International
Tables for Crystallography Volume C Mathematical, physical and chemical tables Edited by E. Prince © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. C. ch. 7.4, pp. 654-655
Section 7.4.2.2.1. Evaluation of J(q)
B. T. M. Willisd
|
The frequencies ωj(q) and polarization vectors ej(q) of the elastic waves in equation (7.4.2.7) can be calculated from the classical theory of Voigt (1910
) [see Wooster (1962
)]. If
,
,
are the direction cosines of the polarization vector with respect to orthogonal axes x, y, z, then the velocity vj is determined from the elastic stiffness constants
by solving the following equations of motion.
Here, Akm is the km element of a 3 × 3 symmetric matrix A; if
,
,
are the direction cosines of the wavevector q with reference to x, y, z, the km element is given in terms of the elastic stiffness constants by
The four indices klmn can be reduced to two, replacing 11 by 1, 22 by 2, 33 by 3, 23 and 32 by 4, 31 and 13 by 5, and 12 and 21 by 6. The elements of A are then given explicitly by
The setting up of the matrix A is a fundamental first step in calculating the TDS correction factor. This implies a knowledge of the elastic constants, whose number ranges from three for cubic crystals to twenty one for triclinic crystals. The measurement of elastic stiffness constants is described in Section 4.1.6
of IT B (2001
).
For each direction of propagation , there are three values of
(j = 1, 2, 3), given by the eigenvalues of A. The corresponding eigenvectors of A are the polarization vectors ej(q). These polarization vectors are mutually perpendicular, but are not necessarily parallel or perpendicular to the propagation direction.
The function J(q) in equation (7.4.2.7) is related to the inverse matrix A−1 by
where
,
,
are the x, y, z components of the scattering vector H, and classical equipartition of energy is assumed [Ej(q) = kBT] . Thus A−1 determines the anisotropy of the TDS in reciprocal space, arising from the anisotropic elastic properties of the crystal.
Isodiffusion surfaces, giving the locus in reciprocal space for which the intensity J(q) is constant for elastic waves of a given wavelength, were first plotted by Jahn (1942a,b
). These surfaces are not spherical even for cubic crystals (unless
), and their shapes vary from one reciprocal-lattice point to another.
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