On the theory of relaxation processes in liquid helium

Abstract

After a brief introduction we discuss a slight generalization of the Brueckner-Sawada theory of the hard-sphere boson gas. This generalization is introduced in order that we can discuss the effect of attractive forces between the bosons. We have used the results obtained by Brueckner and Sawada to compare the properties of the hard-sphere boson gas with the experimental properties of liquid helium. It turns out that the value of the parameter λ 2 ( λ 2 is a measure for the matrix element of the t-matrix corresponding to the scattering of two zero-momentum state particles) chosen by Brueckner and Sawada is somewhat larger than the value which fits the experimental data best. In most of our subsequent calculations we have therefore taken λ 2 to be equal to 30 rather than equal to Brueckner and Sawada's value of 40. In Section III we have investigated the effects due to the depletion of the zero-momentum state, and we show that one should expect about half of all the particles to be in excited states. Section IV is devoted to a discussion of the effects due to the attractive forces, for which we adopt a potential which resembles the Slater-Kirkwood interatomic potential for helium. The agreement between the calculated energy spectrum for the quasi particles and the experimentally obtained excitation spectrum for liquid helium is much poorer than in the case of a hard-sphere boson gas without attractive forces. This shows that the effect of the attractive forces remains a major problem and it sheds considerable doubt on the applicability of the hard-sphere model to liquid helium. In Section V we discuss briefly the dependence of the t-matrix elements on energy, and we show that if this effect is properly taken into account it is impossible to get an energy spectrum which resembles the experimental spectrum. Notwithstanding all the evidence brought forward in Sections III to V against the hard-sphere boson gas model for liquid helium, we have used it to evaluate in Sections VI and VII the mean lifetime of rotons. In Section VI we consider phonon-roton collisions, and especially the phonon + roton → phonon + roton process, as we show that for temperatures below 1.8°K and in the approximation which we are using other processes are less likely to occur. The resulting lifetime decreases steeply with increasing temperature and is about 3 × 10 −12 sec at 1.1°K, about 10 −12 sec at 1.6°K, and about 6 × 10 −13 sec at 1.8°K. In Section VII we consider the roton + roton → roton + roton process and show that other roton-roton processes are less likely to occur. The lifetimes now are about 2 × 10 −11 sec at 1.1°K, 10 −12 sec at 1.6°K, and 6 × 10 −13 sec at 1.8°K. The theoretical results obtained in Sections VI and VII are compared in Section VIII with experimental data about the neutron-scattering line widths, and also with the data about the viscosity of liquid helium. The theoretical lifetimes are up to one order of magnitude smaller than those derived from experimental data. A short discussion is given of the influence of higher order processes on the relaxation times calculated in Sections VI and VII. We then turn to the problem of the isotopic impurity 3He in liquid helium II, and in Section X determine its effective mass using a hard-sphere model for the potential; this mass turns out to be about twice the mass of a free 3He atom, in reasonable agreement with experimental data. In Section XI we study the influence of these impurities on the roton viscosity, and estimate the coefficient for diffusion of 3He atoms through liquid helium II. The calculated lifetimes are between one and two orders of magnitude smaller than the ones following from the experimental data.