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

We present Deep Inelastic Neutron Scattering (DINS) measurements of mixtures of liquid ${^3}$He-${^4}$He in both the superfluid and normal phases. The measurements were performed on the MARI time-of-flight spectrometer at the ISIS pulsed spallation neutron source, at wavevectors $26\le$ Q $\le29$ \AA$^{-1}$ for four different $^{3}$He concentrations $x$. From the data, we extract both the condensate fraction $n_0$ and the single particle kinetic energies $\langle K_3\rangle$ and $\langle K_4\rangle$ of each isotope. We find a relative increase in $n_0$ from 7.25\% for the bulk data ($x=0\%$) to $\sim$12\% for $x=20\%$, in agreement with theoretical calculations but less than that found in the only other DINS measurement of $n_0$. The measured values of $\langle K_3\rangle$ ($\sim$11 K for $x=10\%$) fall in the range of previous DINS measurements. Surprisingly, $\langle K_4\rangle$ is found to be somewhat independent of $x$, in contrast to most calculations and measurements.