Excitations in Liquid Helium: Thermodynamic Calculations

The entropy, specific heat, normal fluid density, and velocity of second sound in liquid helium II have been calculated by applying statistical mechanics to the thermal excitations. The calculations were based on the energy-momentum relation obtained by neutron scattering measurements described by Yarnell, Arnold, Bendt, and Kerr, and were made on an IBM-704 electronic digital computer by numerical integrations over the observed excitation curve. A better approximation than Landau's has been obtained by extending Landau's theory to take account of the temperature dependence of the excitation curve. An expression of the form $E(p, T)=c-d\left(\frac{{\rho }_n}{\rho }\right)$ was used to interpolate the excitation energy between temperatures at which it was measured. Results between 0.2 and 1.8°K are not sensitive to the exact form of the interpolation expression. Agreement of the calculations with experimental measurements is as follows: entropy, ±3% in the temperature range 0.2 to 1.8°K; specific heat, ±4% between 0.2 and 1.7°K; second sound velocity, ±4% between 0.8 and 1.8°K, and ±2% between 1.0 and 1.7°K. The calculated normal fluid density ${\rho }_n$ agrees with experimental values derived from second sound velocity and specific heat measurements within ±8% between 0.7 and 2.0°K, and within ±5% from 1.1 to 1.9°K. These values are, however, higher than those obtained from torsion pendulum measurements, which are 27% below the calculated value at 1.2°K. Also calculated as functions of temperature are the average effective mass (as defined by Landau) of excitations in four momentum intervals, and values of $-\frac{\kappa }{\overline{B}}$, the thermal conductivity $\kappa$ divided by the average over momentum of the Khalatnikov nonequilibrium kinetic coefficient $-B$, and $\frac{\eta }{\overline{C}}$, the viscosity $\eta$ divided by the average value of the Khalatnikov coefficient $C$.