Condensate, momentum distribution, and final-state effects in liquid 4 He

TLDR: In this article, high precision measurements of the dynamic structure factor $J(Q,y)$ of liquid at several temperatures over a wide wave vector transfer range were presented, with the same model of a condensate fraction consisting of a momentum distribution, and a FS broadening function.

Abstract: We present benchmark, high precision measurements of the dynamic structure factor $J(Q,y)$ of liquid ${}^{4}\mathrm{He}$ at several temperatures over a wide wave vector transfer range $15l~Ql~29 {\mathrm{\AA{}}}^{\ensuremath{-}1}.$ $J(Q,y)$ is very different in the superfluid phase below ${T}_{\ensuremath{\lambda}}$ and in the normal phase above ${T}_{\ensuremath{\lambda}}$ where ${T}_{\ensuremath{\lambda}}=2.17 \mathrm{K}.$ Below ${T}_{\ensuremath{\lambda}},$ $J(Q,y)$ contains a pronounced additional contribution near $y=0$ that is asymmetric about $y=0,$ reflecting a condensate contribution modified by asymmetric final-state (FS) effects. The asymmetry in $J(Q,y)$ is direct qualitative evidence of a condensate. We analyze the data at all T using the same model of $J(Q,y)$ consisting of a condensate fraction ${n}_{0},$ a momentum distribution ${n}^{*}(\mathbf{k})$ for states $kg0$ above the condensate, and a FS broadening function $R(Q,y).$ We find a condensate fraction given by ${n}_{0}{(T)=n}_{0}(0)[1\ensuremath{-}{(T/T}_{\ensuremath{\lambda}}{)}^{\ensuremath{\gamma}}]$ with ${n}_{0}(0)=(7.25\ifmmode\pm\else\textpm\fi{}0.75)%$ and $\ensuremath{\gamma}=5.5\ifmmode\pm\else\textpm\fi{}1.0$ for $Tl{T}_{\ensuremath{\lambda}},$ which is 30% below existing observed values, and ${n}_{0}=(0\ifmmode\pm\else\textpm\fi{}0.3)%$ for $Tg{T}_{\ensuremath{\lambda}}.$ We determine $n(k)$ in both phases. The ${n}^{*}(\mathbf{k})$ is significantly narrower than a Gaussian in both superfluid and normal ${}^{4}\mathrm{He}$ and narrowest in the normal phase. The final-state function is determined from the data and is the same within precision above and below ${T}_{\ensuremath{\lambda}}.$ The precise form of $R(Q,y)$ is important in determining the value of ${n}_{0}(T)$ below ${T}_{\ensuremath{\lambda}}.$ When independent, theoretical $R(Q,y)$ are used in the analysis, the ${n}_{0}(T)$ is found to be the same as or smaller than the above value.