- Email: [email protected]
Analytical expression for the energy distribution of Mössbauer transitions in the presence of mixed interactions
Nuclear Instruments and Methods in Physics Research B 171 (2000) 515±527
www.elsevier.nl/locate/nimb
ssbauer Analytical expression for the energy distribution of Mo transitions in the presence of mixed interactions Krzysztof Szyma nski
*
Institute of Experimental Physics, University of Biaøystok, Lipowa 41, 15-421 Biaøystok, Poland Received 30 May 2000
Abstract Based on the analytical results for intensity tensor components, expressions for energy distribution of M ossbauer 3/ 2±1/2 transitions in the case of a uniform magnetic ®eld and randomly oriented electric ®eld gradient (EFG) are presented. The expression obtained contains elliptic integrals. The logarithmic singularities appearing in the integrals are integrable and their origin is discussed. Next, the shape of the 57 Fe M ossbauer spectrum in standard, as well as recently developed, circularly-polarized M ossbauer spectroscopy was derived. The general results obtained agree with previous works in the limiting cases, e.g., with the analytical shape obtained for asymmetry parameter set to zero, and the results obtained by perturbation methods. In the limit of local axial symmetry of the EFG (g 0), expressions simplify to an algebraic form and the singularities are of x ÿ x0 ÿ1=2 type. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In M ossbauer spectroscopy, problems of mixed interactions, e.g., presence of magnetic ®eld and electric ®eld gradient (EFG) acting on the nucleus in the I 3=2 state, appear very often. In the early days of the technique, methods based on the numerical solution of the spin Hamiltonian [1±8] were derived. The transition probabilities were calculated by the use of explicit Hamiltonian eigenstates [9±14]. An important step was made
*
Tel.: +48-243-833518; fax: +48-85-457223. E-mail address: [email protected] (K. SzymanÂski).
when analytical solutions for eigenenergies of the Hamiltonian were obtained [15,16]. When hyper®ne magnetic ®eld vector directions exhibit certain distributions with respect to the principal axes system (PAS) of the EFG tensor, one has to perform integration over all possible transitions. Consequently, instead of sharp transitions, one arrives at a continuous distribution of transitions forming bands and in¯uencing absorption line shape. Many numerical and analytical treatments of this problem involved perturbation methods [17±23]. An analytical expression for the shape of the M ossbauer spectrum was obtained for a special case, namely an axially symmetric EFG [24]. The advantage of analytical results is unquestioned when one is analysing
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 0 ) 0 0 3 1 1 - 6
516
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
global properties of the spectrum or checking the precision of the numerical procedures, particularly those in which approximations are introduced for time saving purposes. Recently, a monochromatic, circularly-polarized M ossbauer source (MCPMS) technique was developed [25±28]. Unfortunately, widely distributed commercial packages do not have suitable options for polarized M ossbauer spectroscopy. This stimulated us to the reanalysis of known methods in order to make use of them in the MCPMS technique [29]. Interestingly, we have found that, using the so-called intensity tensor formalism [30±35], one is able to obtain analytical expressions for transition probabilities without explicit calculation of the eigenfunctions of the Hamiltonian [36,37]. This approach is continued in the present paper. How to get an analytical shape for the energy distribution in the case of a uniform magnetic ®eld and randomly oriented EFG is shown. Singular points which appear in the calculations essentially present no problem. The use of the described method is illustrated on several examples.
For the excited Ie 3=2 levels the total Hamiltonian is conveniently reduced to 3=2
3=2
3=2 Hel Htot Hmag
g3=2 lN B 3=2 H : 2
This Hamiltonian H3=2 in the PAS of the EFG tensor and in the representation of Iz eigenfunctions has a form as given in [16]
2:4 The parameter R is proportional to the Vzz component of the EFG: R eQVzz =2g3=2 lN B. Let us abbreviate eigenvalues of the excited Ie 3=2 Hamiltonian by Ea and introduce dimensionless energies ka 2Ea =g3=2 lN B. Energies ka are roots of the secular equation originating from (2.4) k4 pk2 qk r 0;
2. Intensity tensor for a single site
HelI
eQVzz 2 g 3I z ÿ I 2 I 2 I 2ÿ ; 4I 2I ÿ 1 2
2:5
where
In the following, the Hamiltonian of the ground and excited states is de®ned as in [16,37]; we shall also use abbreviations appearing in the cited papers. The Hamiltonian of the nuclear system with I spin I is the sum of the magnetic Hamiltonian Hmag I and the electric quadrupole Hamiltonian Hel , I ÿgI lN I ~ B; Hmag
2:3
2:1 2:2
where gI is a nuclear g-factor and lN denotes the nuclear magneton, Q is the nuclear quadrupole moment and g is the so-called asymmetry parameter. Cartesian components of the EFG tensor are Vij ÿo2 V =oxi oxj , where V denotes an electric potential at the nucleus. The principal axis system of the EFG tensor can be chosen so that jVzz j P jVyy j P jVxx j and then g Vxx ÿ Vyy =Vzz .
p ÿ10 ÿ 2R
2
g2 1 ; 3
^ m ~; q ÿ16~ mU 1 2 ^2 m ~: mU r p 4 ÿ 16~ 4
2:6
~ , de®ned by spherical angles h The unit vector m and u, is parallel to the direction of the hyper®ne ^ is a second-order tensor whose magnetic ®eld ~ B. U dimensionless components are proportional to the EFG. In its PAS it reads 2 3 1ÿg 0 0 R ^ ÿ 4 0 1 g 0 5: 2:7 U 2 0 0 ÿ2 An analytic form for ka can be found in [15,16]. The transition probabilities, or line intensities, can be expressed in an elegant way within the framework of the intensity tensor formalism. For
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
example, in the thin absorber approximation, the area under an absorption line is equal to [38] Aabf
Cp 1 s ~ ab ~ Tr ^Iab ÿ~ fs t c ^Iab ~ c ÿ 2fG c; 2 2
2:8
3 40k2a ÿ 4qka p 4 p 16 ÿ 4r ; 8 16bwa 2:9
where wa 4k3a 2pka q:
2:10
Parameter wa is equal to the derivative of the secular equation (2.5). Consequently, parameter wa non-equal to zero is equivalent to the condition that excited levels are non-degenerate. The symmetric part of the intensity tensor multiplied right~ is equal to and left-handed by vector m s ~ ~ ^Iab m m
1 b 64k2a 8ka q 128wa 8 p 4 p 16 ÿ q2 ÿ 32r
2:12 We turn the attention of the reader to the fact that the unit vector ~ c in Eq. (2.8) may have any orientation with respect to the PAS of the tensor ^Iab , the latter depending on the spherical angles h and u which de®ne direction of the hyper®ne ~ . In Eqs. (2.11) and (2.12), the magnetic ®eld m s ~ ~ same angles h and u de®ne tensors ^Iab ; G ab and m as well.
3. Randomly oriented EFG 3.1. Average intensity tensor Let us now consider the special, important case of randomly-oriented EFG and well-de®ned direction of the hyper®ne ®eld. The situation is realised experimentally when a powdered sample with negligible local anisotropy is exposed to a uniform external magnetic ®eld. We start with an explicit form of the symmetric part of the intensity tensor which is a function of ^ m ^ and the vector m ~ . ~ : ^Iab ^Iab U; the tensor U Let R R X be a real rotation matrix de®ned by Euler angles X. We are interested in the angular ^ R; m ~ iX over the average of the value h^Iab RT U angles X. From the symmetry arguments it follows that, in the PAS of the averaged intensity tensor, ~ is directed along one of the axes. We may vector m ~ is diassume, without loss of generality, that m rected along the z-axis. Further on, diagonal components xx and yy are equal to each other. Thus, the averaged tensor in the PAS can be written as 2
64k3a 4k2a q 32ka p ÿ 8 2q p 28:
1 ~ ab m ~ G b 2k2a p 16 ÿ8ka q 64wa ÿ 80k2a 8ka q ÿ 2p2 ÿ 40p 8r ÿ 128:
where t is the eective thickness of the absorber, fs the recoilless fraction of the source and C is the natural width. Index f 1 in (2.8) describes two opposite circular polarisations of the photon. ~ c is a unit vector parallel to the direction of the photon. ^Iab is the so-called intensity tensor of the transition s ~ ab are symmetric between states a and b. ^Iab and G ^ and antisymmetric parts of Iab . Indices a run over four excited Ie 3=2 states while b 1; ÿ1 correspond to the ground Ig 1=2 states with higher and lower energy, respectively. The results important for further evaluation are listed below (the reader is referred for details to [36]). In the case of a single site, the trace of the intensity tensor is equal to Tr ^Iab
517
2:11
~ ab and The product of the antisymmetric part G ~ is equal to vector m
Iab?
6 s ^ R; m ~ iX 6 h^Iab RT U 4 0 0
0 Iab? 0
0
3
7 0 7 5: Iabk 3:1:1
518
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
The averaging over three Euler angles results in 1 s ~ ^Iab ~ ihu ; m Iab? hTr ^Iab ÿ m 2 s ~ ihu ; Iabk h~ m ^Iab m
3:1:2
s ~ as well as the tensor ^Iab where the unit vector m depend on the same angles h and u, i.e., Tr ^Iab and s ~ are given by Eqs. (2.9) and (2.11), re~ ^Iab m m spectively. The averaging in (3.1.2) is performed in a standard way: Z 1 sin h dh du: hihu 4p sphere
In a similar way we can calculate the antisymmetric part of the intensity tensor for our problem. The result in the same PAS system is 2 3 0 ~ ab RT U ^ R;~ 5: 0 ez iX 4 3:1:3 hG ~ ~ ihu hG ab m ~ ab m ~ , is described by Eq. (2.12). The product, G Unit vector ~ ez in (3.1.3) belongs to the basis of the PAS of our averaged intensity tensor (3.1.1) and is parallel to the direction of the uniform magnetic ®eld. The reader should not be confused by the ~ in (3.1.2) and (3.1.3), notation of the unit vector m which is only used for performing the average and does not indicate the direction of the uniform magnetic ®eld. Explicit forms of Eqs. (2.9)±(2.12) serve for calculation of the energy distribution of the intensity tensor components. In the next paragraph we show how to get an explicit expression for the energy distribution resulting from averaging (3.1.2) and (3.1.3). 3.2. Energy distribution The energy distributions of the intensity tensor components are important quantities since they are related to the shape of the absorption lines in the spectrum. We are interested in particular in the spectral distribution dhf i=dk connected with each component of the intensity tensor f. By f we mean any component of Eq. (3.1.2) or (3.1.3). Having explicit expressions f k; h; u one can formally write the expression for the energy distribution:
f k; h; u f k h; u; h; u Z f t; h; ud t ÿ k h; u dt;
3:2:1
where d is the Dirac d-function. Thus, Z hf k; h; uihu hf t; h; ud t ÿ k h; uihu dt 3:2:2 and, because Z hf k; h; uihu
dhf i dk; dk
3:2:3
one gets dhf i k0 hf k0 ; h; ud k0 ÿ k h; uihu : dk
3:2:4
Eq. (3.2.4), although written in a short form, needs further evaluation since it involves integration over angles h and u over isoenergetic energy line k h; u k0 . It is shown below that the methods involved in calculating electronic density of states in metals [39] can be applied for ®nding an isoenergetic set C k and integrating the function f over it. We will show next that the energy distribution of all components ((3.1.2) and (3.1.3)) can be expressed by elliptic integrals. Since parameter k is a function of (h,u), which means that f depends, in fact, on the spherical angles (h,u) only, the function f k; h; u can be considered as de®ned on a three-dimensional unit sphere S. The averages (3.2.4) can be considered as integrals of f over the surface and can be calculated in the following way. Consider an element of the surface, dS, located between two lines C k and C k dk, see Fig. 1. C k0 is de®ned as a line on which k has constant value k0 . The width of the element dS dl dh, where element dl is along the ~ so C k contour, is dh dk=jrkj; f k; h; u dS f k; h; udh dl
f k; h; u dk dl: ~ jrkj
3:2:5
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
519
radius while the right-hand side a second-order surface (ellipsoid, hyperboloid, etc.). The intersection of the two surfaces corresponds to the line C k which enters integral (3.2.7) explicitly, see Fig. 2. It follows from Eqs. (2.5) and (3.2.9) that ^ ÿ ^1 m ~ 32 A ~; rk wH and our integral (3.2.7) takes the form Z dhf i 1 jH j jwjf dl p : dk 4p 32 C k m ^2 m ~A ~ÿ1
Fig. 1. A unit sphere S and the element dS located between curves C k and C k dk, see text.
Thus, hf k; h; uihu
1 4p
Z
Z dk
C k
3:2:11
3:2:12
The geometrical form of (3.2.8) is helpful in the determination of the domain of integration. From (3.2.8), the following relation between h and u for points on the C curves can be obtained: cos2 h
bt ; ct
3:2:13
where
f k; h; u dl : ~ jrkj 3:2:6
Comparing Eqs. (3.2.6) and (3.2.2) we conclude that Z dhf i 1 f k; h; u dl : 3:2:7 ~ dk 4p C k jrkj
ÿ2 Axx Ayy ; Axx ÿ Ayy ÿ2Azz Axx Ayy ; c Axx ÿ Ayy b
3:2:14
t cos2u:
It is interesting that, in the case of tensor components (2.9)±(2.12), the integral (3.2.7) can be transformed to a more familiar form. To do that, let us return to the secular equation (2.5) and rewrite it in the equivalent form ^ m ~m ~m ~A ~; m
3:2:8
where ^ H kU ^ U ^ 2 A
3:2:9
and factor H is a rational function of k, H
16 4
2
k pk p=2 2
2
:
3:2:10
Eq. (3.2.8) has a simple geometrical interpretation: the left part of (3.2.8) represents a sphere of unit
Fig. 2. Isoenergetic curve C k resulting from the intersection of a second-order surface and a unit sphere.
520
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
Parameters b and c are important and we will be using them frequently in what follows. The explicit forms of b and c can be obtained from (3.2.9): b c
4
k pk 8R k R p=2 8 ; 8gR ÿk R 6k R 3 ÿ g2 : 2g ÿk R
3:2:15
Z
1
ÿ1
jwjf dt p : b t c t 1 ÿ t2
1 ; ct
3:2:17
1 ; r r0 r1 ct where
K2 ÿ 16 k2 8R2 8Rg b ÿ c ÿ 16R; 4 ÿk R
Rk 9 ÿ g2 K2 ÿ 16 k 2R 8g ÿk R
2
ÿR 9 ÿ g2 K2 ÿ 16 k 2R 8g ÿk R
K 4 2k2 p:
ÿkq1 :
2
ÿ8R 3 gc b ÿ c 3:2:18
3:2:20
Thus, any intensity tensor component f can be expressed as a ®nite series: f
2 1X f n n; w n0 c t
3:2:21
in which coecients f n do not depend on t (f n components depend explicitly only on k, R and g). One easily notices that the value of the parameter |w|, present in the numerator of (3.2.16), is irrelevant. Because w is present in each denominator of the f series (3.2.21), it is only the sign of (w) which matters. On the other hand, w 6 0 means only that the excited levels are non-degenerate. The sign of w is thus constant in each energy band, or in the domain of integration (3.2.16), and can be taken outside the integral. How to determine this sign will be discussed in paragraph 6. Now we may insert tensor component (3.2.21) into (3.2.16) and get the ®nal form of the energy distribution. It should be remembered that Eq. (3.2.16) was derived for a C curve on the unit sphere S, with the loop encircling the z-axis (condition c=b > 1 and ÿ1 < b < 1), see Fig. 1. In a general case, C curve may run around the x- or yaxis; so we have to describe these cases too. The ®nal result is 2 dhf i sign w X Jn b; cf n : dk 4gjR k ÿ Rj n0
2
q1
r1
In Eqs. (3.2.18) and (3.2.19)
The function f in (3.2.16) is one of the types: s ~ ~ or m ~ ^Iab m ~ , see Eqs. (2.9)±(2.12). Tr ^Iab , G ab m The numerators of the functions of interest are linear combinations of 1, q, r and q2 . Parameters q, r and q2 taken at points which belong to the C curve can be expressed as linear combinations of 1, ÿ1 ÿ2 c t and c t because
q0
2
3:2:19
3:2:16
q q0 q1
ÿR K2 ÿ 16k k 8R p 4 4 ÿk R 4 2
If coecients b and c satisfy inequalities: c=b > 1 and ÿ1 < b < 1, the discussed energy distribution will be represented as an integral over two loops around the z-axis (the intersection of the hyperboloid and the sphere results in two contours, located at positive as well as negative values of z, shown as an example in Fig. 2): dhf i 1 1 dk 4p 4gjR k ÿ Rj
r0
ÿk k3 pk q0 ;
2
2
and
3:2:22
The functions Jn are given by integrals, in which the limits of integration result from positions of the C curve on the sphere:
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
8R1 > < Rÿb jn b; c; t dt; ÿb Jn b; c j b; c; t dt; ÿ1 n > :R1 j b; c; t dt; ÿ1 n
where
c; b 2 sx ; c; b 2 sy ; c; b 2 sz ;
where 1 p : 4p c t b t c t 1 ÿ t2 n
3:2:24 Three sets, sx , sy and sz , which de®ne the domain of Jn , are given by the inequalities c; b 2 sx () c ÿ b sign 1 ÿ b2 1 > 0; 2
c; b 2 sy () c ÿ b sign 1 ÿ b 1 < 0; c; b 2 sz () b c ÿ b sign 1 ÿ b2 ÿ 1 < 0: 3:2:25 Function Jn in (3.2.23) can be expressed by elliptic integrals, see Appendix A. 3.3. Spectral density of the intensity tensor With the help of (3.2.17) and (3.2.21), the components of the intensity tensor (3.1.1) can be ÿ1 expressed through powers of c t . The perpendicular component is thus equal to 1 1 s s ~ ~ ^Iab m I? Tr ^Iab ÿ m 2 2 2 1X 1 I? n n; w n0 c t
3:3:1
where q21 b ; 256 ÿq1 K ÿ b q0 ÿ 20k ; I? 1 128 q0 ÿ 32k 2K ÿ b q0 ÿ 8k : I? 0 256 I? 2
3:3:2
Similar calculations lead to ~ ^Iab m ~ Ik m s
2 1X 1 Ik ; w n0 n c tn
ÿq21 b ; 128 q1 K 24 ÿ b q0 ÿ 20k; Ik 1 64 b 8K K 12 ÿ 256k2 ÿ q0 q0 ÿ 40k Ik 0 128 1 K 24 q0 16k ÿ 576k: 64 3:3:4 Ik 2
3:2:23
jn b; c; t
521
3:3:3
For the vector components we have 1 1X 1 ~ ab m ~ ab m ~ ~ n G ; G w n0 c tn
3:3:5
1 q1 b K 12; 64 1 ~ ab m ~ 0 K 12 ÿ2K b q0 ÿ 8k: G 64 3:3:6
~ ab m ~ 1 G
For completeness of the results, we also present expressions for the trace of ^Iab , Tr ^Iab s
1 1X 1 s Tr ^Iab n n; w n0 c t
3:3:7
3 s Tr ^Iab 1 q1 ; 8 3 1 s Tr ^Iab 0 4k3 2pk q0 bK K 12: 8 16 3:3:8 We may now present some analytical results for the spectral distribution of the intensity tensor components. As an example, consider the situation of the isotropic intensity tensor. In this case, the direction of the hyper®ne magnetic ®eld is randomly distributed with respect to the PAS of the EFG tensor, and also the PAS of the latter are randomly distributed in real space. In the considered case, the trace of the averaged intensity tensor is equal to
522
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
dhTr ^Iab i 3 sign w q1 J1 b; c dk 32gjR k ÿ Rj sign w 6 4k3 2pk q0 64gjR k ÿ Rj s
bK K 12J0 b; c:
3:3:9
We remind the reader that parameters q0 and q1 were de®ned in (3.2.18), K in (3.2.20), p in (2.6), b,c in (3.2.15) and w in (2.10); so Eq. (3.3.9) is a function of three parameters: k, R and g. 4. Application: shapes of spectra Taking the sum of (3.3.9) over all transitions a, b and using (2.8), we get the shape of the energy distribution corresponding to the M ossbauer spectrum, shown in Fig. 3(a). The curves are not convoluted with the Lorentzian distribution representing natural broadening. In a similar way, with help of Eqs. (2.8), (3.3.1) and (3.3.6), we construct spectra corresponding to an external ®eld parallel or perpendicular to the direction of observation ~ c, and also other spectra which can be measured using the MCPMS technique, see Figs. 3(b) and (c). 5. Special case: local axial symmetry of the EFG In the case of local axial symmetry of the EFG, g 0, the expressions can be greatly reduced. Eq. (3.2.22) becomes dhf i jwjf q ; dk 2 2 2 4 3R 2k R k ÿ R2 ÿ 3 ÿ 4 ÿk R 5:1 where w
2 3k 2Rk3 ÿ 4 5 R2 k Rk ÿ 2 9 ÿ 2R2 R4 : 2k R 5:2
Function f in Eq. (5.1) is given by any component of the intensity tensor. Explicit expressions for the components of interest are as follows:
Fig. 3. Calculated energy distribution corresponding to the absorption spectrum for R 2:2 and g 0:6. (a) Isotropic intensity tensor, (b) hyper®ne magnetic ®eld perpendicular to the observation direction, (c) hyper®ne magnetic ®eld parallel to the observation direction, (d) hyper®ne magnetic ®eld parallel to the observation direction; measurement with circularly polarized source, (e) hyper®ne magnetic ®eld antiparallel to the observation direction; measurement with circularly polarized source. Horizontal scale corresponds to the dimensionless energy k (Doppler velocity v g3=2 lN Bc=2Ec k).
2
d 3 k2 ÿ R2 ÿ 9 s hTr ^Iab i b ; dk 8 4w
5:3
d d s ~i Ik h~ m ^Iab m dk dk 2 k2 ÿ 2bk ÿ R2 2bR ÿ 3 k ÿ R 3b 3
2
k ÿ R ÿ 3b k R b =ÿ32wb 2k R ; 5:4
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
d d 1 ^ s 1 s ~i ~ ^Iab I? h Tr Iab ÿ 2 m m dk dk 2 k2 2bk ÿ R2 ÿ 2bR ÿ 3 k ÿ R 3b 3
k R ÿ b 4k ÿ 4kR2 12k ÿ w= 128wb 2k R; 5:5 d ~ ~ i k2 2bk ÿ R2 ÿ 2bR ÿ 3 hG ab m dk k ÿ R 3b k R ÿ b k2 ÿ R2 3=ÿ16wb 2k R: 5:6 The spectra constructed with the help of Eqs. (2.8), (5.3)±(5.6) are shown in Fig. 4.
523
6. Discussion The most interesting feature seen in Figs. 3 and 4 is the appearence of singularities at special points on the energy (velocity) scale. Their origin is similar to the one of van Hove's singularities. Let us discuss this problem in greater detail. In Fig. 5, we have drawn an angular dependence of the eigengenvalue ki of the secular equation (2.5). In the special directions, along axes of the PAS system of the EFG, there are extreme values of energy. Since the secular equation is of 4th order, we have 4 3 12 special points on the energy scale. The special points can be obtained easily from the secular equation (2.5). Four minima and four maxima, which de®ne the borders of the energy bands, occur at ki , p k1;2 1 D 4R; p k3;4 1 ÿ D 4R; 6:1 p k5;6 1 D 2R 1 g; p k7;8 1 ÿ D 2R 1 g; where D ÿ p=2 1:
6:2
The bands are shown in Fig. 6 by shaded areas. There are also four solutions of the secular equation for saddle points, p ks1 ÿ1 ÿ D 2R 1 ÿ g; p ks2 1 ÿ D ÿ 2R 1 ÿ g; 6:3 p ks3 ÿ1 D 2R 1 ÿ g; p ks4 1 D ÿ 2R 1 ÿ g; shown in Fig. 6 by dashed lines located on the shaded areas. It follows from Fig. 6 that except for six values of Ri ,
Fig. 4. Same as Fig. 3 for axially symmetric EFG, g 0 and R 2:5.
3 R1;2 p ; 9 ÿ g2 3 R3;4 p ; 2g 3 g 6 ; R5;6 3g
6:4
524
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
Fig. 6. Minima and maxima of solutions of the secular equation for g 0:6, shown by solid lines in the (k,R) plane. The saddle points are shown by dashed lines. Energy bands of the excited levels are shown by shaded areas. The sign of parameter w for each energy band is shown by + or ).
Fig. 5. Angular dependence of the energy k for (a) the two highest and (b) the two lowest energy levels for R 2:2 and g 0:6 represented in the PAS of the EFG. Any vector connecting the origin of the coordinate system with a point on the surface is given by spherical angles h and u (which enter the secular equation (2.5)) and has length k. Maximum, minimum and saddle points of the energy k with respect to h and u are located at intersections of the surface and the axes.
there are four energy bands separated by three energy gaps in the level structure of the excited states. Fig. 6 conveniently presents the behaviour of the sign of the determinant w, mentioned in Section 3.2: the sign of w is constant within each energy.
Let us discuss the type of divergence of function (3.2.22) at the saddle points. Fig. 3 suggests that f ! 1. Indeed, when k is in the vicinity of the saddle point ksi , see Eq. (6.3), the argument v is close to 1, see (A.3), and the elliptic integral K v, see Eq. (A.8), diverges like log jk ÿ ksi j. Thus, there are eight points in the spectrum with logarithmic divergence. In the case of g 0, it follows from Eq. (5.1) that at points ki for which the denominator is zero, the function dhf i=dk behaves like ÿ1=2 . ki ÿ k The physical origin of the singularities in Figs. 3 and 4 can be easily understood. During integration over angles h and u, see Eq. (3.1.1), the energy is varied unless we are at the points where energy does not depend locally on angles. This occurs at the special points (6.1)±(6.3). In Figs. 3 and 4, we have drawn vertical lines at the positions given by Eqs. (6.1)±(6.3). There are 12 singular points for excited levels. Because there are transitions to two ground levels, this results in 2 12 24 singular points in the distribution corresponding to the absorption spectrum (we turn the attention of the reader to the fact that, for the R value used in the presented examples, the dashed lines are very close to the borders of the energy bands; see the lines for R 2:2 at k ÿ1:8 in Fig. 6. This, in
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
turn, results in very close proximity of singular points in Fig 3: one pair of dashed lines is located at k ÿ3:54 and k ÿ3:52, while the second one is located at k ÿ0:02 and k ÿ0:04). In the case of axial symmetry of the EFG, g 0, shapes analogous to those shown in Fig. 5 possess axial symmetry. Therefore, the dashed line in Fig. 6 would overlap with one of the border lines for each energy band. The ®rst consequence of this is the smaller number of singular points in spectrum 16, see the dashed lines in Fig. 4. The second consequence is a qualitative dierence in the shape of the distribution. In the case of g 0, the point of divergence is always located at the side of the peak corresponding to the energy band, resulting in more asymmetric peaks (see the peak at k ÿ6 in Fig. 4(a)). In the case of lack of axial symmetry of the EFG, the divergence point is located somewhere in the middle of the energy band, resulting in more symmetric peaks (see the peak at k ÿ6 in Fig. 3(a)). We see that the shapes of the absorption peaks may be used in the determination of the asymmetry parameter. In order to compare the discussed shapes with any experiment, one has to convolute the functions with Lorentzian distributions describing the natural broadening. After convolution, most of the details of the rich energy distribution, e.g., narrow integrable peaks, will be hidden under wide resolution. We see that natural broadening prevents direct observation of the shapes shown in Figs. 3 and 4. Assume, however, that we are able to achieve precise deconvolution of the experimental spectrum and to localize singularities on the energy scale. Would it be useful information? Note that the positions of the singular points correspond to the well-de®ned orientation of the hyper®ne ®eld in the PAS of the EFG. Thus, using Eqs. (6.1)± (6.3), we would be able to determine the value of the hyper®ne magnetic ®eld and the asymmetry parameter g as well as the Vzz component of the EFG tensor. Secondly, our analysis has been performed assuming random orientation of the EFG axes. If the orientation deviates to some extent from random distribution, the shape of the spectrum would be dierent from those presented in Fig. 3. However, singular points would be located
525
at exactly the same positions on the energy scale. Thus, one can quantitatively analyse properties of powdered absorbers which have unknown texture. Our expressions (3.2.22) are in agreement with the results obtained by perturbation methods [17,22,23], analytical results for g 0 [24] and numerical results [24]. In summary, the analytical results for energy distribution of averaged intensity tensor components have been obtained. The problem was reduced ®rst to integration over curve C, which appears as an intersection of two second-order surfaces in three-dimensional space. Final results contain complete elliptical integrals (in the special case of g 0, expressions for the intensity tensor reduce to an algebraic form). Having determined the intensity tensor and the explicit form of its spectral density, features of M ossbauer spectra corresponding to typical experimental geometries are discussed. The spectrum, before convolution with the Lorentzian distribution, contains generally 24 singular points. In the vicinity of eight of them, logarithmic divergences are found. The remaining singular points correspond to the borders of energy bands of the excited states. The shapes of the absorption peaks depend strongly on the asymmetry parameter. The positions of all singular points on the energy scale correspond to special orientations of the hyper®ne magnetic ®eld with respect to the PAS of the EFG and may have potential application in the analysis of samples with unknown texture. Acknowledgements The author expresses many thanks Prof. L. Dobrzy nski for a lot of stimulating discussions and critical reading of the manuscript. The work was sponsored by the State Council of Science through grant 2P03B11516. Appendix A The aim is to calculate integral Jn b; c, see (3.2.23). Let us de®ne integral In (x),
3 c ÿ 1 c 12 c ÿ b2
2 b 1 1 2bc ÿ 3c2
Z x In x 4p jn b; c; t dt 0 Z x dt p : n b t c t 1 ÿ t2 c t 0
A:1
Changing the variable to u
1 c 1 t 1 ÿ c 1 ÿ t
A:2
ÿ 1 c ÿ b 3 c2
ÿ2 ÿ b2 3c2
2
and introducing the abbreviation
ÿb ÿ 1 c 1 c ÿ b
3 c2 ÿ 12 c ÿ b 3 1 ÿ c 1 c c ÿ b
4 1 2bc ÿ 3c2
2
ÿ2 c2 ÿ 1
ÿ2 ÿ b 3c ÿ 3bc 3c2
2 1 ÿ b 1 2bc ÿ 3c2
3 1 ÿ c2 1 c c ÿ b2
ÿ2 ÿ b 3c ÿ 3bc 3c2
1ÿb 1 ÿ c c ÿ b
3 1 ÿ c2 1 c c ÿ b
e2 k2
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
e1
526
v
1 b 1 ÿ c ; 1 ÿ b 1 c
A:3
the integral (A.1) can be changed to the canonical form [39]. Next, the integral can be expressed by incomplete elliptic functions E and F of complex arguments: Z u0 du q ; F u0 ; v 0 1 ÿ v sin2 u A:4 Z u0 q 2 1 ÿ v sin u du: E u0 ; v
1 b c ÿ 1 2 c ÿ b
1 cÿb 0 1 2 p b 1 c ÿ 1 2 c ÿ b b 1 c ÿ 1
1 cÿ1 0 1
ÿ1 1ÿc 0 1
2 p 1 ÿ b 1 c s 2 c ÿ b 1 b 1 ÿ c 1 ÿ b 1 c
P v
Let us introduce abbreviations, ppp p Fuv 1 ÿ u 1 ÿ uv uF Arcsin u; v; ppp p Euv 1 ÿ u 1 ÿ uv uE Arcsin u; v;
1 6 b 6 c
ÿ1 6 b < 1 6 c
ÿ1 6 b 6 c < 1
A:5
and de®ne in for n 0; 1 and 2 by recurrence formula, 2
1 ÿ c 1 ÿ x Fuv ; 1c 1 1 ÿ b 1 ÿ x2 i0 ÿ Euv i1 ÿ 1ÿc 1 c b ÿ c 2 1 x b x ; ÿ 1 ÿ c b ÿ c i0
i2 ÿ ÿ
b, c
Table 1 The argument v and coecients P, kn and en of Eq. (A.9)
k0
e0
k1
0
A:6
2 1 2bc ÿ 3c2 b ÿ 3c i1 i0 3 b ÿ c 1 ÿ c2 3 b ÿ c 1 ÿ c2 2 1 ÿ x2 b x : 3 1 ÿ c2 b ÿ c c x
The integral In (x) is proportional to in , in In x p : b x c x 1 ÿ x2
A:7
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
The integrals which enter into (3.2.23) can be obtained by taking (A.7) at appropriate values of arguments x. Next, using transformations rules for complete elliptic integrals [40], we arrive at the result, which depends now on the familiar complete elliptic integrals of the ®rst and second kind of real argument, de®ned as p ;v ; K v F p2 A:8 ;v : E v E 2 The ®nal result for Jn b; c reads Jn b; c
P kn K v en E v; 4p
n 0; 1; 2: A:9
The argument v and coecients P, kn and en are functions of b and c, and are listed in Table 1. The domains for b and c, given in the ®rst column of Table 1, cover only part of the set given by inequalities (3.2.25). In order to get Jn b; c, de®ned for the whole domain, one needs to use an identity which follows directly from (3.2.23), n
Jn ÿb; ÿc ÿ1 Jn b; c:
A:10
References [1] [2] [3] [4] [5]
M.H. Cohen, Phys. Rev. 96 (1954) 1278. L.C. Brown, P.M. Parker, Phys. Rev. 100 (1955) 1764. P.M. Parker, J. Chem. Phys. 24 (1956) 1096. J.R. Gabriel, Moessbauer E. Methodol. 1 (1965) 121. J.R. Gabriel, S.L. Ruby, Nucl. Instr. and Meth. 36 (1965) 23. [6] W. K undig, Nucl. Instr. and Meth. 48 (1967) 219. [7] J.R. Gabriel, D. Olson, Nucl. Instr. and Meth. 70 (1969) 209. [8] E. Kreber, U. Gonser, Nucl. Instr. and Meth. 121 (1974) 17.
527
[9] H.H.F. Wegener, F. Obenshain, Z. Phys. 163 (1961) 17. [10] H. Frauenfelder, D.E. Nagle, R.D. Taylor, D.R.F. Cochran, W.M. Visscher, Phys. Rev. 126 (1962) 1065. [11] J.M. Daniels, J.M. Cotignola, Can. J. Phys. 40 (1962) 1342. [12] D. Barb, S. Constantinescu, D. Tarina, Rev. Roum. Phys. 16 (1971) 263. [13] B.L. Chrisman, Nucl. Phys. A 186 (1972) 264. [14] D. Barb, D. Tarina, Rev. Roum. Phys. 18 (1973) 1209. [15] P.G.L. Williams, G.M. Bancroft, M ossbauer E. Methodol. 7 (1971) 39. [16] L. H aggstr om, Report UUIP-851, Uppsala, 1994. [17] R.L. Collins, J. Chem. Phys. 42 (1965) 1072. [18] G.N. Belozerskii, Yu.P. Khmich, V.N. Gitsovich, Phys. Stat. Sol. B (1977) K125. [19] G. Le Caer, J. Phys. E 12 (1971) 1083. [20] M. Eibsh utz, M.E. Lines, Phys. Rev. B 26 (1982) 2288. [21] M.E. Lines, M. Eibsch utz, Phys. Rev. B 25 (1982) 6042. [22] R.A. Brand, Nucl. Instr. and Meth. B 28 (1987) 417. [23] R.A. Brand, Nucl. Instr. and Meth. B 28 (1987) 398. [24] N. Blaes, H. Fischer, U. Gonser, Nucl. Instr. and Meth. B 9 (1985) 201. [25] K. Szyma nski, L. Dobrzy nski, B. Prus, M.J. Cooper, Nucl. Instr. and Meth. B 119 (1996) 438. [26] K. Szyma nski, Nucl. Instr. and Meth. B 134 (1998) 405. [27] K. Szyma nski, Nucl. Instr. and Meth. B 143 (1998) 601. [28] K. Szyma nski, D. Satuµa, L. Dobrzy nski, J. Phys. Cond. Matter 11 (1999) 881. [29] K. Szyma nski, Nucl. Instr. and Meth. B 168 (2000) 125. [30] P. Zory, Phys. Rev. A 140 (1965) 1401. [31] R. Zimmermann, Chem. Phys. 34 (1975) 416. [32] R. Zimmermann, Nucl. Instr. and Meth. 128 (1975) 537. [33] J.M. Daniels, Nucl. Instr. and Meth. 128 (1975) 483. [34] H. Spiering, Hyper®ne Interactions 3 (1977) 213. [35] U. Gonser, H. Fischer, in: U. Gonser (Ed.), M ossbauer Spectroscopy II, Springer, Berlin, 1980, p. 99. [36] K. Szyma nski, Hyper®ne Interactions 126 (2000) 431. [37] K. Szyma nski, J. Phys. C: Condens. Matter (2000) in press. [38] R. Zimmermann, in: B.V. Thosar, P.K. Iyengar (Eds.), Advances in M ossbauer Spectroscopy Applications to Physics, Chemistry and Biology, Elsevier, Amsterdam, 1983, p. 273. [39] N.W. Ashcroft, N.D. Mermin, in: Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [40] H. Bateman, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.
www.elsevier.nl/locate/nimb
ssbauer Analytical expression for the energy distribution of Mo transitions in the presence of mixed interactions Krzysztof Szyma nski
*
Institute of Experimental Physics, University of Biaøystok, Lipowa 41, 15-421 Biaøystok, Poland Received 30 May 2000
Abstract Based on the analytical results for intensity tensor components, expressions for energy distribution of M ossbauer 3/ 2±1/2 transitions in the case of a uniform magnetic ®eld and randomly oriented electric ®eld gradient (EFG) are presented. The expression obtained contains elliptic integrals. The logarithmic singularities appearing in the integrals are integrable and their origin is discussed. Next, the shape of the 57 Fe M ossbauer spectrum in standard, as well as recently developed, circularly-polarized M ossbauer spectroscopy was derived. The general results obtained agree with previous works in the limiting cases, e.g., with the analytical shape obtained for asymmetry parameter set to zero, and the results obtained by perturbation methods. In the limit of local axial symmetry of the EFG (g 0), expressions simplify to an algebraic form and the singularities are of x ÿ x0 ÿ1=2 type. Ó 2000 Elsevier Science B.V. All rights reserved.
1. Introduction In M ossbauer spectroscopy, problems of mixed interactions, e.g., presence of magnetic ®eld and electric ®eld gradient (EFG) acting on the nucleus in the I 3=2 state, appear very often. In the early days of the technique, methods based on the numerical solution of the spin Hamiltonian [1±8] were derived. The transition probabilities were calculated by the use of explicit Hamiltonian eigenstates [9±14]. An important step was made
*
Tel.: +48-243-833518; fax: +48-85-457223. E-mail address: [email protected] (K. SzymanÂski).
when analytical solutions for eigenenergies of the Hamiltonian were obtained [15,16]. When hyper®ne magnetic ®eld vector directions exhibit certain distributions with respect to the principal axes system (PAS) of the EFG tensor, one has to perform integration over all possible transitions. Consequently, instead of sharp transitions, one arrives at a continuous distribution of transitions forming bands and in¯uencing absorption line shape. Many numerical and analytical treatments of this problem involved perturbation methods [17±23]. An analytical expression for the shape of the M ossbauer spectrum was obtained for a special case, namely an axially symmetric EFG [24]. The advantage of analytical results is unquestioned when one is analysing
0168-583X/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 5 8 3 X ( 0 0 ) 0 0 3 1 1 - 6
516
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
global properties of the spectrum or checking the precision of the numerical procedures, particularly those in which approximations are introduced for time saving purposes. Recently, a monochromatic, circularly-polarized M ossbauer source (MCPMS) technique was developed [25±28]. Unfortunately, widely distributed commercial packages do not have suitable options for polarized M ossbauer spectroscopy. This stimulated us to the reanalysis of known methods in order to make use of them in the MCPMS technique [29]. Interestingly, we have found that, using the so-called intensity tensor formalism [30±35], one is able to obtain analytical expressions for transition probabilities without explicit calculation of the eigenfunctions of the Hamiltonian [36,37]. This approach is continued in the present paper. How to get an analytical shape for the energy distribution in the case of a uniform magnetic ®eld and randomly oriented EFG is shown. Singular points which appear in the calculations essentially present no problem. The use of the described method is illustrated on several examples.
For the excited Ie 3=2 levels the total Hamiltonian is conveniently reduced to 3=2
3=2
3=2 Hel Htot Hmag
g3=2 lN B 3=2 H : 2
This Hamiltonian H3=2 in the PAS of the EFG tensor and in the representation of Iz eigenfunctions has a form as given in [16]
2:4 The parameter R is proportional to the Vzz component of the EFG: R eQVzz =2g3=2 lN B. Let us abbreviate eigenvalues of the excited Ie 3=2 Hamiltonian by Ea and introduce dimensionless energies ka 2Ea =g3=2 lN B. Energies ka are roots of the secular equation originating from (2.4) k4 pk2 qk r 0;
2. Intensity tensor for a single site
HelI
eQVzz 2 g 3I z ÿ I 2 I 2 I 2ÿ ; 4I 2I ÿ 1 2
2:5
where
In the following, the Hamiltonian of the ground and excited states is de®ned as in [16,37]; we shall also use abbreviations appearing in the cited papers. The Hamiltonian of the nuclear system with I spin I is the sum of the magnetic Hamiltonian Hmag I and the electric quadrupole Hamiltonian Hel , I ÿgI lN I ~ B; Hmag
2:3
2:1 2:2
where gI is a nuclear g-factor and lN denotes the nuclear magneton, Q is the nuclear quadrupole moment and g is the so-called asymmetry parameter. Cartesian components of the EFG tensor are Vij ÿo2 V =oxi oxj , where V denotes an electric potential at the nucleus. The principal axis system of the EFG tensor can be chosen so that jVzz j P jVyy j P jVxx j and then g Vxx ÿ Vyy =Vzz .
p ÿ10 ÿ 2R
2
g2 1 ; 3
^ m ~; q ÿ16~ mU 1 2 ^2 m ~: mU r p 4 ÿ 16~ 4
2:6
~ , de®ned by spherical angles h The unit vector m and u, is parallel to the direction of the hyper®ne ^ is a second-order tensor whose magnetic ®eld ~ B. U dimensionless components are proportional to the EFG. In its PAS it reads 2 3 1ÿg 0 0 R ^ ÿ 4 0 1 g 0 5: 2:7 U 2 0 0 ÿ2 An analytic form for ka can be found in [15,16]. The transition probabilities, or line intensities, can be expressed in an elegant way within the framework of the intensity tensor formalism. For
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
example, in the thin absorber approximation, the area under an absorption line is equal to [38] Aabf
Cp 1 s ~ ab ~ Tr ^Iab ÿ~ fs t c ^Iab ~ c ÿ 2fG c; 2 2
2:8
3 40k2a ÿ 4qka p 4 p 16 ÿ 4r ; 8 16bwa 2:9
where wa 4k3a 2pka q:
2:10
Parameter wa is equal to the derivative of the secular equation (2.5). Consequently, parameter wa non-equal to zero is equivalent to the condition that excited levels are non-degenerate. The symmetric part of the intensity tensor multiplied right~ is equal to and left-handed by vector m s ~ ~ ^Iab m m
1 b 64k2a 8ka q 128wa 8 p 4 p 16 ÿ q2 ÿ 32r
2:12 We turn the attention of the reader to the fact that the unit vector ~ c in Eq. (2.8) may have any orientation with respect to the PAS of the tensor ^Iab , the latter depending on the spherical angles h and u which de®ne direction of the hyper®ne ~ . In Eqs. (2.11) and (2.12), the magnetic ®eld m s ~ ~ same angles h and u de®ne tensors ^Iab ; G ab and m as well.
3. Randomly oriented EFG 3.1. Average intensity tensor Let us now consider the special, important case of randomly-oriented EFG and well-de®ned direction of the hyper®ne ®eld. The situation is realised experimentally when a powdered sample with negligible local anisotropy is exposed to a uniform external magnetic ®eld. We start with an explicit form of the symmetric part of the intensity tensor which is a function of ^ m ^ and the vector m ~ . ~ : ^Iab ^Iab U; the tensor U Let R R X be a real rotation matrix de®ned by Euler angles X. We are interested in the angular ^ R; m ~ iX over the average of the value h^Iab RT U angles X. From the symmetry arguments it follows that, in the PAS of the averaged intensity tensor, ~ is directed along one of the axes. We may vector m ~ is diassume, without loss of generality, that m rected along the z-axis. Further on, diagonal components xx and yy are equal to each other. Thus, the averaged tensor in the PAS can be written as 2
64k3a 4k2a q 32ka p ÿ 8 2q p 28:
1 ~ ab m ~ G b 2k2a p 16 ÿ8ka q 64wa ÿ 80k2a 8ka q ÿ 2p2 ÿ 40p 8r ÿ 128:
where t is the eective thickness of the absorber, fs the recoilless fraction of the source and C is the natural width. Index f 1 in (2.8) describes two opposite circular polarisations of the photon. ~ c is a unit vector parallel to the direction of the photon. ^Iab is the so-called intensity tensor of the transition s ~ ab are symmetric between states a and b. ^Iab and G ^ and antisymmetric parts of Iab . Indices a run over four excited Ie 3=2 states while b 1; ÿ1 correspond to the ground Ig 1=2 states with higher and lower energy, respectively. The results important for further evaluation are listed below (the reader is referred for details to [36]). In the case of a single site, the trace of the intensity tensor is equal to Tr ^Iab
517
2:11
~ ab and The product of the antisymmetric part G ~ is equal to vector m
Iab?
6 s ^ R; m ~ iX 6 h^Iab RT U 4 0 0
0 Iab? 0
0
3
7 0 7 5: Iabk 3:1:1
518
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
The averaging over three Euler angles results in 1 s ~ ^Iab ~ ihu ; m Iab? hTr ^Iab ÿ m 2 s ~ ihu ; Iabk h~ m ^Iab m
3:1:2
s ~ as well as the tensor ^Iab where the unit vector m depend on the same angles h and u, i.e., Tr ^Iab and s ~ are given by Eqs. (2.9) and (2.11), re~ ^Iab m m spectively. The averaging in (3.1.2) is performed in a standard way: Z 1 sin h dh du: hihu 4p sphere
In a similar way we can calculate the antisymmetric part of the intensity tensor for our problem. The result in the same PAS system is 2 3 0 ~ ab RT U ^ R;~ 5: 0 ez iX 4 3:1:3 hG ~ ~ ihu hG ab m ~ ab m ~ , is described by Eq. (2.12). The product, G Unit vector ~ ez in (3.1.3) belongs to the basis of the PAS of our averaged intensity tensor (3.1.1) and is parallel to the direction of the uniform magnetic ®eld. The reader should not be confused by the ~ in (3.1.2) and (3.1.3), notation of the unit vector m which is only used for performing the average and does not indicate the direction of the uniform magnetic ®eld. Explicit forms of Eqs. (2.9)±(2.12) serve for calculation of the energy distribution of the intensity tensor components. In the next paragraph we show how to get an explicit expression for the energy distribution resulting from averaging (3.1.2) and (3.1.3). 3.2. Energy distribution The energy distributions of the intensity tensor components are important quantities since they are related to the shape of the absorption lines in the spectrum. We are interested in particular in the spectral distribution dhf i=dk connected with each component of the intensity tensor f. By f we mean any component of Eq. (3.1.2) or (3.1.3). Having explicit expressions f k; h; u one can formally write the expression for the energy distribution:
f k; h; u f k h; u; h; u Z f t; h; ud t ÿ k h; u dt;
3:2:1
where d is the Dirac d-function. Thus, Z hf k; h; uihu hf t; h; ud t ÿ k h; uihu dt 3:2:2 and, because Z hf k; h; uihu
dhf i dk; dk
3:2:3
one gets dhf i k0 hf k0 ; h; ud k0 ÿ k h; uihu : dk
3:2:4
Eq. (3.2.4), although written in a short form, needs further evaluation since it involves integration over angles h and u over isoenergetic energy line k h; u k0 . It is shown below that the methods involved in calculating electronic density of states in metals [39] can be applied for ®nding an isoenergetic set C k and integrating the function f over it. We will show next that the energy distribution of all components ((3.1.2) and (3.1.3)) can be expressed by elliptic integrals. Since parameter k is a function of (h,u), which means that f depends, in fact, on the spherical angles (h,u) only, the function f k; h; u can be considered as de®ned on a three-dimensional unit sphere S. The averages (3.2.4) can be considered as integrals of f over the surface and can be calculated in the following way. Consider an element of the surface, dS, located between two lines C k and C k dk, see Fig. 1. C k0 is de®ned as a line on which k has constant value k0 . The width of the element dS dl dh, where element dl is along the ~ so C k contour, is dh dk=jrkj; f k; h; u dS f k; h; udh dl
f k; h; u dk dl: ~ jrkj
3:2:5
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
519
radius while the right-hand side a second-order surface (ellipsoid, hyperboloid, etc.). The intersection of the two surfaces corresponds to the line C k which enters integral (3.2.7) explicitly, see Fig. 2. It follows from Eqs. (2.5) and (3.2.9) that ^ ÿ ^1 m ~ 32 A ~; rk wH and our integral (3.2.7) takes the form Z dhf i 1 jH j jwjf dl p : dk 4p 32 C k m ^2 m ~A ~ÿ1
Fig. 1. A unit sphere S and the element dS located between curves C k and C k dk, see text.
Thus, hf k; h; uihu
1 4p
Z
Z dk
C k
3:2:11
3:2:12
The geometrical form of (3.2.8) is helpful in the determination of the domain of integration. From (3.2.8), the following relation between h and u for points on the C curves can be obtained: cos2 h
bt ; ct
3:2:13
where
f k; h; u dl : ~ jrkj 3:2:6
Comparing Eqs. (3.2.6) and (3.2.2) we conclude that Z dhf i 1 f k; h; u dl : 3:2:7 ~ dk 4p C k jrkj
ÿ2 Axx Ayy ; Axx ÿ Ayy ÿ2Azz Axx Ayy ; c Axx ÿ Ayy b
3:2:14
t cos2u:
It is interesting that, in the case of tensor components (2.9)±(2.12), the integral (3.2.7) can be transformed to a more familiar form. To do that, let us return to the secular equation (2.5) and rewrite it in the equivalent form ^ m ~m ~m ~A ~; m
3:2:8
where ^ H kU ^ U ^ 2 A
3:2:9
and factor H is a rational function of k, H
16 4
2
k pk p=2 2
2
:
3:2:10
Eq. (3.2.8) has a simple geometrical interpretation: the left part of (3.2.8) represents a sphere of unit
Fig. 2. Isoenergetic curve C k resulting from the intersection of a second-order surface and a unit sphere.
520
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
Parameters b and c are important and we will be using them frequently in what follows. The explicit forms of b and c can be obtained from (3.2.9): b c
4
k pk 8R k R p=2 8 ; 8gR ÿk R 6k R 3 ÿ g2 : 2g ÿk R
3:2:15
Z
1
ÿ1
jwjf dt p : b t c t 1 ÿ t2
1 ; ct
3:2:17
1 ; r r0 r1 ct where
K2 ÿ 16 k2 8R2 8Rg b ÿ c ÿ 16R; 4 ÿk R
Rk 9 ÿ g2 K2 ÿ 16 k 2R 8g ÿk R
2
ÿR 9 ÿ g2 K2 ÿ 16 k 2R 8g ÿk R
K 4 2k2 p:
ÿkq1 :
2
ÿ8R 3 gc b ÿ c 3:2:18
3:2:20
Thus, any intensity tensor component f can be expressed as a ®nite series: f
2 1X f n n; w n0 c t
3:2:21
in which coecients f n do not depend on t (f n components depend explicitly only on k, R and g). One easily notices that the value of the parameter |w|, present in the numerator of (3.2.16), is irrelevant. Because w is present in each denominator of the f series (3.2.21), it is only the sign of (w) which matters. On the other hand, w 6 0 means only that the excited levels are non-degenerate. The sign of w is thus constant in each energy band, or in the domain of integration (3.2.16), and can be taken outside the integral. How to determine this sign will be discussed in paragraph 6. Now we may insert tensor component (3.2.21) into (3.2.16) and get the ®nal form of the energy distribution. It should be remembered that Eq. (3.2.16) was derived for a C curve on the unit sphere S, with the loop encircling the z-axis (condition c=b > 1 and ÿ1 < b < 1), see Fig. 1. In a general case, C curve may run around the x- or yaxis; so we have to describe these cases too. The ®nal result is 2 dhf i sign w X Jn b; cf n : dk 4gjR k ÿ Rj n0
2
q1
r1
In Eqs. (3.2.18) and (3.2.19)
The function f in (3.2.16) is one of the types: s ~ ~ or m ~ ^Iab m ~ , see Eqs. (2.9)±(2.12). Tr ^Iab , G ab m The numerators of the functions of interest are linear combinations of 1, q, r and q2 . Parameters q, r and q2 taken at points which belong to the C curve can be expressed as linear combinations of 1, ÿ1 ÿ2 c t and c t because
q0
2
3:2:19
3:2:16
q q0 q1
ÿR K2 ÿ 16k k 8R p 4 4 ÿk R 4 2
If coecients b and c satisfy inequalities: c=b > 1 and ÿ1 < b < 1, the discussed energy distribution will be represented as an integral over two loops around the z-axis (the intersection of the hyperboloid and the sphere results in two contours, located at positive as well as negative values of z, shown as an example in Fig. 2): dhf i 1 1 dk 4p 4gjR k ÿ Rj
r0
ÿk k3 pk q0 ;
2
2
and
3:2:22
The functions Jn are given by integrals, in which the limits of integration result from positions of the C curve on the sphere:
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
8R1 > < Rÿb jn b; c; t dt; ÿb Jn b; c j b; c; t dt; ÿ1 n > :R1 j b; c; t dt; ÿ1 n
where
c; b 2 sx ; c; b 2 sy ; c; b 2 sz ;
where 1 p : 4p c t b t c t 1 ÿ t2 n
3:2:24 Three sets, sx , sy and sz , which de®ne the domain of Jn , are given by the inequalities c; b 2 sx () c ÿ b sign 1 ÿ b2 1 > 0; 2
c; b 2 sy () c ÿ b sign 1 ÿ b 1 < 0; c; b 2 sz () b c ÿ b sign 1 ÿ b2 ÿ 1 < 0: 3:2:25 Function Jn in (3.2.23) can be expressed by elliptic integrals, see Appendix A. 3.3. Spectral density of the intensity tensor With the help of (3.2.17) and (3.2.21), the components of the intensity tensor (3.1.1) can be ÿ1 expressed through powers of c t . The perpendicular component is thus equal to 1 1 s s ~ ~ ^Iab m I? Tr ^Iab ÿ m 2 2 2 1X 1 I? n n; w n0 c t
3:3:1
where q21 b ; 256 ÿq1 K ÿ b q0 ÿ 20k ; I? 1 128 q0 ÿ 32k 2K ÿ b q0 ÿ 8k : I? 0 256 I? 2
3:3:2
Similar calculations lead to ~ ^Iab m ~ Ik m s
2 1X 1 Ik ; w n0 n c tn
ÿq21 b ; 128 q1 K 24 ÿ b q0 ÿ 20k; Ik 1 64 b 8K K 12 ÿ 256k2 ÿ q0 q0 ÿ 40k Ik 0 128 1 K 24 q0 16k ÿ 576k: 64 3:3:4 Ik 2
3:2:23
jn b; c; t
521
3:3:3
For the vector components we have 1 1X 1 ~ ab m ~ ab m ~ ~ n G ; G w n0 c tn
3:3:5
1 q1 b K 12; 64 1 ~ ab m ~ 0 K 12 ÿ2K b q0 ÿ 8k: G 64 3:3:6
~ ab m ~ 1 G
For completeness of the results, we also present expressions for the trace of ^Iab , Tr ^Iab s
1 1X 1 s Tr ^Iab n n; w n0 c t
3:3:7
3 s Tr ^Iab 1 q1 ; 8 3 1 s Tr ^Iab 0 4k3 2pk q0 bK K 12: 8 16 3:3:8 We may now present some analytical results for the spectral distribution of the intensity tensor components. As an example, consider the situation of the isotropic intensity tensor. In this case, the direction of the hyper®ne magnetic ®eld is randomly distributed with respect to the PAS of the EFG tensor, and also the PAS of the latter are randomly distributed in real space. In the considered case, the trace of the averaged intensity tensor is equal to
522
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
dhTr ^Iab i 3 sign w q1 J1 b; c dk 32gjR k ÿ Rj sign w 6 4k3 2pk q0 64gjR k ÿ Rj s
bK K 12J0 b; c:
3:3:9
We remind the reader that parameters q0 and q1 were de®ned in (3.2.18), K in (3.2.20), p in (2.6), b,c in (3.2.15) and w in (2.10); so Eq. (3.3.9) is a function of three parameters: k, R and g. 4. Application: shapes of spectra Taking the sum of (3.3.9) over all transitions a, b and using (2.8), we get the shape of the energy distribution corresponding to the M ossbauer spectrum, shown in Fig. 3(a). The curves are not convoluted with the Lorentzian distribution representing natural broadening. In a similar way, with help of Eqs. (2.8), (3.3.1) and (3.3.6), we construct spectra corresponding to an external ®eld parallel or perpendicular to the direction of observation ~ c, and also other spectra which can be measured using the MCPMS technique, see Figs. 3(b) and (c). 5. Special case: local axial symmetry of the EFG In the case of local axial symmetry of the EFG, g 0, the expressions can be greatly reduced. Eq. (3.2.22) becomes dhf i jwjf q ; dk 2 2 2 4 3R 2k R k ÿ R2 ÿ 3 ÿ 4 ÿk R 5:1 where w
2 3k 2Rk3 ÿ 4 5 R2 k Rk ÿ 2 9 ÿ 2R2 R4 : 2k R 5:2
Function f in Eq. (5.1) is given by any component of the intensity tensor. Explicit expressions for the components of interest are as follows:
Fig. 3. Calculated energy distribution corresponding to the absorption spectrum for R 2:2 and g 0:6. (a) Isotropic intensity tensor, (b) hyper®ne magnetic ®eld perpendicular to the observation direction, (c) hyper®ne magnetic ®eld parallel to the observation direction, (d) hyper®ne magnetic ®eld parallel to the observation direction; measurement with circularly polarized source, (e) hyper®ne magnetic ®eld antiparallel to the observation direction; measurement with circularly polarized source. Horizontal scale corresponds to the dimensionless energy k (Doppler velocity v g3=2 lN Bc=2Ec k).
2
d 3 k2 ÿ R2 ÿ 9 s hTr ^Iab i b ; dk 8 4w
5:3
d d s ~i Ik h~ m ^Iab m dk dk 2 k2 ÿ 2bk ÿ R2 2bR ÿ 3 k ÿ R 3b 3
2
k ÿ R ÿ 3b k R b =ÿ32wb 2k R ; 5:4
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
d d 1 ^ s 1 s ~i ~ ^Iab I? h Tr Iab ÿ 2 m m dk dk 2 k2 2bk ÿ R2 ÿ 2bR ÿ 3 k ÿ R 3b 3
k R ÿ b 4k ÿ 4kR2 12k ÿ w= 128wb 2k R; 5:5 d ~ ~ i k2 2bk ÿ R2 ÿ 2bR ÿ 3 hG ab m dk k ÿ R 3b k R ÿ b k2 ÿ R2 3=ÿ16wb 2k R: 5:6 The spectra constructed with the help of Eqs. (2.8), (5.3)±(5.6) are shown in Fig. 4.
523
6. Discussion The most interesting feature seen in Figs. 3 and 4 is the appearence of singularities at special points on the energy (velocity) scale. Their origin is similar to the one of van Hove's singularities. Let us discuss this problem in greater detail. In Fig. 5, we have drawn an angular dependence of the eigengenvalue ki of the secular equation (2.5). In the special directions, along axes of the PAS system of the EFG, there are extreme values of energy. Since the secular equation is of 4th order, we have 4 3 12 special points on the energy scale. The special points can be obtained easily from the secular equation (2.5). Four minima and four maxima, which de®ne the borders of the energy bands, occur at ki , p k1;2 1 D 4R; p k3;4 1 ÿ D 4R; 6:1 p k5;6 1 D 2R 1 g; p k7;8 1 ÿ D 2R 1 g; where D ÿ p=2 1:
6:2
The bands are shown in Fig. 6 by shaded areas. There are also four solutions of the secular equation for saddle points, p ks1 ÿ1 ÿ D 2R 1 ÿ g; p ks2 1 ÿ D ÿ 2R 1 ÿ g; 6:3 p ks3 ÿ1 D 2R 1 ÿ g; p ks4 1 D ÿ 2R 1 ÿ g; shown in Fig. 6 by dashed lines located on the shaded areas. It follows from Fig. 6 that except for six values of Ri ,
Fig. 4. Same as Fig. 3 for axially symmetric EFG, g 0 and R 2:5.
3 R1;2 p ; 9 ÿ g2 3 R3;4 p ; 2g 3 g 6 ; R5;6 3g
6:4
524
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
Fig. 6. Minima and maxima of solutions of the secular equation for g 0:6, shown by solid lines in the (k,R) plane. The saddle points are shown by dashed lines. Energy bands of the excited levels are shown by shaded areas. The sign of parameter w for each energy band is shown by + or ).
Fig. 5. Angular dependence of the energy k for (a) the two highest and (b) the two lowest energy levels for R 2:2 and g 0:6 represented in the PAS of the EFG. Any vector connecting the origin of the coordinate system with a point on the surface is given by spherical angles h and u (which enter the secular equation (2.5)) and has length k. Maximum, minimum and saddle points of the energy k with respect to h and u are located at intersections of the surface and the axes.
there are four energy bands separated by three energy gaps in the level structure of the excited states. Fig. 6 conveniently presents the behaviour of the sign of the determinant w, mentioned in Section 3.2: the sign of w is constant within each energy.
Let us discuss the type of divergence of function (3.2.22) at the saddle points. Fig. 3 suggests that f ! 1. Indeed, when k is in the vicinity of the saddle point ksi , see Eq. (6.3), the argument v is close to 1, see (A.3), and the elliptic integral K v, see Eq. (A.8), diverges like log jk ÿ ksi j. Thus, there are eight points in the spectrum with logarithmic divergence. In the case of g 0, it follows from Eq. (5.1) that at points ki for which the denominator is zero, the function dhf i=dk behaves like ÿ1=2 . ki ÿ k The physical origin of the singularities in Figs. 3 and 4 can be easily understood. During integration over angles h and u, see Eq. (3.1.1), the energy is varied unless we are at the points where energy does not depend locally on angles. This occurs at the special points (6.1)±(6.3). In Figs. 3 and 4, we have drawn vertical lines at the positions given by Eqs. (6.1)±(6.3). There are 12 singular points for excited levels. Because there are transitions to two ground levels, this results in 2 12 24 singular points in the distribution corresponding to the absorption spectrum (we turn the attention of the reader to the fact that, for the R value used in the presented examples, the dashed lines are very close to the borders of the energy bands; see the lines for R 2:2 at k ÿ1:8 in Fig. 6. This, in
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
turn, results in very close proximity of singular points in Fig 3: one pair of dashed lines is located at k ÿ3:54 and k ÿ3:52, while the second one is located at k ÿ0:02 and k ÿ0:04). In the case of axial symmetry of the EFG, g 0, shapes analogous to those shown in Fig. 5 possess axial symmetry. Therefore, the dashed line in Fig. 6 would overlap with one of the border lines for each energy band. The ®rst consequence of this is the smaller number of singular points in spectrum 16, see the dashed lines in Fig. 4. The second consequence is a qualitative dierence in the shape of the distribution. In the case of g 0, the point of divergence is always located at the side of the peak corresponding to the energy band, resulting in more asymmetric peaks (see the peak at k ÿ6 in Fig. 4(a)). In the case of lack of axial symmetry of the EFG, the divergence point is located somewhere in the middle of the energy band, resulting in more symmetric peaks (see the peak at k ÿ6 in Fig. 3(a)). We see that the shapes of the absorption peaks may be used in the determination of the asymmetry parameter. In order to compare the discussed shapes with any experiment, one has to convolute the functions with Lorentzian distributions describing the natural broadening. After convolution, most of the details of the rich energy distribution, e.g., narrow integrable peaks, will be hidden under wide resolution. We see that natural broadening prevents direct observation of the shapes shown in Figs. 3 and 4. Assume, however, that we are able to achieve precise deconvolution of the experimental spectrum and to localize singularities on the energy scale. Would it be useful information? Note that the positions of the singular points correspond to the well-de®ned orientation of the hyper®ne ®eld in the PAS of the EFG. Thus, using Eqs. (6.1)± (6.3), we would be able to determine the value of the hyper®ne magnetic ®eld and the asymmetry parameter g as well as the Vzz component of the EFG tensor. Secondly, our analysis has been performed assuming random orientation of the EFG axes. If the orientation deviates to some extent from random distribution, the shape of the spectrum would be dierent from those presented in Fig. 3. However, singular points would be located
525
at exactly the same positions on the energy scale. Thus, one can quantitatively analyse properties of powdered absorbers which have unknown texture. Our expressions (3.2.22) are in agreement with the results obtained by perturbation methods [17,22,23], analytical results for g 0 [24] and numerical results [24]. In summary, the analytical results for energy distribution of averaged intensity tensor components have been obtained. The problem was reduced ®rst to integration over curve C, which appears as an intersection of two second-order surfaces in three-dimensional space. Final results contain complete elliptical integrals (in the special case of g 0, expressions for the intensity tensor reduce to an algebraic form). Having determined the intensity tensor and the explicit form of its spectral density, features of M ossbauer spectra corresponding to typical experimental geometries are discussed. The spectrum, before convolution with the Lorentzian distribution, contains generally 24 singular points. In the vicinity of eight of them, logarithmic divergences are found. The remaining singular points correspond to the borders of energy bands of the excited states. The shapes of the absorption peaks depend strongly on the asymmetry parameter. The positions of all singular points on the energy scale correspond to special orientations of the hyper®ne magnetic ®eld with respect to the PAS of the EFG and may have potential application in the analysis of samples with unknown texture. Acknowledgements The author expresses many thanks Prof. L. Dobrzy nski for a lot of stimulating discussions and critical reading of the manuscript. The work was sponsored by the State Council of Science through grant 2P03B11516. Appendix A The aim is to calculate integral Jn b; c, see (3.2.23). Let us de®ne integral In (x),
3 c ÿ 1 c 12 c ÿ b2
2 b 1 1 2bc ÿ 3c2
Z x In x 4p jn b; c; t dt 0 Z x dt p : n b t c t 1 ÿ t2 c t 0
A:1
Changing the variable to u
1 c 1 t 1 ÿ c 1 ÿ t
A:2
ÿ 1 c ÿ b 3 c2
ÿ2 ÿ b2 3c2
2
and introducing the abbreviation
ÿb ÿ 1 c 1 c ÿ b
3 c2 ÿ 12 c ÿ b 3 1 ÿ c 1 c c ÿ b
4 1 2bc ÿ 3c2
2
ÿ2 c2 ÿ 1
ÿ2 ÿ b 3c ÿ 3bc 3c2
2 1 ÿ b 1 2bc ÿ 3c2
3 1 ÿ c2 1 c c ÿ b2
ÿ2 ÿ b 3c ÿ 3bc 3c2
1ÿb 1 ÿ c c ÿ b
3 1 ÿ c2 1 c c ÿ b
e2 k2
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
e1
526
v
1 b 1 ÿ c ; 1 ÿ b 1 c
A:3
the integral (A.1) can be changed to the canonical form [39]. Next, the integral can be expressed by incomplete elliptic functions E and F of complex arguments: Z u0 du q ; F u0 ; v 0 1 ÿ v sin2 u A:4 Z u0 q 2 1 ÿ v sin u du: E u0 ; v
1 b c ÿ 1 2 c ÿ b
1 cÿb 0 1 2 p b 1 c ÿ 1 2 c ÿ b b 1 c ÿ 1
1 cÿ1 0 1
ÿ1 1ÿc 0 1
2 p 1 ÿ b 1 c s 2 c ÿ b 1 b 1 ÿ c 1 ÿ b 1 c
P v
Let us introduce abbreviations, ppp p Fuv 1 ÿ u 1 ÿ uv uF Arcsin u; v; ppp p Euv 1 ÿ u 1 ÿ uv uE Arcsin u; v;
1 6 b 6 c
ÿ1 6 b < 1 6 c
ÿ1 6 b 6 c < 1
A:5
and de®ne in for n 0; 1 and 2 by recurrence formula, 2
1 ÿ c 1 ÿ x Fuv ; 1c 1 1 ÿ b 1 ÿ x2 i0 ÿ Euv i1 ÿ 1ÿc 1 c b ÿ c 2 1 x b x ; ÿ 1 ÿ c b ÿ c i0
i2 ÿ ÿ
b, c
Table 1 The argument v and coecients P, kn and en of Eq. (A.9)
k0
e0
k1
0
A:6
2 1 2bc ÿ 3c2 b ÿ 3c i1 i0 3 b ÿ c 1 ÿ c2 3 b ÿ c 1 ÿ c2 2 1 ÿ x2 b x : 3 1 ÿ c2 b ÿ c c x
The integral In (x) is proportional to in , in In x p : b x c x 1 ÿ x2
A:7
K. Szyma nski / Nucl. Instr. and Meth. in Phys. Res. B 171 (2000) 515±527
The integrals which enter into (3.2.23) can be obtained by taking (A.7) at appropriate values of arguments x. Next, using transformations rules for complete elliptic integrals [40], we arrive at the result, which depends now on the familiar complete elliptic integrals of the ®rst and second kind of real argument, de®ned as p ;v ; K v F p2 A:8 ;v : E v E 2 The ®nal result for Jn b; c reads Jn b; c
P kn K v en E v; 4p
n 0; 1; 2: A:9
The argument v and coecients P, kn and en are functions of b and c, and are listed in Table 1. The domains for b and c, given in the ®rst column of Table 1, cover only part of the set given by inequalities (3.2.25). In order to get Jn b; c, de®ned for the whole domain, one needs to use an identity which follows directly from (3.2.23), n
Jn ÿb; ÿc ÿ1 Jn b; c:
A:10
References [1] [2] [3] [4] [5]
M.H. Cohen, Phys. Rev. 96 (1954) 1278. L.C. Brown, P.M. Parker, Phys. Rev. 100 (1955) 1764. P.M. Parker, J. Chem. Phys. 24 (1956) 1096. J.R. Gabriel, Moessbauer E. Methodol. 1 (1965) 121. J.R. Gabriel, S.L. Ruby, Nucl. Instr. and Meth. 36 (1965) 23. [6] W. K undig, Nucl. Instr. and Meth. 48 (1967) 219. [7] J.R. Gabriel, D. Olson, Nucl. Instr. and Meth. 70 (1969) 209. [8] E. Kreber, U. Gonser, Nucl. Instr. and Meth. 121 (1974) 17.
527
[9] H.H.F. Wegener, F. Obenshain, Z. Phys. 163 (1961) 17. [10] H. Frauenfelder, D.E. Nagle, R.D. Taylor, D.R.F. Cochran, W.M. Visscher, Phys. Rev. 126 (1962) 1065. [11] J.M. Daniels, J.M. Cotignola, Can. J. Phys. 40 (1962) 1342. [12] D. Barb, S. Constantinescu, D. Tarina, Rev. Roum. Phys. 16 (1971) 263. [13] B.L. Chrisman, Nucl. Phys. A 186 (1972) 264. [14] D. Barb, D. Tarina, Rev. Roum. Phys. 18 (1973) 1209. [15] P.G.L. Williams, G.M. Bancroft, M ossbauer E. Methodol. 7 (1971) 39. [16] L. H aggstr om, Report UUIP-851, Uppsala, 1994. [17] R.L. Collins, J. Chem. Phys. 42 (1965) 1072. [18] G.N. Belozerskii, Yu.P. Khmich, V.N. Gitsovich, Phys. Stat. Sol. B (1977) K125. [19] G. Le Caer, J. Phys. E 12 (1971) 1083. [20] M. Eibsh utz, M.E. Lines, Phys. Rev. B 26 (1982) 2288. [21] M.E. Lines, M. Eibsch utz, Phys. Rev. B 25 (1982) 6042. [22] R.A. Brand, Nucl. Instr. and Meth. B 28 (1987) 417. [23] R.A. Brand, Nucl. Instr. and Meth. B 28 (1987) 398. [24] N. Blaes, H. Fischer, U. Gonser, Nucl. Instr. and Meth. B 9 (1985) 201. [25] K. Szyma nski, L. Dobrzy nski, B. Prus, M.J. Cooper, Nucl. Instr. and Meth. B 119 (1996) 438. [26] K. Szyma nski, Nucl. Instr. and Meth. B 134 (1998) 405. [27] K. Szyma nski, Nucl. Instr. and Meth. B 143 (1998) 601. [28] K. Szyma nski, D. Satuµa, L. Dobrzy nski, J. Phys. Cond. Matter 11 (1999) 881. [29] K. Szyma nski, Nucl. Instr. and Meth. B 168 (2000) 125. [30] P. Zory, Phys. Rev. A 140 (1965) 1401. [31] R. Zimmermann, Chem. Phys. 34 (1975) 416. [32] R. Zimmermann, Nucl. Instr. and Meth. 128 (1975) 537. [33] J.M. Daniels, Nucl. Instr. and Meth. 128 (1975) 483. [34] H. Spiering, Hyper®ne Interactions 3 (1977) 213. [35] U. Gonser, H. Fischer, in: U. Gonser (Ed.), M ossbauer Spectroscopy II, Springer, Berlin, 1980, p. 99. [36] K. Szyma nski, Hyper®ne Interactions 126 (2000) 431. [37] K. Szyma nski, J. Phys. C: Condens. Matter (2000) in press. [38] R. Zimmermann, in: B.V. Thosar, P.K. Iyengar (Eds.), Advances in M ossbauer Spectroscopy Applications to Physics, Chemistry and Biology, Elsevier, Amsterdam, 1983, p. 273. [39] N.W. Ashcroft, N.D. Mermin, in: Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [40] H. Bateman, Higher Transcendental Functions, Vol. II, McGraw-Hill, New York, 1953.