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:
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 wavenumber. 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.
2. Parameterization of the recoilless fraction
The recoilless fraction is a smooth function of the wave-vector transfer to the system and it satisfies the following conditions:
(2.1)
One can define symmetric, against the wave-vector transfer inversion, and antisymmetric amplitudes, respectively, in the following way:
(2.2)
For a single "independent" point-like spherical "scatterer" one can obtain the spatial distribution amplitude versus position as:
(2.3)
and a corresponding spatial probability density function as:
(2.4)
The spatial probability density function could be defined in coordinates centered on the average position in the following way:
(2.5)
The resulting smooth function obeys the following relationships:
(2.6)
with being a pseudo wave-function. The function 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 can be interpreted as a matrix element of the density matrix.
The function could be back-transformed into the recoilless fraction by the following set of operations:
(2.7)
Hence, one can obtain the "thermal" spatial probability density function upon having measured corresponding "dynamical" recoilless fraction .
The recoilless fraction could be expressed as:
(2.8)
The smooth function takes on the following form [7]:
(2.9)
where 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 and used to evaluate the recoilless fraction. Hence, the wave-vector transfer to the system is defined as:
(2.10)
Parameters describe isotropic part of the recoilless fraction, while the parameters describe anisotropy of the recoilless fraction. The latter parameters are usually required to satisfy the following condition:
(2.11)
The above mentioned condition is "automatically" satisfied for all odd indices , while for the even indices it leads to the following constraint on the parameters :
(2.12)
The above constraint "removes" one parameter from the even terms.
Functions take on the form:
(2.13)
Hence, the following relationships are obeyed:
(2.14)
except for the all indices , and being even. For the latter case one obtains:
(2.15)
where the symbol denotes the Euler-gamma function.
In particular the function obeys the following relationship in accordance with the expression (2.15):
(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 . Namely, the recoilless fraction could be rewritten in the form where:
(2.17)
Therefore, the amplitude takes on the form:
(2.18)
while the function is expressed as:
(2.19)
The centered coordinate x follows from the expressions:
(2.20)
Hence, the spatial density function along the above axis satisfies the following conditions:
(2.21)
with being a corresponding pseudo wave-function. This function has a similar meaning as the function , however it is defined in a single dimension along the axis pointing in the direction. Hence, the function 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 into the recoilless fraction in the direction by the following set of operations:
(2.22)
It is interesting to note, that the recoilless fraction described solely by terms with results in the following Gaussian function:
(2.23)
where:
(2.24)
Hence, the following deviation function is a good measure of the presence of the terms with :
(2.25)
For a purely Gaussian behavior and . If one neglects all terms with a cubic environment with the local inversion centre has and , and it is described completely by: , , and . The isotropic term is very small for a cubic environment exhibiting local inversion centre [8], and one is left with three parameters: , and 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 tensor, ten independent components of the tensor and fourteen independent components of the tensor in the case of general symmetry. For the most general symmetry the tensor has independent components for odd and such components for even .
The above mentioned cubic system has the following function :
(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 presence of the inversion centre: ; remaining are zero. The above choice satisfies expression (2.11) and causes that the anisotropy is described by a single parameter . Positive values of the parameter cause the largest displacements to be along the main axes, while the negative values give the minimum displacements along the main axes.
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 along the cube diagonal, respectively. The above discrepancy amounts to: . The next most likely to occur and allowed tensor is . It has two non-trivial components for the above environment: and .
The function:
(2.27)
is plotted in Fig. 1 for , and various settings of the parameter ranging here between (-1; 3/2). The function is described here by the expression (2.26).
Fig. 1 Possible shapes of the quartic anisotropy in the cubic system. See, expressions (2.26) and (2.27) for details. The anisotropy vanishes for .
For a strictly spherical symmetry all tensors vanish and one is left with isotropic parameters solely. Usually, and parameters are dominant with being a sole parameter for a spherically symmetrical harmonic environment.
For a general symmetry in a harmonic environment parameter and a tensor survive solely.
The presence of the tensors with odd l is the indication of the absence of the local inversion centre. For such a case neither nor .
One has to note, that parameters and tensors strongly depend upon the temperature.
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 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 has to be replaced by:
(2.28)
where stands for the amplitude of the n-th particle in the s-th site at the position , while 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]:
(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.
Table I
Coefficients of the lattice constant dependence upon temperature.
a0 / 5.5870(4) / [Å]C1 / 1.7∙10-7 (68·10-7) / [Å/K]
C2 / 7.8(4)·10-7 / [Å/K2]
C3 / -1.34(9)·10-9 / [Å/K3]
C4 / 1.25(9)·10-12 / [Å/K4]
C5 / -4.5(4)·10-16 / [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: , and using expressions (2.26) and (2.8) separately for the average sodium cation and chlorine anion versus temperature. The following temperature dependencies were found applying polynomial fits with the minimum number of parameters [9]:
(3.2)
The numerical values of the parameters are listed in Table II [9]. Additionally, the ratio was calculated versus temperature for sodium and chlorine. The functions: , , and are plotted versus temperature in Fig. 3 for sodium and chlorine both [9].
Table II
Parameters describing recoilless fractions versus temperature.
Na / ClA [-] / 1.14(9)·10-4 / 5.6(4)·10-5
B [1/K] / 2.01(2)·10-6 / 1.33(1)·10-6
α [-] / -2.0(4)·10-9 / 0.0
β [1/K3] / 6.51(5)·10-16 / 6.46(3)·10-16
a [-] / 2.2(2)·10-8 / 1.41(7)·10-8
b [1/K] / -1.38(9)·10-10 / -1.06(4)·10-10
c [1/K2] / 4.2(1)·10-13 / 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.
The biggest sources of the systematic experimental errors are the following: 1) ex-tinction, 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.