Measurements of the dynamic structure factor near the lambda temperature in liquid helium

The dynamic structure factor $S(q,\omega )$ was measured by Brillouin scattering in liquid ${}_{}{}^4{}_{}{}^{}\mathrm{He}$ at $q=1.79\mathrm{}\times {10}^5$ ${\mathrm{cm}}^{-1\mathrm{}}$. Results were obtained at two densities: $\rho =0.175\mathrm{}$ g/${\mathrm{cm}}^3$ for which $P_{\lambda }=23.1\mathrm{}$ bars and $\rho =0.179\mathrm{}$ g/${\mathrm{cm}}^3$ for which $P_{\lambda }=28.5\mathrm{}$ bars. Values of the reduced temperature $\epsilon \equiv \frac{(T-T_{\lambda })}{T_{\lambda }}$ varied from -5 × ${10}^{-2\mathrm{}}$ to -5 × ${10}^{-6\mathrm{}}$ and from 1.3 × ${10}^{-5\mathrm{}}$ to 5 × ${10}^{-2\mathrm{}}$. This range involves both the hydrodynamic and the critical regions; values of $q\xi$ fall between 5 × ${10}^{-2\mathrm{}}$ and 21 below $T_{\lambda }$ and between 4.5 and 2 × ${10}^{-2\mathrm{}}$ above $T_{\lambda }$. This article presents the data on the critical mode, which is second sound below the transition and heat diffusion above. The most striking result is that the damping of the critical mode is essentially independent of $\epsilon$ over the entire range of temperatures studied. This behavior is qualitatively different from the strong temperature dependence, consistent with dynamic scaling, which is observed at low frequencies. On the other hand, any dispersion in the velocity of high frequency (∼ several MHz) second sound, if present, is less than 2%. $S(q,\omega )$ retains a two-peaked structure well into the critical region below $T_{\lambda }$; it still appears in the measured spectra (which contain the instrumental width as well) at $q\xi \sim 4$. At $T_{\lambda }$ and in the critical region just above, there is evidence for a non-Lorentzian shape of $S(q,\omega )$. The critical mode contribution to $S(q,\omega )$ is compared with a theoretical calculation of Hohenberg, Siggia, and Halperin based on the planar-spin model of ${}_{}{}^4{}_{}{}^{}\mathrm{He}$. The theory matches both the overall width and the general features of the shape of $S(q,\omega )$ to within our spectral resolution at $T_{\lambda }$. However, closer to $q\xi =1\mathrm{}$ in the critical region, and in the hydrodynamic regions both above and below $T_{\lambda }$, the theory predicts linewidths which are too small.