By Professor Dr. Tasso Springer (auth.)
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Extra resources for Springer Tracts in Modern Physics, Volume 64
54]. The position of the particle r(t) is described by a linear superposition of statistically independent components r which should obey equations of motion with restoring forces -e)~2~v(t) and stochastic driving forces F~(t), namely d 2 Cv/d t 2 + t/~d r t + coy2 r = F~(t). (87) Such an equation does not lead to any diffusive motion. Therefore, it is necesseary to introduce, for a certain fraction of the degrees of freedom, a motion without restoring force as in (70), namely d2 Cv/d t 2 + fl, d~/dt = F~(t).
For interstitial lattices which are not of the Bravais type, the calculation of the scattering law is complicated. For instance, in the case of a bcc host lattice there exist in one unit cell m = 3 non-equivalent sites with an octahedral, and m = 6 sites with a tetrahedral arrangement of the neighbouring host lattice atoms (Fig. 18). As a consequence, we have to introduce sets of jump vectors Sij,k which connect a site of type i(i = 1..... m) with a neighbouring site of type j ( j = 1..... m); k labels these jump vectors.
The scattering law is the space-time Fourier transform of (100) with (108)9 After introducing new time variables t - t 2 , = z 2 . , and analogous conventions for the space coordinates, the integrals in the sum (100) can be factorized yielding a simple geometrical series. Leaving out numerous intermediate steps of the calculation, one finds Sine(K, (D) = "170 A [1 + (U/zo) ] z1 S [1 + (A/zO] Zo + zl 1 - ( A B / % h ) + Zo + z~ 1 - ( A B / % z l ) " (109) Here A and B are the space-time Fourier transforms of the functions p(t) g(r, t) and q(t) h(r, t), respectively9 The appearance of two terms in (109) takes into account that only a certain fraction of the particles has started with an oscillatory motion (as has been tacitly assumed above); the other particles have started in their diffusive state.