Reexamination of evidence for a Bose-Einstein condensate in superfluidHe4

During the last five years, several different neutron scattering experiments have been carried out in search of direct evidence for Bose-Einstein condensation in superfluid ${\mathrm{He}}^4$. Although the structure in the scattering line that was expected if a condensate were present has never been distinctly visible, various methods of analyzing the data have led to estimates for the condensate contribution that range between 2.4 and 17%. The reeexamination begins with a review of this previous work. The next step is a calculation of the line shape using the impulse approximation (IA) and the momentum distribution function determined by McMillan by a Monte Carlo technique. The possibility of simple line broadening due to final-state interactions (FSI) is also considered. This initial treatment is essentially a repetition of one first carried out by Gersch and Smith, except that our work conforms to the conditions of the Mook, Scherm, and Wilkinson (MSW) experiment, which provides the most accurate data presently available for comparison purposes. Direct comparison suggests that the condensate fraction is no greater than about 4%, and that it may be vanishingly small. Observing that a more precise estimate would be possible if one could assess more accurately the FSI effects, particularly with respect to broadening, we then present evidence based on sum rules and other theoretical considerations which supports the idea that corrections to the IA may indeed be negligible under the conditions of the MSW experiment. With this as a basis, it is postulated that the IA is strictly applicable to the MSW data and the consequences are studied. First, it is found that the shape of the peak of the line observed by MSW would be consistent with no condensate at all; but if any were present, it would have to be less than 1%. A discussion of the conflict between this estimate and the (2.4 ± 1)% deduced by MSW is then given. Also a method of checking on the validity of the postulate concerning the applicability of the IA to the MSW conditions is described. Next, neglecting whatever condensate that is possibly present, the IA formula for $S(Q,\omega )$ is inverted and the experimental data of MSW is used to compute $n\left(k\right)\mathrm{}$, the momentum distribution of atoms in liquid ${\mathrm{He}}^4$. This, together with the kinetic energy per particle and the one-particle density matrix in the space-coordinate representation are reported for ${\mathrm{He}}^4$ at 1.2°K and 4.2°K.