Spatial Distribution of Atoms in LiquidHe4

Abstract

The x-ray scattering diagrams of liquid helium, obtained by one of the authors, have been analyzed in this paper using the Los Alamos "701" IBM Electronic Calculator. The liquid ${\mathrm{He}}^{4}$ atomic correlation and pair distribution functions have thus been obtained over a wide range of interatomic separations and at a number of liquid temperatures between 1.25 and 4.20\ifmmode^\circ\else\textdegree\fi{}K.The limitations in the experimental scattering structure factors give rise, among others, to two types of limitations in the correlation functions. One of these concerns their behavior at small atomic separations; this limitation could be compensated for without difficulties. The second limitation, of more elusive character, appears to be connected with the lack in these functions of any manifest tendency toward their asymptotic behavior at large atomic separations. The here derived correlation or pair distribution functions should, nevertheless, be good approximations to their correct values at medium and intermediate separations extending to about fifteen angstroms, whereby the relevant range extends to 10-15 A. This could be verified using two types of integral checks on the correlation functions. In one of these, their space integrals, out to relevant distances, turned out to be quite close to their expected values determined only by the macroscopic properties of the liquid. In the second series of integral checks, they reproduced closely the original experimental structure factors. This verified the internal consistency of the calculations and indicated that the results should be correct at the relevant interatomic separations.At all the liquid temperatures here investigated, in any sphere of given radius, around an origin atom, or in any spherical shell of given thickness, the computed number of atoms was always less than what one would compute neglecting the interatomic correlations.An application, of major physical significance, of the correlation or pair distribution functions obtained in this work was made by deriving the approximate mean potential energies per liquid ${\mathrm{He}}^{4}$ atom at the various temperatures. The mean potential energy, as a function of the liquid temperature, has a cusp at the lambda point. Its temperature derivative, which is the configurational heat capacity, is thus discontinuous at the transition point with an inverted lambda type of discontinuity. The knowledge of this qualitative behavior of the configurational heat capacity was sufficient to prove, on quite general grounds, that the lambda transition of liquid ${\mathrm{He}}^{4}$ is of kinetic energy origin, i.e., it is connected with the momentum space behavior of the liquid atoms.