THE INHOMOGENEITY AND ANISOTROPY OF DEBYE CHARACTERISTIC TEMPERATURES

Abstract

As Debye chiracteristic temperature denotes the average vibration energy of atoms or ions in a costal, while different atoms in the crystal may have different vibration modes, or the same atom in a noncubie crystal may have different vibration modes along different directions, there arises the problem of inhomogeneity and anisotropy of characteristic temperatures.It is discussed in this paper the methods of investigating the inhomogeneity and anisotropy of characteristic temperatures in crystals.If there are two kinds of atoms (a) and (b) in a crystal, then the diffraction lines in the Debye-Scherrer photograph may be classified into two categories: in one of them the structure factor is the sum of the structure factors of respective atoms, F5= mFa + nFb, while in the other, the structure factor is the difference of the structure factors of respective atoms, F = gFa - qFd. Owing to the fact that both Fsobs and Fdobs are functions of sin θ/λ, so, if observed values of Fsobs and Fdobs are plotted against sin θ/λ, two smooth curves should be obtained, where Fobs= KFcalc. In the calculated structure factors, Fa= fa exp(-Ba sin2 θ/λ2) and Fb= fbexp(-Bb sin2θ/λ2), where fa and fb are the atomic scattering factors of the respective atoms, while Ba and Bb are Debye parameters of the respective atoms in the crystal. It is seen that from the two curves one can find the corresponding values of Fobsd or Fobss at the same abscissas as Fobss or Fobsd. If corresponding pairs of values are multiplied by proper coefficients and then be added or subtracted, a series of values of Fobs+ and obs- can be obtained which are now pure functions of Fa and Fb. Then if log (fa/Fobs+) and log (fb/Fobs-) are plotted agains sin2θ respectively, two straight lines can be obtained which should intercept at the same point on the ordinate axis, the slopes of which represent respectively Ba log e/λ2 and Bb log e/λ2, whence one can obtain Ba and Bb. For anisotropic crystals, a series of (h, k, 0) reflexions should be singled out. Just as in the case of isotropic crystals, one may plot log(Icalc/Iobs) against sin2θ, a straight line should be obtained, the slope of which gives 2B⊥ log e/λ2. Then single out another series of (0, 0, 1) reflexions. If one plots log (Icatc/Iobs) against sin2θ, then another straight line should be obtained, the slope of which gives 2B‖ log e/λ2. The two straight lines should intercept at the same point on the ordinate axis. Here B‖ and B⊥ represent the Debye parameters parallel and perpendicular to the principal axis respectively.Having datermined Debye parameters, characteristic temperatures can be obtained in the same way as we have described in our previous paper.The characteristic temperatures of CaF2 have been determined. It is shown that the characteristic temperature of Ca2+ ion in CaF2 crystals is 400K, while that of F-ion is 476K, the difference being 76K.