PHYSICAL REVIEW B VOLUME 57, NUMBER 10 1 MARCH 1998-II Theory of nuclear resonant scattering of synchrotron radiation in the presence of diffusive motion of nuclei. II. V. G. Kohn and G. V. Smirnov Russian Research Centre ``Kurchatov Institute,'' 123182, Moscow, Russia Received 12 March 1997 A general theory of the time dependence of nuclear resonant forward scattering of synchrotron radiation in the presence of diffusive motion of nuclei is further developed. The scattering problem is solved for the two characteristic cases of diffusive motion. The first one is the continuous isotropic localized diffusion of a particle within a cage formed by a drift potential. The second case is the jump anisotropic unlimited diffusion of nuclei on a crystalline lattice. In both cases the frequency dependence of nuclear susceptibility has a complicated shape described by a superposition of Lorentzian functions having different weights and widths. Correspondingly several stages appear in the time evolution of the nuclear forward scattering which are characterized by different decay rates. In the thick absorber case the target can exhibit successively different partial thicknesses in the time evolution of forward scattering. S0163-1829 98 03709-6 I. INTRODUCTION pendence of radiation field can be calculated through a rep- resentation of SR pulse as a homogeneous coherent superpo- It is well known that Mo¨ssbauer spectroscopy provides a sition of monochromatic waves and a subsequent evaluation unique possibility to explore the dynamics of very slow of the response of the nuclear ensemble to each monochro- atomic motions owing to its extremely high-energy resolu- matic wave as a scattering amplitude. The latter depends on tion. First of all, it is applicable to a diffusive motion of the diffusive motion of nuclei represented by the double di- atomic particles which occurs in liquids and solids for a mensional momentum and frequency Fourier image of the review see Refs. 1­4 . The analysis of Mo¨ssbauer spectra Van Hove function. In contrast to the absorption spectra affected by diffusion was done by Singwi and Sjo¨lander5 in where the diffusion influences only an absorption coefficient, terms of the Van Hove correlation function G(r,t). This the forward scattering time dependence is influenced by dif- function describes a probability for finding the nucleus in a fusion in a more complicated way. In Ref. 9 only a case of matter at a position r at time t, if it was at the origin at time free continuous diffusion was analyzed. Here we develop the t 0. The phase shift of rays scattered by the nucleus is theory further to include the continuous localized diffusion and the jump unlimited diffusion. In the next section both the related to the shift of the mean position r(t) of the nucleus. It main ideas and results of the general theory9 are shortly sum- depends on the time interval between the moments of exci- marized. tation and deexcitation. The explicit form of the Van Hove function is different in each particular case of diffusive mo- tion. In any case a diffusion manifests itself in the profile of II. GENERAL FORMULAS the Mo¨ssbauer absorption spectra through the broadening A short pulse of synchrotron radiation can be decomposed and the change in shape of resonance lines. into a continuous set of coherent monochromatic waves Recently a powerful technique for studying nuclear within the frequency interval centered at the resonance resonance was developed with the use of nuclear resonant frequency and well exceeding the width of resonance range. scattering of synchrotron radiation SR . This technique is With a good accuracy one can consider all monochromatic based on measuring the time dependence of intensity reemit- components to be equivalent in weight. To calculate the for- ted by nuclei after an excitation of the nuclear system by ward transmitted wave packet one should integrate all for- very short pulse of synchrotron radiation. The observed co- ward transmitted monochromatic components. As it was herent reemission into the forward direction allows us to shown9 the result can be written as follows: claim a formation of excitation distributed coherently over the entire nuclear system, called a nuclear exciton, which has K an unusual nonexponential character of decay. The coher- E t,z E0 z d 2 exp i t exp i 2 g n z . ent decay of the nuclear exciton is characterized by a 1 speed-up effect accompanied by quantum and dynamical beats of intensity.6,7 Many nuclear and solid-state parameters Here E(t,z) is a time-dependent electric-field amplitude of can be explored by studying the time dependences of the synchrotron radiation transmitted through a nuclear target of nuclear exciton decay.8 thickness z. The function E0(z) has the modulus A general theory of time-dependent nuclear resonant for- (I0 / )1/2 exp( ez/2) with e K being the electron ward scattering of SR pulse by a system of nuclei moving absorption coefficient and I0 being the intensity of SR within diffusively has been developed earlier.9 It was shown that the frequency band as determined by a monochromator similarly to the approach developed in Ref. 10, the time de- system. The wave number K 2 / /c. 0163-1829/98/57 10 /5788 10 /$15.00 57 5788 © 1998 The American Physical Society 57 THEORY OF NUCLEAR RESONANT . . . 5789 The nuclear part of the susceptibility of the target g(n) is ways into the scattering amplitude. The first one enters closely related to the scattering amplitude and can be repre- through the well-known Lamb-Mo¨ssbauer factor f LM , while sented as the second one is described by the universal resonance func- tion in a more complicated form. Mathematically this proce- dure can be done in all cases by simply introducing the Lamb-Mo¨ssbauer factor in the explicit form and considering g n k, i 0 2 Bge k, eg , 2 ge the total dephasing function without the Lamb-Mo¨ssbauer factor. We note that the Lamb-Mo¨ssbauer factor has different where the value behavior in solids and liquids. The most frequent application of SR in the nuclear reso- nance spectroscopy is a measurement of the time dependence 8 f B LM k of the forward scattering intensity. ge g j k e 2 3 2V0 2Ig 1 0 characterizes the strength of nuclear response at the reso- I nance frequency f s t,z E t,z 2. 7 eg . Here the indexes g and e numerate the hyperfine sublevels of the ground and excited states of a nucleus, It is of interest to compare the time dependence with the 0 is the natural width of the excited level, f LM(k) is the Lamb-Mo¨ssbauer factor, I frequency dependence of intensity, which is ordinarily inves- g is the nuclear spin in the ground state, V tigated in the Mo¨ssbauer absorption spectroscopy. Without a 0 is the target volume corresponding to one nucleus, and g j (k) e is the matrix element of the scalar source convolution it is as follows: component of the nuclear current density operator along the polarization vector of the incident wave. In particular, the latter parameter equals zero if the transition is forbidden for Ia ,z E ,z 2 E0 z 2 exp Kz Img n .... a given multipolarity of radiation. 8 The frequency dependence of susceptibility near the reso- nance is determined entirely by the universal resonance func- One can see that the shape of the absorption spectra is di- tion (k, ) which takes into account the diffusive motion. It rectly determined by the real part of the universal resonance is described by the formula function (k, ) . As for the time dependence of the radia- tion field, it is calculated through the frequency representa- tion, Eqs. 1 ­ 3 , where the entire universal function should t k, dt exp i i be used. 0/2 t t Fs k,t t , On the other hand, this frequency-dependent function is 4 determined by the time dependence of the two processes where the correlation function in momentum representation shown in Eq. 4 , namely, by the decay of individual excited is introduced. Such a representation naturally corresponds to nucleus and by the diffusive motion represented by the MT a scattering problem. We term the function F correlation function . s(k,t) as a momentum-time MT correlation function. It is tightly re- The MT correlation function 5 represents the mean lated to the space-time Van Hove correlation function, value of the phase factor over a whole volume of the target. G(r,t), via the Fourier transformation Being a function of time this averaged phase factor reflects a diffusive motion of nuclei. By its nature the correlation func- tion is very similar to the Lamb-Mo¨ssbauer factor which de- scribes a damping of the scattering amplitude owing to the Fs k,t dr exp ikr...G r,t . 5 thermal motion. The difference between them is only in the time scales which are characteristic of the motions involved. For the convenience of the following calculation, we shall The evident property of the Van Hove function use below a more direct representation for the function G(r,0) (r... gives the property of the MT correlation func- (k, ) which is obtained by changing the variable, namely, tion Fs(k,0) 1. Without a diffusive motion of nuclei in the target, this initial value stays constant. Correspondingly the resonance function takes the ordinary form k, dt exp i t (k, ) i/( i 0/2). However, in a presence of diffusive 0t/2 Fs k,t 6 0 motion the correlation function can drop down at a time comparable with the lifetime of the nucleus. As a conse- A derivation of formulas 1 ­ 6 is given in the first part quence, the coherence time in the forward scattered wave of this work.9 Here we emphasize once again that we restrict packet becomes shorter, while the spectral width of radiation ourselves by the cases when polarization mixing is absent in becomes broader. the coherent forward scattering. For example, this takes place It is useful to consider the case of small thickness z of the for the M1 transition of 57Fe nuclei. On the other hand, we target where one can expand the exponential function in Eq. consider the cases where the fast thermal motion of nuclei 1 in a power series. Then considering only the first term of near a temporal equilibrium site and the slow motion of the the expansion, one easily arrives at the following expression equilibrium site itself are factorized and enter in different for the scattering intensity: 5790 V. G. KOHN AND G. V. SMIRNOV 57 I I t,z 0 Kz 0 2 w r Z 1exp U r /T..., Z drexp U r /T.... 4 exp ez 0t Fs k,t 2 12 Bgeexp i egt 2, t 0. 9 The substitution of this formal solution in Eq. 5 and then in ge Eq. 4 yields The time-dependent square modulus of the MT correlation function enters here as a multiplier, i.e., directly influences a k, i dr exp ikr i 0/2 iL 1 decay of the coherent signal. The approximation 9 is usu- ally called a kinematical approximation in which only a scat- exp ikr w r . 13 tering of one photon by one nucleus is taken into account. As it was shown in Eq. 10 the universal resonance func- It was shown16 that it is convenient to consider a Hermit- tion (k, ) can be found by averaging the resonance factor ian form H of the Fokker-Planck operator L which is defined by the relation i k, d H exp U/2T L exp U/2T D 2 V r , 2 Fs k, i 0/2 , 10 14 where F where s(k, ) represents the spectral density of the momentum-time correlation function Fs(k,t). This expres- sion can be interpreted as an average value of standard scat- V r 2T 2 U r ...2 2T 1 2U r . 15 tering amplitude over Doppler shifts of the resonance fre- One can easily see that L w(r) quency caused by diffusive motion of nuclei. 0H 0 if 0 Z 1/2 exp U r /2T..., 16 III. LOCALIZED DIFFUSION IN GENERAL CASE and the resonance function (k, ) can be found as a In the first part of our work9 the case of free diffusion has quantum-mechanical average of the Hermitian operator rep- been analyzed as an example of an application of the general resenting the resonance interaction theory. The free diffusion is understood as the unlimited in space, continuous motion of particles in a medium that is k, i 0 exp ikr i 0/2 iH 1 described by the diffusion coefficient only. The developed exp ikr 0 . 17 theory can be easily extended to more complicated regimes of diffusive motion. Indeed, the main problem here is to find By employing the total set of eigenfunctions of the Her- an explicit form of the Van Hove correlation function. We mitian operator H , Eq. 17 can be transformed with the help note that the same problem should be solved in the Mo¨ss- of the relation bauer absorption spectroscopy. Following the first consider- ation given by Singwi and Sjo¨lander5 one can distinguish two limiting cases: a continuous motion of large particles in 0 AHB 0 0 A n n H n n B 0 , 18 n a medium and a jump motion of nuclei between the sites on the crystal lattice. The first motion can be limited in space to the equation under the influence of a drift potential. The second motion generally is unlimited in space, however there are cases k, i 0 exp ikr n n exp ikr 0 , where it also can be limited.11­14 n i 0/2 iD n Here we consider the localized diffusion which represents 19 the case of spacially restricted diffusive motion of a particle under the influence of a drift potential U(r). It is known where D n is the eigenvalue of the operator H correspond- see, for example, Refs. 15,16 that in this case the Van Hove ing to the eigenfunction n . The sum over n denotes the correlation function is a solution of the Fokker-Planck equa- summation over all eigensolutions. When the operator H has tion both the discrete and the continuous parts of spectrum then this sum must be added by the integral. G r,t The last formula allows us to reveal some general prop- erties of localized diffusion. We rewrite it in a more compact t L G r,t , L D 2 B U r form 11 An k... where L is the Fokker-Planck operator describing the k, i i , 20 Brownian motion in the drift potential U(r) with the diffu- n 0/2 iD n sion coefficient D and the drift coefficient B D/T. Here T with is the absolute temperature in energy units. This equation has a formal solution G(r,t) exp( L t)G(r,0) where G(r,0) can A A be taken in the form G(r,0) exp(ikr)w(r), where w(r) is n(k) dr exp ikr 0* r n r 2, n k... 1. n the standard Boltzmann distribution function 21 57 THEORY OF NUCLEAR RESONANT . . . 5791 It is easy to verify that H 0 0. This means that the eigen- easily deduced from Eq. 22 for the potential considered. value of the ground state is 0 0 independently on the ex- Moreover, the only one state with p k has a nonzero coef- plicit form of the drift potential. The ground state represents ficient Ap 1. Thus, in the limit of an infinitely large cage we the partial state of a nucleus where it does not move diffu- arrive once again at the universal resonance function charac- sively and preserves its average position unchanged during teristic for a free diffusion9 all time. The corresponding resonance has the natural width as for a static nucleus in isolation. The weight of this state is i defined by the coefficient A k, . 24 0 . This coefficient has a property i which is essential for the retrieval of the drift potential. Ac- 0/2 iDk2 cording to Eqs. 12 , 16 , and 21 one can write The most simple case of bounded diffusion within a cage is realized when the cage has the form of a rectangular box A 2 with dimensions X,Y,Z in the directions x,y,z, correspond- 0 k f k f , f 0 k dr exp ikr U r /T.... ingly. This illustrative example might be also of a practical 22 interest in the view of future experiments because it can be Thus the weight of the ground state A realized artificially with the control of parameters such as the 0(k) is directly asso- ciated with the drift potential profile U(r) through the Fou- diffusion coefficient or the size of the cage.17 Owing to the rier transformation. In the case of nuclear resonance scatter- fact that the cages are oriented randomly in different parts of ing the modulus of k is fixed, therefore one cannot probe the a real sample the task is spherically symmetrical. That is potential at different magnitudes of k. However, the mea- why, perhaps, in Ref. 17 the accurate solution of the Fokker- surements at different directions of k can give information Planck equation with a spherical cavity of radius r as a hole about the symmetry of the potential. In addition, the tem- in the Swiss cheese18 has been considered to analyze the perature dependence of A experimental results. Nevertheless, it is useful to consider 0 can provide the potential strength. In particular, when A and analyze in detail the task of the rectangular cavity. 0 does not depend on T a specific case of localized diffusion is realized-a continuous diffusion in- We take the potential U(r) to be equal to zero inside the side a strictly restricted volume. This type of diffusion we box and infinity outside of it. The potential V(r) is also equal shall refer to as the bounded diffusion within a cage. to zero inside the box but it has a singularity on the box Finally we conclude that in the case of localized diffusion walls. Unlike the similar task of quantum mechanics we have the problem is reduced to finding the total set of eigensolu- to find now the eigensolutions n(r) which obey the bound- tions of the Hermitian operator H with a specific potential ary conditions (n )B 0 where n is a normal to the box obtained by the transformation of a drift potential see Eq. boundaries see Ref. 16 for details . The eigensolutions for a 15 . The momentum-time correlation function in this ap- particle diffusing inside the box with the coordinates proach takes the form originating from Eq. 6 as follows: 0 x X, 0 y Y, 0 z Z are found in the form k2 l2 F s k,t An k exp D nt . 23 hkl x,y,z fh x fk y fl z , hkl 2 h2 , n X2 Y2 Z2 We note that this expression has a clear physical sense. One 25 can expect different rates of diffusive movement which lead where h,k,l 0,1,2, . . . . Below we introduce the notation to different speeds of dephasing the coherently scattered ra- j h,k,l, s x,y,z, S X,Y,Z. The solution is factorized diation similar to dephasing due to thermal vibrations de- into the functions independently describing the motion along scribed by the Lamb-Mo¨ssbauer factor . The weights of each of the three main axes. A particular function is as fol- these movements depend on the propagation vector k both a lows: magnitude and a direction and on the form of the drift po- tential. f j s 21/2 CjS 1/2) cos js/S . 26 Here C IV. BOUNDED DIFFUSION WITHIN A CAGE j 1 for all j 0 and C j 2 for j 0. One can readily find the weights of different diffusion Until now the drift potential was assumed to be a finite states in accord with Eq. 21 . These also consist of three one having an arbitrary profile. In this section we regard a independent factors particular case where the potential equals zero inside a finite volume, called a cage, and is infinite outside. It can be shown Ahkl kx ,ky ,kz ah kx ak ky al kz , 27 that if the size of the cage tends to infinity then one arrives at with the case of free diffusion. Let us consider this limit case in view of the general theory described above. 1 The potentials U(r) and V(r) are equal to zero in all a ksS j ksS j 2 j ks 2C j0 2 1 jj0 2 , space actually in a volume with a size being much larger j 28 than the wavelength of radiation . In this case a full set of eigenfunctions can be chosen in the form of plane waves where j0(z) sinz/z is a zero-order spherical Bessel function. p(r) 1/2 exp(ipr) corresponding to eigenvalues It is easy to verify that this formula gives the relation 2 p p2. The ground state is 0 const 1/2. However, aj(0) j0. On the other hand, a0(ks) j0(ksS/2). Therefore this state is not realized because A0 0 for k 0. This can be the ground state 000 is now really existing. 5792 V. G. KOHN AND G. V. SMIRNOV 57 FIG. 1. The weights al of the Lorentzian functions contributing into the universal resonance function in the case of bounded diffu- sion 29 in dependence on the size of the one-dimensional cage Z; a0 is the weight of the partial resonance having the natural width 0. In the model considered a dependence arises on the ori- entation of the cage with respect to the ray propagation direction. Let us take the orientation for which k FIG. 2. The shape of the real part of the universal resonance x ky 0. In this case the radiation can probe only a diffusive motion function, describing the absorption ability of the nuclear target, at along the z direction. Then the general formula 20 trans- different sizes of the one-dimensional cage Z 0,0.01, . . . ,0.2 nm. forms to Each next curve is shifted up on 0.05 relative to the previous one. a q 2 l 1 2Dkz t0. Thus, we see that the widths of the Lorent- k l kz 0 z , l kz , i zians contributing in the function are dependent on Z only l 0 l 0 iql 0/2 , 29 very softly when Z 2 . where the weights al(kz) and the relative widths ql(Z) of the Figure 1 shows that the behavior similar to that of free partial resonances are determined by diffusion should appear already for not so large a box size, namely, about Z 2 where the coefficient a0 becomes 1 1 l 1cos k close to zero. The existence of a zero term having a natural a 1 zZ l kz 4Cl kzZ 2 , l 2 k resonance width makes the situation quite different. Thus zZ 2 2 one should have a significant change of the resonance shape only within the transition region 0 Z 2 . Studying this l 2 q transition region is most informative with respect to the po- l 1 2Dt0 , 30 Z2 tential parameters. The absorption spectral function f where t a( ,Z) 0 / 0 is the natural lifetime of the excited nucleus. Re (k The formula for the weights follows directly from Eq. 28 . z , )/2t0 is displayed in Fig. 2 for this transition range of Z 0,0.01, . . . ,0.2 nm. For a better view each next The maximum value of each term in the sum 29 equals curve is shifted up on 0.05 relative to the previous one. The 1 l(kz,0) 2t0al(kz)ql , i.e., it is determined by both the calculation has been made for 57Fe with 0.086 nm and weight and the width of each Lorentzian contribution. Let us analyze the shape of the function as a function of D 10 14 m 2/s. In this case fd / 0 16 where 2 the box size Z. The coefficients a fd 0 2 Dkz is the resonance width in the case of free l as the functions of Z are shown in Fig. 1 for the case of the 57Fe nucleus with diffusion. As it follows from the calculation, an apparent k decrease of the resonance dips of the function f z 2 / 73 nm 1. Each coefficient has a pronounced a( ,Z) oc- main maximum and side maxima which are of much lower curs there accompanied by a significant broadening of the height. The main maxima for different coefficients are dis- resonance. The resonance shape close to that in the regime of tributed over the Z axis rather regularly. After the initial fast free diffusion is approached already at Z where the drop down within the interval 0 Z the heights of the weight of the unbroadened partial resonance drops to zero next maxima are reducing very slowly farther on. It is clearly the curve 9 . Afterwards the coefficient a0 increases again at seen in the figure that at any value of Z there are only a few about Z 1.4 that results in a narrowing of the resonance one, two, or a maximum three significant coefficients while for this range which is well seen in the figure the curves all others are very small. Hence only a few Lorentzians con- 12­15 . tribute essentially to the universal resonance function at any We turn now to the time response of the nuclear en- box size. The l indexes of the contributing Lorentzians in- semble. The analysis is the most simple in the case of a thin crease with a rise of the size. The l value which corresponds single line target. Here the intensity is proportional to z2, to the main contribution, equals approximately while the time dependence is determined by the following l0 kzZ/ 2Z/ . The relative width of this contribution is forward scattering function: 57 THEORY OF NUCLEAR RESONANT . . . 5793 FIG. 3. The time dependence of nuclear forward scattering of FIG. 4. The time dependence of nuclear forward scattering of synchrotron radiation in the limit of the thin target 31 in the case synchrotron radiation by the thick target in the case of bounded of bounded diffusion at different sizes of the one-dimensional cage diffusion at different sizes of the one dimensional cage Z 0, 0.01, Z 0,0.01, . . . ,0.1 nm. . . . , 0.1 nm. The effective thickness of target nz 10. by multiple scattering of radiation by nuclei in the target. It f fs t,Z exp t/t0 al kz exp l2 2Dt/Z2 2, is of interest that the dynamical beat pattern appears to be l 0 highly sensitive to the box size. With increasing the size the beat minima are shifting towards later times. Then they dis- t 0. 31 appear completely in the observation time window curves 1 5 in Fig. 4 . This effect is related to a redistribution of The entire time dependence of the forward scattering in- weights of contributing Lorentzians, particularly to the drop- tensity is the sum of exponents where each term is charac- ping down of the coefficient a0 which represents the rigidly terized by its own decay region t tl where bound state of the nucleus. The latter leads to decreasing the tl t0(1 2Dt0Z 2 2l2) 1. So the number of contributing effective thickness of the target similar to that due to the fall terms decreases when the delay time increases. Finally only down of the recoilless factor. We shall discuss this effect in the term having the natural decay time becomes dominant. more detail in the next section. The time evolution for the Therefore the contribution of this term given by the coeffi- intermediate range of Z curves 6­8 in Fig. 4 is similar to cient a0 can be well separated in the time dependence. The that for a thin target. The nonmonotonous transformation is function f seen again later on curves 9­11 in Fig. 4 . However, under f s(t,Z) is shown in Fig. 3 for box sizes in the range Z 0,0.01, . . . ,0.1 nm. The uppermost curve corresponds to conditions close to the free diffusion regime, i.e., for the well a rigidly bounded nucleus. It is described by only a zero term size large compared to the wavelength of radiation, the dy- with a namical beat pattern specific for the full effective thickness 0 1 and q0 1, i.e., it exhibits the natural decay. With the increase of the box size the diffusion is activated of the target is restored. and the next terms related to the broader resonances start to The initial slope of all curves seems to be somewhat uni- contribute. This results in a faster decay of the scattering versal. This slope can be easily obtained analytically using intensity observed within the initial time interval. However, the asymptotic behavior of the function. It is known that until the zero term has a noticeable magnitude the range the asymptotic behavior of the universal resonance function 0 Z 0.06 nm in Fig. 1 , the natural decay rate is reached. at far tails of resonance allows us to estimate the temporal It is manifested by the straight segments of the curves ob- nuclear response at the initial time. We obtain directly from served at later times in the logarithmic scale the curves 1 7 Eq. 29 in the case of large that in Fig. 3 . In this range of box sizes the coefficient a0 can be determined directly by extrapolating the straight segments of i fd 2 the curves to zero time. lim kz , 1 i . 2 *** , fd 0 2 Dkz The coefficient a0 drops down sharply with the increase 32 of Z, reaches its minimum and then slightly oscillates Fig. 1 . This behavior is reflected in the nonmonotonous transfor- Here fd just corresponds to the free diffusion case see mation of the time dependences around Z 0.09 nm the above . To derive this result one has to take into account the curves 8 11 in Fig. 3 . relations To illustrate the role of bounded diffusion in a scattering from a thicker nuclear target we consider the time response 2 in the case of a single line sample having an effective reso- al kz 1, al kz l kz . 33 l 0 l 0 nance thickness nz 10, where n K geBge see 3 . The results of the computer calculation are shown in Fig. 4 The first relation follows from the general theory see Eq. for the same range of box sizes. The time dependence con- 21 . The second one is the consequence of the fact that tains now the dynamical beats of intensity which are caused inside the volume of the cage we have the same equation as 5794 V. G. KOHN AND G. V. SMIRNOV 57 in the case of free diffusion the potential is absent . There- lated in the more general case. This function can be written fore l l(z) d2 l(z)/dz2 and the relation is obtained by in a form similar to Eq. 23 , namely, inserting this formula into the integral 21 which defines generally the coefficients al(kz) and then integrating by F a parts. s k,t l k exp[ l k t/2t0], l From Eqs. 1 , 2 , and 32 we obtain after expanding the exponent in the Taylor series and using the residue theorem m a 2 l ci il , al 1, 35 i 1 l limI 2 f s t,z E0 z 2 nz where t t 0 4t0 0 / 0 is the nuclear lifetime as above, i is the index of inequivalent sites inside the primitive unit cell of the crystal lattice. This site has the ith local symmetry. The 1 t 1 2 nz t 2Dkz *** . 34 relative lth decay rate 0 4t0 l(k... and the vector il(k) are the eigenvalue and orthonormalized eigenvector of the probabil- In accordance with this formula the initial slope is deter- ity jump matrix, which is a Hermitian one mined by three physical reasons represented by the three terms in the straight brackets , namely, by natural decay of 2t0 1 1 n the excited state, by diffusive motion of nuclei, and by co- Aij k exp ikR c c i j i j j. q iq nji ji n herent speedup of decay. The Z dependence of the initial i 36 slope is absent. Usually the term related to a diffusion brings the main contribution and this determines the behavior of the Here and above, ci is the probability of the occupation of the curves in Figs. 3, 4. The time dependence of intensity in the ith sublattice, nji is the number of sites of the ith sublattice 1 case of limited diffusion cannot be faster than in the case of which surround the site of the jth sublattice, ij is the jump free diffusion. rate from the site of symmetry i to any nearest-neighbor site This fact has a clear physical sense. In the limit of small of symmetry j, R(n) i j is the nth vector distance of the set of delay time only displacements of nuclei which are smaller distances between nearest-neighbor sites of the ith and jth than the cage size are essential. Therefore the walls of the sublattices. Detailed balance demands that cage cannot influence the time behavior. In the opposite limit of very large delay time, the picture of nuclear motion is ci c j . 37 influenced essentially by the reflections from walls. On av- nij ij nji ji erage, the coherent nuclear scattering is related to the mean If position of a nucleus, as if it is at rest. The strength of this i j ji then c j const 1/m and al m 1 i il 2. Expression 35 leads immediately to the following ex- scattering channel is described by the coefficient a0 which pression for the universal resonance function: plays the same role as the Lamb-Mo¨ssbauer factor which takes into account the thermal motion. Obviously when the al k... mean displacement of the nucleus Z/2 becomes comparable k, i iq with the wavelength of the radiation the coefficient a n l k 0/2 , ql k 1 l k.... 0 tends to zero due to a dephasing of the scattered waves. 38 When the coefficient a0 is not too small one can find in the This formula implies that each resonance becomes a super- case of the thin target a characteristic intermediate time position of several resonances having the same position but range about 80­120 ns in Fig. 3 where the transition from different widths, i.e., like in the case of bounded diffusion the free diffusion regime to the natural nuclear decay occurs. Eqs. 23 and 32 . The time of transition can be estimated as As an example, we consider in detail the case of jump t 2 diffusion in the alloy Fe tr ln(1/a0)/2Dkz 10 ln(1/a0) ns in the case considered 3Si which was investigated recently (57Fe, and D 10 14 m 2/s . It depends significantly on the using the nuclear forward scattering of synchrotron size of cage Z through the coefficient a radiation.22 The time dependences of the forward scattering 0, on the diffusion coefficient D as well as on the wavelength of the radiation. along the 113 crystal direction were measured. Single crys- The real sample may contain different cages with differ- tals of Fe 3Si have a cubic superstructure consisting of four ent sizes and orientation. Therefore the mean value has to be sublattices. In the entirely ordered crystals the three sublat- calculated. This procedure is essential in fitting the experi- tices are occupied by iron atoms while the fourth one is mental data. However the general properties of the time occupied by Si atoms. The diffusion mechanism of iron in spectra considered here will be unchanged. this structure was studied.20 It was proven that in accordance with earlier suggestions iron atoms jump between three iron sublattices only and avoid the fourth silicon sublattice. V. JUMP DIFFUSION Therefore the matrix Aij takes the form The theory developed for the quasielastic Mo¨ssbauer spectroscopy5,19 describes a jump diffusion on simple Bra- 2t vais lattices where all crystal sites are equivalent. The exten- 2 E E* A E* 0 , 0 sion of the theory to the case of non-Bravais crystal lattice , 39 was done in Refs. 20,21. Here we shall use the results of Ref. E 0 21 where the correlation function Fs(k,t) has been calcu- where 57 THEORY OF NUCLEAR RESONANT . . . 5795 E cos kxd cos kyd cos kzd isin kxd sin kyd sin kzd 40 with kx ,ky ,kz 1,1,3 K/ 11, d a/4 . 41 Here K 2 / /c is the wave number of incident radia- tion which fits the nuclear resonance transition in 57Fe in our case , a is a crystal lattice constant of Fe3Si. Taking into account the specific values K 72.98 nm 1, a 0.571 nm one obtains that kxd 3.1411, i.e., very close to . Therefore approximately E E* 1 and with definite accuracy the jump matrix 39 has the following analytical eigensolutions q FIG. 5. The time dependence of nuclear forward scattering of l 3 0 , synchrotron radiation from Fe3Si crystal in the direction 113 in 1 the presence of jump diffusion at different q factors see text . 2 1 0 1 1 Curves1­4correspondtoq 2.2,5.1,11.5,and36,respectively. l 1 1 1 , 6 The effective thickness of target 1 2 1 3 1 nz 3. al 8/9 0 1/9. 42 lows the same law described above but at a later time the It follows from Eq. 42 that only two resonance contribu- dynamical beats appear within the natural lifetime of nuclear tions have nonzero weight, one having the natural width and excitation. It is of interest to note that a position of the beat another broadened. Correspondingly the universal resonance minimum is sensitive to a diffusion rate like in the case of function takes the form having only one parameter bounded diffusion where it is sensitive to the cage size, see Fig. 4 . The minimum is shifting towards the later times with i 1 an increase of the diffusion coefficient. Such a behavior is k, 8 9 iq/2t , q 1 6t0 / . different from that predicted9 for the case of free continuous 0 i/2t0 43 diffusion where the beat pattern position is unchanged. This difference is tightly related to the difference in form of the In the range of high temperatures where diffusion takes resonance universal function. In the case of free diffusion it place the hyperfine splitting of the nuclear levels is absent is a single Lorentzian having variable width. Here and in the and the time-dependent electric field of the scattered wave is case of bounded diffusion it is the sum of several Lorentz- described by the following equation: ians having different weights and widths. The split of the universal resonance function into several terms leads to the fragmentation of the effective resonance E t,z E z d nz 2 exp i t 4t k, 0 , thickness of a target into relevant partial thicknesses which 0 44 are al nz. The contributing Lorentzians turn out to deter- mine essentially the decay of the nuclear exciton within dif- where 0 is a resonance frequency, z is a crystal plate thick- ness, and n N 0 fLM is an absorption coefficient at reso- nance with N being a number of iron atoms in the unit vol- ume, 0 being a nuclear cross section at resonance, being an enrichment by resonant isotope 57Fe and f LM being the Lamb-Mo¨ssbauer factor. We compare the time-dependent forward scattering inten- sity for different diffusion rates, respectively, different q fac- tors. The time dependences for the two effective thicknesses of target nz 3 and 21.5 are displayed in Figs. 5,6. In a thin target limit using approximation 9 one can readily find that the time response is described by the sum of two exponential functions, one of which exhibits the natural decay of the nuclear excitation while another shows the accelerated de- cay. With an increase of the q factor the acceleration of decay is well seen within an initial time interval in Fig. 5. In the limit q 1 the two exponential functions turned out to be FIG. 6. The time dependence of nuclear forward scattering of well separated in time and the two stages of the decay are synchrotron radiation from the Fe3Si crystal in the direction 113 clearly observed. in the presence of jump diffusion at different q factors. Curves 1­4 The time dependences for the thicker sample have a more correspond to q 2.2, 5.1, 11.5, and 36, respectively. The effective complicated character. The initial stage of decay mostly fol- thickness of target nz 21.5. 5796 V. G. KOHN AND G. V. SMIRNOV 57 ferent time intervals in dependence on the Lorentzian widths. thin target for any regime of diffusion. The spectral density This leads, roughly saying, to the fact that the target exhibits of this function F(k, ) is found by the Fourier transforma- its different partial thicknesses in different parts of the time tion of F(k,t). Actually the universal resonance function dependence of the forward scattering. In general, the effect (k, ) represents the averaging of the standard resonance of the thickness split is revealed for any relation between the amplitude describing the scattering process by a static weights and widths of contributing Lorentzians. Obviously nucleus over the spectral density function F(k, ) which re- the dynamical beat pattern should be sensitive to this rela- flects the motion of particles in the nuclear ensemble owing tionship. This is actually the physical reason for the shift of to the Doppler effect. the beat minima with the change of both the potential well The analytical solution for time dependences of the coher- size and the diffusion rate. ent forward scattering of SR can be obtained only for the In our example in the limit of large-q factor q 1, where two contributing Lorentzians have essentially different case of free diffusion where the universal resonance function widths, one can distinguish two characteristic stages in the has Lorentzian shape. The additional width of the resonance entire time dependence. At an early stage of decay the tails line is simply proportional to the diffusion coefficient in this of the resonance are essential and one can consider only the case. Respectively, an additional exponential factor appears broadened contribution. Making use of the result obtained in in the time response, the decrement of which contains the Ref. 9 for the case of one broadened resonance one obtains diffusion coefficient. This yields an accelerated decay of the coherent signal. As to the dynamical beat structure it does I T qt not depend on the resonance broadening. I t,z 0 1t0 2 4t exp ez 2t J1 T1t/t0 , In contrast to the regime of free diffusion in the case of 0 bounded diffusion inside a potential well, and in the case of t t jump diffusion between different sites vacancies in solids 0 /q, 45 the universal resonance function has a more complicated where J1(x) is a Bessel function of first order and shape represented in general by the coherent superposition of T1 8 nz/9. In this time interval the evolution of the for- the Lorentzian functions where the weight and the width of a ward scattered intensity is similar to the case of free diffu- separate Lorentzian are determined by the specific character sion, and via parameter q one can determine the diffusion of the diffusion process. The main physical parameters af- coefficient D a2/32 . fecting the shape of the universal function are the diffusion At later times, on the contrary, the region near the center coefficient, temperature, the drift potential profile, and the of the resonance is essential in the integral 44 . Therefore jump rate. one can neglect in the first term of Eq. 43 and consider Such a shape of the universal function corresponds to a approximately the case with one resonance of natural width. As a result the time evolution takes the form more complicated behavior of the time response. In general, there are several stages of the decay which are characterized I T 8 t by the different decay rates. The initial stage reveals a faster I t,z 0 2t0 n monotonous decay which is more accelerated the larger dif- 4t exp e 9q z 2t0 fusion coefficient and the size of the potential well are. At J2 later times the decay rate becomes slower and the dynamical 1 T2t/t0 , t t0 /q, 46 beats appear in the case of the thick target. The dynamical where T2 nz/9. We note that similar to the case of bound beat pattern is transformed drastically in contrast to the free diffusion the broad resonance effectively provides an absorp- diffusion regime depending either on the size of the poten- tion in addition to the electronic absorption, while the effec- tial well in the case of bounded diffusion or on the tempera- tive thickness of the target turned out to be less in accord ture in the case of jump diffusion. When both parameters are with the weight of the resonance of the natural width. Thus rising, the transition to a beat pattern characteristic for a one and the same target exhibits, in this case, behavior of the thinner target occurs. target with the effective thickness T1 at an early stage and The physical reason for this result is in the split of the with T2 at a later stage of decay. universal resonance function into several terms. It leads to a Finally we want to note that the consideration presented fragmentation of the effective resonance thickness of the tar- above deals with the pure diffusion process, while the relax- get into relevant partial thicknesses. When the contributing ation process is assumed to be the same as in the case with- Lorentzians have essentially different widths the target ex- out diffusion. hibits its partial thicknesses in the time dependence within different time intervals. Hence a time-variable thickness is characteristic for the nuclear exciton decay in these cases VI. CONCLUSION rather than a unique thickness. In general, the effect of the thickness split is revealed for an arbitrary relation between The response of the nuclear ensemble in the presence of the weights and widths of contributing Lorentzians. The dy- diffusive motion of nuclei is described by the universal reso- namical beat pattern is sensitive to this relationship. How- nance function (k, ) which is related to the Van Hove ever, under conditions close to the free diffusion regime, i.e., space-time correlation function G(r,t). While considering a for the well size large compared to the wavelength of the scattering problem it is natural to use the momentum-time radiation in the case of bounded diffusion, or for high tem- correlation function F(k,t) which enters directly into the peratures in the case of jump diffusion, the dynamical beat time dependence of nuclear exciton decay in the limit of a pattern is restored to the full effective thickness of the target. 57 THEORY OF NUCLEAR RESONANT . . . 5797 The time evolution of the coherent forward scattering of complex amplitudes of oscillations of the electromagnetic synchrotron radiation is thus transformed not only quantita- field, while a spectroscopic method resonance absorption tively but also qualitatively in contrast to the relevant Mo¨ss- exhibits only their strengths see Ref. 8 . This makes forward bauer spectra. 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