Pseudospin symmetry

The pseudospin symmetry has been introduced in nuclear structure physics many years ago [105,106], in order to account for ``unexpected'' degeneracies in single-particle spectra. It has been attributed to an accidental cancelation between the spin-orbit field $\vec{\ell}\cdot\vec{s}$ and the ${\vec{\ell}}^2$ term in the Nilsson potential [107], the latter reflecting the fact that the nuclear single-particle field has a flat bottom due to the saturation of nuclear forces. Later, many similar degeneracies were also observed among rotational bands, see, e.g., the pair of bands in ¹⁸⁷Os studied in Ref. [108] and references cited therein.

Recently, a very elegant and natural explanation was suggested [109,110,111], which is based on simple relativistic arguments and properties of the RMF Dirac equation for nucleons. Indeed, when the Dirac Hamiltonian is solved with the vector and scalar potentials, V_V and V_S, respectively,

$$H= \vec{\alpha}\cdot\vec{p}+\beta(M+V_S)+V_V,$$ (3)

and when V_V+V_S=const, solutions can be classified according to a new SU(2) group, called the pseudospin group generated by $\vec{\tilde{s}}$= $U_p\vec{s}U_p$ for U_p= $2(\vec{s}\cdot\vec{p})/p$. Then, the pseudo-orbital symmetry $\vec{\tilde{\ell}}$=$\vec{j}$- $\vec{\tilde{s}}$ is also conserved, and the lower-component radial wave functions of the pseudospin partner states are identical to one another.

  

Figure 8: Upper (left panel) and lower (right panel) components of the Dirac wave functions for the 2s_1/2 and 1d_3/2 states in ²⁰⁸Pb. (From Ref. [112]. Reprinted from Nuclear Physics A, Vol 690, 2001, pp 41-51, J.N. Ginocchio, ``A relativistic symmetry in nuclei: its...'', Copyright (2001), with permission from Elsevier Science.)

In finite nuclei, condition V_V+V_S=const is, of course, violated because the potentials have finite range, and hence, they must depend on the distance from the center of the nucleus. However, it turns out that in realistic situations the lower Dirac components of the pseudospin partners are still fairly close to one another, and their single-particle energies are almost degenerate. This is illustrated in Fig. 8 [112], where the upper (left panel) and lower (right panel) components of two bound states, 2s_1/2 and 1d_3/2, in ²⁰⁸Pb are plotted. These two states are the $\tilde{p}$ pseudospin partners corresponding to $\tilde{\ell}$=1, i.e., to the 1 $\tilde{p}_{1/2}$ and 1 $\tilde{p}_{3/2}$ orbitals, and their lower Dirac components are very similar indeed.

Nevertheless, the utility of the idea of the pseudospin symmetry crucially depends on the symmetry breaking schemes in real nuclei, and this aspect of the proposed relativistic explanation requires further study, see the analyses presented in Refs. [113,114] and references cited therein. In particular, an explanation of the fact that the pseudospin symmetry does not hold in light nuclei is still lacking (relativity arguments should hold irrespective of the number of particles). Similarly, it would be very interesting to see if the pseudospin symmetry still holds in neutron rich nuclei, where changes in the surface diffuseness may act against it [76].