We present benchmark, high precision measurements of the dynamic structure factor $J(Q,y)\mathrm{}$ of liquid ${}^4\mathrm{He}$ at several temperatures over a wide wave vector transfer range $15<~Q<~29\hspace{0.5em}{\mathrm{\AA }}^{-1}.$ $J(Q,y)\mathrm{}$ is very different in the superfluid phase below $T_{\lambda }$ and in the normal phase above $T_{\lambda }$ where $T_{\lambda }=2.17\mathrm{}\hspace{0.5em}\mathrm{K}.$ Below $T_{\lambda },$ $J(Q,y)\mathrm{}$ contains a pronounced additional contribution near $y=0\mathrm{}$ that is asymmetric about $y=0,$ reflecting a condensate contribution modified by asymmetric final-state (FS) effects. The asymmetry in $J(Q,y)\mathrm{}$ is direct qualitative evidence of a condensate. We analyze the data at all T using the same model of $J(Q,y)\mathrm{}$ consisting of a condensate fraction $n_0,$ a momentum distribution $n^{*}\left(\mathbf{k}\right)$ for states $k>\mathrm{}0\mathrm{}$ above the condensate, and a FS broadening function $R(Q,y).$ We find a condensate fraction given by $n_0{\mathrm{}\left(T\right)=n}_0\mathrm{}\left(0\right)\mathrm{}\left[1-{(T/T}_{\lambda }{)}^{\gamma }\right]$ with $n_0\mathrm{}\left(0\right)=(7.25\mathrm{}\pm 0.75)\%$ and $\gamma =5.5\mathrm{}\pm 1.0$ for $T<\mathrm{}{T}_{\lambda },$ which is 30% below existing observed values, and $n_0=(0\mathrm{}\pm 0.3)\%$ for $T>\mathrm{}{T}_{\lambda }.$ We determine $n\left(k\right)\mathrm{}$ in both phases. The $n^{*}\left(\mathbf{k}\right)$ 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 $T_{\lambda }.$ The precise form of $R(Q,y)\mathrm{}$ is important in determining the value of $n_0\mathrm{}\left(T\right)\mathrm{}$ below $T_{\lambda }.$ When independent, theoretical $R(Q,y)\mathrm{}$ are used in the analysis, the $n_0\mathrm{}\left(T\right)\mathrm{}$ is found to be the same as or smaller than the above value.

• Received 15 February 2000

DOI:https://doi.org/10.1103/PhysRevB.62.14337