Statistical Thermodynamics ofλTransitions, Especially of Liquid Helium

The thermodynamic properties of systems in the immediate neighborhood of a locus of $\lambda$ transitions have been investigated. It is assumed that the transitions arise from spin orientation (or some other order-disorder phenomenon), and that the partition function can be broken down into a product of a lattice part and a spin part, the latter dependent only on $I=\frac{J}{\mathrm{kT}}$, where $J$ is an energy parameter. If $J$ depends only on the volume $V$, then the specific heat at constant volume $C_v$ tends to become infinite along the $\lambda$ line, but as is well known, an instability sets in before this point is reached. It is shown that this instability occurs only very close to the $\lambda$ line, and $C_v$ and ${\left(\frac{\partial P}{\partial T}\right)}_V$ may be expected to parallel each other much farther from the $\lambda$ line. If an intrinsic volume change is associated with the ordering phenomenon, $J$ is more approriately taken as an enthalpy parameter, and may be supposed to depend on the pressure $P$ rather than on $V$. Isothermal-isobaric partition functions are used. $C_p$ tends to become infinite and to parallel ${\left(\frac{\partial V}{\partial T}\right)}_P$ a considerable distance from the $\lambda$ line. Only very close to the $\lambda$ line does ${\left(\frac{\partial V}{\partial P}\right)}_T$ become negatively infinite. The results are applied to liquid helium under pressure, and shown to accord with the data. The behavior of $C_v$ is discussed. A possible generalization of the theory is suggested.