Isospin symmetry breaking in the mirror pair 73 Sr- 73 Br
,, Isospin symmetry breaking in the mirror pair Sr- Br S. M. Lenzi, A. Poves, and A. O. Macchiavelli Dipartimento di Fisica e Astronomia, Universit`a degli Studi di Padova,and INFN, Sezione di Padova, I-35131 Padova, Italy Departamento de F´ısica Te´orica and IFT-UAM/CSIC,Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA (Dated: July 9, 2020)The recent experimental observation of isospin symmetry breaking (ISB) in the ground states ofthe T = 3 / Sr - Br is theoretically studied using large-scale shell model calculations.The large valence space and the successful PFSDG-U effective interaction used for the nuclear partof the problem capture possible structural changes and provide a robust basis to treat the ISBeffects of both electromagnetic and non-electromagnetic origin. The calculated shifts and mirror-energy-differences are consistent with the inversion of the I π = 1/2 − , / − states between Sr - Br, and suggest that the role played by the Coulomb interaction is dominant. An isospin breakingcontribution of nuclear origin is estimated to be ≈
25 keV.
I. INTRODUCTION
In a recent article entitled “Mirror-symmetry viola-tion in bound nuclear ground states” [1], Hoff and col-laborators reported the results of an experiment carriedout at the National Superconducting Cyclotron Labo-ratory (NSCL), in which the decay of the proton-rich, T = 3 / T z =-3/2, isotope Sr was studied. Follow-ing a detailed and convincing analysis of the experi-mental data they conclude that its ground state has I π = 5/2 − . This observation is at odds with its mir-ror T = 3 / T z =3/2 partner Br which has a I π =1/2 − ground state, and thus the topic of their work.The theoretical interpretation, which accompanies thepaper, cannot reproduce the inversion and the authorsconclude with two main points, one related to the wellknown Thomas-Ehrman shift [2, 3]: (sic) Such a mech-anism is not immediately apparent in the case of Sr/ Br, and it may be that charge-symmetry-breakingforces need to be incorporated into the nuclear Hamilto-nian to fully describe the presented results , and the otherone related to possible structural effects: (sic) (the) in-version could be due to small changes in the two compet-ing shapes, particularly their degree of triaxiality, andthe coupling to the proton continuum in the (IsobaricAnalogue State) IAS of Rb .Besides the fact that the I π = 1/2 − in Sr is an ex-cited state, there is no information available about itslocation. On the contrary, the level scheme of Br isbetter known with an I π = 5/2 − state at 27 keV, an I π =3/2 − at 178 keV and another I π = (3/2 − , 5/2 − ) stateat 241 keV. Given the above it seems opportune to com-ment already that the Mirror Energy Difference (MED)of the 1/2 − arising from the mirror symmetry viola-tion can be as low as ∼
30 keV. Notice that MED’s aslarge as 300 keV have been measured for the 2 + statesof the Ca- S mirror pair, which can be understoodwithout invoking threshold effects [4]. Even further, inthe same mirror pair, a prediction of a huge MED of 700 keV for the first excited 0 + states has been madein Ref. [5], again without the need of threshold effects.There is abundant experimental and theoretical workon the subject of the MED’s which we believe providesa natural framework to interpret the new data. Actu-ally, Ref. [6] places the new result within the contextof the extensive body of available data and the authorsconcluded that, being entirely consistent with normalbehavior, the inversion does not provide further insightinto isospin symmetry breaking (ISB).Here, in line with the findings of Refs. [7, 8], we pro-pose an explanation based on the configuration interac-tion shell model (SM-CI) to treat the nuclear (isospinconserving) part of the problem, plus a detailed analy-sis of both Coulomb and other ISB effects. The largevalence space and the well established effective inter-action we use allow us to describe deformed nuclei inthe laboratory frame without the restriction to axiallysymmetric shapes as considered in Ref. [1]. II. THE SHELL MODEL FRAMEWORKA. The nuclear input
We describe the A = 73, T = 3 / pf -shell to the N = 40 and N = 50 islands ofinversion. Recently applied to the structure of Ni [10],it can be considered as an extension of the LNPS in-teraction [11] which encompasses nuclei at and beyond N = 50.The PFSDG-U interaction, defined for the full pf + sdg shells, is here used in the valence space givenby the orbits: 0 f / , 1 p / , 0 f / , 1 p / , 0 g / and 1 d / with the single particle energies (SPE) taken directlyfrom the experimental spectra of Ca as summarizedin Table I. In the present calculation an inert core of a r X i v : . [ nu c l - t h ] J u l TABLE I: Valence space and single particle energies used inthe present SM-CI calculations.Orbit 0 f / p / f / p / g / d / SPE (MeV) -8.363 -5.93 -1.525 -4.184 -0.013 0.937 Ni is adopted and the number of excitations across N = Z = 40 are limited to four, to achieve convergencefor the states of interest which have dimension ≈ .The isospin conserving (nuclear only) calculation pro-duces a ground state I π = 5/2 − and the first excitedstate, I π = 1/2 − at 21 keV as shown schematically inFigure 1 and in agreement with the new measurementfor Sr. A I π = 3/2 − is found at 288 keV. With this asour starting point, we shall next turn our attention tothe role of the different ISB effects, responsible for theinversion of states in Br.
B. Isospin symmetry breaking analysis
In the following we consider two methods to accountfor the ISB effects.
Method 1:
The Coulomb interaction, V C , is an-ticipated to be the most important mechanismcontributing to the isospin breaking. In this firstapproach, it is simply added to the nuclear one in theSM-CI calculation: H = H N + V C (1)We have verified that non-perturbative and perturbativetreatments give almost identical results. In the former,the Hamiltonian in Eq. 1 is directly diagonalized foreach of the two mirror nuclei: H | Sr, I π (cid:105) = E I π ( Sr ) | Sr, I π (cid:105) H | Br, I π (cid:105) = E I π ( Br ) | Br, I π (cid:105) In the latter, the eigenstates of H N , | A, T, I π (cid:105) , are usedto compute the expectation value of the Coulomb inter-action for each nucleus: δE pert ( Sr, I π ) = (cid:104) , / , I π | V C ( Sr ) | , / , I π (cid:105) δE pert ( Br, I π ) = (cid:104) , / , I π | V C ( Br ) | , / , I π (cid:105) V C can be divided into three terms: core, one-body andtwo-body. With the indexes m, n representing the pro-tons in the core and i, j the valence protons, we have V C,Core = (cid:88) n,m e /r n,m Br Sr - (1/2 - ) 5/2 - (5/2 - )
27 keVX keV
SM Nuclear only - - - - SM Nuclear and Coulomb
63 keV 53 keV21 keV
FIG. 1: (Color online) Shell model results (light-blue andblue levels), compared to the experimental data (black anddashed levels). The SM isospin conserving result with onlythe nuclear interaction is shown in the middle of the panel. V C, B = (cid:88) j n j ( (cid:88) n e /r n,j ) V C, B = (cid:88) i,j e /r i,j The first term is the same for both nuclei and is notconsidered further. The one-body term affects only thesingle particle energies of the proton orbits. We adoptthe experimental spectrum of Sc, where a lowering of225 keV of the energies of the p orbits relative to the f orbits is observed. The two-body Coulomb matrixelements are calculated with harmonic oscillator (HO)wave functions using ¯ hω = 45 A − / − A − / MeV.Their expectation values are denoted by C C TABLE II: Method 1. Isospin symmetry breaking contribu-tions to the excitation energies of the lower states in Sr and Br, C1 (1B) and C2 (2B) (in keV). They are added to thenuclear only values to produce the Total and MED columns,the ones to be eventually compared with experiment. I π Nuclear Sr Br MED C C C C / − / −
21 25 17 63 -27 6 0 1163 / −
288 3 -79 212 55 18 361 -104
It is clearly seen that: i) this approach produces thedesired inversion in Br, and, ii) it is the one-body partof the Coulomb interaction, i.e. the shift in the protonsingle particle energies of the p − orbits relative to the f − orbits, which is responsible for this phenomenon. Ifwe shift the proton SPE’s of the g − and d − orbits by thesame quantity than the p − orbits we obtain qualitativelythe same results, and do not change appreciably evenif we double the SPE correction. It is important tonote that the difference of SPE between protons andneutrons, taken from the experimental data, may notbe only of electromagnetic origin.The MED’s are defined as the difference between theexcitation energy of analogue states, thus putting theMED for the ground states to zero [7], which have ingeneral the same spin and parity. As this is not thecase here, we calculate the MED with respect to the5 / − state, that is the lowest state for the pure nuclearfield. We report in the last column of Table II the MEDobtained asMED I π = E ∗ I π ( Sr) − E ∗ I π ( Br)where E ∗ I π = E I π − E / − . Method 2:
Here we follow the approach discussed inthe review article [8] that considers several contributionsto the MED:
Multipole Coulomb C M . It is constructed as theCoulomb 2B in Method 1, the only difference is thatonly the multipole part of the two-body Coulomb ma-trix elements is considered. It is sensitive to microscopicfeatures such as the change of single-particle spin recou-pling and alignment. Single-particle energy corrections C (cid:96)s and C (cid:96)(cid:96) . Startingfrom identical single-particle orbits for protons andneutrons, given in Table I, relative shifts due to theelectromagnetic spin-orbit interaction C (cid:96)s [12] and theorbit-orbit term C (cid:96)(cid:96) [13] are introduced.The electromagnetic spin-orbit interaction is: V (cid:96)s = ( g s − g (cid:96) ) 12 m N c (cid:18) − r dV C dr (cid:19) (cid:126)(cid:96).(cid:126)s where g l and g s are the g-factors and m N the nucleonmass. The correction is given by C (cid:96)s (cid:39) . g s − g (cid:96) )( ZA )[ (cid:96) ( (cid:96) + 1) + s ( s + 1) − j ( j + 1)] keVwhich, although ∼
50 times smaller than the nuclearspin-orbit interaction, its effect on the excitationenergies can be of several tens to hundreds of keV. Itis clear that this interaction contributes differently onprotons and neutrons.The C (cid:96)(cid:96) energy correction has been deduced in Ref. [13]and is given by C (cid:96)(cid:96) = − . Z / cs [2 (cid:96) ( (cid:96) + 1) − N ( N + 3)] A / ( N + 3 /
2) keVwith Z cs the atomic number of the closed shell. For A = 73, Z cs =20 and the corresponding HO principal quantum numbers N = 3 and N = 4, the energy shiftsto be added to the bare energies in Table I are reportedin Table III. TABLE III: Method 2. Energy corrections introduced by theelectromagnetic C (cid:96)s and C (cid:96)(cid:96) terms to the SPE’s of neutronsand protons (in keV)0 f / p / f / p / g / d / neutrons ( (cid:96)s ) 52.5 17.5 -70 -35 70 35protons ( (cid:96)s + (cid:96)(cid:96) ) -100 65 47 128 -144 38 The corrections of electromagnetic origin introducedso far have no free parameters and affect the excitationenergy of the analogue states in each of the mirror nu-clei. In the following we discuss two additional correc-tions purely of isovector character. Therefore we knowtheir contributions to the MED’s, but ignore their ef-fect on each mirror partner separately. Both terms areempirical and schematic.
Radial term C r . Of Coulomb origin, it takes into ac-count changes of the nuclear radius for each excitedstate. These changes are due to differences in the nu-clear configuration that depend on the occupation num-ber of the orbits. Low- (cid:96) orbits have larger radius thanthe high- (cid:96) orbits in a main shell. This has a sizableeffect in the MED: Protons in larger orbits suffer lessrepulsion than those in smaller orbits, which reflects inthe binding energy of the nuclear states. Originally in-troduced in [14], the halo character of low- (cid:96) orbits hasbeen recently discussed in detail in [15]. The isovec-tor polarization effect in mirror nuclei tends to equalizeproton and neutron radii. Thus, the contribution of theradial term to the MED at spin I π can be parametrizedas a function of the average of proton and neutron radii,considering the change in the occupation of low- (cid:96) orbitsbetween the ground state (gs) and the state of angularmomentum I π [8]: C r ( I π ) = 2 | T z | α r (cid:18) n π ( gs ) + n ν ( gs )2 − n π ( I π ) + n ν ( I π )2 (cid:19) . The value α r =200 keV, has been used in extensivestudies of MED’s in the pf -shell [8]. In the present case,since we are also filling the shell g / and d / orbits,we have to include them as they have larger radii thanthe f orbits as well. We adopt the same value α r = 200keV for the p / orbit, α r = 100 keV for the p / orbitthat is almost full [16], and a larger value of α r = 300keV for the N = 4 g / and d / orbits. The estimatedradial contribution is C r (1 / − ) = −
16 keV.
Isospin–symmetry breaking interaction V B . This is anisovector correction deduced from the A = 42, T = 1mirrors in Ref. [7] and more recently modified and gener-alized in Ref. [17]. It consists of a difference of -100 keVbetween the I = 0 , T = 1 proton-proton and neutron-neutron matrix elements. Originally introduced for the f / shell, here we apply it to all orbitals in the modelspace.Taking into account all the corrections above we com-pute the MED’s for the Sr and Br mirror pair infirst order perturbation theory as,MED I π = E ∗ I π ( Sr ) − E ∗ I π ( Br )= ∆ (cid:0) (cid:104) C M (cid:105) ( I π ) + (cid:104) C (cid:96)s + (cid:96)(cid:96) (cid:105) ( I π ) (cid:1) + C r ( I π ) + V B ( I π ) (2)where the first two terms are obtained as the difference(∆) of the expectation values of C M , C (cid:96)s and C (cid:96)(cid:96) be-tween the two mirrors. The third and forth terms cor-respond to the radial and ISB terms respectively. Theindividual corrections and the total MED’s are given inTable IV. TABLE IV: Method 2. MED’s between Sr and Br andthe contribution of each term in Eq. 2 (in keV). I π C M C (cid:96)s + (cid:96)(cid:96) C r V B MED5 / − / −
11 23 -16 25 433 / − -97 -130 6 -29 -250 Since the excitation energy of the 1 / − state in Sris not yet known we just have a lower limit for the MEDof this state, which has to be greater than 27 keV. TheMED value reported in Table IV is compatible with thislimit but there is room for further explorations usingdifferent values of α r for the p / , p / , g / and d / orbits. A V B contribution > ∼
10 keV in Eq. 2 is neededto account for the MED experimental lower limit.
III. CONCLUSION
We have studied the inversion of the I π = 1/2 − , / − states between the mirror pair Sr - Br within the framework of large-scale shell model calculations usingthe PFSDG-U effective interaction for the nuclear partof the problem. The Coulomb force and other isospin-symmetry breaking effects were analyized using twowell established methods which, not surprisingly, pointto the prominent role played by Coulomb effects to ex-plain the observed inversion. In Method 1 the Coulombinteraction is added to the nuclear Hamiltonian andtreated both pertubatively and non-perturbatively withthe calculated shifts in agreement with experiment. Inthis approach, possible nuclear ISB contributions mightbe included in the difference between neutron andproton SPE’s which are empirically derived from thespectra of Ca and Sc. In Method 2, electromagneticand non-Coulombic effects on the MED’s are evaluated.Within the anticipated contributions of electromagneticorigin, this second approach suggests the need for anisospin breaking nuclear contribution, in line with ourestimate of V B ≈
25 keV, to explain the inversion.
Acknowledgments
This material is based upon work supported by theU.S. Department of Energy, Office of Science, Of-fice of Nuclear Physics under Contract No. DE-AC02-05CH11231(LBNL). AP acknowledges the support ofthe Ministerio de Ciencia, Innovaci´on y Universidades(Spain), Severo Ochoa Programme SEV-2016-0597 andgrant PGC-2018-94583. AOM would like to thank theRivas family for their hospitality during the course ofthis work. [1] E. M. Hoff, A. M. Rogers, et al.
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