The nuclear shell model assumes, as a first approximation, that the particles move independently in a common spherically symmetric average potential. For a so-called "doubly magic" nucleus, in which the numbers of neutrons and protons are just sufficient to fill certain shells, the result is a spherically symmetric "closed shell core" of positive parity and zero total angular momentum. Because closed shell nuclei are spherical, the shells are characterized by the orbital angular momentum quantum number l, and because the spin orbit interaction is strong in nuclei, the total angular momentum, j = l ± 1/2, of the particle (including its intrinsic spin) is also a good quantum number. The traditional spectroscopic notation s, p, d, f, ... is used to denote l values of 0, 1, 2, 3, ... respectively.
Particles in shells outside closed shells are called valence particles. By colliding, valence particles can easily be scattered from one shell into another, especially if the two shells have similar energies. In modeling states of two or more valence particles, it is therefore necessary to take into account all basis states formed by distributing the particles among a set of available shells that lie close together in energy.
As an example, consider two valence particles distributed among the close-lying d5/2 (l = 2, j = 5/2) and s1/2 (l = 0, j = 1/2) shells. The total angular momentum J of the two particles, and its z component M, will be good quantum numbers because of rotational invariance. For given values of J and M the possible basis states can be denoted by
Each of these states has a different configuration: the first has
both particles in the d5/2 shell, the second
has both particles in the s1/2 shell, and
the third has one particle in each shell. For some J values not all of these
configurations are allowed; thus for J=0, M=0 the only possibilities are
(M)
This set M of two states forms a suitable model space for the low-lying J = 0, M = 0 states of the nucleus consisting of two particles in addition to the closed-shell core. Normal quantum mechanical procedure would be now to calculate all possible matrix elements of the inter-particle interaction V between states in the basis M, add in the contributions of the single-particle energies, and then diagonalize the resulting matrix (a 2 by 2 matrix in this case). The eigenvalues of this matrix will, for a weak enough interaction, be good approximations to the energies of the corresponding physical states of the actual nucleus. Unfortunately this procedure fails for the very strong inter-particle potentials that are known from fitting nucleon-nucleon scattering data.
The procedure can be saved by a formal trick: one introduces an effective interaction Veff, defined so that it gives exactly correct physical energies when used in place of V, and so that the eigenvectors of the model-space problem agree with the portions of the physical states that lie in the model space. Of course this replaces the original problem with the problem of determining Veff. This is not a trivial problem. The replacement of V by Veff must compensate for all the configurations that are absent from the set M, including those involving omitted valence shells and those involving excitation of one or more particles out of the closed shells into valence shells. The close relationship of Veff to physical energies is brought out by considering the quite common simple case in which M is one-dimensional. For example, suppose we are interested in J = 4, M = 4 states for two particles distributed among the d5/2 and s1/2 shells. Here the only possible basis state is
so the matrix to be diagonalized is 1 by 1 and its eigenvalue is the only matrix element of Veff (plus twice the d5/2 single-particle energy).
There are two main approaches to finding effective interactions: ab initio theoretical calculations based on the Brueckner G matrix, and empirical extraction of Veff from experimental data.
