Anisotropy of the recoilless fractions in Fm3m crystals
K. Ruebenbauer and U.D. Wdowik
Mössbauer Spectroscopy Division, Institute of Physics, Pedagogical University
PL-30-084 Cracow, ul. Podchorążych 2, Poland
E-mail: sfrueben@cyf-kr.edu.pl
Short title: Anisotropy of the recoilless fractions
Keywords: anisotropy, recoilless fraction, quartic, sodium chloride
Abstract
A general formalism describing recoilless fraction in terms of spherical tensors
conforming to the crystal symmetry is outlined. The above formalism has been already
applied to describe successfully previously obtained data on the recoilless fractions in single
crystals of sodium chloride. The original data were obtained by using Rayleigh scattering of
the Mössbauer radiation. It was found that above the quantum region and at a temperature
high enough to neglect anisotropy of the ionic form-factors there is some anisotropy of the
thermal motion of the entirely quartic character. On the other hand, quadratic terms in the
wave-vector transfer are completely quasi-harmonic. A departure from the Gaussian thermal
distribution is quite significant for both cations and anions. The anisotropy tends to diminish
at high temperatures.
1. Introduction
Recoilless fractions are very good probes of the local atomic motions around the
equilibrium positions in solids [1-3]. They could be measured either by using coherent
methods, i.e., X-ray or neutron diffraction, or by incoherent methods like the Mössbauer
spectroscopy. Incoherent methods do not suffer from the static disorder contributions, but
they usually operate at the constant wave-number. On the other hand, coherent methods have
generally low energy resolution resulting in the significant thermal diffusive contamination
particularly at high temperatures. The exception is the Rayleigh scattering of the Mössbauer
radiation either in the energy or time domain. It is characterized by the very good energy
resolution.
Nuclear methods seem superior as the scatterers are truly point-like. However, the
standard nuclear neutron diffraction has generally poor energy resolution at the wave-number
transfers required, while the Mössbauer spectroscopy operates at the constant wave-number
and for the limited range of scatterers. One has to realize, that the nuclear (Mössbauer)
forward or Bragg scattering of the synchrotron radiation is a two step process, and hence, it
suffers from the constant wave-number restriction as well.
The paper is aimed at the analysis of the recoilless fractions behavior in sodium
chloride based on the data obtained by the Rayleigh scattering of the Mössbauer radiation
from single crystals [4-5]. Very high intensity 183W Mössbauer sources (about 70 Ci at the
termination of irradiation in the high flux reactor) called super-sources were used for the
purpose [6].
Section 2 deals with the general formalism used to describe the recoilless fraction and
its anisotropy. Particular attention is paid to the optimal parameterization involving the
minimum number of parameters. This section is devoted to the reconstruction of the spatial
distribution functions from the recoilless fractions as well. Section 3 is devoted to the
application of the formalism to the NaCl data, and finally Section 4 summarizes results.
1
2. Parameterization of the recoilless fraction
The recoilless fraction f (q ) is a smooth function of the wave-vector transfer q to the
system and it satisfies the following conditions:
0  f (q )  1 , f (q  0)  1 and lim [ f (q )]  0 .
q
(2.1)
One can define symmetric, against the wave-vector transfer inversion, R (q ) and
anti-symmetric X (q ) amplitudes, respectively, in the following way:
R(q )  12  [ f (q )]1/ 2  [ f (q )]1/ 2  and X (q )  12  [ f (q )]1 / 2  [ f (q )]1 / 2  .
(2.2)
For a single "independent" point-like spherical "scatterer" one can obtain the spatial
distribution amplitude A(r ) versus position r as:




A(r )   d 3 q R(q ) cos( q  r )   d 3 q X (q ) sin( q  r ) ,
(2.3)
and a corresponding spatial probability density function  (r ) as:

 (r )  N A(r ) A (r ) with N   d 3 r A(r ) A* (r ) .
1
*

(2.4)
The spatial probability density function could be defined in coordinates x centered on
the average position r  in the following way:

x  r   r  with  r    d 3 r r  (r ) .

(2.5)
The resulting smooth function  ( x ) obeys the following relationships:

 ( x )  0 ,  d x  ( x )  1 and
3
-

 d x x  (x)  0
3
-
(2.6)
with ( x)  N
A(x) being a pseudo wave-function. The function  ( x ) is not a true wave
function as it cannot be associated with a particular quantum state except the ground state at a
temperature being close to null. On the other hand, the product (x) * (x' ) can be interpreted
as a matrix element of the density matrix.
The function  ( x ) could be back-transformed into the recoilless fraction f (q ) by the
following set of operations:
1 / 2

  3


f (q )    d x  R ( x ) cos( q  x )   d 3 x  X ( x ) sin( q  x ) 


 
 R ( x )  12  [  ( x )]1 / 2  [  ( x )]1 / 2  and
2
 3

  d x  R ( x )  , where


 

 X ( x )  12  [  ( x )]1 / 2  [  ( x )]1 / 2  with
 

x  x    d 3 x x [  ( x )]1 / 2 
 

 3

  d x [  ( x )]1 / 2  .


 

(2.7)
2
Hence, one can obtain the "thermal" spatial probability density function  ( x) upon
having measured corresponding "dynamical" recoilless fraction f (q ) .
The recoilless fraction f (q ) could be expressed as:
f (q )  exp[ W (q )] with W (q )  0 , W (q  0)  0 and W (q ) q   .
(2.8)
The smooth function W (q ) takes on the following form [7]:

l l m


(l )
(l )
W (q )   q l   R (l )    bmk
mk
( )  ,
l 2
m 0 k 0


(2.9)
where q  0 stands for the wave-number transfer to the system,  and  stand for the polar
and azimuthal angles of the wave-vector transfer to the system, respectively, and they are
defined in the orthogonal right-handed Cartesian reference frame centered on the average
position r  and used to evaluate the recoilless fraction. Hence, the wave-vector transfer to
the system q is defined as:
 sin  cos  


q  q  sin  sin   .
 cos  


(2.10)
Parameters  R  describe isotropic part of the recoilless fraction, while the
(l )
parameters bmk
describe anisotropy of the recoilless fraction. The latter parameters are usually
required to satisfy the following condition:
(l )
l

l m
2
 b  d sin   d 
m 0 k  0
(l )
mk
(l )
mk
0
( )  0 .
0
(2.11)
The above mentioned condition is "automatically" satisfied for all odd indices l , while
for the even indices l it leads to the following constraint on the parameters b2( 2ml''2k2') :
2 l ' 2 2 l ' 2 2 m '
 
2 m ' 0
2 k ' 0
( 2 l ' 2 )
2 m'2 k '
b

2
0
0
 d sin   d 
( 2 l ' 2 )
2 m'2 k '
( )  0
with l '  2,3, ; m'  0,1,  l '1; k '  0,1,  , l 'm'1 .
(2.12)
The above constraint "removes" one parameter from the even terms.
(l )
( ) take on the form:
Functions mk


l!
(l )
 (sin  ) mk (cos  )l mk (sin  ) k (cos  ) m .
mk
( )  
 m! k! (l  m  k )! 
(2.13)
Hence, the following relationships are obeyed:

2
 d sin   d 
(l )
mk
0
( )  0
0
(2.14)
except for the all indices l , m and k being even. For the latter case one obtains:
3

2
 d sin   d 
(l )
mk
0
( ) 
0
2 l! (l  m  k  1) 2 (m  1) 2 (k  1) 2
,
m! k! (l  m  k )! (l  3) 2
(2.15)
where the symbol  denotes the Euler-gamma function.
In particular the function 00( 2l '2) ( ) obeys the following relationship in accordance
with the expression (2.15):

2
4
( 2l ' 2 )
d

sin

0
0 d 00 ( )  2l '1 .
(2.16)
The above outlined formalism allows to calculate spatial density function along the
axis pointing in the ( ) direction and going through the average position r  . Namely, the
recoilless fraction f (q ) could be rewritten in the form f (q )  f (q |  ) where:
R(q |  ) 
X (q |  ) 
1
2
1
2
[ f (q |  )]
[ f (q |  )]
1/ 2
1/ 2

.
 [ f (q |    ,   )]1 / 2 and
 [ f (q |    ,   )]
1/ 2
(2.17)
Therefore, the amplitude A(r |  ) takes on the form:




A(r |  )  2   dq R(q |  ) cos( qr )   dq X (q |  ) sin( qr )  ,
0
0

(2.18)
while the function  (r |  ) is expressed as:

 (r |  )  N ( ) A(r |  ) A (r |  ) with N ( )   dr A(r |  ) A* (r |  ) .
1
*

(2.19)
The centered coordinate x follows from the expressions:

x  r   r  with  r    dr r  (r |  ) .

(2.20)
Hence, the spatial density function along the above axis  ( x |  ) satisfies the
following conditions:
 ( x |  )  0 ,




 dx  (x |  )  1 and
 dx x  (x |  )  0
(2.21)
with  ( x |  )  N ( ) A( x |  ) being a corresponding pseudo wave-function. This
function has a similar meaning as the function  ( x ) , however it is defined in a single
dimension along the axis pointing in the ( ) direction. Hence, the function  ( x |  ) is the
probability density function for finding a particle in the plane perpendicular to the above axis
and being offset from the average position by x.
One can again back-transform the function  ( x |  ) into the recoilless fraction in the
direction ( ) by the following set of operations:
1 / 2
4

 


f (q |  )    dx  R ( x |  ) cos( qx)   dx  X ( x |  ) sin( qx) 


 
 R ( x |  )  12  [  ( x |  )]1/ 2  [  ( x |  )]1/ 2  ,
2


  dx  R ( x |  )  with


 

 X ( x |  )  12 [  ( x |  )]1/ 2  [  ( x |  )]1/ 2  and
 
x  x    dx x [  ( x |  )]1/ 2
 





  dx [  ( x |  )]1/ 2

 

 .


(2.22)
It is interesting to note, that the recoilless fraction f (q |  ) described solely by terms
with l  2 results in the following Gaussian  ( x |  ) function:
 (x |  )  G(x |  )  [( )]1/ 2 exp  [x   r  ]2 [ ( )] ,
(2.23)
where:
2 2 m
( 2)
( 2)
 ( )   R ( 2)    bmk
mk
( ) and  ( )  0 .
m 0 k 0
(2.24)
Hence, the following deviation function D ( x |  ) is a good measure of the presence
of the terms with l  2 :
D( x |  )   ( x |  )  G ( x |  ) with

 dx D(x |  )  0 .

(2.25)
For a purely Gaussian behavior r   0 and r   0 . If one neglects all terms with
l  4 a cubic environment with the local inversion centre has r   0 and r   0 , and it is
( 4)
described completely by:  R ( 2)  ,  R (3)  ,  R ( 4)  and b40
. The isotropic term  R (3)  is very
small for a cubic environment exhibiting local inversion centre [8], and one is left with three
( 4)
parameters:  R ( 2)  ,  R ( 4)  and b40
in the coordinates coinciding with the main crystal axes
(for a strict cumulant approximation odd isotropic terms always vanish).
There are five independent components of the b̂ ( 2 ) tensor, ten independent
components of the b̂ (3) tensor and fourteen independent components of the b̂ ( 4 ) tensor in the
case of general symmetry. For the most general symmetry the bˆ ( l ) tensor has 12 (l  1)(l  2)
independent components for odd l and 12 (l  1)(l  2)  1 such components for even l .
The above mentioned cubic system has the following function W (q ) :
( 4)
5 (sin  ) 2 [1  (sin  ) 2{1  (sin  ) 2  (sin  ) 4 }]  1  .
W (q )  q 2  R ( 2)   q 4  R ( 4)   b40
(2.26)
The expression (2.26) follows from the general expressions [(2.9)-(2.15)] taking into account
isotropic terms up to the quartic one with the exception of the cubic term, and considering
solely quartic anisotropy with the following constraints due to the cubic symmetry and
( 4)
( 4)
( 4)
( 4)
( 4)
( 4)
 b04
 b00
 2b22
 2b20
 2b02
presence of the inversion centre: b40
; remaining are
zero. The above choice satisfies expression (2.11) and causes that the anisotropy is described
( 4)
( 4)
by a single parameter b40
. Positive values of the b40
parameter cause the largest
displacements to be along the main axes, while the negative values give the minimum
displacements along the main axes.
5
The anisotropy is solely due to the quartic (and eventually higher even) terms. For a
constant q maximum discrepancy occurs between directions pointing along the cube edge and
( 4)
along the cube diagonal, respectively. The above discrepancy amounts to: 53 q 4 b40
. The next
most likely to occur and allowed tensor is b̂ ( 6 ) . It has two non-trivial components for the
(6)
(6)
above environment: b60
and b40
.
The function:
W (q )  q 2  R ( 2)  q 4  R ( 4 ) 
(2.27)
is plotted in Fig. 1 for 0     / 2 , 0     / 2 and various settings of the parameter
( 4)
b40
/  R ( 4 )  ranging here between (-1; 3/2). The function W (q ) is described here by the
expression (2.26).



Fig. 1 Possible shapes of the quartic anisotropy in the cubic system. See, expressions (2.26)
( 4)
/ R ( 4 )   0 .
and (2.27) for details. The anisotropy vanishes for b40
For a strictly spherical symmetry all bˆ ( l ) tensors vanish and one is left with isotropic
parameters  R (l )  solely. Usually,  R ( 2)  and  R ( 4)  parameters are dominant with  R ( 2) 
being a sole parameter for a spherically symmetrical harmonic environment.
For a general symmetry in a harmonic environment parameter  R ( 2)  and a tensor b̂ ( 2 )
survive solely.
The presence of the bˆ ( l ) tensors with odd l is the indication of the absence of the local
inversion centre. For such a case neither r   0 nor r   0 .
One has to note, that  R (l )  parameters and bˆ ( l ) tensors strongly depend upon the
temperature.
6
The above parameterization applies to a single particle located at the relatively well
defined position, and in the framework of the independent vibrational dynamics. Observed
quantities are time averaged stationary observables. The particle itself is considered as a
point-like spherical entity.
It seems that currently available experimental methods are able to reach up to the
quartic terms inclusive. The parameter  R (3)  is likely to be too small to be observed by
currently available experimental methods within systems exhibiting local inversion centres
[8].
The above outlined formalism could be easily extended to the cases with poorly
localized and/or overlapping particles which are hard to distinguish one from another.
Namely, for such cases the amplitude A(r ) has to be replaced by:
A(r )   Ans r   rs   ,
ns
(2.28)
where Ans r   rs   stands for the amplitude of the n-th particle in the s-th site at the position
r , while  rs  is the average position of the s-th site. The above substitution generally leads to
a dependent dynamics.
3. Recoilless fractions in NaCl
In order to illustrate the usefulness of the above outlined formalism recoilless fractions
have been calculated versus temperature in the main axis and the chemical unit cell diagonal
directions for the average sodium cation and chlorine anion, respectively, [9] basing upon the
results published in the ref. [4], and for various Bragg reflections. The latter results have been
derived from the Rayleigh scattering of the Mössbauer radiation (RSMR) measurements
performed upon single crystals at various temperatures and for various Bragg reflections
having wave-vector transfer either along the main crystal axis or along the cell diagonal. The
former reflections had even Miller indices, while the latter had either even or odd indices. The
46.5 keV γ-ray line of 183W was used as the radiation source. Experimental details are
described in ref. [4]. The energy resolution of the RSMR method assures that the intensity
under the Bragg reflection is entirely due to the elastic component with the negligible thermal
diffusive contribution.
The lattice constant a(T) evolution with the temperature T has been approximated by
the relationship [9]:
5
a(T)  a 0   C k T k ,
k 1
(3.1)
where the parameters a0 and Ck have been fitted to the data of the refs. [10-12]. The numerical
results are summarized in Table I [9], while the lattice constant versus temperature is plotted
in Fig. 2 including data of the refs. [10-12]. The function a(T) is further abbreviated by the
symbol a. Hence, the "natural" units of length were used in further calculations, i.e., fractions
of the lattice constant at a given temperature. Such an approach is possible in the cubic
systems with the unique length scale.
7
Table I
Coefficients of the lattice constant dependence upon temperature.
a0
C1
C2
C3
C4
C5
5.5870(4)
1.7∙10-7 (68·10-7)
7.8(4)·10-7
-1.34(9)·10-9
1.25(9)·10-12
-4.5(4)·10-16
[Å]
[Å/K]
[Å/K2]
[Å/K3]
[Å/K4]
[Å/K5]
Interpolation valid between 50 K and 950 K. Note irrelevance of the linear term in agreement
with the quantum mechanical expectations (error for the C1 parameter value is much bigger
than the value itself, and hence the parameter becomes irrelevant).
Fig. 2 Evolution of the lattice constant versus temperature at "zero" pressure.
Calculated recoilless fractions have been fitted to the parameters: R ( 2)  / a 2 ,
( 4)
/ a 4 using expressions (2.26) and (2.8) separately for the average sodium
R ( 4)  / a 4 and b40
cation and chlorine anion versus temperature. The following temperature dependencies were
found applying polynomial fits with the minimum number of parameters [9]:
 R ( 2 )  / a 2  A  BT ,
 R ( 4)  / a 4     T 3 ,
( 4)
b40
/ a 4  a  bT  cT 2 .
(3.2)
The numerical values of the parameters are listed in Table II [9]. Additionally, the ratio
( 4)
b40
/  R ( 4 )  was calculated versus temperature for sodium and chlorine. The functions:
8
( 4)
( 4)
/ a 4 and b40
/  R ( 4 )  are plotted versus temperature in Fig. 3 for
R ( 2)  / a 2 , R ( 4)  / a 4 , b40
sodium and chlorine both [9].
Table II
Parameters describing recoilless fractions versus temperature.
A [-]
B [1/K]
α [-]
β [1/K3]
a [-]
Na
1.14(9)·10-4
2.01(2)·10-6
-2.0(4)·10-9
6.51(5)·10-16
2.2(2)·10-8
Cl
5.6(4)·10-5
1.33(1)·10-6
0.0
6.46(3)·10-16
1.41(7)·10-8
b [1/K]
c [1/K2]
-1.38(9)·10-10
4.2(1)·10-13
-1.06(4)·10-10
3.50(5)·10-13
Temperature range covered: 300 K – 950 K for Na and 220 K – 950 K for Cl.
Fig. 3 Parameters of the recoilless fractions plotted as functions of the temperature at "zero"
pressure.
9
The biggest sources of the systematic experimental errors are the following: 1) extinction, 2) static disorder, and 3) non-local character of the scattering ionic amplitudes. It
seems that neither extinction nor the static disorder have any meaning for the samples
investigated [4]. Isotropic contributions to the non-local scattering amplitudes have been
accounted for during the original data evaluation [4]. However, the anisotropic contributions
due to the solid state effects cannot be accounted for reliably leading to the mixing between
genuine recoilless fractions and the ionic scattering amplitudes (distributions of scattering
electrons). This effect becomes stronger at low temperatures where the "thermal" motion is
small, while the electronic distribution remains practically unaffected for the sodium chloride.
It is particularly strong for light atoms with few almost unperturbed core electrons
contributing to the scattering. Hence, the data could be treated as reliable above 220 K for
chlorine and above 300 K for sodium.
Fig. 4 Functions  ( x |  ) and D ( x |  ) plotted versus x/a for sodium and chlorine at various
temperatures and for the angles ( ) corresponding either to the <100> or <111>
crystal directions. One has to note, that the above functions are dimensionless in the x/a
coordinates.
10
Fig. 4 shows reconstructed functions  ( x |  ) and D ( x |  ) plotted in the "natural"
coordinates x/a for various temperatures and two crystal directions, i.e., <100> and <111>,
both for sodium and chlorine [9].
Fig. 5 Potentials U ( x |  ) plotted versus x/a for sodium and chlorine at various temperatures
and for the angles ( ) corresponding either to the <100> or <111> crystal directions.
Corresponding isotropic Gaussian potentials U G ( x |  ) are shown for comparison.
Pseudo wave-functions  ( x |  ) could be used to calculate the effective temperature
dependent and time averaged scalar interaction potential between vibrating particle and the
rest of the crystal. Such a potential (energy) calculated for ( ) direction versus x and gauged
to zero for x = 0 takes on the following form for a single point-like spherical non-relativistic
particle in accordance with the Schrödinger equation:
  2  1
  ( x |  ) (d 2 / dx 2 ) ( x |  )   1 (0 |  ) (d 2 / dx 2 ) ( x |  ) x0 ,
U ( x |  )  
 2M 
(3.3)
where M stands for the mass of the particle and  for the Planck constant divided by 2π. The
above potential approximates interaction between particle and the rest of the crystal in the



11

 
vicinity of the x = 0 point. For a purely Gaussian function (x |  )  G(x |  ) the above
potential takes on a very simple form:

 2
2
 x .
U G ( x |  )  
2
 2M ( ) 
(3.4)
( 2)
The Gaussian potential is isotropic for cubic symmetries as  ( )   R  in such cases.
Fig. 5 shows the above potentials plotted in the "natural" coordinates x/a.
4. Conclusions
One can conclude that quadratic terms are isotropic as expected in the cubic
environment. They are linear versus temperature above the quantum region and maybe the
vicinity of the melting point provided they are plotted in the "natural" coordinates. Hence,
they represent quasi-harmonic motion. Residua at zero temperature are small indicating a
small static disorder at all temperatures within the range investigated. The crystal is strongly
ionic, and hence, the static disorder is expected to be small.
Isotropic quartic terms are practically the same for sodium and chlorine within the
whole temperature range considered. Such a behavior indicates that these terms are due to
collective motions, i.e., interaction between acoustic phonons.
The anisotropy of the motion has the same character for both ions leading to bigger
displacements along the main crystal axes in comparison with the displacements along the
cell diagonals. However lighter ions exhibit bigger anisotropy. It seems that all phonons
contribute to the anisotropy as it depends upon temperature in a more complex way than
quartic isotropic terms. It has to be stressed that the entire anisotropy is of the quartic
character. The motion becomes more isotropic at high temperatures. The data at very low
temperatures cannot be reliably evaluated due to the admixture of the scattering amplitudes
anisotropy.
Departures from the Gaussian "thermal" distributions are quite significant at all
temperatures accessible to the data evaluation. Distributions are flatter than Gaussians. This is
an indication that bottoms of the potential wells are flatter than parabolic. The above flatness
depends upon direction being the biggest in the main axes directions.
Acknowledgments
Professor James G. Mullen, Purdue University, West Lafayette, Indiana, U.S.A. and
Dr. Carmen K. Shepard, Maysville Community College, Maysville, Kentucky, U.S.A. are
warmly thanked for calling our attention to their experimental work, suggesting that we carry
out a theoretical analysis, and giving unrestricted access to their experimental data [4].
Appendix
Tensors bˆ ( l ) describing anisotropy of the recoilless fraction constitute a sub-set of the
more general tensors describing material texture, i.e., a directional distribution of rigid bodies
having arbitrary shape. A texture density distribution function T ( ) depends upon three
Eulerian angles ( ) and obeys the following general rules:
12
T ( )  0 ; 0    2 , 0     , 0    2 ;
2

2
1
 d  d sin   d T ( )  1 ; T ( )  N T0 ( ) ; T0 ( )  0 ;
0
0
0
2

2
0
0
0
 d  d sin   d T ( )  N
0
; N 0 .
(A.1)
A particular body orientation could be described by two mutually orthogonal unit vectors 
and  generated with the help of the Eulerian rotation operators as follows:
cos   sin  0
 cos  0 sin  
cos   sin  0




Rˆ   sin  cos  0 ; Rˆ    0
1
0  ; Rˆ   sin  cos  0 ;
 0
 sin  0 cos  
 0
0
1
0
1
 0   sin  cos     1 
  
  
  Rˆ Rˆ   0    sin  sin      2  ;
 1   cos     
  
  3
 1   cos  cos  cos   sin  sin    1 
  
  
  Rˆ Rˆ  Rˆ  0    cos  sin  cos   cos  sin      2  .
 0 
  
 sin  cos 
  
  3
(A.2)
Subsequently one can use components of the above generating vectors to calculate angular
distribution functions in term of the powers of their components. Such procedure leads to the
following expressions:


l!
(l )
 (sin  ) mk (cos  )l mk (sin  ) k (cos  ) m ;
mk
(  )  
 (l  m  k )! m! k! 
l  0,1,2, ; m  0,1,2,, l ; k  0,1,2,, (l  m) ;


!
 ( sin  cos  )   
 ( ) ( )  
 (     )! !  ! 
(cos  sin  cos   cos  sin  ) (cos  cos  cos   sin  sin  )  ;
  0,1,2, ;   0,1,2,,  ;   0,1,2,, (   ) .
(A.3)
Finally, one can expand a texture function into angular functions using double upper index
traceless spherical tensors cˆ (l ) components as the coupling coefficients obtaining the
following expressions:
  l l m     (l ) (l )

T0 ( )  exp   cmk mk (  )  ( ) ( )  ;
 l 0 m  0 k  0   0   0   0

2

2
 d  d sin   d
0
0
0
l
l m    
c  

 
m 0 k 0
 0 0
(l )
mk
(l )
mk
(  )  ( ) ( )  0 .
(A.4)
Due to the fact that the recoilless fraction neither depends upon the third Eulerian angle  nor
upon the vector components, and taking into account the property that the scalar term could
be factorized one obtains the following sub-set for the recoilless fraction anisotropy:
13
(l 0)
l (l )
cmk
00   q bmk ; q  0 ; l  2 : remaining components are 0 .
(A.5)
One has to note, that single upper index tensors could be used for bodies having axial
symmetry as for the latter bodies orientation dependence upon vector  vanishes, and hence
upper index  could be dropped. Angles (  ) have the same meaning as the angles ( ) .
References
[1]
[2]
D.C. Champeney and G.W. Dean, J. Phys. C8, 1276 (1975).
C.K. Johnson, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr.
25, 187 (1969).
[3] B.V. Thompson, Phys. Rev. 131, 1420 (1963).
[4] C.K. Shepard, J.G. Mullen, and G. Schupp, Phys. Rev. B 57, 889 (1998).
[5] C.K. Shepard, J.G. Mullen, and G. Schupp, Hyperfine Interact. 110, 151 (1997).
[6] W.B. Yelon, G. Schupp, M.L. Crow, C. Holmes, and J.G. Mullen, Nucl. Instrum.
Methods Phys. Res. B 14, 341 (1986).
[7] K. Ruebenbauer, U.D. Wdowik, and M. Kwater, Physica B 229, 49 (1996).
[8] A.A. Maradudin and P.A. Flinn, Phys. Rev. 129, 2529 (1963).
[9] K. Ruebenbauer and U.D. Wdowik, Phys. Rev. B 61, 11416 (2000).
[10] P.D. Pathak and N.G. Vasavada, Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor.
Gen. Crystallogr. 26, 655 (1970).
[11] P.D. Pathak, J.M. Trivedi, and N.G. Vasavada, Acta Crystallogr., Sect. A: Cryst. Phys.,
Diffr., Theor. Gen. Crystallogr. 29, 477 (1973).
[12] K.E. Salimäki, Ann. Acad. Sci. Fenn., Ser. AI: Math. 56, 7 (1960).
14