It is shown that the moment of inertia of a noninteracting many-body fermion system moving under periodic boundary conditions has the classical or rigid value when calculated on the "cranking" model of Inglis. By investigating the analogous "pushing" case for the inertial mass we show that the rigid moment can be associated with rigid rotation in spite of apparent surface currents. The effect of particle-particle forces is investigated in the lowest order of perturbation theory. The terms corresponding to a level shift or effective mass are just compensated by other terms and there is no change in the moment of inertia. The possible general validity of these results and their consequence is discussed.
The effects of possible superfluidity on the cranking moment of a large many-fermion system moving under periodic boundary conditions are investigated within the framework of the theory of superconductivity recently formulated by Bogolyubov. The Hamiltonian is initially subjected to Bogolyubov's general unitary transformation. The collective excitations of the fermions are then considered in the usual pair approximation; the appropriate cranking terms are linear in the boson pair operators. On performing a unitary transformation which transforms away these linear terms, one obtains an expression for the moment of inertia of the system which includes both the effects of possible superfluidity and collective excitation. This expression, by virtue of its being stationary with respect to arbitrary variations in the amplitude associated with the latter unitary transformation, is then utilized as a variational principle for the moment of inertia. For the normal state, the result previously obtained by the author, that the moment of inertia has the rigid value, is rederived in more compact form. For the superfluid state, one finds that collective excitations effect a marked increase in the superfluid moment at intermediate coupling strengths although the resulting moment is still quite small compared to the rigid value. In contrast to the normal state case, where particle-hole pairs play a major role, this increase is almost entirely due to excitations consisting of particle-pairs or hole-pairs. The precise magnitude of the apparent resonance in the moment produced by the
d-wave part of the cranking interaction is dependent to some extent on the features of the particle-particle potential which leads to the superfluid state. Variational expressions for the moment are exhibited for both Yukawa and delta-function shell potentials. These results are identical in charged and neutral Fermi systems. A calculation of the cranking moment at finite temperatures is presented in an Appendix along with an interpretation of it in terms of Bardeen's two-fluid model of superconductivity.
The interaction effects on the moment of inertia of a large many-body fermion system moving under periodic boundary conditions have been explicitly derived in the second order of particle-particle coupling for the case of nonsingular repulsive interparticle forces. The derivation is facilitated by the use of a new Hamiltonian constructed by canonical transformation. This transformation is determined by requiring that it remove the compensating effects of pair excitation and that the new interaction operators of first and second order in the cranking field be also of first order in particle-particle coupling. A graphical analysis is presented in which those classes of diagrams which yield corrections vanishing relative to the rigid moment in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$ are distinguished. Some remarks on the related problem of diamagnetism are made. The possible application of the method of canonical transformation to related many-body problems is indicated.
The response of a many-fermion system to dilatation is studied through the introduction, in close analogy with the "cranking" and "pushing" models, of a "squashing" model characterized by the Hamiltonian, $H={H}_{0}\ensuremath{-}\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\epsilon}}{D}_{x}$, where
Dx is the generator of dilatations in the
x direction. The evaluation of the "rigid" dilatational moment of such a noninteracting system is carried out for both standing wave and periodic boundary conditions in a cubic box of side
L. As in the pushing model, the dilatation moment can be evaluated formally using the relation between
Dx and the commutator [
x2,
H0]. For independent fermions moving in a harmonic oscillator one-body potential, this relation is shown to lead to further simplification after the introduction of perturbed one-body wave functions. In the case of periodic boundary conditions, it is shown that the rigid dilatational moment is not altered by an interparticle interaction to first order in the interaction strength; by comparison with the analogous cranking moment problem, this result extends to all orders in the interaction strength. The response to a symmetric dilatation,
D=Dx+Dy+Dz, is studied in a soluble Hartree model of "nuclear" correlations wherein a separable monopole-monopole interaction acts between "nucleons." In this model the dilatation moment in the presence of pair correlations is found to be the rigid moment, with the kinetic energy of dilatation $\ensuremath{\propto}{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\epsilon}}}^{2}$, there being no higher order corrections.NUCLEAR STRUCTURE Approximate many-body methods applied to solvable models.
This is the second of a series of papers aimed at exploiting a generalized self-consistent approach to the study of collective excited states in finite many-particle systems. It contains: (i) A restatement of the equations of the method with further discussion of the basic approximations involved; (ii) a study, through the medium of the exactly soluble problem of uniform translational motion, of the inherent limitations in accuracy of the method when applied to the "strong-coupling" limit for which it was designed. A mode of approximation which can be made self-consistent is indicated. (iii) This approximation is applied to the problem of rotational motion in a plane. In the further limit in which one assumes the spectrum of the rigid rotator, one regains the results of the self-consistent cranking model. In contrast to previous work, the possibility of removing this limitation is indicated, but remains to be carried through.