Suppression of the stellar enhancement factor and the reaction 85Rb(p,n)85Sr
T. Rauscher, G. G. Kiss, Gy. Gyürky, A. Simon, Zs. Fülöp, E. Somorjai
aa r X i v : . [ a s t r o - ph . S R ] A ug draft Suppression of the stellar enhancement factor and the reaction Rb(p,n) Sr T. Rauscher, ∗ G. G. Kiss, Gy. Gy¨urky, A. Simon, Zs. F¨ul¨op, and E. Somorjai Department of Physics, University of Basel, CH-4056 Basel, Switzerland Institute of Nuclear Research (ATOMKI), H-4001 Debrecen, POB.51., Hungary (Dated: October 29, 2018)It is shown that a Coulomb suppression of the stellar enhancement factor occurs in many endother-mic reactions at and far from stability. Contrary to common assumptions, reaction measurementsfor astrophysics with minimal impact of stellar enhancement should be preferably performed forthose reactions instead of their reverses, despite of their negative Q value. As a demonstration, thecross section of the astrophysically relevant Rb(p,n) Sr reaction has been measured by activationbetween 2 . ≤ E c . m . ≤ .
96 MeV and the astrophysical reaction rates at p process temperaturesfor (p,n) as well as (n,p) are directly inferred from the data. Additionally, our results confirm apreviously derived modification of a global optical proton potential. The presented arguments arealso relevant for other α - and proton-induced reactions in the p , rp , and νp processes. PACS numbers: 26.50.+x Nuclear physics aspects of novae, supernovae, and other explosive environments,24.60.Dr Statistical compound-nucleus reactions, 27.50.+e 59 ≤ A ≤ I. INTRODUCTION
Astrophysical reaction rates are central to tracingchanges in the abundances of nuclei by nuclear reactions.They provide the temperature- and density-dependentcoefficients entering reaction networks, the large sets ofcoupled differential equations required to study nucle-osynthesis and energy generation in astrophysical envi-ronments. The reaction rates are computed from reac-tion cross sections which, in turn, may be predicted intheoretical models or extracted from experiments. In ad-dition to the difficulties arising in the determination ofthe cross sections, the conversion to reaction rates is fur-ther complicated by modifications of the rates in a hotplasma and the fact that the rates of forward and reverserate for the same reaction have to be consistent to ensurenumerical stability and proper equilibrium abundances athigh temperature. Both issues can be addressed at onceby accounting for the thermal population of target stateswhich leads to stellar rates obeying a reciprocity relationbetween forward and reverse rate. Using this reciprocity,knowledge of the rate in only one direction is neededbecause the other reaction direction can be directly com-puted from that rate, thus ensuring consistency.For numerical reasons, further elaborated in Sec. II B,it is usually preferable to start from the rate of a reactionwith positive reaction Q value when computing the ratefor its inverse reaction. Even more importantly, experi-mentalists want to determine rates as close as possible tothe actual stellar rates, i.e. rates with minimal thermalpopulation effects of the target. Again, it can be ar-gued that this is the case for exothermic reactions. Thisled to the commonly applied rule that measurements ofexothermic reactions are more important than those ofendothermic ones. In this paper we show that there is a ∗ Electronic address: [email protected] considerable number of reactions for which a suppressioneffect brings the stellar rate of an endothermic reactioncloser to the laboratory value than its exothermic coun-terpart.As an example of how to exploit this suppression ef-fect and to obtain stellar rates from a measurement ofan endothermic reaction, we experimentally studied thereaction Rb(p,n) Sr, having Q = − .
847 MeV. Theimportance of the reaction is manifold. In the last sev-eral years a number of proton capture cross section mea-surements with relevance for γ process studies have beencarried out (see, e.g., [1] and references therein). The γ process was shown to synthesize p nuclides (proton-rich isotopes not accessible to the s and r processes) bya series of photodisintegrations of stable nuclides in hotlayers of massive stars [2, 3, 4, 5]. Recently, systematic γ process simulations found not only that photodisintegra-tion reactions are important but also that (p,n) reactions,and in particular Rb(p,n) Sr, strongly influence the fi-nal p abundances [6]. Additionally, this reaction is wellsuited to test the optical potential used for calculatingthe interaction between protons and target nuclei.We commence by outlining the theoretical backgroundregarding stellar rates and the suppression effect in Secs.II A and II B. The results of a large-scale study of the ef-fect in the full extension of the nuclear chart are discussedin Sec. II C. Focusing on the reaction Rb(p,n) Sr,their relevance is discussed in Sec. III A, the experimentaldetails are provided in Secs. III B–III D, and the astro-physically relevant rates are derived in Sec. III G. Ad-ditionally, Sec. III F discusses implications of our newexperimental results for the proton optical potential. Fi-nally, a summary is given in Sec. IV.A brief account of our findings was already given in[7]. The present follow-up paper expands the discussionand also provides additional results in all parts of thisinvestigation. π π π J J J ρ A A π kCkC J π ρ Fcap Q F Q nF J π nF C F J FIG. 1: Schematic view of the transitions (full arrows de-note particle transitions, dashed arrows are γ transitions) ina compound reaction involving the nuclei A and F, and pro-ceeding via a compound state (horizontal dashed line) withspin J k C and parity π k C in the compound nucleus C. The reac-tion Q values for the capture reaction (Q cap ) and the reactionA → F (Q F ) are given by the mass differences of the involvednuclei. The effective cross section σ eff (Eqs. 4 and 6) for areaction type is a sum over all energetically possible transi-tions to bound states (capture: in nuclei A and C; otherwise:in nuclei A and F) from the compound level as shown here(see text for details). In each nucleus, a number of low-lyingstates with given spin J and parity π is explicitely specified.Above the last state, transitions can be computed by inte-grating over nuclear level densities (shaded areas). In stellarcross sections σ ∗ all transitions are additionally weighted by aBoltzmann distribution factor depending on the stellar tem-perature, spin, and the excitation energy of the involved state(see Eq. 3). II. SUPPRESSION OF THE STELLARENHANCEMENTA. Stellar reaction rates
The stellar enhancement factor (SEF) f is defined asthe ratio of the stellar rate r ∗ relative to the ground staterate r g . s . [8] f = r ∗ r g . s . = r ∗ r lab . (1)The rate r lab derived from cross sections σ lab measuredin the laboratory is the same as r g . s . because so far allexperiments use target nuclei in their ground states. TheSEF is a measure of the influence of the thermally excitedtarget states in the hot plasma.Astrophysical reaction rates are usually defined as giv-ing the number of a specific reaction occurring per time.Here, we constrain ourselves to two-body reactions ofnuclei and nucleons. The concept of the stellar ratessuppression introduced below is easily extended to otherreaction types. Reaction cross sections are folded withthe energy distribution of the interacting nuclei to ob-tain the reaction rate. The energy distributions of nuclei and nucleons in an astrophysical plasma follow Maxwell-Boltzmann distributions in most applications, thus yield-ing [8] r i = n n δ F ( kT ) / Z ∞ σ i Ee − EkT dE (2)= n n δ R i for reactions proceeding from target state i with reactioncross section σ i , where n , n are the number densitiesof the interacting nuclei, T is the plasma temperature, k denotes the Boltzmann constant, and F is a renormal-ization factor F = p / ( πµ ) with µ = A A / ( A + A )being the reduced mass number A .When nuclei are in thermal equilibrium with their en-vironment, their excited states are populated accordingto a Boltzmann factor [8] P i = (2 J i + 1) e − EikT P n (2 J n + 1) e − EnkT , (3)with P i , J i , E i denoting the relative population, spin,and excitation energy of state i , respectively. Each ofthe states is bombarded with Maxwell-Boltzmann dis-tributed projectiles which would require to have a sepa-rate rate for each target state weighted by the populationfactor of the state i from which the reaction proceeds.It was shown in [9] (see also [10]) that by making use ofthe reciprocity theorem for nuclear reactions and detailedbalance (assuming thermalization of both initial and fi-nal states of a reaction), for compound reactions the rateequation can be simplified to R ∗ = F ( kT ) / X i (cid:18)Z ∞ P i ( T ) σ i (cid:0) E i (cid:1) E i e − EikT dE i (cid:19) = (cid:0) J + 1 (cid:1) F ( kT ) / G Z ∞ σ eff ( E ) Ee − EkT dE , (4) r ∗ = n n δ R ∗ . (5)In order to avoid additional computations caused by thepopulation coefficients and also to avoid having a temper-ature dependent stellar cross section, the effective crosssection σ eff was introduced above, which sums over allbound states in the initial and final system (denoted by i and j , respectively; the energetics of the transitions isshown in Fig. 1) [10]: σ eff = X i X j σ ij . (6)This is a theoretical construct (as any measurementwould always proceed on a certain initial state and thusneglect the sum over target states) but it is useful in tworespects. Firstly, it simplifies the computation of the rateand therefore is utilized in all astrophysical compound re-action codes. Secondly, it allows us to easily find a reci-procity relation between forward and inverse rate by re-membering that Eσ eff obeys reciprocity between forwardand inverse reaction due to detailed balance. It should benoted that only stellar reactivities R ∗ (and thus stellarrates r ∗ ) obey reciprocity (as long as detailed balance isapplicable) whereas rates derived from ground state crosssections σ lab = P j σ j do not, unless the SEF is equalto unity in the given direction.For reactions 1 + 2 → R ∗ = (2 J + 1)(2 J + 1) (cid:18) A A A A (cid:19) / G G e − Q kT R ∗ , (7)where Q is the reaction Q value, J denote ground statespins, and G are nuclear partition functions summingover states i and integrating over a level density ρ abovethe last discrete state m included [10, 11]: G ( T ) = m X i =0 (2 J i + 1) e − EikT (8)+ E max Z E m X J,π (2 J + 1) e − EkT ρ ( E, J, π ) dE . This partition function also appears in Eq. 4 where it issufficient to compute it once and separately from the rateintegration.Stellar capture reactions 1 + 2 → γ are related tostellar photodisintegration by [10, 11] R ∗ γ = (2 J +1) (cid:18) A A A (cid:19) / (cid:18) kT π ~ (cid:19) / G G e − Q kT R ∗ . (9) B. Reaction Q value and stellar enhancementfactor Figure 1 shows a sketch of the energetically allowedtransitions included in the effective cross section definedby Eq. (6). It is obvious that there are more transitionspossible to and from states of the nucleus being the finalnucleus in a reaction with positive Q value. Therefore,assuming a similar level structure in all involved nuclei, itis expected that the SEF (see Eq. 1) of a given reactionwill be smaller for the exothermic direction f forw thanfor the endothermic one f rev (here we define the forwardreaction to be the one with positive Q value and thereverse reaction having negative Q value): f rev > f forw . (10)This is especially pronounced in photodisintegration re-actions due to the many possible γ transitions [12, 13].In consequence, aiming at performing a measurement asclose as possible to the stellar value, an exothermic reac-tion should be chosen. Another impact of the Q value is found by inspectionof Eqs. (7) and (9) where the Q value appears in an expo-nential. For numerical consistency and to obtain properequilibrium abundances when forward and reverse reac-tion are both fast and in equilibrium, reaction networkcodes avoid employing separate rates for the two direc-tions but rather make use of these equations. Taking anendothermic reaction as starting point for application ofthe equations would lead to a large value of the exponen-tial term, amplifying any numerical errors inherent in theoriginal rate and in the Q value. This is mainly impor-tant when dealing with rate fits. In many astrophysicalreaction network codes, the rates are implemented notas large tables but as fits with a smaller number of pa-rameters per reaction. Any deficiency in the fit would beamplified when computing an exothermic rate from anendothermic one.For the above reasons, it was commonly assumed thatit is always preferable to determine the cross section andrate of an exothermic reaction and not those of an en-dothermic one. Here, we want to correct that notion byshowing that there are cases for which f rev < f forw . (11)The basic idea is to realize that although there are moretransitions energetically possible to the final states ofexothermic reactions, some of them may be suppressedand thus not contributing. Of course, it is obvious thatnot all transitions are of equal strength. Quantum me-chanical spin and parity selection rules and centrifugalbarriers (or lack thereof) may prefer certain transitionsover others. This will be important in reactions withsmall | Q | and in nuclei with large level spacings. In bothcases, only a small number of transitions will be possibleand the spins can give larger weight to an even smallersubset. However, for reactions with sizeable Q values orinvolving nuclei with high level densities this suppressiondue to spins will not be sufficient because there will al-ways be a number of states with matching spins.Transitions to higher lying excited states of a nucleusproceed at lower relative energy. Except for s-wave neu-trons, also transitions at lower relative energy will beweaker than those at larger relative energy. If the sup-pression of transitions with smaller relative energy is dif-ferent in the entrance and exit channel of the reaction,this may also result in f rev < f forw . The strongest sup-pression for charged particles is due to the Coulomb bar-rier. Having different Coulomb barriers in the entranceand exit channel, e.g. in (n,p) or (p, α ) reactions, canmore strongly suppress the transitions to the nucleuswith higher Coulomb barrier than to the one with lowerCoulomb barrier. With respect to Fig. 1 and assuming,e.g., a reaction A(n, α )F this means that most transitionsto states in nucleus F are suppressed and the contribut-ing transitions may be fewer than those accessing statesin nucleus A.This Coulomb suppression of the SEF is a general prin-ciple almost independent of nuclear structure and will act -10-8-6-4-2 0 10 20 30 40 50 60 70 80 R ea c t i on Q V a l ue [ M e V ] Target Charge Number Z(p,n)( α ,n) FIG. 2: Reaction Q values for (p,n) and ( α ,n) reactions with f rev < f forw . for a large range of nuclei. Whether the suppression isstrong enough to yield f rev < f forw depends on the sizeof the Q value relative to the Coulomb barrier, i.e. theeffect can occur in a reaction provided that there are dif-ferent Coulomb barriers in the entrance and exit channeland | Q | is low compared to the Coulomb barrier. Thestrongest impact is to be expected when the forward reac-tion involves neutrons in the entrance channel which forma compound state by s-waves on excited target states andcharged particles experiencing a high Coulomb barrier inthe exit channel. As discussed in Sec. III G the reaction Rb(p,n) Sr is an excellent example for such a case. Aquantitative exploration of the suppression effect acrossthe nuclear chart is given in the following section.Not only theoretically interesting, this Coulomb sup-pression effect is also important for experiments becauseit allows to directly determine an astrophysically relevantrate by measuring in the direction of suppressed SEF.The above mentioned complication of fitting rates withnegative Q values can be circumvented by directly apply-ing detailed balance and numerically computing the ratefor the forward reaction before performing a fit. This ispossible when f rev ≈
1. Subsequently, fits for both ratescan be obtained in the standard way. As an example, anapplication of this procedure to the rate of Rb(p,n) Sris shown in Sec. III G.
C. Exploration of the SEF suppression across thenuclear chart
In this section, we quantitatively study the SEF sup-pression introduced and discussed in the previous section.Using NON-SMOKER results [11, 14] we compared f forw and f rev for reactions involving light projectiles (nucle-ons, α ) and targets from Ne to Bi between the protonand neutron driplines. To avoid trivial cases, only reac-tions with f forw /f rev ≥ . T ≤ . Z N natural+longlived γ n αγ n γ p γγ p FIG. 3: (Color online) Targets for endothermic reactions with f rev < f forw in the nuclear chart, where charge is denoted by Z and neutron number by N . The reaction type is given bythe label. Only capture or photodisintegration reactions areshown. Also printed for orientation are stable and longlivednuclides. important in most nucleosynthesis environments and toeliminate cases only occurring at high temperature. Be-cause of our aim to provide guidance for experiments, wefurther only focus on examples with f rev ≤ .
5. Evenwith these restrictions we find 1200 reactions exhibitingsuch a strong suppression effect that f rev < f forw .To check the dependence on the Coulomb barrier, Fig.2 shows the obtained range of Q values still yielding f rev < f forw as a function of target charge Z for (p,n) and( α ,n) reactions with negative Q values. It can be clearlyseen that larger maximal | Q | is allowed with increasingcharge Z . The different increase in permitted maximal | Q | is different for the two reactions, reflecting the differ-ence in the height of the acting Coulomb barriers. Beloweach maximally allowed | Q | for each given charge, thereis a range of other values. This scatter is mainly causedby the available Q values (as defined by the masses of thenuclei) and not by other effects such as spins and paritiesof the involved nuclei. Although the strengths of the in-volved transitions also depend on spin and parity of theinitial and final state, Coulomb repulsion dominates thesuppression when the interaction energy is small, as it isthe case for astrophysically relevant energies.Tables I − V list the reactions found to have f rev As an example of the suppression effect and for thederivation of the astrophysical rates for an endother-mic reaction, we experimentally studied the reaction Rb(p,n) Sr.Reactions of the (n,p) type have been shown to be im-portant in the γ process [6]. This nucleosynthesis processcreates proton-rich isotopes of elements beyond Fe whichare not made in the s and r processes. It was shownto occur in hot O/Ne layers of massive stars, either ina core collapse supernova explosion when the shockfrontis passing these layers or already pre-explosively depend-ing on the initial mass of the star [4]. At temperatures T > γ ,n) reactions.Charged-particle emitting ( γ , α ) and ( γ ,p) reactions candeflect the reaction path to lower charge number. The-oretical investigations show that ( γ ,n)/( γ ,p) branchingsplay a key role in the production of the lighter p nucleiwhereas ( γ ,n)/( γ , α ) branchings are important at highermasses [5, 6]. Further reactions with the emitted neu-trons are mainly important in the freeze-out phase whenphotodisintegration ceases [6, 22]. The flow back to sta-bility is sped up by (n,p) reactions which are faster than β decays close to stability [6]. Even at stability, (n,p) re-actions act to push material to lower proton numbers. Inthis context, Rb(p,n) Sr is directly important because Sr m85 Sr IT 86.6%9/2 + ε+β + Rb -– -– -– -– 514 keV96% 231.86 keV84.4% Rb(p,n) Sr FIG. 5: Simplified decay scheme of the products of the Rb(p,n) Sr reaction. The half-lives of the reaction prod-ucts, the branching ratios, the spin and parity of the levelsand the transitions used to determine the reaction cross sec-tion are indicated [23]. it is the inverse reaction to Sr(n,p) and we found thatits SEF is smaller than the one of its inverse, despite ofits negative Q value (see Sec. III G).The reaction Rb(p,n) Sr is also important to testthe predictions of astrophysical rates and their under-lying nuclear properties. Although many (n,p) and(p, γ ) reactions important in the γ , rp , and νp processes[6, 15, 16] occur far from stability, the models and as-sumptions used in the prediction of the rates can bechecked at stability. Especially suited for testing the re-liability of the optical potential used for the calculationof transitions involving protons are (n,p) and (p,n) reac-tions because the proton width is smaller than the neu-tron width at practically all energies (except very closeto the neutron threshold) and thus determines the crosssection.We measured Rb(p,n) Sr using the activationmethod. Thin RbCl targets were bombarded by protonbeam provided by the Van de Graaff and cyclotron accel-erators of ATOMKI [7]. The (p,n) reaction on Rb canpopulate both the ground and metastable states of Sr[23]. To determine the cross section of the Rb(p,n) Sr g reaction the 514.01 keV γ line was used, in the case ofthe Rb(p,n) Sr m reaction cross section the yield of the231.84 keV transition was measured. In the followingSecs. III B–III D a detailed description of the experimen-tal technique is given, while the experimental results aregiven in Sec. III E. A comparison to theory and the finalastrophysical reaction rates are provided in Secs. III Fand III G. 200 250 300 350 400 450 500 550 60010 E p = 2.4 MeVt w = 540 min 511 keV 514.01 keV Rb(p,n) Sr388.51 keV Rb(p,n) Sr231.64 keV Rb(p,n) Sr c oun t s / c hanne l 200 250 300 350 400 450 500 550 60010 E p = 3.8 MeVt w = 30 min 511 keV 514.01 keV Rb(p,n) Sr388.51 keV Rb(p,n) Sr231.64 keV Rb(p,n) Sr 480 490 500 510 520 530 54001020304050 E p = 2.4 MeVt w ~ 30 days 511 keV 514.01 keV Rb(p,n) Sr c oun t s / c hanne l γ -energy [keV] 480 490 500 510 520 530 5400306090120150 E p = 3.8 MeVt w ~ 30 days 511 keV 514.01 keV Rb(p,n) Sr γ -energy [keV] 480 500 520 54001530456075 480 500 520 540050100150200250300 FIG. 6: Typical γ -spectra taken after the irradiation of RbCl targets with 2.4 (left panel) and 3.8 MeV (right panel) protonbeams. The 514.01 keV peak from the Rb(p,n) Sr g reaction can be well separated from the annihilation peak as can beseen in the insets. The length of the waiting time (t w ) between the end of the irradiation and the start of the γ -countingswere 540 (E p = 2.4 MeV) and 30 min (E p = 3.8 MeV). The lower panels show typical spectra taken in the repeated activitymeasurement approximately one month after the irradiations (for details see text). B. Target properties and the determination of thenumber of target atoms The targets were made by evaporating chemically pure(99.99%) RbCl onto two different kinds of Aluminumfoils: the thicker one had a chemical purity of 99.999%and thickness of 50 µ m, while the purity and the thick-ness of the thinner one was 99% and 2.4 µ m, respectively.The distance between the evaporation boat and the tar-get backing was 10 cm, therefore it was possible to as-sume that the evaporated layer is homogeneous. Thisassumption was proved using Rutherford BackscatteringSpectroscopy (RBS, see later). Targets with differentthicknesses were used, thicker ones (on thicker backings)were employed for irradiations at lower and thinner ones(on thinner backing) at higher energy. Owning to thistreatment, the yield of the investigated 514.01 keV peakwas always higher than, or comparable to that of the 511keV annihilation peak – and this way the separation ofthe peaks was achieved – as it is demonstrated in theupper part of Fig. 6.The number of the target atoms was determined withRutherford Backscattering Spectrometry (RBS) at theNuclear Microprobe facility of ATOMKI [25, 26, 27]. Asa consistency check, in the case of the targets evaporatedonto the thinner backing, weighing was also used to de-termine the number of target atoms. The weight of theAl foil used as backing was measured before and after the evaporation and from the difference — assuming that ourtarget is uniform — the number of target atoms was cal-culated. The results of the two different methods used todetermine the number of target atoms are in very goodagreement ( ≤ 3% difference). C. Irradiation The RbCl targets were bombarded with a proton beamprovided by the Van de Graaff and cyclotron acceleratorsof ATOMKI. The energy of the proton beam was between2 and 4 MeV, this energy range was covered in 200 keVsteps. The beam current was typically 600 nA. Eachirradiation lasted approximately 7 − p = 2.6 MeV was measured with bothaccelerators and no difference was found.An ion implanted Si detector was built into the irradi-ation chamber at θ = 150 ◦ relative to the beam directionto measure the yield of the backscattered protons. Thebackscattering spectra were taken continuously and wereused to monitor the target stability. Having a beam re-stricted to 600 nA, no target deterioration was found.For calculating the reaction cross section the properknowledge of the incident particle flux is necessary. Toobtain this, the collected charge was measured in a cham-ber similar to the one in [1]. After the beam definingaperture, the whole chamber served as Faraday cup tocollect the accumulated charge. A secondary electronsuppression voltage of − 300 V was applied at the en-trance of the chamber. The beam current was kept asstable as possible but to follow the changes the currentintegrator counts were recorded in multichannel scal-ing mode, stepping the channel in every minute. Thisrecorded current integrator spectrum was used for theanalysis solving the differential equation of the popula-tion and decay of the reaction products numerically. D. Activity determination Figure 5 shows the simplified decay scheme of the Sr g,m isotopes. To determine the cross section ofthe Rb(p,n) Sr g reaction the 514.01 keV, for the Rb(p,n) Sr m reaction the 231.84 keV gamma line wasused.For measuring the induced γ -activity a lead shieldedHPGe detector was used as in our previous (p,n)-study[1]. After each irradiation, a cooling time of one hour wasinserted in order to let the disturbing shortlived activitiesdecay. The γ spectra were taken for 12 hours and storedregularly in order to follow the decay of the short-livedreaction product.Figure 6 shows typical spectra collected after irradi-ating RbCl targets with the 2.4 MeV (left panel) andthe 3.8 MeV (right panel) proton beams. The yield ofthe 511 keV peak was always less than or comparable tothe investigated 514.01 keV transition, as shown in theinsets. The Sr has a relatively long halflife ( T / =64.84 d). Due to this, the activity measurement could berepeated for each target after approximately one month,when the intensity of the 511 keV radiation was substan-tially reduced. The spectra taken in the repeated ac-tivity measurement for the 2.4 and 3.8 MeV irradiationsare shown in the lower panels of Fig. 6. The two mea-surements yielded consistent cross sections proving theproper separation of the 511 keV and 514.01 keV peaks. E. Experimental results and comparison withliterature data In the case of the Rb(p,n) Sr g reaction, two sepa-rated analysis were done. The agreement between thecross sections derived in the γ -counting after the irradi-ation and the ones from the repeated activity measure-ment was always better than 4%. The final results werecalculated from the average weighted by the statisticaluncertainty of the two γ countings. The halflife of the Sr m is shorter, therefore the yield of the 231.64 keV γ radiaton was measured only after the irradiation. The fi-nal experimental result can be found in Table IX. Partialcross sections leading to the ground and isomeric state of S f a c t o r [ M e V ba r n ] Energy [MeV] Experimental data Modified potential Standard potential FIG. 7: Experimental (full triangles) and theoretical (lines)astrophysical S factors of Rb(p,n) Sr. The solid line is the S factor calculated with the modified proton optical potentialintroduced in [1] and the dashed line shows the result usingthe standard proton optical potential from [31] with low en-ergy modifications by [32] (see text). Sr can be found in [7]. The error of the cross sectionvalues is the quadratic sum of the following partial errors:efficiency of the HPGE detector system (6%), number oftarget atoms ( ≤ ≤ ≤ σ and the S factors, the latterbeing defined as [8] S ( E ) = σE e − πη , (12)with the Sommerfeld parameter η accounting for theCoulomb barrier penetration.The cross section of the Rb(p,n) Sr reaction was al-ready investigated by [24] between E c.m. = 3.1 and 70.6MeV. However, their accuracy is not sufficient for astro-physical applications, mainly because of the large uncer-tainty of the c.m. energies. Moreover, there is only onedata point in the relevant energy region for the γ processand it bears an uncertainty of ± F. Comparison to theory and implications for theproton optical potential The measured S factors are compared to theoreti-cal predictions obtained with the code NON-SMOKER[11, 14] in Fig. 7. The standard calculation applied a pro-0 TABLE IX: Details of the irradiations and the resulted cross sections (astrophysical S factors).E lab. [ MeV ] E c.m. [ MeV ] Accelerator Collectedcharge [ mC ] Total σ [ mbarn ] S factor ˆ MeV barn ˜ ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ton optical potential widely used in astrophysical applica-tions, based on a microscopic approach utilizing a localdensity approximation [31]. Low-energy modifications,which are relevant in astrophysics, have been providedby [32]. As can be seen in Fig. 7, the theoretical energydependence of the resulting S factor is slightly steeperthan the data, although there is general agreement inmagnitude. In the energy range covered by the measure-ment, the proton width is smaller than the neutron width(except close to the threshold) and thus uncertainties inthe description of the proton width (and proton transmis-sion coefficient) will fully impact the resulting S factor.A recent investigation [1] suggested that the strength ofthe imaginary part of the microscopic potential should beincreased by 70%. We find that the energy-dependenceof the theoretical S factor is changed in such a way asto show perfect agreement with the new data, as seen inFig. 7. This independently confirms the conclusions ofprevious work [1]. G. Astrophysical reaction rates Regarding the Coulomb suppression effect, a compar-ison of 1 . ≤ f pn ≤ . 08 and 2 . ≤ f np ≤ . Sr are moreimportant than those to states in Rb in the relevantplasma temperature range of 2 ≤ T ≤ f pn is due to the suppres-sion of the proton transmission coefficients to and fromthe excited states of Rb for small relative proton ener-gies because of the Coulomb barrier. There are only fewtransitions able to contribute due to the low Q value. Asshown by the small f pn , the transition from the groundstate of Rb dominates the proton channel. Obviously, aCoulomb suppression is not present in the neutron chan-nel. On the contrary, for this reaction f np is even moreenhanced due to the spin structure of the available nu-clear levels and especially the large spin of Sr g . Be-cause of its large spin, it is connected to the (dominat-ing) low spin states in Rb through higher partial wavesthan the excited states, such as the isomeric state, which have lower spins. Thus, the transitions from the groundstate are suppressed by the centrifugal barrier relative totransitions from excited states and the latter will quicklybecome important, even at low temperature. As a conse-quence of the enhancement of f np and the suppression of f pn , it is more advantageous to measure the (p,n) direc-tion. Important transitions to states in Sr are includedin our data and the small impact of transitions from ex-cited states in Rb is within the experimental error.Applying Eq. 2 directly with the experimental crosssections already yields the stellar rate because the SEFis small in the (p,n) direction. The stellar rate of theexothermic (n,p) reaction can then be computed usingEq. (7). By computing the forward rates directly fromthe backward rates without using fits, the complicationwith the negative Q value in fitted data is also avoided.Table X gives the stellar reactivities (as defined byEq. 4) for Rb(p,n) Sr as well as for Sr(n,p) Rb.Our data covers an energy range sufficient to computethe rates between 2 and 4 GK. Because of the excellentagreement of theory with experiment (using the newlymodified potential of [1]), we supplement the data withthe theoretical values to compute the reactivities at lowerand higher temperatures, applying the same errors as forthe data.It is to be noted that fits of the rates should be ob-tained by first fitting the (n,p) rate and then derivingthe (p,n) rate fit by modifying the fit coefficients accord-ing to detailed balance as given in Eq. (7) (see [11] fordetails). For convenience, we provide the fit coefficients(including a 10% error) for the (n,p) reactivity in thewidely used REACLIB format [11, 17] N A R ∗ = exp a + a T + a T / + a T / + a T + a T / + a ln T (cid:17) , (13)where N A is Avogadro’s number and the plasma tem-perature T = T / , with T in K. Using the usual di-mension of cm s − mole − for N A R ∗ the fitted coeffi-cients evaluate to a = 33 . +ln 1 . . , a = − . a = 40 . a = − . a = 6 . a =1 TABLE X: Astrophysical reactivities N A R ∗ of the reactions Rb(p,n) Sr and Sr(n,p) Rb computed from experimen-tal data. The values in italics are at temperatures where theexperimental data mostly contribute to the rate. The othervalues are computed by supplementing theoretical cross sec-tions using the modified optical potential.Temperature Rb(p,n) Sr Sr(n,p) Rb[10 K] [cm s − mole − ] [cm s − mole − ]0.10 (1.72 ± × − (1.19 ± × ± × − (1.49 ± × ± × − (1.74 ± × ± × − (2.15 ± × ± × − (2.55 ± × ± × − (2.99 ± × ± × − (3.49 ± × ± × − (4.09 ± × ± × − (4.80 ± × ± × − (5.62 ± × ± × − (6.57 ± × ± × − (1.35 ± × ± ± × ± × (4.57 ± × ± × (7.81 ± × ± × (1.28 ± × ± × (2.04 ± × ± × (3.17 ± × ± × (4.76 ± × ± × (9.52 ± × ± × (1.54 ± × ± × (2.01 ± × ± × (2.18 ± × ± × (2.05 ± × r a t i o T new/stdnewfit/new FIG. 8: The newly derived stellar reactivity of Sr(n,p) Rbis compared to the one given in [33] (area labelled “new/std”).The temperatures T are stellar plasma temperatures in GK.The shaded area accounts for an error of ± ≤ T ≤ 4. Also shown is the reactivity from a fit ofour new result compared to the actual value (curve labelled“newfit/new”). This shows that the fit accuracy is high. − . a = 31 . a . The coefficients for the(p,n) direction are the same, except a pn0 = 33 . +ln 1 . . and a pn1 = − . N A R ′ pn ( T ) = exp a pn0 + a pn1 T + a T / + a T / + a T + a T / + a ln T (cid:17) , (14) N A R ∗ pn ( T ) = G Sr ( T ) G Rb ( T ) N A R ′ pn ( T ) . (15)The required partition functions G ( T ) are provided in[11] as a function of temperature.Figure 8 shows a comparison of the new Sr(n,p) Rbreactivity to the “standard” one of [33]. At temperaturesabove 3 GK, we see an increase of 10 − 30% comparedto the previous values. Below 2 GK, the new reactivityis 20 − 30% lower than previously. The change in thetemperature dependence is due to the different protonoptical potential used. At very low temperature, the re-activity becomes less sensitive to the proton potential.This explains the ratio becoming almost unity towardszero temperature. Also shown in Fig. 8 is a comparisonbetween the fit of the new reactivity with the parametersabove and the reactivity itself. This ratio stays close tounity for all temperatures. The deviations between thereactivity and its fit are very small and negligible com-pared to the other uncertainties involved. IV. SUMMARY We showed that – contrary to common wisdom – alarge number of endothermic reactions exhibit smallerstellar enhancement than their exothermic counterpartsand are thus preferable for experimental studies. Themain cause of suppression of the SEF in an endother-mic reaction is the Coulomb suppression of transitionswith low relative energy. This Coulomb suppression ofthe SEF was found to act for reactions with Q < | Q | and charged projectiles. Allowing only nucleons, α particles, and photons as projectiles or ejectiles, andrestricting the results to experimentally useful values ofthe SEFs, this effect still appears in 1200 reactions, in-cluding α captures relevant in the p process [5, 6] andproton captures relevant in the rp process [15] and the νp process [16]. A large number of cases was also foundfor (p,n) reactions which allow to determine astrophysicalreaction rates relevant to the γ process [6].As an example, we measured the astrophysically im-portant reaction Rb(p,n) Sr close above the thresh-old in the energy range relevant for the γ process. It2was shown that in this case it is possible to derive astro-physical reaction rates for the (n,p) as well as the (p,n)direction directly from the (p,n) data despite of the neg-ative reaction Q value. Additionally, our measurementconfirms a previously derived modification of the globalproton optical potential used in theoretical predictions. Acknowledgments This work was supported by the European ResearchCouncil grant agreement no. 203175, the Economic Com- petitiveness Operative Programme GVOP-3.2.1.-2004-04-0402/3.0., OTKA (K68801, T49245), and the SwissNSF (grant 2000-105328). Gy. Gy. acknowledges supportfrom the Bolyai grant. [1] G. G. Kiss, Gy. Gy¨urky, Z. Elekes, Zs. F¨ul¨op, E. 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