Agrawal, BK

De, JN

Samaddar, SK

Shlomo, S

Sil, Tapas

Physical Review C

Texas A&amp;M Repository

PHYSICAL REVIEW C, VOLUME 64, 024305Coulomb energy differences in mirror nuclei revisitedB. K. Agrawal,1 Tapas Sil,1 S. K. Samaddar,1 J. N. De,2,3 and S. Shlomo31Saha Institute of Nuclear Physics, 1/AF Bidhannagar, Calcutta 700 064, India2Variable Energy Cyclotron Centre, 1/AF Bidhannagar, Calcutta 700 064, India3Cyclotron Institute, Texas A&M University, College Station, Texas 77843~Received 13 March 2001; published 29 June 2001!We calculate the Coulomb displacement energies ~CDEs! of mirror nuclei using the recent parameter set~NL3! in the relativistic mean-field ~RMF! model which includes self-coupling of the scalar meson. The resultsobtained are compared with the available ones calculated in the nonrelativistic Skyrme-Hartree-Fock ~SHF!approach that have the best fit to the experimental data. When adjusted to reproduce the charge root-mean-square ~rms! radius rc and the rms radii of the valence orbits, the results of the RMF model for the CDEs agreewith those of the SHF model within ;1%. Our investigation also shows that, although the RMF with the NL3parameter set reproduces the kink in the isotope variation of rc , the values obtained for CDEs are too small toaccount for the experimental values without the addition of the contribution due to long-range correlationeffects.DOI: 10.1103/PhysRevC.64.024305 PACS number~s!: PACS: 21.10.Sf, 21.60.2nCoulomb displacement energies ~CDEs! of analoguestates ~mirror nuclei!, DEC , provide a stringent test fornuclear models. Nolen and Schiffer pointed out for the firsttime @1,2# that a calculation of DEC within the framework ofthe independent particle model ~mean-field approach! leadsto disagreement with experimental values. Using a Woods-Saxon potential well which reproduced the experimentalcharge distribution, Nolen and Schiffer ~NS! found DEC tobe ;7% smaller than the experimental values. It was alsopointed out that attempts to remove this discrepancy in DECby adjusting the parameters of the potential well leads tosurprisingly smaller values of the charge radii. This discrep-ancy between the experimental and theoretical evaluations ofDEC in mirror nuclei, referred to as the NS anomaly, hasbeen the subject of many investigations in which variouscorrection terms, namely, the exchange term, vacuum polar-ization, electromagnetic spin-orbit interaction, proton-neutron mass difference, finite size effect of the proton, cen-ter of mass motion, Auerbach-Kahana-Weneser ~AKW!effect ~isospin impurity of the core!, polarization of the coreby the valence particle, and the Thomas-Ehrman effect ~thedifference between the neutron and proton wave functions!,were considered @1,3–10#. It has been established that the netcontribution of all these correction terms is too small to ac-count for the anomaly @4,11#. The contribution of chargesymmetry breaking ~CSB! interaction was found @4,11,12# tobe ;3% of DEC , i.e., accounting for only half of the dis-crepancy between theory and experiment.It was first pointed out by Shlomo and Love @13# that,contrary to earlier estimates, the effect of long-range corre-lations ~LRCs! on DEC is not negligible. Using the particle-vibration model and taking into account multipole excita-tions up to L55, significant contributions of ;1 –3 % toDEC were obtained. Therefore, to explain the NS anomaly, itis necessary to go beyond the mean-field approximation andinclude the contributions due to the LRC effects and the CSBinteraction; see also Refs. @14–16#. It is important to pointout that long-range ground-state @random phase approxima-tion ~RPA!# correlation effects can account for the discrep-0556-2813/2001/64~2!/024305~4!/$20.00 64 0243ancies between Skyrme-Hartree-Fock ~SHF! predictions andexperimental results for the charge root-mean-square ~rms!radii rc associated with anomalous kinks in the mass depen-dence of rc ~fluctuation in the isotope shift! @17,18#. There-fore a consistent description of DEC and the anomalouskinks in rc is achieved by including the LRC contributions tothe mean-field results obtained within the SHF approach.In recent years, relativistic mean-field ~RMF! @19# theoryhas been extremely successful in describing various facets ofnuclear structure properties. The RMF theory with a smallnumber of parameters is able to give a quantitative descrip-tion of the ground-state properties @20–22# of spherical anddeformed nuclei. Recently, a series of nonlinear RMF param-eter sets has been proposed in order to explain the finer de-tails of atomic nuclei. In particular, they have proved to bevery successful in reproducing the anomalous kink in theisotope shift of Pb nuclei @23# and the first ever microscopicdescription of the anomalous isotopic shift in Sr and Krchains @24#. The anomalous charge radii of Cr isotopes thatrequired invocation of LRCs ~zero-point oscillations! for anexplanation in the nonrelativistic theory @17# also finds anatural explanation @25# in the RMF approach. Because ofthese successes of the RMF theory, in the present communi-cation, we reexamine the problem of the Coulomb energydifference in mirror nuclei and calculate the CDEs within theRMF theory using recent parameter sets that proved success-ful in reproducing the anomalous kink in rc . We also com-pare our results with earlier calculations performed in thenonrelativistic SHF formalism @9,10# and in the RMF for-malism @26#.The Coulomb energy difference in mirror nuclei is givenbyDEC5B~N11,Z !2B~N ,Z11 !, ~1!where B(N ,Z) is the binding energy for a nucleus with Nneutrons and Z protons. In this work the calculation is based©2001 The American Physical Society05-1AGRAWAL, SIL, SAMADDAR, DE, AND SHLOMO PHYSICAL REVIEW C 64 024305on the self-consistent determination of the energy E(A) ofthe core nucleus with A nucleons (N5Z) and then determin-ing the energy differenceD5Ep2En , ~2!where Ep and En are the single-particle energies of the protonand the neutron that have to be added to or removed from thecore. The energy E ~say, of a particle state! according toKoopman’s theorem @27# satisfies the relationE5E~A11 !2E~A !. ~3!Note that D forms the major part of DEc . As a result of theself-consistent calculation of E(A), D includes the contribu-tions of the AKW and the Thomas-Ehrman effects @9#. Anumber of small but significant contributions like the ex-change term, vacuum polarization, electromagnetic spin-orbit interaction, proton-neutron mass difference, finite sizeeffect of the proton, center of mass motion, and polarizationof the core by the valence particle are to be added to D inorder to compare it with DEc . Suzuki et al. @10# includedthese correction terms to the SHF evaluation of D with theparameter set SGII. With the inclusion of the contributionsdue to CSB interaction, they found that the theoretical valuesare smaller than the experimental one by ;2%. Marcos et al.@26# carried out RMF calculation of DEc , taking D from theSHF evaluation as a reference model for comparison withpredictions from the RMF theory. Their results ~with param-eter set labeled R2! underestimate the SHF values consider-ably, typically by ;5%.We perform calculations with the NL3 parameter set@25,28# that includes nonlinear self-coupling of the s meson.We have also repeated calculations with the NL-SV2 param-eter set @29#; here the self-coupling of vector meson @30# isfurther included. It has been shown @29# that vector self-coupling might be important for an accurate description ofnuclear shell effects, particularly for nuclei near the drip line.We have carried out relativistic Hartree mean-field calcu-lations for six pairs of mirror nuclei, with 16O, 40Ca, and56Ni as the core. The Lagrangian density used to obtain theRMF equations is given asL5c¯ ~ igm]m2M !c112 ]ms]ms2U~s!2gsc¯ sc214 VmnVmn112 mv2 vmvm2gvc¯ gmvmc214RW mnRW mn112 mr2rW mrW m2grc¯ gmrW mtWc214 FmnFmn2ec¯ gm~12t3!2 Amc114 g4~vmvm!2, ~4!which contains nucleons ~c! with mass M, s, v, and rmesons, the electromagnetic field, and nonlinear self-interactions of the s field:02430U~s!512 ms2 s2113 g2s3114 g3s4. ~5!For the NL3 parameter set, the last term in Eq. ~4! is absent.The third component of isospin t3 equals 21 for protons and1 for neutrons. Recourse to the variational principle followedby the mean-field approximation treating the fields as c num-bers results in the Dirac equation for the nucleon and Klein-Gordon–type equations for the mesons and the photon ~seeRef. @21# for details!.The value of DEC is determined predominantly by therms value of the charge distribution of the core and that ofthe valence proton, in a particle or hole state. To facilitate thediscussions, in Table I we present the calculated charge radiifor a host of spherical nuclei over the periodic table andcompare them with the experimental data @31,32#. Since thecharge rms radius rc of 56Ni is not known, we assess itsvalue using the experimental values of 58Ni, 56Fe, and 54Fe@31#. We take rc(56Ni)5rc(58Ni)2@rc(56Fe)2rc(54Fe)#53.73 fm, assuming isotope shifts due to the two 2p3/2 neu-trons in the Ni isotopes and in the Fe isotopes to be the same.The charge radii are reproduced to within ;0.5% accuracywith the NL3 set and ;1% accuracy with the NL-SV2 set.The R2 parameter set also explains the experimental dataextremely well. The SHF calculation for the nuclei 16O and40Ca with the SGII parameter set @10# also reproduces theirexperimental charge radii reasonably.TABLE II. The RMF and Skyrme-Hartree-Fock results for D5Ep2En and the experimental Coulomb energy differences DEC inmirror nuclei ~MeV!.System NL3 NL-SV2 R2 a SGII a DEC~expt. b!15O-15N 3.925 3.876 3.667 3.808 3.53617F-17O 3.575 3.525 3.290 3.665 3.54339Ca-39K 7.352 7.381 7.249 7.450 7.31341Sc-41Ca 7.011 7.090 6.868 7.289 7.27855Ni-55Co 9.504 9.500 – — 9.42257Cu-57Ni 8.895 9.376 – — 9.499aReference @26#.bReference @35#.TABLE I. Comparison of the calculated values of charge radii~in fm! with the experimental data.Nucleus NL3 NL-SV2 R2 a SGII a Expt. b16O 2.674 2.684 2.75 2.75 2.69332S 3.282 3.291 – – 3.26340Ca 3.469 3.438 3.48 3.47 3.47848Ca 3.471 3.456 3.48 – 3.47956Ni 3.709 3.716 – – 3.73 c90Zr 4.269 4.254 4.27 – 4.270208Pb 5.513 5.513 5.53 – 5.504aReference @26#.bReference @31#.cAn estimate; see text.5-2COULOMB ENERGY DIFFERENCES IN MIRROR NUCLEI . . . PHYSICAL REVIEW C 64 024305In Table II, the values of the Coulomb energy differenceD5Ep2En , calculated in our model, are compared withthose from the R2 set and also from the SGII set; as ex-plained earlier, the results from the latter set are taken to bea good reference for comparison. In the earlier RMF calcu-lation with the R2 parameter set, it is seen that the discrep-ancy with the SGII results is ;5%. With the NL3 parameterset, the said discrepancy is reduced to ;2.5%. Thus one canfind that there is an overall improvement by approximately afactor of 2 with the NL3 parameter set. One also notes thatwith the inclusion of self-coupling of vector mesons ~NL-SV2 set!, except for the Cu-Ni pair, the results for D are onlymarginally changed.Information on the valence neutron and proton wave func-tions can be deduced from the magnetic form factor obtainedin backward electron scattering. Since their extension has aseminal role in the determination of the Coulomb energy,their calculations and comparison with the available experi-mental data may throw more light on the intricate nature ofthe Coulomb energy anomaly. In Table III, results for theradii of the valence neutron orbits are presented for the Oand Ca isotopes. With the NL3 parameter set, the rms radii ofthe valence neutron orbitals in 17O and 41Ca are larger thanthe experimental values @33,34# by ;4%, leading to a de-crease of D by ;2%. Therefore, increasing the values of Dfor the A517 and 41 mirror nuclei obtained in the RMFcalculation with the NL3 parameter set by ;2%, we obtainTABLE III. Comparison of the calculated values of the rms radii~in fm! for the valence neutron orbitals with the experimental data.Nucleus NL3 NL-SV2 SGII a Expt. b15O 2.517 2.814 2.825 –17O 3.482 3.408 3.275 3.3639Ca 3.689 3.718 3.606 –41Ca 4.156 4.105 3.989 3.99aReference @26#.bReferences @33,34#.02430an agreement within ;1% with the results obtained with theSGII interaction; see Table II. The net contributions @4,10#due to the exchange term, vacuum polarization, electromag-netic spin-orbit interaction, proton-neutron mass difference,finite size effect of the proton, center of mass motion, andpolarization of the core by the valence particle decrease thecalculated CDEs by about 0.45, 0.20, 0.40, and 0.30 MeV forthe A515, 17, 39, and 41 mirror nuclei, respectively. Thus,the discrepancy between the mean-field ~SHF or RMF! re-sults for the CDEs and the corresponding experimental val-ues is reduced to about 3–5 %. As pointed out earlier, thisremaining discrepancy can be accounted for by including thecontributions due to CSB interaction and LRC effects @13#.To conclude, we have calculated the Coulomb energy dif-ferences in some pairs of mirror nuclei with recent nonlinearparameter sets ~NL3 and NL-SV2! in the relativistic mean-field theory. These results are compared to nonrelativisticSkyrme-Hartree-Fock calculations with the SGII interactionand also with the earlier RMF results ~R2!. Compared to thecalculations with the R2 parameter set, the present results arein closer agreement with the SGII values. Inclusion of theself-coupling of the vector meson improves the results fur-ther. When adjusted to reproduce rc and the rms radius of thevalence orbits, the RMF model ~with the NL3 parameter set!results for D agree with those of the SHF model within;1%. An agreement with experimental data is obtained byadding the contributions due to the CSB interaction and LRCeffects. 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Coulomb energy differences in mirror nuclei revisited

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Coulomb energy differences in mirror nuclei revisited

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Coulomb energy differences in mirror nuclei revisited

Abstract

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Texas A&M Repository