Modification of the Brink-Axel Hypothesis for High Temperature Nuclear Weak Interactions
MModification of the Brink-Axel Hypothesis for High Temperature Nuclear WeakInteractions
G. Wendell Misch , George M. Fuller , and B. Alex Brown Department of Physics, University of California, San Diego, La Jolla, CA 92093, USA and National Superconducting Cyclotron Laboratory, and Department of Physics and Astronomy,Michigan State University, East Lansing, Michigan 48824, USA (Dated: September 13, 2018)We present shell model calculations of electron capture strength distributions in A=28 nucleiand computations of the corresponding capture rates in supernova core conditions. We find that inthese nuclei the Brink-Axel hypothesis for the distribution of Gamow-Teller strength fails at low andmoderate initial excitation energy, but may be a valid tool at high excitation. The redistribution ofGT strength at high initial excitation may affect capture rates during collapse. If these trends whichwe have found in lighter nuclei also apply for the heavier nuclei which provide the principal channelsfor neutronization during stellar collapse, then there could be two implications for supernova coreelectron capture physics. First, a modified Brink-Axel hypothesis could be a valid approximationfor use in collapse codes. Second, the electron capture strength may be moved down significantly intransition energy, which would likely have the effect of increasing the overall electron capture rateduring stellar collapse.
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I. INTRODUCTION
The Brink-Axel hypothesis posits that the electromag-netic giant dipole resonance in nuclei resides at the samerelative energy from excited states as it does from theground state [1, 2]. That is, if a given nucleus in itsground state has the resonance at 10 MeV excitation,then that same nucleus in an excited state would havethat resonance at 10 MeV above the excited level, andindeed experiment bears this out [3]. The idea that theresponse properties of excited states might be similar tothose of the ground state is alluring, especially for astro-physical weak interaction calculations, in part because itcan be difficult to measure or calculate excited state prop-erties and relatively easier to study ground state prop-erties. Moreover, in stellar collapse environments, nucleican reside in highly excited states, and an approximationlike Brink-Axel for Gamow-Teller (we will frequently ab-breviate this to GT) transitions is widely applied [4–19].(In fact, for isovector Fermi transitions, the Brink-Axelhypothesis holds exactly insofar as isospin is a good quan-tum number of the nucleus.) In this paper we examineelectron capture strength on nuclei with high initial exci-tation energy and its effect on the electron capture rate,with a particular emphasis on adapting the Brink-Axelhypothesis for use in this channel.Neutronization of the collapsing core through electroncapture is pivotally important in the supernova problem,as electrons provide pressure support within the core.During infall, the mass of the homologous inner core(that portion which collapses subsonically) is set by theelectron-to-baryon ratio Y e . This mass, which acts as asort of piston at core bounce, sets the initial post-bounceshock energy. Moreover, Y e figures into the nuclear com-position of the outer core, which dissipates much of theshock energy through photodissociation of its nuclei and affects neutrino transport through coherent interactionwith nuclei [20–32].During supernova core collapse, the density is veryhigh, starting at around 10 g/cm at the onset of col-lapse and proceeding to > g/cm at bounce. Thetemperature is very high at ∼ ≈ E ≈ a ( k B T ) (1)where a ≈ A MeV − is the level density parameter. Witha typical nuclear mass of ∼ e ≈ .
32, which implies neutron rich nuclei. Inorder to understand neutronization during core collapse,we must therefore consider the capture of electrons ontolarge, highly excited, and eventually neutron rich nuclei.Large, highly excited, neutron rich nuclei are, unfortu-nately, problematic to understand both experimentallyand theoretically. Experimental data on these nucleiis sparse [40], and while large nuclei certainly exist inabundance, there are as of yet no experimental means bywhich to put them into high energy states without utterlydestroying them. The (n,p), (p,n), ( He,t), (d, He), andsimilar charge exchange channels give information on theGamow-Teller structure [41–43], but these experimentscan only probe nuclei in the ground state, whereas low en- a r X i v : . [ a s t r o - ph . H E ] A ug tropy, high temperature environments favor much higherexcitations. The Extreme Light Infrastructure may even-tually be able to provide some insight into the structureand behavior of highly excited nuclei through the use ofmultiple MeV laser light, but it is not yet in operation[44]. Of course, even when high energy states becomereadily attainable, we still face the problem that nucleiof the appropriate neutron richness are highly unstablein the laboratory; it is the high density and low entropyof the supernova core that allows them to exist in thatenvironment.From the theoretical direction, we should look fortrends in the Gamow-Teller electron capture strength dis-tribution, as the Brink-Axel hypothesis has had experi-mental success in the electromagnetic channel. Fuller,Fowler, and Newman [4, 45–47] (hereafter FFNI, FFNII,FFNIII, and FFNIV, respectively, for those specific pa-pers, and FFN for the body of work as a whole) wasthe first to adopt the Brink-Axel hypothesis for use inthe Gamow-Teller charged current channel (we will callthis and similar techniques the GT Brink-Axel hypothe-sis to distinguish it from the experimentally verified giantdipole electromagnetic phenomenon). This approach andmodifications thereof have since been widely used to com-pute weak rates. Variations include essentially copyingthe FFN approach [5], using a broad GT resonance that isthe same for all excited states [6, 7], computing in detailonly the lowest few states in the parent and/or daughternuclei and employing the GT Brink-Axel hypothesis totreat the bulk of the strength at high excitations or ne-glecting highly excited states entirely [8–16], and usingthermal averaging techniques [17, 18]. Recently, electroncapture rates have been tabulated using combinations ofthese approaches over a wide range of nuclear masses andstellar conditions [19].However, there is mounting evidence that we would beunwise to take the Brink-Axel hypothesis at face value.Angell et al [48] have shown experimentally that Brink-Axel does not hold for the pygmy dipole resonance, andNabi & Sajjad [49–51] have observed in their theoreticalcalculations the failure of Brink-Axel for the Gamow-Teller interaction even at modest excitation energies.Thus, whenever it is computationally feasible, we shouldavoid use of the GT Brink-Axel hypothesis. Oda et al[52] performed full sd shell model computations of thefirst 100 excited states in each sd shell nucleus, whileothers have taken to the random phase approximationto examine heavier nuclei [53, 54]. But the Oda et alapproach of neglecting states higher than the 100 th exci-tation may miss some important features of higher-lyingstates, and while RPA does well at determining the over-all strength distribution, it is unable to accurately repro-duce the detailed distributions to which electron capturerates are sensitive. We are therefore well served by scru-tinizing detailed strength distributions up to very highinitial excitation to learn in what ways the distributionevolves. We will show that at least in the sd shell, a mod-ified form of the GT Brink-Axel hypothesis derived from large scale shell model calculations can both be compu-tationally tractable and capture features of the strengthdistribution at low and high excitation with consequencesfor core collapse.Computationally, large nuclei are difficult to study sim-ply because of the large number of nucleons involved; thesheer combinatorics of so many nucleons rapidly drivesup the computational requirements. In practice, this dif-ficulty is usually circumvented by holding most of thenucleons fixed and only allowing a few to occupy sin-gle particle states above the lowest energy. While thisapproach works reasonably well for the lowest-lying nu-clear states, it’s efficacy breaks down at higher energies(higher nuclear energies imply more nucleons above thelowest single particle energies) and when the model hastoo few single particle states, i.e. is restricted, allowingtoo few basis states to yield a realistic set of total nucleareigenstates.Because of these computational obstacles and the factthat we want to understand the GT structure of very highly excited nuclei, we are relegated in this work tostudying relatively light nuclei. The biggest drawback ofthis approach is that although light nuclei are abundantprior to the onset of collapse, they are disfavored duringinfall. In our favor, reference [38] found that in some re-spects, heavy nuclei and light nuclei exhibit similar weaktransition characteristics. In any case, light nuclei areat present the only option for computing highly excitedstates, and we will ideally learn something that will shedlight on the behavior of all nuclei, including heavier, moreneutron rich species.In section II, we provide a brief overview of the nuclearshell model and GT transitions, as it will be convenientin later sections to have that picture in mind. Section IIIoutlines the historical approach to the problem at handand discusses its weaknesses. The results of our electroncapture strength computations are in section IV, and us-ing those results, we show calculations of electron capturerates in section V. We give discussion and conclusions insection VI. II. NUCLEAR SHELL MODEL AND GTTRANSITIONS
In the shell model, individual nucleons are consideredto occupy non-interacting single-particle states, with thesets of occupied states (configurations) coupled to havegood spin J and isospin T. Energy, angular momentum,and isospin eigenstates can be constructed by diagonal-izing a residual nucleon-nucleon Hamiltonian in the con-figuration basis. This mixes many configurations into asingle nuclear state: | Ψ J,T i i = X k A ik | C J,T i k (2)where | Ψ J,T i i is nuclear eigenstate i with spin J andisospin T, the A ik are complex amplitudes, and | C J,T i k is the k th configuration with spin J and isospin T.One-body nuclear transitions – such as the Gamow-Teller transition – consist of a single nucleon changingits single particle state. There are three qualitatively dif-ferent single particle GT transitions: spin flip transitions(from an l + state to an l − state), back spin flip tran-sitions (from l − to l + ), and lateral transitions (nochange in total angular momentum). Respectively, theserepresent a net gain, loss, and no change in single particleenergy up to differences in energy between neutron andproton single particle states. If a nuclear state has as oneof its components a configuration resulting from a singleparticle transition in a particular initial state, then thenucleus can transition to that final state. The strengthof the transition from an initial nuclear state | i i to a finalstate | f i is given by |h f | X k b o k | i i| (3)where b o k is a single body operator on the k th nucleon.Throughout this paper, “GT strength” or “electron cap-ture strength” will refer to the reduced nuclear transitionprobability B(GT) if , given by |h f || Σ k ( ~στ − ) k || i i| J i + 1 (4)where ~στ − is the one-body Gamow-Teller lowering oper-ator and the sum is over nucleons. III. PREVIOUS ADAPTATION OF GTBRINK-AXEL HYPOTHESIS
FFNII [45] approached the problem of GT transitionstrength distributions by using experimental values ofthe strength where known, supplementing that with esti-mated allowed and forbidden strength to known states inthe daughter nucleus, and placing the remainder of theGT strength computed from a zero-order shell model intoa single narrow resonance at an energy also computed us-ing a zero-order shell model. Using two simple assump-tions, FFNII took the strength and relative energy of theresonance to be the same for all excited states as it is forthe ground state. First, assume that the individual nucle-ons are distributed among the single particle states in away that is on average independent of nuclear excitationenergy. Second, assume that the transition energy of theGT resonance is principally due to a single nucleon un-dergoing a spin flip, and thus is similar in excited statesto that of the ground state. To the extent that theseapproximations are valid, they are extremely useful, asthe partition function becomes algebraically irrelevant indetermining the resonant electron capture rate. FromFFNII, the total electron capture rate through resonanttransitions is given byΛ res = X i P i λ resi (5) where P i is the probability that the nucleus is in state | i i (given by the product of the degeneracy and the Boltz-mann factor, divided by the partition function) and λ resif is the resonant transition rate from state | i i to state | f i ,itself a function of nuclear structure and electron distri-bution in the supernova core. But under the assumptionthat the GT resonances are the same–irrespective of nu-clear excitation energy–all of the λ resi are identical; weshall call them λ res . We now haveΛ res = X i P i λ res = λ res X i P i = λ res (6)So, the total resonant transition rate is simply the res-onant transition rate of any single state, which we take tobe the ground state. Of course, highly excited states inthe parent would be in the GT resonances of lower energystates in the daughter, leading to “back-resonant” tran-sitions. Accounting for the fact that the P i will not sumto unity for back-resonant transitions (low-lying initialstates have no back-resonance) and otherwise treatingthem identically to resonant transitions, we eventuallyarrive at Λ backres = λ backres G d G p e − RkT (7)where G p (G d ) is the partition function of the parent(daughter) nucleus and R is the characteristic transitionenergy of the GT resonance from the daughter nucleus tothe parent. Finally, Fermi transitions are handled in anidentical manner to the GT transitions, and the rates aresummed along with the rates from known and estimatedtransitions to get the total capture rate.A priori, we might expect the GT Brink-Axel hypothe-sis to fail. If we keep the assumption that single particlesare distributed roughly independently of nuclear excita-tion energy, we should be unsurprised if the GT resonancemoves dramatically or is redistributed in transition en-ergy, since at sufficiently high initial excitation, there willbe strength for the daughter nucleus to be at many ener-gies relative to the parent, without any particular singleparticle transition dominating the strength. By assump-tion the single particles in all of these daughter states arealso arranged similarly, so we would rather expect the GTstrength to be broadly distributed in transition energy.The question, then, is in what way does the hypothesisfail? Do strength distributions evolve in some character-istic way as initial excitation energy increases, or must weabandon the hypothesis completely and replace it with athermal mean strength distribution? IV. GT STRENGTH COMPUTATIONS
Using the shell model code Oxbash [55], we performedshell model calculations of A = 28 nuclei using a closed O core and 12 valence nucleons in the sd shell. Al-though A = 28 is unrealistically light for the supernovacore environment, we chose to use it because it providesa good balance of complexity and computability; that is,we have many valence nucleons and holes (implying manysingle particle configurations), but there are few enoughconfigurations that we can compute nuclear eigenstatesin a reasonable time. We also performed a computationof Mg using the same interaction, but with 8 valencenucleons.The sd shell consists of the single particle sates 0d / ,1s / , and 0d / . In these computations, we used theUSDB Hamiltonian [56], with single particle energies − . − . . / to 2s / , and from either d sub-orbital to either d sub-orbital.In order to make a comparison with the FFN results,we need to address quenching [57]. FFN implementedquenching as follows: (1) experimentally-determined and“guessed” relatively low-lying vector and axial vectortransitions are, of course, already “quenched”; (2) cal-culated Gamow-Teller resonance transitions were delib-erately not quenched. We follow the same procedurehere to facilitate comparison with FFN. As detailed inFFNIV, inspection of the effective log-ft values indicate ifstellar weak rates are dominated by low-lying transitions(regime 1) or resonance transitions (regime 2). Since weare mostly concerned here with high density and tem-perature, usually the stellar weak rates are resonance-dominated, i.e. in regime 2. Here we do not quenchour calculated rates where unmeasured, calculated GTstrength is involved - again, simply to facilitate compari-son with FFN. However, we recommend quenching wher-ever sd and fp shell model strength is used to computerates for astrophysical or any other use. A. Si We first examined Si. Although this nucleus is neu-tron poor by supernova collapse standards, it has themost single particle configurations among sd-shell nucleiand therefore computationally is the most realistic. Fig-ure 1 shows the density of states per 1 MeV for thisnucleus broken down by isospin. The inflection pointsmark the regions where the density of states departs rad-ically from an exponential form, indicating a departurefrom our expectation for reality, in turn implying thatresults for states with energies near and above the inflec-tion point may be significantly impacted by the modelspace restriction. The inflection points on the T = 0 and T = 1 curves occur a little above 30 MeV, so we willtreat states with energies above the mid-20s of MeV withcircumspection.Figure 2 shows the electron capture strength distri-bution in 0.5 MeV transition energy bins as a functionof excitation energy and nuclear transition energy (that Excitation Energy (MeV) S t a t e s p e r M e V T=0.0 T=1.0 T=2.0 T=3.0
FIG. 1: Si density of states per 1 MeV as functions ofisospin. The inflection points for T = 0 and T = 1 near 30MeV–which indicate a marked departure from the expectedexponential growth–suggest we should be wary of results forstates with energies above the mid-20s. is, the total energy input required make the transition,including the change in nuclear mass); the distributionsare averaged over the indicated number of states in eachparent nucleus excitation energy bin. We found that theGT Brink-Axel hypothesis as originally formulated doesnot obtain in that the strength distributions of excitedstates bear no resemblance to the ground state. How-ever, the GT strength distribution is almost independentof initial state energy for transitions proceeding from ini-tial excitations greater than 12 or 16 MeV. There appearsto be some energy dependence above 24 MeV excitation,though this may be due to the limitations of the modelspace. Figure 2 also shows fits of the strength distribu-tions to a double Gaussian of the form C e − (∆ E − ∆ E ) / σ + C e − (∆ E − ∆ E ) / σ . (8)The fit parameters for strength density are shown in ta-ble I. Note that the computed strength distributions infigure 2 are histograms, so the fit curves are scaled verti-cally to account for the effect of the particular choice oftransition energy bin width. This analysis confirms thatthe distributions are weak functions of parent nucleusenergy at high excitation.Figure 3 shows the total GT strength vs. excitationenergy for our shell model states, with each point corre-sponding to a single initial state. The vertical stripes aredue to sampling; all shell model states up to 20 MeV areincluded, as are many states near 24 and 28 MeV. Theblack line shows the total strength for all included statesaveraged over 1 MeV bins. Where the sampling is dense,there is an overall positive trend in total strength withexcitation energy.Decomposing the strength distributions into contribu-tions from states with specific initial spin and isospin B ( G T ) Nuclear transition energy (MeV)
FIG. 2: Average Gamow-Teller strength distribution in Si asa function of initial excitation energy E i . Strength is binnedin 0.5 MeV increments of transition energy. Also shown arefits of the distributions to the double Gaussian in equation 8.The dependence on initial excitation energy becomes small athigh excitation.E C ∆E σ B(GT) C ∆E σ B(GT) Si. E is the initial excitation energy inMeV, and the parameters are as shown in equation 8. TheC i are dimensionless, and the other parameters are in unitsof MeV. reveals that the trend of figure 2 holds; that is, regard-less of choice of a particular initial spin and/or isospin,the Brink hypothesis fails at low excitation, but is recov-ered at high excitation. Furthermore, nuclear spin is notan important contributor in determining either the shapeor total strength of the distribution. Figure 4 shows thedistributions for a representative selection of spins withinitial isospin T i = 0 in the E i = 20 −
24 MeV bin. We
Initial excitation energy (MeV) T o t a l B ( G T ) FIG. 3: Total GT strength in Si as a function of excitationenergy. Each point corresponds to an individual state com-puted from the shell model, giving a GT sum rule for thatstate. The black line shows the average total strength in 1MeV bins. observed this pattern of J i -independence at all excitationenergies and isospins.Turning our attention now to isospin, we find thatisospin does play a role in the distribution of GT strength.In figure 5 we show strengths for the E i = 20 −
24 MeVbin. Each panel gives the distribution for a different T i ,but because nuclear spin does not strongly affect the dis-tribution, we include in figure 5 all values of J i . Asisospin increases, the locations and strengths of the peaksand plateaus in the distribution shift.The shapes of the single initial isospin strength dis-tributions can be partially understood by decomposingthem into contributions from final states with specificisospin. Although the level density is dominated by T = 0 and T = 1 states in the energy range of interest forthe supernova problem, we examine here T i = 2 becauseit has a greater number of final isospins and therefore ismore illustrative of the effect. Figure 6 shows the T f -decomposition for T i = 2 states in the E i = 20 −
24 MeVbin. Evidently, the strength distribution is strongly de-pendent on the final isospin with distinct consequenceson the shape of the full distribution. For example, thelarge peak in the total strength distribution at ∆ E ≈ T i = 2 is due to transitions to finalstates with T f = 2, and the small peak at ∆ E ≈
13 MeVis due to transitions to final states with T f = 3.Finally, we sought an understanding of the similarityof the strength distributions in the high excitation energyregime. To this end, we examined single particle distri-bution as a function of nuclear excitation energy. Figure7 shows the average single particle state occupations asfunctions of spin and excitation energy for T = 0 states
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.20 B ( G T ) J i =0.0, T i =0.0 Ei: 20.0-24.0, 10 states
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.20 B ( G T ) J i =2.0, T i =0.0 Ei: 20.0-24.0, 10 states
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.20 B ( G T ) J i =3.0, T i =0.0 Ei: 20.0-24.0, 12 states
FIG. 4: Gamow-Teller strength distribution in Si at initialexcitation energy E i = 20 −
24 MeV with initial isospin T i = 0as a function of initial spin J i . The strength distribution isnot strongly dependent on J i . We saw this trend in all nucleiwe studied. in Si. The most salient features are that the occupa-tions have no clear dependence on nuclear spin, and the1d state occupations have a linear dependence on nuclearexcitation with slopes of roughly 1 particle per 12 MeV(which is approximately the spin-orbit splitting energy +particle-hole repulsion energy in this sub-shell), while the2s / occupation is independent of excitation; this is incontrast to FFNII [45], which assumed that the averageoccupations of all single particle states were independentof nuclear excitation energy. While figure 7 shows onlyT= 0 states, the trends are consistent for all isospins,with the exception that the intercept of the 1d / (1d / )occupation gradually shifts by -1 (1) particle as T goes
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.200.250.30 B ( G T ) T i =0.0 Ei: 20.0-24.0, 108 states
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.200.250.30 B ( G T ) T i =1.0 Ei: 20.0-24.0, 97 states
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.200.250.30 B ( G T ) T i =2.0 Ei: 20.0-24.0, 34 states
FIG. 5: Gamow-Teller strength distribution in Si at initialexcitation energy E i = 20 −
24 as a function of initial isospin T i . The strength distribution has an apparent dependence oninitial isospin. from 0 to 3, and shifts an additional -1 (1) particle asT goes from 3 to 4, but this shift may be due to modelspace restrictions.Since we did the computations in this paper withisospin as a good quantum number, we can take the singleparticle occupations in figure 7 to be split proportionatelybetween the valence protons and neutrons. In the case of Si, then, the proton and neutron single particle occupa-tion numbers are each 1/2 the total occupation. This im-plies that the 1 particle per 12 MeV slope in fig. 7 is splitevenly between protons and neutrons, giving a slope foreach species of 1 particle per 24 MeV. Perhaps, then, theassumption in [45] that the single particle distributionsare all similar to the ground state can be simply revised
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.200.250.30 B ( G T ) T f = Ei: 20.0-24.0, 34 states
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.200.250.30 B ( G T ) T f = Ei: 20.0-24.0, 34 states
20 15 10 5 0 5 10 15 20Nuclear transition energy (MeV)0.000.050.100.150.200.250.30 B ( G T ) T f = Ei: 20.0-24.0, 34 states
FIG. 6: Strength distribution for Si with initial isospin T i =2 as a function of final state isospin. Comparison with figure 5shows that certain features of the total strength distributionare consequences of the distributions to specific final stateisospins. to say that above a certain nuclear excitation energy,the single particle distributions change only very slowlywith excitation energy, resulting ultimately in similarlyslowly changing strength distributions. This leaves us tochallenge the second assumption in that work: that thetransition energy of the GT resonance does not changewith nuclear excitation energy.The spin flip single particle transition dominates inelectron capture from the ground state, resulting in theresonance at the observed energy. However, at higherexcitation energies there is an abundance of final nu-clear states that are reachable by the other single par-ticle transitions that can leave the daughter nucleus at A v e r a g e o cc u p a t i o n J=0J=1J=2J=3 J=4J=5J=6J=7 A v e r a g e o cc u p a t i o n A v e r a g e o cc u p a t i o n FIG. 7: Si single particle state occupation for nuclear stateswith isospin T = 0. The occupation numbers are the sumof protons and neutrons. The linear dependence on excita-tion energy of the d orbital occupation numbers is understoodto arise from the spin-orbit splitting and particle-hole repul-sion energies of those orbitals. This dependence is consistentacross all values of T, although the intercepts of the d orbitalsdo shift as T increases. similar or lower excitation. Thus, the GT strength distri-bution changes with increasing initial excitation, spread-ing to lower transition energy. Comparing the relativepositions of the peaks in the strength distributions of allnuclei considered in this paper suggests there may be acorrelation between the single particle state and particle-hole repulsion energies with the locations of the peaks inthe strength distributions, but we will not further explorethat in this paper.Ultimately, given that single particle state occupationsvary slowly at high excitation energy, it is unsurprisingthat over a broad range of energy (above the rapid varia-tion at low energy and below where the density of statesfalls below exponential growth), the strength distribu-tions are largely independent of excitation. B. Mg For the sake of connecting our Si results with ear-lier work, we computed the Gamow-Teller strengths for Mg. Frazier et al [58] examined total strength as a func-tion of excitation and found results similar to ours, givenin figure 8. The gradual increase of total strength withinitial excitation corroborates the result for Si (figure3) and suggests it is a general feature of nuclei, at leastin the sd shell.
Initial excitation energy (MeV) T o t a l B ( G T ) FIG. 8: Total GT stregth in Mg as a function of excita-tion energy. The black line shows the average total strength,computed from 1 MeV bins.
The density of states for Mg (figure 9) has an in-flection point near 28 MeV, so strength distributions forstates with initial energies above the low 20s are suspect.The strength distributions for Mg behave qualitativelythe same as Si. That is, above 12 MeV (and below themodel space restriction range), the distributions (figure10) are not strong functions of initial excitation. TableII shows that where the distribution is stable (between12 and 20 MeV excitation), the fit parameters agree withthose for Si. This bodes well for extrapolating to othernuclei.
Excitation Energy (MeV) S t a t e s p e r M e V FIG. 9: Mg density of states. The inflection point is near28 MeV, indicating that we cannot be confident of strengthdistributions for initial state energies above the low 20s. B ( G T ) Nuclear transition energy (MeV)
FIG. 10: Computed average Gamow-Teller strength distribu-tion in 0.5 MeV transition energy bins and fits to a doubleGaussian in Mg as a function of initial excitation energy E i . As in the analysis of Si, the fit curves are scaled to ac-count for the choice of transition energy bin width. Betweenthe E i = 12 −
16 MeV and 20 −
24 MeV bins, the strengthvaries slowly with excitation. E C ∆E σ B(GT) C ∆E σ B(GT) Mg. The good agreement with Si meansextrapolation to other nuclei may be tenable.
Excitation Energy (MeV) S t a t e s p e r M e V FIG. 11: Mg density of states. The inflection point at ∼ C. Mg Figure 11 shows the density of states for Mg; theinflection point occurs at E ≈
22 MeV. The low energyof the inflection really crowds the region where we ex-pect the strength distribution to be stable, and this in-deed manifests out in figure 12. The high transition en-ergy peak in the distribution appears to stabilize brieflyaround 8-16 MeV, but it rapidly falls off and the lowtransition strength grows as the initial excitation goesinto the model space restricted region. D. Na With only 3 protons and 3 neutron holes in the sd shell, Na really pushes the limits of the model space; we seein figure 13 that the density of states is very low, withthe inflection point at 12 MeV or less. While the sd shellmay yield acceptable results for low-lying states in Na,we cannot rely on it to get the high initial excitation dis-tributions. Figure 14 shows that there is essentially no B ( G T ) Nuclear transition energy (MeV)
FIG. 12: Mg strength distribution. The distribution sta-bilizes briefly between 8 and 16 MeV before model space re-striction impacts the results. initial energy region with a stable strength distribution,as per expectation from the low-lying inflection point inthe density of states, though we might speculate at someobservable stability between the E i = 4 − −
12 MeV bins. This observation coupled with the re-sults from the other nuclei make it clear that when modelspace restrictions severely limit the density of states, thestrength distribution is not independent of initial excita-tion energy.
V. COMPUTATION OF TRANSITION RATE
Throughout this section, we will use natural units suchthat (cid:126) = c = k B = 1. Following FFNI, the electroncapture rate for a given initial nuclear state is λ if = ln(2) f if ( T, µ e )( f t ) if (9)where ( f t ) if is the ft-value appropriate for the transitionfrom parent nucleus state i to daughter nucleus state f .Here, ( f t ) if is computed from the corresponding Gamow-0 Excitation Energy (MeV) S t a t e s p e r M e V FIG. 13: Na density of states. Model space restrictionsmake the density of states very low, with the departure fromexponential growth occurring at low excitation.
Teller ( M GTif ) and Fermi ( M Fif ) matrix elements bylog( f t
GTif ) = 3 . − log( | M GTif | ) (10)log( f t Fif ) = 3 . − log( | M Fif | ) (11)1( f t ) if = 1 f t GTif + 1 f t
Fif . (12)The factor f if ( T, µ e ) is the phase space integral for theincoming electron and outgoing neutrino. T is the tem-perature, and µ e is the electron Fermi energy, includingrest mass. The numerical values “3.596” and “3.791” cor-respond to choices of axial vector and vector couplingschosen to match those used in FFN, to facilitate com-parison. The phase space integral is f if = Z ∞ w l w ( w − q ) G ( Z, w ) f e ( w, µ e , T )(1 − f ν ) dw (13)where w is the total electron energy in units of electronmass, q is the change in total nuclear energy M f + E f − M i − E i in units of electron mass, Z is the nuclear charge,and f e and f ν are the electron and neutrino occupationprobabilities. The lower limit w l is a function of q, asthe incoming electron must supply enough energy to the nucleus to make the transition; if q <
1, then w l =1 (cor-responding to zero electron kinetic energy), while if q > l =q. G is related to the Coulomb barrier factor and isdetailed in FFNI; rather than use the limiting approxi-mations described in that work, we use the form givenby eqn. 5b therein. Note that that work defines q in thenegative sense of its use here; that is to say, q in thatwork is defined as the parent energy minus the daughterenergy.Up until neutrino trapping sets in at ρ ∼ g/cm ,we may take f ν ≈
0. Here f e ( w, µ e , T ) is the Fermi-Dirac B ( G T ) Nuclear transition energy (MeV)
FIG. 14: Strength distribution for Na. There is no obviousenergy regime where the strength is independent of initialexcitation. distribution (1 + e ( wm e − µ e ) /T ) − . Using this and ourdefinition of w l and integrating over final states, we atlast arrive at λ i = ln(2) Z −∞ (cid:18) B GTi ( q )10 . + B Fi ( q )10 . (cid:19) dq Z ∞ f e ( w, µ e , T ) w ( w − q ) G ( Z, w ) dw +ln(2) Z ∞ (cid:18) B GTi ( q )10 . + B Fi ( q )10 . (cid:19) dq Z ∞ q f e ( w, µ e , T ) w ( w − q ) G ( Z, w ) dw (14)1where B GTi ( q ) ≡ P f ∈{ q } | M GTif | and B Fi ( q ) ≡ P f ∈{ q } | M Fif | , where the sums are over final states f with dimensionless (units of electron mass) Q-value q .To compute the total capture rate, we sum over popula-tion index weighted initial states.Λ = X i λ i (2 J i + 1) e − E i /T G ( T ) (15)where G is the partition function. Recall, however, thatabove ∼
12 MeV, the strength distributions look similar.Therefore, we propose a modification to the GT Brink-Axel hypothesis by applying a cutoff energy below whichall states are included and weighted by their populationindex, and with all remaining statistical weight carried bya single high energy average state. This is in contrast tothe FFN approach of placing the bulk of the strength ina single resonant transition that is identical for all states.In other words, where FFN treated all states as havingan identical giant GT resonance, we treat all states abovethe cutoff energy as having exactly the same distribution.The difference in these two treatments is profound; fig-ure 15 shows the strength distributions in the groundstate for the FFN approach and the shell model; the largepeak in the FFN distribution is the GT resonance. Thetwo major differences are that the shell model result hasless total strength, and the strength is spread to lowertransition energies; the former will have the effect of de-creasing the capture rate, while the latter will tend toincrease it.
Nuclear transition energy (MeV) B ( G T ) Shell modelFFN
FIG. 15: Si ground state strength distribution. The solidline shows the distribution using our shell model calculations,and the dotted line shows the strength from the FFN pre-scription. The large peak in the FFN distribution is the GTresonance used in those works.
Despite the overestimate of the total strength and themisplacement of the resonance, the power of FFN is that it used experimental strengths wherever they were avail-able, and any other technique of computing rates wouldbe well-served by following that example. Therefore, thestrength distributions that we ultimately use to computecapture rates are defined as follows. (1) We take thesame experimentally measured strength distribution asFFN. (2) We sum the experimental strength and (3) re-move that much total strength from our correspondingshell model state by subtracting an equal amount of thelowest-lying strength in the shell model distribution. (Inthis procedure we do not correct for quenching.) (4)We then sum the experimental distribution with whatremains of the shell model distribution for that state.This gives a better estimate of both the capture strengthsum rule and the (non-experimental) strength distribu-tion, but is only applicable to initial states with experi-mentally measured energies. For transitions from higher,unmeasured initial states, we simply used our shell modeldistributions; we do not include any shell model parentstates with an excitation energy lower than the highestused experimental state.We now require the nuclear partition function to ob-tain appropriate initial state occupation indexes. Thereare a few approaches to the partition function problem,but in our case, the simplest and most self-consistent is toinclude only the sd shell states, i.e. only include in thepartition function those states that can be constructedfrom configurations in the sd shell. The biggest weak-nesses of this approach are that at high enough energies,the density of shell model states actually decreases tozero, and all negative parity states are neglected, as wellas any other states that include configurations with oneor more particles promoted into or out of the sd shell. Bythe same token, those states will also not be consideredto contribute to the electron capture rate, thereby com-pensating for the overestimate of the included states’soccupation indexes. With the partition function in hand,we can compute the total capture rate from eqn. 15.The electron occupation probability consists of twoqualitatively different domains: when 1 ≤ w ≤ µ e /m e ,it varies slowly from a maximum of at most 1 at w = 1down to a minimum of 0.5 at w = µ e /m e (we will callthis the “shoulder”), and when w > µ e /m e , it is ex-ponentially damped (“tail”). We numerically integratedthe inner integrals of eqn. 14 using a combination oftwo methods, one for each domain. When the shoulderwas part of the integration domain ( i.e. , q < µ e /m e ), weintegrated the shoulder with a 64-point Gauss-Legendrequadrature. Some or all of the tail is aways in the in-tegration domain, and we integrated it with a 64-pointGauss-Laguerre quadrature.Figure 16 shows electron capture rates for Si as afunction of electron Fermi energy and temperature. Thesolid lines were computed using a cutoff energy of 12MeV and a high energy average state strength distribu-tion computed from the spin-weighted (2 J + 1) averageof every state between 12 and 14 MeV, and the dashedlines are the rates computed using the FFN resonance2prescription. At sufficiently high Fermi energy, there areenough electrons above the GT resonance used in theFFN approach for the rates to outstrip those of our shellmodel results, as the large amount of strength in theresonance outcompetes the shell model. However, at lowFermi energy, the spread of strength to low transition en-ergies found in the shell model approach serves to boostthe rates above the FFN estimates. -3 -2 -1 E l e c t r o n c a p t u r e r a t e ( s − ) T=0.8, cutoff=12.0T=1.0T=1.5T=2.0 T=0.8, FFNT=1.0T=1.5T=2.0
FIG. 16: Electron capture rates for Si as a function of elec-tron Fermi energy and temperature. Solid lines show rateswith all states up to 12 MeV considered individually and therest of the statistical weight carried by a single high energyaverage state, while dashed lines correspond to the rates whenall states are assumed to have the same narrow GT resonance,in accordance with the FFN approach to the GT Brink-Axelhypothesis.
Figure 17 compares the shell model capture rates witha cutoff of 12 MeV against a GT Brink-Axel approach(as in fig. 16), but with the single resonance in theFFN model replaced by the shell model strength distri-bution for the ground state. That is, in the “Brink”approach here, we used experimental values of the tran-sition strength for each initial state where known, andthe rest of the strength in each excited state is carriedby the ground state distribution. In contrast to the be-havior of the FFN approach, the shell model Brink-Axelcurves lack the marked jump above the more comprehen-sive shell model rates as Fermi energy increases, and theyeventually converge. It is notable that the GT Brink-Axel results are not uniformly greater or lesser than themore comprehensive shell model rates; in the Fermi en-ergy region between 5 and 15 MeV, the T=0.8 and 1.0MeV Brink-Axel rates just peek above the correspondingshell model rates.In light of the apparent sensitivity to how excitedstates are handled in rate calculations, we compare in fig-ure 18 the thermodynamically unweighted (meaning thepopulation factor is not included) capture rates of thehigh energy average states corresponding to several cut- -3 -2 -1 E l e c t r o n c a p t u r e r a t e ( s − ) T=0.8, cutoff=12.0T=1.0T=1.5T=2.0 T=0.8, BrinkT=1.0T=1.5T=2.0
FIG. 17: Electron capture rates for Si as a function of elec-tron Fermi energy and temperature. Solid lines show rateswith all states up to 12 MeV considered individually and therest of the statistical weight carried by a single high energyaverage state, while dashed lines correspond to the rates whenall states are assumed to have the same bulk GT strength dis-tribution as our shell model calculation of the ground state. off energies (the HEA state being that which carries all ofthe statistical weight above the cuttoff). The solid linesshow rates for an HEA state including all shell modelstates between 12 and 14 MeV, as in the previous calcu-lations. The dashed lines give the rates of an HEA statecomputed from all states between 15 and 16 MeV, andthe dotted lines are for an HEA state comprised of statesbetween 20 and 20.3 MeV. The widths for the averagingwere chosen such that each HEA state was comprised ofat least 50 individual states.The rates for all three HEA states differ from one an-other by at the most a factor of 3 in the range considered,which is offset by the reduction in statistical weight car-ried by the HEA state as the cutoff energy increases. TheHEA statistical weight is simply the remaining probabil-ity after the occupation indexes of all lower-energy statesare accounted for: w HEA = 1 − G ( T ) X E i 10 percent in the extreme case of T= 2 MeVand E f < E l e c t r o n c a p t u r e r a t e ( s − ) T=0.8, cutoff=12.0T=1.0T=1.5T=2.0T=0.8, cutoff=15.0T=1.0T=1.5T=2.0T=0.8, cutoff=20.0T=1.0T=1.5T=2.0 FIG. 18: Thermodynamically unweighted electron capturerates for high energy average states in Si. The solid lines arethe rates for an HEA state with a cutoff energy of 12 MeV,the dashed lines show a cutoff of 15 MeV, and the dotted linesare for a cutoff of 20 MeV. by a particular choice of cutoff energy will ultimately bewashed out by other uncertainties, including the eventualtreatment of quenching T (MeV) Cutoff = 12 MeV 15 MeV 20 MeV0.8 2 . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − . × − TABLE III: Statistical weights of the high energy averagestate as a function of temperature and cutoff energy. VI. DISCUSSION AND CONCLUSIONS The three principle observations from this work arethat 1) at high excitation energies the GT strength dis-tribution does not depend sensitively on nuclear excita-tion energy (though it is a function of isospin), 2) the GTstrength distribution spreads to low and negative transi-tion energies, and 3) the spreading of the strength tendsto increase the electron capture rate, as not only does itdecrease the electron capture energy threshold, but for agiven incoming electron, it also increases the phase spaceof the outgoing neutrino.As seen in figures 16 and 17, point 3 above is contra-dicted in some regimes of temperature and Fermi energy.In order to understand why the shell model rates some-times fall short of other approaches, we return to thetotal strength, i.e. sum rule, as a function of excitationenergy. Considering first the FFN approach, comparingfigures 15 and 3 reveals that the strength in the GT res-onance employed by FFN is about twice the average to- -3 -2 -1 E l e c t r o n c a p t u r e r a t e ( s − ) T=0.8, cutoff=12.0T=1.0T=1.5T=2.0 T=0.8, cutoff=20.0T=1.0T=1.5T=2.0 T=0.8, ratioT=1.0T=1.5T=2.0 R a t i o FIG. 19: Electron capture rates for Si comparing twochoices of cutoff energy. The solid lines correspond to a cutoffenergy of 12 MeV, while the dashed lines are for a cutoff of 20MeV. The dotted lines show the ratio of the cutoff = 12 ratesto the cutoff = 20 rates. That the rates are nearly identicallends credence to the technique of using a high energy averagestate. tal strength at all excitation energies computed from theshell model, resulting in an overestimate of the capturerate at high Fermi energies.The sources of the deviations in the shell model Brink-Axel approach are a little more subtle. There are regimesof temperature and density where the rates derived fromour shell model treatment are greater than those de-rived using the Brink-Axel assumption. This stems inpart from transitions from the parent nucleus to rela-tively low-lying discrete states in the daughter nucleus.These low-lying daughter states have more favorable Q-values (see figure 2). Furthermore, figure 3 shows thaton average, the total strength increases slowly with par-ent nucleus excitation energy (roughly, from B(GT) ∼ ∼ all excited states as hav-ing the same bulk GT strength distribution as the groundstate, but the more comprehensive model includes con-tributions from those states that have less total strength.Importantly, some of those states are at low initial excita-tion. Hence, they do not have the low-lying strength (lowQ-value) seen in higher states, and they have a compar-atively large population factor. The combination of lowtotal strength, no low-lying strength, and a large popula-tion factor yield temperature and Fermi energy regimeswhere the Brink-Axel approach overestimates the rate.Ultimately, we must conclude that the GT Brink-Axelhypothesis as it has been traditionally used is likely inap-4propriate for obtaining accurate electron capture rates –and by extension, all nuclear weak rates – at the hightemperatures and densities characteristic of collapsingsupernova cores. We must be circumspect, however, asthe nuclei examined here are very light by supernova corestandards. If later work is able to demonstrate that thetrends found here are applicable to larger nuclei, thenwe will have found a useful technique for simplifying theaccurate computation of weak rates in those nuclei.The analysis of Si in this work is essentially a cruderversion of the work of Oda et al. We performed no carefulmatching of the energies of the daughter states relative tothe parent states, meaning that where experimental datawere not used, the distributions shown here will not haveprecise transition energies. This imprecision is unimpor-tant for the sake of our goal here, which was to demon-strate the failure of the GT Brink-Axel hypothesis andhow it can be modified for use at high initial excitation.With these results and the 20 or so years of experimentaldata collected since the Oda et al rate survey, though,it is worth re-examining the weak rate calculations for sd-shell nuclei, which are important in the late phases ofstellar evolution leading up to core collapse.This leaves us with two major directions to follow up.First, we will recompute the weak rates for all sd-shellnuclei over a wide range of temperatures and densitiesrelevant to late stellar evolution and core collapse usingour modification to the GT Brink-Axel hypothesis andthe most recent experimental data. 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