Progress in Particle and Nuclear Physics 59 (2007) 226–242

www.elsevier.com/locate/ppnp

Review

Shell model and nuclear structure

Etienne Caurier ∗

Institut Pluridisciplinaire Hubert Curien (IN2P3-Universit´ Louis Pasteur), F-67037 Strasbourg Cedex 2, France

e

Abstract

The present possibilities, problems and limitations of the description of the nuclear structure in the

framework of the Shell Model are discussed. Some applications are presented, in particular to the calculation

of the nuclear matrix elements for ββ decays.

c 2007 Elsevier B.V. All rights reserved.

Keywords: Nuclear structure; Shell model; Double beta decay

1. Introduction

The nuclear shell model (SM) was introduced [1,2] some 55 years ago to explain the

regularities of the nuclear properties associated with magic numbers. Its authors proposed an

independent particle model, assuming that the main effect of the two body nucleon–nucleon

interaction was to generate a mean ﬁeld. They modelled this mean ﬁeld as the sum of a harmonic

oscillator potential plus a spin–orbit force.

1

mω2r 2 − C ls.

2

The success of the independent particle model strongly suggests that the very singular free NN

interaction can be regularized in the nuclear medium. Starting with a regularized interaction, the

exact solution of the secular problem, in the (inﬁnite) Hilbert space built on the mean ﬁeld orbits,

is approximated in the large scale shell model calculations by the solution of the Schr¨ dinger

o

equation in the valence space, using an effective interaction such that:

h(r ) =

H Ψ = EΨ −→ Heff. Ψeff. = EΨeff. .

∗ Tel.: +33 03 88 10 64 84; fax: +33 03 88 10 62 02.

E-mail address: etienne.caurier@ires.in2p3.fr.

0146-6410/$ - see front matter c 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.ppnp.2006.12.012

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E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242

In general, effective operators have to be introduced to account for the...