Splitting Sensitivity of the Ground and 7.6 eV Isomeric States of 229Th
aa r X i v : . [ nu c l - t h ] J u l Splitting Sensitivity of the Ground and 7.6 eV Isomeric States of Th A. C. Hayes, J. L. Friar and P. M¨oller
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (Dated: November 2, 2018)The lowest-known excited state in nuclei is the 7.6 eV isomer of
Th. This energy is withinthe range of laser-based investigations that could allow accurate measurements of possible temporalvariation of this energy splitting. This in turn could probe temporal variation of the fine-structureconstant or other parameters in the nuclear Hamiltonian. We investigate the sensitivity of thistransition energy to these quantities. We find that the two states are predicted to have identicaldeformations and thus the same Coulomb energies within the accuracy of the model (viz., withinroughly 30 keV). We therefore find no enhanced sensitivity to variation of the fine-structure constant.In the case of the strong interaction the energy splitting is found to have a complicated dependenceon several parameters of the interaction, which makes an accurate prediction of sensitivity to tem-poral changes of fundamental constants problematical. Neither the strong- nor Coulomb-interactioncontributions to the energy splitting of this doublet can be constrained within an accuracy betterthan a few tens of keV, so that only upper limits can be set on the possible sensitivity to temporalvariations of the fundamental constants.
PACS numbers: 23.20.-g,06.20.Jr,27.90.+bb,42.62.Fi,21.10.Sf,21.60.-n,21.60.Ev
The isotope
Th has recently become of interest be-cause of its unusually low-lying (7.6 eV, 3/2 + ) [1] first ex-cited state, which is an isomeric state with an estimatedhalf-life of 5 hours. The 5 / + − / + ground-state-to-isomer transition energy is within the range of atomictransitions, and it has been suggested that this almostdegenerate doublet in Th could be used as a nuclearclock [2] and as a sensitive probe of possible temporalvariation of fundamental constants, including the fine-structure constant ( α ) and the quark mass[3]. The sen-sitivity of the transition energy to temporal changes inthe fundamental constants varies considerably depend-ing on the assumptions made[3, 4, 5]. For example, inrefs. [3, 5] temporal variation of the fine structure con-stant ( ˙ α ) was related to constants in the Nilsson Hamil-tonian, whereas in [4] ˙ α was shown to be proportional tothe Coulomb energy difference between the two states,which requires detailed information about the deforma-tions involved. Thus it is important to understand thenuclear-structure issues giving rise to this doublet.In the present work we examined this doublet using thefinite-range microscopic-macroscopic model (FRDM)[6],which describes many nuclear-structure properties (suchas ground-state masses and deformations) over a broadrange of nuclei. Our goal is to examine the sensitivityof the energy splitting between these states to the un-derlying components of the effective nuclear interaction,including the single-particle potential and the pairing,spin-orbit, and Coulomb interactions. Knowledge of thissensitivity is essential for determining the sensitivity ofthe transition energy to possible temporal variation offundamental constants.In our macroscopic-microscopic model[6] the macro-scopic terms give the smooth variation of the nuclear po-tential energy (mass) with proton number Z , neutronnumber N , and deformation. The dependence of nuclearstructure properties on microscopic quantum-mechanicaleffects is obtained from a deformed single-particle poten- tial through the use of Strutinsky’s method[7, 8]. Nineconstants of the macroscopic part have been determinedin a least-squares adjustment to 1654 measured nuclearmasses with Z ≥ N ≥
8; the details are given in[6]. For these 1654 nuclei ranging from O to ≥
65 thecorresponding accuracy is 0.448 MeV. Values of othermodel constants (such as the depth and diffuseness of thesingle-particle potential, and the strengths of the spin-orbit and pairing interactions) are determined from otherglobal considerations, as discussed in [6]. Ground-statemasses and shapes have been calculated for 8979 nucleiand tabulated in [6]. The shape parameters tabulated arequadrupole ( ǫ ), octupole ( ǫ ), hexadecapole ( ǫ ), andhexacontatetrapole ( ǫ ) deformation (shape) degrees offreedom. Strong evidence of reliability of the model isgiven by its now well-established predictive capabilitiesfor new nuclear-mass regions, including unstable nucleiand super-heavy nuclei.In the present calculations we carried out a high-accuracy determination of the ground-state deformationand quasi-particle energy by minimizing the potential en-ergy for Th on a fine deformation grid. Taking ad-vantage of enhanced present-day computational power,we varied all four shape parameters in steps of 0.001,which is to be compared to the 0.05 grid step used in [6].We predict three very closely lying states with identicaldeformation: the 5 / + ground state, the 3 / + isomer,and a 5 / − state. The three lowest (almost degenerate)neutron quasi-particle states are predicted to have de-formations: ǫ = 0 . ǫ = 0 . ǫ = − . ǫ = − . / − state, with asymptotic quan-tum numbers [ N n z Λ , K π ] = [732 , / − ], lies lowest inenergy. The ground-state and isomer doublet of inter-est have asymptotic quantum numbers [6 3 3 , / + ] and[6 3 1 , / + ], respectively, and are predicted to lie at 13.6keV and 21.6 keV above the 5 / − .Since the three lowest neutron quasi-particle configura-tions are predicted to have the same deformation (whichdetermines their proton distribution), they also have thesame Coulomb energy (within the model uncertainties).This is not the case for other states in Th, for whichthe energy was found to be minimized by different valuesof the shape parameters.Although minor changes to the model parameterscould move this triplet of states relative to one another,the exact position of the 5/2 − state is not essential toour present interest, since the predicted energy of thatstate does not directly affect the sensitivity of the split-ting between the 5/2 + − / + doublet to the underlyinginteraction. Given the global accuracy of the model inthis mass region, we expect that the 5/2 − state shouldlie somewhere within 50 keV of the ground state.The calculated positions of the three lowest-lyingstates as functions of deformation are shown in Fig. 1,and the three states are seen to track very closely in en-ergy for values of ǫ between roughly 0.12 and 0.22. Asdiscussed below, we find a similar close tracking for all ofthe deformation parameters: ǫ , · · · , ǫ . For comparison,we also show in Fig. 2 the [6 3 1 , / + ] and [6 3 3 , / + ]doublet together with two other low-lying states of Th.Note the very different energy scales used in Figs. 1 and2, and that the latter two states do not show a minimumat the same deformation as the three lowest states, whichis the more typical situation seen in deformed nuclei. Theenergies of different quasi-particle configurations for a nu-cleus are not generally minimized by the same values ofdeformation parameters.The predicted energy separation of the [6 3 1 , / + ] and[6 3 3 , / + ] doublet (viz., 8.3 keV) is large compared tothe actual excitation energy of the isomer (7.6 eV), butit nonetheless means that on a “normal” nuclear-energyscale the two states are predicted to be almost degener-ate. For comparison, typical quasi-single-particle energysplittings in this mass region are a few hundred keV. Forcompleteness we note that the odd-neutron wave func-tion for the Ω π = 3 / + isomer is calculated to have thefollowing asymptotic Nilsson components: | / + i = 0 . . . − . − . − . , (1)while that of the Ω π = 5 / + ground state has compo-nents: | / + i = 0 . . . − . − . − . . (2)Those components in the wave functions whose squaredamplitudes are less than 1% are not listed.When quasi-particle configurations have the same de-formation, the energy difference between them is givenby∆ E g . s . − iso = p ( ǫ iso − λ ) + ∆ − q ( ǫ g . s . − λ ) + ∆ , (3) where λ is the Fermi energy, ∆ is the pairing gap obtainedby solving the pairing equations, and ǫ iν are the single-particle energies for the two states; all of these energiesdepend on deformation. The near degeneracy betweenthe ground state and isomer of Th arises because theFermi surface energy is about midway between ǫ iso and ǫ g . s . . The isomer and the ground state will be exactlydegenerate when ( ǫ iso − λ ) = − ( ǫ g . s . − λ ) (i.e., for λ =( ǫ iso + ǫ g . s . ) / ǫ ν depend on the deformedsingle-particle potential well and on the spin-orbit in-teraction. These in turn depend on parameters govern-ing the depth and diffuseness of the potential well andthe strength parameter of the spin-orbit interaction. Al-though the single-particle energies depend on the shapeof the potential, the shape parameters are not in the cat-egory of adjustable parameters, and single-particle ener-gies are calculated by minimizing the energy of the chosenconfiguration.The dependence of the doublet splitting on the fine-structure constant is particularly straightforward to ex-amine in our model, as we discuss below. On the otherhand, expressing that splitting in terms of more fun-damental parameters of the strong interaction (such asthe quark mass, for example) would require a very de-tailed and non-trivial analysis of the relation between themodel parameters and fundamental sub-nucleon degreesof freedom[9]. Our analysis is therefore necessarily re-stricted to a study of the sensitivity of the predicted dou-blet splitting to the effective interactions in our model.The similarity in the shape of the ground and isomericstates of Th implies that (within the accuracy of themodel) they have the same charge (i.e., proton) distri-bution and therefore the same Coulomb energy. Thishas unfortunate implications with respect to experimen-tal searches for a temporal variation in the fine-structureconstant ( ˙ α ) obtained from measuring a temporal vari-ation in the energy splitting between these two states(denoted by ˙ ω ). The latter variation is proportional tothe Coulomb-energy difference of the two states [4]:˙ ω = hh V C ii ˙ αα , (4)where hh V C ii = h iso | V C | iso i − h g . s . | V C | g . s . i .We therefore find essentially no sensitivity to ˙ α , since hh V C ii = 0 for our chosen mesh-parameter step size. Al-lowing ǫ to vary by the mesh-parameter step size (0.001)in this calculation produces a variation in hh V C ii of ap-proximately 30 keV, and this sets the upper limit forvariations in ˙ ω relative to ˙ α/α . Uncertainties in the ef-fective nuclear interaction prevent any nuclear-structurecalculation from being predictive on an eV scale.With the exception of accidental degeneracies, nearlydegenerate doublets in deformed nuclei reflect a strongsimilarity in the deformation of the states involved. Ifthe deformations of two states are not similar their en-ergy splittings are typically at least several tens of keV.To illustrate this and to investigate the accuracy of the ∆ E g . s . − iso Best V (+10% , − a (+10% , − λ n (+10% , − G (+10% , − + -5/2 + doublet energy splitting for Th induced by a ±
10% variation of four of theglobal parameters of the model (see Ref.[6]). The column labeled “Best” is the predicted splitting with the standard values ofthese parameters, V is the depth of the single-particle potential, a is the range or diffuseness of the single-particle potential, λ n is the strength of the neutron spin-orbit interaction, and G is the pairing strength. Splittings that are listed with a negativesign mean that the [6 3 3 , / + ] and [6 3 1 , / + ] states were inverted. [6 3 3 5/2 + ] [6 3 1 3/2 + ] [7 5 2 5/2 − ] Th ε −
50 0 50 100 Q ua s i - P a r t i c l e E ne r g y ( k e V ) FIG. 1: [Color online] Calculated quasi-particle energiesfor the three lowest quasiparticle states (relative to the[6 3 3 , / + ] ground state, which therefore has zero energy forall deformations) in Th as functions of ǫ . All three statesremain almost degenerate at all values of ǫ near the commonminimum at ǫ = 0 . . model for very close-lying (eV) doublets we examined theground and isomeric states in U, where the observedenergy splitting is 76 eV. We calculated this (7 / − , / + )doublet, again allowing all of the deformation parame-ters to vary independently for each state. The results arevery similar to those for Th, with the ground-stateand isomer energies being minimized by the same valuesof the deformation parameters, and lying just below andabove the Fermi surface, respectively. The deformations [6 3 1 3/2 + ] [6 3 3 5/2 + ] [7 4 3 7/2 − ] [6 0 6 13/2 + ] Th ε − − Q ua s i - P a r t i c l e E ne r g y ( M e V ) FIG. 2: [Color online] Calculated quasi-particle energies ofthe 3/2 + -5/2 + doublet and of a 7 / − state and a 13 / + statein Th as functions of ǫ . The doublet states remain almostdegenerate for a wide range of values of ǫ . Note the changein energy scale from Fig 1. for these two states in U were found to be ǫ = 0 . ǫ = 0, ǫ = − .
07 and ǫ = 0 . Th doublet to several of the globally determinedstrong-interaction parameters of the model, namely, thedepth and diffuseness of the single-particle potential,and the strength of the pairing and spin-orbit interac-tions. We varied each of these four global parameters by ± A : λ n , p = k n , p A + l n , p . (5)With this parameterization the constraints on k n , k p , l n , l p are considerably tighter than 10%. Nonethe-less, in order to gain a better understanding of the originof the near degeneracy in Th we varied the strengthof this and the other three parameters by ± Th. Within the presentmodel the two states are predicted to be different quasi-single-particle neutron states, but corresponding to thesame nuclear deformation. The Coulomb energy is pre-dicted to be the same for the two states (within an uncer-tainty of roughly 30 keV induced by our mesh-step size),which suggests no enhanced sensitivity to ˙ α . The originof the near degeneracy appears to be accidental and isdifficult to parameterize in terms of any one componentof the effective interaction. Nevertheless, this doublet re-mains of interest in possible searches for time variationsin fundamental physics because of the small energy split-ting involved. I. ACKNOWLEDGMENTS
We wish to thank Naftali Auerbach, John Becker, Vic-tor Flambaum, Steve Lamoreaux, Jerry Wilhelmy, BobWiringa, and Xinxin Zhao for useful comments and/ordiscussions. [1] B.R. Beck, J.A. Becker, P. Beiersdorfer, G.V. Brown, K.J.Moody, J.B. Wilhelmy, F.S. Porter, C.A. Kilbourne, R.L.Kelley, Phys. Rev. Lett. (2007) 142501.[2] E. Peik and Chr. Tamm, Europhys. Lett. (2003) 181.[3] V.V. Flambaum, Phys. Rev. A (2006) 034101; V.V.Flambaum, Phys. Rev. Lett (2006) 092502.[4] A.C. Hayes and J.L. Friar, Phys. Lett. B 650 (2007) 229.[5] X.-t. He, Z.-z Ren, J. Phys.
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