This paper presents results of a new and highly accurate technique for measuring low-energy phonon dispersion in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The technique is based on the behavior of ultrasonic second-harmonic generation in a lossless, dispersive medium. By using frequencies in the low gigahertz range and measuring second-harmonic intensity as a function of propagation distance, the coherence length for the harmonic generation can be determined. The coherence length is, in turn, related to the phonon dispersion curve in a simple way. The results are interpreted in terms of the series expansion $\epsilon \mathrm{}\left(k\right)=c_0\hslash k(1+{\alpha }_{1}k+{\alpha }_2{k}^2+{\alpha }_3{k}^3+\cdots )$, where $\epsilon$ and $k$ are phonon energy and wave number, respectively. By using measurements taken at two different fundamental frequencies, we find $\left|{\alpha }_1\right|<\mathrm{}{10}^{-3\mathrm{}}$ Å at saturated vapor pressure (SVP) and 6.3 bars, and ${\alpha }_2=(1.56\mathrm{}\pm 0.06)$ ${\mathrm{\AA }}^2$ at SVP. If ${\alpha }_1$ is assumed to be zero, the ${\alpha }_2$ can be determined from a measurement at a single frequency, and we find ${\alpha }_2=(1.55\mathrm{}\pm 0.01)$ ${\mathrm{\AA }}^2$ at SVP. At higher pressures, ${\alpha }_2$ decreases. Since the excitation spectrum is probed with such low-momentum phonons ($k<\mathrm{}0.011\mathrm{}$ ${\mathrm{\AA }}^{-1\mathrm{}}$), the analysis is insensitive to assumed values of ${\alpha }_4$, ${\alpha }_5$, etc., and is only slightly sensitive to the assumed value of ${\alpha }_3$.

• Received 17 April 1984

DOI:https://doi.org/10.1103/PhysRevB.30.2595