Interacting Boson Approximation

The Interacting Boson Approximation ( IBA ), often also referred to as the Interacting Boson Model (IBM) , is an approximation method used to describe the structure of atomic nuclei , especially those with an even number of nucleons from cerium to lead .

In IBA, all nucleons outside a core shell are coupled in pairs to form bosons - in this case to form particles with angular momentum 0 (s boson) or 2 (d boson). The model originally developed by Akito Arima and Francesco Iachello in 1974 couples two neutrons to form a boson and two protons to form a boson. A similar method was developed almost simultaneously by Janssen, Jolos and Dönau. Accordingly, this IBA-1 model is only suitable for describing nuclei with an even number of neutrons and an even number of protons (gg nuclei).

The range of application to the nuclei from cerium to lead is mainly given by the fact that there is a very large mass range between the shell with neutron number 82 and the corresponding proton shell for lead with Z = 82, where a large number of these bosons are taken into account got to. In the model, the number of bosons is counted from the closest shell closure, their coupling to one another is done by a simple 2-body force.

Algebraic structure

In the following, the formalism of the second quantization is used. For the d boson we define the so-called d boson generator operators with as well as the s boson generator and the corresponding annihilation operators. We consider the 36 combinations , , and . This set of so-called generators forms a U (6) -Lie algebra (U for unitary). Several physically meaningful subalgebras can be found for this algebra. These are designated by U (5), O (6) (O for orthogonal) and SU (3) (SU for specifically unitary). These three subalgebras in turn contain physically relevant subspaces: ${\ displaystyle d_ {m} ^ {\ dagger}}$ ${\ displaystyle m = -2, -1,0,1,2}$ ${\ displaystyle s ^ {\ dagger}}$ ${\ displaystyle s ^ {\ dagger} s}$ ${\ displaystyle s ^ {\ dagger} d_ {m}}$ ${\ displaystyle d_ {m} ^ {\ dagger} s}$ ${\ displaystyle d_ {m_ {1}} ^ {\ dagger} d_ {m_ {2}}}$

${\ displaystyle U (6) \ supset U (5) \ supset SO (5) \ supset SO (3) \ supset SO (2)}$

${\ displaystyle U (6) \ supset SO (6) \ supset SO (5) \ supset SO (3) \ supset SO (2)}$

${\ displaystyle U (6) \ supset SU (3) \ supset SO (3) \ supset SO (2)}$

Often the three subalgebras U (5), SO (6) and SU (3) by the so-called. Casten graphed -Dreieck. The corners correspond to these 3 chains of embedded algebras. Such a mapping often also contains further points, which are denoted by X (5) and E (5). However, these are not IBM's algebras.

Applied to atomic nuclei, the U (5) limit corresponds to a vibrator, the SO (6) limit to a soft core and the SU (3) limit to a rotor. ${\ displaystyle \ gamma}$

Extensions to the model

An extension introduces bosons with higher angular momentum, the g bosons. Another possibility is to consider more complicated interactions as 2-body forces. However, it could be shown that both extensions are partially mathematically equivalent.

An important extension was the distinction between proton and neutron bosons. This extension is also known as the IBA-2 model.

There is also the possibility of coupling individual nucleons with bosons, e.g. B. If the number of neutrons is odd, the remaining neutron is coupled with the bosons. This extension is known as IBMF, the coupling with two unpaired nucleons as IBMFF.

The consideration of fermions leads to other very interesting aspects, e.g. B. on the boson-fermion symmetry, d. H. the supersymmetry , which also plays an important role in particle physics .

Fonts

with Francesco Iachello : Collective nuclear states as representations of a SU (6) Group. In: Physical Review Letters. Vol. 35, 1975, ISSN 0031-9007 , pp. 1069-1072, doi : 10.1103 / PhysRevLett.35.1069 .
with Francesco Iachello: The interacting boson model. Cambridge University Press, Cambridge et al. 1987, ISBN 0-521-30282-X .
Arima, Iachello Interacting boson model of collective states , part 1 (the vibrational limit) Annals of physics Vol. 99, 1976, pp. 253-317, part 2 (the rotational limit) ibid. Vol. 111, 1978, pp. 201-38, part 3 with Scholten (the transition from SU (5) to SU (3)), ibid. Vol. 115, 1978, pp. 325-66, part 4 (the O (6) limit) ibid. Vol. 123, 1979, pp. 468-92
with Francesco Iachello: The Interacting Boson Model. In: Annual Review of Nuclear and Particle Science. Vol. 31, 1981, ISSN 0163-8998 , pp. 75-105, doi : 10.1146 / annurev.ns.31.120181.000451 .