Neutron-proton pairing collectivity

Subject of the neutron-proton (np) pairing in nuclei has been studied since the early sixties, however, recently it witnessed an impressive renewal of interest which was due to increased experimental possibilities of studying medium heavy N$\approx$Z nuclei. High-spin aspects of the np pairing are discussed in Ref. [10], therefore, here we concentrate on the question of collective ground-state correlations of this type. Despite the long history, the subject is plagued with contradicting opinions and results, and very often even the definitions of basic quantities are under debate. Without attempting a review of the whole issue, let us present here several important recent steps in this domain.

Question of the np pairing is very often discussed in connection to the so-called Wigner energy term in the mass formula, see Ref. [87] and references cited therein. Such a term represents an additional binding of N$\approx$Z nuclei with respect to a smooth-reference dependence on N and Z. The same additional binding is observed when the experimental masses are compared to mean-field mass predictions. It turns out that both, smooth part of experimental masses and mean-field masses, are in a perfect agreement with one another, and can be described by a term proportional to (N-Z)²=4T_z²=4T², where T_z and Tare the ground-state values of the projection on the third axis and total isospin, respectively. On the other hand, the additional binding (the Wigner energy) is a function of |N-Z|=2|T_z|=2T, and therefore has a cusp at N=Z. Moreover, when added together both contributions to the binding energy give approximately a term proportional to T(T+1), i.e., the numerical coefficients in front of the smooth term proportional to T², and the cusp proportional to T, are almost equal.

This simple observation has been known since many years [88], and recently was reanalyzed in Refs. [89,90]. A controversy, however, arises at the level of interpretation of this result, namely, the authors of Refs. [89,90] consider the T(T+1) term as a simple symmetry energy that has nothing to do with pairing correlations, and hence claim that there is no experimental indication for the collective np pairing in nuclei. On the other hand, the authors of Refs. [87,91] consider the mean-field term T² as representing the symmetry energy, and attribute the remaining contribution (proportional to T) to the T=0 np pairing correlations.

The dispute, however, is far from being an academic discussion triggered by a misunderstanding of definitions. Indeed, what is usually meant by correlations is the question whether an experiment that probes for a simultaneous presence of a neutron and a proton gives the probability higher or not than the product of probabilities to find a neutron and a proton in two independent experiments. Since the mean-field approximation gives by definition no correlations (in the sense of the above definition) it is reasonable to attribute all the effects, or contributions to energy, that are beyond the mean-field approximation, to correlations.

  

Figure 6: The Wigner energy (left top panel) and pairing coupling constants (left middle panel) together with the T=0, 1, and 2 excitation energies in N=Z even-even and odd-odd nuclei. In the left bottom panel, experimental data (stars) are compared with the results of calculation with (full symbols) and without (open symbols) the T=0 np interaction included. From Ref. [92]

As was proved in Ref. [87], the full shell-model calculations in the sd and fp shells give the binding energies roughly proportional to T(T+1), and hence they perfectly reproduce the Wigner energy. This result is, of course, not a trivial result of the isospin invariance of the interaction. Indeed, the isospin invariance requires that energies are functions of the total isospin T, and not necessarily functions of the quadratic invariant T(T+1). [Similarly, not all rotationally-invariant Hamiltonians give rotational spectra proportional to J(J+1).] The fact that the nuclear binding energies do behave as T(T+1) is a non-trivial result of a specific structure of the nuclear (shell-model) interaction. In Ref. [87], it was also shown that the Wigner term disappears when all T=0 matrix elements are removed from the Hamiltonian. However, this observation does not constitute a proof that the T=0 pairing correlations are at the origin of the Wigner term. The shell-model calculations take into account all sorts of correlations, and it is rather difficult to disentangle and quantify the np pairing correlations in the final result. On the other hand, an approach that starts from the mean-field approximation and than adds correlations on top of it, is tailored towards such a quantification.

The first consistent description of several experimental facts observed in N$\approx$Z nuclei was recently obtained [92] within the mean-field plus pairing approach. The authors were able to include all pairing channels simultaneously on top of the standard mean-filed solutions, and showed that the T=0 np pairing is essential to obtain agreement with several types of observables. Their results are summarized in Fig. 6. After an adjustment of the T=0 pairing coupling constant to the Wigner energy term (top left panel) they can reproduce (without any additional fitted parameters), (i) the T=2 states in even-even N=Z nuclei (bottom left panel), (ii) the T=1 states in even-even N=Z nuclei (middle panel), and (iii) the T=1 and T=0 states in odd-odd N=Z nuclei (right panel). There are two key elements of their description, namely, the idea of the iso-cranking, i.e., a consistent application of the broken-symmetry treatment of nuclear excitations to the isospin channel, and (ii) identification of two-quasiparticle excitations among states of a given isospin, that is based on analyzing their time-reversal and iso-signature properties. Although these results are not a direct proof of existence of the collective np pairing correlations in nuclei, they are the best circumstantial evidence thereof, that is available to date.