CHAPTER II. The Angular and Polarization Distributions of the Mossbauer Radiation 3/2 + 1/2 + in Single Crystals *

Transcription

1 Chapter II 7 CHAPTER II The Angular and Polarization Distributions of the ossbauer Radiation / + / + in Single Crystals *. Introduction When a gamma-ray emitting or absorbing nucleus is placed in the presence of magnetic and electric fields, the magnetic dipole moment of the nucleus interacts with the surrounding magnetic field, and the quadrupole moment of the nucleus interacts with the electric field gradient set up by the ligands around it []. Attempts have been made to construct a theory, assuming asymmetry parameter () non-zero, to evaluate hyperfine field parameters in single crystals using the ossbauer resonance. In the nuclear transition / /, eight lines are obtained. The coherency matrix [], the intensity and the degree of polarization [] have been calculated for each line using the multipole radiation-field theory [] and densitymatrix approach [4-5]. Extensive use of D-matrix [6] has been made to transform the results from the principal axis system (PAS) to the crystal fixed axis system (CFAS) in single crystals. When the magnetic and quadrupole interactions are equal and unequal in magnitudes, a remarkable change in some of the results is observed. These * We have published this work in the Journal: Nuclear Instruments and ethods in Physics Research B (004) Please see the reprint at the end of thesis.

2 Chapter II 8 observations are very much useful for the angular and polarization measurements as shown below.. Theory The interesting and important part of the hyperfine structure is the magnetic part arising from the interaction of the nuclear magnetic dipole moment with the magnetic field H due to the atom s own electrons. The magnetic field is supposed to be parallel to the z-axis. The Hamiltonian for the magnetic interaction [7] is H m =.H = g N I.H, (.) and the energy levels are E m = H m I / I = g N Hm I (.) with m I = I, I,..., I and N is the nuclear magneton and g is the nuclear g-factor. According to Eq. (.) there are (I + ) equally spaced energy levels. In general, a gamma transition between the ground and the excited states of spins I g and I e must conserve the z-component of angular momentum, i.e., the angular momentum, L, carried off by the gamma ray must satisfy I g I e L I g + I e with L 0 A transition with L = is called an electric dipole E transition, if it is accompanied by a change in parity otherwise it is a magnetic dipole transition.

3 Chapter II 9 The effective magnetic field acting at the nucleus arising from the atom s own electrons is usually called the internal field. When an atom is placed in a regular crystalline environment, it loses its spherical symmetry and the electrostatic field interacts with the atom at the lattice site. If the surrounding crystal symmetry is less than cubic symmetry then electric field gradient at the lattice site becomes non-zero and interacts with the quadrupole moment of the nuclear state. The nuclear projection quantum number states get split partially, depending upon nuclear spin. The quadrupole interaction Hamiltonian can be expressed as [8, 9] H Q e Qq η [I z I(I ) (I I )] 4I(I ), (.) where Q, q, I and are quadrupole moment of the nucleus, electric field gradient, spin of the state under consideration and asymmetry parameter, assuming non zero, respectively. If both magnetic and electric quadrupole interactions are simultaneously present then the complete Hamiltonian [0] is H gμ N e Qq η I '. H [I z I(I ) (I I )] 4I(I ) z, (.4) where ' I z and I z are defined with respect to two different axis systems.

4 Chapter II 0 For a single photon, the transition probability for an emission or absorption process depends on the interaction energy H' which can be given by the scalar product of current density j of the source and the vector potential A of the field. Since the interaction energy is a zero-rank tensor, H' must be given as [] H' 0 b m ( ) m A m T m ( N), (.5) where T m (N) is a tensor of rank l in the co-ordinates of the nucleus and related to its multipole moments. The tensor T m (N) can be obtained from the multipole solution of the axwell s electromagnetic field equations []. In the ossbauer resonance, the initial state i of the nucleus having the angular momentum I i and other quantum numbers i can be given as Ii i mi a γ, I, m, (.6) mi i i i and final state of the nucleus is If f b γ, I, m, (.7) mf mf f f f where Ii a mi and b If mf can be obtained from the diagonalization of the hyperfine interaction Hamiltonian. In the radiative nuclear transition the possible multipole radiations generally encountered in the ossbauer resonance are of the dipole and quadrupole character []. The nature of the electric or magnetic transition radiation is contained in the definition of A m. Hence, the H' for the nuclear radiative transition is

5 Chapter II m m m ] A )T ( δ A )T ( Const[ H'. (.8) The transition probability for the nucleus to decay form state i to the state f is proportional to the square of the matrix element f H' i. The intensity of radiation in a given direction n can easily be computed using Eqs. (6)-(8) and it is proportional to the expression given by ' f i f i f i f i ',' I m' I m I m' I m,m' m,m' m m' -' m' I I m m I I ) ( b b a a f f f f i i i i f f i f A A m A A f i f i m, f i f i m m' m m' I I m m I I ) ( Re ', ' f i f i f i f i m' m m' m' m' I I m m m I I ) ( m m m A m A. (.9) The amplitudes i i I m a and f f I m b can be evaluated by the first order perturbation theory because H Q is much smaller than the energy of the ossbauer transition and the admixture coefficient, the ratio of E to, becomes zero [4, 5]. When a gamma-ray emitting or absorbing ossbauer nucleus is placed in the presence of magnetic and electric fields [6], the new eigen energies and eigen states come into existence.

6 Chapter II Let us consider radiation obtained between the two spins I e = / (first excited state spin) and I g = / (ground state spin). When the nucleus decays from the first excited state to the ground state, eight lines are obtained.. Polarization and Angular Distribution of the Radiation The ossbauer radiation measurements of angular and polarization distribution of a particular energy should give hyperfine field parameters. There are some unique features in the measurement of the degree of polarization in the radiation emitted from a single crystal, complications due to anisotropic Debye-Waller-factor [7] which always affect the angular distribution of the ossbauer radiation in single crystals, will be absent. It is almost impossible to separate the hyperfine field parameters from unavoidable crystal orientation dependent Debye-Waller-factor if one is to measure only the angular distributions. These complications do not exist in polarization measurements. Therefore, the polarization determination has several advantages over the angular distribution measurements. An attempt has been made by Housley et al [8] to understand the polarization effects in ossbauer absorption or emission of radiation in single crystals. Here they have considered that the principal axis system of the EFG tensor coincides with crystal fixed axis system. However, the principal axis system of the EFG tensor has no a priori relation to the crystal axis system. Hence, the proper calculations become essential to understand the polarization measurements.

7 Chapter II The polarization measurements are also useful for the determination of mean square displacement tensors in a single crystal. The polarization of the electromagnetic radiation can be calculated by two methods. The Stokes method [9] is ideally suited when the measurements allow the determination of the relative phase of the electric field vectors, which are orthogonal at a given point, specifying the eletromagnetic radiation [0]. The Stokes method is applicable when the measurements allow the introduction of the desired relative phase difference between the orthogonal electric-field vectors specifying the radiation []. Such polarization analysers are not available in the region of the X-ray wavelengths. Thus, the Stocks method for the polarization measurements of the ossbauer gamma-rays is not accessible. The density matrix formalism [4,5] gives the degree of polarization which is measurable for the gamma-radiation. In the determination of polarization it is essential that we should consider the coherent properties of the radiation, the coherence in this context is to be interpreted as that property which retains degree of polarization of the electromagnetic wave as it propagates in space from the source to the detector. In many nuclear transitions the ossbauer- radiation is highly coherent, for example for 57 Fe, coherent length 0 m. Since the nuclear hyperfine interaction admixes the m-projection quantum number states, hence the radiation emitted from such admixed states should carry in its degree of polarization the information about hyperfine-field parameters. When the radiation is emitted in the direction n ()

8 Chapter II 4 with respect to a co-ordinate system, the electric field may be represented as E ê + E ê, where E and E are the field amplitudes in the direction of the unit vectors ê and ê. The coherency matrix by definition is [] ρ ρ ρ ρ ρ E θ * E θ.e * E θ.e, (.0) E and the intensity = E + E = trace of the matrix. (.) The degree of polarisation P is defined as the ratio of the intensities of the polarized portion to the total intensity P I Polarized 4 ρ /(ρ ρ ), (.) ITotal where is the determinant of matrix. P = for the monochromatic radiation since = 0, and the wave is said to be completely polarized, for P = 0, the wave is said to be completely unpolarized. In all other cases (0<P<), the radiation is partially polarized []. In order to get intensity of the different ossbauer lines we need to evaluate matrix elements for each transition and square of the total transition amplittude should give the radiation intensity. Consider and to be the polar angles, specifying direction of the out coming gamma-ray in the PAS. The unit

9 Chapter II 5 vectors xˆ, ŷ, and ẑ in rectangular co-ordinate system can be related to ê, ê and ê in the spherical co-ordinate system, as given in Fig... Z ê N ê ê O Y X Fig.. Unit vectors ê, ê, and ê are mutually orthogonal and ê is perpendicular to the plane defined by OZ,O or ON. xˆ = ê sin cos + ê cos cos ê sin, ŷ = ê sin sin + ê cos sin + ê cos (.) ẑ = ê cos ê sin. The radiation fields are given as [] /6π[cos (xˆ iŷ) sinθ exp ( i )ẑ], A A 0 i /8πsin [sin xˆ cos ŷ]. (.4) The degree of polarization, coherency matrix and intensity of each line in a single crystal have been calculated in both the systems, namely; (Section..) the

10 Chapter II 6 principal axis system (unrotated system) and (section..) the crystal fixed axis system (rotated system or laboratory system)... The Principal Axis System (Unrotated System) The transition amplitudes for all allowed eight lines are calculated according to the selection rule m = ±,0. The results are given in the Table-., where Cs are the Clebsch-Gordan Coefficients []. Table-.. Transition amplitudes in terms of Clebsch-Gordan Coefficients of each line. Line No. Transition C( ) m A m. ' ' C A C A. ' '. ' ' C A 0 C A 0 4. ' ' C A C 4 A 5. ' ' C 5 A C A 6 6. ' ' 7. ' ' C 6 A 0 C 7 A 0 8. ' ' C 7 A C 8 A

11 Chapter II 7 From Eqs. (.) and (.4) the values of A ± and A 0 are given by A /6π[ê θcos θcos - êcosθs in iêθcos θsin iêcosθ cos êθ s in cos iê sin θsin ],.5) θ 0 A i /8π sin [êθcosθcos sin - êsin êθ sin cos cos êcos ]. Now let us collect and dependent parts of the electric field E. This can be obtained just by taking the coefficients of ê and ê in Eq. (.5). Thus, E (A ) /6 [cos i sin ], (.6) 0 E (A ) 0, and E (A ) /6 cos [sin i cos ], (.7) E 0 (A ) / 8 sin. i Calculation of the coherency matrix (), trace of the matrix and degree of polarization (P) for all eight line transitions in radiation [4]. For the transition +/ +/: Using Eq. (.6), E θ C A C A C (cos isin) C (cos isin) 6 6

12 Chapter II 8 E ( C C )cos i( C C 6 ) sin 6π K cos il sin, where K C C, L C C, 6 6 K cos il sin K cos sin E. E * il ρ K cos Lsin, 6π Using Eq. (.7), E C A CA C cos (sin icos ) ( C) cos (sin icos ) 6 6 E cos K sin ilcos, 6π cos 6 6 K sin il cos cos K sin cos E. E * il ρ cos θ K sin Lcos, 6π Using Eqs. (.6,,7), = E.E * = ( )[ K cos il sin ] ( cos )[ K sin il cos ] 6 6 cosθ (K 6π L )cos sin ik L

13 Chapter II 9 and ) 6 6 K cos il sin ( cos ) K sin cos * E. E ( il cosθ (K L )cos sin ik L. 6π Now using Eq.(.0), [ K cos 6 cos L sin ] cos ( K L ) cos sin ik L 6 ( K L ) cos sin ik L cos K sin L cos cos ( K L )(cos sin ) K L K L. 9 L K 6 cos 0 (.8) 56 Trace of the matrix = + K (cos cos θsin ) L(sin cos θcos ), (.9) 6π and the degree of polarization, 4 P. (.0) ( ) Similarly, the coherency matrices, trace of the matrices and degrees of polarization have been calculated for all eight lines. The results are given in Table-.. The graphical representations are given in Figs..(a).(c). When, (a) the magnetic and the quadrupole interactions are equal, i.e. H m = H Q, (b) the magnetic field is five times of the quadrupole interaction, i.e. H m = 5H Q, (c) the quadrupole interaction is five times of the magnetic field, i.e. H Q = 5H m. These combinations have been taken for computer calculations.

14 Chapter II 40

15 Chapter II 4 where, 8 C 7 C 4 K, 6 C 5 C,K 4 C C,K C C K 8 C 7 C 4 L, 6 C 5 C,L 4 C C,L C C L η ] a b / η ) a b ( a b [ η C C b a b a b a b a b a b a R S T U V W L N O Q P P P R S T U V W L N O Q P P P ( ) / [ ( ) / ] /, C b a b a b a R S T U V W L N O Q P P P [ ( ) / ] /, C b a b a b a b a b a b a 4 R S T U V W L N O Q P P P R S T U V W L N O Q P P P ( ) / [ ( ) / ] /,

16 Chapter II 4 C5 L N L N R T U W b b / b S ( ) a V a a O Q P O b b R U/ / b S a V [ ( ) ] P a a T W Q P, C6 C7 b b R U/ O/ b S a V [ ( ) ] P a a L N L N T L N R T W U W b b / b S ( ) a V a a Q P O Q P O b b R U/ / b S a V [ ( ) ] P a a T W Q P,, C8 b b R U/ O/ b S a V [ ( ) ] P a a L N T e Qq a and b = g 4I(I ) N H. W Q P,

17 Chapter II 4

18 Chapter II 44

19 Chapter II 45

20 Chapter II 46

21 Chapter II 47

22 Chapter II 48

23 Chapter II 49

24 Chapter II 50

25 Chapter II 5

26 Chapter II 5

27 Chapter II 5

28 Chapter II 54

29 Chapter II 55

30 Chapter II 56

31 Chapter II 57

32 Chapter II 58

33 Chapter II 59

34 Chapter II 60

35 Chapter II 6... Transformation of Radiation Field Amplitudes from the PAS to the CFAS Since there is no a priori reason that the CFAS and the PAS should coincide with one another. Therefore, the transformation of observations from the PAS to the CFAS becomes essential. Such transformations have been carried out through rotation matrices [0], D(,, ), where, and are Euler angles, as given in Fig... (a) (b) (c) Fig... Representation of the direction of gamma-ray emission in the co-ordinate system. (a) The Euler angles, and and their corresponding rotation which carry the (x, y, z) co-ordinate system into the (x', y', z') co-ordinate system. (b) Orthogonal co-ordinate system denoted as the crystal fixed axis system. (c) Principal axis system defined by the nuclear hyperfine interaction gives [6] The rotation operator R (,,) operating on spherical harmonics A l m (,) l j m m' R A (, ) R D (,, ) A (, ) (.) l ml m', m l c c with [5]

36 Chapter II 6 j D (,,) = exp [ im'] exp [ im] m ',m s s ( ) [( j m)!( j m')!( j m)!( j m')!] / jmm' s (cos / ) (.) s!( j s m')!( j m s)!( m' s m)! ( sin/) m' m+s. The field amplitudes for the radiation from Eq (.) are R A R = D, A + D, A + D 0, A0, R A R = D, A + D, A + D 0, A0, (.) R A 0 R = D,0 A + D,0 A + D 0,0 A0, as given in the appendix A. Thus, E [R(A ± )R - ] = /6 [f ± i f ], E [R(A 0 ) R - ] = i /8 f5, (.4) E [R(A ± ) R - ] = /6 [f f 4 ], and E [R(A 0 ) R - ] = i /8 f6, (.5) using the Eqs (.6), (.7) and (.), where

37 Chapter II 6 f ( cos ) (cos cos( ) sin sin( )) ( cos ) (cos cos( ) sin sin( )), (.6) f ( cos ) (sin cos( ) cos sin( )) ( cos ) (cos sin( ) sin cos( )), (.7) f ( cos ) cos (sin cos( ) cos sin( )) ( cos ) cos (sin cos( ) cos sin( ) (sin sin sin )), (.8) f 4 ( cos ) cos (cos cos( ) sin sin( ) ( cos ) cos (cos cos( ) sin sin( ) (sin cos sin )), (.9) f 5 = sin sin( ) (.0) and f 6 = [cos sin cos ( )], (.) as given in the appendix B. For the transition +/ +/: Using Eq (.4), L N E R C A C A R ( ) 6 ( C C ) f i(c C ) f O QP

38 Chapter II 64 K 6 f il f, L N O Q P L N E R( C A C A ) R E * R ( C A C A ) R K 6 f L f, O Q P Using Eq. (.5), L N E R ( C A C A ) R 6 ( C C ) f i(c C ) f 4 O QP K f 6 il f 4, L N O Q P L N E R( C A C A ) R E * R ( C A C A ) R K 6 f L f4, O Q P Using Eqs. (.4,.5), L N O Q P L N E R( C A C A ) R E * R ( C A C A ) R K 6 f f L f f ik L (f f f f ), 4 4 O Q P

39 Chapter II 65 and E * R C A C A R E R C A C A R ( ) ( ) L N O Q P L N O Q P K 6 f f L f f ik L (f f f f ). 4 4 Now using Eq. (.0), L N L N O QP K f L f K f f L f f ik L (f f f f ) K f f L f f ik L (f f f f ) K f L f O QP = 0, (.) Trace of the matrix (intensity) = +. K (f f ) L (f f ), 6π 4 (.) and the degree of polarization, 4 P. (.4) ( ) Similarly, the coherency matrices, trace of the matrices (intensities) and degrees of polarization have been calculated for all 8 lines (transitions). The results are given in the Table-.. The graphical representations are given in Figs..(a).0(c). When, (a) the magnetic and the quadrupole interactions are equal,

40 Chapter II 66 i.e. H m = H Q, (b) the magnetic interaction is five times of the quadrupole interaction, i.e. H m = 5H Q, (c) quardrupole interaction is five times of magnetic interaction, i.e. H Q = 5H m. These combinations have been taken for computer calculations. The symmetries of each line are calculated as shown in Table.4.

41 Chapter II 67

42 Chapter II 68

43 Chapter II 69

44 Chapter II 70

45 Chapter II 7

46 Chapter II 7

47 Chapter II 7

48 Chapter II 74

49 Chapter II 75

50 Chapter II 76

51 Chapter II 77

52 Chapter II 78

53 Chapter II 79

54 Chapter II 80

55 Chapter II 8

56 Chapter II 8

57 Chapter II 8

58 Chapter II 84

59 Chapter II 85

60 Chapter II 86

61 Chapter II 87.4 Result and Discussion The degrees of polarization, coherency matrices, intensities, ratio of intensities w.r.t. first line and symmetries of each line in a single crystal have been calculated in the systems namely; (section.4.) unrotated system and (section.4.) rotated system..4. Unrotated System (the Principal Axis System) When magnetic and quadrupole interactions are simultaneously present then eight lines (transitions) are obtained. The degree of polarization is calculated separately and it is found unity for each line. The wave is said to be completely polarized and the components E x and E y of electric vector are mutually coherent and its phase is equal to the difference between the phases of the two components. The coherency matrix, degree of polarization, intensity and ratio of intensities are shown in Table.. The intensities of lines,, 6 and 7 are found to depend upon the angle and that of the lines, 4, 5 and 8 are found to depend upon the angles and. The symmetries of each line are calculated as shown in the Table.4. (Assuming, I I I I I I I I a, b, 4 c, 5 d, 6 e, 7 f and 8 g for all I I I I I I Figs..(a).0(c) where I, I,.I 8 are the intensities of the lines,,,.8, respectively).

62 Chapter II When the agnetic and the Quadrupole Interactions are Equal: (i) In Fig..(a) (c), the graphs a, b, e and f are maximum at 00, 80, e>f>b> a, and minimum at 0, 90. The graphs c, d and g have opposite trends with respect to each other. The two fold symmetry is found in each case. (ii) In Fig..4(a)-(c), the graphs a, b, d, e, f and g are maximum at 00, 80, e>f>g>d>b>a, and minimum at 0 and 90. The graph c has opposite trend with respect to other graphs. It has maxima at 00, 80. The symmetry of is seen in each graph. (iii) In Fig..5(a)-(c), the graphs a, b, c, e and f are maximum at 00, 80, e>f>c>b>a, and minimum at 0, 90. The graphs d and g have opposite trends with respect to other graphs. They have maxima at 0, 90 and minima 00, When agnetic Interaction is Five Times of Quadrupole Interaction: In Figs..6-.8, the trend of the graphs a-g is found to be the same as in Figs..-.5, respectively. The graphs b, d and g show sharp rise whereas a and c fall off rapidly. The behaviours of c and d are entirely different from each other..4..when the Quadrupole Interaction is Five Times of agnetic Interaction In Figs..9-., the trend of the graphs a-g is found to be the same as in Figs..-.5, respectively. The graphs a, c and f show sharp rise whereas the b and e show sharp fall.

63 Chapter II Rotated System (Crystal Fixed Axis System) The transformation of calculations is carried out from the PAS to the CFAS. It is found that degree of polarization is unity for each line in this system also. The intensity depends upon the angles, and Euler angles, and. The determinant of the coherency matrix, trace of the matrix, degree of polarization and relative intensities have been calculated as shown in Table.. The symmetries of each line have been determined in various parameters,,, and. They are shown in Table When the agnetic and Quadrupole Interactions are Equal (i) In Fig..(a) (c), the graphs a, b, e and f are maximum at 5, 5, e>f>b>a, and minimum at 45, 5. The graphs c and d & g have opposite trends with respect to each other. The two fold symmetry is found in each case. (ii) In Fig..(a)-(c), the graphs a, b, e, f, d and g are maximum at 0, 80, 60 e>f>g>d>b> a, and minimum at 90, 70. The graph c has opposite trend with respect to other graphs. It has maxima at 90, 70 and minima at 0, 80, 60. The symmetry of is seen in each graph. The peak of the graphs are shifted by 45 in forward direction. (iii) In Fig..4(a)-(c), the graphs a, b, c e and f are maximum at 90, 70, e>f>c>b> a, and minima at 0, 80, 60. The graphs d and g have opposite trends with respect to other graphs. They have maxima at 0,

64 Chapter II 90 80, 60, g>d, and minima at 90, 70. The rest of the behaviour is like that of case (ii)..4..when the agnetic Interaction is Five Times of the Quadrupole Interaction In Figs..5-.7, the trend of the graphs a-g is found to be the same as in Figs..-.4, respectively. The graphs b and g rise sharply whereas the a, c and f show sharp fall. The behaviour of the graphs c and d are again entirely different from each other..4.. When the Quadrupole Interaction is Five Times of the agnetic Interaction In Figs..8-.0, the trend of the graphs a-g is found to be the same as in Figs..-.4, respectively. The graphs c and f show sharp rise whereas the graph b shows sharp fall. By looking at all the graphs it turns out that the relative intensities is very strong functions of various parameters viz.,, and in the case of unrotated system and,,,, and in the case of rotated system. It is seen that the relative intensities becomes most sensitive when the magnetic interaction is greater than quadrupole interaction. It is also found that the ratio of intensities becomes more sensitive when the asymmetry parameter () approaches unity.

65 Chapter II 9.5 Conclusion An inspection of the calculation and graphs reveals that the degree of polarization remains unity, i.e., the electromagnetic radiation remains plane polarized in unrotated and rotated systems. But the relative intensities show strange behaviour with various crystal parameters, viz., and when H m =H Q, H m =5H Q and H Q =5H m are in unrotated and rotated systems. It is also seen that the relative intensities becomes more sensitive when the electric field gradient is highly asymmetric. This sharp change in relative intensities leads to accurate measurement of the parameters which are involved in these interactions. Certain symmetries have been found in both the cases, namely unrotated and rotated systems for each line.

66 Chapter II 9 References [] N.N. Green Wood, T.C. Gibb, ossbauer Spectroscopy, Chapman and Hall, London, 97, p. 6. []. Born, E. Wolf, Principles of Optics, London, 959, p.490. [] Alimuddin, Asharfi Lal, K. Rama Reddy, IL Nuovo Cimento B(N-) (976) 89. [4] E. Wolf, Nuovo Cimento (954) 884. [5] J. Pancharatnam, Proc. Ind. Acad. Sci. 44A (956) 98. [6] A.R. Edmunds, Angular omentum in Quantum echanics, Princeton, NJ, 957, p.5. [7] G.K. Wertheim, ossbauer Effect, Academic Press, New York, 965, p.7. [8] C.P. Slichter, Principles of agnetic Resonance, Harper and Row, New York, 96, p.74. [9] E.L. Hahn, T.P. Das, Solid State Physics, Academic Press, Inc. Publishers, New York, 958, Suppl., p.5. [0] N.N. Green Wood, T.C. Gibb, ossbauer Spectroscopy, Chapman and Hall, London, 97, p. 65. [].E. Rose, Elementary Theory of Angular omentum, New York, 96, p. 70. [] J.D. Jackson, Classical Electrodynamics, New York, 96, p.98. [] A. H. uir Jr., K. J. Ando, H.. Coogan (Eds.), ossbauer Effect Data Index , New York, 966.

67 Chapter II 9 [4] V. S. Shirley, J. O. Rasmussen, Phys. Rev. 09 (958) 09. [5] W.T. Achor, W.E. Phillips, J.I. Hopkins, S. K. Haynes, Phys. Rev. 4 (959) 7. [6] G. K. Wertheim, ossbauer Effect, Academic Press, New York, 965, p. 74. [7] R.. Housley, U. Gonser, R.W. Grant, Phys. Rev. Lett. 0 (968) 79. [8] R.. Housley, R. W. Grant, U. Gonser, Phys. Rev. 78 (969) 54. [9] G.G. Stokes, Trans. Cambr. Phil. Soc. 9 (85) 99, reprinted in his athematical and Physical Papers Vol., London, 90, p.. [0] U. Fano, J. Opt. Soc. Am. 9 (949) 859. [] R.C. Brown, J.A.R. Griffith, O. Karbon, S. Roman. in Proc. of the rd Int. Symp. on Polarization Phenomena in Nuclear Reactions, adison, Wis., 970, p.78. [] S.W. arshall, R.. Wilenzick, Phys. Rev. Lett. 6 (966) 9. [] S. De. Benedetti, G. Lang and R. Ingalls, Phys. Rev. Lett. 6 (96) 60. [4] Sikander Ali, Alimuddin, J. Phys. Soc. of Jpn. 69 (7) (000) 05. [5].E. Rose, Elementary Theory of Angular omentum, New York, 96, p. 4.