Excitations in Liquid Helium: Neutron Scattering Measurements

Abstract

The energy-momentum spectrum of the excitations in liquid helium II has been measured in a neutron scattering experiment. At $T=1.1\mathrm{}°$K, data were obtained in the momentum range $\frac{p}{\hslash }=0.55\mathrm{} \mathrm{to} 2.36$ ${\mathrm{A}}^{-1\mathrm{}}$. The results bear a striking resemblance to the spectrum proposed by Landau in 1947. If a phonon spectrum of the form $E=v_{1}p$ is fitted to the point at $\frac{p}{\hslash }=0.55\mathrm{}$ ${\mathrm{A}}^{-1\mathrm{}}$, a value of 239±5 meters/sec is obtained for $v_1$. The measured spectrum passes through a maximum of $\frac{E}{k}=13.92\mathrm{}\pm 0.10°$K at $\frac{p}{\hslash }=1.11\mathrm{}\pm 0.02$ ${\mathrm{A}}^{-1\mathrm{}}$. There is a minimum at $\frac{p}{\hslash }=1.92\mathrm{}\pm 0.01$ ${\mathrm{A}}^{-1\mathrm{}}$, which may be represented by Landau's roton expression, $E=\frac{\Delta +{(p-p_0)}^2}{2\mu }$, with $\frac{\Delta }{k}=8.65\mathrm{}\pm 0.04°$K, $\frac{p_0}{\hslash }=1.92\mathrm{}\pm 0.01$ ${\mathrm{A}}^{-1\mathrm{}}$, $\mu =(0.16\mathrm{}\pm 0.01)m_{\mathrm{He}}$. Above $\frac{p}{\hslash }=2.18\mathrm{}$ ${\mathrm{A}}^{-1\mathrm{}}$, the spectrum rises linearly with a slope corresponding to the velocity of first sound. Data were obtained in the region of the minimum at $T=1.6\mathrm{}°$K and at $T=1.8\mathrm{}°$K. The spectrum has the same general shape observed at $T=1.1\mathrm{}°$K, shifted downward by 0.22°K at 1.6°K, and by 0.50°K at 1.8°K. The temperature variation may be represented by the empirical expression $\frac{\Delta }{k}=8.68-0.0084\mathrm{}{T}^7°$K. At the higher temperatures, the energy spread of the excitations was larger than the energy resolution of the apparatus. The observed full width at half maximum was ∼1°K at $T=1.6\mathrm{}°$K, and ∼2°K at $T=1.8\mathrm{}°$K.