A contribution to the n-p capture amplitude which results from the modification of the nucleon current operators by the nuclear interaction is calculated. The two-nucleon states are represented by Heitler-London states and the capture amplitude is related to single-nucleon matrix elements by means of an expansion corresponding to the exchange of mesons between the nucleons. These single-nucleon matrix elements are evaluated using the fixed-source theory. The one-meson excited Heitler-London state gives a contribution similar to the isobar model of Austern; the role of this state in the nuclear potential is also discussed. If a deuteron wave function with a relatively large Estate probability is used, the interaction effect is about 3% of the phenomenological cross section.* Based on a thesis submitted to Carnegie Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy.f Supported in part by a grant from the U. S. Air Force Office of Scientific Research.

Girardeau has shown that a particular class of variational approximations for the free energy of a Fermi-Dirac system may be expressed in terms of the exact free energy of a certain soluble "model Hamiltonian." This exact formula for the partition function associated with the model system involves the solution of a certain nonlinear integral equation, and in the present work it is shown in detail that there is a correspondence between phase transitions in the system and the existence of multiple solutions of this equation. The connection between the existence of phase transitions and the properties of the interaction is studied, and it is shown that these phase transitions may occur for a very wide class of interactions. It is established that there is an upper bound to the temperature, above which no phase transitions will occur at any density. The extension of these results to a classical and a Bose-Einstein system is indicated, and the lattice-gas model previously studied by Mermin is re-examined in the light of the present results.

The first two terms in the high-density expansion for the energy of a degenerate electron gas imbedded in a uniform background of positive charge are given correctly by the Hartree-Fock theory. 1 In this formulation, one neglects all correlations except those due to the Pauli principle, and thus the effective interaction, which acts only between particles of parallel spin, is purely attractive. The purpose of the present note is to point out that for moderately high densities-for which it is often assumed that the HF theory is a convenient starting point for the analysis of such a system-the electron gas as described by the HF theory undergoes a phase transition analogous to the familiar liquid-gas transition. The density at which this occurs is of order 10 22 /cc and is not related to the melting of the Wigner lattice 2 which occurs at much lower densities. 3 Further, since we assume no polarization of the spins, this transition is not connected with the spin-density waves studied by Overhauser. 4 Mathematically speaking, the phase transition occurs because, as will be shown below, the particle density «(/i), when considered as a function of the chemical potential i±, has a finite discontinuity for a certain value Lt 0 and this implies 5 the existence of a first-order phase transition.In units in which all lengths are expressed in terms of the radius of the first Bohr orbit and all energies in Rydbergs, the single-particle energy of an electron of momentum k is given in the HF theory by £(fe)=fe2~r J^(q-k^+A 2e(M " e m (1)where, for generality, we have introduced a shielding parameter A, and where 6 is the step function which vanishes for negative values of its argument and is unity for positive ones. The relationship in Eq. (1) is, of course, well known, although customarily the 6 function which appears under the integral is not present, but rather the domain of integration is restricted to be the interior of a sphere of radius k 0 . The implied one-to-one relationship between the chemical potential /x and k 0 is always assumed although rarely spelled out in detail. For our purposes, we start with the "more basic" Eq. (1) and note that it is a nonlinear integral equation and that in general, for any given values for A and ii, it may have more than one solution. Physically, this possibility of having a multiplicity of solutions of Eq. (1) offers no difficulty; and on reflection of the meaning and the origin of this equation, one concludes that we must select that solution for which the free energy (equivalently, the total energy) is a minimum. Because of this minimization criterion, it turns out that for a certain value of \i one must switch from one of these multiple solutions of Eq. (1) to another, and this change induces a discontinuity in W(LI) at this point, and thus 5 we have a phase transition. 341

A study is made of a certain model Hamiltonian which describes a system of fermions. Making use of the variational results of Girardeau one can obtain, in the limit of infinite volume, an explicit formula for the free energy in terms of the solution of a nonlinear integral equation. It is established that for certain values of the coupling constant the model exhibits condensation, and that qualitatively at least, the isotherms for the model reproduce those for actual physical systems. The model is applied to He 3 and it is shown that by a suitable choice for the parameters of the model, reasonable agreement with the low-temperature properties of He 3 is obtained. The second virial coefficient for He 4 is also reproduced with the same parameter values which give the correct virial coefficient for He 3 .

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