Dilatometric and X-Ray Thermal Expansion in Noncubic Crystals. I. Role of Dislocation Climb

Comparison of macroscopic (dilatometric) and microscopic (x-ray) precision thermal-expansion measurements at elevated temperatures has been widely used to obtain the equilibrium concentration of vacancies in cubic crystals. For noncubic crystals these measurements must be made in more than one orientation (e.g., in axial crystals, along both the $c$ and $a$ axes). Such measurements give information not only on the equilibrium vacancy concentration, but also on the relative rates of climb of the various dislocations in the crystal, which act as sources and sinks for the vacancies. Quantitative expressions are obtained for the ratio $\frac{{\Delta }_c}{{\Delta }_a}$, where $\Delta$ is the difference between fractional macroscopic and microscopic linear expansions along the $c$ or $a$ direction in an axial crystal. Consideration is given to the two extreme cases of dislocation climb, viz., the diffusion limited and climb-rate limited cases. An important point is that the measurements, in general, will be sample dependent, so that the samples of different orientation must be taken from the same single crystal.