X-Ray Scattering from Liquid Helium

We have measured the liquid structure factor $S\left(k\right)\mathrm{}$ for liquid ${\mathrm{He}}^4$ and for liquid ${\mathrm{He}}^3$ by x-ray diffraction. Our results for liquid ${\mathrm{He}}^4$ agree with the Feynman formula $\frac{\hslash k}{2mc}$ in the range $0.3<k<\mathrm{}0.6\mathrm{}$ ${\mathrm{\AA }}^{-1\mathrm{}}$, in contrast to previous measurements of $S\left(k\right)\mathrm{}$ which fell about 25% below $\frac{\hslash k}{2mc}$ in this range. The peak in the pair distribution function $g\left(r\right)\mathrm{}$ derived from the scattering data occurs at greater interatomic distance and is lower, broader, and less symmetrical for liquid ${\mathrm{He}}^3$ than for liquid ${\mathrm{He}}^4$, in agreement with predictions based on variational calculations. However, the experiments indicate slightly more structure in both liquids than the calculations predict. We have also measured $S\left(k\right)\mathrm{}$ for ${\mathrm{He}}^4$ vapor at 4.2 K, 0.98 atm. For ${\mathrm{He}}^4$ vapor, $g\left(r\right)\mathrm{}$ shows a distinct nearest-neighbor shell and probably a second-nearest-neighbor shell.