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Phonon-phonon interactions in the theory of fluids Lokken, John Erwin


This thesis is devoted to the effect of phonon-phonon interactions on the energy of a non-viscous fluid, and hence to its specific heat. It extends the work of previous authors by taking into account the terms in the Hamiltonian of higher than the second order (lowest order) in the field variables. It is shown that the term of third order in the field variables, contributing in second order, and the term of fourth order, contributing in first order, may give significant contributions if the theory is applied to liquid helium II. The phonon energy in this approximation is linear in the momentum, Ek = (1 + ∝-δ)c₀kk- Here, c₀kk is the contribution of the fourth order term and -δc₀kk the contribution of the third order term. c₀kk is the value obtained by previous authors by considering the lowest order term only. It is shown that for liquid helium II the contribution to the energy of the non-linear terms is smaller than that of the linear terms. In this expression for the energy (1 + ∝ - δ)c₀ is interpreted as the measured velocity of first sound. Thus the cubic term in the specific heat is unchanged in this approach, the effect of the higher order terms in the Hamiltonian being to change the velocity of sound. Because the non-linear terms have been found to be small the conclusion has been reached that the terms of higher order in the temperature than the third cannot be attributed to phonons, and that therefore this theory is only valid for liquid helium II below 0.6°K.

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