be excited by photon exchange from an excited nucleus j by the interaction c-2jµ(i) +ij jµ(j) (see, e.g., [10, appendix B]). The equations of motion are obtained by taking the Fourier transform of the decaying exciton G(t - III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 237 t0)| e . The excited state propagator G(t - t0) is obtained using the Green function approach of [2,10,12]. The resulting coupled quantum equations of motion, which determine the expo- nentially decaying semi-stationary states Xn and the complex mode frequencies n, are of the same general form as the classical equations, hX = X, (6.10) where now the state vector X is the nuclear exciton, as given by eq. (6.9), and again i hii = 0 - , (6.11) 2 eik0Rij h ij(i = j) = - , (6.12) 2 ij k0Rij but where now eik0Rij 2 exp(ik 0Rij) ij = - jba jab - . (6.13) k µ k µ k 0Rij Rij Here k = -i i and k = i i, where the gradient is with respect to Ri, the equilibrium position of atom i, and the gradients operate on the factors to the right or left as indicated. jba µ (k) = c-1 dx b|eik·xjµ(x)| a is the Fourier transform of the nuclear current density. In the far field region, Rij 0, the longitudinal and scalar parts of the current­ current contraction drop out, and the expression simplifies to eik0Rij 2 exp(ik 0Rij) ij = jba · jab - , R k k k ij 0. (6.14) 0Rij Rij In this limit, the operators k and k can be replaced by kij = k0nij (but often it is convenient to first take sums over sites j before applying the gradient operators). To compare with the classical case, we take a system of "linear" E1 oscillators (i.e., Jz = 0), each with ground state |a = |j0 = 0, m0 = 0 and excited state |a = |j1 = 1, m1 = 0 , and local quantization axis zJ. Then jab (-kij) = 0 Y(e) 10 (nij), with Y(e) 10 (nij) given by eq. (A.22), giving ij = (3/2)[1 - (nij · zJ)2], which agrees with the classical far zone result of eq. (6.8). 6.2. Eigenmodes For either the classical or quantum systems, the determinant equation Det h - 1 = 0, (6.15) where 1 is the N ×N unity matrix, determines the complex frequencies for the normal modes, m = m - i m/2, m = 1, . . . , N. m = m - 0 gives the frequency shift of the mth normal mode from the natural resonance frequency of a single oscillator; 238 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 and m gives the decay rate of the mode, which will generally be quite different from the decay rate of an isolated nucleus. Substituting m for in the eigenvalue equation (6.10), and applying the trans- pose normalization condition of eq. (6.17) below, determines the mth eigenmode cm 1 N cm 2 | m = cm l |bl G0(l) = Xm = . . (6.16) . l=1 . cm N 6.3. Orthogonality and sum rules The Hamiltonian h given by eqs. (6.5)­(6.7) or (6.7)­(6.13) is symmetric rather than Hermitian ­ i.e., hji = hij. As a consequence, the eigenmodes will be transpose orthogonal, rather than Hermitian orthogonal, T m n = XTm · Xn = mn, (6.17) where Tm| = XTm is the transpose state, N T m cm l bl| G0(l) = XTm = cm 1 cm 2 . . . cm N . (6.18) l=1 The Hamiltonian h can be diagonalized by the similarity transformation, U h U-1 = , (6.19) where is the diagonal eigenvalue matrix, mn = m mn. The N ×N transformation matrix U is obtained from the eigenvectors Xm in the usual manner, XT 1 XT U = 2 . . (6.20) .. XTN Because of the orthogonality condition (6.17), U-1 = UT. Since hXm = mXm, eq. (6.19) immediately follows. Since similarity transformations leave the trace invariant, we have Tr h = Tr , (6.21) which gives i m = N 0 - . (6.22) 2 m III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 239 Taking the real and imaginary parts of eq. (6.22) gives the "sum rules" for the eigenfrequency shifts and normal mode decay widths: m = m - 0 = 0, (6.23) m m and 1 N m = . (6.24) m The frequency shift sum rule, eq. (6.23), shows that for some modes the frequencies increase, while for others the frequencies decrease, such that the frequency shifts of all modes average to zero. The decay width sum rule, eq. (6.24), shows that the total decay widths of the various normal modes average to the decay width = + of a single oscillator. The decay width of the mth mode can be expressed as m = (m) c + + , where (m) c is the partial width for coherent radiative decay in mode m, and + gives the total width for spatially incoherent radiative and nonradiative decay. Equation (6.24) can now be rewritten as a sum rule for the coherent decay widths, 1 (m) N c = - . (6.25) m For a classical system, and for a nondegenerate quantum system, = 0, and we assume this in all subsequent discussions. Thus, some modes will have (m) c < , while for some (m) c > , such that the coherent radiative decay widths average to . For planar and crystalline arrays of nuclei, there will be superradiant modes, for which (m) c , and subradiant modes, for which (m) c . 6.4. Radiative decay of a normal mode As a function of time, the mth mode develops as m(t - t0) = G(t - t0)| m = 1(t - t0) e-i m(t-t0)| m , (6.26) and the (relative) probability of excitation decays exponentially, 2 m(t - t0) m(t - t0) = Xm(t - t0) = e- m(t-t0)|Xm|2. (6.27) In the normal mode approach, all multiple scattering interactions between the resonators have been taken into account in the normal mode equation of motion, eq. (6.10), and the photon emitted from the system to an external point (z, t) is then simply a freely propagating photon, with no further multiple scattering, A(m) µ (z, t) = 0|c-1 dx +(z, x)jµ(x)G(tx - t0)| m . (6.28) 240 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Taking t0 = 0, z = R, and assuming R is large compared to the dimensions of the system, gives |R - R exp[-i A(m)(R, t) = 1 t - j| m (t - |R - Rj|/c)] jab - cm c |R - R k j j j| exp(ik 1 t exp -i mR) mt jab XT(-k . (6.29) R (-km) m) · Xm Here t = (t - R/c) is the retarded time, k m = ( 0 + m)/c, km = kmR, and we have introduced the "phased vector" X(k), with jth component exp(ik · Rj), so that XT(-km) · Xm = e-ikm·Rjcm j . (6.30) j Equation (6.28) is the quantum version of Huygens' principle of classical optics: given the actual motions xi(t) of a system of dipoles, the radiation field at any point (z, t) is the superposition the freely propagating waves emanating from each source. 6.5. Radiative decay of a general exciton state The N semi-stationary normal mode states | m are a complete set of states for a single excitation in the system. Within this manifold, the resolvent operator is I = | m T m . (6.31) m Any single-exciton state | e can then be resolved into normal mode states, | e = am| m , (6.32) m where the expansion coefficient is a m = T m e . (6.33) As a function of time, | e then develops as e(t) = ame-i mt| m , (6.34) m and the photon potential for the emitted radiation is A(R, t) = amA(m)(R, t), (6.35) m with the normal mode field A(m)(R, t) given by eq. (6.29). The expected flux (probability/cm2/s) for finding a photon at (R, t) is then given by ( 0/2 c)|A(R, t)|2, eq. (3.11). For a system of classical oscillators, |A(R, t)|2 is proportional to the expected energy flux, S = (c/4 )Re(E × B ). III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 241 Since the semi-stationary states | m are transpose orthogonal, rather than Her- mitian (i.e., generally n| m = 0), there can be interference effects between the modes in the decay of the excitation probability, e(t) e(t) = a namei( n- m)t n| m . (6.36) m,n These mode interference effects can lead to a "dephasing" of e(t)| e(t) and to the emitted signal |A(R, t)|2. In the case of a synchrotron pulse exciton | e(k0) , this dephasing effect contributes to the initial superradiant decay if k0 is off-Bragg. These interference effects can also lead to dynamical beats in the emitted signal |A(R, t)|2. When this occurs, e(t)| e(t) will "flatten out" (i.e., the rate of de- cay d/dt e(t)| e(t) 0) whenever |A(R, t)|2 has a sharp minimum. In effect, the excitation is momentarily "trapped" within the system. 6.6. Normal mode excitation by coherent elastic scattering For coherent elastic scattering of an incident photon A0 by the system, the scat- tered photon potential As is, in the notation of (3.5), As = +MA0, (6.37) where M is the coherent elastic scattering operator for the system. The basic problem is to obtain M. When treating M¨ossbauer -ray optics in its full complexity, the simplest proce- dure is to develop M in terms of single atom scattering, which leads to the simple multiple scattering eqs. (3.16) and (3.17), which determine the coherent elastic scat- tered wave. For a system of two level nuclei, however, we can use the normal mode approach and calculate M directly for the system. The scattered wave is Asµ(z, t) = 0| - ic-2 +(z - x)eiH0txJµ(x)G(tx - ty)J (y)A0 (y)e-iH0tx dx dy| 0 , (6.38) where Jµ(x) is the current density operator for the system and G(t) is the corrected propagator for exciton states. Inserting the resolvent operator I = m | m T m| to the left of J (y), the resulting scattered photon potential, at a point R far removed from the system, is simply eikR As(R, ) = 0|J (-k)| m Tm|J(k0) · e0| 0 a R [ 0 m m - ] eikR [XT(-k) · X = jab m][XTm · X(k0)] , (6.39) R (-k)jba(k0) · e0 a0 Q m m( ) 242 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 where the resonance response denominator for the mth mode is (m) Q c + m( ) = [ m - ] = 0 + m - - i . (6.40) 2 In the second line of (6.39), j(k) is the Fourier transform of the nuclear current density operator for a single nucleus. X(k) is the notation introduced in eq. (6.29) for a "phased vector", with jth component exp(ik · Rj). The wave vector coupling of the incident photon to the mth mode is then XTm · X(k0) = cjeik0·Rj, (6.41) j and the coupling to the emitted photon k = /c R is XT(-k) · Xm = cje-ik·Rj. (6.42) j From this perspective, the incident photon | k0, scatters by the virtual excitation/de-excitation of the exciton normal mode intermediate states | m , with the usual resonant frequency response 1/[ m - ]. The photon interacts with the system through the distributed system current Jµ(x), with the coupling to the system determined both by the polarization coupling to the individual nuclei, j · e0, and by the wave vector coupling to the mode, [XTm · X(k0)]. The wave vector coupling changes with the orientation of k0, and in some instances, such as in the enhancement effect at Bragg, it is possible to have selective excitation of a normal mode in the scattering. 6.7. Two nuclei For two nuclei, h is 2 × 2, and the solution for the normal mode frequencies is immediate: 12eik0R12 (±) = 0 - i , (6.43) 2 2 k0R12 so that the frequency shifts of the two modes are 12 cos(k0R12) (±) = , (6.44) 2 k0R12 and the coherent decay widths for the two modes are (±) 12 sin(k0R12) c = 1 ± . (6.45) k0R12 The frequency shifts and coherent widths satisfy the sum rules (+) + (-) = 0, and [ (+) c + (-) c ]/2 = . III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 243 The corresponding eigenvectors are the symmetric and antisymmetric states 1 X 1 (±) = . (6.46) 2 ±1 Since all vector components are real, X(±) are Hermitian orthogonal as well as transpose orthogonal. For k0R12 1, +c 2 and -c 0. In this limit, the symmetric mode has an enhanced decay rate (superradiant), while the antisymmetric mode becomes nonradiative (subradiant). This behavior is easy to understand classically. The symmetric mode corresponds to two oscillators oscillating together in-phase, which, for k0R12 1, doubles the dipole moment. The radiation fields are doubled and the radiation power is increased to four times that of a single oscillator. Since the total mechanical energy T is also doubled, the decay rate for the system is -(d T/dt)/ T = 2 . In contrast, in the antisymmetric mode, the oscillators move 180 out of phase, and the radiation fields from the two oscillators interfere destructively, giving a very weak (electric quadrupole) radiative decay. For coherent elastic scattering of an incident photon | k0, , As is given by (6.39), with the wave vector coupling to the incident and scattered photons now given by XT(±) · X(k0) = eik0·R1 ± eik0·R2 2, (6.47) XT(-k) · X(±) = e-ik·R1 ± e-ik·R2 2. We note, in particular, that the X(+) mode can be selectively excited by taking k0 R12, i.e., for this orientation, [XT(-) · X(k0)] = 0. For this case, the incident wave drives the two oscillators in phase, which is the symmetric mode. This solution for As, using the normal mode approach, agrees with the multiple scattering solution worked out in [10, appendix B], as it should. 6.8. Planar array For a planar array, we reverse the solution procedure, first obtaining the eigen- modes Xq by symmetry, and then solving for q directly from the Hamiltonian equation hXq = qXq. Two-dimensional translational symmetry dictates that the eigenmodes should be 2-D Bloch waves 1 e(q) = Xq = X(q), (6.48) N where, as before, X(q) is the phased vector with jth component exp(iq · Rj), and the N 2-D vectors q uniformly fill the first Brillouin zone. For simplicity, taking a square array with lattice spacing a, we have q = (2 /L)[nx x+ny y], with L = N a, and nx, ny - N/2, . . . , + N/2. Or, in a continuum limit, the q's fill the first Brillouin zone (- /a qx,y /a) with mode density (q) = (L/2 )2. 244 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Now these state vectors are, in fact, Hermititian orthogonal rather than transpose orthogonal, X q · Xq = q,q , (6.49) or, in the continuum limit, X q · Xq = (2 /L)2 (q - q ). However, we will see that the states Xq and X-q are degenerate, i.e., -q = q. Thus, we can take the standing combinations [Xq ± X-q]/ 2, which are transpose orthogonal. For most purposes, however, it is convenient to use the traveling wave solutions Xq. The proper resolvent operator is then I = q |Xq Xq|, rather than eq. (6.31). Substituting Xq into the Hamiltonian equation, and carrying out the resulting planar lattice sum as in (3.23), the complex mode frequency is + (q) c q = 0 + q - i = i , (6.50) 2 0 - i - s - S s where n s(q) = - jba . (6.51) g µ (ks+)jab µ (-ks+) + jba µ (ks-)jab µ (-ks-) s(q) Here, ks± = q + s ± gs(q) z, (6.52) gs(q) = k20 - q + s)2, (6.53) where the s are the planar reciprocal lattice vectors, n = 1/a2 is the planar density of the nuclei, and S indicates that the real part of the self-action (radiation reaction of a single nucleus) is to be subtracted from i s. The sum over s breaks into two regions: those s for which gs(q) is real, which give open radiation channels to the normal mode field A(q), given in eq. (6.57) below; and those s for which gs(q) is imaginary, gs(q) = i|gs(q)|, which give a nonradiative contribution to A(q). The sum over all closed channels gives an exponentially damped contribution to A(q) on the order of e-k0|z|/k0|z|. We denote the open channel sum as s<, and the closed channel sum as s>. The partial width for coherent elastic decay in mode q is then (q) c = (q) c (s), (6.54) s< where the partial width for emission into the symmetric open channels ks+ and ks- is 2 n (q) c (s) = jba . (6.55) g (ks+) · jab (-ks+) + jba (ks-) · jab (-ks-) s(q) To compare with the classical case, we take a system of linear E1 oscillators (i.e., Jz = 0), each with ground state |a = |j0 = 0, m0 = 0 and excited state |a = III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 245 |j1 = 1, m1 = 0 , and local quantization axis zJ. If zJ lies in the plane of the nuclei (the x, y plane), then (q) c (s) = 2 n ¯ 2 k20 - (( s + q) · zJ)2 k coh, (6.56) 0 gs(q) where coh = [(2j1 + 1) /(4j0 + 2) = (3/2) . This result agrees with the classical treatment of [11]. The normal mode radiation field (Feynman photon potential) A(q), obtained from eq. (6.29) by first carrying out the planar lattice sum before applying the gradient operator k = i R, is given by i2 n A(q)(R, t) = 1 t e-i qt 1 eigs(q)|z|ei(q+ s)·Rxyjab -k . (6.57) N g s s(q) This is a superposition of a finite number of open radiation channels, ks±, and a nonradiative contribution from the closed channels on the order of e-k0|z|/k0|z|. The open channels ks+ and ks- are a symmetric pair above and below the x, y plane, each channel making an angle s with the plane, where sin s(q) = gs(q)/k0 (see figure 8). The expected flux (probability/cm2/s) for finding a photon at (R, t) is ( 0/2 c) × |A(q)(R, t)|2 (eq. (3.11)). For the planar system, coherent emission goes into one of a finite number of open channels ks ( = ±). The cross-sectional area for the ks channel is L2 sin s. Using just the ks± contributions to A(q)(R, t), the probability/s that the state | e(q), t de-excites by coherent -ray emission into either of the ks± channels at time t is then given by (q,s) c (t) = (q) c (s)e-( + + (q) c )t, (6.58) where (q) c (s) is the partial width for emission into the symmetric open channels ks±, as given by (6.55), and (q) c = s (q) c (s) is the partial width for emission into all open channels. Integrating (6.58) over t, the probability that | e(q) decays by coherent photon emission into the ks± channels is (q) P (q) c (s) c (s) = . (6.59) + + (q) c For coherent elastic scattering of an incident photon A0(z, t) = a0e0 exp[i(k0 × z - t)], using eq. (6.38), the scattered wave is i jab As(R, ) = 0n eig k)jba s(q)|z|ei(q+ s)·Rxy (- (k0) · e0a0 , (6.60) sin s s 0 + q - - i( (q) c + )/2 where q = k0xy (reduced to the first Brillouin zone) in the denominator, and the notation for gs and s is now that of eq. (3.19), where the planar scattering was treated in the Born approximation. Equation (6.60) gives the exact result for elastic 246 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 scattering from a planar array of nuclei, including the planar self-action. We note that the incident k0 selectively excites the q = k0xy planar normal mode. Planar surface mode and total transmission. The most interesting feature of the planar solution (6.60) is that total transmission can occur [11]. From (6.53) and (6.55), we see that for well defined directions of k0, there will exist a s such that gs = 0 and c(s ) = . The coherent width for emission into the s channel is infinite and the nuclear excitation amplitudes are identically zero. The scattered wave is a single channel, parallel to the surface, with intensity equal to that of the incident wave, but the ratio of the total energy of this wave to the incident energy is zero (because of the sin s / sin 0 ratio of the cross-sectional areas). There is no forward scattering, so the total transmitted wave is A0 and the layer becomes perfectly transparent to the incident radiation. This phenomenon of surface mode channels in single layers, and in M-layer crystals, is discussed further in [96]. Exact planar vs. Born approximation. In our development of the generalized Darwin­ Prins dynamical theory of X-ray and -ray optics, the scattering from each plane is treated in the Born approximation, and then the planes are coupled through their scattered radiation fields. The only approximation involved is the neglect of planar self-action. This approximation is in fact very good, except for the special case when a planar surface mode channel is just opened or just closed. For a system of two level nuclei, we can use the exact planar result of eq. (6.60), and give an exact Darwin­Prins development, as carried out in [11]. However, the only difference between (6.60) and the Born approximation is the resonance denominator Q( ): for the Born approximation, Q( ) = 0 - - i /2, while, with planar self- action, Q( ) = 0 + q - - i( + (q) c )/2, with q = k0xy. Now from (6.56), (q) c (s) 2 n coh/ sin s 2 × 10-2 coh/ sin s, where we have taken a 2.5 A and 0.86 A . Thus, s < 10-2 rad may double the coherent width (compared to an isolated nucleus), but since this is only a small fraction of the total width, s less than about 10-3 rad is required to double the total width. Similarly, the frequency shift q will normally be a negligible fraction of . Thus, for most orientations of k0, the Born approximation is quite accurate for treating the scattering of a single plane. 6.9. M-layer crystal For an M-layer crystal, the invariance of the system under x, y translations by planar lattice vector displacements dictates that the normal modes will be 2-D Bloch waves Xq in the x, y direction. Each plane layer sends out plane waves ks±(q) which couple the layers together, and scatter into the same set of open channels. Since there are M plane layers, there will be M different linear combinations of the planar III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 247 solutions, giving M different normal modes within each q-manifold. Thus, the general form of the normal mode states is Xq,l = Zq,l Xq, (6.61) where Zq,l is an M-component vector c(q,l) 1 c(q,l) Z 2 q,l = . (6.62) .. . c(q,l) M The new index l (l = 1, 2, . . . , M) specifies the M different normal modes within each q-manifold. In mode Xq,l, the exciton state is | q,l = j cq,l j |bj |G0(j) , and if the nucleus j lies in the mth layer, at the position Rj, then cq,l j = c(q,l) m exp(iq · Rj)/ Nxy. The Bloch wave states Xq factor out, and in the q-manifold, the Hamiltonian equation reduces to an M × M matrix equation h(q)Zq,l = Zq,l, (6.63) where now h + (q) c nn = 0 + q - i , (6.64) 2 i hmn(m = n) = - (q) ig . (6.65) 2 c (s) exp s|n - m|d s In (6.65), for the closed channel contributions (s>), (q) c (s)/2 = s(q), as given by (6.51). The determinant equation, Det[ h(q) - 1] = 0, determines the complex fre- quencies q,l = [ q,l - i q,l/2] = [ 0q + q,l - i( + (q,l) c )/2] of the M normal modes, l = 1, 2, . . . , M. The sum rules (6.23) and (6.25) hold globally, and within the q-manifold, we have the sum rules M M q,l = q,l - 0q = 0 (6.66) l=1 l=1 and 1 M (q,l) (q) M c = (q) c = c (s). (6.67) l=1 s< The Hamiltonian h(q) is again symmetric, rather than Hermitian, so the eigen- vectors Zq,l satisfy the transpose orthogonality condition within the q-manifold, ZTq,l · Zq,l = ll . An exciton state | e = Z(0), which lies in the q-manifold (such as 248 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 state | e(k0) created by a synchrotron pulse, which lies in the q = k0xy manifold), will then develop in time as M Z(t) = ale-i q,ltZq,l, (6.68) l=1 where al = ZTq,l · Z(0). (6.69) The field emitted by the crystal in the (q, l) normal mode is i2 n A(q,l)(R, t) = e-i q,lt eiks+·R j g (-ks+) ZTq,l · Z(-gs), z > (M - 1)d, s< s(q) (6.70) i2 n = e-i q,lt eiks-·R j g (-ks-) ZTq,l · Z(gs), z < 0. s< s(q) Here Z(±gs) is the M-component phased vector with mth component exp(±igsmd), with m = 0, 1, . . . , M - 1, so that ZTq,l · Z(±gs) = m c(q,l) m exp(±igsmd). For an initial exciton state | e = Z(0), the subsequent emission field is a superposition of normal mode fields, M A(R, t) = alA(q,l)(R, t), (6.71) l=1 with al given by (6.69). Symmetric two-wave approximation. We are particularly interested in the radiative decay of nuclear exciton states | e(k0) created by synchrotron pulses. Two important cases are for k0 off-Bragg, and for k0 satisfying the condition for a symmetric Bragg reflection. To treat these cases, it is sufficient to only include the coupling provided by the k0+ and k0- channels, omitting the contributions of all other open channels ks±. This is essentially the usual two-wave approximation made in the dynamical theory of X-ray optics, restricting our attention to the symmetric case. We take the scattering plane defined by k0+ and k0- to be the (z, x)-plane, so that q = q x = k0 cos 0 x, (6.72) g0 = k20 - q2 = k0 sin 0, (6.73) k0± = q ± g0 z. (6.74) This is the geometry shown in figure 8. We will furthermore assume maximum coupling of the radiation fields to the nuclei. Assuming a linear Jz = 0, E1 nuclear transition, the quantization axis zJ III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 249 should lie in the = y direction, perpendicular to the scattering plane. The partial width for coherent decay, eq. (6.56), then becomes 2 n (q) ¯ 2 c (0) = sin coh, (6.75) 0 and the sum rule (6.67) becomes M (q,l) c = M (q) c (0) = 2 n¯ 2l coh, (6.76) l=1 where n = 1/(a2d) is now the volume density of nuclei in the crystal, and l = Md/ sin 0 is the thickness of the crystal in the k0± directions. We note the right- hand side of (6.76) is precisely the superradiant decay width at Bragg as calculated by the Fermi Golden Rule in eq. (4.22). For any particular number of layers M, we can now solve the determinant equa- tion Det[ h(q) - 1] = 0 for the M eigenfrequencies q,l, and then substitute q,l into the eigenvalue equation h(q)Z(q,l) = q,lZ(q,l) to determine the eigenmodes Z(q,l). However, there is a better procedure, in which we first deduce the general form of the eigenstates, and then determine the eigenfrequencies. We first rewrite the Hamiltonian equation (6.63) in component form, i M-1 Q0( ) Z(q,l) m = (q) Z(q,l) ig , (6.77) 2 c (0) n exp 0|n - m|d n=0 where Q0( ) = 0 - - i . We then try a trial solution of the form Zm = eik dm ± eik d(M-1-m) = Km ± KM-1-m, m = 0, 1, . . . , M - 1, (6.78) where k is a complex parameter yet to be determined, and in the second line, we have introduced the notation K = exp(ik d). We choose this form of trial solution based on the physical thinking that there exist two propagating waves inside the crystal. The eigenmodes are the stationary states formed by the superposition of the two waves, similar to the standing waves on a beaded string. Noting that Km + KM-1-m = 2K(M-1)/2 cos[(M - 1 - 2m)k d/2] and Km - KM-1-m = -2iK(M-1)/2 sin[(M - 1 - 2m)k d/2], we call the "+" solutions "cosine solutions" and the "-" solutions "sine solutions". Substituting (6.78) into (6.77) yields a consistency equation for Zm to be a solution, KM+1 - eig0dKM ± 1 - eig0dK = 0, (6.79) and a dispersion equation, i 1 1 Q0( ) = (q) - . (6.80) 2 c (0) 1 - K exp(ig0d) 1 - K exp(-ig0d) 250 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Figure 25. (a) Plot of the normal mode frequency shifts l as a function of g0d for an M = 3 layer crystal. (b) The normal mode coherent decay widths (l) c vs. g0d. For both figures, the vertical scale is in units of (q) c (0), the coherent decay width for a single plane; and g0d varies from 0 to . The consistency equation (6.79) constitutes two (M + 1)-order equations, with the (+) equations giving the cosine solutions and the (-) equations giving the sine solutions. There are then 2M + 2 solutions to (6.79). Two of these solutions are "null" solutions, giving the null "mode" Z 0. The null solutions are K = 1, which is always a solution to the (-) equation, and K = -1, which is a solution to the (+/-) equation if M is (odd/even). The remaining 2M solutions are paired, with each pair being an identical solution. If M is even, there are M/2 distinct cosine and sine solutions; while if M is odd, there are (M + 1)/2 cosine solutions and (M - 1)/2 sine solutions. In either case, there are M distinct eigenmodes Zq,l for an M-layer crystal in the q subspace, which we designate by the index l = 1, 2, . . . , M. The eigenmode frequency q,l is obtained from the dispersion equation (6.80), by substituting the solution Kl of (6.79) which generates Zq,l into the right-hand side of (6.80). The solutions Kl of (6.79) are functions of g0d, or equivalently, of the mode angle 0. Changing g0 changes the q-manifold, q = k0 cos 0 x. As a function of g0d, the configuration of the mode Zq,l changes, as does the frequency shift of the mode, q,l, and the partial width for coherent radiative decay, (l) c . As g0d varies from 0 , k0 varies from a symmetric Bragg condition ( 0 = 0), to off-Bragg (g0d = /2), and back to the next higher order symmetric Bragg reflection (g0d = ). In figure 25, we show the variation of the mode frequency shifts and coherent radiative widths vs. g0d for the simplest nontrivial case of an M = 3 layer crystal. In (a) we show the frequency shifts l of the three modes. The shifts satisfy the sum rule l l = 0. We note in particular that at Bragg, the frequency shifts all vanish; but off-Bragg, the frequencies are spread over a region 3 (q) c (0) = 2 n¯ 2l coh. In (b) we show the coherent decay widths (l) c . The widths satisfy the sum rule l (l)c = 3 (q)c(0) = 2 ¯ 2l coh. The most interesting feature is that at Bragg, two of the modes become nonradiative, and one mode becomes superradiant with the coherent decay width being the full 3 (q) c (0) of the sum rule. For this mode, all the planar fields add in-phase, giving enhanced emission, while for the other two modes, the fields of the three planes exactly cancel. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 251 * These are general features for any M: at Bragg, one mode becomes superradi- ant, with enhanced coherent decay width c = 2n ¯ 2l coh, and all other modes are nonradiant. Off-Bragg, the widths are spread, consistent with the sum rule l (l)c = 2 ¯ 2l coh, with a few modes being (weakly) superradiant, and most modes subradiant. The frequency shifts satisfy the sum rule l l = 0, and at Bragg, all frequency shifts are zero. Off-Bragg, there is a broad spread of frequen- cies, with the total spread being c = 2 ¯ 2l coh. The remarkable behavior at Bragg is the essential point of the coherent enhancement effect at Bragg [11]. As a simple example, for M = 2 layers, the eigenmodes are the symmetric state (with corresponding oscillators Rj in each plane oscillating in-phase), and the anti- symmetric state (with corresponding oscillators 180 out-of-phase), in direct analogy to two nuclei. At the g0d = Bragg, the symmetric state is nonradiative: the self field of each plane is exactly canceled by the field from the other plane, the radia- tion reaction on each oscillator then vanishes, and the oscillators have free undamped motion. Correspondingly, the emitted radiation fields from the two layers cancel and the mode is nonradiative. For the antisymmetric mode, the fields add, so the radiation reaction on each oscillator is doubled, and the mode decays superradiantly, at 2 × the rate of a single layer. For g0d = 2 , the roles are reversed ­ the symmetric mode is superradiant and the antisymmetric mode is nonradiative. 6.10. Superradiance and dynamical beats in the decay of a nuclear exciton Of particular interest is the decay of an exciton | e(k0) created by a synchrotron pulse. If k0 satisfies the symmetric Bragg condition, then a single superradiant eigen- mode is excited, and the subsequent decay is a simple enhanced exponential decay. If k0 is off-Bragg, then | e(k0) is a superposition of normal modes, and the spread of frequencies and Hermitian nonorthogonality play an important role both in the initial superradiance and subsequent dynamical beats. The exciton | e(k0) created by a synchrotron pulse is a Bloch wave, 1 e(k0) = eik0·Rj b , (6.81) N j |G0(j) j where k0 = ( 0/c) n0, with 0 = (Eb - Ea)/ , and n0 the direction of the incident pulse. Hence, | e(k0) lies in the q = k0xy manifold of eigenmodes of an M-layer crystal. Factoring out the xy-Bloch states Xq, and suppressing the q index on all states and eigenfrequencies, the initial exciton state is 1 eig0d 1 1 e(k0) = Z(g . . (6.82) M 0) = M .. eig0d(M-1) 252 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Expanding in terms of eigenmodes, M e(k0) = alZl, (6.83) l=1 where ZT a l · Z(g0) l = . (6.84) M The state then develops as M e(k0, t) = e-i ltalZl, (6.85) l=1 and the excitation probability decays as e(k0, t) e(k0, t) = a l alei( l - l)tZ l · Zl. (6.86) l,l The eigenmodes are transpose orthogonal rather than Hermitian orthogonal, so gener- ally Z l · Zl = 0, and there are interference effects between the modes in the decay of e(k0, t)| e(k0, t) . The radiation field (Feynman photon potential) emitted by the crystal in the state | e(k0, t) is M A(R, t) = alA(l)(R, t), (6.87) l=1 where al is given by (6.84), and, in the symmetric two-wave approximation, the field of the lth normal mode is i A(l)(R, t) = - C (q) i(k ZT 2 c (0) exp 0+ · R - lt) l · Z(-g0), z > (M - 1)d, (6.88) i = - C (q) i(k ZT 2 c (0) exp 0- · R - lt) l · Z(g0), z < 0. Here = y is the polarization perpendicular to the (k0+, k0-) scattering plane, C = 2/¯ coh, (q)c(0) is the partial width for coherent decay per plane, as given by (6.75), and ZTl · Z(±g0) = m Z(l) m exp(±ig0d m), m = 0, . . . , M - 1. In figure 26(a), we show the coherent photon signal |A(+)(t)|2 vs. t emitted by a synchrotron pulse exciton | e(k0, t) in the k0+ direction, for k0 on Bragg (g0d = ), and for k0 off-Bragg (g0d = /2), for an M = 3 layer crystal; and in (b) we show the corresponding decay of the excitation probability e(k0, t)| e(k0, t) . In these figures, we have "turned off" the incoherent processes, i.e., have set + = 0. The Bragg decay has a simple exponential decay, both for |A(+)(t)|2 and for e(k0, t)| e(k0, t) , at the enhanced rate c = 2 n¯ 2l coh. The off-Bragg is more complex: there is an initial superradiant phase, decaying at half the Bragg rate, c/2, indicated by the dashed III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 253 Figure 26. (a) The coherent photon signal log[|A(+)(t)|2] vs. t emitted by a synchrotron pulse exciton | e(k0, t) in the k0+ direction, for k0 on Bragg (g0d = ), and for k0 off-Bragg (g0d = /2). We have normalized the curves to unity at t = 0. The Bragg decay is superradiant, at the enhanced rate c = 2 n¯ 2l coh. The dashed line shows the initial off-Bragg superradiant decay at the enhanced rate c = n¯ 2l coh. The number of crystal layers is M = 3. Time t is in units of [ (q) c (0)]-1. (b) The decay of the excitation probability log[ e(k0, t)| e(k0, t) ] vs. t for k0 on- and off-Bragg, for a 3-layer crystal. lines in (a) and (b). This is followed by a slowly decaying phase, in which the photon signal |A(+)(t)|2 exhibits dynamical beats, periodically vanishing. Correspondingly, e(k0, t)| e(k0, t) "flattens out" (i.e., the rate of decay d/dt e(k0, t)| e(k0, t) 0) whenever |A(R, t)|2 has a sharp minimum. In effect, the excitation is momentarily "trapped" within the system. This periodic behavior in the decay of e(k0, t)| e(k0, t) off-Bragg is due to the interference effects between the modes, which arise because the eigenmodes | m are transpose orthogonal, rather than Hermitian, as noted in eq. (6.36). This interference also gives a "dephasing" effect due to the spread of frequencies, which plays a major role in the initial superradiant decay of e(k0, t)| e(k0, t) . At Bragg, a single superradiant eigenmode is excited, and the decay of e(k0, t)| e(k0, t) is a simple exponential, at the enhanced rate c = 2 n¯ 2l coh. Correspondingly, for Bragg excitation, the photon emission field is an eigenmode field, with the coherent photon signal having two equal strength channels, |A(+)(t)|2 and |A(-)(t)|2, in the k0+ and k0- directions, respectively. Carrying out the calculation as in (6.58), the probability/s that the state | e(k0), t de-excites by coherent -ray emission into either the k0+ or the k0- channel at time t is given by c(t) = ce-( + + c)t, (6.89) 254 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 and integrating over t, the probability that | e(k0) decays by coherent photon emission is P c c = . (6.90) + + c Off-Bragg, | e(k0) is a superposition of normal modes, and correspondingly, A(R, t) is a superposition of normal modes fields A(l)(R, t), as given by (6.87). The combination of the modes, determined by the expansion coefficients al, is such that initially |A(-)(t)|2 |A(+)(t)|2 ­ i.e., the system is phased so that there is initial constructive interference between the planar waves in the "forward" k0+ direction, but destructive interference in the k0- direction. The decay is initially superradiant, but at half the Bragg rate c/2 because the k0- decay channel is shut down. As can be seen from figure 26(b), with about 80% probability, | e(k0) de-excites during this initial superradiant phase, so the probability for coherent decay into the k0+ channel is, approximately, P c/2 c . (6.91) + + c/2 Following this initial superradiant phase, the off-Bragg signal decays slowly, with the onset of "dynamical beats". This is a natural consequence of adding wave contributions spread over a frequency range c. For example, if the normal mode wave contributions entered with equal amplitude over a frequency range 0 ± c/2, and if all modes decayed at the natural , then the emitted signal would have dynamical beats of frequency c/2 and a weakly enhanced long time decay |A(+)(t)|2 exp(- t)/t2: + c/2 2 sin( A(+)(t) d e-i( 0+ -i /2)t = ct/2) e-i( 0-i /2)t. (6.92) - t c/2 This example is much too simple, but it illustrates the beats and inverse power law type decay which arise from Fourier synthesis of bands of frequencies. Making the correct resolution of | e(k0) into normal mode contributions, and using the correct complex mode frequencies, the signal |A(+)(t)|2 is very close in form to the dynamical theory off-Bragg transform solution, I(T) c (t) = exp(- t)[J1( 4 ct )]2/ ct. This is illustrated in figure 27(a), for an M = 10 layer crystal, again with the incoherent processes turned off ( + = 0). The reason for the discrepancy is twofold: I(T) c (t) is a continuum approximation to a discrete problem, and, more importantly, the off-Bragg dynamical theory treatment is a sin- gle channel approximation which includes only forward scattering ­ i.e., only the k0+ channel is kept. The normal mode solution correctly accounts for the "back cou- pling" in the k0- channel. As a consequence, I(T) c (t) is a poor approximation for a thin crystal, but the agreement becomes progressively better the larger the number of layers M. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 255 Figure 27. (a) Comparison of the exact normal mode solution (solid line) for |A(+)(t)|2 for an off-Bragg exciton in an M = 10 layer crystal, and the approximate solution exp(- t)[J1( 4 ct)]2/ ct (dashed line). (b) The asymmetric probability distribution within the crystal for an off-Bragg exciton | e(k0, t) : the top solid line shows the relative probability |Z0(t)|2 that the top layer (m = 0) is excited at time t; the bottom solid line, the relative probability |ZM-1(t)|2 that the bottom layer (m = M - 1) is excited at t. The upper dashed line shows the free decay of a single plane. The lower dashed line shows the excitation probability e(k0, t)| e(k0, t) . Here M = 3. In figure 27(b), we show another very interesting feature of the off-Bragg solu- tion, the asymmetric probability distribution within the crystal for an off-Bragg exciton | e(k0, t) . Here the top solid line shows the relative probability |Z0(t)|2 that the top layer (m = 0) is excited at time t; the bottom solid line, the relative probability |ZM-1(t)|2 that the bottom layer (m = M - 1) is excited at t. For a system of classi- cal oscillators, the lines correspond to the total mechanical energy E(t) of oscillators in these planes. The upper dashed line shows the decay of an isolated single plane. The lower dashed line shows the excitation probability e(k0, t)| e(k0, t) , as also shown in figure 26(b). Here we have taken M = 3. The first layer initially decays like an isolated single layer, but back reaction from the lower layers, although small, then slows the decay. The last layer, on the other hand, initially decays superradiantly, at the Bragg decay rate, c = 2 n¯ 2l coh. Initially, the waves from all the "upstream" layers add constructively to act on the last layer, just as at Bragg. (In the Bragg case, each layer in the crystal sees the same enhanced field, and all layers decay superra- diantly together.) As discussed in section 4.2.3, this marked asymmetry of the planar decay rates across the film can easily be detected by observing the incoherent resonant processes (conversion electrons, incoherently scattered -radiation, . . . ) occurring at different depths of the film. 256 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Acknowledgement Partial support for this work has been provided by the National Science Foun- dation under grant No. DMR-90-13058, and by the Welch Foundation under grant C-1049. Appendix A A.1. Electronic scattering For X-ray scattering from the atomic electrons, the coherent elastic scattering amplitude (correct to second order in HI) is given by [5,7,118] fe = f0 + fm + fr, (A.1) where f0 is the Thomson contribution, fm is the nonresonant magnetic scattering, and fr is the resonance dispersion (or "anomalous") scattering contribution. The Thomson scattering contribution is given by f0 = -r0fD(Q) n(Q) e f · e0, (A.2) where Q = k0 - kf is the momentum transfer, fD is the Debye­Waller factor, and the form factor n(Q) is the ensemble average of the Fourier transform of the electron density, n(Q) = p 0 0 exp(iQ · rj) 0 , (A.3) 0 j where p0(T ) gives the statistical probabilities for the various possible initial states | 0 . The nonresonant magnetic scattering contribution to the coherent elastic scattering can be expressed as 1 f 0 m = i r L(Q) · A + S(Q) · B , (A.4) mc2 0fD(Q) 2 where S(Q) and L(Q) are the ensemble averaged Fourier transforms of the spin and orbital magnetization densities, S(Q) = p 0 0 sj exp(iQ · rj) 0 , (A.5) 0 j 1 L(Q) = p f(Q · r , (A.6) 2 0 0 j)lj + ljf (Q · rj) 0 0 j where (ix)n f(x) = 2 . (A.7) (n + 2)n! n=0 III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 257 The polarization vectors are given by B = e f × e0 + kf × e f kf · e0 - k0 × e0 k0 · e f - k f × e f × k0 × e0 , (A.8) A = A - A · Q Q, (A.9) where A = -4 e f × e0 sin2 , (A.10) with the scattering angle. Finally, the resonance dispersion contribution to the coherent elastic scattering amplitude is 1 f 0|J(-kf) · e f|I I|J(k0) · e0| 0 r = 2 ¯ fD(Q) p0 E 0 I I0 I - E0 - - i I /2 - 0|J(k0) · e0|I I|J(-kf ) · e f| 0 , (A.11) EI - E0 + where J(k) = c-1 dx eik·xJ(x) is the Fourier transform of the current density operator given by e J(k) · e = e · p exp(ik · r mc j + i (sj × k) j). (A.12) j In all cases the appropriate phonon factor is the Debye­Waller factor. This is an im- mediate result for all nonresonant contributions, but also holds to good approximation for resonant X-ray scattering because the resonance widths I, typically eV, are large compared to vibrational energies , typically 0.01 eV (hence, resonant X-ray scatter- ing is a fast process, / I 10-16 s, compared to vibrational times, / 10-14 s). A.1.1. Isotropic limit For many cases the M¨ossbauer frequency is large in comparison to the atomic absorption edges. For example, for 57Fe, the M¨ossbauer resonance energy is = 14.4 keV, while the highest energy atomic edges are K = 7.1 keV, L1 = 0.85 keV, L2 = 0.79 keV and L3 = 0.78 keV. In situations like this, where the frequency of the radiation is large compared to any of the atomic absorption edge frequencies, the dominant real contribution to the scattering amplitude f is the Thomson contribution f0, and the dominant imaginary contribution comes from the photoelectric poles in the sum over intermediate states in fr corresponding to photoemission of electrons from the various shells. In this limit, the coherent elastic X-ray scattering amplitude is given to good approximation by f e e = e f · e0fD(Q) - n(Q) r0 + i , (A.13) 4 ¯ 258 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 where e( ) is the photoelectric cross-section at the frequency , arising primarily from the tightly bound K- and L-shell electrons (so the form factor for scattering is near unity). In this approximation the polarization response is that of an isotropic electric dipole oscillator and gives no polarization mixing. For frequencies near an atomic absorption edge however, the response is generally much more complex and interesting, with strong anisotropic scattering sensitive to the directions of the local fields (B and EFG). A.1.2. Multipole expansion for the resonance dispersion contribution fr Near absorption edges, very large resonant enhancements to the magnetic sen- sitive X-ray scattering occur, with amplitudes ranging from 0.1r0 to 100r0 [3,6,116]. This strongly enhanced "magnetic" X-ray scattering is actually electric multipole res- onance scattering with the magnetic sensitivity arising from the effects of exchange and spin­orbit correlation [3]. To treat these interesting cases, it is necessary to make a multipole expansion of the currents in fr. The multipole development given here carries over directly for the nuclear resonant scattering amplitudes. The interaction current J(k)·e, given by eq. (A.12), can be expanded in multipole contributions following the standard treatments [8], J(k) · e = J(q) LM Y(q) LM k · e, (A.14) q=e,m LM where q = e or m designates an electric or magnetic multipole; L = 1, 2, . . . gives the order of the multipole, and the azimuthal index M = -L, -L + 1, . . . , L. The multipole moment operator J(q) LM for the QL multipole (Q = E or M) is a spherical tensor operator of order L given by 4 iLkL L + 1 J(e,m) LM = - Q(e,m) (2L + 1)!! L LM . (A.15) For an electric 2L-pole, Q(e) LM = ELM + ELM (s). (A.16) EL and EL(s) denote, respectively, the charge and spin contributions of parity (-1)L, leading to electric radiation, which to lowest order in r are given by ELM = erLjYLM rj , (A.17) j s E j LM (s) = el rL r · . (A.18) mc2 j j YLM j L + 1 j III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 259 For a magnetic 2L-pole, l Q(m) j LM MLM = 2µB j rLjYLM rj · + s , (A.19) (L + 1) j j where µB = e /2mc is the Bohr magneton. The vector spherical harmonics which enter eq. (A.14) are defined as 1 Y(e) LM k = k , (A.20) L(L + 1) kYLM Y(m) LM k = k × kYLM k . (A.21) For the spherical harmonics YLM we use the conventions of Rose [117]. For reference we give the explicit forms of the vector spherical harmonics Y(q) LM ( k) for E1, M1, and E2 transitions: for E1, 3 3 Y(e) 10 = - sin( )e e±i cos( )e . (A.22) 8 , Y(e) 1±1 = 16 - ie Here e and e are the usual spherical polar unit vectors, and and are the polar and axial angles specifying the photon direction k, with the z axis coinciding with the quantization axis zJ determined by the direction of the internal field at the nucleus. For an M1 transition, 3 3 Y(m) 10 = i sin( )e e±i e , (A.23) 8 , Y(m) 1±1 = 16 ± i cos( )e and for an E2 transition, 15 Y(e) 20 = - sin(2 )e 32 , 5 Y(e) 2±1 = e±i cos(2 )e , (A.24) 16 - i cos( )e 5 Y(e) 2±2 = e±i2 (1/2) sin(2 )e . 16 ± i sin( )e The polarization dependence for the QL multipole of the current expansion (A.14) is determined by the Lth rank spherical tensor, T (q) LM e, k = e · Y(q) LM k , (A.25) made up of the vectors e and k, and being linear in the components of e. For an EL transition, this is expressed as products of the components of e and k by 4 (2L + 1) +1 e · Y(e) LM k = C(1, L - 1, L; µ, M - µ, M)Y k Y 3(L + 1) L-1,M-µ 1µ(e). µ=-1 (A.26) 260 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 For an ML transition, the polarization factor is 4 (2L + 1) +1 e · Y(m) LM k = C(1, L - 1, L; µ, M - µ, M)Y k Y 3(L + 1) L-1,M-µ 1µ(b), µ=-1 (A.27) where b = k×e is the B-vector of the radiation field. Alternatively, using the circularly polarized bases eµ (µ = ±1) transverse to k, the polarization factors are given in terms of the rotation matrices D(L) µM by 2L + 1 e µ · Y(e) LM k = D(L) k , (A.28) 8 µM 2L + 1 e µ · Y(m) LM k = (µ) D(L) k . (A.29) 8 µM Using the current multipole expansion, fr is then given by f r = 4 ¯ e f · Y(q ) L M kf Y(q) LM ( k0) · e0 F(q q) L M ;LM , (A.30) q L M qLM where 1 F (q q) 0|J(q ) L M |I I|J(q) LM | 0 L M ;LM = fD(Q) p0 E 0 I I0 I - E0 - - i I /2 - 0|J(q) LM |I I|J(q ) L M | 0 . (A.31) EI - E0 + This result is still a completely general expression for fr. An important point to note is that the dependence on the photon polarizations e0 and e f and the photon directions k0 and kf, given by the factor [e f · Y(q ) L M ( kf)][Y(q) LM ( k0) · e0], has now been separated out from the current matrix elements and frequency response given by the factor F (q q) L M ;LM . The polarization dependence gives an important "handle" for analyzing scattering experiments. In most situations, a near resonant transition is a pure multipole transition, so q = q and L = L. Due to crystal field effects however (local symmetry, bonding orbitals, . . . ), Jz is generally not conseved, and it is possible to have M = M. A.1.3. QL-multipole transition with M = M An important special case is coherent elastic scattering from a pure multipole transition with M = M. This will be the situation when crystal field effects can be ignored, and even in the presence of crystal fields, this condition will often hold (e.g., E1 or M1 transitions in a cubic symmetry field). The resonance contribution is then f QL = 4 ¯ e f · Y(q) LM kf Y(q) LM k0 · e0 F (q) LM , (A.32) qLM III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 261 where F (q) LM = F (qq) LM;LM as given by eq. (A.31), where now the sum over intermediate states is just over the near resonant excited state sublevels. As noted in section 2.4, the polarization response [e f · Y(q) LM ( kf)Y(q) LM ( k0) · e0] is precisely the polarization response of a qL, Jz = M classical oscillator. Further- more, the overall form of the polarization response is dictated by symmetry consider- ations and can depend only on e0, k0, e f, kf and the local quantization axis zJ, since these are the only defined directions in the scattering (in the absence of crystal fields). For a QL transition, the explicit Y(q) LM expressions give the scattering amplitude [3] 2L fQL = Al( )Pl e fkf; e0k0; zJ , (A.33) l=0 where the polarization factors Pl contain zJ to the lth power, and the amplitudes Al are combinations of the F (q) LM . For E1 transitions, the polarization factors are e f · e0, l = 0, Pl = -i e · z J , l = 1, (A.34) f × e0 e f · zJ (e0 · zJ), l = 2, and the corresponding resonant amplitudes are F (e) 3 11 + F (e) 1-1, l = 0, Al = (A.35) 16 F (e) 11 - F (e) 1-1, l = 1, 2F (e) 10 - F (e) 11 - F (e) 1-1, l = 2. For a magnetic dipole transition (M1), the Pl are given by (A.34) with the E-polarization vectors e replaced by the B-polarization vectors b = k × e, and the Al are given by (A.35) with the F (e) 1M replaced by the F (m) 1M . For an E2 transition, the polarization factors are k e + e , l = 0, f · k0 0 k0 f · e0 -i k e · z e f · k0 f × e0 J + 0 k0 + e + e , l = 1, 0 k0, e f kf f kf k e e + e P f · k0 f · zJ 0 · zJ 0 k0 l = (A.36) + e + e , l = 2, 0 k0, e f kf f kf -i k k e · z e f · zJ 0 · zJ f × e0 J + 0 k0 + e + e , l = 3, 0 k0, e f kf f kf kf · zJ k0 · zJ e f · zJ e0 · zJ , l = 4. 262 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 The corresponding resonant amplitudes are F (e) 22 + F (e) 2-2, l = 0, 1 F (e) , l = 1, 5 2 22 - F (e) 2-2 Al = (A.37) 16 F (e) - F (e) , l = 2, 21 + F (e) 2-1 22 + F (e) 2-2 1 2 F (e) - F (e) , l = 3, 2 21 - F (e) 2-1 22 - F (e) 2-2 6F (e) 20 - 4 F (e) 21 + F (e) 2-1 + F (e) 22 + F (e) 2-2 , l = 4. In a spiral antiferromagnet, each order l > 0 will give rise to a distinct magnetic satellite about the central Bragg peak [3]. The appearance of two harmonic satellites is a characteristic signature of a dipole resonance (E1 or M1). The first order E1 harmonic does not allow scattering, while the first order M1 harmonic does not allow scattering. Quadrupole resonances (E2 or M2) will have four harmonics, and, generally, allow all the possible combinations of , and , scattering. A.1.4. Effective spin-orbital operators and quasi-elastic resonant scattering The expressions (A.33)­(A.37) are useful for determining the local magnetic field direction, but are not in a useful form for relating the observed scattering cross- sections to the expected moments L , S , . . . of the valence shell being probed. This information is available though: As shown by Luo et al. [5,6], for the important case of quasi-elastic scattering, if the "fast collision" approximation can be made, then indeed there is a simple relation between the scattering and the spin-orbital moments. For a pure EL transition, the scattering amplitude (A.30) becomes 2L k f r = 4 ¯ mkT (k) q e fkfe0k0 M(k) , (A.38) EL f q (li, si) 0 k=0 q=-k where the M(k) q (li, si) are kth rank spin-orbital multipole moments depending only on the li and si operators of the valence electrons for the valence shell involved in the resonance, with each contribution M(k) q (li, si) having a distinct polarization and wave vector dependence T (k) q (e fkfe0k0). The factor mk contains products of Clebsch­ Gordan coefficients, a squared radial matrix element for the transition involved, and a complex resonance denominator. Thus, for an E1 transition, the scattering amplitude operator contains up to quadrupole spin-orbital moments, and for an E2 transition up to hexadecapole moments. The odd moments are odd under time reversal T and, hence, purely magnetic, while the even-order moments are even under T , and include the charge multipole moments (giving nonmagnetic Templeton effects). "Quasi-elastic" resonant scattering means that after the scattering, the total num- ber of valence electrons in the final state is the same as that of the initial state, but the two states may have different quantum numbers (e.g., different L, S, J, . . .). Quasi- III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 263 elastic resonant scattering includes elastic scattering (| f = | 0 ) as well as spin-wave and higher multipole-wave excitations. The "fast collision" approximation means that the collision time T = | - i /2 |-1 is short compared to / EI, the time scale over which the torques exerted amongst the core hole and the intermediate state valence electrons affect the evolution of the intermediate state. Here = EI - E0 - is the deviation of the inci- dent radiation from resonance, and EI is the spread of the intermediate levels EI. Under this condition, the spread EI can be ignored, and there is a single resonance denominator for the transition. We note that, even if EI, the fast collision approximation can be assured by simply tuning off-resonance several EI. The polarization factors T (k) q (e fkfe0k0) are the spherical tensors made by cou- pling e0·Y(e) LM ( k0) and e f·Y(e) LM ( kf). For an E1 transition, T (0) 0 e f·e0, T(1) e f ×e0, and the T (2) q (q = -2, . . . , +2) are the second rank tensors made up from products of e f and e0. The expressions for the spin-orbital multipole moment operators M(k) q (li, si) are given by Luo [6,5]. If total L, S are conserved for the valence shell being probed (for the ground state and low lying excitations), then the operators can be expressed in terms of L and S, M(k) q = M(k) q (L, S). The simplest situation occurs for quasi-elastic scattering within a manifold of good total J, in which case M(k) q = M(k) q (J) T (k) q (J), the kth order spherical tensor obtained from J. This is often a good approximation for transitions to the 4f valence shell of rare earth ions. For an E1 transition, such as the 3d3/2,5/2 4f transitions at the M4 and M5 edges, the vector part of the effective scattering operator is (e f × e0) · J, and the quadrupole part is T(2) -q(e f e0)T (2) q (J). The expressions for M(k) q (L, S) and M (k) q (J) have been worked out by Hu [118]. The simple relation between fr and the spin-orbital operators M(k) q (li, si) is of considerable importance for the interpretation of resonant X-ray scattering experiments. The coherent elastic scattering gives sharp Bragg peaks and satellites which measure the long range order of the thermal expectation values of these moments, and the diffuse inelastic scattering gives information about coherent and incoherent excitations, and short range correlations. Of particular interest, below the ordering temperature TN, the inelastic resonant scattering can determine the dispersion curves (q) for spin wave excitations, just as for inelastic neutron scattering, but moreover, because of the higher spin-orbital moments entering fr, the inelastic resonant X-ray scattering couples to quite new types of spin-charge wave excitations ­ quadrupolar, octupolar, and hexadecupolar waves. Just above TN, the frequency integrated diffuse critical scattering depends upon the instantaneous moment­moment correlations M(k) q (i)M(k) q (j) between the ith and jth sites. This includes the short range spin­spin correlations sisj , just as for neutron scattering, but also gives quite new information about the correlations of higher order moments above TN. 264 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 A.2. Resonant nuclear scattering For the sharp resonant -ray scattering by the nucleus, the coherent elastic scat- tering amplitude is f 0|J(-kf) · e f| I I |J(k0) · e0| 0 n = fM p0 , (A.39) E 0 I I - E0 - - i I /2 where J(k) is the Fourier transform of the nuclear current density operator and fM is the M¨ossbauer phonon factor. Here represents the state of the crystal, excluding the vibrational state, and E represents its energy (e.g., includes the magnetic state of the crystal and the internal state of the nucleus). Finally, I is the total width of the excited state I, including the radiation width and the internal conversion width . In the fast relaxation cases (relaxation times very much smaller than the nuclear Larmor precession times L), we may take and E in eq. (A.39) as the state and energy of the nucleus i in the external fields and the "static effective fields" of the surrounding medium. Similarly, in the slow relaxation limit, and E may be taken as the state and energy of the atom i, but unless the hyperfine energy is negligible relative to the electronic Larmor frequency, one can no longer consider the transitions between nuclear states in an effective field but must allow for the dynamic effects of the nucleus plus atomic electrons as a coupled quantum-mechanical system. The modifications necessary when the relaxation times are on the order of L are discussed briefly in section A.2.5. The usual situation is the fast relaxation limit. The states 0 and energies E0 are then the hyperfine levels of the nuclear ground state, produced by the interaction of the ground state magnetic dipole moment of the nucleus with the effective magnetic field Bint arising from the s-electron density at the origin, and the coupling of the quadrupole moment of the nucleus in the ground state with the electric field gradient (EFG). Similarly, the I and EI are the hyperfine levels of the excited state of the nucleus involved in the M¨ossbauer transition. The ground and excited states will be states of good angular momentum, j0 and j1, but Jz need not be conserved. Because the ground state hyperfine splittings are typically 10-8 eV 10-4 K, the initial state is normally equally likely to be in any of the (2j0 + 1) hyperfine substates, and p0 = C/(2j0 + 1), where C is the enrichment fraction for the M¨ossbauer isotope. A.2.1. Coaxial B and EFG If the effective fields acting on the nucleus have a common axis of symmetry ( z), then the ground and excited nuclear states are states of good Jz. M¨ossbauer transitions occur with well-defined multipolarity E1, M1, E2, and in a few cases, M1­E2. The multipole expansion of the Fourier transform nuclear current density J(k) is given by (A.14). The matrix elements of the current multipole moment operator J(q) LM are evaluated in terms of the radiative width (QL), i.e., the width associated with the emission of (QL)-multipole radiation. (The ratio of the radiative III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 265 widths gives the mixing ratio 2, i.e., 2(E2/M1) = (E2)/ (M1).) Denoting the spin of the states explicitly, | 0 = |a, J0, m0 and | I = |n, J1, m1 , we have 0 J(q) LM I = C(j0Lj1; m0Mm1) (QL) ei (QL), (A.40) where (QL) is the reduced matrix element (for emission). Substituting (A.14) into the width expression 2 = -1 0 d k 0|J(-k) · e | I , (A.41) e m0 we obtain (QL) = 0 (QL). (A.42) The phase (QL) of the reduced matrix element is only important for mixed multipole transitions. For such transitions, Lloyd has shown that if T invariance is valid for emission/absorption, then (QL) - (Q L ) = 0 or [119]. For a pure QL M¨ossbauer transition with conserved Jz, the coherent elastic scattering amplitude is L f n = 4 ¯ e f · Y(q) LM kf Y(q) LM k0 · e0 F (q) LM ( ), (A.43) M=-L where now F (q) LM is given by Cf j0 C(j F (q) M 0Lj1; m0M )2 LM ( ) = , (A.44) 2j0 + 1 x(m m 0M ) - i 0=-j0 with x(m0M) = 2[E(j1; m0 +M)-E(j0; m0)- ]/ . Alternatively, we can express fn as 2L fn = Al( )Pl e fkf; e0k0; zJ , (A.45) l=0 where the amplitude factors Al and the polarization factors Pl are given by eqs. (A.34)­ (A.37) for E1, M1 and E2 transitions, with the F (q) LM now given by eq. (A.44). For a mixed M1­E2 transition, f n = 4 ¯ e f · Y(q ) L M kf Y(q) LM k0 · e0 F (q q) L M;LM , (A.46) q L qLM where 266 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 Cf F (q q) M (Q L ) (QL) L M;LM = ei[ (Q L )- (QL)] (2j0 + 1) j0 C(j × 0L j1; m0M )C(j0Lj1; m0M ) . x(m m 0M ) - i 0=-j0 Here both (qQL) and (q Q L ) are summed over (mM1) and (eE2). A.2.2. Isotropic limit An important special case is the "isotropic limit" (magnetically disordered state for which the splittings are negligible compared to , e.g., 57Fe). In this case all amplitudes Al( ) in eq. (A.45) have the same resonance denominator, and fn is in- dependent of the direction of the quantization axis zJ which enters the expressions for Pl(e fkf; e0k0; zJ). The calculation is simplified by averaging over all directions of zJ. The resulting coherent elastic scattering amplitudes for E1, M1, and E2 in the isotropic limit are 2j f 1 + 1 E1 = ¯ CfM e , (A.47) [x - i] 4j f · e0 0 + 2 2j f 1 + 1 M1 = ¯ CfM k · k , (A.48) [x - i] 4j f × e f 0 × e0 0 + 2 2j f 1 + 1 E2 = ¯ CfM k e + k k , (A.49) [x - i] 4j f · k0 f · e0 f · e0 0 · e f 0 + 2 where x = 2[E1 - E0 - ]/ . A.2.3. Jz not conserved Here we discuss the modifications (for the fast relaxation limit) when Jz is not conserved. This can arise from crystal field effects, e.g., an EFG tensor which is not axially symmetric, or an axially symmetric EFG tensor which is not coaxial with Bint. It is then possible to have cross terms M = M in the absorption and emission matrix elements in the scattering process. (But we note again that even in the presence of crystal fields, the condition M = M will often hold, e.g., E1 or M1 transitions in a cubic symmetry field.) The asymmetry terms will mix the states of good Jz, and the eigenstates for the level |n, Jn are given by | n |n, Jn, µn = K(µn, mn)|n, Jn, mn . (A.50) mn The index µn takes on the same values as mn, and the K(µn,mn) are the elements of the unitary transformation matrix which diagonalizes the interaction Hamiltonian. The (QL) multipole component of the current matrix element for emission becomes 0 J(q) LM I = GLM (µ0µ1) (QL), (A.51) III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 267 where GLM(µ0µ1) = K(µ0m0) K(µn; M + m0)C(j0Lj1; m0M). (A.52) m0 The coherent elastic scattering amplitude for a pure QL resonance is then f QL = 4 ¯ e f · Y(q) LM kf Y(q) LM k0 · e0 F (q) LM ;LM , (A.53) M M where Cf G F (q) M LM (µ0µ1)GLM (µ0µ1) LM ;LM = . (A.54) 2j0 + 1 x(µ µ 0µ1) - i 0 µ1 Here x(µ0µ1) = 2[E(j1µ1) - E(j0µ0) - ]/ . As a simple example we consider a (j0 = 1/2) (j1 = 3/2) M1 transition where the effective Hamiltonian of the nucleus is given by 1 H = H0 + g BJz + Q 3J2z - J2 + J2 . (A.55) 2 (+) + J2(-) The asymmetry term, (J2(+) + J2(-)), mixes Jz states which differ by Jz = ±2. Thus, for the j0 = 1/2 ground state, Jz is still conserved and the eigenstates are | 0, j0, m0 , m0 = ±1/2, while for the excited level, there is mixing between m1 = 3/2 and m1 = -1/2, and between m1 = 1/2 and m1 = -3/2. The resulting transformation matrix is a 3/2 0 b3/2 0 0 a K(µ 1/2 0 b1/2 1m1) = b , (A.56) -1/2 0 a-1/2 0 0 b-3/2 0 a-3/2 where a -1/2 = 1 + 2 , b = a , [(3Q ± Z)2 + 3Q2 2]1/2 - (3Q ± Z) ±3/2 = , Q 3 ±1/2 = - 3/2, with Z = g B. We note that 0 as 0. Thus, the polarization for a photon emitted by the |3/2µ = 3/2 |1/2, 1/2 transition, which would be Y(m) 11 ( k) for Jz conserved, now contains an admixture of Y(m) 1-1( k), i.e., the polarization is proportional to C(1/2, 1, 3/2; 1/2, 1)Y(m) 11 k + 3/2C(1/2, 1, 3/2; 1/2, -1)Y(m) 1-1 k . For the |3/2, µ = 3/2 |1/2, -1/2 transition, which is not allowed for Jz conserved, the amplitude and polarization of the emitted photon is proportional to [ 3/2C(1/2, 1, 3/2; 1/2, -1)Y(m) 1-1( k)]. 268 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 A.2.4. Screening effects When a photon is emitted from a nucleus, the scattering of the photon by the surrounding electron cloud of the atom gives rise to a complex index of refraction for passage of the photon through the electron cloud. In effect, the radiation emitted by the nuclear transition induces multipole currents in the electron cloud, and the radiation emitted by these induced electronic currents interferes with the primary radiation from the nucleus. The effect of the electronic scattering is to replace the bare nuclear current jfn by the "screened" nuclear current Jfn, Jfn (-k) = jfn (-k)[1 + (QL) + i (QL)], (A.57) Jfn being just the bare nuclear current multiplied by the screening factor n(QL) = 1+ (QL)+i (QL). The parameter +i will be referred to as the screening parameter, or, alternatively, as the interference parameter. The contribution (QL) gives the effect of the induced electronic currents which are in phase, or 180 out of phase, with the nuclear current. Typically, (QL) values are small ( 10-5­10-2 1), and in emission give only a small correction to the nuclear radiation width, which can generally be neglected. On the other hand, the imaginary part i (QL), although small ( 10-3­10-1), gives a phase shift to the emitted multipole wave which can lead to observable effects, as discussed below. Screening also affects resonant -ray scattering from the nucleus, which is a combined absorption­emission process. The effect of screening again modifies the "emission current" jfn (-k) by the factor [1 + (QL) + i (QL)] as given in eq. (A.57), and similarly, the screening modifies the "absorption current" jna (k0) by the same screening factor. The overall effect of the screening processes in -ray scattering is then to modify the nuclear scattering amplitude fn by the factor (1 + + i )2 1 + 2 (QL) + i2 (QL). In particular, for coherent elastic scattering by a pure QL M¨ossbauer transition, the modified scattering amplitude is fn, as given by eq. (A.43), times the factor 1 + 2 (QL) + i2 (QL). The total cross-section for resonant scattering and absorption is then given by the optical theorem, n = (4 ¯ ) [fn(k0, e0; k0, e0)]. For each (m0M) resonance, [(1+i2 )/(x(m0M)-i)] = (1+2 x)/(1+x2). In the limit of no hyperfine splitting, the resonant cross-section becomes 1 + 2 x n = 0 , (A.58) x2 + 1 where 0 is the total cross-section at exact resonance. Thus the screening gives a small dispersion term [2 x/(x2 + 1)] 0 to the total cross-section. The effect of this contribution is to give a small asymmetry to the M¨ossbauer absorption spectrum, and to shift the minimum of the spectrum from x = 0 to x . Although the screening effect is small, the induced dispersion can be important in isotope shift measurements. III-1.2 J.P. Hannon, G.T. Trammell / Coherent -ray optics 269 For absorption experiments, the imaginary part of the screening parameter, (QL), which gives the dispersion term, arises from the interference between coherent com- peting processes, and for this reason we refer to as the interference parameter. The dominant contribution to arises from the interference between nonreso- nant photoelectric absorption and resonant absorption followed by internal conversion. Since both processes have the same initial and final states (assuming no nuclear spin flip), they are coherent and, hence, interfere. This contribution to (QL) we call conversion screening. A second contribution to arises from the interference between resonant scatter- ing from the nucleus and nonresonant scattering from the atomic electrons. We refer to this contribution as Rayleigh screening. In most cases the primary contribution to is from conversion screening, but in a few isolated cases Rayleigh screening is dominant. The magnitude of the screening effect is dependent on the multipolarity of the M¨ossbauer transition, but not as strongly as one might expect. In the case of Rayleigh screening the strongest effect occurs if the M¨ossbauer transition is E1. This is because the scattering from the atomic electrons is dominantly E1, so if the resonant transition is M1 or higher, the final photon states are almost orthogonal, i.e., f nfe d 0, and it is only the small admixture of higher multipole orders in the Rayleigh scattering which lead to a nonzero Rayleigh screening contribution R. In the case of conversion screening we might expect a similar result since photo- electric absorption is also primarily E1. However, this conclusion is not correct, and in particular, the screening parameters for E2 transitions are often larger than those for E1 transitions. This occurs because for a QL multipole nuclear transition, the interfer- ence parameter (QL) is proportional to pe(QL) ic(QL), where pe(QL) is the QL multipole contribution to the photoelectric absorption cross-section, and ic(QL) is the cross-section for internal conversion­absorption at resonance. For an E2 transition, the photoelectric partial cross-section pe(E2) is much less than the E1 contribution, i.e., pe(E2) pe(E1), but, on the other hand, internal conversion is much stronger for an E2 transition than for a corresponding E1 transition, so that ic(E2) ic(E1). This amplification of the internal conversion cross-section just offsets the reduction of the photoelectric partial cross-section, so that generally (E2) (E1). Various aspects of this problem are treated in [2,120­122], and extensive calcu- lations of the screening parameters are given by Goldwire [123]. A.2.5. Relaxation effects Up to now, we have assumed the "fast relaxation limit", which is the usual situation, in which the atomic relaxation processes are very fast relative to the lifetime of the excited M¨ossbauer state and to the nuclear Larmor precession period, L. In this limit, we can take and E in eq. (A.39) as the state and energy of the nucleus i in the external fields and the "static effective fields" of the surrounding medium. 270 J.P. Hannon, G.T. Trammell / Coherent -ray optics III-1.2 If the relaxation times are on the order of L, then it is convenient to write (A.39) in the form fn = i dtei( +i /2 )t 0 × 0 0|e-ikf·r(t)G 0(t)e f · J(-kf)G0(t)J(k0) · e0eik0·r(0)| 0 0 , (A.59) where G0(t) = exp(-iH0t). Here we are concerned only with coherent elastic scat- tering, and the brackets indicate that the ensemble average is to be taken. To calculate the resonance response of the scattering operator, it is necessary to calculate the current­current correlation function. 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