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

We present benchmark, high precision measurements of the dynamic structure factor J(Q,y) of liquid He-4 at several temperatures over a wide wave vector transfer range 15 less than or equal to Q less than or equal to 29 Angstrom (-1). J(Q,y) is very different in the superfluid phase below T-lambda and in the normal phase above T-lambda where T-lambda = 2.17 K. Below T-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*(k) for states k > 0 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 - (T/T-lambda)(gamma)] with n(0)(0) = (7.25 +/- 0.75)% and gamma = 5.5 +/- 1.0 for T < T-, which is 30% below existing observed values, and n(0) = (0 +/- 0.3)% for T > T-lambda. We determine n(k) in both phases. The n*(k) is significantly narrower than a Gaussian in both superfluid and normal He-4 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) is important in determining the value of n(0)(T) below T-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.