On the inversion of isobaric-analogue states in nuclei
OOn the inversion of isobaric-analogue states in mirrornuclei
J. Henderson ∗ and S. R. Stroberg † Lawrence Livermore National Laboratory, Livermore CA, USA Department of Physics, University of Washington, Seattle WA, USA
Isospin is an approximate symmetry in atomic nuclei, arising from the rather similarproperties of protons and neutrons. Perhaps the clearest manifestation of isospin withinnuclei is in the near-identical structure of excited states in mirror nuclei: nuclei withinverted numbers of protons and neutrons [1]. Isospin symmetry, and therefore mirror-symmetry, is broken by electromagnetic interactions and the difference in the massesof the up and down quarks. A recent study by Hoff and collaborators [2] presentedevidence that the ground-state spin of Sr is different from that of its mirror, Br,due to an inversion of the ground- and first-excited states, separated by only 27 keVin the Br system. In this brief note, we place this inversion within the necessarycontext of the past half-century of experimental and theoretical work, and show thatit is entirely consistent with normal behaviour, and affords no new insight into isospin-symmetry breaking. The essential point is that isospin-breaking effects due to theCoulomb interaction frequently vary from level to level within a given medium-mass ∗ [email protected] † [email protected] a r X i v : . [ nu c l - e x ] M a y ucleus by as much as 200 keV. Any level splitting smaller than this is liable to manifesta level inversion in the mirror partner which, absent disagreement with an appropriatenuclear model, does not challenge our understanding. While we note the novelty of aninversion in nuclear ground states, we emphasize that in the context of isospin there isnothing specifically illuminating about the ground state, or a level inversion.A great deal of experimental work has been performed in mid-mass nuclei [3–7],in an effort to understand isospin-symmetry breaking through energy shifts in mirrornuclei. We employ the data generated from that work to understand the magnitudeof normal isospin-breaking effects. (By “normal” we mean consistent with the scaleof the leading isospin-breaking terms in the Hamiltonian.) We define E as the en-ergy difference between two states in a given nucleus. The relevant quantity for stateinversion in the mirror nucleus is the difference δE = E N>Z − E N
N
50 MeV, V = −
33 MeV, V (cid:96)s = − r = 1 .
25 fm, a = 0 .
65 fm, R = r A / . As with the experimental data, thetheoretical data were required to have excitation energies below the proton-separationenergy. It is clear that the Coulomb interaction is sufficient to produce the distributionof δE values. From a point-by-point comparison the WS choice has a root-mean-squared (rms) deviation from experiment of 53 keV, as compared to 88 keV for the HOchoice, while the rms deviation between the two methods is 74 keV. This highlightsthe difficulty of extracting information about isospin-symmetry breaking forces fromindividual level shifts (especially small ones) due to ambiguities in the single-particlebehaviour. On the other hand, both methods produce very similar δE distributions.In the A=73 pair [2], the ground- and first-excited states are described as belongingto different, near-degenerate intrinsic shapes. It is not unreasonable to expect that thebehaviour of these states will more closely resemble that of IAS-pairs with differingparity in Fig. 1 than those of like-parity. To estimate the likelihood of a pair of IASundergoing an energy inversion, Fig. 2 shows the probability that a pair of states will in-vert based on the data in Fig. 1. Approximately 35 % of like-parity IAS-pairs separatedby 27 keV would be expected to invert. The same probability distribution is calculatedbased on the shell-model calculations and reproduces the like-parity distribution well.Figure 2 also shows two famous cases of additional isospin-symmetry breaking phe-nomena, both arising in unbound states [14–16], with one resulting in the only otherknown case in which the ground states of a mirror pair differ. Both of these casessignificantly exceed the distribution of data shown in Fig. 1, demonstrating that thereis a need to invoke additional isospin-symmetry-breaking effects (weak binding in thesecases). 6n summary, we have analysed energy shifts in pairs of isobaric analogue states(IAS) in mirror nuclei. We find that differences in shifts between pairs of IAS of greaterthan 27 keV are common, occurring in approximately 35 % of cases. The inversion ofground- and first-excited-state observed in the recent study [2] of Sr- Br lies wellwithin the bounds of “normal” Coulombic isospin-symmetry-breaking behaviour and,in the absence of reliable model-predictions contradicting the inversion, requires noadditional, exotic symmetry-breaking effects. Finally, we emphasise that state inver-sions (ground-state or otherwise) provide no more information about isospin-symmetrybreaking than any other mirror-energy shift, of which many hundreds have been ob-served.
Acknowledgements:
Discussions with P. Adsley, G. F. Bertsch, A. Gade, R. V. F. Janssens,B. P. Kay, T. D. Morris, and A. Ratkiewicz are gratefully acknowledged. Work at LLNLwas performed under contract DE-AC52-07NA27344. SRS is supported by the DOEunder contract DE-FG02-97ER41014.
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