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Based on the relativistic mean field ͑RMF͒ approach the existence of the broken pseudospin symmetry is investigated. Both spherical RMF and constrained deformed RMF calculations are carried out employing realistic Lagrangian parameters for spherical and for deformed sample nuclei. The quasidegenerate pseudospin doublets are confirmed to exist near the Fermi surface for both spherical and deformed nuclei. ͓S0556-2813͑98͒51107-7͔PACS number͑s͒: 21.60. Cs, 21.10.Ϫk, 24.10.Jv Pseudospin symmetry was discovered in nuclear physics nearly 30 years ago ͓1-3͔. The recent claim ͓4͔ that pseudospin symmetry may arise because of the near equality in magnitude of attractive scalar and repulsive vector fields in relativistic mean theory has revived the activity related to the understanding of the origin of this symmetry in real nuclei. The concept of pseudospin symmetry ͓1,2͔ is based on the experimental observation of the existence of quasidegenerate doublets of normal parity orbitals (n,l, jϭlϩ 1 2 ) and ͑nϪ1, lϩ2, jϭlϩ 3 2 ͒ such as (4s 1/2 ,3d 3/2 ), (3d 5/2 ,2g 7/2 ), etc., in the same major shell. Since for spherical systems the quantum numbers j are conserved, the pseudospin angular momenta ( l,sϭ1/2) satisfy jϭ jϭ lϮ 1 2 . In order to interpret this near degenerate pair of jϭl ϩ1/2 and jϭlϩ3/2 states as pseudospin doublets corresponding to m s ϭϮ1/2, l has to be lϩ1. It then follows for the major oscillator quantum number: Ñ ϭNϪ1, for the radial quantum number ñ ϭ(Ñ Ϫ l)Ϫ1 and for the parity ϭϪ. For zero pseudospin orbit splitting, the pseudospin multiplet will be degenerate. Thus the pair of orbitals (4s 1/2 ,3d 3/2 ) and (3d 5/2 ,2g 7/2 ) can be viewed as the (2p 1/2 ,2p 3/2 ) and (1 f 5/2 ,1f 7/2 ) pseudospin doublets. The symmetry can also be investigated in deformed nuclei. In the asymptotic Nilsson scheme one finds the pseudospin quantum numbers ͑Ñ ϭNϪ1, ñ 3 ϭn 3 , ⌳ ϭ⌳ϩ1, and ⍀ ϭ⍀͒. Therefore, the Nilsson orbitals ͓N,n 3 ,⌳,⍀ϭ⌳ϩ1/2͔ and ͓N,n 3 ,⌳ϩ2,⍀ϭ⌳ϩ3/2͔ can be viewed as the pseudospinorbit doublets ͓Ñ ,ñ 3 ,⌳ ,⍀ ϭ⌳ Ϯ1/2͔ ͓5͔.Apart from the rather formal relabeling of quantum numbers various proposals for an explicit transformation from the normal scheme to the pseudospin scheme have been made in the last 20 years and several nuclear properties have been investigated in this scheme ͓6-9͔. However, the origin of pseudospin symmetry remained unknown until the recent observation of Ginocchio ͓4,10͔, where for the first time the origin of this symmetry is claimed to be revealed as due to the near equality in magnitude of the attractive scalar and repulsive vector fields in relativistic theories. Here in this Rapid Communication we follow this idea and investigate to what extent the pseudospin symmetry is broken for realistic cases. For this purpose we concentrate on spherical as well as on deformed nuclei and we use the framework of relativistic mean field ͑RMF͒ theory ͓11͔. It has been shown that this phenomenological approach is very successful in describing the ground state nuclear properties of spherical,...

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The effect of interband transitions of electrons on the linear as well as the bilinear polarization induced in a metal by a light wave of (circular) frequency w has been calculated. The calculation of the linear polarization or linear current density leads to the familiar expression for the dielectric constant e(w). The part of the bilinear polarization varying as e~2 iat for free conduction electrons with a potential barrier at the surface is known to have the formwhere E (cd) and H (w) are, respectively, the electric and magnetic fields varying as e~i wt . It is shown that the introduction of a periodic potential leads to the same form for P< 2c0 > (NL) for isotropic metals, with a and fi now containing both the intraband and interband contributions. Except near a resonance for interband transitions involving at least 3 bands, it is found that in the long-wavelength limit for the light wave the general expression for P< 2a>) may be approximated by a form which may be completely specified in terms of e (w) and e (2co). By solving Maxwell's equations for the fundamental as well as the second-harmonic fields for arbitrary e, a, and /3, general expressions for both linear and bilinear reflection coefficients have been derived. These phenomenological solutions can be used to determine experimentally the residual 3-band contributions to P (2w) (NL) which cannot be expressed in terms of the linear dielectric constant alone.

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