We use the Quasiparticle Random Phase Approximation to include the effects of
low-lying Gamow-Teller and first forbidden strength in neutrino capture by very
neutron-rich nuclei with N = 50, 82, or 126. For electron neutrinos in what is
currently considered the most likely r-process site the capture cross sections
are two or more times previous estimates. We briefly discuss the reliability of
our calculations and their implications for nucleosynthesis.Comment: 9 pages, 4 figure

A. Hosaka

C. Gaarde

D. Cha

D.J. Horen

F. Krmpotic

G.C. McLaughlin

G.F. Bertsch

G.M. Fuller

H. Homma

H.G. Janka

J. Engel

J.A. Halbleib

J.D. Walecka

Jonathan Engel

K. Nakayama

R.D. Smith

Rebecca Surman

S.E. Woosley

Y.-Z. Qian

Yong-Zhong Qian

English

arXiv

Crossref

Physical Review C

Neutrino capture by<b><i>r</i></b>-process waiting-point nuclei

Surman, Rebecca

Engel, Jonathan

Carolina Digital Repository

arXiv:nucl-th/9805007v1  4 May 1998Neutrino capture by r-process waiting-point nucleiRebecca Surman and Jonathan EngelDept. of Physics and Astronomy, CB3255, University of North Carolina, Chapel Hill, NC 27599April 16, 2018AbstractWe use the Quasiparticle Random Phase Approximation to include the ef-fects of low-lying Gamow-Teller and first forbidden strength in neutrino captureby very neutron-rich nuclei with N = 50, 82, or 126. For electron neutrinosin what is currently considered the most likely r-process site the capture crosssections are two or more times previous estimates. We briefly discuss thereliability of our calculations and their implications for nucleosynthesis.1The r process, which is responsible for the formation of half of all elements withA > 70, is thought to take place in the “hot bubble” that expands off a proto-neutronstar during a type II supernova[1]. If that is the case the nuclei involved are subject to anintense neutrino flux while they are made. This fact has been used to try to constrain thedistance of the r-process site from the neutron star[2] and to argue that the time scale forthe process is less than usually believed[3].Serious investigation of these and related issues will require knowledge of cross sectionsfor neutrino capture by the very neutron-rich nuclei that lie along the r-process path. Al-though several groups have recently estimated the cross sections[2, 4, 5], none has tried tobe very precise, for two reasons. First, precision is difficult unless one focuses on a few nu-clei, and the r-process involves a huge number of neutron-rich isotopes. Second, conditionsin the hot bubble are so uncertain that precise estimates are not really warranted. Here,nonetheless, we attempt to calculate charge-changing neutrino-nucleus scattering rates1a little more carefully than before in particularly important “waiting-point” isotopes atneutron closed shells. The reason is not so much to be precise as it is to investigate effectsthat systematically increase cross sections above current estimates.Uaing the neutrino-nucleus scattering formalism presented in Ref. [6], we examine twokinds of nuclear transitions not yet considered in this context that can be prompted byneutrinos and therefore add to the cross section: Gamow-Teller (GT) and first-forbiddentransitions to low-lying excited states. Since most of the GT transition strength, inducedby the operator ~στ+, lies in a single broad resonance the low-energy strength has usuallybeen neglected; Ref. [5], for example, approximates the entire distribution by a Gaussian ofwidth 5 MeV. But the energy of the GT resonance in very neutron-rich nuclei is probablyhigh enough to prevent excitation by most hot-bubble neutrinos, the average energy ofwhich is only about 11 MeV. The same is true of the isobar analog (IA) resonance, whichis excited even less because J = 0 states have only a single M -substate. The low-lyingGT strength may therefore contribute nearly as much to cross sections as the resonances,even though it is small in comparison.Forbidden transitions induced by the operators ~rτ+ and [~r~σ]J=0,1,2τ+ have not beenconsidered either, except briefly and without definite conclusion in Ref. [4], because in sta-ble nuclei they too are concentrated in high-lying (dipole-like) resonances and, moreover,are further suppressed by a factor of (qR)2, where R is the nuclear radius. In very neutron-rich nuclei, however, it is possible that some forbidden strength lies low. The reason is thatthe forbidden operators, which change parity and ordinarily must a create a proton oneoscillator shell (roughly speaking) above the neutron they destroy, can actually create aproton in an oscillator shell below the destroyed neutron if there are many more neutronsthan protons. The diagram in figure 1 shows that in the unstable nuclei with N = 82, forexample, operators with Jπ = 1− and 2− can transform a neutron near its Fermi surfaceto a proton close to its Fermi surface. As a result, some forbidden strength may lie lowenough to overcome the (qR)2 suppression (which is really not so great in heavy nuclei)and contribute significantly to the neutrino-nucleus cross sections. The likely magnitude1The corresponding antineutrino rates are suppressed in neutron-rich nuclei and won’t be consideredhere.2of this unusual contribution, and of that due to low-lying GT strength, is what we examinehere.0 h 11/20 i 13/21 g 9/2p n140Ce140Pr0 h 11/20 g 9/2p nfilled levels122Zr122NbFigure 1: Single-particle picture of forbidden charge-changing (neutron −→ proton) tran-sitions in a stable nucleus and a neutron-rich nucleus, both with N = 82. The changein parity restricts neutrons to move up an oscillator shell as they become protons in thestable nucleus, but in the neutron-rich nucleus they can move down an oscillator shell aswell.Assessing the effects of states outside a giant resonance requires a calculation sophis-ticated enough to represent the competition between single-particle structure, which isresponsible for the existence of low-lying strength, and the residual nucleon-nucleon in-teraction, which gathers strength into the higher-lying resonance. At the same time, andfor reasons mentioned above, it is probably not worthwhile to aim for high precision.We therefore use a relatively simple microscopic description based on the neutron-protonQuasiparticle Random Phase Approximation (QRPA)[7, 8]. We restrict ourselves to nucleiwith closed neutron shells (N = 50, 82, 126) in part to avoid dealing with deformation butalso because the closed-shell nuclei form “bottlenecks” in the r-process flow and thus de-termine the time scale of nucleosynthesis. The nuclear-structure effects that are significanthere probably play a role in deformed nuclei as well.3Our calculations proceed as follows: We obtain single-particle energies from the pa-rameterized Wood-Saxon potential in Ref. [9]. Where possible we include all neutronand proton levels that participate with reasonable probability in transitions induced bythe Gamow-Teller and forbidden operators. Those that are unbound we take to be reso-nances; to obtain their wave functions we neglect the Coulomb interaction, making thembound. (We include Coulomb effects in perturbation theory when calculating their en-ergies.) This procedure results in single-particle spaces of up to 20 levels for each kindof particle. Within these large spaces we use two different residual two-body interactionsin the QRPA. The first is a δ-function with independent strengths in the particle-holeand particle-particle channels as described in Ref. [10], and the second a combination of 7Yukawa potentials fit in Ref. [11] to a Paris-potential G-matrix.To reproduce systematics in stable nuclei and extrapolate to neutron-rich isotopes wemodify both forces. First we adjust the strength of each interaction in the pairing channelto obtain pairing gaps ∆p = ∆n = 12/√A MeV[12] from the BCS equations. In theRPA we then adjust 2 parameters, the strengths of the particle-hole interactions in the0+ and 1+ channels, so as to place the IA and GT resonances at appropriate energies.Unfortunately, although the energy EIA of the analog state follows from the Coulombenergy difference between parent and daughter nuclei, the appropriate value of EGT invery neutron-rich nuclei is less certain. (It is also more important, since the GT resonancecontributes more to the cross section.) The usual way to estimate EGT is through a relationof the form EGT = EIA + δ, where, δ has a linear dependence on neutron excess. Thoughone can fit this parameter to data in the valley of stability[13], it is not obvious how bestto extend it to unstable nuclei. Fuller and Meyer[2] take δ to be 0 for Z/A < 0.377, whileQian, et al. [5] extrapolate the fit from Ref. [13], δ = 26A−1/3 − 18.5(N − Z)A−1 MeV,with the result that EGT < EIA far from stability. Here we try both prescriptions. Wealso use gA = 1 for the GT transitions to account for missing strength.Not surprisingly, as is apparent from the top half of figure 2, the δ-function and G-matrix based interactions produce similar GT distributions2 once they have been modi-fied/fit as just described, and in particular both predict significant amounts of low-lyingstrength (though noticeably less than in stable nuclei for N = 82 and 126). The reasonwe use two interactions is really the uncertainty in the location of the forbidden strength,which is harder to measure, decompose, and fit than allowed strength. Instead of rely-ing on the sketchy systematics that do exist[15], we use the two forces without furthermodification to get a handle on forbidden transitions. For the δ-function interaction, thestrengths in the IA and GT channels determine the strengths of the S = 0 and S = 1parts of the force, and we fix these once they have been fit to the allowed resonances. Forthe G-matrix based interaction, we use the original negative-parity proton-neutron matrixelements. As shown in the bottom of the figure, the 2− strength distributions producedby the two interactions are quite similar. Both forces in the end do predict some low-lyingforbidden strength; the large neutron excess prevents the giant resonances from swallow-ing everything up. As a result the forbidden transitions as well as the GT transition to2The large resonance widths are due to a prescription for spreading taken from Ref. [14]; they may notbe realistic but altering them has little effect on the neutrino cross sections.4low-lying states can change the reaction rates substantially.Figure 2: Calculated GT (a) and forbidden 2− (b) strength distributions for 124Mo. Thesolid lines come from the δ-function interaction and the dashed lines from the G-matrixbased interaction. The dot-dashed line in (a) is from a Gaussian GT resonance centeredat the energy of the isobar analog state and with a width of 5 MeV, normalized to givethe correct total strength.To see this quantitatively, we use the (approximate) spectrum of supernova electronneutrinos[5],f(Eν) =1F2(ην)T 3νE2νexp[(Eν/Tν)− ην ] + 1, (1)where T is the temperature, ην the chemical potential, and F2 a normalizing factor, toobtain spectrum-averaged neutrino cross sections〈σν〉 =∫f(Eν)σ(Eν)dEν . (2)The capture rates are directly proportional to 〈σν〉. Several values for the chemical po-tential ην appear in the literature, but while changes in that parameter affect the overallrates, they don’t change the relative importance of low-lying strength with respect to theresonant strength considered previously. We therefore simply use ην = 3 and adjust T sothat the average neutrino energy is about 11 MeV.5Contributions of the low-lying GT and forbidden strength to the spectrum-averagedcross sections for two representative nuclei appear in figures 3 and 4. These particularplots result from the δ-function interaction and the prescription EGT = EIA, but thenotable features are always the same. Figure 3 shows the contribution of low-lying GTstrength in 78Ni and 190Gd to f(Eν)σ(Eν) as a function of neutrino energy. The solidline represents the cross sections obtained with the full QRPA GT strength distributioncalculated here, while the dashed line represents those calculated as in Ref. [5], with aGaussian GT strength distribution centered at the energy of the isobar analog state andhaving a width of 5 MeV (the GT distributions themselves are compared in figure 2). Thelow-lying strength increases the cross sections for neutrinos below 15 MeV substantially,especially when the neutron excess is very large. We may even be underestimating theextent of the increase. The GT distribution shown in figure 1 has considerably less low-lying strength than do most stable nuclei. The reason is that near the drip line theresonance is higher in energy and particle-hole force must therefore be stronger, with theside effect that more GT strength is pulled into the resonance. If the GT distributionsat N = 82 and 126 looked more like those in stable nuclei the allowed contribution toneutrino rates would go up even further.Figure 3: GT contributions to spectrum-averaged neutrino-nucleus scattering cross sec-tions for (a) 78Ni and (b) 190Gd. The solid line is the full GT contribution calculated hereand the dashed line the contribution of the Gaussian GT distribution.6Figure 4 shows the role played by forbidden transitions in the same two nuclei; itcompares the total GT contribution to f(Eν)σ(Eν) just discussed with the contributionsof the 1− and 2− transitions. The low-lying forbidden strength ought to increase withneutron excess and, indeed, figure 4 shows that the forbidden portion of the cross sectionin 190Gd is clearly larger than that in 78Ni. In general, the unusual low-lying forbiddenstrength provides about 5–10% of the total spectrum-averaged cross section when N = 50,10–20% when N = 82, and 20–35% when N = 126.Figure 4: Contributions to spectrum-averaged neutrino-nucleus scattering cross sectionsfor (a) 78Ni and (b) 190Gd. The solid line is the GT contribution and the dashed anddot-dashed lines are the 1− and 2− first-forbidden contributions.Table 1 compares 〈σν〉total, our total cross section, with 〈σν〉0, that calculated underthe assumption that only the IA state and GT resonance (with width 5 MeV) contribute,in several nuclei. The factors in the table are calculated with EGT = EIA; rates withEGT < EIA are all of course larger, but the ratio of 〈σν〉total to 〈σν〉0 remains roughlyconstant for the N = 50 and N = 82 nuclei. (For the N = 126 nuclei, it drops by 10–20%because the parameter δ, which lowers the GT resonance, grows rapidly with neutronexcess for large A).More detailed results for three representative nuclei and the severalprescriptions described above appear in table 2, where 〈σν〉total is broken up into allowedand forbidden parts. In the heaviest nuclei our estimates are more than twice as large asthose made before, and we believe it more likely that they are too small than too large.7Table 1: Comparison of total spectrum-averaged νe cross sections as calculated in thiswork, 〈σν〉total, to allowed-only cross sections, 〈σν〉0, calculated with the Gaussian GTstrength distribution (see text). All cross sections are in units of 10−41cm2.Z N A 〈σν〉total/〈σν〉026 50 76 1.628 50 78 1.630 50 80 1.640 82 122 1.842 82 124 1.846 82 128 1.848 82 130 1.762 126 188 2.364 126 190 2.266 126 192 2.268 126 194 2.1Table 2: Total spectrum-averaged νe cross sections for three representative nuclei, withtwo different forces and two prescriptions for the position of the GT centroid. All crosssections are in units of 10−41cm2.Z N A prescription force 〈σν〉0 〈σν〉allowed 〈σν〉forbidden 〈σν〉total28 50 78 EGT = EIA δ-function 13.6 19.9 1.5 21.4G-matrix 13.6 22.4 1.4 23.8EGT = EIA + δ δ-function 16.1 24.1 1.8 25.9G-matrix 16.1 26.6 1.4 28.042 82 124 EGT = EIA δ-function 19.6 29.4 4.9 34.3G-matrix 19.6 30.1 7.0 37.1EGT = EIA + δ δ-function 29.7 41.3 6.3 47.6G-matrix 29.7 42.0 7.0 49.064 126 190 EGT = EIA δ-function 20.6 34.3 11.9 46.2G-matrix 20.6 38.9 21.3 60.2EGT = EIA + δ δ-function 35.0 49.0 14.7 63.7G-matrix 35.0 49.4 21.3 70.7To sum up, we find that low-lying GT and first forbidden strength increases the rate ofneutrino scattering from very neutron-rich nuclei, usually by factors of two or more. Ourcross sections are still uncertain, in part because of our relatively poor understanding ofnuclei far from stability. More careful and accurate calculations are possible even withoutmore data, but in our view the effort is not warranted yet because of the still quite largeuncertainty in, e.g., the flux of hot-bubble neutrinos. No matter what the sources andamounts of uncertainty, however, our results point to a uniform and significant increasein capture rates over previous estimates. Any serious calculation must include the effects8we describe, and neutrino capture will play a larger role in the r-process than one wouldotherwise believe.References[1] S.E. Woosley, J.R. Wilson, G.J. Mathews, R.D. Hoffman, and B.S. Meyer, Astrophys.J. 433, 229 (1994).[2] G.M. Fuller and B.S. Meyer, Astrophys. J. 453, 792 (1995).[3] Y.-Z. Qian, Nucl. Phys. A 621, 363 (1997).[4] G.C. McLaughlin and G.M. Fuller, Astrophys. J. 455, 202 (1996).[5] Y.-Z. Qian, W.C. Haxton, K. Langanke, and P. Vogel, Phys. Rev. C 55, 1532 (1997).[6] J.D. Walecka, in: Muon Physics, Vol. 2, eds. V.W. Hughes and C.S. Wu (Academic,New York, 1975), p. 113.[7] J.A. Halbleib and R.A. Sorensen, Nucl. Phys. A 98, 542 (1967).[8] D. Cha, Phys. Rev. C 27, 2269 (1983).[9] G.F. Bertsch, The Practitioner’s Shell Model (American Elsevier, New York, 1972).[10] J. Engel, P. Vogel, and M.R. Zirnbauer, Phys. Rev. C 37, 731 (1998).[11] A. Hosaka, K.-I. Kubo, and H. Toki, Nucl. Phys. A 444, 76 (1985).[12] H. Homma, E. Bender, M. Hirsch, K. Muto, H.V. Klapdor-Kleingrothaus, and T.Oda, Phys. Rev. C 54, 2972 (1996).[13] K. Nakayama, A. Pio Galeao, and F. Krmpotic, Phys. Lett. B 114, 217 (1982).[14] R.D. Smith and J. Wambach, Phys. Rev. C 38, 100 (1988).[15] D.J. Horen, C.D. Goodman, D.E. Bainum, C.C. Foster, C. Gaarde, C.A. Goulding,M.B. Greenfield, J. Rapaport, T.N. Taddeucci, E. Sugarbaker, T. Masterson, S.M.Austin, A. Galonsky, W. Sterrenburg, Phys. Lett. B 99, 383 (1981).9

Neutrino capture by r -process waiting-point nuclei

Neutrino capture by r-process waiting-point nuclei

Neutrino capture by r-process waiting-point nuclei

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