Pseudospin as a Relativistic Symmetry

Abstract

We show that pseudospin symmetry in nuclei could arise from nucleons moving in a relativistic mean field which has an attractive scalar and repulsive vector potential nearly equal in magnitude.[S0031-9007(96)02176-X] PACS numbers: 21.60. Cs, 21.30.Fe, 24.10.Jv Almost 30 years ago a quasidegeneracy was observed in heavy nuclei between single-nucleon doublets with quantum numbers ͑n r , ᐉ, j ᐉ 1 1 2 ͒ and ͑n r 2 1, ᐉ 1 2, j ᐉ 1 3 2 ͒ where n r , ᐉ, and j are the single nucleon radial, orbital, and total angular momentum quantum numbers, respectively [1,2]. These authors defined a "pseudo" orbital angular momentumᐉ ᐉ 1 1; for example, ͑ ͑ ͑n r s 1͞2 ͑n r 2 1͒d 3͞2 ͒ ͒ ͒ will haveᐉ 1, ͑ ͑ ͑n r p 3͞2 , ͑n r 2 1͒f 5͞2 ͒ ͒ ͒ will haveᐉ 2, etc. Then these doublets are almost degenerate with respect to "pseudo" spin,s 1 2 , since j ᐉ 6s for the two states in the doublet. This symmetry has been used to explain a number of phenomena in nuclear structure [3] including most recently the identical rotational bands observed in nuclei [4]. Despite this long history of pseudospin symmetry [5,6], the origin of this symmetry has eluded explanation. Recently it was shown [7] that relativistic mean field theories predict the correct spin-orbit splitting [8]. In this paper we identify a possible reason for this; namely that the symmetry arises from the near equality in magnitude of an attractive scalar, 2V s , and repulsive vector, V y , relativistic mean fields, V s ϳ V y , in which the nucleons move. Such a near equality of mean fields follows from relativistic field theories with interacting nucleons and mesons [8], with nucleons interacting via Skyrme-type interactions [9], and from QCD sum rules [10].A nucleon moving in a spherical field has the total angular momentum j, its projection on the z axis, m, and k 2b͑ŝ ?L 1 1͒ conserved, whereb is the Dirac matrix [11]. The eigenvalues ofk are k 6͑j 1 1 2 ͒; 2 for aligned spin ͑s 1͞2 , p 3͞2 , etc.͒ and 1 for unaligned spin ͑p 1͞2 , d 3͞2 , etc.͒. Hence we use the quantum number k since it is sufficient to label the orbitals. The Dirac equation for the single-nucleon radial wave function ͑g k , f k ͒ in dimensionless units is given by [11] ∑ d drwhere r is the radial coordinate in units of length hc͞mc 2 , V ͑r͒ ͓V y ͑r͒ 1 V s ͑r͔͒͞mc 2 , D͑r͒ ͓V s ͑r͒ 2 V y ͑r͔͒͞mc 2 , and E is the binding energy ͑E . 0͒ of the nucleon in units of the nucleon mass mc 2 . First we show that, in the limit of equality of the magnitude of the vector and scalar potential D͑r͒ 0, pseudospin is exactly conserved. To do this we solve for g k in (2) and substitute into (1), obtaining the second order differentialwhich agrees with the original definition of the pseudoorbital angular momentum [1,2]. For example, for ͓n r s 1͞2 , ͑n r 2 1͒d 3͞2 ͔, k 21 and 2, respectively, givingᐉ 1 in both cases. Futhermore, the physical significance ofᐉ is revealed; it is the "orbital angular momentum" of the lower component of the Dirac wave function. Eq.(3) is a Schrödinger equation with an attractive potential V and bind...

Negative parity 70-plet baryon masses in the1/Ncexpansion

“…Understanding the smallness of the spin-orbit interaction from QCD is an important open dynamical problem. In the constituent quark model picture a possible explanation has been given in terms of the approximate cancellation between the spin-orbit pieces due to the Lorentz four-vector and scalar components of the effective potential [46,47].…”

Shell-model-like Approach (SLAP) for the Nuclear Properties in Relativistic Mean Field Theory

“…During the past two decades, it has received wide attention due to its success in describing many nuclear phenomena for the stable nuclei [2,3] as well as nuclei even far from stability [4,5]. It has been shown that the relativistic Brueckner theory can reproduce better the nuclear saturation properties (the Coester line) in nuclear matter [6], present a new explanation for the identical bands in superdeformed nuclei [7] and the neutron halo [8], predict a new phenomenon -giant neutron halos in heavy nuclei close to the neutron drip line [9], give naturally the spin-orbit potential, the origin of the pseudospin symmetry [10,11] as a relativistic symmetry [12,13,14] and spin symmetry in the anti-nucleon spectrum [15], and present good description for the magnetic rotation [16] and the collective multipole excitations [17], etc.…”

Consequences of a relativistic pseudospin symmetry for radial nodes and intruder levels in nuclei

“…The pseudospin symmetry itself has been shown to arise from a relativistic symmetry of a Dirac Hamiltonian in which the sum of the scalar, V S , and vector, V V , potentials cancel, V S + V V = 0 [10,11]. An attractive scalar and repulsive vector potentials of nearly equal magnitudes, V S ∼ −V V , is an inherent feature of realistic relativistic mean fields in nuclei.…”

“…All of the above results are relevant for understanding properties of states in the relativistic pseudospin scheme [10,11]. The generators for the relativistic pseudospin SU (2) algebra,Ŝ µ , which commute with the Dirac Hamiltonian, [ H ,Ŝ µ ] = 0, for the case when…”

“…In the relativistic pseudospin scheme [10,11], the non-relativistic wave functions of Eq. (1) are associated with the upper components of Dirac wave functions which are eigenstates of a Dirac Hamiltonian with V S ∼ −V V .…”