Bose-Einstein Condensation and atomic kinetic energies in liquid $^3$He-$^4$He mixtures

We present neutron scattering measurements of the momentum distribution of liquid ${^3}$He-${^4}$He mixtures. The experiments were performed at wavevectors $Q$, 26 $\leq$ $Q$ $\leq$ 29 \AA$^{-1}$, on the MARI time-of-flight spectrometer at the ISIS pulsed spallation neutron source. Mixtures with $^3$He concentrations $x$ between 0 and 20\% were investigated both in the superfluid and normal phases. From the data, we extract, to new accuracy, the Bose-Einstein condensate fraction $n_0$ and the momentum distributions of $^3$He and $^4$He atoms. We find an increase in $n_0$ above the pure $^4$He value; from 7.25$\pm$0.75\% ($x=0\%$) to 11.2$\pm$1.85\% at $x=15\%$, in agreement with theoretical calculations but in disagreement with the only other measurement. The $^4$He kinetic energy, $\backslash$kef, is found to be largely independent of $x$. The $^3$He momentum distribution $n(\bf{k})$ is not well fitted with a Fermi step function alone. A high momentum tail in $n(\bf{k})$ is needed to get a good fit - a tail that is consistent with calculated tails in $n(\bf{k})$. The $^3$He atomic kinetic energy, $ K_3$, is determined almost entirely by this tail. It is therefore not a well determined single property for comparing theory and experiment. This finding resolves a long-standing discrepancy on $ K_3$ between theory and experiment.