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Physical Review

Quantum-Mechanical Calculation of the Third Virial Coefficient of He4

Sigurd Yves Larsen

Published May 15, 1963• 29 citations
Publisher

Abstract

Using the method of binary collisions, the third virial coefficient of ${\mathrm{He}}^{4}$ at low temperatures is calculated for a potential consisting of an attractive square well and a repulsive core. The shape of this potential is chosen so as to optimize the fit of the second virial coefficient with experimental data. From a comparison of the results with the third virial data the existence of a three-body bound state is inferred. A binding energy of 0.26\ifmmode^\circ\else\textdegree\fi{}K, calculated from the potential, brings the adjusted results and the experimental data into agreement.

DOI
10.1103/physrev.130.1426

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The quantum theory of the third virial coefficient CCC is discussed. Three types of intermolecular pair forces must be distinguished. (1) No bound or low-lying two- and/or three-body states exist. The first four terms of the low-temperature expansion of CBE{C}_{\mathrm{BE}}CBE​ are obtained. They depend on the scattering length, the effective range, and a third length which cannot be inferred from scattering data. The limitations of the applicability of such expansions are discussed, both for He4{\mathrm{He}}^{4}He4 and He3{\mathrm{He}}^{3}He3, by means of a comparison of the corresponding expansion for the second virial coefficient BBB with detailed numerical results known for specific potentials. (2) Existence of a near zero energy level both for the two- and the three-body system. It is shown how in this case the actual potentials may be replaced by suitably matched boundary conditions on the two- and the three-body wave functions near the respective coordinate origins. It is first explained how the method applies to BBB. Then the leading term of CCC is explicitly determined. (3) Existence of strongly bound two- and three-body states. An approximate expression for CCC is given by treating the single atoms and the binary and ternary compounds as a system of three ideal gases in chemical equilibrium.
The binding energy of three He4{\mathrm{He}}^{4}He4 atoms, subject to pair forces, is investigated numerically for a number of proposed intermolecular potentials and for a series of square wells. For some potentials a 3-body bound state is found, for others not. However, a direct correlation is found between 2-body and 3-body binding. The need for more accurate experimental data is discussed.
Two distinct approaches have been followed to produce an approximate expression for the third virial coefficient of a quantum system in terms of the pertinent two-particle quantities (standard T matrix elements and bound state eigenvalues and eigenstates): One approach is based on the T approximation for the self-energy in the theory of temperature Green functions, and the other uses the diagrams which represent the various terms in the multiple-scattering expansion of the three-particle T operator. It is shown that the result given by the former method is reproduced exactly by the contribution of a small number of selected diagrams. The Green function method requires the evaluation of the Kadanoff and Baym generalized T matrix to one order in the activity higher than its familiar low-density limit. The common result can be written entirely in terms of the lowest order self-energy and should give a reasonable approximation to the third virial coefficient for systems with short-range forces.
A quantum calculation of the third virial coefficient of a two-dimensional gas obeying fractional statistics is performed by diagonalizing the three-body Hamiltonian in a harmonic-oscillator potential with a large fermionic basis. The numerical results for the virial coefficient A3{\mathit{A}}_{3}A3​ as a function of the flux strength \ensuremath{\alpha} (\ensuremath{\alpha}=0 for fermions, \ensuremath{\alpha}=1 for bosons) are reliable in the range 0\ensuremath{\le}\ensuremath{\alpha}\ensuremath{\le}0.25. $A sub 3--- increases quadratically with \ensuremath{\alpha} for very small \ensuremath{\alpha}, but higher powers contribute substantially for large \ensuremath{\alpha}.
The virial expansion method is applied within a harmonic approximation to an interacting N-body system of identical fermions. We compute the canonical partition functions for two and three particles to get the two lowest orders in the expansion. The energy spectrum is carefully interpolated to reproduce ground-state properties at low temperature and the noninteracting high-temperature limit of constant virial coefficients. This resembles the smearing of shell effects in finite systems with increasing temperature. Numerical results are discussed for the second and third virial coefficients as functions of dimension, temperature, interaction, and transition temperature between low- and high-energy limits.
We present a field theoretic method for the calculation of the second and third virial coefficients b2{b}_{2}b2​ and b3{b}_{3}b3​ of two-species fermions interacting via a contact interaction. The method is mostly analytic. We find a closed expression for b3{b}_{3}b3​ in terms of the two- and three-body TTT matrices. We recover numerically, at unitarity, and also in the whole Bose-Einstein-condensate--BCS crossover, previous numerical results for the third virial coefficient b3{b}_{3}b3​.