X-ray diffraction methods have been used to study the structural and thermal properties of CdTe epitaxial layers on GaAs substrates. The technique of precession photography is introduced as an ex- tremely useful method for qualitative characterization of thin films, and reveals twin faults in (1,1,1) oriented CdTe epilayers. Measure- ments of the Bragg reflections as a function of temperature were made between 10 K and 360 K. These measurements reveal anoma- lous behavior in the lattice parameters of both the substrate and the epilayer in a (0,0,1) oriented sample deposited at 410(DEGREES)C. The overall change in the CdTe lattice parameter in this temperature range was about five times larger than in bulk CdTe, and the change in the sub- strate lattice parameter was about three times the bulk value. These anomalies were absent in a similar sample deposited at 380(DEGREES)C. Simi- lar measurements were made on GaAs and CdTe single crystals for comparison. The Debye temperatures extracted from the integrated intensities of the reflections perpendicular to the surface are (THETA)(,M)(TURN) 250 K for the GaAs substrates and (THETA)(,M)(TURN)142 K for the CdTe epilayers. These are the same as the corresponding values in single crystals of GaAs and CdTe, revealing no significant change in the overall vibra- tional properties. The Debye temperatures extracted from reflections not perpendicular to the surface are significantly lower. A degree of mosaicity parallel to the surface is the probable cause of this in the GaAs substrate. In the epilayer, a proposed explanation is a static displacement of the Cd and Te atoms from their crystallographic positions, on the order of 0.01 (ANGSTROM). The Debye theory and the One-Particle Potential model are used in the above analyses. A compari- son shows that these two models are identical in the classical tem- perature regime. In addition, a model incorporating both the Debye and Einstein theories is derived, and applied to x-ray diffraction. Although it is physically more realistic, the Debye-Einstein theory does not explain the observed behavior any better than the Debye theory. Finally, the relationship between the Debye temperatures;from x-ray measurements and specific heats is shown for poly- atomic solids; ('1)DOE Report IS-T-1250. This work was performed under contract No. W-7405-Eng-82 with the U.S. Department of Energy.