Relativistic Pseudospin Symmetry in Nuclei

Abstract

Peter Ring was one of the first to really grasp the significance of pseudospin symmetry as a relativistic symmetry [1, 2, 3, 4]. Originally, pseudospin doublets were introduced into nuclear physics to accommodate an observed near degeneracy of certain normal-parity shell-model orbitals with non-relativistic quantum numbers (n _r , ℓ, j = ℓ + 1/2) and (n _r − 1,ℓ + 2, j = ℓ + 3/2) where n _r , ℓ, and j are the single-nucleon radial, orbital, and total angular momentum quantum numbers, respectively [5, 6]. The doublet structure, is expressed in terms of a “pseudo” orbital angular momentum ~ℓ = ℓ + 1 coupled to a “pseudo” spin, ~s = 1/2. For example, (n _r s _1/2, (n _r − 1)d _3/2) will have ~ℓ = 1, (n _r p _3/2, (n _r − 1)f _5/2) will have ~ℓ = 2, etc. Since j = ~ℓ ± ~s, the energy of the two states in the doublet is then approximately independent of the orientation of the pseudospin. Some examples are given in Table 1. In the presence of deformation the doublets persist with asymptotic (Nilsson) quantum numbers [N, n ₃, Λ, Ω = Λ +1/2] and [N, n ₃, Λ + 2, Ω = Λ + 3/2], and can be expressed in terms of pseudoorbital and total angular momentum projections ~Λ = Λ + 1, Ω = ~Λ ± 1/2. This pseudospin “symmetry” has been used to explain features of deformed nuclei [7], including superdeformation [8] and identical bands [9, 10, 11]. While pseudospin symmetry is experimentally well corroborated in nuclei, its foundations remained a mystery and “no deeper understanding of the origin of these (approximate) degeneracies” existed [12].