Enhancement of the Triple Alpha Rate in a Hot Dense Medium
aa r X i v : . [ nu c l - t h ] A ug Enhancement of the Triple Alpha Rate in a Hot Dense Medium
Mary Beard ∗ , Sam M. Austin † , and Richard Cyburt ‡ Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USAJoint Institute for Nuclear Astrophysics National Superconducting Cyclotron LaboratoryJoint Institute for Nuclear AstrophysicsMichigan State University, 640 South Shaw Lane,East Lansing, MI 48824-1321, U.S.A. (Dated: October 6, 2018)
Abstract
In a sufficiently hot and dense astrophysical environment the rate of the triple-alpha (3 α ) reactioncan increase greatly over the value appropriate for helium burning stars owing to hadronicallyinduced de-excitation of the Hoyle state. In this paper we use a statistical model to evaluate theenhancement as a function of temperature and density. For a density of 10 gm cm − enhancementscan exceed a factor of one-hundred. In high temperature/density situations, the enhanced 3 α rateis a better estimate of this rate and should be used in these circumstances. We then examinethe effect of these enhancements on production of C in the neutrino wind following a supernovaexplosion and in an x-ray burster.
PACS numbers: 26.20Fj;26.30.-k;25.40.-h ∗ Deceased † [email protected] ‡ Presently at Concord University, Athens, WV; [email protected] α ) process that converts He into C is one of the fundamental reac-tions in astrophysics; its rate influences the stellar production of many elements [1–3]. Forthe stellar conditions typically encountered in helium burning stars, the 3 α rate is propor-tional to the radiative width of the 7.65 MeV 0 + state (the Hoyle state) in C and is knownto within about 10%. This will be true whenever the triple-alpha reaction proceeds throughresonant processes. Recent theoretical calculations, [4, 5] have shown that this is the casefor
T > K . At large values of the temperature ( T ) and density ( ρ ), however, the widthof the Hoyle State is increased by particle induced de-excitation leading either to the groundstate or to the first excited 2 + state of C at 4.44 MeV as shown in Fig. 1. This increases(enhances) the rate of the 3 α process. A principal motivation of this paper is to determinethese enhancements using presently available techniques and investigate whether they mightbe large enough to influence other astrophysical phenomena that occur at high T and ρ . Forexample, might the enhanced rates produce sufficient seeds in the neutrino driven wind ofa core-collapse supernovae to make a successful r process less likely.These enhancement processes have been studied theoretically in the past. Shaw andClayton [6] examined electromagnetic effects: Coulomb excitation by alpha particles andelectron induced processes. The effects were found to be much smaller than those for thenuclear reaction induced effects considered here, and unimportant for densities less than 10 gcm − . They will not be discussed further in this paper. Truran and Kozlovsky [7] considerednuclear induced processes but before there were reliable measurements or estimates of therelevant cross sections.Following these theoretical estimates, experimental studies of inelastic proton [8] and al-pha scattering [9] from the ground state of C to the Hoyle state were carried out. Thecorresponding enhancements were calculated from the inverse rates that correspond to thesecross sections. These preliminary studies indicated that the enhancements could be signif-icant at the temperatures and densities encountered in supernovae. There were, however,no reliable estimates of cross sections for neutron inelastic scattering that because of theabsence of Coulomb effects would be expected to dominate the enhancements. Nor couldexperiments yield estimates of inverse rates for processes that lead from the Hoyle state tothe first excited 2 + state in C. Such cross sections are not experimentally measurable andmust be obtained from theoretical estimates.In this paper we attempt to deal with these deficiencies. Although any particle present2
IG. 1: (Color on line). The radiative width for typical helium burning stars results from thegamma ray cascade from the Hoyle state through the the 4.44 MeV 2 + state and the (muchweaker) pair decay of the Hoyle state. The additional contributions we estimate, shown as soliddownward arrows, are mediated by inelastic scattering of protons, neutrons and alpha particlesleading from the Hoyle state to the 2 + state and to the ground state. can cause an enhancement, the particle densities and temperatures required for significantenhancements are large, so that in practice it is necessary to consider only the effects ofneutrons, protons, and alpha particles.We provide the relevant reaction rate background; describe the experimental inelasticcross sections and the theoretical calculations used to generate cross sections not availablefrom experiment; and present the enhancements obtained as a function of temperature ata density of 10 g cm − . Since the enhancements are directly proportional to the density,these results are sufficient for applications to astrophysical phenomena. Finally we presentpreliminary estimates of the effects of the enhancements in astrophysical applications anddiscuss the experimental and theoretical advances that could improve the accuracy of theseinitial results.The procedures used follow those described in Davids and Bonner [8]. For the two bodyreactions induced by neutrons, the reaction rate of C with number density N C , andneutrons with number density N n , is given by r = N n N C < σv > cm − sec − (1)where σ is the total reaction cross section for exciting the Hoyle state from the initial state(g.s. or 4.44 MeV 2 + state), v is the relative velocity, and the average is over the Maxwellian3elocity distribution of the two species. For inelastic neutron scattering on C < σv > nn ′ = (cid:18) πµ (cid:19) / (cid:18) kT (cid:19) − / Z ∞ E ′ σ n,n ′ ( E ′ ) exp ( − E ′ /kT ) dE ′ (2)For our purposes we need the inverse of the reaction exciting the Hoyle state, namely < σv > n ′ n = (cid:18) I + 12 I ′ + 1 (cid:19) exp ( − Q/kT ) < σv > nn ′ (3)Here I and I ′ are the spins of the initial and final states for forward excitation of C: 0 + and 0 + for the ground state to Hoyle state transition, and 2 + and 0 + for transitions fromthe 4.44 MeV 2 + state to the Hoyle state, resp. Q = -7.654 (-3.215) for excitation of theHoyle state, from the g.s.(2 + state). The life time for inelastic neutron de-excitation is τ n ′ n ( C Hoyle ) = ( N n < σv > n ′ n ) − sec. (4)We define R as the ratio of the radiative lifetime to the particle-induced lifetime. R = τ γ /τ n ′ n = τ γ N n < σv > n ′ n . (5)Inserting the experimental value τ γ = 1 . × − sec [10, 11], the values of the relevantconstants, and expressing the energy as kinetic energy above threshold, one obtains R = k n ρ n T − . C spin Z ∞ σ nn ′ ( E )( E − Q ) exp ( − . E/T ) dE (6)where E is the c.m. energy above threshold, ρ n is the neutron density in g cm − , T = T / ,and σ nn ′ ( E ) is the cross section in mb. For transitions to the Hoyle state from the groundstate (4.44 MeV 2 + state), C spin = 1(5).In all these equations, for proton inelastic scattering substitute p for n and p ′ for n ′ ; foralpha particle inelastic scattering substitute α for n and α ′ for n ′ . The multiplying constantsare: k n = 6 . × − ; k p = 6 . × − ; k α = 9 . × − .Experimental values are available in the literature for a few of the inelastic cross sections,but they are sparse and often do not extend low enough in energy toward the reactionthresholds. Most important, for the neutron induced reactions expected to dominate atrelatively low temperatures there are no results in the relevant energy range of up to a fewMeV above threshold. Nor are there estimates, for any projectile, of the cross sections fromthe 4.44 MeV state to the Hoyle State. 4or the important energies near threshold, compound nuclear processes are expected todominate and one might first consider employing an R-matrix description. Unfortunately, alarge number of levels, some narrow and some broad, influence these cross sections [12, 13] sothat any analysis will be complex, and will, at the moment, lack the necessary experimentalinformation. The best of the available analyzes [13], presently does not extend to thecompound nucleus energies we require, partially because data is insufficient or contradictory.It appears that significant improvements in the R-matrix approach will take significant effortand time [14]. Many of these comments apply to other possible approaches.The best available option is to follow the standard approach in astrophysics (see, forexample, a description of the JINA REACLIB database [15]), of obtaining unmeasuredcompound nuclear cross sections from the statistical Hauser-Feshbach (HF) model [16, 17].For this purpose we use the reaction code TALYS (version 1.8) [18], a widely acceptedmodern implementation of the HF model. The underlying principle of this statistical modelpicture is that the interaction of a target and projectile result in the formation of a compoundnucleus at a sufficiently high excitation energy that individual nuclear levels can be treatedin an average manner and that the system is fully equilibrated before it decays into the finalreaction channels. The probabilities for the creation and decay of the compound nucleus areexpressed in terms of the transmission functions for its formation and break up. For particlechannels, the transmission functions are obtained from optical model calculations. Asidefrom the transmission functions, one requires level densities, width fluctuation correctionsand other descriptive details of the target.For this light system one cannot expect a priori that the basic assumptions outlinedabove are well fulfilled; we use the HF approach because there are no realistic alternatives. Arelated uncertainty lies in the choice of a particular optical potential for the HF calculations.We used the default models of TALYS (version 1.8) [18]: for protons and neutrons, a globaland local potential based on the Koning and Delaroche model [19]; and for alpha particles,the potentials of Avrigeanu, et al. [20].To evaluate the resulting uncertainties in a conservative manner, we calculate all crosssections in a systematic fashion, using the TALYS default parameter values, compare theresults to the available data and, thereby, assess the reliability of the model. Cross sectionswere also calculated for n, p and α inelastic reactions using different optical model parameters(three for protons and neutrons and nine for alpha particles, as cited in [18]) The default5 C R O SS S E C T I O N ( m b ) NEUTRONS
PROTONSE lab (MeV) C R O SS S E C T I O N ( m b ) gs - 2 + gs - HS2 + - HSgs - 2 + gs - HS2 + - HS FIG. 2: (Color on line) Inelastic scattering cross sections for neutrons (top panel) and protons(bottom panel). Results for the default OMPs described in the text are shown as solid lines,and the maximum and minimum cross sections for other potentials noted in the text are includedwithin the shaded areas. For each projectile results are shown for scattering leading from: theground state to the 4.44 MeV 2 + state, the ground state to the 7.65 MeV Hoyle State, and the4.44 MeV 2 + state to the Hoyle state. Available experimental results are shown as discrete points,for neutrons from [21, 22], for protons from [8, 23–25]. results and the spread of results for the other models are shown in Fig. 2. Although they arenot directly relevant to our enhancement calculations, we also show the cross sections forthe transition from the ground state to the 2 + state at 4.44 MeV since more experimentaldata are available for this strong transition.Cross sections calculated with the various OMPs differ by less than 30% up to approxi-mately 20 MeV; cross sections within 2 MeV of threshold generally dominate the enhance-ments. The calculations generally lie within about a factor of two to three of the sparse6xperiential data, sometimes higher and sometimes lower; the energy dependence of thecross sections is generally reproduced. The single exception is for the resonance in protonscattering to the Hoyle state. This overall level of agreement is similar to that obtained forheavier nuclei.The energy dependencies of the cross sections for neutrons have well known behaviors thatdiffer from those of the charged particles because of the absence of a coulomb barrier. Theforward, endothermic, cross sections shown here vary approximately as E ′ / . The inverse,exothermic, cross sections exhibit the well known 1 /E ′ / or 1 /v behavior. As we shall see,this can lead, for neutrons, to large 3 α enhancements even at relatively low temperatures.The enhancements were calculated for the inverse of each of the contributing transitionrates: gs → .
65 (Hoyle), and 4 . → .
65 state for incident neutrons, protons and alphaparticles, using equation (6). For the case of the proton inelastic scattering to the Hoylestate we used the experimental data up to 2.30 MeV; at lower energies the cross sectionhas strong resonances that are not reproduced by the TALYS calculations. Otherwise thedefault TALYS cross sections were used. The cross sections were (accurately) fitted withcubic splines and the integrals performed with the MathCad routine over the energy rangefrom near threshold to 10 MeV above threshold.Most contributions to the enhancements are for energies less than 2 MeV above threshold;except for the small alpha enhancements, all have converged to within 3% by 5 MeV, even inthe most demanding case: T = 10. Thus, factor of two or three cross section uncertaintiesat higher energies shown in Fig. 2 have small effects on the ratios. The calculated ratios areshown in Fig. 3.As expected, the neutron induced enhancements are largest, proton induced enhance-ments are smaller, and the alpha induced enhancements smaller still. Except for the protoninduced transitions, where the larger ground state enhancement owes to the large resonantcross sections at low energies, the enhancements from from 4.44 MeV 2 + state to the Hoylestate are larger because of the influence of a spin factor of five.For applications, it is the sums of the two cross sections of each projectile that arerelevant. We see from Fig. 3 that these enhancements can be large for sufficiently large T and ρ . For neutrons only, enhancements are larger for small T . Based on the cross sectionuncertainties, it seems a fair summary to conclude that the enhancements are known towithin about a factor of 2 to 3. Thus for example, the enhancement factor for neutrons at7 R aa R pp , R nn T9Enhancement Factors R
Rnn
Rpp R aa FIG. 3: (Color on line) Ratios of the rate induced by the indicated transitions to the measured(gamma + pair decay) rate. The ratios were calculated for a particle density of 10 gm cm − . Thealpha ratios are plotted on the expanded scale on the right hand ordinate. T = 1 .
0, is 115, the sum of the two values for neutrons shown in Fig. 3. For a factor of threeuncertainty, it would lie between 38 and 345. Such large enhancements should be taken intoaccount in calculations at high T , ρ .To investigate the magnitude of these effects in an astrophysical scenario, we calculatedthe enhancements for an adiabatic model [26] as implemented by Schatz, et al. [27]. Inthis model the initial protons and neutrons are in nuclear statistical equilibrium and arelater incorporated into alpha particles, then into C and eventually into heavier seeds. Theresults in Fig. 4 show that these enhancements are large.It has usually been found that in this model the ααn process leading to Be dominatesthe flow into C, but if the 3 α process is sufficiently enhanced and competes strongly, theoverall flow into heavy seeds may increase, leading to a larger number of seeds, a smallerneutron to seed ratio, and a less robust r-process. We have made a preliminary estimatebased on the above adiabatic model as summarized in Fig. 4, and find that the enhanced3 α rate dominates the production of C, presumably leading to a larger seed abundance.This would remain true if the enhanced rates are a factor of three smaller than calculatedhere. On the other hand, if there were a strong resonance at low neutron energies, as thereis for the proton channel, the enhancements might be still larger. We intend to investigatethese possibilities systematically in future work using more realistic models [28].Another site where significant enhancements of the 3 α rate might be important is ac-8 R pp + R nn + R a l ph a N n , N p , N a l ph a Time (sec)
N protonsN neutronsN alphasRpp+ Rnn + Ralpha
Rpp + Rnn +Raa + FIG. 4: (Color on line) On the left hand ordinate are shown, as a function of time, the neutron,proton and alpha particle densities calculated following the adiabatic model. Initially the tempera-ture was set at T = 9 to insure nuclear statistical equilibrium, the entropy at S = 90 k B , Y e = 0 . − . The calculated overall enhancement, owing to induced de-excitationof the Hoyle state by protons, neutrons and alpha particles is shown on the right hand ordinate. creting neutron stars and the resulting x-ray bursts. An increased formation of C couldincrease energy generation during the giant outbursts seen in some x-ray bursters. However,the densities and temperatures seen in two possible models of the process [29] indicate thatthe enhancement would reach a maximum of 30% during the onset of the burst.One might ask whether the enhancement processes considered here could affect otherreaction rates. Indeed, any process involving gamma decay of a state with a larger particledecay width will have a rate proportional to the radiative width and be susceptible toenhancement. But since radiative widths are usually large (compared to that of the Hoylestate) enhancements are less likely to be important. That is the case for the ααn processdiscussed above.The situation for the present then appears to be that, in situations where the densitiesand temperature are large, a reasonable estimate of the triple alpha reaction rate is givenby the enhancement factor of Fig. 3. Uncertainties are probably a factor of three, andthe enhancements could presumably be larger or smaller. If such enhancements cause asignificant change in calculated astrophysical phenomena, a significant experimental andtheoretical effort would then be warranted to better constrain the enhancements.The neutron cross section from the ground state to the Hoyle state can, in principle, be9easured, but the cross sections are relatively small and the measurements will be difficult.If successful, in the context of the present calculations the major uncertainty would thenarise from the uncertainties in the input cross sections for the transitions from the 2 + stateto the Hoyle state where there is no constraint from experiment. Statistical approaches suchas those implemented in the TALYS code are typically assumed to be uncertain by factorsof two to three, which is consistent with the differences from experiment observed in Fig. 2.A detailed R-Matrix approach, coupled with better understanding of the relevant levelstructure of the mass-13 nuclei, could probably yield better results, and is highly desirable.But such a detailed model is not available at present, nor probably, because of the largeeffort that will be involved, for some time in the future.We thank Hendrik Schatz for providing details of his calculations of adiabatic expansionsand x-ray bursters, and acknowledge communications and conversations with James deBoer,Christian Diget, Hans Fynbo, Alex Heger, Luke Roberts, Hendrik Schatz, Henry Weller, andMichael Wiescher. Research support from: US NSF; grants PHY08-22648 (JINA), PHY-1430152 (JINA-CEE), PHY11-02511. [1] Falk Herwig, Sam M. Austin, and John C. Lattanzio, Phys. Rev. C , 025802 (2006).[2] C. Tur, A. Heger, and S. M. Austin, Astrophys. J. , 821 (2007).[3] C. West, A. Heger and S. M. Austin, Astrophys. J. , 2(2013).[4] N. B. Nguyen, F. M. Nunes, and I. J. Thompson, Phys. Rev. C , 054615 (2013) .[5] Hiroya Suno, Yasuyuki Suzuki, and Pierre Descouvemont, Phys. Rev. C , 054607 (2016).[6] Peter B. Shaw and Donald D. Clayton, Phys. Rev. , 1193 (1967).[7] J. W. Truran and B.-Z. Kozlovsky, Astrophys. J. , 1021 (1969).[8] Cary N. Davids and T. I. Bonner, Astrophys. J. , 405 (1971).[9] J. F. Morgan and D. C. Weisser, Nucl. Phys. A151 , 561 (1970).[10] M. Freer and H. O. U. Fynbo, Prog. Part. Nucl. Phys. , 1 (2014).[11] Sam M. Austin, unpublished compilation; values differ slightly from [10] because Seeger andKavanagh, their reference 76, is excluded from this calculation and because the input valuesare not rounded. The difference is much smaller than the uncertainties.[12] F. Ajzenberg-Selove, Nucl. Phys. A , 1 (1991).
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