» Pfeifer W. - An Introduction to the Interacting Boson Model of the Atomic Nucleus
Pfeifer W. - An Introduction to the Interacting Boson Model of the Atomic Nucleus
||An Introduction to the Interacting Boson Model of the Atomic Nucleus
||A model of the atomic nucleus has to be able to describe nuclear properties such as spins and energies of the lowest levels, decay probabilities for the emission of gamma quantas, probabilities ( spectroscopic factors ) of transfer reactions, multipole moments and so forth. In this chapter those models are outlined from which the IBM comes.
The IBM is mainly rooted in the shell model, which has proved to be an excellent instrument for light nuclei ( up to 50 nucleons ). The larger the number of nucleons becomes the more shells have to be taken into account and the number of nuclear states soon becomes so colossal that the shell model will be intractable. For example the 2+ state ( spin 2 and positive parity ) of 154Sm shows 3-1014 different configurations ( Casten, 1990, p. 198 ). The interacting boson model (sometimes named interacting boson approximation IBA) reduces the number of states heavily. It constitutes only 26 configurations for the 2+ state mentioned above.
The shell model reveals that the low-lying states of the even-even nuclei are made up predominantly by nucleon pairs with total spin 0 or 2. Higher spins of such pairs are rare for energy reasons ( Hess, 1983, p. 55 ). Particularly the spins of pairs of identical nucleons are even numbers because they constitute an antisymmetric state ( appendix A2 ). Furthermore, in the case of two identical nucleon pairs the total spin is strictly even, which follows from the fact that the pairs behave like bosons ( see appendix A2 ). This theoretical result is not far from the real situation of even-even nuclei, from which it is known that their total spin predominantly is even.
These and other arguments led to the basic assumption of the IBM which postulates that the nucleon pairs are represented by bosons with angular momenta / = 0 or 2. The multitude of shells which appears in the shell model is reduced to the simple s-shell (/ = 0 ) and the d-shell (1=2) which is composed vectorially by d-bosons analogously to the shell model technique. The IBM builds on a closed shell i.e. the number of bosons depends on the number of active nucleon ( or hole ) pairs outside a closed shell. Each type of bosons, the s- and the d-boson, has its own binding energy with regard to the closed shell. Analogously to the standard shell model, the interacting potential of the bosons acts only in pairs.
As a peculiarity of the IBM there exist special cases in which certain linear combinations of matrix elements of this interaction potential vanish (chapters 10 and 14 ). In these cases the energies of the nuclear states and the configurations can be expressed in a closed algebraic form. These special cases are named "dynamic symmetries". They correspond to the well-known "limits" allocated to the vibration, the rotation et cetera of the whole nucleus. However most nuclei have to be calculated by diagonalising the Hamilton matrix as is usual in quantum mechanics (chapter 12 ).