Lambda Transformation of Liquid Helium Four

Abstract

It will be shown that the extrapolation of empirical logarithmic expressions of a group of thermal properties of liquid ${\mathrm{He}}^4$ at the approaches of the lambda transformation, implying their singular behavior, is without physical justification. These properties do not even start to become large until one invokes temperature separations from the lambda temperatures, the argument of the logarithmic expressions, which are very many orders of magnitude smaller than the root-mean-square temperature fluctuations of the liquid-${\mathrm{He}}^4$ samples investigated. Consideration of temperature intervals smaller than the root-mean-square temperature fluctuations is excluded by statistical thermodynamics. On analyzing the lambda transformation at melting, first an exact thermodynamic relation will be proved through the geometrical properties of a class of isochores in the pressure-temperature diagram. Provided that the thermodynamic characteristics of the lambda transformation remain invariant all along the transformation locus, the thermodynamic relation proved at melting has the following corollaries: If the transformation volume line is not parabolic in temperature at the approaches of its end point at melting, singularities of a group of interrelated thermal properties must exist at the lambda transition. If this approach is parabolic, the existence of singularities neither is necessary nor can it be ruled out. In the latter case, new physical approaches would be needed for an unambiguous determination of the existence or absence of singularities of thermal properties at the lambda transformation.