In some circumstances the nucleus acts as a liquid and in others like an elastic solid. In general it responds elastically to sudden forces, and it flows plastically over longer periods of time (Bertsch (1980, 1988)). Examples of this behaviour are giant resonances and low-lying collective surface vibrations respectively. In the first case, as we shall see in Section 8.3, pairing plays no role, at least in the case of nuclei lying along the valley of stability. The nuclear single-particle states change their shape but the occupation numbers do not change. The energy of a giant resonance in a nucleus is of the order of the energy difference between major shells (ħω ≈ 41/A1/3 MeV, ≈ 7 MeV, for medium heavy nuclei), a quantity which is much larger than the pairing gap Δ ≈ 1–1.5 MeV. Giant resonances are fast modes, the collective motion is dominated by mean-field effects and the rigidity is provided by the mean field (Bortignon, Bracco and Broglia (1998)). On the other hand, low-energy surface modes are associated with particle–hole excitations which are of the order of the pairing gap. Pairing plays a dominant role and the collective states are coherent linear combinations of two-quasiparticle excitations. The situation is, however, different in the case of exotic nuclei, where the last nucleons are very weakly bound. Nucleon spill out makes these systems particularly polarizable leading to ‘pigmy resonances’, whose properties can be influenced by pairing (Frascaria et al. (2004), see also last paragraph of Chapter 6).

General background

As already mentioned in previous chapters, the nuclear structure exhibits many similarities with the electron structure of metals. In both cases, one is dealing with systems of fermions which may be characterized in a first approximation in terms of independent particle motion. However in both systems, important correlations in the particle motion arise from the action of the forces between particles. In particular, it is well established that nucleons moving close to the Fermi energy in time-reversal states have the tendency to form Cooper pairs which eventually condense (Bohr, Mottelson and Pines (1958), Bohr and Mottelson (1975)). This phenomenon, which has its parallel in low-temperature superconductivity, modifies the structure of nuclei in an important way. In particular it influences the occupation numbers of single-particle levels around the Fermi surface (Chapter 3), the moment of inertia of deformed nuclei (Chapter 3), the lifetime of alpha and cluster decay and fission processes (Chapter 7), the depopulation of superdeformed configurations (Chapter 6) and the cross-sections of two-nucleon transfer reactions (Chapter 5).

While one does not expect the transition between the normal and the superfluid phases of the atomic nucleus to be sharp because of finite size effects and the central role played by fluctuations (see Chapter 6), there is a strong analogy between phenomena in nuclei and the corresponding phenomena in bulk superconductors. Spontaneous symmetry breaking is important in both nuclei and superconductors.

Evidence for pairing correlations

The nucleus 208Pb is well described by the independent particle model in terms of closed shells in both neutrons and protons. The absence of low-lying states supports this hypothesis. The shell model would then describe the low-lying levels of 207Pb in terms of the states of a single neutron hole. The observed energies, angular momenta and parities are in good accord with this picture, although small admixtures of more complicated configurations must be invoked to account for some fast electromagnetic decays (see Sections 9.1 and 9.2). The next step is to interpret the levels of 206Pb in terms of two holes, which are combinations of the single-hole states of 207Pb, interacting through a residual interaction. The dramatic effect of the internucleon force is shown by the fact that there is only one excited state below 1.2 MeV, compared with the five we would get in the independent hole approximation (see Fig. 2.1). This becomes even more striking in 204Pb, where the independent hole picture predicts about thirty levels within 1 MeV of the ground state, whereas again only one is observed (Mottelson (1996)).

Another indication of the importance of the residual interaction among nucleon pairs is the well-known difference in physical properties between even and odd nuclei. For example, studies of cosmic abundances show that nuclei with even proton and neutron numbers are much more abundant, and thus more stable, indicating stronger binding energies.

Josephson (1962) proposed that there should be a contribution to the current through an insulating barrier between two superconductors which would behave like direct tunnelling of condensed pairs from one condensed gas of bound pairs at the Fermi surface to the other. The measurement of such an effect has provided a beautiful scenario where the collective rotational degree of freedom in gauge space manifests itself, let alone some of the most accurate measurements of the electron charge (Anderson (1964)).

Before proceeding further let us briefly discuss a technical detail which, aside from being essential to microscopically understanding the mechanism which is at the basis of the effect, also clarifies the long-range order induced by pairing correlations. Because one is interested in calculating the tunnelling of Cooper pairs across the barrier separating the two superconductors, it is natural to start by assuming that it is the pairing interaction that is the source of this transfer, by annihilating a pair in superconductor 2 and creating a pair in superconductor 1 (see Fig. L.1). Although this is what effectively happens, it can be shown that the pairing interaction leads to a negligible contribution to pair transfer, and that essentially all the transfer proceeds through the single-particle mean field acting twice. Note that this reaction mechanism leading to a (successive) two-particle tunnelling does not destroy the correlation existing between the pair of fermions of a Cooper pair participating in the condensate.

As discussed in the last section of Chapter 9 a pair of nucleons can interact with each other through the nuclear surface in a process in which one nucleon excites a vibrational mode which is then absorbed by the other nucleon (see inset Fig. 10.1). This process leads to a renormalization of the nucleon–nucleon interaction which, for nucleons close to the Fermi energy, is controlled by the exchange of low-lying surface collective vibrations. This is because low-energy surface vibrations match the frequencies of these nucleons and are very collective. This argument is the same as that used to explain the central role of surface vibrations in renormalizing the single-particle motion. The contribution of surface vibrations to the single-particle self-energy and to the ω-mass was analysed in Chapter 9 and a simplified version of the particle-vibration coupling model was introduced in Section 9.3. It gave explicit expressions for both the ω-mass and the induced pairing interaction and pairing gap due to phonon exchange. They both have a simple dependence on the coupling strength gp-v which is defined in equation (9.31). The present chapter extends the discussion of the induced interaction and presents the results of microscopic calculations. Section 10.3 presents results in a slab model, where the simplicity of the infinite system is retained (absence of shell structure), without losing surface effects.

When the strength G of the pairing interaction is greater than a critical value Gc, the gap equation has a non-zero solution for the gap parameter Δ and the BCS ground state of a system of nucleons is stable. Single nucleon levels are partially occupied in an energy range Δ around the Fermi energy λ. The BCS state is not an eigenstate of nucleon number and violates gauge invariance. Pairing vibrations, which are fluctuations about the BCS state, were studied in Chapter 4 and it was shown that gauge invariance was restored within the framework of the random phase approximation (RPA). In this chapter we study the question of pairing vibrations within a more general context, considering also pairing vibrations in normal nuclei which have pairing strengths G < Gc and Δ = 0. To a first approximation single-particle levels are occupied with unit probability up to the Fermi energy and with zero probability for states above the Fermi level. Pairing vibrations modify this simple picture and are associated with fields which change the number of particles by 2. They produce correlations which enhance or modify pair transfer amplitudes. Parts of this chapter is based on Broglia and Riedel (1967a,b) and Broglia et al. (1973) (see also Anderson (1958), Högaasen-Feldman (1961), Bes and Broglia (1966), Bohr and Mottelson (1975), Ring and Schuck (1980), Wölfle (1972, 1978), Schmidt (1972)).

The present monograph aims to provide an account of the basic results obtained in the exploration of the subject of nuclear superfluidity, placing special emphasis on recent developments coming out from ongoing research, in particular medium polarization and pairing in exotic nuclei.

The marked mass dependence of the abundance of nuclear species testifies to the fact that nucleons in nuclei move essentially independently of each other in an average potential produced by the effect of all the other nucleons. Special stability is ascribed to the closing of shells in correspondence with magic numbers. Adding nucleons to a closed shell system polarizes the shells, leading eventually to deformations. The best studied of these phase transitions are associated with deformations in: (a) three-dimensional space (violating angular momentum conservation, i.e. rotational invariance), (b) gauge space (violating particle number conservation, i.e. gauge invariance).

The phenomenon of spontaneous symmetry breaking in gauge space (i.e. the fact that the Hamiltonian describing a system is symmetric with respect to gauge transformations while the state is not) is intimately connected with nuclear superfluidity. This is the subject of Chapters 1, 2 and 3.

While potential energy always prefers special arrangements and thus leads to the spontaneous symmetry-breaking phenomena, fluctuations favour symmetry, leading to collective modes (intimately connected with symmetry restoration): pairing rotations and pairing vibrations, analogues in gauge space to rotations in three-dimensional space of the system as a whole, as well as to (multipole) surface vibrations respectively. This is the subject of Chapters 4, 5, and 6.