On the thermodynamic phase diagrams of helium three

Abstract

The object of the present paper is an analysis of the thermodynamic consequences resulting from the observed anomaly at the temperature T a of the melting pressure line P( T) of He 3 through the formation of a minimum at this temperature. One is led to prove first the anomalous character of the volume line V l ( T) of the liquid, in equilibrium with the solid, on the basis of data available on the high temperature side of T a and at T a. This volume anomaly consists in a maximum of V l ( T) at T a. The similar anomaly of the solid volume line V s ( T) at T a will be also to result from the temperature double-valuedness of the isobaric and isochoric states of the liquid and solid along the phase equilibrium lines. The saddle point type character of the states [ P( T a), V l ( T a)] and [ P( T a), V s ( T a)], combined with the existence of the locus of vanishing expansion coefficients of the liquid, lead to recognizing the oscillatory character of the isobar of pressure P( T a) and isochore of volume V l ( T a) of the liquid. The oscillations result from the vanishing of the second partial temperature derivatives of these isobar and isochore of anomaly at three different temperatures, the temperature T a being one of these. These changes in sign of the curvature of these phase diagram lines impose the occurrence of probably observable anomalies in the constant pressure and constant volume heat capacities of higher pressure, p ∼ P( T a), and smaller volume V ∼ V( T a). The anomalous isobaric lines yield a fairly complete description of the associated expansion coefficients of the liquid. The vanishing expansion coefficient of both the liquid and the solid at the states of anomaly will be shown to result directly from the saddle point type character of these states of the two phases. Various features of the isothermal compressibility of the liquid will be derived, together with its evaluation at isolated states of these two phases in equilibrium from the sole knowledge of the local behavior of the equilibrium lines P( T), V l ( T), and V h ( T). The series of anomalous properties of He 3 in a new pressure and temperature range disclosed here appear to warrant their detailed experimental investigations.