PHYSICAL REVIEW B VOLUME 53, NUMBER 10 1 MARCH 1996-II Coherent forward-scattering amplitude in transmission and grazing incidence Mo¨ssbauer spectroscopy L. Dea´k,* L. Bottya´n, and D. L. Nagy KFKI Rsearch Institute for Particle and Nuclear Physics, P.O. Box 49, H-1525 Budapest, Hungary H. Spiering Institut fu¨r Anorganische und Analytische Chemie, Johannes Gutenberg Universita¨t Mainz, Staudinger Weg 9, D-55099 Mainz, Germany Received 25 July 1995 The theory of both transmission and grazing incidence Mo¨ssbauer spectroscopy is reanalyzed. Starting with the nuclear susceptibility tensor a common concise first-order perturbation formulation is given by introducing the forward-scattering amplitude into an anisotropic optical scheme. Formulas of Blume and Kistner as well as those of Andreeva are rederived for the forward-scattering and grazing incidence geometries, respectively. Limitations of several previously intuitively introduced approximations are pointed out. The grazing incidence integral propagation matrices are written in a form built up from 2 2 matrix exponentials which is particularly suitable for numerical calculations and practical fitting of both energy domain conventional source experi- ment and time domain synchrotron radiation experiment Mo¨ssbauer spectra. I. INTRODUCTION 3 3 index of refraction matrix, they accepted Lax's intui- tive suggestion,1 and used a complex 2 2 index of refrac- A great majority of Mo¨ssbauer experiments are performed tion matrix n, corresponding to the two possible independent on polycrystalline samples without applying an external states of polarization of the radiation. n was then related to magnetic field. In such cases, the polarization of the rays the coherent forward scattering amplitude.2 plays no role, the Mo¨ssbauer spectrum can be described in Beside the conventional forward scattering case, grazing terms of resonant and nonresonant absorption, and the reso- incidence Mo¨ssbauer spectroscopy GIMS has gained con- nant absorption cross section can be calculated from the pa- siderable recent attention in studying stratified media: sur- rameters of the hyperfine interaction. This naive approach faces, interfaces, and multilayers.3­7 This method utilizes a fails if the Mo¨ssbauer experiment is performed on a single geometry such that the rays are incident on the flat surface crystal or a textured sample and/or in an external magnetic of the sample at glancing angles of a few mrad close to the field. The resonant cross section in these latter cases depends critical angle of the electronic total external reflection. The on the polarization and the full polarization-dependent scat- detected scattered particles are specularly reflected pho- tering problem has to be treated. The numerical difficulties of tons, conversion electrons, conversion x rays, and incoher- the scattering approach stem from the great number of ran- ently scattered photons. A general treatment of GIMS was domly distributed scattering centers. These difficulties can be published by Andreeva et al. in several papers.6­9 Starting circumvent if, akin to classical optics, a continuum model from the nucleon current density expression of the suscepti- rather than a microscopic scattering theory can be used. It is bility tensor given by Afanas'ev and Kagan10 and using a by no means trivial, however, that such an optical approach covariant formalism of anisotropic optics11 first introduced for rays in condensed matter is feasible since the mean by Fedorov12 these authors take into account that both the distance of scattering centers is usually greater than the elements of the susceptibility tensor and the glancing angle wavelength of the scattered radiation. It has been shown, are small in GIMS and calculate the reflectivity. The however, by Lax1 that, at least for scalar waves, a close to method of calculation, however, especially for the higher unity index of refraction n can be defined and simply related multipolarity nuclear transitions, is rather cumbersome, since to the coherent forward scattering length f , provided that the the nucleon current densities are directly calculated resulting momentum of the scatterers is small compared to that of the in quite complex tensor expressions. In view of the extreme incident wave. Since Lax's paper,1 the refraction index ap- requirements to beam divergence, GIMS is certainly more proach has been used extensively in neutron and x-ray op- suited for synchrotron radiation than for conventional radio- tics. The heuristic generalization of this approach to polar- active source experiments. ized waves and for an anisotropic medium, although claimed Another general description of specular reflection of graz- to be trivial by Lax is by no means straightforward and needs ing incidence Mo¨ssbauer radiation was given by Hannon further elucidation. et al.13­16 Starting from the quantum theory of radiation, In the forward scattering geometry the polarization depen- they formulated the dynamic theory of Mo¨ssbauer optics. dence of the Mo¨ssbauer absorption of radiation was theo- Unfortunately, the dynamic theory provides rather slow algo- retically studied by Blume and Kistner.2 Instead of using a rithms for calculating reflectivity spectra, therefore it is inef- 0163-1829/96/53 10 /6158 7 /$10.00 53 6158 © 1996 The American Physical Society 53 COHERENT FORWARD-SCATTERING AMPLITUDE IN . . . 6159 ficient in spectrum fitting. In the grazing incidence limit, an II. THE NUCLEAR SUSCEPTIBILITY optical model was derived from the dynamical theory,14,16 which has recently been implemented in numerical Let us consider the collective system of the nuclei and the calculations.17 Without using a covariant formalism, how- electromagnetic field. The effect of the electromagnetic field ever, this latter approach also results in quite sophisticated will be treated as a perturbation on the randomly distributed algorithms since, in a layered medium, the eigenpolarizations nuclei. The interaction Hamiltonian H between the nucleus vary from layer to layer. and the electromagnetic field may be written as Our aim is to rigorously derive general formulas for the transmittivity and the reflectivity of radiation in both the 1 forward scattering and the grazing incidence case. Moreover, H c j ri *A ri , 2.1 we shall try to obtain these formulas in such form that is i suitable for fast numerical calculations in order to fit the experimental data. Like Andreeva et al.,6­9 we start from the where j(ri) is the current density of the ith nucleon and Afanas'ev-Kagan nucleon current density expression of the A(ri) is the vector potential of the electromagnetic field at dielectric tensor10 and use a covariant anisotropic optical the point ri : formalism.11,12 Instead of calculating the susceptibility tensor from the current densities of the nucleons, however, we re- duce the problem to the calculation of the transmittance for- A r 2 c 1/2 i ck,pu k,pexp ik­ri H.c. . 2.2 ward scattering case and the reflectivity grazing incidence k,p Vk case from the coherent forward scattering amplitude. We show that, in the case of forward scattering, this approach is In this formula, ck,p denotes a photon annihilation operator equivalent to the theory of Blume and Kistner.2 The present and u k,p a unit polarization vector. treatment is based on no intuitive assumption and represents, The matrix elements of the interaction Hamiltonian H are thereby, a firm basis of the Blume-Kistner theory2 and of the scalar products of the current density matrix elements Andreeva approximation.6­9 Jm and the polarization vectors u gme k,p : 1 1 Hk,p 1/2 2 c 1/2 m 2 c I j r exp ik­r u , 2.3 gme Vk gmgkp i *u k,pck,p i Ieme k,p*Jm i c c Vk gme with 4 N Jm k Jm* k k gme emg c2k2 2I i , 2.7 J g 1 memg m k I j r , 2.4 E gme gmg i exp ik­ri Ieme k Em i emg 2 where Ig and Ie are the nuclear spin quantum numbers in the excited and ground state with the corresponding magnetic where N is the number of resonant nuclei per unit volume, quantum numbers mg and me , respectively. J(k) is the k Ek is the energy of the photon, Em Em Em is the representation of the current density produced by a single emg e g energy difference between the nuclear excited and ground nucleus. Throughout the calculations we shall use the same states, is the natural width of the excited state, and is the letters for physical quantities in r and k representation letting the argument make evident which representation is meant. dyadic vector product sign. The susceptibility tensor (k) In first order perturbation of the electromagnetic field the depends on the propagation vector k of the unperturbed average nucleon current density is10 wave. Instead of Eq. 2.5 a J(1)(k) K (k,k K)E(k K) expression is obtained for nonran- i dom distribution of the scatterers,10 with (2 ) 1K being a J 1 k k E k c k A k , 2.5 reciprocal lattice vector. Only the random scatterer case will be further considered here. with (k) being the conductivity tensor, which in turn de- Equation 2.7 is the starting equation of Andreeva in cal- fines the susceptibility tensor of the medium by culating grazing incidence Mo¨ssbauer spectra.8 In order to calculate (k) for an arbitrary orientation of the hyperfine 4 i fields with respect to k , the currents J are expanded in k mgme k . 2.6 terms of irreducible tensors resulting in a sophisticated formalism.8 For cases like transitions of higher multipolarity, Afanas'ev and Kagan10 calculated the susceptibility ten- mixed multipole transitions, variation of hyperfine fields sor in first order of the vector potential for randomly distrib- within the medium, texture, etc., the formalism therefore be- uted nuclei in terms of the change of the average nucleon comes cumbersome and numerically intractable. Having cal- current density: culated the dielectric tensor of the Mo¨ssbauer medium, An- 6160 L. DEA´K, L. BOTTYA´N, D. L. NAGY, AND H. SPIERING 53 dreeva et al. apply a very elegant covariant formalism11 and reduce to 2 2 ones both in forward scattering and in graz- solve the problem of grazing incidence nuclear scattering by ing incidence geometry. stratified media. The numerical difficulties of the higher multipolarity terms, hyperfine field distributions, texture, etc. have been A. The Borzdov-Barskovskii-Lavrukovich formalism overcome years ago by Spiering18 in treating the thick ab- sorber case in the Blume-Kistner formalism.2 The Hamil- We may write the basic equation for the tangential com- tonian, the scalar product of the current density J ponents of the electric and magnetic fields q E(q *r) and m and the gme polarization vector u Ht(q *r) q q H(q *r) at the point r as follows:11 k,p have simpler transformation proper- ties than Jm , therefore, unlike Andreeva et al.,8 the for- gme ward scattering amplitude q *" Ht q *r ikM q *r Ht q *r , 3.1 q E q *r q E q *r kV 1 Hk,p k,p m Hm f k,p k,p gme emg were q represents the unit normal vector of the surface. The 2 c 2I 2.8 g 1 m i emg material parameters are allowed to vary only in the q direc- Ek Em emg 2 tion stratified medium and the fields depend only on the q *r scalar product. M is the differential propagation matrix rather than Jm is calculated for an arbitrary k direction.18 gme defined by In what follows we shall show that for quanta in the physically relevant representation the 3 3 properties of the dielectric tensor are not fully used by the optical theory. M A B , 3.2 Since the k directions involved in the scattering problem are C D either equivalent forward scattering or extremely close to each other grazing incidence case , the relevant block of the with dielectric tensor is fully described by the four components of the forward scattering amplitude. This latter ensures that the A q * q 1q q a q * q 1b q I, present theory remains valid for nuclear transitions of any multipolarity. Indeed, expressing the susceptibility tensor in the polarization vector system P =(e B q * q 1I ¯I q * q 1b b, , u , ,e 3 k/ k ) of the unperturbed incident radiation the significant matrix ele- ments are C q * q 1a a q * q 1q ¯q , 4 N pp k k2 fk,p k,p p,p , ; 2.9 D q * q 1a q q q * q 1I q b. and being arbitrary polarizations. Once the susceptibil- ity the refractive index or the dielectric tensor of the me- Here 1 is the dielectric tensor, v denotes the dual dium is defined the problem of calculating the propagation of tensor of an arbitrary vector v, and the tilde sign stands for electromagnetic field in the medium becomes an optical the transpose of a tensor. The I (q )2operator projects a problem. Since the nuclear dielectrics is anisotropic, a vector into the plane of the sample surface. The tangential polarization-dependent optical formalism will be used. component of the incident wave vector is b Ik/k, and a: b q is a vector perpendicular to the reflection plane, ¯ det( ) 1, ¯ det( ) 1. Strictly speaking, A, B, C, III. COVARIANT ANISOTROPIC OPTICS OF A NUCLEAR and D are three-dimensional tensors acting only, as it can be DIELECTRICS seen, in the a,b plane. Consequently, M can be properly represented by 4 4 matrices. The permeability tensor The covariant optical formalism of stratified anisotropic me- will play no further role.8 The solution of Eq. 3.1 relates dia developed by Borzdov, Barskovskii, and Lavrukovich11 Ht and q E to each other at the lower and upper surfaces of and applied by Andreeva et al.6­9 will be introduced here for the layered medium. In a homogeneous film of thickness d, three reasons. the solution is given by the so-called integral propagation 1 Approximations made by Andreeva et al. are based on matrix L exp(ikdM), by the matrix exponential of the dif- the assumption that the square of the scattering angle is of ferential propagation matrix. For an n-layer system, the total the order of the susceptibility tensor elements. The border- integral propagation matrix is the product of the individual line of the Andreeva approximation will be specified here. integral propagation matrices L(l) of layer l, thus 2 The Blume-Kistner theory2 will be derived from the covariant optical formalism. L L n . . . L 2 L 1 . 3.3 3 In a practical application of the Blume-Kistner theory one calculates the exponentials of 2 2 complex matrices. The covariant optics uses 4 4 matrices in the exponentials The expression of the planar reflectivity, r defined by leading to rather time-consuming calculations. It will be Hr 0 r 0 t rHt , where Ht and Ht are the tangential amplitudes of shown that in a suitably chosen basis, the 4 4 matrices the reflected and incident waves, respectively, writes as 53 COHERENT FORWARD-SCATTERING AMPLITUDE IN . . . 6161 K (1,2,3,4), viz. K (2,3,4,1), shall also be used. The r t, I 1 2 L I2 r t,I2 L I2 0 . 3.4 differential and integral propagation matrices M and L will be denoted by M and L , respectively, in the K system. Here I2 is the 2 2 unity matrix and the 0, r, and t tensors are the impedance tensors for the incident, specularly reflected, and transmitted waves, respectively, defined by the B. The forward scattering case: The Blume-Kistner equation 0,r,tH0,r,t The 3.2 differential propagation matrix for the case of t q E0,r,t 3.5 normal incidence2 in the K system has the following simple equation. Since the tensors act in the plane perpendicular form: to q , they can be represented by 2 2 matrices.11 The elements of the reflectivity matrix R are geometri- cally related to the elements of the planar reflectivity, r, i.e., 0 0 22 21 0 0 M 12 11 . 3.7 R 1 0 0 0 r22 , R r21sin 1 , 0 1 0 0 R r12sin , R r11 , 3.6 where and are polarizations corresponding to E perpen- The transmittivity may be expressed in terms of the inte- dicular and parallel to the plane of incidence, respectively. gral propagation matrix, L exp(ikdM) as11 For numerical calculations we shall choose different appro- priate coordinate systems. The laboratory system S will be 1 defined so that the x, y, and z axes are parallel to a, b, and t 2 I2,I2 L 1 I2 3.8 I2 q , respectively. The field components in Eq. 3.1 define a natural permutation K (1,2,3,4) basis of the four- defined by Ht 0 t tHt can be explicitly elaborated to obtain the component field vectors Hx ,Hy ,(q E)x ,(q E)y with re- Blume-Kistner formulas.2 Indeed, using the identity which spect to the S system. A convenient permutation of can easily be proved by expanding the exponentials exp 02 B cosh BC 1/2 B CB 1/2sinh CB 1/2 , 3.9 C 02 C BC 1/2sinh BC 1/2 cosh CB 1/2 with 02 being the 2 2 zero matrix, the matrix exponential 2 N of M can be expressed in terms of a 2 2 submatrix npp pp k2 fk,p k,p , 3.13 B ( 22 21 ) of Eq. 3.7 so that the 3.8 transmittivity 21 11 where is the Kronecker symbol and p,p , . 1 C. The grazing incidence case: The Andreeva approximation t cosh ikdB1/2 2sinh ikdB1/2 B1/2 B 1/2 1. 1. The differential propagation matrix 3.10 In order to see which elements in Eq. 3.6 are of the same order of magnitude, we eliminate the explicit dependence Making use of the smallness of the susceptibility one can of R by applying a linear transformation T in the K sys- easily write B1/2 B 1/2 2I2 and the transmittivity: tem of the form: t exp ikdB1/2 . 3.11 sin 1 0 0 0 0 sin 1 0 0 In order to compare the result 3.11 with those of Blume and T . 3.14 0 0 1 0 Kistner,2 now we define the transmission coefficient for the electric field by Et t 0 0 0 1 EE0. Expressing H with E we obtain the Blume-Kistner equation2 It can be easily seen that only the integral propagation matrix t L l TL l T 1 3.15 E q tq exp ikdn , 3.12 depends on , and the reflectivity matrix depends on the where n 1 . Comparing Eq. 2.9 with Eq. 3.12 we elements of L(l ) only. The transform of the differential propa obtain the Lax formula1 as generalized by Blume and gation matrix M(l ) TM(l )T 1 of layer l is obtained with the Kistner2 same similarity transformation: 6162 L. DEA´K, L. BOTTYA´N, D. L. NAGY, AND H. SPIERING 53 0 0 1 0 0 0 0 1 1 0 0 l 11 l 13 0 0 0 l 12 0 0 0 M l 31 l 33 l 32 0 0 l sin , 3.16 1 0 0 0 sin 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 l 22sin l 21 l 23 (l)ij (i,j 1,2,3) being the matrix elements of the susceptibility tensor of layer l. The three matrices in Eq. 3.16 are of the order of magnitude of , / , and , respectively. Without a rigorous explanation, Andreeva et al.9 intuitively drop the third term containing only those elements of the tensor which are not related to the forward scattering amplitude. This approximation is obviously valid if is small compared to and / . Since typically 10 5 the interval for in order for the third term to remain below 1% of the first two is 10 3 10 2 which is, indeed, the typical region of a grazing incidence experiment. From the 3.16 form of M it is clearly seen what conditions have to be fulfilled for the Andreeva approximation to be valid. Note that there is not only an upper but also a lower bound for .Returning to the covariant notation, the differential propagation matrix M(l) of layer l with a 1% accuracy in the K system is of the form M l a­ l q b a I b b 1 a­ l a , 3.17 I a a 1 q * l q q * l a a b which is identical to the form suggested by Andreeva et al.9 In the grazing incidence case the a and b vectors are approximate unit vectors a b cos 1. This approximation is equivalent to neglecting terms of the order of sin2 as compared to 1. In this limit k b . We can choose the two polarization vectors so that u 1 a and u 2 q . Transforming the matrix given in the polarization vector system P by Eq. 2.9 into the S system and substituting into Eq. 3.17 the differential propagation matrix can be expressed in terms of the forward scattering amplitude: 0 01 0 4 N l 4 N k, k, 0 0 l k, k, k2 f l k2 f l sin2 M l , 3.18 4 N l 4 N k, k, l k, k, k2 f l sin2 0 0 k2 f l 0 1 0 0 which, as we shall see, is a particularly suitable form for where x(l) ikd(l)sin , with d(l) being the thickness of layer numerical calculations (N 2 (l) is the number of resonant nuclei l. (l) 4 N(l)d(l)f(l) is proportional to the forward scat- per unit volume and f k,p k,p tering amplitude f (l) is the coherent forward scat- (l) . tering amplitude in layer l . Starting with Eq. 3.18 a time- To evaluate the integral propagation matrix 3.19 one effective numerical algorithm is derived in the following may notice that the differential propagation matrix is block- subsection. antidiagonal. We show that the problem, like in the Blume- Kistner case in Sec. III B reduces to the calculation of a 2. Numerical calculations single 2 2 matrix exponential of a small quantity. Indeed, using again the identity 3.9 , with B(l) x(l)I2 (1/x(l)) (l) and C The matrix 3.18 contains small quantities of the order of (l) x(l)I2 the integral propagation matrix of Eq. 3.19 with F sin2 10 4 and the much larger number unity. The calcu- (l) (x(l)B(l))1/2 is given by lation of the exponential of M(l) to a sufficient accuracy is 1 rather time consuming. For each energy channel the expo- coshF l F l sinhF l nential of M should be calculated thus typically 210 times per L x l l . 3.20 Mo¨ssbauer spectrum. x 1 l F l sinhF l coshF l The corresponding transformed integral propagation ma- trix in the K system cf. Eq. 3.16 can be written as Equation 3.20 is well suited for numerical calculations since it contains only the 2 2 matrix exponential expF(l) . By the present method the large matrix elements are sepa- L l exp ikd l TM l T 1 rated from the small ones. If the argument of the exponential 1 is of the order of 10 4 the expy (1 y/2n)2n approximation 0 exp 2 x l I2 x l gives a sufficient accuracy with n as small as 2. l , 3.19 Using Eqs. 3.4 , 3.6 , 3.3 , and 3.20 the reflectivity in x l I2 02 the , basis is given by 53 COHERENT FORWARD-SCATTERING AMPLITUDE IN . . . 6163 R L 11 L 12 L 21 L 22 1 L 11 L 12 L 21 L 22 , 3.21 where the L APPENDIX: DERIVATION OF THE REFLECTIVITY ab s (a,b 1,2) are 2 2 submatrices of the integral propagation matrix L see Appendix . Since Eq. FORMULA 3.21 gives the reflected amplitude rather than the reflected The integral propagation matrix L of Eq. 3.4 is ex- intensity it is equally applicable in calculating spectra in con- pressed by L of Eq. 3.15 : ventional source i.e., energy domain and in synchrotron radiation i.e., time domain experiments. Using the present method a computer program was developed capable of fitting r t, I 1 2 V 1T 1L TV I2 experimental spectra both in the energy and in the time do- r main. t,I 2 V 1T 1L TV I2 , A1 IV. SUMMARY 0 The goal of the present paper was twofold. First, to establish where T is given in Eq. 3.14 ; t,r,0 are the impedance ten- a working theory of Mo¨ssbauer spectroscopy by specularly sors for the transmitted, reflected, and incident radiation as reflected rays for both the conventional source and for the defined in Eq. 3.5 . Assuming vacuum on both sides of the synchrotron radiation experiment, and second, deriving the stratified sample which - by allowing for a thick enough corresponding formulas in a computationally tractable form. substrate - imposes no further restriction the 's are of the Starting from the nucleon current density expression of the form11 susceptibility tensor of Kagan and Afanas'ev10 we use a co- variant formalism11 of anisotropic optics. Both in the trans- mission and in the grazing incidence geometry the suscepti- 0 t r sin 0 . A2 0 sin 1 bility is expressed in terms of the coherent forward scattering amplitude. The Blume-Kistner formula2 of the perpendicular V is the matrix of the (1,2,3,4) (2,3,4,1), (K K ) transmittivity and the Andreeva approximation9 for the graz- transformation of the form ing incidence reflectivity are rederived in a rigorous manner. In the grazing incidence case a concise 2 2 block-matrix exponential expression for the differential propagation ma- 0 1 0 0 0 0 1 0 trix is obtained for transitions of arbitrary multipolarity and V . A3 in a computationally convenient way. This latter allows for 0 0 0 1 fast numerical calculation and practical fitting of Mo¨ssbauer 1 0 0 0 spectra both in energy and in time domain. Performing the calculations in Eq. A1 with the above ma- ACKNOWLEDGMENTS trices the planar reflectivity Fruitful discussions with Dr. M. A. Andreeva are gratefully 1 acknowledged. This work was partly supported by the r 0 sin 1 L L L L 1 1 0 11 12 21 22 PHARE ACCORD Program under Contract No. H-9112- 0522 and by the Hungarian Scientific Research Fund OTKA under Contract Nos. 1809 and T016667. The au- L 11 L 12 L 21 L 22 0 sin 1 . A4 1 0 thors also thank the partial support by the Deutsche For- schunsgemeinschaft and the Hungarian Academy of Sciences From Eqs. A4 and 3.6 we obtain the 3.21 reflectivity in frames of a bilateral project. formula. *Electronic address: deak@rmki.kfki.hu 7 M. A. Andreeva, S. M. Irkaev, and V. G. Semenov, Sov. Phys. 1 M. Lax, Rev. Mod. Phys. 23, 287 1951 . JETP 78, 965 1994 . 2 M. Blume and O. C. Kistner, Phys. Rev. 171, 417 1968 . 8 M. A. Andreeva and R. N. Kuz'min, Messbauerovskaya Gamma- 3 J. C. Forst, B. C. C. Cowie, S. N. Chapman, and J. F. Marshall, Optika Moscow University, Moscow, 1982 . Appl. Phys. Lett. 47, 581 1985 . 9 M. A. Andreeva and K. Rosete, Poverkhnost' 9, 145 1986 ; Vest- 4 D. L. Nagy and V. V. 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