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Physics of Particles and Nuclei

Second- and Third-Order Anharmonic Effects within the Quantum Many-Body Theory

Шитов Михаил Игоревич, S. P. Kamerdzhiev

Published Sep 1, 2019• 0 citations
Publisher

Abstract

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.

DOI
10.1134/s1063779619050216

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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.
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 ).
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.
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.
A nonlinear generalization of the quasiparticle random phase approximation is constructed with the aim of describing anharmonic effects in the low-lying spectra of vibrational nuclei. Angular momentum zero pairing correlations of the ground state are described by a generalized BCS calculation based on a variational method. The dynamics of the vibrational states are described by the Heisenberg equations of motion for fermion pair operators; matrix elements of products of pair operators are treated by sum rule methods. Kinematical constraints are expressed by the algebra of the pair operators and by boson-like polynomial expansions for the quasiparticle conserving multipole operators in terms of the quasiparticle pair changing operators, these expansions enforcing the content of the Pauli principle. The physical basis for the choice of approximations is described. The final algorithm is organized so as to involve little more than repeated BCS-like and RPA-like calculations, converging to a self-consistent solution of the full nonlinear scheme. Detailed discussion is carried out for a modified form of the pairing plus quadrupole interaction Hamiltonian, but the method is not confined to this model.NUCLEAR STRUCTURE Approximate many-body methods applied to modified pairing plus quadrupole model. Description of vibrational spectrum of single closed shell nuclei. Equations of motion and algebraic methods.
Abstract The article discusses nuclear ground state correlations (GSCs), which are particularly clearly manifested when using the formalism of quantum Green’s functions (GFs) and Feynman diagrams. In addition to the one-particle-one-hole ground state correlations in the well-known random phase approximation (RPA) method, there also exist and make a noticeable quantitative contribution GSCs that arise from using more complex configurations than in RPA, specifically, configurations containing phonons and those containing only three quasiparticles, phonon–GSCs and GSC3s, respectively. The latter case of GSC3 is discussed in detail in this paper. It has been shown that GSC3s make a significant quantitative contribution to problems related to explaining certain characteristics of low-lying excited states (phonons) and transitions between them in even–even nuclei.
A new method for systematic treatment of the anharmonicity effect in the many-phonon states in fermion space is proposed. The most important part of the method is a recursion formula for the norm of many-phonon states in fermion space. This recursion formula is applied to the calculation of every matrix element of an arbitrary operator. A successive approximation procedure to calculate the recursion formula is also proposed. This new method is rewritten so as to be suitable for the case of collective-phonon truncation of fermion space. The recursion formula in this case is expressed in a form expanded with respect to to what extent the anharmonicity effect due to the Pauli principle is taken into account. It is shown that the cutoff factor appearing in the boson representation of the SU(6) algebra is obtained by the lowest order of the successive approximation procedure for this expansion and the values of the cutoff factor estimated in this approximation show good fit to those obtained from the exact solution of the many-quasi-particle states which were previously calculated by the present authors. This implies that the new method proposed in this paper gives good results even in the lowest order approximation.