Second-Harmonic Generation of Microwave Phonons in Quartz and Sapphire

The phonon-phonon volume interaction due to the nonlinear elastic properties of solids has been investigated together with nonlinearities in the end-surface generation of microwave phonons. The experimental method consisted of generating a microwave-ultrasonic fundamental at one end of a rod and detecting the second harmonic at the opposite end by means of the piezoelectric effect. For $Z$-cut quartz and sapphire, where the phonon-phonon interaction dominates, experimental observations prove that the original flow of energy from the fundamental to the second harmonic is completely reversed after longitudinal waves are reflected from a stress-free surface or from a half-wavelength-thick transducer. Thus, the second harmonic in an almost lossless medium vanishes upon arrival at the generating transducer because the reflection reverses the phase angle $2{\phi }_1$ relative to ${\phi }_2$, where ${\phi }_1$ and ${\phi }_2$ are the phase angles, respectively, of the fundamental and the second harmonic. When the thickness of the CdS transducers was made less than one-half wavelength, the smaller phase shift produced a correspondingly smaller energy reversal. For transverse waves in $\mathrm{AC}$-cut quartz, the original increase of the second harmonic due to volume nonlinearities was unaffected by the presence of the stress-free boundary, thereby confirming that there is no phase shift for transverse waves. These phase-shift phenomena, together with the frequency and rod-length dependence of the harmonic generation, were used to separate surface from volume nonlinearities. By measuring the coupling constants of the phonon-phonon interaction at 4.2°K, the following third-order elastic coefficients (in units of ${10}^{11}$ N/${\mathrm{m}}^2$) were obtained: $c_{111}=-2.6\mathrm{}\pm 0.5$ for $X$-cut natural quartz, and -38±3 for 0.01% Cr-doped $a$-oriented sapphire grown by the Verneuil process; $c_{333}=-14\mathrm{}\pm 4$ for $Z$-cut quartz, and -21±1 for both undoped and 0.01% Cr-doped $c$-oriented sapphire.