Pairing correlations affect properties of atomic nuclei in a profound way [1,2,3,4]. They impact nuclear binding, properties of nuclear excitations and decays, and dramatically influence the nuclear collective motion. In particular, pairing plays a crucial role in exotic, weakly bound nuclei in which the magnitude of the chemical potential is close to that of the pairing gap. In such systems, a naive independent single-particle picture breaks down and the pair scattering, also involving the continuum part of the phase space, can determine the very nuclear existence [5].
Many aspects of nuclear superfluidity can be successfully treated within the independent quasiparticle framework by applying the Bardeen-Cooper-Schrieffer (BCS) [6] or Hartree-Fock-Bogoliubov (HFB) approximations [2]. The advantage of the mean-field approach to the pairing problem lies in its simplicity that allows for a straightforward interpretation in terms of pairing fields and deformations (pairing gaps) associated with the spontaneous breaking of the gauge symmetry. However, this simplicity comes at a cost. In the intrinsic-system description, the gauge angle associated with the particle-number operator is fixed; hence, the particle-number invariance is internally broken [1,2,3]. Therefore, to relate to experiment, the particle-number symmetry needs to be, in principle, restored.
Some observables, like masses, radii, or deformations are not very strongly affected by the particle-number-symmetry restoration, while some other ones, like even-odd mass staggering or pair-transfer amplitudes are influenced significantly. Moreover, quantitative impact of the particle-number projection (PNP) depends on whether the pairing correlations are strong (open-shell systems) or weak (near closed shells). Therefore, methods of restoring the particle-number symmetry must be implemented in studies of pairing correlations. This can be done on various levels [2,7], including the quasiparticle random phase approximation, Kamlah expansion [8,9], Lipkin-Nogami (LN) method [10,11,12,13,14,15,16,17,18], the particle-number projection after variation (PNPAV) [2,19], the projected LN method (PLN) [20,21,19,18,22], and the variation after particle-number projection (VAPNP) [23,24,25,26,19].
In this work, we concentrate on the VAPNP method. Recently, it has been shown [25] that the total energy in the HFB+VAPNP approach can be expressed as a functional of the unprojected HFB density matrix and pairing tensor. Its variation leads to a set of HFB-like equations with modified self-consistent fields. The method has been illustrated within schematic models [27], and also implemented in the HFB calculations with the finite-range Gogny force [26,19]. Here, we adopt it for the Skyrme energy-density functionals and zero-range pairing forces; in this case the building blocks of the method are the local particle-hole and particle-particle densities and mean fields. In the present study, the HFB equations are solved by using the Harmonic Oscillator (HO) basis, but the formalism can be straightforwardly applied with the Transformed Harmonic Oscillator (THO) basis [18,28], which helps maintain the correct asymptotic behavior of the single-quasiparticle wave functions.
The paper is organized as follows. Section 2 gives a brief overview of the HFB theory and defines the densities and fields entering the formalism. Section 3 extends the VAPNP method of Ref. [27] to the case of the HFB theory with Skyrme interaction. The technical details pertaining to the Skyrme HFB+VAPNP method are given in Sec. 4, while Sec. 5 contains an illustrative example of calculations for the Ca and Sn isotopes. In particular, the LN and PLN approximations are compared to the VAPNP results. Summary and discussion are given in Section 6. Preliminary results of our VAPNP calculations were presented in Ref. [29].