ЯДЕРНАЯ ФИЗИКА, 2014, том 77, № 8, с. 1024-1037
= ЯДРА ^^
CONTINUUM SHELL MODEL AND NUCLEAR PHYSICS AT THE EDGE OF STABILITY
© 2014 A. Volya1), V. Zelevinsky2)*
Received July 30, 2013
Studies of nuclei far from the valley of stability are currently in the center of modern nuclear physics. For such loosely bound systems, the continuum effects are vitally important. We develop the continuum shell model based on an effective non-Hermitian Hamiltonian. This rigorous quantum-mechanical method is powerful for description of open quantum systems unifying their structure and reactions. The formalism is explained and examples of its application are given; the results are in a very good agreement with recent experiments on exotic nuclei. We show also how this approach can be successfully applied to a general problem of a signal transmission through an open quantum system.
DOI: 10.7868/S0044002714070186
1. INTRODUCTION
A quantum system perfectly isolated from its surrounding is an idealization convenient for theoretical modeling but not always appropriate for studying the real physics. A harmonic oscillator is an idealized example where the system keeps the intrinsic structure independently of its level of excitation. Realistic many-body systems have both a discrete spectrum of bound states and, starting from some threshold energy when the constituents of the system can separate, a continuum spectrum of scattering states. In nuclear physics loosely bound systems are currently at the center of interest, both theoretical and experimental. This interest is related to the general problem of stability of nuclear matter and to the question of the nucleosynthesis in the universe that supposedly proceeds far from the valley of stability.
Interest to unstable quantum systems extends beyond nuclei: all families of the so-called elementary particles are in fact groups of resonances with very different lifetimes; in mesoscopic physics many promising applications, including quantum informatics, are based on the signal transmission which is a temporary excitation and then deexcitation of a small quantum aggregate of interacting particles.
All such phenomena carry a lot of similarity. The system is referred to as open when there are qua-sistationary states excited in some reaction. These
^Department of Physics, Florida State University, Tallahassee, USA.
2)National Superconducting Cyclotron Laboratory and Department of Physics and Astronomy, Michigan State University, East Lansing, USA. E-mail: zelevins@mscl.msu.edu
states are then decaying through a certain channel that may or may not coincide with the channel used for excitation. There are unitarity requirements that the total probability of all processes started in any entrance channel equal to one. Relatively long-lived quasi-stationary states are seen in reactions as resonances with the width r ~ h/r inversely proportional to the lifetime t. The typical time evolution of a quasi-stationary state is given by exp[—iEt — (r/2)t], that can be formulated in terms of complex energy,
S = E-l-T. (1)
In the case of overlapping resonances, the cross sections of reactions going through unstable intermediate states can reveal complicated fluctuations so that the only possible approach is that of statistical correlational analysis.
The phenomenological description of complex reaction processes was developed in early years of quantum theory. However, the theoretical connection between the external description that includes various reactions and their observables, directly defined by the experiment, and the intrinsic structure came much later. The theory of open quantum systems is a cross-disciplinary subject that is relevant to many-body quantum systems, such as complex nuclei, atoms, molecules, or small condensed matter devices.
The nuclear shell model, also known as configuration interaction method, is one of the best practical tools for describing the properties of stable nuclei. Here one uses a truncated orbital space for nucleons in a mean field and an effective interaction between the particles. The interaction is roughly derived from the nucleon—nucleon scattering data
and simplest nuclei and renormalized, usually in a phenomenological way, to account for the medium effects and truncation of the space. The large-scale diagonalization of the Hamiltonian matrix (exact if the dimensions allow that or approximate) produces the stationary wave functions, their energies and observables (multipole moments and transition rates). One of the best available examples is given by the sd space (nuclei from 16O to 40Ca). The input here contains three effective single-particle energies and 63 two-body interaction matrix elements allowed by the symmetries (spin, isospin, and parity). The output provides thousands of nuclear characteristics in good agreement with data (for level energies in stable nuclei typically ±200 keV). All approaches of this type assume that the states under consideration are stationary in the discrete spectrum.
For open and marginally stable systems one has to go beyond the standard shell-model description. Loosely bound nuclei have very few bound states, or like the helium isotopes 5'7He are unbound being seen only as resonances in reactions. Instead of the discrete spectrum, we have to deal with continuum, either real or virtual. Among various approaches to the problem of open quantum systems, one of the main and convenient formulations is the use of the effective non-Hermitian Hamiltonian. This method is based on the projection formalism [1, 2], see also a recent review article [3] and references therein. The approach is quite flexible in specific modifications, preserves unitarity and is convenient for applications reducing the part of the continuum problem to the diagonalization based on the same mean-field theory that is used practically always in many-body quantum physics.
In what follows we explain the essential features of the method and give some examples of how it works including realistic applications to the physics of exotic nuclei. Very briefly we will also touch the open non-nuclear mesoscopic systems.
2. EFFECTIVE HAMILTONIAN
The formal introduction of the effective Hamilto-nian is quite general being based on an arbitrary decomposition of the total space by two projectors, Q + + P = 1. The stationary states with real energy E can be now described, with the use of the projection onto one subspace, let say Q, by the effective Hamiltonian
H{E) = Hqq + HQP
1
E — Hpp
Hpq. (2)
Here the subscripts symbolically indicate transitions between the classes of states.
The Hamiltonian (2) depends on running energy E. Now we assume that the class P contains the
states with continuum asymptotics so that they can be characterized by the channel index, a, b, c,..., and energy E. Each channel opens at its threshold energy E(c), so that at E > E(c) the homogeneous equation Hpp^c;E = E$c;E has non-zero solutions $c;E which belong to the continuous spectrum. If the total Hamiltonian H would not connect classes Q and P, the intrinsic states Q would be stationary with no access to the reaction channels. These states can be described by a shell model based on the mean-field picture or, in case of intrinsic chaos, modeled by random matrix theory. In the presence of connecting matrix elements Hqp, such states acquire access to the continuum and finite lifetime. To handle the singularity related to the existence of solutions $c; E, we require that the intrinsic states have a component of outgoing waves which is achieved through the substitution E E (+) = E + i0. Using, for simplicity, the eigenchannel representation [4] for the states in subspace P, we separate in Eq. (2) the principal value (off-shell) and imaginary delta-function part (on-shell) and come to the convenient form of the effective Hamiltonian, that is non-Hermitian and energy-dependent,
n = H-%-W, H = Hqq + A(E). (3)
Written in the form of a matrix in Q space, this Hamiltonian contains the energy shift operator A(E), added to the original Hermitian operator Hqq,
Ai2(E) = (4)
= P.v.^ i(dT')dE'{1\Hqp\c,r',E') x
c ■'
where the principal value of the integral includes all kinematic variables of a channel. The anti-Hermitian part W is factorized in terms of the effective amplitudes ^1(E) as matrix elements of Hqp taken on-shell, with all kinematic factors coming from the integral over channel variables included into their definition,
wu(e) = 2k y; a1(e)ac2*(e).
(5)
c(open)
The real part A of the effective Hamiltonian includes the sum over all channels, open and closed, while the imaginary part W contains only contributions of the channels open at given energy. The kinematic factors (the density of states in the continuum) guarantee that any amplitude ^1(E) vanishes at the threshold energy E(c). The factorized structure in Eq. (5) is a consequence of the unitarity of
the scattering matrix [5]. The numerical indices in Eqs. (4), (5) run over the whole intrinsic space that can be identified with the truncated space of the standard shell model.
The diagonalization of the effective Hamilto-nian (3) leads to the spectrum of its complex eigenvalues (1) and corresponding eigenfunctions which form a biorthogonal complete set that reduces to the normal orthogonality for bound states when r = 0. The same Hamiltonian determines the part of the scattering matrix S that corresponds to the processes going through the intrinsic states of subspace Q,
S(E) = 1 - iT(E) = -iTp +
1 - (i/2)K(E) l + {i/2)K{EY
(6)
where the hats accompany the matrices in the channel space. The amplitude Tp describes the direct reactions, or potential scattering, without virtual involvement of intrinsic states. The matrix K is an
analog of the standard R matrix in reaction theory [6], = (7)
it includes all propagations inside the system with virtual admixture of the off-shell excursions to the continuum through the operator (2
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