Temperature variation of elastic constants of quartz across the α-β transition

The alpha-beta transition of quartz was successfully observed with using a single sample by means of the rectangular parallelepiped resonance (RPR) method. An oriented rectangular parallelepiped of alpha-quartz single crystal was prepared and the resonant frequencies of 30-11 vibrational modes were measured from room temperature to 700 degrees C. The softening of quartz crystal was observed as the significant reduction of resonant frequencies near the alpha - beta transition. The present study is the first application of the RPR method to the study of phase transition. The complete set of elastic constants of alpha- and beta-quartz were determined as a function of temperature by the least-squares inversion of the measured frequency data obtained by a single run. This is a merit yielded by the RPR method. It is shown near the alpha-beta transition in both alpha- and beta-quartz that the elastic parameters decrease proportionally to | T - T-0|(-n), where T is temperature and T-0 is the transition temperature, 573.0 degrees C for alpha-quartz and 574.3 degrees C for beta-quartz. It was also seen that linear incompressibilities K-1 = (C-11 + C-12 + C-13)/ 3 and K-3 = (C-33 + 2C(13))/ 3 decrease rapidly toward the transition, whereas, shear moduli C-44, C-S1 = (C-11 + C-33 - 2C(13))/ 4 and C-S3 = (C-11 - C-12)/ 2 = C-66 decrease only slightly. The shear modulus C-S3 = C-66 increased slightly in a- quartz. The elastic properties of isotropic aggregate of quartz were calculated, and it is shown that the longitudinal wave velocity significantly decreases at the alpha-beta transition, whereas, the shear wave velocity decreases only slightly.