The Nature of Higher-Order Phase Transitions with Application to Liquid Helium

A general discussion is given of the relation between transitions of various order, especially those which are not concerned with solid lattices. It is shown that third-order transitions bear much the same relation to anomalous first-order transitions as second-order transitions bear to ordinary first-order transitions. The conditions under which higher-order transitions can occur are considered, and it is shown that they may be associated with a dispersed phase. If the transition is second order, the dispersed phase must undergo a transition in itself. The ideas concerning second-order transitions are applied to the $\lambda$ transition in liquid helium. An argument is given, based on energetic grounds, which indicates that the superfluid in liquid helium is in the form of clusters which are separated in ordinary space as well as in momentum space. If this view is correct, the $\lambda$ transition corresponds to the appearance of much larger clusters, essentially marking the beginning of long-range orders. It is shown that the Bose-Einstein statistics is quite essential for the formation of the clusters; they cannot appear in ${\mathrm{He}}^3$, and no $\lambda$ transition is to be expected in ${\mathrm{He}}^3$. Finally, a possible explanation is given for Taconis' hypothesis that ${\mathrm{He}}^3$ is not soluble in the superfluid part of ${\mathrm{He}}^4$.