Few-phonon structure functions for liquid helium II

Abstract

A semiphenomenological method is formulated for calculating the Fourier transforms of the few-body distribution functions—the few-phonon liquid structure functionsS _s(q ₁,...,q _s−1)—on the basis of a functional Ansatz independent of a superposition approximation. The two-phonon functionS ₂ (q)=S(q) is calculated and compared with the experimentalS(q) to determine the parameters of the functional Ansatz. The procedure is to writeS _s in second-quantized form, to transfer from particle operatorsa _k to operatorsα _k , and then to take expectation values with respect to the ground state |0) of theα _k 's. The transformation is, fork≠0, a _k =(1+v ² _k )^1/2α _k +v ² _k α ⁺_−k . Fork=0, a ⁺₀ a₀|0) is written in terms of the othera _k 's as effectively the most general possible expression. This expression is nonlinear and involves arbitrary parametric functions subject to the restrictions that(0|a ⁺₀ a₀|0)=N₀, and that theS _s have the proper physical properties. Thes-body functionS _s is then expressed as a functional ofv _k , ands−1 parametric functionsc ₂,...,c_s−1 arising from the expression fora ⁺₀ a₀|0). The functionsc ₃, c₄,..., contribute only for smallq. A simple algebraic expression forv _k , containing only a few parameters, is found which yields agreement with experiment forS(q) to within the 10% calculational error. The relative number of particles in the zero-momentum staten ₀ , calculated as a consistency check, turns out to be equal to 0.2. Having determinedv _k andc ₂, S₃ can then be evaluated in a straight-forward manner.S ₃ is explicitly displayed as a functional ofv _k , c₂, andc ₃. It is shown thatc ₃(k, k′) can be chosen so thatS ₃(s,p) has the correct known behavior for smalls.