Origin of the p-nuclei...puzzle of the origin of the p-nuclei. It is an excellent example of the multifaceted, interdisciplinary approaches required to understand nucleosynthesis

IOP PUBLISHING REPORTS ON PROGRESS IN PHYSICS

Rep. Prog. Phys. 76 (2013) 066201 (38pp) doi:10.1088/0034-4885/76/6/066201

Constraining the astrophysical origin ofthe p-nuclei through nuclear physics andmeteoritic dataT Rauscher1,2,3, N Dauphas4, I Dillmann5,6, C Frohlich7, Zs Fulop2

and Gy Gyurky2

1 Department of Physics, University of Basel, 4056 Basel, Switzerland2 MTA Atomki, 4001 Debrecen, POB 51, Hungary3 Centre for Astrophysics Research, University of Hertfordshire, Hatfield AL10 9AB, UK4 Origins Laboratory, Department of the Geophysical Sciences and Enrico Fermi Institute, TheUniversity of Chicago, Chicago, IL 60637, USA5 GSI Helmholtzzentrum fur Schwerionenforschung GmbH, Darmstadt, Germany6 II. Physikalisches Institut, Justus-Liebig-Universitat, Gießen, Germany7 Department of Physics, North Carolina State University, Raleigh, NC 27695, USA

E-mail: [email protected]

Received 15 February 2011, in final form 11 March 2013Published 10 May 2013Online at stacks.iop.org/RoPP/76/066201

AbstractA small number of naturally occurring, proton-rich nuclides (the p-nuclei) cannot be made inthe s- and r-processes. Their origin is not well understood. Massive stars can produce p-nucleithrough photodisintegration of pre-existing intermediate and heavy nuclei. This so-called! -process requires high stellar plasma temperatures and occurs mainly in explosive O/Neburning during a core-collapse supernova. Although the ! -process in massive stars has beensuccessful in producing a large range of p-nuclei, significant deficiencies remain. Anincreasing number of processes and sites has been studied in recent years in search of viablealternatives replacing or supplementing the massive star models. A large number of unstablenuclei, however, with only theoretically predicted reaction rates are included in the reactionnetwork and thus the nuclear input may also bear considerable uncertainties. The currentstatus of astrophysical models, nuclear input and observational constraints is reviewed. Afteran overview of currently discussed models, the focus is on the possibility to better constrainthose models through different means. Meteoritic data not only provide the actual isotopicabundances of the p-nuclei but can also put constraints on the possible contribution ofproton-rich nucleosynthesis. The main part of the review focuses on the nuclear uncertaintiesinvolved in the determination of the astrophysical reaction rates required for the extendedreaction networks used in nucleosynthesis studies. Experimental approaches are discussedtogether with their necessary connection to theory, which is especially pronounced forreactions with intermediate and heavy nuclei in explosive nuclear burning, even close tostability.

(Some figures may appear in colour only in the online journal)

This article was invited by Robert E Tribble.

Contents

1. Introduction 22. The case of the missing nuclides 23. Processes and sites possibly contributing to the

production of p-nuclei 43.1. General considerations 4

3.2. Shells of exploding massive stars 4

3.3. White dwarf explosions 6

3.4. Thermonuclear burning on the surface ofneutron stars 6

0034-4885/13/066201+38$88.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA

http://dx.doi.org/10.1088/0034-4885/76/6/066201

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Rep. Prog. Phys. 76 (2013) 066201 T Rauscher et al

3.5. Neutrino-wind and accretion disk outflows 63.6. Galactic chemical evolution 7

4. Meteoritic constraints on p-isotope abundances andnucleosynthesis 74.1. Cosmic abundances of p-nuclides 74.2. Clues on the synthesis of Mo and Ru p-isotopes

from extinct nuclides 92Nb and 146Sm 84.3. p-Isotope anomalies in meteorites 10

5. Reaction rates and reaction mechanisms 125.1. Introduction 125.2. Definition of the stellar reaction rate 125.3. Reaction mechanisms 13

6. Nuclear aspects of the p-nucleus production 14

6.1. Important reactions 146.2. Nuclear physics uncertainties 176.3. Challenges and opportunities in the

experimental determination of astrophysicalrates 20

7. Experimental approaches 257.1. ! -induced measurements 257.2. Charged-particle induced measurements 267.3. "-elastic scattering 297.4. Neutron-induced measurements 30

8. Conclusion 32Acknowledgments 32References 32

1. Introduction

The origin of the intermediate and heavy elements beyond ironhas been a long-standing, important question in astronomyand astrophysics. The neutron capture s- and r-processessynthesize the bulk of those nuclei. While low-mass,asymptotic giant branch (AGB, M ! 8 M!) and massivestars (M " 8 M!) were found to contribute to the s-process,the site of the r-process still remains unknown. Moreover,a number of naturally occurring, proton-rich isotopes (thep-nuclei) cannot be made in either the s- or the r-process.Although their natural abundances are tiny compared withisotopes produced in neutron-capture nucleosynthesis, theirproduction is even more problematic. The long-time favoredprocess, photodisintegration of material in the O/Ne-shell of amassive star during its final core-collapse supernova explosion,fails to produce the required amounts of p-nuclei in severalmass ranges. Several alternative sites have been proposed butso far no conclusive evidence has been found to favor one or theother. Further important uncertainties stem from the reactionrates used in the modeling of the thermonuclear burning.Investigations in astrophysical and nuclear models, togetherwith various ‘observational’ information (obtained fromstellar spectra, meteoritic specimens and nuclear experiments)comprise the pieces which have to be put together to solve thepuzzle of the origin of the p-nuclei. It is an excellent exampleof the multifaceted, interdisciplinary approaches required tounderstand nucleosynthesis.

This review attempts to provide a general overview ofthe conditions required to produce p-nuclei and a summaryof the commonly discussed production processes and sites.It then focuses on the possibilities to better constrainastrophysical models through measurements, from meteoriticisotopic abundances to nuclear experiments. The derivationof astrophysical reaction rates for intermediate and heavynuclei from experiments necessitates the use of theoreticalmodels of nuclear reactions, due to the nuclear properties (suchas binding energies and cross sections) demanding extremethermonuclear conditions to allow the synthesis of proton-richnuclides.

We start with a brief historic overview and definition of thep-nuclei in section 2, followed by an overview of the suggestedastrophysical processes and sites (section 3), also outlining

the problems encountered. Section 4 presents the informationwhich can be extracted from the analysis of meteoritic material,from the actual solar p-abundances (section 4.1) to constraintsfrom extinct radionuclides (section 4.2) and isotopic anomalies(section 4.3). The discussion of the relevant nuclear physicsstarts with basic definitions in section 5. Important reactions,the main nuclear uncertainties and the ways in which nuclearexperiments can help are examined in sections 6.1, 6.2 and 6.3.Experimental approaches are reviewed in section 7, morespecifically photodisintegration reactions and their limitations(section 7.1), charged-particle induced reactions (section 7.2),elastic scattering (section 7.3) and neutron-induced reactions(section 7.4).

2. The case of the missing nuclides

In the first detailed analysis of solar abundances published by[1], it was already indicated that at least two types of processesmay be required to produce the abundance distribution aboveiron, one leading to neutron-rich isotopes and a differentone for neutron-deficient nuclides. Only one year later [2](B2FH) and [3] made detailed studies on suitable processesand their constraints, based on the data by [1] and additionalastronomical observations and nuclear data. It turned outthat two types of neutron-capture processes were requiredto explain the abundance patterns of intermediate and heavynuclei, the so-called s- and r-processes [4–6]. It was alsorealized that a number of proton-rich isotopes can never besynthesized through sequences of only neutron captures and#" decays (figure 1) and required the postulation of a thirdprocess. This was termed p-process because it was initiallythought to proceed via proton captures at high temperature,perhaps even reaching (partial) (p, ! )–(! , p) equilibrium. Thisnucleosynthesis process was tentatively placed in the H-richenvelope of type II supernovae by B2FH but it was later realizedthat the required temperatures are not attained there [7, 8]. Thisalso shed doubts on the feasibility to use proton captures forproducing all of the nuclides missing from the s- and r-processproduction.

It is somewhat confusing that in the literature the name‘p-process’ is sometimes used for a proton-capture processin the spirit of B2FH, but also sometimes taken as a tokensubsuming whatever production mechanism(s) is/are found to

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Se 74 Se 76Se 75

Kr 79

Sr 85

Kr 81 Kr 82

Se 77

Kr 83

Se 78

Kr 84

Se 80 Se 81

Kr 85

Br 82

Rb 86

Se 79

Br 80

Kr 78

Br 79

Se 82

Rb 87

Sr 88Sr 87

Kr 86Kr 80

Br 81

Sr 84 Sr 86

Rb 85

processs

-processisotopesp

r processdecay chains

Figure 1. The p-isotopes are shielded from r-process decay chainsby stable isotopes and are bypassed in the s-process reaction flow.

Figure 2. Comparison of the solar abundances for thep-nuclei [9, 10]; the connecting lines are drawn to guide the eye.

be responsible for the p-nuclides. For easier distinction ofthe production processes, here we prefer to adopt the modernnomenclature focusing on naming the nuclides in questionthe p-nuclides (they were called ‘excluded isotopes’ by [3])and using different names to specify the processes possiblyinvolved.

Historically there were 35 p-nuclides identified (figure 2and table 1), with 74Se being the lightest and 196Hg theheaviest. It is to be noted, however, that this assignmentdepends on the state-of-the-art of the s-process models (justlike the ‘observed’ r-abundances depend on them) and also onestimates of r-process contributions (e.g. to 113In and 115Sn[11, 12]). Almost all p-isotopes are even–even nuclei, withthe exception of 113In (Z = 49), 115Sn, 138La, and 180Tam.The isotopic abundances (table 1) are 1–2 orders of magnitudelower than for the respective r- and s-nuclei in the same massregion, with the exception of 92,94Mo and 96,98Ru.

The two neutron-magic p-isotopes 92Mo (neutron numberN = 50) and 144Sm (N = 82), and the proton-magic(charge number Z = 50) Sn-isotopes 112,114Sn exhibit largerabundances than the neighboring p-nuclei (figure 2). Theabundance of 164Er also stands out and already B2FH realizedthat it may contain considerable contributions from the s-process. It was indeed found that there are large s-processcontributions to 164Er, 152Gd and 180Ta [13], thus possiblyremoving them from the list of p-isotopes. If the abundances

Table 1. Contribution of p-isotopes to the isotopic composition ofelements [14] and solar p-abundances (relative to Si= 106) fromAnders and Grevesse [9] and Lodders [10].

p-isotopecontribution Solar abund. Solar abund. Change

Isotope (%) [14] (2003) [10] (1989) [9] (%)

74Se 0.89 (4) 5.80 # 10"1 5.50 # 10"1 5.4578Kr 0.355 (3) 2.00 # 10"1 1.53 # 10"1 30.7284Sr 0.56 (1) 1.31 # 10"1 1.32 # 10"1 "0.6192Mo 14.53 (30) 3.86 # 10"1 3.78 # 10"1 2.1294Mo 9.15 (9) 2.41 # 10"1 2.36 # 10"1 2.1296Ru 5.54 (14) 1.05 # 10"1 1.03 # 10"1 2.2398Ru 1.87 (3) 3.55 # 10"2 3.50 # 10"2 1.43102Pd 1.02 (1) 1.46 # 10"2 1.42 # 10"2 2.82106Cd 1.25 (6) 1.98 # 10"2 2.01 # 10"2 "1.49108Cd 0.89 (3) 1.41 # 10"2 1.43 # 10"2 "1.40113In 4.29 (5) 7.80 # 10"3 7.90 # 10"3 "1.27112Sn 0.97 (1) 3.63 # 10"2 3.72 # 10"2 "2.55114Sn 0.66 (1) 2.46 # 10"2 2.52 # 10"2 "2.38115Sn 0.34 (1) 1.27 # 10"2 1.29 # 10"2 "1.94120Te 0.09 (1) 4.60 # 10"3 4.30 # 10"3 6.98124Xe 0.0952 (3) 6.94 # 10"3a 5.71 # 10"3 21.54a

126Xe 0.0890 (2) 6.02 # 10"3a 5.09 # 10"3 18.27a

130Ba 0.106 (1) 4.60 # 10"3 4.76 # 10"3 "3.36132Ba 0.101 (1) 4.40 # 10"3 4.53 # 10"3 "2.87138La 0.08881 (71) 3.97 # 10"4 4.09 # 10"4 "2.93136Ce 0.185 (2) 2.17 # 10"3 2.16 # 10"3 0.46138Ce 0.251 (2) 2.93 # 10"3 2.84 # 10"3 3.17144Sm 3.07 (7) 7.81 # 10"3 8.00 # 10"3 "2.38152Gd 0.20 (1) 6.70 # 10"4 6.60 # 10"4 1.52156Dy 0.056 (3) 2.16 # 10"4 2.21 # 10"4 "2.26158Dy 0.095 (3) 3.71 # 10"4 3.78 # 10"4 "1.85162Er 0.139 (5) 3.50 # 10"4 3.51 # 10"4 "0.28164Er 1.601 (3) 4.11 # 10"3 4.04 # 10"3 1.73168Yb 0.123 (3) 3.23 # 10"4 3.22 # 10"4 0.31174Hf 0.16 (1) 2.75 # 10"4 2.49 # 10"4 10.44180Tam 0.01201 (32) 2.58 # 10"6 2.48 # 10"6 4.03180W 0.12 (1) 1.53 # 10"4 1.73 # 10"4 -11.56184Os 0.02 (1) 1.33 # 10"4 1.22 # 10"4 9.02190Pt 0.012 (2) 1.85 # 10"4 1.70 # 10"4 8.82196Hg 0.15 (1) 6.30 # 10"4 4.80 # 10"4 31.25

a Abundances by [17]: 124Xe: 6.57 # 10"3 (+5.63%); 126Xe:5.85 # 10"3 (+2.91%).

of 113In and 115Sn can also be explained by modificationsof the s-process and/or contributions from the r-process [12],this would leave only 30 p-isotopes to be explained by otherprocesses.

The nuclei 138La and 180mTa do not fit the local trendwell and have much lower abundance than their neighbors.This indicates a further process at work, also because thestandard photodisintegration process cannot synthesize themin the required quantity (see section 3).

Table 1 lists also the isotopic composition as given in [14].The quoted uncertainties are on the abundance relative toother isotopes of the same element. They introduce additionaluncertainties in reaction measurements using natural samples(see section 6.3). In general the composition uncertainties arebelow 5%. It should be noted that there have been recentmeasurements with higher precision which were not beenincluded in [14], e.g. [15, 16] for Os. The modern compositionuncertainties for their p-isotopes are below 1%.

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A special obstacle not found in the investigation of the s-and r-processes is the fact that there are no elements dominatedby p-isotopes. Therefore, our knowledge of p-abundances islimited to solar system abundances derived from meteoriticmaterial (section 4.1) and terrestrial isotopic compositions.Before astronomical observations of isotopic abundances atthe required discrimination level become feasible (if ever), itis impossible to determine p-abundances in stars of differentmetallicities and thus to obtain the galactic chemical evolution(GCE) picture directly. On the other hand, depending on theactual p-production mechanism, this may also be problematicfor determining early s-process contributions. If the p-nuclides(or some of them) turn out to be primary (i.e. independent ofmetallicity) or have a different dependence of production onmetallicity than the s-process (perhaps by initially originatingfrom r-nuclei), they may give a larger contribution to elementalabundances in old stars than the s-process. Observations of Geand Mo in metal-poor stars may cast further light on the originof the p-nuclei.

3. Processes and sites possibly contributing to theproduction of p-nuclei

3.1. General considerations

There are several possibilities to get to the proton-rich side.Sequences of proton captures may reach a p-isotope fromelements with lower charge number. They are suppressedby the Coulomb barriers, however, and it is not possibleto arbitrarily compensate for that by just requiring higherplasma temperatures. At high temperature (! , p) reactionsbecome faster than proton captures and prevent the build-upof proton-rich nuclides. Only a very proton-rich environmentallows fast proton captures, see (5.8). Photodisintegrationsare an alternative way to make p-nuclei, either by directlyproducing them through destruction of their neutron-richerneighbor isotopes through sequences of (! , n) reactions (theseare the predominant photodisintegration processes for moststable nuclei), or by flows from heavier, unstable nuclides via(! , p) or (! ,") reactions and subsequent #-decays.

There are three ingredients influencing the resultingp-abundances and these three are differently combined in thevarious sites proposed as the birthplace of the p-nuclides.The first one is, obviously, the temperature variation as afunction of time, defining the timescale of the process and thepeak temperature. This already points to explosive conditionswhich accommodate both the necessary temperatures forphotodisintegrations (or proton captures on highly chargednuclei) and a short timescale. The latter is required becauseit has to be avoided that too much material is transformedin order to achieve the tiny solar p-abundances (assumingthat these are typical). The second parameter is the protondensity. While photodisintegrations and # decays are notsensitive to the proton abundance, proton captures are. With ahigh number of protons available, proton captures can prevailover (! , p) reactions even at high temperatures. Last but notleast, the seed abundances, i.e. the number and composition ofnuclei on which the photodisintegrations or proton captures act

s/r seednucleideflection pointZ ( ,n)!

(n, )!

( ,p)!

( )!,"

#+/EC

Figure 3. Reaction flow in the ! -process.

initially, are also highly important but not well constrained. Inmost suggested production mechanisms (see below), the finalp-abundances depend sensitively on these seeds and thereforeare secondary. Thus, they depend on some s- and/or r-processnuclides being already present in the material, either becausethe star inherited those abundances from its proto-stellar cloudor because some additional production occurred within the sitebefore the onset of p-nucleus production.

So far it seems to be impossible to reproduce the solarabundances of the remaining 30 p-isotopes by one singleprocess. In our current understanding several (independentlyoperating) processes seem to contribute to the p-abundances.These processes can be realized in different sites, e.g. the! -process discussed below will occur in any sufficiently hotplasma. It was first discovered in simulations of massive starexplosions but also appears in type Ia supernovae.

3.2. Shells of exploding massive stars

For a long time, the favored process for production of p-nucleihas been the ! -process occurring during explosive O/Ne-shellburning in massive stars [18–23]. It was realized early thatthe abundances of most p-nuclei are inversely correlated withtheir photodisintegration rates [3, 18], pointing to an importantcontribution of photodisintegration. At temperatures of 2 #T # 3.5 GK, pre-existing seed nuclei in the p-nuclearmass range can be partially photodisintegrated, starting withsequences of (! , n) reactions and creating proton-rich isotopes.Several mass units away from stability, the (! , n) reactionscompete with #-decays but also with (! , p) and/or (! ,") (seefigure 3 and section 6.1).

Massive stars provide the required conditions oftransforming s- and r-process material already present inthe proto-stellar cloud or produced in situ in the weaks-process, during the He- and C-burning phase. The ! -process occurs naturally in simulations of massive stars anddoes not require any artifical fine-tuning. During the finalcore-collapse supernova (ccSN) a shockwave ejects and heatsthe outer layers of the star, by just the right amount neededto produce p-nuclei through photodisintegration. It is crucialthat a range of temperatures be present as nuclei in the lowermass part of the p-nuclides require higher temperatures forphotodisintegration (2.5–3.5 GK) whereas the ones at highermass are photodissociated more easily and should not beexposed to very high temperature (T < 2.5 GK), as otherwiseall heavy p-nuclei would be destroyed. Stars with higher mass

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10.0

1.0

1

01

08 001 021 041 061 081 002

prod

uctio

n fa

ctor

rel

ativ

e to

16O

rebmun ssam

51S91S12S52S

10.0

1.0

1

01

08 001 021 041 061 081 002

prod

uctio

n fa

ctor

rel

ativ

e to

16O

rebmun ssam

ddol51Sddol52S

Figure 4. Production factors Fi relative to 16O of p-nuclei formassive star models taken from [22] with initial solar metallicity, forprogenitors with 15 (S15, S15lodd), 19 (S19), 21 (S21) and 25 (S25,S25lodd) solar masses. The initial metallicity is based on [9] (top)and [10] (bottom). The shaded area gives a factor of 2 around the16O production factor as acceptable range of co-production. Thelines are drawn to guide the eye.

may reach ! -process temperatures already pre-explosively[22] although some or all of those pre-explosive p-nuclei maybe destroyed again in the explosion.

All p-nuclei are secondary in this production mechanism,i.e. depending on the initial metallicity of the star. Figure 4shows the production factors relative to 16O for stellar modelswith initially solar composition. These models were the firstto self-consistently follow the ! -processes through the pre-supernova stages and the supernova explosion. The productionfactors Fi relative to 16O are defined as

Fi = fi

f16O= Y final

i /Y initiali

Y final16O /Y initial

16O

, (3.1)

with the initial and final abundances Y initiali , Y final

i , respectively.The nuclide 16O is the main ‘metal’ produced in massive starsand its production factor is often used as a fiducial point todefine a band of acceptable agreement in production.

As seen in the upper part of figure 4, the p-nucleusproduction not only depends on the initial metallicity but alsoon details of the stellar evolution, which are also determined by

the stellar mass. Similar trends, however, can be found in allmodels. The p-nuclei in the mass ranges 124 # A # 150 and168 # A # 200 are produced in solar abundance ratios withinabout a factor of 2 relative to 16O. Below A < 124 and between150 # A # 165 the p-isotopes are severely underproduced.The S19 model shows special behavior due to the partialmerging of convective shells. In general, the total productionof the proton-rich isotopes increases for higher entropy in theoxygen shell, i.e. with increasing mass, but also depends ondetails of stellar structure and the composition of the star atthe time of core collapse. To consider the total contribution ofmassive stars to the Galactic budget of p-nuclei, the individualyields have to be averaged over the stellar mass distribution,giving more weight to stars with less mass. Since the ! -processyields of stars with different mass can vary wildly, a fine gridof masses has to be used.

The results shown in the upper part of figure 4 usedthe solar abundances of [9]. Recent new abundancedeterminations have brought considerable changes especiallyin the relative abundances of light nuclei (see section 4.1).The lower part of figure 4 shows models calculated withabundances by [10] (with amendments as given in [24]).The differences in the final p-nuclei production comparedwith the older solar abundances are not due to differentsolar intermediate and heavy element abundances, as can alsobe verified by inspection of table 1. Rather, the differentlight element abundances (including 16O) and their impact onthe hydrostatic burning phases (in particular helium burning[25]) lead to a different pre-supernova structure of the star,affecting the ! -process nucleosynthesis later on. Obviously,the normalization to the 16O production factor is also affected(section 4.1).

The very rare 138La cannot be produced in a ! -processbut it was suggested to be formed through neutrino reactionson stable nuclei (the $-process) [26, 27]. This was shownto be feasible with neutrinos emitted by the nascent neutronstar emerging from the core collapse of the massive star, thusproducing this isotope at the same site as the other p-nuclidesbut in a different process. This $-process is included in theresults shown in figure 4. The equally rare 180mTa probablyalso received a large contribution from the $-process. Theprediction of its yield from massive stars, however, suffersfrom the problem of accurately predicting the final isomericstate to ground state (g.s.) ratio after freeze-out of nuclearreactions (see, e.g., [22, 28, 29]). It entails not only thepopulation of these states through neutrino- and ! -inducedreactions but also following internal ! -transitions throughoutthe nucleosynthesis phase.

The long-standing shortcomings in the production of thelight p-nuclei with A < 124 and also those in the mass range150 # A # 165 have triggered a number of investigationsin astrophysics and nuclear physics aimed at resolving thesedeficiencies. It was already realized early on that it isunfeasible to produce the most abundant p-isotopes, 92,94Moand 96,98Ru, by photodisintegration in exploding massive starsdue to lack of seed nuclei in their mass region [18, 30, 31].Therefore, a different production environment has to be found.The underproduction at higher masses, on the other hand,

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may still be cured by improved astrophysical reaction rates(see section 6.2).

3.3. White dwarf explosions

A crucial problem in obtaining solar p-production in the! -process in massive stars is the seed distribution which isstrongly constrained. Thermonuclear explosions of stronglys-process enriched matter provide an alternative. If theappropriate temperature range is covered in such explosions,a ! -process ensues but with a different seed distributioncompared with the one found in massive stars. Such anenvironment is provided in the thermonuclear explosion ofa mass-accreting white dwarf (WD), mainly composed of Cand O [30]. Exploratory, parameterized calculations for thecanonical type Ia supernova (SNIa) model—the explosion of aWD after it has accreted enough mass from a companion starto reach the Chandrasekhar limit—also found underproductionof light p-nuclei, even when assuming a seed enrichment offactors 103–104 in s-process nuclei from an AGB companion[8]. Recent studies, based on the carbon deflagration modelof [32], found similar problems [33, 34]. In contrast, post-processing of high-resolution 2D models considering two typesof explosions, deflagration and deflagration–detonation, findthat they can co-produce all p-nuclei (with the exceptionof 113In, 115Sn, 138La, 152Gd and 180Ta) when using strongenhancements in the assumed s-process seeds [35]. It wasconcluded in [34, 35] that a high-resolution treatment of theouter zones of the type Ia supernova is crucial to accuratelyfollow the production of p-nuclides.

Another alternative is a subclass of type Ia supernovaewhich is supposed to be caused by the disruption of a sub-Chandrasekhar WD due to a thermonuclear runaway in aHe-rich accretion layer [36]. High neutron fluxes are built up inthe early phase of the explosion and a weak r-process ensues.Once the temperature exceeds T9 $ 2, photodisintegrationstake over and move the nucleosynthesis to the proton-rich sidewhere two processes act: the ! -process, as described above,and additionally rapid proton captures on proton-rich unstablenuclei at 3 < T9 # 3.5. The latter is somewhat similar tothe rp-process but at much lower proton densities and thuscloser to stability. The proton captures are in equilibriumwith (! , p) reactions but nuclei with low capture Q-valuecannot be bridged efficiently within the short timescale of theexplosion. Large numbers of neutrons, however, are releasedin the reactions 18O(", n)21Ne, 22Ne(", n)25Mg and 26Mg(",n)29Si during the detonations [8]. The waiting points withlow (p, ! ) Q-value thus can be bypassed by (n, p) reactions[8, 36]. This so-called pn-process can efficiently produce thelight p-nuclides from Se to Ru but it overproduces them inrelation to the heavier ones. Again, a strong increase inthe photodisintegration seed abundances would be required toproduce all p-nuclei at solar relative abundances. (It has to benoted that here the nuclei produced in the pn-process would beprimary whereas the others are secondary.) It was concluded,nevertheless, that subChandrasekhar He-detonation models arenot an efficient site for p-nucleus synthesis [36, 37].

Both WD scenarios (canonical and sub-Chandrasekharmass type Ia supernovae) suffer from the fact that they

are difficult to simulate and self-consistent hydrodynamicalmodels including accretion, pre-explosive burning, explosion,and explosive burning in turbulent layers are missing. Acomplete model would also allow one to study the actualamount of seed enhancement (if any) in these environments,through strong s-processing either in the companion star orduring accretion. Another open question is the actual p-nucleus contribution of type Ia supernovae during the chemicalevolution of the Galaxy. It is still under debate whatfraction of type Ia supernova events is actually comprisedof the single-degenerate type (with a companion star) andhow much double-degenerate events (collision of two WDs)contribute.

3.4. Thermonuclear burning on the surface of neutron stars

Explosive H- and He-burning on the surface of a mass-accreting neutron star can explain a type of burst observed ingalactic point-like x-ray sources on timescales of a few to a fewtens of seconds [38–42]. Such a rp-process involves sequencesof proton captures and #-decays along the proton dripline[38, 41, 42]. How far the burning can move up the nuclear chartdepends on details of the hydrodynamics (convection) and thenumber of accreted protons. There is a definitive endpoint,however, when the rp-process path runs into the region ofTe "-emitters [43]. Therefore, if the rp-process actuallyruns that far, only p-nuclides with A < 110 can be reachedthrough decays of very proton-rich progenitors. This wouldconveniently allow one to only account for the underproductionof the light p-nuclides, which would be primary.

Currently it is unclear whether the produced nuclides canbe ejected into the interstellar medium or whether they aretrapped in the gravitational field and eventually only modifythe surface composition of the neutron star. The fact thatp-nuclides are only produced at the bottom of the burningzone in each burst and have to be quickly moved outwardby strong convection complicates the ejection of significantamounts [44].

3.5. Neutrino-wind and accretion disc outflows

Conditions suited for the synthesis of nuclides in the range ofthe p-nuclei may also be established by strong neutrino flowsacting on hot matter, moving out from ccSN explosions or fromaccretion discs around compact objects. They would give riseto primary production of p-nuclides.

For example, very proton-rich conditions were found inthe innermost ejected layers of a ccSN due to the interactionof $e with the ejected, hot and dense matter at early times. Thehot matter freezes out from nuclear statistical equilibrium andsequences of proton captures and #-decays ensue, similar tothe late phase of the rp-process (section 3.4) and the pn-process(section 3.3). The nucleosynthesis timescale is given by theexplosion and subsequent freeze-out and is shorter than inthe rp-process. The flow toward heavier elements would behindered by nuclides with low proton capture Q-value andlong #-decay half-lives. Similar to the pn-process, (n, p)reactions accelerate the upward flow. The neutrons stem fromthe reaction $e +p % n+e+. Although at early times the $e flux

6


is small, the sheer number of free protons guarantees a constantneutron supply. Thus, this process was termed the $p-processby [45]. It was confirmed by the calculations of [46, 47].Like the rp- and pn-processes it may contribute to the lightp-nuclides via decays of very proton-rich species, although itis not known up to what mass. The details depend sensitivelyon the explosion mechanism, the neutrino emission from theproto-neutron star, and the hydrodynamics governing the ejectamotion. Nuclear uncertainties are discussed in section 6.1 andin [48, 50, 51].

Accretion discs around compact objects have also beendiscussed as a viable site for the production of p-nuclei. Suchdiscs can be formed from fallback of material in ccSN or inneutron star mergers. The production strongly depends on theassumed accretion rate, which determines the neutrino trappinginside the disc. Fully consistent hydrodynamic simulationshave not been performed yet and the models have to makeassumptions on accretion rates, disc properties, and especiallythe amount of ejected material in wind outflows. Productionof p-nuclei comparable to that in the O/Ne-shell ! -processhas been found in parameterized, hot ccSN accretion discs,including the underproduction problem for light p-nuclei [52].Conversely, it was shown that only light p-nuclei up to 94Mocan be synthesized in parameterized black hole accretion discsunder slightly neutron-rich QSE (quasi-statistical equilibrium)conditions at high accretion rates [53], but that a full $p-processcan ensue at lower accretion rates [54]. The production of thelightest p-nuclei under QSE conditions was previously alsodiscussed in the directly ejected matter in a ccSN [48, 55, 56].For a detailed discussion of QSE conditions, see, e.g., [56].

Recently, magnetically driven jets ejected from rapidlyrotating, collapsing massive stars have been studied regardingtheir nucleosynthesis. Obviously, ejection of the producednuclei is guaranteed in this case. Again, axisymmetric jetsimulations show light p-nucleus production up to 92Mothrough QSE proton captures, but it was also suggested that113In, 115Sn, and 138La can be synthesized through fissionin very neutron-rich zones of the jet [57]. The conclusionsregarding the fission products are highly uncertain, however,because they sensitively depend on unknown fission properties,such as fission barriers and fragment distributions. Conversely,full 3D magneto-hydrodynamic simulations of such jets foundvery neutron-rich conditions, closer to neutron star mergerconditions than to the ones found in ccSN neutrino-windoutflows, precluding the production of p-nuclei [58].

3.6. Galactic chemical evolution

The yields of different sources add up throughout the history ofthe Galaxy. Therefore, the evolution of chemical enrichmenthas to be followed in order to fully understand the distributionof p-nuclides in the Galaxy. This is hampered by severalproblems. Not only have the frequency, spatial distribution,and yields of the different sources to be known but alsotheir dependence on metallicity (if the p-nucleus production issecondary) and how the products are mixed in the interstellarmedium. Clearly, this poses great challenges both for GCEmodels and for the simulations of each site, including the

determination of the ejection of the produced nuclei. ForccSN, it has been shown that the deficiencies in the ! -processessentially remain also when integrating over a range of stellarmasses [21, 59]. Although this was derived using stars of solarmetallicity, it is not expected to be different when includingstars of lower metallicity, as the p-production in the ! -processis secondary and scales with the amount of seed nuclei presentin the star.

As pointed out above, in the case of the p-nuclides there arenot enough observables to constrain well GCE models or evensingle production sites, as the isotopic abundances of p-nucleicannot be separately determined in stellar spectra. Thisunderlines the importance of analyzing meteoritic materialas explained in the following section 4. Combining theisotopic information, e.g. of extinct radioactivities, with GCEallows one to put severe constraints on the possible productionprocesses. For example, section 4.2.4 shows that processesmaking 92Mo but not 92Nb cannot have contributed much tothe composition of the presolar cloud. This would rule out asignificant contribution of p-nuclides at and above Mo fromvery proton-rich environments.

4. Meteoritic constraints on p-isotope abundancesand nucleosynthesis

Unlike large planetary bodies, which have had theircompositions modified by core/mantle segregation and silicatemantle differentiation, meteorites provide a minimally alteredrecord of the composition of the dust present in the solarprotoplanetary disc. The study of meteorites has helped definethe cosmic abundance of the nuclides, and has revealed thepresence of presolar grains and extinct radionuclides in theearly solar system. In the past decade, significant progressin mass spectrometric techniques has put new constraintson p-nucleosynthesis, in particular on the roles of rp- and$p-processes in the production of light p-nuclides 92Mo, 94Mo,96Ru and 98Ru.

4.1. Cosmic abundances of p-nuclides

Solar spectroscopy provides critical constraints on the cosmicabundance of key elements such as H, He, C, N and O [60].However, remote techniques do not allow one to establish theabundance of p-isotopes. A class of meteorites known as CIchondrites (CI stands for carbonaceous chondrite of Ivuna-type) have been shown to contain most elements in proportionsthat are nearly identical to those measured in the solarspectrum [9, 10, 61]. For this reason, CI chondrites have beenextensively studied to establish the relative abundances andisotopic compositions of heavy elements that cannot be directlymeasured in the solar photosphere. The virtue of this approachis that CI chondrite specimens (e.g. the Orgueil meteoritethat fell in France in 1864 and weighted 14 kg total) can bemeasured in the laboratory by mass spectrometry, providinghighly precise and accurate data. The only elements for whichCI chondrites do not match the present solar composition areLi, which is burned in the Sun, and volatile elements (e.g. H, C,N, noble gases) that did not fully condense into solids and were

7


removed from the protoplanetary disc when nebular gas wasdissipated. Thus, the relative proportions and isotopic ratiosof p-nuclides are well known from meteorite measurements(table 1).

Silicon is most commonly used to normalize meteoritic tosolar photosphere abundances. For the purpose of comparingmeteoritic p-isotope abundances with nucleosynthetic modelpredictions, it is more useful to normalize the data to16O. The ratios of p-isotopes to 16O are still uncertainbecause of uncertainties in the abundance of O in the solarphotosphere, which was drastically revised downward [60, 62].A difficulty persists, however, as helioseismology requires alarger abundance of O (and other metals) to account for theinferred sound speed at depth, as well as other observables[63]. It is not known at present what is the cause of thisdiscrepancy but the abundances inferred by one of thesemethods (helioseismology versus solar spectroscopy) must beincorrect. The solar O abundance was revised from 8.93 dex in1989 [9] to 8.69 dex in 2009 [60], i.e. a factor of 1.74. This issignificant with respect to p-nucleosynthesis as a factor of twomismatch in predicted to measured abundances of p-nuclidesrelative to 16O is often taken as the cutoff between success andfailure (see section 3.2).

In p-isotope abundances Yp, there is an overall decreasewith increasing atomic mass (figure 2), reflecting thedecreasing abundance of seed s- and r-process nuclides athigher masses [23]. This trend can be fitted by an empiricalformula Yp/Y16O = 4.86 # 108 # A"8.65. A second empiricalrelationship is also found between pairs of p-isotopes separatedby two atomic mass units, such as 156Dy and 158Dy. For suchpairs, the ratio of their abundances increases with atomic massfollowing approximately, YA+2/YA = 0.019 # A"1.409. Anempirical relationship of this kind was used to estimate therelative abundance of the short-lived p-isotope 146Sm to thestable 144Sm [64].

4.2. Clues on the synthesis of Mo and Ru p-isotopes fromextinct nuclides 92Nb and 146Sm

Meteorites contain extinct radionuclides with short half-lives(i.e. relative to the age of the solar system) that were presentwhen the solar system was formed but have now decayed belowdetection level [65]. An example of an extinct radionuclideis 26Al (t1/2 = 0.7 Myr), which was present in sufficientquantities when planetesimals were accreted (26Al/27Al &5 # 10"5) to induce melting and core segregation. Althoughextinct nuclides have since long completely decayed, theirpast presence can be inferred from measurement of isotopicvariations in their decay products. Some phases formedwith high parent-to-daughter ratios, inducing variations inthe daughter nuclide by decay of the parent nuclide. Theseisotopic variations are nearly always small and discoveries ofnew extinct radionuclides depend on developments in massspectrometry to measure isotopic ratios precisely, as well assample selection to find phases with high parent-to-daughterratios. Two documented extinct radionuclides in meteoritesoriginate from a process also making p-nuclei: 92Nb and 146Sm.Other short-lived radionuclides of such origin may have been

present in the solar protoplanetary disc when meteorites wereformed but have not been found yet.

4.2.1. Samarium-146 (t1/2 = 68 My). It was first suggestedby [64] that this nuclide might be present in the solar systemand that it could be a useful nuclear cosmochronometer. Earlyefforts to estimate the solar system initial 146Sm/144Sm ratioyielded uncertain results [73, 74]. The initial abundance of146Sm in meteorites was first solidly established by measuring142Nd/144Nd (isotope ratio of the decay product of 146Sm to astable isotope of Nd) and 147Sm/144Nd (ratio of stable isotopesof Sm and Nd) in achondrite meteorites [72, 67, 68]. Manystudies followed, confirming the 146Sm/144Sm value, albeitwith improved precision and accuracy [70, 71]. To estimate theinitial abundance of 146Sm, one has to establish what is knownas an extinct radionuclide isochron. If 146Sm was present whenmeteorites were formed, one would expect to find a correlationbetween the ratios 142Nd/144Nd and 147Sm/144Nd measured bymass spectrometry,!142Nd

144Nd

"

present=

!142Nd144Nd

"

0+

! 146Sm144Sm

"

0#

! 147Sm144Nd

"

present,

(4.1)

where 0 subscripts correspond to the values at the time offormation of the meteorite investigated. The slope of thiscorrelation gives the initial 146Sm/144Sm ratio (figure 5). Allsamples formed at the same time from the same reservoir willplot on the same correlation line, which is called an isochronfor this reason. While most meteorites formed early, theremay have been some delay between collapse of the molecularcloud core that made the Sun and formation of the meteoritesinvestigated. For this reason, one often has to correct initialabundances inferred from meteorite measurements to accountfor this decay time. Extinct radionuclides with very shorthalf-lives (e.g. 36Cl, t1/2 = 0.30 Myr) are very sensitive tothis correction. Time zero is often taken to be the time ofcondensation from nebular gas of refractory solids knownas calcium–aluminum-rich inclusions (CAIs), which mustcorrespond closely to the formation of the solar system [75].The most important development with 146Sm in the past severalyears with respect to p-nucleosynthesis is a drastic revisionof its half-life from 104 to 68 Myr [70]. Using this newhalf-life and the most up-to-date meteorite measurements, theinitial 146Sm/144Sm ratio at CAI formation is estimated to be(9.4 ± 0.5) # 10"3. Note that both 144Sm and 146Sm arep-nuclides.

4.2.2. Niobium-92 (t1/2=34.7 Myr). This nuclide, whichdecays into 92Zr, was first detected by [69] who measuredthe Zr isotopic composition of rutile (TiO2) extracted froma large mass of the Toluca iron meteorite. The correction tothe time of CAI formation was uncertain, yet they were ableto estimate an initial 92Nb/93Nb ratio of & 1 # 10"5. Thisresult was questioned by several studies that reported higherinitial 92Nb/93Nb ratios of &10"3 [76–78]. The later studiesmight have suffered from unresolved analytical artifacts andthe most reliable estimate of the initial 92Nb/93Nb ratio is(1.6 ± 0.3) # 10"5 [66]. It is customary in meteoritic studies

8


Estacado (H6)

Vaca Muerta (Mes)

(92Nb/92Mo)tCAI=(2.8±0.5) 10-5

92Zr

93Nb/90Zr

-3

-2

-1

0

1

2

3

4

0 0.02 0.04 0.06 0.08

142 N

d

144Sm/144Nd

(146Sm/144Sm)tCAI=(9.4±0.5) 10-3

LEW 86010

!!

Figure 5. Meteoritic evidence for the presence of 92Nb [66] and146Sm [67] in the early solar system (also see [68–72]). In bothdiagrams, the variations in the daughter isotope (92Zr or 142Nd)correlate with the parent-to-daughter ratio (Nb/Zr or Sm/Nd),demonstrating the presence of short-lived nuclides 92Nb and 146Smin meteorites (Estacado is an ordinary chondrite, Vaca Muerta is amesosiderite, and LEW 86010 is an angrite). The slopes of thesecorrelations give the initial abundances of the extinct radionuclides,see (4.1). The % notation is explained in figure 7.

to normalize the abundance of an extinct radionuclide tothe abundance of a stable isotope of the parent nuclide, i.e.93Nb (a pure s-process nuclide) for 92Nb. For the purposeof examining p-nucleosynthesis and comparing meteoriticabundances with predictions from GCE, it is more useful tonormalize 92Nb to a neighbor p-nuclide such as 92Mo [69]. Theearly solar system initial 92Nb/92Mo ratio is thus estimated tobe (2.8 ± 0.5) # 10"5.

4.2.3. Technetium-97 (t1/2 = 4.21 Myr) and technetium-98(t1/2 = 4.2 Myr). These two nuclides have the same originas p-nuclides and may have been present at the birth of thesolar system but they have not been detected yet (i.e. onlyupper limits could be derived). This stems from severaldifficulties; their expected abundances in meteorites are low,little fractionation is expected between parent and daughternuclides (i.e. Tc/Mo and Tc/Ru ratios), and no stable isotopeof the parent nuclide exists (one has to rely on a proxy elementsuch as Re). Available Mo and Ru isotopic analyses yieldthe following constraints on 97Tc and 98Tc abundances at CAIformation: 97Tc/92Mo < 3 # 10"6 (or 97Tc/98Ru < 4 # 10"4)

[79] and 98Tc/98Ru < 2#10"5 [80]. Note that 92Mo and 98Ruare both pure p-nuclides.

4.2.4. Galactic chemical evolution. The abundances ofextinct p-radionuclides in the early solar system can becompared with predictions from models of the chemicalevolution of the Galaxy. Meteoriticists have often used simpleclosed-box GCE or uniform production models, which predictthat the ratio of an extinct radionuclide in the interstellarmedium RISM (e.g. 146Sm/144Sm) should be related to theproduction ratio Rproduction through RISM = Rproduction&/TG,where & is the mean life of the nuclide (t1/2/ ln 2) and TG is thetime elapsed between Milky Way formation and solar systembirth. The closed-box model, however, fails to reproducefirst order astronomical observables such as the metallicitydistribution of G-dwarfs [82, 83]. Open-box models involvinggrowth of the Galaxy by infall of low-metallicity gas are morerealistic. In such models, the abundance of a short-lived p-nuclide in the average interstellar medium (ISM) at the birthof the solar system becomes

RISM/Rproduction = (k + 2)&/TG, (4.2)

where k is a constant [81, 84, 85]. Early work estimated itsvalue to be between 1 and 3 [84]. A value of k = 1.7 ± 0.4was obtained using a non-linear infall GCE model constrainedby recent astronomical observations [81]. The time elapsedbetween the formation of the Galaxy and the formation of thesolar system can be estimated, using the same infall GCEmodel and the U/Th ratio, to be 14.5–4.5 ' 10 Gyr [86].Therefore, the only unknown in the above equation is theproduction ratio, which can be estimated using nucleosynthesiscalculations. However, a complication remains to comparethe predicted abundances and the measured ones as (4.2)only gives the predicted abundance averaged over all ISMreservoirs while the solar system must have formed frommaterial that was partially isolated from fresh nucleosyntheticinputs. The earliest models used a free-decay interval toaccount for this isolation period, so the ratio in the early solarsystem would be the ratio in the ISM decreased by somefree decay, RESS = RISMe"('free"decay/& ). More realistically,isolation from fresh nucleosynthetic inputs was not complete.To address this issue, [87] devised a three-phase ISM mixingmodel between (1) dense molecular clouds from which stellarsystems form, (2) large H–I clouds, and (3) smaller H–Iclouds that can be evaporated by supernova shocks. Usingthe same parameters as those used by [87], the expectedratio in the molecular cloud core from which the Sun wasborn is

RESS = RISM/[1 + 1.5Tmix/& + 0.4(Tmix/& )2], (4.3)

where & is the nuclide mean-life and Tmix is the three-phase ISM mixing timescale. Realistic values for the mixingtimescale are probably on the order of 10–100 Myr. This isthe only free parameter in the model and there are two extinctp-radionuclides to uniquely constrain its value.

It was shown by [81] how 92Nb can put importantconstraints on the role of the rp- and $p-processes in the

9


Table 2. Extinct p-radionuclides in the early solar system; updated from [81, 65].

Ratio t1/2 (Myr) Rmeteorite Rproduction RISM 4.5 Ga (model)

97Tc/98Ru 4.21 <4 # 10"4 (4.4 ± 1.3) # 10"2 (1.1 ± 0.3) # 10"4

98Tc/98Ru 4.2 <8 # 10"5 (7.3 ± 2.5) # 10"3 (1.8 ± 0.6) # 10"5

92Nb/92Mo 34.7 (2.8 ± 0.5) # 10"5 (1.5 ± 0.6) # 10"3 (3.0 ± 1.3) # 10"5

146Sm/144Sm 68 (9.4 ± 0.5) # 10"3 (1.8 ± 0.6) # 10"1 (7.1 ± 2.4) # 10"3

nucleosynthesis of Mo and Ru p-isotopes. The calculationsgiven below are updated with more recent data (table 2). Asdiscussed in section 3, the site and exact nuclear pathwayfor the nucleosynthesis of light p-nuclides is still a matterof debate. The production ratios of extinct p-nuclidesin the ! -process are 97Tc/98Ru = (4.4 ± 1.3) # 10"2,98Tc/98Ru = (7.3 ± 2.5) # 10"3, 92Nb/92Mo = (1.5 ±0.6) # 10"3, and 146Sm/144Sm = (1.8 ± 0.6) # 10"1 [22].The corresponding ratios in the average ISM 4.5 Gyr ago,see (4.2), are 97Tc/98Ru = (1.1 ± 0.4) # 10"4, 98Tc/98Ru =(1.8 ± 0.7) # 10"5, 92Nb/92Mo = (3.0 ± 1.3) # 10"5, and146Sm/144Sm = (7.1 ± 2.5)# 10"3. The upper limits on 97Tcand 98Tc from meteorite measurements, 97Tc/98Ru < 4#10"4

and 98Tc/98Ru < 8 # 10"5, are consistent with the inferredISM ratio from GCE and ! -process modeling. The predictedvalues for 92Nb and 146Sm are also in excellent agreement withmeteorite data; in meteorites 92Nb/92Mo = (2.8±0.5)#10"5,while we predict in the ISM 92Nb/92Mo = (3.0±1.3)#10"5;in meteorites 146Sm/144Sm = (9.4 ± 0.5) # 10"3, whilewe predict in the ISM 146Sm/144Sm = (7.1 ± 2.5) # 10"3

(table 2). We are comparing here the abundances measuredin meteorites with those in the average ISM 4.5 Gyr agopredicted from GCE modeling. Partial isolation of ISMmaterial from fresh nucleosynthetic inputs can be taken intoaccount using (4.3). We find that the three-phase ISMmixing timescale must be small, less than about 20 to 30 Myr.Otherwise the predicted abundances would not match themeasured values in meteorites. For example, adopting Tmix =30 Myr would decrease the expected ratios in the early solarsystem to 92Nb/92Mo = 1.5 # 10"5 and 146Sm/144Sm =4.7 # 10"3; factors of 2 lower than meteorite values. Tosummarize, 92Nb/92Mo and 146Sm/144Sm production ratiosfrom the ! -process can reproduce extremely well the measuredabundances of these extinct radionuclides in the early solarsystem. However, it is well documented that state-of-the-art! -process calculations underproduce 92Mo, 94Mo, 96Ru and98Ru (section 3.2), isotopes that are in similar abundance to s-process isotopes of the same elements. What if other processes,such as the rp-process or the $p-process, had produced themissing p-isotopes of Mo and Ru? This implies that &10%of 92Mo would be produced by the ! -process while &90%would be produced by the rp- or the $p-process. However,92Nb cannot be produced in these processes because it isshielded from a contribution by proton-rich progenitors duringfreeze-out by the stable 92Mo (figure 6). If 90% of 92Mo hadbeen made by processes that cannot produce 92Nb, this wouldhave decreased the effective 92Nb/92Mo production ratio bya factor of &10. The predicted 92Nb/92Mo ratio in the earlysolar system would also be lower than the ratio measured inmeteorites by a factor of 10. A significant contribution to Mo–Ru by any process that does not make 92Nb can therefore be

excluded [81]. Independently, another study concluded thata significant contribution to Mo p-isotopes from a $p-processwas unlikely, based on the inferred 94Mo/92Mo isotope ratioin this process [88].

The calculation outlined above can also be performed theother way around, using the 92Nb/92Mo ratio in meteorites tocalculate its production ratio. The 146Sm/144Sm ratio indicatesthat the three-phase ISM mixing timescale must be small; forthe purpose of simplicity we assume that it is zero. Taking anon-zero value would result in a higher inferred 92Nb/92Moproduction ratio, therefore our estimate corresponds to aconservative upper limit. We find the 92Nb/92Mo productionratio to be 0.0015+0.0012

"0.0009 (or higher). Models should takethis value as a fundamental constraint on p-nucleosynthesisin the Mo–Ru mass region. For instance, it remains to betested whether the recently reported ! -process predictions forSNIa [35], which do not experience a Mo–Ru underproductionproblem, can reproduce the abundances of 92Nb and 146Sm inmeteorites.

4.3. p-Isotope anomalies in meteorites

When the solar system was formed, temperatures in the innerpart of protosolar nebula were sufficiently high to inducevaporization of the dust present. However, some presolargrains survived heating and these grains can be retrievedfrom primitive meteorites. They are found by measuringtheir isotopic compositions, which are non-solar, indicatingthat the grains condensed in the outflows of stars that livedbefore the solar system was formed. Several comprehensivereviews have been published on this topic [89–91]. A varietyof phases from various types of stars have been documented.Six types of grains have a supernova origin; nanodiamonds,silicon carbide (SiC) of type X, low-density graphite, siliconnitride (Si3N4), a small number of presolar corundum (Al2O3)grains, and nanospinels. In the case of nanospinels, it is stillunknown whether the grains condensed in the outflows of ccSNor SNIa, as these grains are characterized by large excesses inthe neutron-rich isotope 54Cr, which can be produced by bothkinds of stars [92, 93]. Because heavy elements are presentat low concentrations and presolar grains are small (i.e. upto a few tens of micrometres but most often much less thanthat), it is analytically challenging to measure their isotopiccompositions. However, the development of the technique ofresonant ionization mass spectrometry (RIMS) has allowedcosmochemists to measure the isotopic composition of traceheavy elements in single presolar grains [94]. The advantagesof this technique over secondary ionization mass spectrometry(SIMS) for this type of measurement are that it selectivelyionizes the element of interest using tunable lasers, so isobaric

10


Figure 6. Reaction flows in the ! -process producing 92Mo and the extinct radionuclide 92Nb. Size and shading of the arrows show themagnitude of the reaction flows f on a logarithmic scale, nominal p-nuclides are shown as filled squares. The nuclide 92Nb can be producedby the ! -process but it cannot be produced by the rp- and $p-processes (or any process involving a decay of proton-rich nuclei contributingto 92Mo) as it is shielded from contributions by these processes by the stable 92Mo. The presence of 92Nb in meteorites indicates thatproton-rich processes did not contribute much to the nucleosynthesis of Mo and Ru p-isotopes [81].

interferences are almost non-existent, and it has high yield,meaning that a significant fraction of the atoms in the samplemake it to the detector. The Sr, Zr, Mo and Ba isotopiccompositions of presolar X-type SiC grains of supernova originwere measured by [95]. Notably, they reported large excessesof 95Mo and 97Mo relative to other Mo isotopes. It wasshown that these signatures could be explained by an episodeof neutron-burst that took place in a He-shell during passageof the supernova shock, which produced 95Y, 95Zr and 97Zrradioactive progenitors that rapidly decayed into 95Mo and97Mo [96]. This is expected to occur in a very localizedregion of the ccSN and the reason why the grains record sucha signature is not understood but it must reflect a selection biasin dust formation/preservation.

To this day, no ! -process signature has been documentedin heavy trace elements in presolar grains of supernovaorigin. However, the types and numbers of grains thathave been studied by RIMS are limited and further work isrequired to document isotope signatures in grains of supernovaorigin. The next generation of RIMS instruments may havethe capability to tackle this question in a more systematicmanner [97].

Isotopic anomalies for heavy elements can also be presentin macroscopic objects, in some cases reaching the scaleof bulk planets [79]. The isotopic anomalies are muchmore subdued than those documented in presolar grains butthese anomalies can be measured with other instruments(TIMS: thermal ionization mass spectrometers, and MC-ICPMS: multi-collector inductively coupled plasma massspectrometers) at greater precision, reaching a few parts permillion on isotopic ratios (i.e. &0.001% for TIMS or MC-ICPMS versus &1% for SIMS or RIMS). Isotopic variations(1 % deficit, where % = 0.01%) in the isotopic abundanceof the p-isotope 144Sm in bulk meteorites of carbonaceoustype are documented [99]. Similar results were found usinga more aggressive sample digestion technique known as fluxfusion, ensuring that the 144Sm isotopic variations did not result

142Nd = 4 % p + 96 % s

142Nd = 20 % p + 80 % s

or (Nd/Sm)p-carrier phase

=5 (Nd/Sm)p-solar

p-process excess

p-process deficit

-2 -1 0

144Sm (0.01 % deviation relative to terrestrial)

-0.6

-0.4

-0.2

0.2

0

142 N

d (0

.01

% d

evia

tion

rela

tive

to te

rres

tria

l)

1

Figure 7. Correlation in bulk chondrites between 142Nd and 144Smisotopic variations [98]. The correlation probably corresponds tomixing between a solar component and a presolar end-memberenriched in p-isotopes. The slope of the correlation cannot beexplained by assuming a solar mixture of 4% p and 96% s for 142Ndin the presolar end-member. Instead, the grains that carry thoseanomalies may have a fractionated Nd/Sm p-isotope contributionratio resulting from chemical fractionation of those two elementsupon condensation in a circumstellar environment.

from incomplete digestion of the samples [100]. Variationsin 144Sm were also resolved by [98] and a correlation withisotopic variations in 142Nd was found (figure 7). This issignificant because isotopic variations in 142Nd in planetarymaterials have been ascribed to decay of 146Sm, with importantimplications for planetary differentiation processes in the earlysolar system [101–103].

It was calculated that in order to explain the 142Nd–144Smcorrelation by variations in the ! -process component, a 20%! -process contribution to 142Nd would be needed [98]. This isat odds with the predominant s-process nature of this nuclide.

11


Such a high contribution can most likely be ruled out basedon several lines of evidence. Firstly, the s-process in thismass region is well understood and reproduces well the cosmicabundance of 142Nd [13]. If anything, the s-process producestoo much of this isotope rather than too little. Secondly, a20% ! -process contribution would plot far off the empiricaltrend of abundance versus mass for p-isotopes. The empiricalrelationship gives 142(Nd " p)/16O = 0.000 12, while a 20%! -process contribution would mean 142(Nd"p)/16O = 0.003,i.e. a factor of &30 higher than predicted. Thirdly, the142Nd produced in the ! -process comes from the "-decayof 146Sm. The 146Sm/144Sm ratio in the ! -process dependsstrongly on the relative strength of the reactions 148Gd(! ,n)147Gd and 148Gd(! ,")144Sm but is independent of the seednuclei. Nuclear physics experiments (see section 6.2) ruleout a ratio which would allow one to contribute significantlyto 142Nd.

If confirmed, the cause for the correlation between 142Ndand 144Sm remains to be explained. A likely interpretationis that the presolar phase controlling the ! -process Sm–Ndisotopic anomalies in planetary materials does not containthese elements in solar proportions. In fact, chemicalfractionation of Sm and Nd during grain condensation canproduce a correlation with a steeper slope than expected (see,e.g., [104] for a similar discussion in the context of correlatedMo–Ru isotope anomalies). Isotope anomalies also havebeen reported for the p-isotopes 184Os [15] and 180W [105]but further work is needed before these variations can beunderstood.

5. Reaction rates and reaction mechanisms

5.1. Introduction

The previous sections discussed the astrophysical sitesand observational constraints for the production of p-nuclei. Important for modelling nucleosynthesis is areliable foundation in the nuclear physics required to predictastrophysical reaction rates. It is important to note thatthere are fundamental differences between experimental andtheoretical studies of reactions on intermediate and heavynuclei, as appearing in p-nucleosynthesis, and reactions onlighter nuclei up to Si. Different nuclear properties areimportant at low mass than at high mass and new challengesarise, due to the higher nuclear level density (NLD) andthe higher Coulomb barriers encountered in heavier nuclei.Experimental and theoretical approaches well suited forreactions on lighter nuclei are not directly applicable to heavynuclei involved in explosive nucleosynthesis. Because of thehigh Coulomb barriers and the higher temperatures, giving riseto pronounced stellar effects not encountered in light nucleito such an extent, experimental investigations are not able tocompletely determine the astrophysical reaction rate in mostcases and have to be supplemented by theory. In the rest ofthis review, we discuss the challenges arising and the methodscurrently available to address them in the quest for providingreliable and accurate reaction rates to study the origin of thep-nuclei.

5.2. Definition of the stellar reaction rate

The astrophysical reaction rate r( for an interaction betweentwo particles or nuclei in a stellar environment is obtainedby folding the Maxwell–Boltzmann energy distribution (,describing the thermal center-of-mass (c.m.) motion of theinteracting nuclei in a plasma of temperature T , with a quantity) ( which is related to the probability that the reaction occurs,and by multiplying the result with the number densities na , nA,i.e. the number of interacting particles in a unit volume,

r( = nanA

1 + *aA

# )

0) ((E)((E, T ) dE = nanA

1 + *aA

R(. (5.1)

The stellar reactivity (or rate per particle pair) is denotedby R(. To avoid double counting of pairs, the Kroneckersymbol *aA is introduced. It is unity when the nucleia and A are the same and zero otherwise. The asterisksuperscript indicates stellar quantities, i.e. including the effectof thermal population of excited nuclear states in a stellarplasma. Depending on temperature and nuclear level structure,a fraction of nuclei is present in an excited state in the plasma,instead of being in the g.s. This has to be considered whencalculating the interactions and rates using the stellar crosssection ) ( [106, 107]

) ((E, T ) = ) eff(E)

G0(T )

= 1G0(T )

$

i

$

j

2Ji + 12J0 + 1

Wi)i%j (E " Ei), (5.2)

which involves a weighted sum over transitions from all initialexcited states i, with spin Ji and energy Ei , up to the interactionenergyE, leading to all accessible final states j . As usual, crosssections for individual transitions ) i%j are zero for negativeenergies. The quantity

G0 =$

i

(2Ji + 1) exp%" Ei

kT

&

2J0 + 1(5.3)

is nothing else than the partition function of the target nucleusnormalized to the g.s. spin, and ) eff is usually called effectivecross section [108]. The weights

Wi(E) = E " Ei

E= 1 " Ei

E(5.4)

of the contributions of excited states to the effective crosssection depend linearly on the excitation energy becauseMaxwell–Boltzmann energy-distributed projectiles act oneach excited state [106]. The relevant energies E are given bythe energy range contributing most to the integral in equation(5.1), see section 5.3.1.

Although not discussed in further detail here, it is worthmentioning that weak interactions and decays are also affectedby the thermal population of excited states. This is importantbecause it changes the decay lifetimes of nuclei such as,e.g., 92Nb and 180Ta, while temperatures are still high duringnucleosynthesis.

It is straightforward to show that the stellar reactivitiesdefined with the effective cross section—connecting all initial

12


states to all final states—also obey reciprocity, just as theindividual transitions ) i%j do [106, 108, 109]. The reciprocityrelations for a reaction A(a, b)B and its reverse reactionB(b, a)A are [106, 108, 110]

R(Bb

R(Aa

=(2JA

0 + 1)(2Ja + 1)

(2JB0 + 1)(2Jb + 1)

GA0 (T )

GB0 (T )

!mAa

mBb

"3/2

e"QAa/(kT )

(5.5)

when a, b are particles, and

R(B!

R(Aa

=(2JA

0 + 1)(2Ja + 1)

(2JB0 + 1)

GA0 (T )

GB0 (T )

!mAakT

2+h2

"3/2

e"QAa/(kT )

(5.6)

when b is a photon. The normalized partition functions GA0

and GB0 of the nuclei A and B, respectively, are defined as

before. The photodisintegration reactivity

R(B! =

# )

0) (! (E)(Planck(E, T ) dE (5.7)

includes a stellar photodisintegration cross section ) (! (E)

defined in complete analogy to the stellar cross section ) (

in equation (5.2). In relating the capture rate of A(a, ! )Bto the photodisintegration rate ,(

B! = nBR(B! , however, it

has to be assumed that the denominator exp(E/(kT )) " 1of the Planck distribution (Planck for photons appearing inequation (5.7) can be replaced by the one from the Maxwell–Boltzmann distribution exp(E/(kT )). The validity of thisapproximation has been investigated independently severaltimes [106, 110–113]. The contributions to the integral in (5.7)have to be negligible at the low energies where ( and (Planck

differ considerably. This is ensured by either a sufficientlylarge and positive QAa , which causes the integration overthe Planck distribution to start not at zero energy but ratherat a sufficiently large threshold energy, or by vanishingeffective cross sections at low energy due to, e.g., a Coulombbarrier. It turns out that the change in the denominator is agood approximation for the calculation of the rate integrals,especially for the temperatures and reactions encountered inp-nucleus production.

The photodisintegration rate of a nucleus only depends onplasma temperature T , whereas rates of reactions with particlesin the entrance channel depend on T and the number densityna of the projectile. The number density na scales with theplasma matter density - and the abundance of the projectileYa . A variation of plasma density, on the other hand, will notaffect photodisintegration of a nucleus. Therefore, the ratio oftemperature and density sets the ratio of photodisintegration tocapture rates (for a fixed Q-value)

r(B!

r(Aa

* YB

YA

T 3/2e"QAa/(kT )

NA-Ya

. (5.8)

This is the reason why under conditions with large -Ya

(e.g. high proton densities), capture reactions can balancephotodisintegrations even at high T , leading to equilibriumvalues for the abundances YA, YB .

It has to be emphasized that (5.5) and (5.6) only holdwhen using the stellar and effective cross sections ) ( and ) eff ,

respectively, in the calculation of the reactivity, not with theusual laboratory cross sections

)i =$

j

) i%j , (5.9)

where i = 0 if the target is in the g.s. In reactions involvingintermediate and heavy nuclei at high plasma temperatures,transitions on excited states of the target will be significant inmost cases (see section 6.3).

5.3. Reaction mechanisms

5.3.1. Relevant energy windows. In order to decide whichtype of reactions have to be considered in the prediction ofthe stellar cross sections, the relative interaction energies Eappearing in the astrophysical plasma have to be known. Theseenergies and the location of the maximum of the integrand canbe found in [114]. They have to be derived by inspection of theactual integrand in (5.1). The simple, frequently used formulafor estimating the Gamow window (see, e.g., [115, 116]) fromthe charges of projectile and target is only applicable when theenergy dependence of the cross section is fully given by theentrance channel. This has been found inadequate for manyreactions except those involving light nuclei [110, 114, 117]and thus is not applicable for the reactions appearing in theproduction of the p-nuclei.

The effective temperature range for the formation ofp-nuclei is 1.5 # T # 3.5 when considering both a! -process and a proton-rich environment, such as requiredfor the $p-process. This translates into relative interactionenergies of about 1.5–3.5 MeV for protons (at the high endof the temperature range and for the light p-nuclei) in the ! -process. The relevant energies shift more strongly with chargefor "-particles (at the low end of the temperature range andfor the heavier p-nuclei) and energies are in the ranges of,e.g., 6–9, 7–10 and 9–11 MeV, respectively, at 2 GK and forcharges Z $ 62, Z $ 74, and Z $ 82, respectively. Protoncaptures are only of limited importance in the $p-processbecause a (p, ! )–(! , p) equilibrium is upheld most of thetime [45, 49, 51]. Non-equilibrium proton captures at the lowend of the temperature range yield interaction energies of about1–2 MeV. Crucial for $p-processing is the acting of (n, p)reactions at all temperatures. This results in neutron energiesup to 1 MeV in (n, p). Of further importance in both ! -processand proton-rich nucleosynthesis are (n, ! ) reactions and theirinverses, also for the whole temperature range. This impliesneutron energies of up to 1400 keV for neutron captures.

5.3.2. Compound reactions. The compound formation andexcitation energy Eform = E + Q depends on the interactionenergy E and the reaction Q-value. The NLD integrated overthe compound excitation energies obtained with the relevantenergy ranges from section 5.3.1 determines which reactionmechanism will dominate. With only a few levels within theenergy range appearing in the integration of the reactivity,individual resonances have to be considered. With a high NLD,individual resonances are no longer resolved and the sum overindividual resonances can be replaced by averaged resonances

13


which should reproduce the integrated properties (widths) ofall resonances at all energies [118, 106]. This is the statisticalHauser–Feshbach model of compound nuclear reactions. Itsastrophysical application has been discussed in many recentpublications [106, 108, 119–121] and we do not want to repeatall the details here.

Averaged properties can be predicted with higher accuracythan individual resonances in most cases. It is important tonote, however, that there is a difference in the applicationof the statistical model to the calculation of cross sectionsand to reactivities. Since the calculation of the reactivityinvolves additional averaging over the Maxwell–Boltzmanndistribution, it is more ‘forgiving’ when fluctuations in thecross sections are not reproduced as long as the average valueis correct across the relevant energy range. Therefore, theHauser–Feshbach model can be applied at lower NLD atcompound formation energy for the calculation of the ratethan for the calculation of the cross section. Put differently,even when fluctuations due to resonances are seen in theexperimental cross sections but not in the Hauser–Feshbachpredictions, the model may still yield a reliable rate providedit gives the same Maxwell–Boltzmann average. See section 6.3for further details.

Combining the above with the typical interaction energiesquoted in section 5.3.1, it is easy to see that the statistical modelis applicable for all reactions involved in p-nucleosynthesis(see sections 3 and 6.1 for the important reactions). Problemsmay arise for (p, ! ) and (! , p) on nuclei with magic protonnumber or very proton-rich nuclei. The proton separationenergy decreases for proton-richer isotopes and this shifts thecompound formation energy to regions of lower NLD. Both the! - and the $p-processes, however, do not involve nuclei closeto the driplines where it is known that the statistical modelcannot be applied anymore [41, 120, 122]. Moreover, thereis a (p, ! )–(! , p) equilibrium in the $p-process, diminishingthe importance of individual reactions [49, 51]. This leavesproton reactions at charge Z = 28 and Z = 50, the latter onlybeing important in the ! -process, as possible cases where thestatistical model may not be fully applicable at all temperaturesand individual resonances may have to be considered.

Reactions involving "-particles should not be problematicfor the statistical model even though they are importantin a region of "-emitters with negative (", ! ) Q-values(see section 6.1), due to their much higher astrophysicallyrelevant interaction energies.

5.3.3. Direct reactions. It is well known that direct reactionsare important at interaction energies above several tens ofMeV because of the reduced compound formation probability[118, 123, 124]. In light nuclei with widely spaced energylevels, direct reactions can give important contributions tothe cross sections between resonances. In the p-nucleus massrange, however, direct reactions are not expected to contributeto the astrophysical reaction rates due to the generally higherNLD at the compound formation energy Eform, as discussed inthe previous section 5.3.2.

It has been realized recently, however, that low-energydirect inelastic scattering has to be included in the analysis

of (", ! ) laboratory cross sections [125, 126]. With highCoulomb barriers, Coulomb excitation [127] can becomenon-negligible at low interaction energies, modifying theexperimental yield. Data are scarce close to the astrophysicallyrelevant energy region (see section 7.2) but an overpredictionof "-induced cross sections relative to the experimental valueswas observed for several cases while standard predictionsworked well for others. Many attempts to consistently describethe data with modified global"+nucleus optical potentials havefailed. Accounting for Coulomb excitation at low energies canexplain the deviations at least partially and will pave the wayfor improved global understanding of reaction rates involving"-particles. Section 6.2 gives an example of how strongthe Coulomb excitation effect can be. Nuclear excitation isnegligible at the low energies relevant for astrophysics as theCoulomb scattering takes place far outside the nucleus.

Obviously, there is no Coulomb excitation for (! ,")rates which are needed for the ! -process. Therefore, anoptical potential describing the "-emission has to be used.Due to detailed balance considerations, this has to be similarto the one describing compound formation in the absenceof Coulomb excitation. Nevertheless, even when usingcharged-particle induced reactions, direct reactions not leadingto compound formation can be considered in the Hauser–Feshbach model quite generally by simply renormalizing the"-transmission coefficients T. in the entrance channel for eachpartial wave . [125, 126]

T +. = f.T. =

!T.

T. + T direct.

"T., (5.10)

where the transmission coefficientT direct. into the direct channel

can be derived from the direct cross section ) direct. . Here,

) direct. is the Coulomb excitation cross section calculated in

a fully quantum mechanical approach as, e.g., shown in[127]. In this approach, the transmission coefficients T. forcompound formation are computed using an optical potentialnot including the direct reactions, or more specifically theCoulomb excitation, in its imaginary part. This opticalpotential thus only describes the absorption into the compoundchannel and not into all inelastic channels. Only with sucha potential can the stellar reactivity of the (! ,") reaction becomputed by applying (5.6).

6. Nuclear aspects of the p-nucleus production

6.1. Important reactions

Although various astrophysical sites were presented insection 3, the actual types of participating reactions arelimited. In environments producing the light p-nuclei underQSE conditions, individual proton captures in the vicinity ofthese p-nuclei only play a role in the brief freeze-out phase.While the reactions in a given mass region are in equilibrium,the abundances are determined by the known nuclear massdifferences [56, 106, 121]. A similar situation is encounteredfor the proton captures in the $p-process which are in (p, ! )–(! , p) equilibrium most of the time [48, 49, 51]. Therefore, avariation of the proton capture rates only has limited impact

14


but may locally redistribute isotopic abundances [49–51]. Notin equilibrium, however, are the (n, p) reactions required toovercome waiting points with low proton-capture Q-value. Inthe mass region relevant for the production of light p-nuclei,they occur at the N = Z line, starting at 56Ni and 2–3 massunits toward stability [49, 51]. They crucially determine theonset of the $p-process as well as how quickly matter can beprocessed toward higher masses.

The schematic reaction path for the ! -process has beenshown in figure 1. The nuclei and reactions involved in the! -processes in various hot environments do not largely differ.The permitted temperature range is rather tightly constrainedby the fact that photodisintegrations must be possible butnot so strong as to completely destroy the p-nuclei or theirradioactive progenitors. The main difference between the sitessuggested in section 3 is the amount and distribution of seednuclei to be photodisintegrated. This will change the flow ina given reaction sequence accordingly but it should be notedthat abundance ratios of nuclei originating from the same seedare not affected by this, such as the 146Sm/144Sm productionratio discussed in sections 4.2.4 and 4.3. Another differencearises from the temperature evolution which is not necessarilythe same in, say, ccSN and SNIa. Since different types ofreactions exhibit different temperature dependence, deflectionsand branchings in the ! -process may shift with temperature.The time spent at a certain temperature is weighted by thetemperature evolution and thus also how rates compete at agiven nucleus. For instance, the (! , n)/(! ,") branching at148Gd is temperature sensitive. A higher temperature favors (! ,n) with respect to (! ,") and thus increases 146Sm production[18, 129]. The competition between (! , n), (! , p), and (! ,") atdifferent temperatures has been studied in detail by [130, 131].

Not all reactions are equally important in all sections ofthe nucleosynthesis network. Figures 8 and 9 show the zonalproduction factors for all p-nuclides as a function of the peaktemperature reached in the zone of the ccSN model by [132].In a ! -process, light p-nuclei are predominantly produced athigher temperatures (allowing efficient photodisintegration ofthe nuclei around mass A $ 100) whereas the productionmaximum of the heavy species lies toward the lower end ofthe temperature range. Neutron captures and especially (! ,n) reactions are important throughout the ! -process networkas the photodisintegration of stable nuclides commences with(! , n) reactions until sufficiently proton-rich nuclei have beenproduced and (! , p) or (! ,") reactions become faster. Thereleased neutrons can be captured again by other nuclei andpush the reaction path back to stability in the region of thelight p-nuclides.

In each isotopic chain we commence with initial (! , n)on stable isotopes and move toward the proton-rich side. Theproton-richer the nucleus, the slower the (! , n) rate while (! ,p) and/or (! ,") rates increase. At a certain isotope within theisotopic chain, a charged-particle emission rate will becomefaster than (! , n) and thus deflect the reaction sequence to alower charge number. Historically, this endpoint of the (! , n)chain has been called ‘branching’, inspired by the branchingsin the s-process path [18, 130]. A more appropriate termwould be ‘deflection (point)’, though, because—unlike in the

10.0

1.0

1

01

001

8.1 2 2.2 4.2 6.2 8.2 3 2.3 4.3

Yi/Y

ini

T kaep

47 eS87 rK48 rS

29 oM49 oM

69 uR89 uR201 dP601 dC

10.0

1.0

1

01

001

8.1 2 2.2 4.2 6.2 8.2 3 2.3 4.3

Yi/Y

ini

T kaep

801 dC311 nI

211 nS411 nS511 nS

021 eT421 eX621 eX031 aB

Figure 8. Production factors Yi/Yini relating the final abundance Yi

to the initial abundance Yini as a function of peak temperatureattained in a burning zone. Shown are the production factors forp-nuclides with mass numbers 74 ! A ! 106 (top) and108 ! A ! 130 (bottom). Initial solar abundances were used, thetrajectories were similar to the ones from [21] but reaction rateswere taken from [128].

s-process—the reaction path does not necessarily split intotwo branches. The relative changes of the photodisintegrationrates from one isotope to the next are so large that at eachisotope only one of the emissions dominates. As an example,a comparison of photodisintegration rates within a chain of Moisotopes is shown in figure 10. It can be seen how rapidly (! , n)rates are decreasing with decreasing neutron number, whereas(! , p) is strongly increasing because it becomes easier to emita proton than a neutron. The (! , n) rates exhibit strong odd–even staggering. Since in a reaction sequence, the timescaleof the whole sequence is determined by the slowest reactionlink(s), it is apparent that the (! , n) sequences will be mostlysensitive to a variation in the slow (! , n) rates and these haveto be studied preferentially.

There are very few cases of two (or even three) types ofemissions being comparable at a single isotope or over a shortseries of isotopes and these strongly depend on the opticalpotentials used. In these cases, true branchings would appear.They are mostly found in the range of the heavier p-nuclides,

15


10.0

1.0

1

01

001

8.1 2 2.2 4.2 6.2 8.2 3 2.3 4.3

Yi/Y

ini

T kaep

261 rE461 rE

861 bY471 fH081 aT

081 W481 sO091 tP

691 gH

10.0

1.0

1

01

001

8.1 2 2.2 4.2 6.2 8.2 3 2.3 4.3

Yi/Y

ini

T kaep

231 aB631 eC831 eC831 aL

441 mS251 dG651 yD851 yD

Figure 9. Same as figure 8 but for p-nuclides with mass numbers132 ! A ! 158 (top) and 162 ! A ! 196 (bottom). The isotopes138La and 180Ta are underproduced because no $-process wasincluded here and the population of the 180mTa isomer was notfollowed separately. For 148Gd(! ,")144Sm the rate from [119]was used.

where (! ,") competes with (! , n) as explained below, becausethe (! ,") rates do not vary as systematically as the rates forneutron or proton emission.

Examination of the deflection points easily shows thatat higher mass (! ,") deflections are encountered whereas atlower mass most deflections are caused by (! , p) due to thedistribution of reaction Q-values and Coulomb barriers [130].This is a well-understood nuclear structure effect and it canbe seen in figures 11 and 12 that "-emissions compete withneutron emission above N = 82. The rate field plots in thetwo figures give the dominant destruction reaction for eachnuclide, derived from a comparison of the rates per nucleus.For photodisintegration reactions the rate per nucleus is justthe reactivity R(

! as defined in section 5.2. It should be notedthat these are not reaction flows as they neglect the abundancesof the interacting nuclei. In the ! -process, however, this isa good approximation because, except for neutron capturesmentioned below, there are no reactions counteracting thephotodisintegration of a single nucleus. Therefore, the rateper nucleus is very suitable to also study the competition ofdifferent reaction channels. The possible competition pointsare marked by diamond shapes in figures 11 and 12. They are

01 21-01 01-01 8-01 6-01 4-01 2-01 001 201 401 601 8

01 01

64 84 05 25 45 65 85 06

rate

(1/

s)

rebmun nortuen

(!!!,"

)n,( )p,( )

Figure 10. Comparison of photodisintegration rates for Mo isotopesat 2.5 GK. The (! , n) rate per nucleus is shown as a solid line, the(! , p) rate per nucleus as a dashed line, and the (! ,") rate pernucleus as a short-dashed line.

defined by two or more reaction channels having comparablerates within the assumed uncertainties in the prediction. Theplots are based on calculations with the SMARAGD code[106, 133] using the standard optical potentials of [134] forneutrons and protons, and [135] for "-particles. The assumeduncertainties were factors of two and three (up and down)for (! , n) and (! , p), respectively, while one fifth of thepredicted value was assumed to be the lower limit for the rangeof possible (! ,") rates. Since the reaction flow is comingfrom stable isotopes, the first competition point in each (! , n)sequence is of major importance. Only if it is shown to bedominated by (! , n), will the second competition in a chainalso be of interest, and so on.

The above considerations explain the results of earlierstudies with systematic variations of the (! , p) and (! ,") ratesby constant factors [136] or using a selection of different opticalpotentials for the (! ,") rates [130, 131]. They also found thatuncertainties in the (! , p) rates mainly affect the lower half,whereas those in the (! ,") rates affect the upper half of thep-nuclide mass range.

The higher the plasma temperature the further into theproton-rich side (! , n) reactions can act. At low temperatures,even the (! , n) reaction sequence may not be able to movemuch beyond the stable isotopes because it becomes very slowcompared with the explosive timescale (and neutron capturemay be faster). Obviously, (! , p) and (! ,") are even slowerin those cases and the photodisintegrations are ‘stuck’, justslightly reordering stable abundances. Because of the tightlimits on permissible temperatures to successfully producep-nuclei, the competition points will not shift by more than1–2 mass units, if at all. More detailed investigations ofthe temperature dependence of deflections, competitions andbranchings can be found in [18, 130, 131], where each laterwork supersedes the previous one with updated reaction ratecalculations and further studies of the dependence on opticalpotentials.

The free neutrons released by (! , n) affect the finalabundances of light p-nuclei (up to Sn) in the ! -process inseveral ways. The major effect is that of destruction of

16


55 60 65 70 75 80neutron number

46

48

50

52

54

56

58

60

prot

onnu

mbe

r

35 40 45 50 55neutron number

34

36

38

40

42

44

46

prot

onnu

mbe

r

Figure 11. Reactivity field plots for 33 ! Z ! 46 (bottom) and 45 ! Z ! 61 (top) at 3 GK. The arrows give the dominant destructionreaction for each nucleus. Competition points are marked by red diamonds. The first competition point in a sequence of (! , n) reactionsfrom stability is mainly important. The N = Z line is shown by the straight blue line.

light p-nuclei by neutron captures in zones with sufficientlyhigh temperatures to photodisintegrate heavier nuclei but notenough for the lighter species. No change in the abundances oflight p-nuclei would then be expected when only consideringphotodisintegration but such zones show strong destruction ofthese nuclei, as can be seen in figures 8 and 9. This is becausethe neutrons released in the destruction of the heavier speciesdestroy the pre-existing p-nuclei (if non-zero initial metallicitywas assumed). Since all ejected zones are added up, this mayaffect the total yield of light p-nuclei. Neutron captures alsocounteract (! , n) at temperatures at which the light p-nucleiare destroyed [8]. This may prevent the (! , n) flow movingfar into the proton-rich side. It was pointed out in [136] that(n, p) reactions can push the ! -process path back to stabilityas well. Obviously, the availability of neutrons depends onthe assumed seed abundance distribution and especially on theabundances in the region of heavier nuclei, which are alreadydestroyed at a lower temperature. Therefore, this will requirespecial attention in models assuming strongly enhanced seeds,such as in the single-degenerate SNIa model (section 3.3).

Finally, neutron captures before the onset of the ! -processcan indirectly influence the p-production by modifying theseed abundances. In massive stars, the weak s-processingsensitively depends on the 22Ne(", n) rate, acting as thedominant neutron source (see, e.g., [22, 137] for the effect ofa variation of this rate). For s-processing in AGB stars, thisrate and 13C(", n) are important. Since the seed enhancement

in thermonuclear burning of SNIa is supposed to come fromthe matter accreted from a companion star (or s-processingduring the accretion), these rates would also affect the seedsand ! -processing in these models were they simulated self-consistently.

6.2. Nuclear physics uncertainties

Before the main nuclear uncertainties in the synthesis ofp-nuclei can be summarized, it is necessary to define thesensitivity of reaction rates and cross sections. While thedefinition of an ‘error bar’ for theory is complicated byfundamental differences to attaching an experimental error (see[107] for details), properly defined sensitivities immediatelyallow one to see the impact of various uncertainties in nuclearproperties and input. This also implies that it is easy to seewhich properties are in need of a better description in order tobetter constrain the astrophysical rate. In order to quantify theimpact of a variation of a model quantity q (directly taken frominput or derived from it) on the final result / (which is eithera cross section or a reactivity), the relative sensitivity s(/, q)is defined as [106, 107]

s (/, q) = v/ " 1vq " 1

. (6.1)

It is a measure of a change by a factor of v/ = /new//old in/ as the result of a change in the quantity q by the factor

17


100 105 110 115 120 125neutron number

72

74

76

78

80

82

prot

onnu

mbe

r

75 80 85 90 95 100neutron number

60

62

64

66

68

70

72

prot

onnu

mbe

r

Figure 12. Reactivity field plots for 59 ! Z ! 72 (bottom) and 71 ! Z ! 83 (top) at 2 GK. The arrows give the dominant destructionreaction for each nucleus. Competition points are marked by red diamonds. The first competition point in a sequence of (! , n) reactionsfrom stability is most important.

vq = qnew/qold, with s = 0 when no change occurs ands = 1 when the final result changes by the same factor asused in the variation of q, i.e. s(/, q) = 1 implies v/ = vq .Further information is encoded in the sign of the sensitivitys. Since both v/ > 0 and vq > 0 for the quantities studiedin this context, a positive sign implies that / changes in thesame manner as q, i.e. / becomes larger when the value ofthe quantity q is increased. The opposite is true for s < 0,i.e. / decreases with an increase in q. The varied quantitiesq in reaction rate studies are neutron-, proton-, "- and ! -widths. Sometimes also the NLD is varied although it canbe shown that it mainly affects the ! -width in astrophysicalapplications. This is due to the fact that the particle widthsare dominated by transitions to low-lying levels whereas ! -transitions to a continuum of levels at higher excitation of thecompound nucleus determine the ! -width [138].

Extended tables of sensitivities for reactions on targetnuclei between the driplines and with 10 # Z # 83 havebeen published in [107], for rates as well as cross sections.These tables are used to determine which nuclear propertieshave to be known to accurately determine a rate, once ithas been identified as relevant for nucleosynthesis. It isimportant to note that, according to (5.5) and (5.6), the samesensitivities apply to forward and reverse stellar rates, e.g. forthe stellar capture rates as well as the stellar photodisintegrationrates.

Despite the number of suggested sites, the nuclear physicsunderlying the p-production and its uncertainty is similar

in all of them, except for the contribution of rapid protoncapture processes far from stability. It has been argued insection 4.2 that the latter cannot contribute significantly tothe p-nuclei from Mo upward. Since proton captures are inequilibrium in those models, the main uncertainties lie in thereactions bridging the waiting points close to N = Z. Asmentioned in sections 3.5 and 6.1, (n, p) reactions are of highestimportance in the $p-process. Because of their large Q-value,the compound nucleus is formed at high excitation energy andthe statistical model is expected to apply well (section 5.3.2).In contrast to astrophysical neutron captures at stability, noisolated resonances are contributing and the rates are onlysensitive to the neutron width, whereas in neutron capturesat stability the ! -width is also important. The calculation ofthe neutron width depends on the optical potential used and onknowledge of low-lying levels in both target and final nucleus.In the unstable region, the lack of accurately determined low-lying levels introduces the largest uncertainty (see below,however, for considerations regarding the optical potential).

The nuclei involved in the! -process are the stable nuclidesand moderately unstable, proton-rich nuclei. The masses havebeen measured and thus the reaction Q-values are well known.Half-lives are known in principle but the electron captures and#+-decays need to be modified in the stellar plasma by theapplication of theoretical corrections for ionization and thermalexcitation. This has not been sufficiently addressed so far, witha few exceptions (see, e.g., [8]).

18


What is the actual impact of uncertainties in thephotodisintegration rates on the production of the p-nuclides?Although the ! -process is not an equilibrium process and areaction network with a large number of individual reactionshas to be employed, it has become apparent in section 6.1 thatnot all possible reactions in the network have to be known withhigh accuracy. Rather, only the dominant reaction sequenceshave to be known accurately, and within such a sequence theslowest reaction as it determines the amount of processingwithin the given short timescale of the explosive process.Charged-particle rates are important only at or close to thedeflection points because they are many orders of magnitudesmaller than the (! , n) rates on neutron-richer isotopes.

As a general observation it was found that the ! -widthis not relevant in astrophysical charged particle captures onintermediate and heavy mass nuclei [107]. This is due to thefact that the Coulomb barrier suppresses the proton- and "-widths at low energy and makes them smaller than the! -widthsat astrophysical interaction energies. The cross sections andrates will always be most sensitive to the channel with thesmallest width. It also follows from this that whenever an"-particle is involved, the reaction will be mostly sensitiveto its channel. The situation for neutron captures is morediverse. For most nuclei close to stability, the rates are stronglysensitive to the ! -width. In between magic neutron numbersalong stability and especially in the region of deformed nuclei,however, they are also or even more sensitive to the neutronwidth (see figures 14 and 15 in [107]). This can be explained bythe fact that the compound nuclei in these regions have higherlevel densities and this leads to comparable sizes of the neutron-and ! -widths at astrophysical energies. A comparison oftheory to experimental data along stability revealed that theuncertainties are within a factor of two, with an averagedeviation of better than 30% [106, 120]. Since the ! -processdoes not go far into the unstable region, similar uncertaintiesare expected.

Charged particle captures and photodisintegrations areonly sensitive to the optical potential at the astrophysicallyrelevant low energies. These optical potentials are usuallyderived from elastic scattering at higher energies and are thusnot well constrained around the Coulomb barrier. In particular,the imaginary part should be energy-dependent (see [106]for more details on optical potentials in astrophysical ratepredictions and the influence of other nuclear properties).Particular problems persist with "-captures at energies andin the mass region relevant for the ! -process. Comparisonsof theoretical predictions with the few available data atlow energies (see section 6.3.1) revealed a mixed patternof good reproduction and some cases of maximally 2–3times overprediction of the (", ! ) cross sections when usingthe ‘standard’ potential of [135] (see, e.g., [140–144] andreferences therein; see also section 7.2). So far, the only knownexample of a larger deviation was found in 144Sm(", ! )148Gd(determining the 144Sm/146Sm production ratio in the ! -process, see section 4.2) where the measured cross sectionis lower than the standard prediction by more than an orderof magnitude at astrophysical energies [139], inexplicable byglobal potentials and also not reproduced using a potential

01 62

01 72

01 82

01 92

9 5.9 01 5.01 11 5.11 21 5.21 31 5.31

S-f

acto

r (M

eVba

rn)

E mc )VeM(

pxe)SFcM(FHxeluoC htiw )SFcM(FH3/)xeluoC htiw )SFcM(FH(3/)SFcM(FH

Figure 13. Astrophysical S-factor for 144Sm(", ! )148Gd as afunction of c.m. energy. Data are from [139]. The astrophysicallyrelevant energy is about 8–9 MeV. Shown are the standard predictionwith the potential by [135] (McFS, full line), the same but correctedfor Coulomb excitation (dashed–dotted line), and the Coulexcorrected prediction with the "-width divided by a constant factor of3 (dashed line). Also shown is the standard prediction withoutcorrection but with the "-width divided by 3 (dotted line). See textfor an interpretation.

independently derived from elastic "-scattering at higherenergies [145].

As pointed out in section 5.3.3 and by [125, 126], theinclusion of Coulomb excitation may alleviate the putativeproblem in the prediction of the (", ! ) laboratory cross sectionsand provide a more consistent picture. The 144Sm(", ! )148Gdcase is shown in figure 13. The prediction using the standard"+nucleus potential by [135] is a factor of about 4 higher thanthe data at the upper end of the measured energy range. Ithas a completely different energy dependence, though, andyields a value higher by almost two orders of magnitude thanthe extrapolation of the data to the astrophysically relevantenergy of 9 MeV. Applying (5.10) to correct the predictionof the laboratory value for the fact that part of the "-flux isgoing into the direct inelastic channel, which is not included inthe optical potential, leads to calculated cross sections whichreproduce the energy dependence of the data but are very highby a factor of 3. The optical potential should be corrected onlyto account for this factor, i.e. the "-width should be divided bythis factor as shown in the figure. This corrected "-potential—but without further consideration of the Coulomb excitation (asexplained in section 5.3.3)—has to be employed to calculatethe stellar"-capture reactivity from which the actually relevant(! ,") rate is derived. The S-factor is lower than the standardvalue by about a factor of two at 9 MeV. Elastic "-scatteringexperiments at low energies, if feasible, may help to constrainthe "-width renormalization, see section 7.3, because theS-factor and cross sections at the upper end of the measuredenergy range are not only sensitive to the "-width but alsoto the neutron and the ! -width. Other reactions may noteven need changes to the optical potential and the apparentdiscrepancies may be explained by the laboratory Coulombexcitation alone [125, 126]. The uncertainties involved are therequirement to know precise B(E2) transition strengths (whichis challenging for odd nuclei with non-zero g.s. spin) and the

19


eX I

stnemirepxe noitavitcA eT bS

nS nI

dC gA

mT dP rE hR oH uR

yD cT bT oM

dG bN uE rZ

mS Y mP rS dN bR rP rK

eC rB aL eS

aB sA sC eG

stnemirepxe maeb-nI

Figure 14. Isotopes on which (p, ! ) cross sections relevant for the ! -process have been measured. The upper part of the p-isotope massregion is not shown since there are no data available there. The measured cross section data can be found in [144, 150, 151, 155–167].

need to accurately calculate Coulomb barrier penetrabilities atvery low "-energies.

While the standard "+nucleus potential most widely usedin astrophysical applications is a purely phenomenologicalone, the optical potentials for protons and neutrons used inthe prediction of astrophysical rates and also the interpretationof nuclear data are based on a more microscopic treatment,a Bruckner–Hartree–Fock calculation with the local densityapproximation including nuclear matter density distributions,which are in turn derived from microscopic calculations[106, 134, 146–148]. It has been pointed out, however, thatthe isovector part of the potential is not well constrainedby scattering experiments [149]. So far, this has not beenfound to give a noticeable effect in neutron captures atstability but may introduce an additional uncertainty bothat the far neutron- and the proton-rich side, wherever theneutron-width is dominating the rate. Also in comparisonwith measured low-energy (p, ! ) cross sections, calculationsusing these potentials often have been in very good agreementwith the data, sometimes being off by a maximal factor oftwo (see, e.g., [106, 150, 151] and references therein). Ingeneral, the uncertainty in the astrophysical rates caused bythe nucleon optical potentials seems to be much lower thanfor the "-capture rates. Recent (p, ! ) and (p, n) data ofhigher precision close to the astrophysically relevant energywindow, however, revealed a possible need for modification ofthe imaginary part of the nucleon potentials [106, 151–154].Consistently increased strength of the imaginary part at lowenergies improves the reproduction of the experimental datafor a number of reactions.

6.3. Challenges and opportunities in the experimentaldetermination of astrophysical rates

6.3.1. Status. It has become apparent in the previoussections that several hundreds of reactions contribute to

the synthesis of the comparatively few p-nuclides. Thevast majority of the astrophysical reaction rates has onlybeen predicted theoretically. Experimental data close toastrophysically relevant energies are very scarce, especiallyfor charged-particle reactions. Figures 14 and 15 show thoseisotopes for which (p, ! ) and (", ! ), respectively, crosssection measurements relevant for the ! -process are available.Only those isotopes are included where the motivation of theexperiments was the origin of the p-nuclides. As can be seen,currently proton-capture measurements are available for about30 isotopes along the line of stability. The measurementsare concentrated mainly in the lower mass region of thep-isotopes, which is in line with the fact that (! , p) reactionsplay the more important role in the lower mass range of a! -process network (see section 6.1). Data for (", ! ) reactionsare even more scarce, leaving the theoretical reaction ratecalculations largely untested. Moreover, the important highermass region is almost completely unexplored, especially closeto astrophysical energies.

Figures 14 and 15 show only those isotopes whereradiative capture cross sections have been measured. Onthe other hand, particle emitting reaction cross sections, suchas (p, n), (", n) or (", p), have also been measured alongwith the (p, ! ) or (", ! ) reactions, or in their own right.These reactions can provide valuable additional information toconstrain the theoretical description of astrophysically relevantnuclear properties as explained below.

The situation is somewhat better regarding neutron-induced reactions along stability as the low-energy crosssections for neutron captures have been determined fors-process studies. The KADoNiS database [179–181]centrally compiles data and provides recommended valuesfor Maxwellian averaged cross sections (MACS) at 30 keVand reactivities up to 100 keV. Unfortunately, this is only oflimited use for the ! -process as peak temperatures of 2–3 GKcorrespond to Maxwellian energies of kT = 170–260 keV

20


eX I

stnemirepxe noitavitcA eT bS

nS nI

dC gA

mT dP rE hR oH uR

yD cT bT oM

dG bN uE rZ

mS Y mP rS dN bR rP rK

eC rB aL eS

aB sA sC eG

stnemirepxe maeb-nI

Figure 15. Isotopes on which (", ! ) cross sections relevant for the ! -process have been measured. The upper part of the p-isotope massregion is not shown since there are no data available there with the exception of the 197Au(", ! )201Tl [168]. The measured cross section datacan be found in [139–143, 169–178].

but require measured neutron capture cross sections up to1400 keV, which is beyond the measured range for mostnuclei. The standard way to deal with this is to renormalizeHauser–Feshbach calculations to the measured value at oneenergy (usually at 30 keV, but 100 keV has been discussed asa possible alternative) in order to get an energy dependencefor extrapolation to higher (and lower) energies. This canbe problematic when the Hauser–Feshbach model is notapplicable at the renormalization energy. Moreover, it hasbeen pointed out recently that the renormalization procedureapplied so far has not properly included the stellar effect ofthermal population of excited states (see below) [182]. Theproper renormalization is discussed in section 6.3.2.

Summarizing the available data for studying explosiveburning it becomes apparent that, with few exceptions,almost no measurements in the relevant energy region areavailable. Although, as discussed in section 6.3.2, high-temperature environments severely limit the possibility todirectly determine astrophysical reactivities, there is an urgentneed to experimentally cover the relevant energy range toobtain data for the improvement of the theoretical predictions.Given the shortcomings and diversity of processes suggestedfor the production of p-nuclei (section 3), a reliable databaseand reliable predictions are needed as a firm basis for futureinvestigations. The accurate knowledge of reaction rates or,at least, their realistic uncertainties may allow one to rule outcertain astrophysical models on the grounds of nuclear physicsconsiderations.

6.3.2. Challenges and opportunities in the experimentaldetermination of astrophysical rates. There are severalchallenges hampering the direct determination of stellarreactivities by experiments. Although such challenges wouldappear in many astrophysically motivated studies, they aremore pronounced in high-temperature environments and for

intermediate and heavy nuclei. Reactions involved in theproduction of p-nuclei thus pose special problems, not or ona much smaller level encountered in the experimental studyof reactions with light nuclei and in hydrostatic burning at alower temperature. The main challenges include (i) reactionson unstable nuclei, (ii) tiny cross sections at astrophysicallyrelevant energies, (iii) different sensitivities of the crosssections inside and outside the astrophysical energy window,(iv) large differences between the stellar cross section and thelaboratory cross section. In this section, we briefly addresseach of these points and its implications for experiments.

Small cross sections and unstable target nuclei. The relevantenergies for calculation of the astrophysical reactivity havebeen summarized in section 5.3.1 and can be found in[114]. In spite of high plasma temperatures, the interactionenergies are small by nuclear physics standards. Althoughthe energy windows for reactions involving charged particlesin the entrance or exit channel are shifted to a higherenergy with increasing charge, the cross sections are alsostrongly suppressed by the Coulomb barrier. This resultsin low counting rates in laboratory experiments and requiresefficient detectors as well as excellent background suppression.Although there has been recent experimental progress(section 7), the astrophysically relevant energy window hasonly been reached in a few cases due to the tiny charged-particle cross sections, and full coverage of such a window for! -process conditions has not been achieved yet. Obviously,this does not apply to neutron captures but high accuracymeasurements covering the full ! -process energy windoware also scarce. Low nuclear absorption cross sections arealso an obstacle for scattering experiments with chargedparticles. Such experiments are required to improve the opticalpotentials, which are not well constrained at astrophysicalenergies (see sections 6.2 and 7.3). At such energies, however,the obtained cross sections are almost indistinguishable

21


1-

5.0-

0

5.0

1

0 1 2 3 4 5 6 7 8 9

sens

itivi

ty o

f cro

ss s

ectio

n

]VeM[ E

!np"

Figure 16. Sensitivities s of the laboratory cross section of 96Ru(p,! )97Rh to variations of nucleon-, "- and ! -widths, plotted asfunctions of c.m. energy [107]. The shaded region is theastrophysically relevant energy range for 2 ! T ! 3 GK [114]. Thesensitivity is defined in (6.1).

from pure Coulomb scattering (Rutherford scattering). Thesituation is worsened by the fact that the majority of reactionsfor p-nucleosynthesis proceeds on unstable nuclei. Althoughfuture facilities will allow the production of such nuclei itremains to be seen what types of reactions can be studiedat which energies (see also section 7.2.4). Since reactionswith short-lived nuclei must be studied in inverse kinematics,neutron-induced reactions cannot be addressed because noneutron target exists.

Sensitivities. The sensitivity of a cross section or rate toa variation of nuclear properties is defined in (6.1). Itis very important to measure in the correct energy rangeor at least close to it. The reason is that cross sectionsabove the astrophysical energies may exhibit a completelydifferent sensitivity to nuclear properties, which complicatesextrapolation. For the same reason, comparisons withtheoretical predictions outside the astrophysical energy rangeare only helpful when the measured cross sections show thesame sensitivities as the astrophysical reactivity. Otherwiseneither agreement nor disagreement between theory and dataallows one to draw any conclusion on the quality of theprediction of the rate. Figure 16 shows a striking exampleof how different the sensitivities can be in the astrophysicalenergy range and above it. If a measurement of 96Ru(p, ! )97Rhshowed discrepancies to predictions above, say, 5–6 MeV itwould be hard to disentangle the uncertainties from differentsources. Moreover, such a discrepancy would imply nothingconcerning the reliability of the prediction at astrophysicalenergies, as the cross section is almost exclusively sensitiveto the proton width there, whereas the proton width does notplay a role at a higher energy. If good agreement was found,on the other hand, between experiment and theory at a higherenergy, this does not constrain the uncertainty of the reactionrate. Such sensitivity considerations and plots have become anessential tool in the planning and interpretation of experiments.Further examples regarding the application of sensitivity

plots to the astrophysical interpretation of experimentaldata are found in [106, 142–144, 151, 155, 177, 183, 184] andreferences therein.

Also the impact of uncertainties in the (input) quantities onthe final rate or cross section can be studied using the relativesensitivities defined in (6.1). When an uncertainty factor Uq isattached to a quantity q, it will appear as an uncertainty factor/U = |s (/, q)| Uq in the final result.

Laboratory cross sections )0 as defined in (5.9) exhibitdifferent sensitivities in forward and reverse reactions, whereasit follows from the reciprocity relations (5.5), (5.6) that thesensitivities of the stellar reaction rates are the same for forwardand reverse reaction. It is also to be noted that even in theastrophysical energy range laboratory cross sections do notnecessarily show the same sensitivity as the stellar rate. Thisdepends on the g.s. contribution X0 to the stellar rate (seebelow) and/or the sensitivities of the transitions from excitedstates in the target nucleus. Usually the cross section in thereaction direction with larger X0 will behave more similar tothe stellar rate. See [107] for a more detailed discussion.

Section 5.3.3 pointed out a further complication, namelythe change in reaction mechanism or appearance of furtherreaction mechanisms at low energy. This applies both toreaction and scattering experiments, as the absorptive part of anoptical potential derived from scattering includes all inelasticchannels but cannot distinguish between them. For the stellarrate and the application of the reciprocity relations, however,different mechanisms may have to be accounted for differently.

Stellar effects. Under the conditions relevant to p-nucleosynthesis, all constituents of the stellar plasma,including nuclei, are in thermal equilibrium. This impliesthat a fraction of the nuclei will be present in an excitedstate. Thus, reactions not only proceed on target nuclei inthe g.s. (as in the laboratory) but also on nuclei in excitedstates. This is automatically accounted for when using thestellar cross section ) ( as defined in (5.2). The relativecontribution of transitions from g.s. and excited states to thestellar cross section, and thus to the stellar rate, depend onspin and excitation energy of the available levels, and on theplasma temperature. Again, nuclei with intermediate massand heavy ones behave differently from light species. Theyshow pronounced contributions of excited states already forneutron captures at s-process temperatures [182, 185]. Sincethe weights (5.4) of the excited state contributions also includea dependence on the energy E, which is confined to theastrophysically relevant energy window, even more levels athigher excitation energy can contribute for charged-particlereactions than for neutron captures at the same temperature.This is because the energy window is shifted to higher energiesby the cross section dependence on the Coulomb barrier.And since the nuclear burning processes synthesizing p-nucleioccur at considerably higher temperature than the s-process,strong contributions of transitions from excited states areexpected.

Even when measurements are possible directly atastrophysical energies, only the g.s. cross section )0 (or thecross section )i of a long-lived isomeric state), as defined in

22


0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1

06 08 001 021 041 061 081 002 022

X

rebmun ssam

Figure 17. G.s. contributions X0 (and isomeric state contributionX2 for 180mTa) to the stellar neutron capture rate at 2.5 GK fornatural isotopes and their uncertainties, taken from [107]. The filledsymbols indicate p-isotopes.

(5.9), can be determined in the laboratory. Therefore, it isnecessary to know the fraction of the stellar rate it actuallyconstrains. The relative contribution Xi of a specific level i tothe total stellar rate r( is given by [182]

Xi(T ) = 2Ji + 12J0 + 1

e"Ei/(kT )

')i (E)((E, T ) dE') eff(E)((E, T ) dE

, (6.2)

where )i =(

j )i%j , as before, and ) eff is the effective cross

section as given in (5.2). For the g.s., this simplifies to [185]

X0(T ) =')0(E)((E, T ) dE') eff(E)((E, T ) dE

= R0') eff(E)((E, T ) dE

.

(6.3)

It is very important to note that this is different from the simpleratio R0/R

( of g.s. and stellar reactivity, respectively, whichhas been called stellar enhancement factor in the past.

The relative contribution Xi has several convenientproperties. It only assumes values in the range 0 # Xi # 1.The value of X0 decreases monotonically with increasingplasma temperature T . It was shown in [185] that themagnitude of the uncertainty scales inversely proportionallywith the value of X0 (or generally Xi), i.e. X0 = 1 has zerouncertainty as long as G0 is known (this is the case close tostability), and that the uncertainty factor UX $ 1 of X0 is givenby max(uX, 1/uX), where

uX = u (1 " X0) + X0, (6.4)

and u is an averaged uncertainty factor in the predicted ratiosof the Ri . These ratios are believed to be predicted with betteraccuracy than the rates themselves and so it can be assumedu # Uth, with Uth being the uncertainty factor of the theoreticalprediction. In any case, the uncertainties are sufficiently smallto preserve the magnitude of X0, i.e. small X0 remain smallwithin errors and large X0 remain large, as can also be seen infigures 17–19.

Complete tables of g.s. contributions X0 for reactionson target nuclei between the driplines from 10 # Z # 83

0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1

56 07 57 08 58 09 59 001 501 011 511 021

X0

rebmun ssam

Figure 18. G.s. contributions to the stellar proton-capture rate at3 GK for natural isotopes and their uncertainties, taken from [107].The filled symbols indicate p-isotopes.

0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

1

011 021 031 041 051 061 071 081 091 002 012

X

rebmun ssam

Figure 19. G.s. contributions (and isomeric state contribution X2for 180mTa) to the stellar "-capture rate at 2 GK for natural isotopesand their uncertainties, taken from [107]. The filled symbolsindicate p-isotopes.

are given in [107]. Figures 17–19 show g.s. contributions(and isomeric state contributions X2 for 180mTa) for (n, ! ),(p, ! ) and (", ! ) reactions, respectively, on natural isotopesat ! -process temperatures, including their uncertainties. Itimmediately catches the eye that excited state contributionsare non-negligible for the majority of cases and that g.s.contributions are especially small in the region of the deformedrare-earth nuclei. This is because they have an inherently largerNLD. Near shell closures, on the other hand, NLDs are lowerand the g.s. contributions larger. An additional effect actingfor some of the (", ! ) cases is the Coloumb suppression effectof the excited state contributions, which is explained furtherbelow.

Laboratory measurements can only determine the stellarreactivity when the contribution Xi of the target level (in mostcases this is the g.s.) is close to unity. The stellar rate has tobe derived by combining experimental data with theory whenXi < 1. Strictly speaking, the experimental cross section canonly replace one of the contributions to the stellar rate while

23


the others remain unconstrained by the data, unless furtherknowledge is present. Knowing Xi and the theory valuesfor Ri and R(, the proper inclusion of a new experimentallyderived reactivity R

expi into a new stellar rate is performed by

modifying the theoretical stellar reactivity R( to yield the newstellar reactivity [182]

R(new = f (R(, (6.5)

with the renormalization factor

f ( = 1 + Xi

!R

expi

Ri

" 1"

(6.6)

containing the experimental result. Note that therenormalization factor is, of course, temperature dependent.

Also the uncertainty in the experimental cross sections(the ‘error bar’) can be included. Since the ultimate goalof a measurement is to reduce the uncertainty inherent in apurely theoretical prediction, it is of particular interest to knowthe final uncertainty of the new stellar reactivity R(

new. It isevident that the new uncertainty will only be dominated bythe experimental one when Xi is large. Using an uncertaintyfactor Uexp $ 1 implies that the ‘true’ value of R

expi is in

the range Rexpi /Uexp # Rtrue

i # Rexpi Uexp, and analogous

for the theoretical uncertainty factor Uth = U ( of the stellarreactivity R(. For example, an uncertainty of 20% translatesinto Uexp = 1.2. Experimental and theoretical uncertainty arethen properly combined to the new uncertainty factor

U (new = Uexp + (U ( " Uexp)(1 " Xi) (6.7)

for R(new. Here, Uexp # U ( is assumed because otherwise the

measurement would not provide an improvement. Obviously,the uncertainty factor is also temperature dependent becauseat least Xi depends on the plasma temperature. It is furtherpossible to consider the uncertainty of Xi in U (

new. Its impact,however, is small with respect to the other experimentaland theoretical uncertainties. The fundamental differencesbetween error determinations in experiment and theory arediscussed in [107] and appropriate choices are suggested insection 6.2 and in [182].

It should be kept in mind that the corrections for thermallyexcited nuclei have to be predicted even if a full database ofexperimental (g.s.) rates were available. Therefore, designingan experiment it should be taken care to measure a reactionwith the largest possible Xi . This also implies that the reactionshould be measured in the direction of largest Xi . Sincethe astrophysically relevant energies of the reverse reactionB(b, a)A are related to the ones of the forward reaction byErev = E + Q (see section 5.3.2), it is obvious that transitionsfrom excited states contribute more to the stellar rate in thedirection of negative Q-value. This is due to the fact that theweights given in (5.4) depend on E and Erev, respectively,and decline more slowly with increasing excitation energyEi when E is larger. This is especially pronounced inphotodisintegrations because of the large |Q|. Table 3 givesexamples for g.s. contributions to (! , n) rates of intermediateand heavy target nuclides at typical ! -process temperatures.These numbers have to be compared with the ones for the

Table 3. G.s. contributions X0 for selected (! , n) reactionsat 2.5 GK.

Target X0 Target X0 Target X0

86Sr 0.000 59 186W 0.000 49 198Pt 0.001 890Zr 0.000 34 185Re 0.000 21 197Au 0.000 3596Zr 0.006 1 187Re 0.000 24 196Hg 0.000 4394Mo 0.004 3 186Os 0.000 16 198Hg 0.000 84142Nd 0.002 8 190Pt 0.000 069 204Hg 0.008 8155Gd 0.001 2 192Pt 0.000 11 204Pb 0.005 9

neutron captures shown in figure 17. It is obvious that theyare tiny in comparison. (This is also the reason why we do notshow full plots similar to the ones for the capture reactions.All g.s. contributions and further plots can be found in [107].)

As argued above, the smallest stellar correction is usuallyfound in the direction of positive Q-value. Figure 19and the tables in [107], however, contain a few exceptionsto the Q-value rule. For example, (", ! ) reactions withnegative Q-values appear in the ! -process network and showsmaller corrections than their photodisintegration counterparts.Among them is, for example, the important case of144Sm(", ! )148Gd with X0 $ 1. These cases can be explainedby a Coulomb suppression effect, first pointed out in [153]and studied in detail in [154]. Since the relative interactionenergies Erel

i = E " Ei of transitions commencing on excitedlevels decrease with increasing Ei , cross sections )i stronglydepending on Erel

i will be suppressed for higher lying levels.For large Coulomb barriers this can result in a much fastersuppression of excited state contributions to the stellar ratethan expected from the weights Wi . If there is a much higherCoulomb barrier in the entrance channel of a reaction withnegative Q-value than in its exit channel (e.g. when there is nobarrier present as for (", ! ) or (p, n) reactions), this can yield alarger X0 in the direction of negative Q-value than in the otherdirection. On these grounds it can be shown that it is generallymuch more advantageous to measure in the capture directionfor any type of capture reaction, regardless of Q-value, becausethe X0 will be larger than for the photodisintegration andthus closer to the stellar value. This stellar value can thenconveniently be converted to a stellar photodisintegration rateby applying (5.6).

How experiments can help. Due to the high plasmatemperatures encountered in explosive nucleosynthesis and thelarge number of possible transitions between levels in target,compound and final nucleus, an experimental determination ofthe stellar rate is impossible for the majority of intermediateand heavy nuclei. The few exceptions can be selected bysearching for reactions with large g.s. contribution X0. Afurther constraint is the fact that most of the reactions involveunstable nuclei. Nevertheless, measurements can provideimportant information on specific transitions and parts ofthe stellar rate, which can be compared with predictionsof theoretical models. A good example is the past (", ! )and (p, ! ) measurements at low energies which have showndeficiencies in the description of low-energy charged-particlewidths (see section 6.2). Another example would be studies ofsingle transitions to better constrain excited state contributions.

24


Essential for the development of reaction models, which canalso be extended into the region of unstable nuclei, is to havesystematic measurements across larger mass ranges. Suchsystematics are lacking even at stability for astrophysicallyimportant reactions and transitions at astrophysical energies.

The following section 7 provides an extensive overviewof present and future experimental approaches to improvereaction rates for studying the synthesis of p-nuclides.

7. Experimental approaches

7.1. ! -induced measurements

As explained in section 6.3.2 and shown in table 3, theexperimental study of ! -induced reactions has only limitedrelevance for the direct astrophysical application. From alaboratory ! -induced cross section measurement with thetarget nucleus being in its g.s., no direct information can beinferred for the stellar reaction rate. Nevertheless, as pointedout in section 6.3.2, the experimental study of ! -inducedreactions can provide useful information for certain nuclearproperties relevant to heavy element nucleosynthesis. Theexperimentally determined (! , x) cross section (where x can bea neutron, a proton, or an "-particle) probes particle transitionsinto the final nucleus initiated by a well-defined ! -transitionfrom the g.s. of the target nucleus. Theoretical predictions forthe relevant transitions can then be tested by comparing themwith the measured photodisintegration cross section.

Again, the sensitivity of the reaction (see sections 6.2and 6.3.2) has to be carefully checked to see what can beextracted from such a measurement. The sensitivities dependon the initial ! -energy and the particle separation energies.The latter define the relative interaction energies Erel

i (i.e.the energies of the emitted particles) and thus the particlewidths. As mentioned in section 6.2, the channel withthe smallest width determines the sensitivity of the crosssection. For laboratory (! , n) reactions, this is always the! -width. A comparison of predicted laboratory (! , n) crosssections and measured ones, however, does not test a quantitythat is of relevance in the astrophysical application for tworeasons. Firstly, astrophysical charged-particle captures andphotodisintegrations depend solely on the charged-particlewidths as can be verified by inspection of the figures and tablesgiven in [107]. Secondly, although astrophysical neutroncaptures and (! , n) do partially depend on the ! -widths, theyare sensitive to a different part of the ! -strength function thancan be tested in a laboratory experiment. The relevant ! -transitions are those with 2–4 MeV downward from states afew tens to a few hundreds of keV above the neutron threshold[138]. According to the reciprocity relation (5.6), this appliesto stellar capture rates as well as photodisintegration rates.Assuming the validity of the Brink–Axel hypothesis [186, 187]and using low ! -energies on the nuclear g.s. does not helpbecause the strength function cannot be probed below theparticle emission threshold. A more promising approach isto study partial particle emission cross sections, such as (! ,n0), (! , n1), and so on. Their ratio depends, apart from spinselection rules, on the particle emitting transitions and can

thus be used to test the prediction of the ratios of g.s. particletransitions to transitions on excited states. This is then relevantto astrophysical applications, to study the interaction potentialsand the stellar excitation effects (see, e.g., section 6.3.2). On-line detection of the outgoing particle (see below) is requiredfor this.

The same can be carried out for charged-particle emission.In addition, depending on the nucleus and the ! -energy used,photodisintegration with charged-particle emission is sensitiveto the particle channel and can thus be used to test, e.g., opticalpotentials. Comparing "-emission data with predictionswould be of particular interest regarding the action of theCoulomb excitation effect described in section 5.3.3. It wouldimply different transmission coefficients for "-capture and "-emission and should not show up in the photodisintegrationresults. Partial cross sections are helpful also in this casebecause they allow testing at different relative interactionenergies.

In the remainder of this section the available experimentaltechniques and those ! -induced reaction cross sectionmeasurements are reviewed for which the ! -process wasspecified as motivation. Experimental data on ! -inducedreactions are mainly available around the giant dipoleresonance, i.e. at much higher energy than important forastrophysics. Owing to the small cross section and othertechnical difficulties (see below), only very few measured crosssections are available at lower energies.

Different experimental approaches can be used to measurelow-energy ! -induced reaction cross sections. The commonrequirement of these techniques is a high ! -flux in order tomeasure low cross sections. If the aim is to measure excitationfunctions (i.e. the cross section as a function of energy),a monoenergetic ! -beam is needed. Quasi-monoenergetic! -rays can be obtained with the laser Compton scatteringtechnique. In this method ! -rays are produced by head-oncollision of laser photons with relativistic electrons. Thismethod has been successfully applied for several reactions,e.g. at the National Institute of Advanced Industrial Scienceand Technology (AIST), Tsukuba, Japan [188–190]. Thereare proposals for performing ! -process studies also at the HighIntensity ! -Ray Source (HI!S) at Duke University, USA.

Another method for ! -induced reaction studies is theapplication of bremsstrahlung radiation. If a high-energyelectron beam hits a radiator target, continuous energybremsstrahlung radiation is produced with an endpoint energyequal to the energy of the electron beam. Since the ! -energy spectrum is continuous, no direct measurement ofthe excitation function is possible. By the superposition ofseveral bremsstrahlung spectra of different endpoint energies,however, the high-energy part of a thermal photon flux canbe approximated well [191]. Therefore, the reaction rate ofa photodisintegration reaction can be measured instead of thecross section itself. Since the target is always in its g.s., onlythe g.s. contribution to the reaction rate can, of course, bemeasured. Several ! -induced reactions have been studied withthis method at the S-DALINAC facility in Darmstadt, Germany[191–195], and at the ELBE facility in Dresden-Rossendorf,Germany [196–198] (see table 4).

25


Table 4. Experimentally studied ! -induced reactions relevant for the ! -process. Abbreviations: act. = activation, br. = bremsstrahlung,CD = Coulomb dissociation, LCS = laser Compton scattering, n.c. = neutron counting.

Target Reaction Method Facility Reference

91,92,94Zr (! , n) LCS, n.c. AIST [190]92,100Mo, 144Sm (! , n), (! , p), (! ,") br., act. ELBE [196–198]92,93,94,100Mo (! , n) CD, n.c. GSI-LAND [201–203]148,150Nd, 154Sm, 154,160Gd (! , n) br., act. S–DALINAC [194]181Ta (! , n) LCS, n.c. AIST [188]186W, 187Re, 188Os (! , n) LCS, n.c. AIST [189]192Os, 191,193Ir (! , n) br., act. S–DALINAC [195]190,192,198Pt (! , n) br., act. S–DALINAC [191, 192]196,198,204Hg, 204Pb (! , n) br., act. S–DALINAC [193]

The bremsstrahlung radiation method can be improvedtoward a quasi-monochromatic ! -beam by applying the so-called tagger technique. In this method, the electron beamhits a thin radiator target where it can be guaranteed that onlyone bremsstrahlung photon is produced by one electron. Ifthe remaining energy of the electron is then measured, the! -energy can be inferred and the required ! -energy withrelatively low-energy spread can be selected. Such a methodhas been recently developed at the S-DALINAC facility [199].

Intense ! -beams created by ultra-high intensity lasers mayopen a new horizon in the study of ! -induced reactions inthe near future. One of the four pillars of the European ELI(Extreme Light Infrastructure) facility to be built in Magurele,Romania, will be devoted to laser-based nuclear physics (ELI-NP). The white book of the ELI-NP project [200] includesthe study of the ! -process as one of the objectives of thefacility. The very brilliant, intense ! -beam of up to 19 MeV,0.1% bandwidth, and 1013 ! /s intensity is hoped to enable themeasurement of (! ,") and (! , p) reaction cross sections onmany isotopes.

Two distinct methods can be applied to determine thenumber of reactions taking place during the ! -irradiation.The first method is based on the on-line detection of theoutgoing particle. This method has been successfully appliedonly for (! , n) reactions because charged-particle emittingreactions have typically lower cross section and the detectionof the resulting low yields in a high ! -flux environmentrequires special experimental technique. The other methodis photoactivation, where the cross section is determinedfrom the off-line measurement of the induced activity ofthe irradiated target. This method is, of course, onlyapplicable when the product nucleus is radioactive, but owingto its technical advantages the majority of the ! -processrelated photodisintegration measurements have been carriedout with this technique. Further details of the in-beam andactivation methods will be discussed in connection with thecharged-particle induced reaction cross section measurementsin section 7.2.

All the above mentioned experimental techniques can onlybe applied to stable target nuclei. For the ! -process, however,photon-induced reactions on proton-rich unstable isotopes arealso important. These reactions can in principle be studiedin inverse kinematics by the Coulomb dissociation method.Coulomb dissociation is a well-known technique, and is, e.g.,being studied at the GSI Helmholtz Center for Heavy Ion

Research in Darmstadt, Germany, with the LAND (LargeArray Neutron Detector) setup. Results for 92,93,100Mo(! , n)have been published recently, the analysis for 94Mo(! , n) isongoing [201–203].

Table 4 lists some of the ! -induced reactions studiedexperimentally in recent years. It is not an exhaustive listsince only those measurements are listed where the ! -processwas mentioned as a motivation of the work. As one can see,with the exception of the 92Mo and 144Sm isotopes, only (! , n)reactions have been studied. The reason for this is the typicallyvery low cross section of charged-particle emitting reactionsat astrophysically relevant energies. The fast development ofthe different experimental methods, however, will most likelyallow for the extensive study of (! ,") and (! , p) reactions inthe near future.

The photodisintegration cross sections are usuallycompared with the predictions of statistical model calculations.Calculations using different input parameters such as ! -raystrength functions are also considered and compared with thedata in several works. It can be stated that in general themodel calculations are able to reproduce the measured datawithin about a factor of two. For further details, the reader isreferred to the original publications, for example the detailedoverview on the direct determination of photodisintegrationcross sections related to the ! -process in [204] and referencestherein.

7.2. Charged-particle induced measurements

Owing to the huge effect of the stellar enhancement factorin the case of ! -induced reactions, it is preferable tostudy experimentally the inverse capture reactions (seesection 6.3.2). As shown in section 6.2, nuclear physicsuncertainties influence strongly the result of a ! -processnetwork calculation, therefore the cross section measurementof capture reactions in the relevant mass and energy range isof high importance. The case of neutron-induced reactionsis discussed in section 7.4. In this section the crosssection measurements of charged-particle induced reactionsare reviewed.

The fundamental difficulty of charged-particle inducedreaction studies is the low values of the cross sections. Table 5shows the case of two p-isotopes, the lowest mass 74Se andthe highest mass 196Hg. The table shows the location of therelevant energy window for proton- and "-capture reactions at

26


Table 5. Location of the Gamow window and the relevant crosssections of proton and "-capture reactions on two p-isotopes.

GamowTemperature window Cross section

Reaction (109 K) (MeV) (barn)

74Se(p, ! )75Br 3.0 1.39–3.07 2 # 10"6–1 # 10"3

74Se(", ! )78Kr 3.0 4.65–7.15 5 # 10"8–3 # 10"4

196Hg(p, ! )197Tl 2.0 2.56–4.64 1 # 10"10–6 # 10"6

196Hg(", ! )200Pb 2.0 6.93–10.03 1 # 10"18–1 # 10"10

some relevant temperatures for these two isotopes as well asthe range of cross section within the energy window obtainedwith the NON-SMOKER code [119]. As one can see, thecross sections range from the millibarn region (in the case ofproton capture on light nuclei) down to the 10"18 barn region(for "-capture on heavy isotopes). Unfortunately only theupper part of this cross section range is measurable, thus theexperiments need to be carried out at higher energies and theresults must be extrapolated to astrophysical energies. Thisis carried out based on theoretical cross section curves, sothe involvement of some theory is inevitable. To minimizeits effect, however, the experiments should be carried out atenergies as low as possible and hence low cross sections needto be measured.

Due to the low cross sections encountered, experimentaldata on proton and "-capture reactions are scarce at lowenergies in the mass region relevant to the ! -process.Experimental data started to accumulate only in the last15 years. Still, the number of studied reactions remainsrelatively small (see figures 14 and 15 in section 6.3.1)compared with the huge number of reactions involved in a! -process network. There are two different methods for crosssection measurements: the in-beam ! -detection technique andactivation. In the following some features of these two methodsare discussed.

7.2.1. In-beam ! -detection technique. The natural way ofmeasuring a capture cross section is the detection of the prompt! -radiation. In the relevant mass and energy range capturereactions mainly proceed through the formation of a compoundnucleus which then decays to its g.s. by the emission of ! -radiation. The excitation energy of the formed compoundnucleus is typically above 10 MeV in ! -process relatedexperiments and thus the level density is very high. Therefore,the particle capture can populate many nuclear levels whichresults in a complicated ! -decay scheme involving manyprimary and secondary transitions. In order to determine thetotal capture cross sections, practically all these transitionsneed to be detected. Any unnoticed transitions can result inan underestimation of the cross section (unless one dominanttransition exists through which all de-excitation ! -cascadeshave to pass). This means that the in-beam ! -detectiontechnique is very sensitive to laboratory background, andeven more to beam-induced background. Weaker transitionsin the investigated reaction can be buried under the peaksand Compton continuum of higher cross section reactionson target impurities, therefore an effective reduction of the

background is crucial. Moreover, transitions during the decayof the compound nucleus can proceed between states withvarious and often unknown spin and parity. Therefore, theangular distribution of the measured ! -radiations is usuallynot isotropic. The measurement of these angular distributionsfor all the studied transitions is thus also necessary in order toobtain total, angle-integrated cross sections.

Despite the experimental obstacles, some cross sectionsrelevant for the ! -process have successfully been measured.These measurements are dominated by proton-capturereactions since in the case of (", ! ) reactions the beam-inducedbackground compared with the signal from the studied reactionis usually much stronger. First experiments with this techniquehave been carried out using a few HPGe detectors mounted ona turntable allowing for angular distribution measurements.Proton-capture cross sections on 93Nb [161], 88Sr [158] and74Ge [144] isotopes have been measured with this techniquein Athens, Greece and Stuttgart, Germany.

The detection efficiency can be increased and themeasurement of angular distributions sped up using a detectorarray consisting of many single ! -detectors arranged in aspherical geometry around the target. Additionally, such aconfiguration allows a substantial reduction of the backgroundby requiring coincidence conditions between the singledetectors in the array. Such a method has been developedrecently at the University of Cologne, Germany [171, 175]

Most of the disadvantages of the in-beam technique canbe avoided by using a 4+ summing crystal for ! -detection[164]. If the target is completely surrounded by, e.g., alarge scintillator detector with a time resolution longer thanthe typical time interval between the successive ! -emissionsduring the compound nucleus decay, then one single ! -peakfor all capture events will appear in the spectrum. The energyof this so-called sum-peak is the sum of the reaction Q-value and the c.m. energy. There is, of course, no needfor angular distribution measurements in this case. Caremust be taken, however, to remove some possible sources ofuncertainty. The energy resolution of a scintillator is poorcompared with a HPGe detector which results in a relativelywide sum-peak. The sum-peak may also contain unwantedevents from reactions on target impurities, if those have similarQ-values than the reaction to be studied. This would lead toan overestimated cross section. The condition that all captureevents generate a signal in the sum-peak is only valid if thedetector has a 100% efficiency for all ! -energies. Althoughthe efficiency of a big summing crystal can be fairly large,it is never 100%. Thus, the sum-peak efficiency must bedetermined experimentally which requires the knowledge of! -ray multiplicities. Several reactions have been studied with thismethod in Athens, Greece [159], in Bochum, Germany [150],and in Bucharest, Romania [167]. The analysis of most of theBochum measurements is still in progress [170].

7.2.2. Activation method. The overwhelming majority of! -process related charged-particle capture cross sections hasbeen measured with the activation technique. In this method,the total number of reactions having taken place is determinedthrough the number of product nuclei, instead of detecting

27


the prompt ! -radiation following the capture process. This isfeasible when the final nucleus is radioactive, decays with aconvenient half-life, and the decay can be measured throughthe detection of a suitable high-intensity radiation.

When a target with areal number density T is bombardedby a proton or "-beam with 0 projectiles per second withenergy E for an irradiation time tirr , then the number ofproduced isotopes N at the end of the irradiation is given by

N = ) (E)T 01 " e",tirr

,, (7.1)

where ) (E) is the reaction cross section and , is the decayconstant of the produced radioactive isotope. If the beamintensity is not constant in time, the formula becomes

N = ) (E)

n$

i=1

T 0i

1 " e",ti

,e",&i , nti = tirr. (7.2)

Here the irradiation time is segmented inton shorter intervals ti .During the shorter intervals the beam intensity 0i is consideredto be constant. Supposing the produced isotope emits ! -radiation with relative intensity 1, then the number of gammasdetected during a subsequent counting interval of tc is

n! = Ne",tw (1 " e",tc )2! 1, (7.3)

where 2! is the detection efficiency for the studied ! -line andtw is the time elapsed between the end of the irradiation andthe start of counting. The cross section can be deduced fromthe above equations when the other quantities are known. Ifthe final nucleus has long-lived isomeric state(s), the aboveformulae become more complicated, see, e.g., [162].

The applicability of the activation method is, of course,limited to reactions leading to radioactive isotopes and noinformation about the details of the ! -transitions during thecompound nucleus decay can be obtained. These limitationsare, however, compensated by the relative easiness of theactivation experiments compared with in-beam measurements.The total cross section is naturally obtained without anyproblems of possibly missed ! -transitions. No angulardistribution effects have to be taken into account. The mostimportant advantage is the typically much lower background.If target impurities—on which long-lived radioactive isotopeswould be produced—are avoided, the background canessentially be reduced to the laboratory background whichcan be effectively shielded. Some beam-induced radioactivityof the target often cannot be completely avoided but thebackground level is always much lower than for in-beamexperiments.

Owing to the lower background, it is possible to studymore than one reaction in a single activation since differentisotopes are characterized by different decay signatures. Usinga target with natural isotopic composition target, the capturecross section of several isotopes of the same element can bemeasured simultaneously (see, e.g., [157]). Also, in additionto the radiative capture, some other reaction channels of thesame isotope can be measured at the same time, such as (", n)and (", p) reactions along with (", ! ) (see, e.g., [142]).

In the mass range relevant for the ! -process the createdradioisotopes are almost always #-radioactive and the decayis often followed by ! -radiation. Therefore, the majority ofthe activation measurements has been based on ! -detection.One exception was the study of the 144Sm(", ! )148Gd reaction,where the produced 148Gd is"-radioactive and the cross sectionwas measured via "-detection. In some cases the #-decay isnot followed by ! -radiation or its intensity is very low. Ifthe decay, however, proceeds through electron capture, thedetection of the emitted characteristic x-ray radiation can beused to determine the cross section. This method has beenused for the first time in the case of the 169Tm(", ! )173Lureaction [140].

It is worth noting that an activation experiment requiresknowledge of the decay parameters of the produced isotope,such as the relative ! -intensities or the decay half-lives.Dedicated half-life measurements of several isotopes havebeen carried out recently to aid the activation experiments[205–208].

7.2.3. Indirect measurements with (d, p) or (d, n) reactions.There is considerable experience in performing indirectmeasurements of reaction cross sections with (d, p) and (d,n) reactions. With light target nuclei and at sufficiently highenergy these can probe states and transitions also importantin, say, capture reactions. The prerequisite for this is thatdirect reactions dominate and can be described, e.g., by thedistorted wave Born approximation (DWBA) [123]. Giventhe success in the realm of light nuclei, such reactions at highenergies—several tens to hundreds of MeV, for instance—havealso been suggested to be used in studying reactions at highermass for explosive thermonuclear burning, including those faroff stability. This method has already been applied to studyproperties of nuclei at neutron shell closures in the r-processpath [209, 210].

The benefits of this method for the ! - and $p-process,however, are more limited because the NLD at the compoundformation energy is much higher in this case and the directreaction mechanism does not contribute significantly at theastrophysically relevant energies. Therefore, many moretransitions are involved and the compound reaction mechanismdominates at astrophysical energies. Such measurements,however, could be used for spectroscopic studies of proton-rich, unstable nuclei at radioactive ion beam facilities. Theextracted information on excited states and spectroscopicfactors is useful in the calculation of the widths appearing in thetreatment of compound reactions at low energies. Important isthe determination of low-lying states, as these are significant forthe calculation of the particle widths and the stellar excitationeffects (see sections 5.2, 6.2, 6.3.2 and [106, 107]). It remainsto be seen whether the optical potentials required in theDWBA analysis of the experimental data can be predicted withsufficient reliability to extract the properties of the discretestates. (It should be noted that these potentials are differentfrom the ones used in the statistical model at low energies.)

7.2.4. Novel approaches. One serious limitation of theactivation method occurs when the half-life of the reaction

28


product is very long and/or its decay is not followed by anyeasily measurable radiation. Then the methods as describedabove cannot be applied and novel techniques have to bedeveloped. The number of produced isotopes can also bemeasured by accelerator mass spectrometry (AMS). However,also in this case there is a restriction to reactions with stabletarget isotopes. To study reactions on unstable, proton-richisotopes special experimental techniques are required. One ofthese new techniques is to use reactions in inverse kinematicsin storage rings such as, e.g., the Experimental Storage Ring(ESR) at the GSI Helmholtz Center for Heavy Ion Researchin Darmstadt, Germany [211]. Some details of these twoapproaches are discussed below.

Measurements using AMS. AMS is an ultra-sensitive andultra-selective analytical method for the detection of traceamounts (sub-ng range) of long-lived radioactive isotopes[212]. It is most commonly used for dating of archaeologicaland geological samples, e.g. with 14C (radiocarbon dating).The method relies on the ability of counting atoms rather thanthe respective decays and is thus superior to most other methodsfor the detection of long-lived isotopes (t1/2 > 104 y) or forthose isotopes which emit no or only weak ! -radiation. One ofthe major challenges for AMS measurements is the suppressionand separation of (stable) isobaric interferences. For a moredetailed description, see, e.g., [213].

AMS is a relative method which measures the amountof a radionuclide versus the ion current of a stable isotope,ideally of the same element. It is up to now the most sensitivedetection method and can reach isotope ratios down to 10"16.However, it requires standards with well-known isotope ratios(of the same order as the measured samples). The productionof these calibration standards via nuclear reactions is notstraightforward. Preparations for non-standard AMS isotopes,such as the ones discussed in this section, involve additionalcomplications.

A few years ago AMS was successfully combinedwith astrophysical activation measurements, mainly forthe determination of (n, ! ) cross sections for s-processnucleosynthesis (see, e.g., [214–216]), and for the 25Mg(p,! )26gAl and 40Ca(", ! )44Ti cross sections [217, 218]. Alsofirst attempts to use AMS to determine the photodissociationcross section of the 64Ni(! , n)63Ni reaction have been made[219]. An overview of cross section measurements for nuclearastrophysics performed with AMS is given in [220].

Searching the chart of nuclides on the proton-rich sideof the valley of stability for radioactive isotopes with half-lives larger than 250 days and masses in the range 68 #A # 209 provides 32 matches. Discarding isotopes forwhich a measurement of the "- or ! -activity is possiblewith normal efforts and those posing severe problems for theAMS technique leaves 68Ge, 93Mo, 146Sm, 179Ta, 194Hg, 202Pband 205Pb.

First tests for AMS have been already performed with93Mo [221], 146Sm [222] and 202Pb [223]. The activity of68Ge could probably be deduced from its short-lived daughter68Ga. However, the small cross section of the 64Zn(", ! )68Gereaction prevented measurements at astrophysically relevant

energies (E" < 7.5 MeV) up to now [224–226]. Tests arecurrently being carried out to determine this cross section withAMS at lower energies [227].

Rare-earth elements are known to poison the ionizer andlead to strongly declining beam currents. Only 146Sm has thepotential for further investigations, using special cathodes. Thereaction 142Nd(", ! )146Sm is planned to be measured in thenear future [228].

The reaction 190Pt(", ! )194Hg is also in reach of AMSbut applications in other fields are missing and the samplematerial is expensive. The two long-lived isotopes 202Pb and205Pb are produced in lead-cooled fast (generation IV) reactors.However, lead sputters quickly in the ion sources, preventingthe production of ion beams that are stable for long periods oftime. The measurement of the 198Hg(", ! )202Pb cross sectionis under consideration [227].

AMS could be an alternative for some cross sectionmeasurements in the ! -process region. However, the effortsneeded to develop the method for non-standard isotopes arecomplicated and time-consuming. Unfortunately they can onlybe justified with additional requests from other research fields,e.g. geology, archeology and fission or fusion technology.

Measurements using a storage ring. All the charged-particlecapture experiments discussed so far are restricted to reactionson stable target isotopes. For explosive nucleosynthesisprocesses, however, reactions on unstable, proton-rich isotopesare also important. The deflection points in the ! -process path lie off stability (see section 6.1). Crosssection measurements on radioactive isotopes require specialexperimental techniques. Radioactive beam facilities havebeen developed over the last years which are suitable for suchmeasurements.

A pioneering experiment was carried out at the storage ringESR in 2009 [229]. Proton-induced reactions were measuredin inverse kinematics involving fully stripped 96Ru44+ ionswhich had been injected into the storage ring and slowed down.This stable beam experiment was an important step for futureinvestigations of charged-particle reactions with radioactivebeams in inverse kinematics. First results at high energywere published recently [229], yielding an upper limit of 4 mbfor 96Ru(p, ! )96Rh, which is in good agreement with thepredictions from the Hauser–Feshbach code NON-SMOKER[119]. This first test measurement was limited to energiesabove the astrophysically interesting energy range for technicalreasons. For future experiments, improved detectors are beingdeveloped to allow measurements at or close to astrophysicalenergies.

7.3. "-elastic scattering

As was emphasized in section 7.2 and also earlier inthis paper, one of the most important input parameters ofthe statistical model calculation in the case of "-inducedreactions is the "-nucleus optical potential. Therefore, thedirect experimental investigation of this potential is of highimportance. The optical potential can be studied by measuringelastic "-scattering cross sections. The measurement of

29


Table 6. Some parameters of elastic "-scattering experimentsrelevant for the ! -process. All measurements have been carried outat ATOMKI with the exception of the ones on the Te isotopes whichhave been studied at the University of Notre Dame.

"-energies AngularIsotope(s) (MeV) range (deg) Reference

144Sm 20 15–172 [145]92Mo 13.8, 16.4, 19.5 20–170 [230]112,124Sn 14.4, 19.5 20–170 [231]106Cd 16.1, 17.7, 19.6 20–170 [232]89Y 16.2, 19.5 20–170 [233]110,116Cd 16.1 19.5 20–175 [234]120,124,126,128,130Te 17, 19, 22, 24.5, 27 22–168 [235]

elastic scattering cross sections at several tens or hundredsof MeV is a well established experimental tool in nuclearphysics and high-energy optical potentials for many stablenuclei are thus quite well known. At low, astrophysicallyrelevant energies, however, elastic scattering is dominated bythe Coulomb interaction (Rutherford cross section), makingthe experimental study of the nuclear part of the potential ratherchallenging.

In order to find substantial deviation from the Rutherfordcross section, elastic scattering experiments for ! -processpurposes are carried out at energies somewhat higher thanastrophysically relevant. Angular distributions in a wideangular range need to be measured in order to see thetypical oscillation pattern of the cross section as a functionof scattering angle. The experiments must be carried out withhigh precision in order to measure the tiny deviations fromthe Rutherford cross section. The precise determination of thescattering angle is also of high importance since the Rutherfordcross section, to which the scattering data are normalized,depends very sensitively on the angle.

Several elastic "-scattering experiments have been carriedout at the Institute of Nuclear Research (ATOMKI) inDebrecen, Hungary. A similar research program has beeninitiated recently at the University of Notre Dame, USA.The studied isotopes, as well as some parameters of theexperiments, are listed in table 6.

The measured angular distributions can be compared withpredictions using different optical potential parameterizations.Global optical potentials (i.e. potentials that are designed forbroad mass and energy regions) are preferred since they canbe used in extended ! -process networks. Several globalpotentials are available in the literature, for a list see, e.g., [233].It is found that different potentials often lead to largely differentangular distributions and by comparison with the measureddata the one with the best match can be selected. More sensitiveanalyses can be performed when the ratio of the measuredcross sections of two isotopes studied at the same energy iscalculated and compared with the corresponding ratio givenby global potentials. Such a comparison has been carriedout, e.g. for the 112,124Sn (Z = 50) and the 106,110,116Cd(Z = 48) isotopes, and the 89Y, 92Mo (N = 50) isotones. Noneof the available global potentials seem to describe well thecross section ratios, clearly indicating the need for improvedpotential parameterizations at low energies.

In addition to the comparison with global potentials, themeasured angular distributions can also be used to constrainsome parameters relevant for the given isotope. By fittingthe measured cross sections using various approaches, such asWoods–Saxon parametrizations or double folding potentials,local optical potential parameters can be obtained. Bystudying several isotopes, the evolution of the best fit potentialparameters can also be investigated. If the angular distributionis measured in an almost complete angular range, total reactioncross sections can be easily obtained by simply calculating themissing flux from the elastic channel [236]. The calculatedtotal cross section can then be compared with experimentaldata, if they are available, or with model predictions [237].

The possible appearance of additional reaction channelsat low energies, such as the Coulomb excitation introduced insection 5.3.3, complicates the interpretation of the scatteringdata for "-particles. If an optical potential is derivedfrom scattering data at an energy where compound nucleusformation is the dominant reaction mechanism, its absorptivepart will only account for this loss of "-flux from the elasticchannel. When the extrapolated potential is then used at lowerenergies at which low-energy Coulomb excitation (or any otheradditional mechanism) acts, it will underestimate the inelastic(reaction) cross section at these energies. Nevertheless, it maybe correctly describing the compound formation probabilityand thus will be appropriate for calculating stellar (! ,") rates,as explained in section 5.3.3. If an optical potential, on theother hand, is derived from scattering data for energies at whichthe additional mechanism is non-negligible, its absorptive partwill include this additional mechanism. Unfortunately, thisdoes not by itself help in the application to the stellar rate,as the compound formation cross section is not constrainedseparately. This would only be possible using additionaltheory, i.e. by calculating the expected cross section for theadditional mechanism (e.g. Coulomb excitation) and thenadjusting the absorption in the optical potential in such a waythat it yields a flux into inelastic channels that is the original onesubtracted by the one going into the direct channel. Most likely,such a modification will introduce additional parameters.

Regardless of the possible complications at low energies,precise low-energy elastic "-scattering experiments canprovide useful information to better understand the opticalpotential behavior at astrophysical energies and furtherexperiments are needed.

7.4. Neutron-induced measurements

7.4.1. Neutron captures. Neutron captures in astrophysicshave been comprehensively studied along stability for s-process nucleosynthesis. A series of publications has compiledthe data and provided recommended values [180, 238, 239].These focused, however, on the energy range relevant for thes-process, which is at lower energies than required in the ! -process. For many of the quoted reactions cross sections arenot even available across the s-process energy range, as onlyMACS at 25 keV were studied. Quasi-stellar neutron spectraat kT = 25 keV (which is close to the dominant s-processtemperature) were produced. When an energy dependence

30


was required, renormalized theoretical cross sections wereused. As mentioned in section 6.3.1, it was shown onlyrecently how to correctly account for stellar effects in therenormalization. The newly renormalized reactivities as afunction of kT , obtained by application of (6.6), are givenin [182].

As has been pointed out in section 6.3.1, almost no dataare available for ! -process energies. This includes crosssections from major libraries (e.g. ENDF/B [240], JEFF [241],JENDL [242]) which are based on theoretical values. A cross-comparison is made difficult by the fact that different, partlyundocumented calculations were adopted. Even within a givenlibrary, different reaction codes were used. Renormalizedtheory values could in principle also be used to cover theenergy range up to several hundreds of keV, as required forthe ! -process. This has two disadvantages, however. Firstly,it is not always clear whether the statistical model, usedfor cross section predictions, is applicable at low energies,where data are available. If not, the renormalization shouldrather be performed at a higher energy. Secondly, accordingto (6.7) and realizing that g.s. contributions (6.3) are smallat ! -process temperatures (see, e.g., figure 17), it turns outthat reactivities at higher temperature are not constrainedstrongly by the experimental data. This may also discouragedirect measurements at ! -process energies but neverthelesscan such data be used to test theoretical reaction models forg.s. reactivities, as is carried out for reactions with chargedparticles?

The only way to obtain laboratory reactivities forhigher temperatures is to provide cross section measurementscovering the astrophysical energy region. Similar to thestudy of charged-particle reactions described in section 7.2,activation of target material can be used or in-beammeasurements at time-of-flight facilities can be performed.Also, a combination of AMS (section 7.2.4) and activationcan be applied. When using activation techniques it is crucialnot to apply Maxwellian neutron spectra. Such neutron spectraare only useful when the g.s. contribution (6.3) is large and thestellar rate can be measured directly. This is not the case forneutron captures in the ! -process.

A number of neutron time-of-flight facilities havecontributed to astrophysical measurements in the past, amongthem the Oak Ridge Electron Linear Accelerator (ORELA),USA; a similar facility, GELINA, at the IRMM in Geel,Belgium; the DANCE setup at the Los Alamos NeutronScience Center (LANSCE). Also dedicated to astrophysicalmeasurements, the nTOF facility at CERN has been veryproductive in recent years. The facility uses high-energyprotons impinging on a lead spallation target to produce apulsed neutron beam. A large neutron energy range and a highinstantaneous neutron flux combined with high resolution dueto the long neutron flightpath are among the key characteristicsof the facility. Since 2010, the experimental area has beenmodified to allow the extension of the physics program toinclude neutron-induced reactions on radioactive isotopes[243]. This facility would also be well suited to provide thedata required for calculating laboratory reactivities for neutroncaptures in the ! -process.

A new facility with focus on nuclear astrophysics, theFrankfurt Neutron Source (FRANZ), is under construction atthe University of Frankfurt [244]. It will provide the highestneutron flux in the keV region worldwide and therefore againbe best suited for neutron capture in the s-process.

The study of neutron captures on unstable nuclei atradioactive ion beam facilities using inverse kinematics is notpossible due to the unavailability of a neutron target. Seesection 7.2.3 for a discussion of the application of (d, p)reactions instead.

7.4.2. Inelastic neutron scattering. It has been pointed outin section 7.1 that it is useful to study transitions to excitedstates in the final nucleus through particle emission. This way,transitions appearing due to the thermal excitation of statesin the stellar plasma can be investigated. Inelastic neutronscattering, i.e. (n,n+), provides another approach to achieve this.

For example, inelastic neutron scattering has been usedto probe neutron transitions in 187Os. Measurements withastrophysical motivation were performed, e.g. at ORELA [245]and at nTOF [246]. This nucleus is of interest because of itsimportance in the Re–Os cosmochronometer [247]. Due tolow-lying excited states, it has a non-negligible contribution ofexcited state transitions to the stellar rate already at s-processtemperatures.

Using (n,n+) at higher energies for the ! -process wouldbe even more important (and an interesting complement toneutron capture measurements) because of the even largerstellar effects.

7.4.3. Studying the optical potential via (n, ") reactions. Aninteresting alternative to studying low-energy "-transitions isthe use of (n, ") reactions, as suggested by [248, 249]. There isa number of reasons for this. Except at threshold, the reactioncross sections are sensitive to the "-width and therefore to theoptical "+nucleus potential [107]. It is also important to notethat, when using stable target nuclei, the Q-values of (n, ")reactions on nuclei in the mass range relevant to the ! -processare such that the relative energies of the emitted "-particlesare in the astrophysically interesting energy range. It is furtheradvantageous that the neutron energy can be varied to probethe energy dependence of the "+nucleus optical potential.Measuring partial cross sections, i.e. (n, "0), (n, "1), (n, "2),. . . , also allows the probing of this energy dependence and thetesting of the prediction of stellar excitation effects.

The small cross sections expected for low neutronenergies, from a few to a few hundred keV, may be problematic.Using predicted cross sections and scaling sample sizes ofprevious measurements, it was estimated that as many as 30nuclides across a wide range of masses should be accessibleto measurements [248].

So far, (n, ") on 143Nd [249] and on 147Sm [248–250]have been measured at neutron energies below 1 MeV. Neutronenergies of several MeV have been applied in the recentmeasurements of (n, ") on 143Nd [251], 147Sm [251] and149Sm [252, 253].

Such experiments also provide an excellent possibilityfor complementary studies to clarify the Coulomb excitation

31


effect suggested to explain the result of the 144Sm(", ! )148Gdmeasurement (see sections 5.3.3, 6.2 and figure 13). Noadditional reduction of the cross section should be observedin "-emission. Although a measurement of 147Gd(n, ")144Smis impossible, it is interesting that the measured (n, ") crosssections of the Nd and Sm isotopes are consistently about afactor of 2–3 below the predictions obtained with several codes.This is in line with the factor of three discrepancy remaining inthe 144Sm(", ! ) case after correcting for Coulomb excitationand can be attributed to deficiencies in the description of theoptical potential alone.

The facilities listed in section 7.4.1, also those focusingon low neutron energies, could also be used for such (n, ")studies in the future.

8. Conclusion

We have come a long way since [2, 3] but the mystery ofthe origin of the p-nuclides is still with us. Modern modelsof low-mass stars show strong s-process contributions toseveral nuclei previously considered to be p-nuclei. Detailedmodels of nucleosynthesis in massive stars, coupling largereaction networks to the hydrodynamic evolution of the star,have confirmed the working of the ! -process, synthesizingp-nuclei through photodisintegration. Supplementing suchphotodisintegration with neutrino processes required in theproduction of 138La and 180Ta explains the bulk of the p-abundances. Deficiencies are found at higher mass, 150 #A # 165, and for light p-nuclei with A < 100. Whilenuclear reactions are uncertain in the high mass part and furtherdevelopments of theory and additional experimental data areneeded there, the severe underproduction of the light p-nuclei isa long-standing problem and may point to a principle difficultyencountered when trying to obtain light p-nuclei from massivestars. Therefore, a number of alternatives have been suggestedand some have been studied in detail recently. Consistenthydrodynamical and nucleosynthetic treatment is still missing,however. Meteoritic specimens provide strong constraintsfor any new process under investigation and thus partiallycircumvent the problem that p-isotopic abundances cannot bedetermined from stellar spectra. They also allow one to invokeGCE models providing further constraints. Although thereare considerable uncertainties in the astrophysical modelingof the sites possibly producing p-nuclei, a sound base ofnuclear reaction rates is essential for all such investigations.As long as an experimental determination of the rates aroundthe deflection points is impossible, measurements of low-energy cross sections of stable nuclides are essential to testand improve the theoretical calculations. This has beenunderlined by the recent results regarding optical potentialsfor the interaction of charged nuclei. Further measurements (ateven lower energy) are highly desirable but have to be designedcarefully, taking into account the astrophysically relevantenergy ranges, sensitivities of stellar rates to nuclear input,and the principally possible contribution of the laboratorycross section to the stellar rate. Combining continuing nuclearphysics efforts, both in experiment and theory, with improved,self-consistent hydrodynamic simulations of the possible

production sites will gradually improve our understanding ofp-nucleosynthesis.

Acknowledgments

The authors thank T Davinson, M Paul, G Rugel, K Sonnabend,A Wallner and R Gallino for useful discussions, and A Hegerfor providing stellar models considering the latest solarabundance determinations. TR is supported by the HungarianAcademy of Sciences, the MASCHE collaboration in theESF EUROCORES programme EuroGENESIS, the THEXOcollaboration within the 7th Framework Programme of theEU, the European Research Council, and the Swiss NSF. Thiswork was supported in part by NASA (NNX09AG59G), NSFEAR Petrology, and Geochemistry (EAR-1144429), and bya Packard Fellowship to ND. ID is supported by the GermanHelmholtz association with a Helmholtz Young Investigatorproject VH-NG-627. CF acknowledges support from the DOETopical Collaboration ‘Neutrinos and Nucleosynthesis in Hotand Dense Matter’ under contract DE-FG02-10ER41677. GGis supported by the ERC StG 203175, and through grantsOTKA K101328 and NN83261 (EuroGENESIS).

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