expected away from the valley of stability. For example, observations of unstable isotopes have shown shifting and even a reordering of the single particle levels of which the shell structure is composed. This is sometimes observed as the creation of an island of inversion or in the reduction of excitation energy gaps above the traditional magic numbers. Basic hypotheses Some basic hypotheses are made in order to give a precise conceptual framework to the shell model: The atomic nucleus is a quantum n-body system. The internal motion of nucleons within the nucleus is non-relativistic, and their behavior is governed by the Schrödinger equation. Nucleons are considered to be pointlike, without any internal structure. Brief description of the formalism The general process used in the shell model calculations is the following. First a Hamiltonian for the nucleus is defined. Usually, for computational practicality, only one- and two-body terms are taken into account in this definition. The interaction is an effective theory: it contains free parameters which have to be fitted with experimental data. The next step consists in defining a basis of single-particle states, i.e. a set of wavefunctions describing all possible nucleon states. Most of the time, this basis is obtained via a Hartree–Fock computation. With this set of one-particle states, Slater determinants are built, that is, wavefunctions for Z proton variables or N neutron variables, which are antisymmetrized products of single-particle wavefunctions (antisymmetrized meaning that under exchange of variables for any pair of nucleons, the wavefunction only changes sign). In principle, the number of quantum states available for a single nucleon at a finite energy is finite, say n. The number of nucleons in the nucleus must be smaller than the number of available states, otherwise the nucleus cannot hold all of its nucleons. There are thus several ways to choose Z (or N) states among the n possible. In combinatorial mathematics, the number of choices of Z objects among n is the binomial coefficient C. If n is much larger than Z (or N), this increases roughly like nZ. Practically, this number becomes so large that every computation is impossible for A=N+Z larger than 8. To obviate this difficulty, the space of possible single-particle states is divided into core and valence, by analogy with chemistry (see core electron and valence electron). The core is a set of single-particles which are assumed to be inactive, in the sense that