Progress in Particle and Nuclear Physics 59 (2007) 226–242
Shell model and nuclear structure
Etienne Caurier ∗
Institut Pluridisciplinaire Hubert Curien (IN2P3-Universit´ Louis Pasteur), F-67037 Strasbourg Cedex 2, France
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
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.
mω2r 2 − C ls.
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
equation in the valence space, using an effective interaction such that:
h(r ) =
H Ψ = EΨ −→ Heff. Ψeff. = EΨeff. .
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0146-6410/$ - see front matter c 2007 Elsevier B.V. All rights reserved.
E. Caurier / Progress in Particle and Nuclear Physics 59 (2007) 226–242
In general, effective operators have to be introduced to account for the...