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Physics of Atomic Nuclei

Anharmonic Effects in the Theory of Finite Fermi Systems

S. P. Kamerdzhiev, Шитов Михаил Игоревич

Published Jan 1, 2020• 0 citations
Publisher

Abstract

Some results obtained by E.E. Saperstein and his coauthors in the theory of the ground state and low-lying excited states of magic and semimagic nuclei are discussed briefly. Corrections to the creation amplitude $$g$$ for one low-lying phonon are considered for the first time. These corrections, treated on the basis of the existence of a small parameter, include the tadpole-induced coupling of the amplitudes for the creation of one and two phonons.

DOI
10.1134/s1063778820010068

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Some results obtained by E.E. Saperstein and his coauthors in the theory of the ground state and low-lying excited states of magic and semimagic nuclei are discussed briefly. Corrections to the creation amplitude ggg for one low-lying phonon are considered for the first time. These corrections, treated on the basis of the existence of a small parameter, include the tadpole-induced coupling of the amplitudes for the creation of one and two phonons.
Abstract The paper presents a brief review of the results in the self-consistent theory of the anharmonic effects of the second and third orders in the phonon production amplitude, based on the quantum many-body theory. Numerical (for second-order effects) and theoretical analyses of new, i.e., three- and four-quasiparticle correlations in the ground state (backward-going graphs) in magic nuclei and in nuclei with pairing are given. A formula for the probability of transition between one- and two-phonon states is obtained and compared to the solution of a similar problem within the quasiparticle-phonon model. It is shown that the approach under consideration contains a number of new effects.
Anharmonic effects in the capillary branch of excitations of a Fermi-liquid drop are considered. The limiting transition to classical hydrodynamics is found. It exists when the orbital angular momenta L/sub i/ of the collective excitations are large. The situation of the classical conservation law for the volume of the system is analyzed in the quantum case. It is found that when the frequency of a collective oscillation is small compared with the characteristic single-particle frequencies, the results obtained on the basis of the phenomenological scheme with a phonon-phonon Hamiltonian are reproduced. It is shown that no new parameters are required to calculate the constants of this Hamiltonian in the theory of finite Fermi systems.
The quantum theory of third order anharmonic effects in the photon production amplitude developed by V.A. Khodel has been generalized for nuclei with pairing and for a more accurate description of low-energy collective phonons. An expression for the transition amplitude between two- and one-phonon states has been obtained. This expression has been compared to the solution of a similar problem considered within the quasiparticle–phonon model. It has been shown that this approach describes a number of new effects, including three- and four-quasiparticle correlations in the ground state ( backward-going graphs ).
A new method which enables us to transcribe the dynamics of the system of even number of fermions into that of the boson system is developed, with the purpose of analysing the “anharmonic effects” on the collective oscillations in spherical even nuclei from the standpoint of the microscopic theory of the collective excitations. In the “harmonic approximation”, a pair of bound quasi-particles is replaced by the “phonon” as an ideal boson. This replacement inevitably leads to neglecting two main effects; the dynamical effect due to the residual interactions omitted under the “harmonic approximation” and the effect of the Pauli-principle among the quasi-particles composing the different pairs. According to our theory, the two effects which cause “anharmonic effects” can be evaluated correctly, in principle. It is not our purpose to go into detailed quantitative calculations, but rather to develop the basic idea.
Many physical systems can be described by one-dimensional (1D) models. It is the case of ultra-cold atoms: under certain circumstances their dynamics occurs only in one dimension. During my PhD I studied some aspects of 1D integrable systems. First, I present a study on the ground state of a system of 2-component repulsive fermions in 1D under harmonic confinement. I use the Bethe ansatz solution to calculate the phase diagram of the system in the homogeneous case. Adding a harmonic confinement I show that the atoms are distributed in a two-shell structure: the partially polarised phase in the inner shell and the fully polarised phase at the edges of the trap. Next I study the finite size effects for the gap of the quasiparticle excitation spectrum in the 1D Hubbard model. Two type of corrections to the result of the thermodynamic limit are obtained: a power law correction inversely proportional to the size of the system L, due to gapless excitations, and an exponential correction on L related to the existence of gapped excitations. In the weakly interacting regime this last correction can become important. Finally I study the response of a highly excited 1D gas to a periodic modulation of the coupling constant. I consider the Lieb-Liniger model and the non-integrable model of a single mobile impurity in a Fermi gas. I show that the non-integrable system is sensitive to excitations with frequencies as low as the mean level spacing, whereas the threshold frequency in the integrable case is much larger. This effect can be used as a probe of integrability for mesoscopic 1D systems, and can be observed experimentally by measuring the heating rate of a parametrically excited gas.
A systematic development of the theory of finite Fermi systems for non-zero temperatures (Tnot =0) is undertaken on the basis of the method of temperature Green's functions. The properties of compound nuclear states are considered. The variations of the density matrix deltarho, the self-energy part delta..sigma.., the mean square radius delta , the quadrupole moment delta, etc, are studied in a transition from the ground state (T=0) to an excited state with temperature T. The relationship connecting the excitation energy with temperature is obtained. Consistently taken into account is the strong interaction of the nucleons, in particular, the change in the one-particle nuclear potential as the temperature T is varied. Qualitative quasi-classical estimates are given for the effects considered, and the limits on the applicability of the statistical description of nuclei at large T are discussed.