As mentioned in the earlier section discussing the Liquid Drop Model, protons and neutrons exhibit a tendency to form pairs [14] (represented by δ symbol in
the LDM). This tendency bares further discussion as not only does it explain some empirically observed features that fail to be described by the Shell Models thus far, but also, in the development of a systematic treatment of these features in the capacity of the Shell Models, introduces an enormous simplification to how the behaviour of valence particles are described. This is of great benefit in the analysis of experimental data obtained from heavier nuclei.
The empirically observed behaviours associated with pairing are rooted in the existence of an extremely short ranged attractive component to the nuclear force. This component will naturally be most prevalent when nucleons are closest to one another and would be strongest between nucleons that share the same spatial co- ordinates. However, the Pauli exclusion principle restricts nucleon behaviour, preventing identical fermions from sharing the exact same quantum numbers. Therefore the greatest degree of overlap occurs between nucleon pairs moving in time reversed orbits.
Each pair of nucleons possessj and K values that are of the same magnitude but opposing signs, and their combined spins will cancel to zero. This cancellation of spin explains why all even-even nuclei have a groundstate spin and parity of
Iπ = 0+. Another phenomena present in even-even nuclei that can be explained by the pairing interaction is the typically large amount of energy required to reach the first excited state above the groundstate, a so-called pairing gap of 1-2 MeV. This systematic is not present in odd A nuclei, thereby giving an indication of the strength of the pairing interactions.
The increased binding energy between paired nucleons also ties in with the pre- viously noted differences in the binding energies between nuclei, with even-even nuclei being more strongly bound than odd-even nuclei which are in turn more strongly bound than odd-odd nuclei.
There is a mass dependence on the pairing strength. In heavier nuclei the out- ermost nucleons are generally further apart meaning that the spatial coordinates between paired nucleons generally will overlap less and thus the pairing interac- tion will be weaker. A common expression for this dependence is
Gp = 17
Gn= 23
AM eV, (3.24)
where Gp and Gn are the strength of the pairing force for protons and neutrons respectively. The repulsive Coulomb force present for protons lowers the effective strength of the pairing force for these nucleons. Close to the Fermi surface, λ,
Occupied
Unoccupied
Δ
ε − λ
i 0V
2 1 ε0 -1 -2 -3 1 2 λFigure 3.9: Without pairing the orbitals will fill each level consecutively, with a distinct Fermi level defined by the highest occupied orbital. With pairing the Fermi surface is diffuse, producing partially occupied states that are described in terms of quasiparticles. Adapted from reference [14].
there is a ”smearing” of the occupied nucleon energy levels, as depicted on the right in Fig. 3.9. This results from nucleons scattering into higher orbitals fol- lowing a collision, which is quite probable for two nucleons travelling in the same orbital space. Therefore at any given time there will be a number of nucleons occupying states above the Fermi surface, with each nucleon possessing a corre- sponding hole below that surface.
If nucleons could not form into time reversed orbitals then such collisions would not take place and any new nucleon added to the system would simply raise the
Fermi surface by occupying the next available orbit(See Fig. 3.9). Furthermore the scatterings of paired nucleons do not take place for orbits far below the Fermi surface as the Pauli exclusion principle blocks them from scattering into the al- ready occupied neighbouring orbitals, therefore these nucleons may be discounted in the treatment of the behaviour of the nucleons on the surface.
The difference between the groudstate and the first excited band is known as the pair gap parameter
∆ =GX
i,j
UiVj, (3.25) which is summed over orbits i, j. G is the strength of the pairing interaction and U and V are the so-called emptiness and fullness factors, which, respectively squared, correspond to the probability that an orbit, i, is empty or full. The pairing gap parameter ∆ may be estimated from the empirical mass difference between adjacent nuclei with odd and even nucleon numbers.
In the fore mentioned scenario of no time reversed orbits, (i−λ) would be the excitation energy required to excite one nucleon from the Fermi orbit to a higher orbit, where i is the single particle energies, with 0 reserved for the level closest to the Fermi level. However, with pairing the single particle excitation energy is replaced by a quasiparticle energy, Ei given by
Ei =
p
(i−λ)2+ ∆2. (3.26) In this manner the individual nucleons discussed so far (and their hole counter- parts) are replaced by a pair of quasiparticle [46][47] representatives describing partially filled levels.