tables of nuclear cross sections and reaction rates: an addendum to the paper “astrophysical...

Atomic Dataand Nuclear DataTables 79, 47–64 (2001)doi:10.1006/adnd.2001.0863, availableonlineat http://www.idealibrary.com on

TABLE S OF NUCLEA R CROSS SECTIONS AND REACTIO N RATES: AN ADDENDUMTO THE PAPER “AST ROPHYSICAL REACTIO N RATES FROM

STATISTICA L MODEL CALCUL ATIONS”

THOMAS RAUSCHER and FRIEDRICH-KARL THIELEMANN

Departement fur Physik und Astronomie, Universitat Basel, Klingelbergstr. 82, CH-4056 Basel, Switzerland

In apreviouspublication (ATOMIC DATA AND NUCLEAR DATA TABLES 75, 1 (2000)), wegaveseven-parameteranalytical fits to theoretical reaction rates derived from nuclear cross sections calculated in the statisticalmodel (Hauser–Feshbach formalism) for targetswith 10≤ Z ≤ 83 (Ne to Bi) and for amass rangereachingthe neutron and proton driplines. Reactions considered were (n,γ ), (n,p), (n,α), (p,γ ), (p,α), (α,γ ), and theirinversereactions. Here, wepresent thetheoretical nuclear crosssectionsand astrophysical reaction ratesfromwhich thoseratefitswerederived, and weprovide thesedataason-lineelectronic files. Corresponding to thefitted rates, two completedatasetsareprovided, oneof which includesaphenomenological treatment of shellquenching for neutron-rich nuclei. C© 2001 Academic Press

Data files associated with this article may be found on IDEAL at http://www.idealibrary.com/links/doi/10.1006/adnd.2001.0863/dat.

0092-640X/01 $35.00Copyright C© 2001 by Academic PressAl l rights of reproduction in any form reserved. 47 Atomic Dataand Nuclear DataTables, Vol. 79, No. 1, September 2001

http://www.idealibrary.com/links/doi/10.1006/adnd.2001.0863/dat


T. RAUSCHER and F.-K. THIELEMANN Nuclear Cross Sections and Reaction Rates

51

CONTENTS

1. INTRODUCTION ............................................................... 48

2. TABULATED INFORMATION ............................................... 482.1. Laboratory Cross Sections for Capture Reactions and Exothermic

Reactions................................................................. 482.1.1. Instructions for the Use of the Cross Section Files......... 49

2.2. Astrophysical Reaction Rates for Capture Reactions and Exother-mic Reactions............................................................ 502.2.1. Instructions for the Use of the Rate Files.................... 50

2.3. Reverse Stellar Rates and Endothermic Laboratory Cross Sections2.4. Experimental Nuclear Level Schemes................................ 51

2.4.1. Instructions for the Use of the Nuclear Level File.......... 522.5. Caveats.................................................................... 52

2.5.1. Applicability of the Statistical Model........................ 522.5.2. Validity of the Predictions..................................... 52

3. SUMMARY ...................................................................... 52

EXPLANATION OF TABLES.................................................... 54

TABLESI. Sample of Cross Section Tables....................................... 56

II. Sample of Rate Tables.................................................. 61III. Sample of Nuclear Level Table........................................ 64

1. INTRODUCTION

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In a previous paper [1] (hereafter referred to as RATHwe presented fits to nuclear reaction rates covering a wrange of isotopes and most of the nuclide chart. The dsets of reaction rate fits in that paper were mainly suiteddirect application in astrophysical simulations. However,certain computational approaches it is preferable to haverates in the original tabular format without any further fittinprocedure which, on the one hand, may be advantageouthe speedy calculation of a rate at a given temperature wan economical use of computer memory but, on the othand, may introduce additional inaccuracies. Furthermofor nuclear physics applications it is often preferable to hathe tables of nuclear cross sections directly. Thereforethought it worthwhile to also publish the original cross setions and reaction rates from which the fits given in RATwere derived; we provide these extensive data in the formelectronic tables. (It should be noted that “data” in this contrefers to results of theoretical calculations; wherever thera reference to experimental data, it is stated explicitly.)

48

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As in RATH, we provide here two sets of calculations with input from two different mass models: thenite range droplet model (FRDM) [2] and an extendThomas–Fermi approach with Strutinski integral (ETFSQ) [3]. Experimental masses and level properties wused in the calculations where available. In additioncross sections and reaction rates, particle separationergies are given. In order to facilitate a comparisonother calculations, all the experimental level informtion used in our calculations is shown in a separtable.

Here, we do not want to repeat the detailed denitions of the statistical model and of reaction ratesready given in RATH. Therefore we limit ourselvesthe most concise form and refer the reader to RATfor more details. In the following, we concentrate on eplaining how to extract values from the on-line tabland on giving a few caveats when using the tabulainformation.

Atomic Data and Nuclear Data Tables, Vol. 79, No. 1, September 2001


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2. TABULATED INFORMATION

The data set consists of seven tables provided as sepon-line files. The four files described in Sections 2.1 andcontain the raw data used to generate the fits given in RAFurther information which may be useful for comparisonexperiments or other calculations is given in three additiofiles (see Sections 2.3 and 2.4). All files are formatted in suway as to be machine readable without losing their accessity to the human reader. In this way, the additional informatprovided here can best serve the needs of several speciafields of nuclear physics as well as astrophysics. Definitiof cross sections and reaction rates as well as further detathe statistical model calculations are already given in RAT

2.1. Laboratory Cross Sections for Capture Reactionsand Exothermic Reactions

Reaction cross sections involving neutrons, protoandα particles as projectiles or ejectiles have been calculafor targets from Ne up to, and including, Bi. The resultstwo different sets of calculations are given here in two filThey differ in the mass model used, which enters intocomputation of the separation energies andQ values as wellas into the microscopic input to the level density calculatiOne set was calculated employing the well-known FRDmass model [2]; the other set employed the ETFSI-Q mo[3], which includes a phenomenological approximationshell-quenching effects (see RATH for further details onmass models).

The laboratory cross sections for capture reactionsfor exoergic particle reactions are tabulated. These assthat the target is in the ground state and cannot, in genbe used for computation of an astrophysical rate as gin RATH (cf. Eqs. (1), (2), (10) in RATH). However, thecan be directly compared to experimentally measured dThe isotope range is as specified in Table A of RATH, whis repeated here again for convenience. Cross sectionexothermic (Q ≥ 0) reactions with particles in the exit channel are given here, as well as all capture reactions regardletheQ value. All those reactions are contained in one file efor each mass model, sorted by target (charge and massber) and projectile (neutron, proton,α particle), in that order.The energy grid is different for each combination of targand projectile as it was chosen in such a way as to providebest grid for the numerical integration involved in the comptation of the astrophysical reaction rate. This means that,the energies are densely clustered around channel opeand more widely spaced in regions of “well-behaved” crosections. The energies are given in MeV; the cross sectare given in barn (1 barn= 10−28 m2 = 100 fm2).

49

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TABLE AIsotope Range of the On-Line Cross Section

and Reaction Rate Files

FRDM ETFSI-Q FRDM ETFSI-Q

Z Nmin Nmax Nmin Nmax Z Nmin Nmax Nmin Nmax

8 5 10 47 41 113 41 1119 5 28 48 42 115 42 112

10 5 31 49 43 117 43 11311 6 33 50 44 119 44 11412 7 35 51 46 121 46 11513 8 38 52 47 124 47 12414 8 40 53 48 126 48 12615 8 42 54 49 128 49 12816 8 44 55 51 130 51 13017 9 46 56 52 133 52 13218 9 49 57 53 135 53 13319 10 51 58 55 137 55 13420 10 53 59 56 139 56 13521 11 55 60 58 141 58 13622 12 58 61 59 144 59 13723 13 60 62 61 146 61 13824 14 62 18 62 63 62 148 62 13925 15 64 18 64 64 64 150 64 15026 16 66 19 66 65 65 153 65 15227 17 69 19 67 66 67 155 67 15428 18 71 20 68 67 69 157 69 15529 19 73 21 69 68 70 159 70 15630 21 75 22 70 69 72 161 72 15731 22 77 23 71 70 73 164 73 15832 23 80 24 72 71 75 166 75 15933 24 82 25 73 72 77 168 77 16034 25 84 26 84 73 78 170 78 16135 26 86 27 86 74 80 173 80 16236 27 88 28 88 75 81 175 81 16337 29 91 29 89 76 83 177 83 17738 30 93 30 90 77 85 179 85 17939 31 95 31 91 78 87 182 87 18240 32 97 32 97 79 88 184 88 18441 33 99 33 99 80 90 186 90 18642 35 102 35 102 81 92 188 92 18843 36 104 36 104 82 93 191 93 19144 37 106 37 106 83 95 193 95 19345 38 108 38 108 84 98 193 98 19346 40 110 40 110 85 101 195 101 195

Note.The charge numberZ and the lower and upper limitsNmin andNmax of the neutron number in the isotopic chain are given.

Provided as additional information are the particle searation energies in the compound nucleus, from whichreactionQ values given in RATH were derived.

2.1.1. Instructions for the Use of the Cross Section Files

The files can be browsed using either a text editing pgram or a code specifically written for reading the provid

Atomic Data and Nuclear Data Tables, Vol. 79, No. 1, September 2001


data. Table I shows an example of a segment from theFRDM file. The data blocks are sorted first by target nu-cleus(charge, massnumber) and then by projectile. For eachtarget–projectile combination, the capture cross sections arelisted for all calculated projectile energies regardless of Qvalue. Additionally, cross sections are given for those reac-tions with particles in the exit channel having Q ≥ 0. Thus,each data block wil l have a different number of lines, de-pending on the number of calculated energies and exoergicreaction channels. The lineswithin each block areorganizedasfollows. Each datablock for aspecific combination of tar-get and projectile starts with an identifying line specifying,in order, the charge and mass number of the target; a digitspecifying the projectile (0 for neutron, 1 for proton, 2 forα particle); the number L of calculated energies; indices band c indicating whether or not a reaction with particles inthe exit channel has Q ≥ 0 and is therefore listed (=1) ornot listed (=0); the charge Zc and mass number Ac of thecompound nucleus; and the particle separation energies Sn,Sp, Sα of the compound nucleus. The end of the line againgives the target isotope (T), the projectile (x), and the threepossibleexit channelsdenoted by roman letters (g= γ , n, p,and a=α). Following the first line are the L center-of-massenergiesin MeV at which thecrosssectionswerecalculated.Thereareat most eight entriesper line. Then, L capturecrosssectionsareprinted, regardlessof the Q value. Valuesfor theother reaction channels are only given if Q ≥ 0. If b = 0and/or c = 0, then no crosssection valuesareprinted for therespectiveexit channel 1, 2. Thus, thecapturecrosssectionsare followed by L × (b+ c) cross sections and therefore atotal of L × (1+ b+ c) cross sections is given. A new datablock startswith thenext identification lineafter the last lineof cross sections.

Asan exampleweconsider thereaction70Ga(p,n)70Ge.Since Q = Sp(71Ge)− Sn(71Ge) > 0, thecrosssectionscanbe found in the regular cross section file. Table I shows asmall part of that table. Looking for 70Ga, e.g. by atext searchfor “ga70,” one first encounters the data block for 70Ga+ n.Farther down one finds the identification line for 70Ga+ psince for the same target isotope the data blocks are sortedby reaction. With a computer code it would be more eco-nomical to search for the numeric codes for charge, massnumber, and projectile at the beginning of the line. The rel-evant identification line tells us that the cross sections werecalculatedat 61energiesand that thevaluesfor both the(p,n)and the(p,α) reactionareprinted. Thislineisfollowedby the61 center-of-mass energies, then 61 capture cross sections,61 (p,n) cross sections and, finally, 61 (p,α) cross sections.Thus, for instance, thevalueof the70Ga(p,n)70Gecrosssec-tion at 2.3022 MeV is given as 2.964 millibarn. The RATH

50

tablesshould alwaysbeconsulted to determineif thestatisti-cal model issuited for predicting crosssectionsfor aspecificreaction at agiven rate (seeSect. 2.5.1).

Thevaluesfor theinversereaction70Ge(n,p)70Gacannotbe extracted from the same file as Q < 0 for this reaction.An inspection of the identification line for 70Ge+ n fartherdown shows that b = 0 and that therefore no cross sectionsare listed for this reaction. Particle reactionswith Q < 0 aregiven in aseparatefile, seeSec. 2.3.

The above instructions are relevant to the posted crosssection files cs_frdm.asc, cs_etfsiq.asc, cs_frdm_endo.asc,and cs_etfsiq_endo.asc.

2.2. Astrophysical Reaction Rates for CaptureReactions and Exothermic Reactions

Reaction rates are given in two separate files, one foreach mass model, for the same reactions as included in thecross section listings. The rates were computed on agrid of24 temperatures: T9 = 0.1, 0.15, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7,0.8, 0.9, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 6.0, 7.0, 8.0,9.0, and 10.0 (T9 isgiven in 109 K). Thereactionsaresortedas before. For each reaction, the laboratory rate NA 〈σv〉lab,as well as the stellar rate NA 〈σv〉∗, is given. The laboratoryratewascomputed directly from thecrosssectionsdiscussedin Section 2.1 and can be used for comparison with ratesderived from experiment. Only the stellar rate considers thethermal excitation in astellar plasma and it should be usedexclusively in astrophysical calculations. It was computedmaking useof thestellar transmission functions(seeEqs. (2)and(3) inRATH). Thestellar reactionrateswerethebasisfortheRATH fits. Theratesaregiven in unitsof cm3 s−1 mol−1.

2.2.1. Instructions for theUseof theRateFiles

Anexamplefor asegment of theFRDM ratefileisgivenin Table II . The structure of the table is somewhat similar tothat of the cross section table as described in Section 2.1.1.Thedatablocksaresortedfirst by target nucleus(charge,massnumber) andthenby projectile.For eachtarget andprojectile,thecaptureratesareprinted regardlessof Q value. Theratesare also printed for those particle exit channels for whichQ ≥ 0. Thus, each data block wil l have adifferent numberof total lines, depending on the number of exoergic particlechannels. A data block starts with an identification line giv-ing the charge and mass number of the target, the projectile(0 for neutron, 1 for proton, 2 for α particle), and indicesb and c specifying whether a particle exit channel is listed.The digits b and c can only assume the values 0(not listed)or 1 (listed). The end of the line gives again target isotope,

Atomic Dataand Nuclear DataTables, Vol. 79, No. 1, September 2001






projectile, and the three possible exit channels. This is fol-lowed by 24 laboratory rates (three lines with eight entrieseach) calculatedat thetemperaturesgivenintheprevioussec-tion. A blank lineseparatesthelast lineof thelaboratory ratesfrom the following three lines with stellar rates. Laboratoryand stellar ratesaregiven for thecapture reaction regardlessof the Q value. This is followed by the laboratory and stel-lar rates of theparticleexit channels for which b = 1 and/orc = 1, i.e. with Q ≥ 0. Thus, thereare2× 24× (1+ b+ c)rates printed in total.

Asanexample, thereaction70Ga(p,n)70Geisagaincon-sidered. As before in Section 2.1.1, since Q = Sp(71Ge)−Sn(71Ge) > 0 the reaction rates can be found in the regularrate file. Table II shows asmall part of that table. Lookingfor 70Ga, for instance, by a text search for “ga70,” one firstencounters the data block for 70Ga+ n. Farther down onefinds the data block 70Ga+ p since for the same target iso-tope thedatablocksaresorted by reaction. With acomputercodeit may bemoreconvenient to search for thenumbersforcharge, mass number, and projectile at the beginning of theline. The relevant identification line tells us that the valuesfor the (p,n) as well as the (p,α) reaction are printed. Unlikethecrosssection table, which startswith L projectileenergyvalues, the temperature grid is not given here because therates are always calculated at the same 24 stellar tempera-tures given above. The first line is immediately followed bythe rates for the capture reaction, first 24 values of labora-tory rates, then 24 stellar rates. These are followed by pairsof 24 laboratory and stellar rates for the (p,n) and the (p,α)reactions. Thus, for instance, the value of the70Ga(p,n)70Gelaboratory rate at T9 = 9 (next to last position in the thirdline) is given as 7.27× 106 cm3 s−1 mol−1. This should becomparedto thestellar rateof 6.94× 106 cm3 s−1 mol−1. TheRATH tablesshould alwaysbeconsulted to determine if thestatistical model is suited for predicting cross sections for aspecific reaction at agiven rate (seeSection 2.5.1).

Thevaluesfor theinversereaction70Ge(n,p)70Gacannotbe extracted from the file as Q < 0 for this reaction. Aninspectionof theidentification linefor 70Ge+ n farther downshows that b = 0 and therefore no rates are printed for thisreaction. However, reverse stellar rates for particle reactionscanbecomputedeasily asexplained in thefollowingsection.

The above instructions are relevant to the posted ratefiles rates_frdm.asc and rates_etfsiq.asc.

2.3. ReverseStellar Rates and EndothermicLaborator y Cross Sections

The data presented in Sections 2.1 and 2.2 correspondto the RATH tables. Since reverse astrophysical rates canbecalculated easily by detailed balance, thepreviousRATH

5

work only gavesuch ratesexplicitly for exothermic reactionsand capture reactions and provided the means to derive thefit parameters for the reverse rates. Thereversestellar rate isalso not explicitly given here but can be derived easily foreach temperaturewith the following formulas. For reactionswith particles in all channels, use

NA 〈σv〉∗rev = NA 〈σv〉∗ Gi (T)

Gm(T)ef. (1)

In order to compute a photodisintegration rate λ∗γ (in s−1)from acapture rate, use

λ∗γ =Gi (T)

Gm(T)T3/2

9 NA 〈σv〉∗ ef. (2)

Theexponent f is defined as

f = arev0 − a0− 11.6045Q

T9. (3)

ThequantitiesGi , Gm arethepartition functionsof thetargetand residual nucleus, respectively, in the forward reaction,i.e., the reaction x + i → m+ y, where x can be aparticleand y can be aparticleor aγ -ray (compareEqs. (12)–(14) inRATH). Theparametersa0 and arev

0 and thereaction Q value(in MeV) for the forward reaction, as well as the requiredpartition functions, canbelookedup in theRATH tables. TheQ value can also be calculated from the particle separationenergies given with thecross sections.

It has to be emphasized that detailed balance can onlybeapplied to stellar ratesand not to laboratory ratesor crosssections with the target in the ground state. In comparisonstoexperimentsit cansometimesbeuseful toalsohaveaccessto cross sections for reactions with negative Q values. Forthat purpose, calculated cross sections for such endothermicreactions are given in two additional files: cs_frdm_endo.ascand cs_etfsiq_endo.asc (one file for each mass model). Theformat is the same as for the cross sections discussed inSection 2.1. Here, only the energies are given for which thecrosssectionsarenot zero, i.e., abovetheparticlethresholds.Cross sections at lower energies can beassumed to bezero.

Photodisintegration cross sections for targets in thegroundstatearenot givenhere. They arediscussedelsewhere[4, 5] and atablewil l bepublished separately [6].

2.4. Experimental Nuclear Level Schemes

The experimental level information utilized in the sta-tistical model calculations is provided in the last file. Upto 20 experimental levels (including the ground state) were

1 Atomic Dataand Nuclear DataTables, Vol. 79, No. 1, September 2001






considered. Thestatesare taken from [7] up to thefirst levelfor whichthespinor parity assignment wasnot known.Abovethis cut-off a theoretical nuclear level density [8] was em-ployed. Ground state spins and parities are known for manyunstable nuclei. Far from stability, where experimental val-uesarenot known, groundstatespinsandparitiesarederivedfrom theneutron and proton single-particlespinsgiven in [9]by applying Nordheim rules. However, only the experimen-tal spin and parity values used are tabulated here becausethe complete list of ground state spins is already given inRATH. This information is important for comparisons withother statistical model calculations.

2.4.1. Instructions for theUseof theNuclear Level File

An example is shown in Table III . The isotopes are or-deredfirst by chargeandthenby massnumber.Thedatablockfor each isotope starts with an identification line specifying,in thisorder, thechargeand massnumber of the isotope, thenumber L of experimental nuclear levels given, and finallytheisotopeagainwritten inelement notation. Thesucceedinglines specify the L nuclear levels, giving for each level theexcitation energy (in MeV) aswell as theparity and thespinassignment. The parity is coded as a sign before the spin.There are at most five levels per line. A new isotope entryagain starts with an identification line.

As an example, consider the nucleus70Ga in Table III,which in turn shows asmall fraction of the full file. It can beseenthat fiveexperimental levelswereusedinthecalculation,including the ground state. The list starts with the Jπ = 1+

groundstate, whilethethirdexcitedstateof 70Gaislocatedatanenergy of 0.6909MeV andhas aspinandparity assignmentof Jπ = 2−.

The above instructions are relevant to the posted filelevels.asc.

2.5. Caveats

The cross sections and reaction rates provided hereshould only be used keeping in mind the fundamental limi-tations of themodel as discussed below.

2.5.1. Applicability of theStatistical Model

The statistical model can be applied provided that theuse of averaged transmission coefficients (Eq. (3) in RATH)ispermitted.Thiswil l bethecasefor high-level densitieswithcompletely overlapping resonances, typical of thecompoundnucleus reaction mechanism. For light nuclei or decreasingparticleseparationenergiesor at shell closures, level densities

52

wil l eventually become too low for the application of thestatistical model at astrophysical temperatures. Inthosecases,single resonances and contributions from the direct reactionmechanism have to be taken into account [10]. Based ona level density description, a quantitative criterion for theapplicability wasderivedrecently [8]. Thelower temperaturelimi t of thevalidity of thestatistical model for thecalculationof reaction rates is already given in the RATH tables. It isadvisabletocheck thoselimitsbeforeusing thecrosssectionsand rates given in this paper.

In principle, a similar criterion can also be derived forcross sections when comparing to experimental data. In thecase of a nuclear experiment, the energy distribution overwhich one has to average is determined not by a Maxwell–Boltzmann velocity distribution but by theuncertainty in thebeam energy and the energy straggling in the target. Sincethese differ depending on the experimental setup, it is notpossibletogive aglobal criterion. However, as afirst approx-imation onecan use the temperature limits given in RATH.

2.5.2. Validity of thePredictions

It should be noted that only purely theoretical rateswhichdonot useany direct experimental information(exceptfor nuclear massesand ground and excited state informationwhere available) are given here. The methods to predict nu-clear properties needed in the statistical model calculationsare chosen to be as reliable as possible in order to retainpredictive power. Nevertheless, the uncertainties and someknownweaknesses(e.g., of theFRDM aroundshell closures)of the input wil l be reflected in the predicted values. This isacompromisewhich may lead to locally enhanced inaccura-cies but it emphasizes the importance of reliable predictionsof ratesfar fromstability. If oneusedexperimental data(suchas level densities) and locally tuned parametrizations of nu-clear properties (optical potentials, Giant Dipole Resonancewidths) as input for our statistical model calculation, a fur-ther increase in accuracy could be achieved. Since the mainscope of this work is the prediction of astrophysical nuclearreactionratesfor experimentally inaccessiblenuclei, aglobalapproach isbetter suited. Therefore, in comparisonswith ex-periment, deviations within a factor of 1.5–2 could appear,although the average deviation wil l be smaller. For neutroncapture it hasbeen shown that theaveragedeviation isabouta factor of 1.3–1.4 [8].

3. SUMMA RY

Nuclear crosssectionsand thermonuclear reaction ratesfor neutron-, proton-, and α-induced reactions and their




inverses have been calculated in the statistical model inconnection with the previous paper, RATH [1]. Al l crosssections and rates from Ne to Bi are given from the protondripline to theneutron dripline, thuscovering aconsiderablepart of the nuclide chart. Fits to stellar reaction rates havebeen provided in RATH for two setsof rates, calculated withinput from two different massmodels. Here, wemakeacces-sible the input data for the original calculation, containingnuclear cross sections as well as stellar and laboratory reac-tion rates. Excitation energy, spin, and parity are quoted fortheexperimental level informationused in thecalculationstofacilitate comparison with other models. Further details ontheunderlying models aregiven in RATH.

In real applications, theseratesshould besupplementedor replaced by experimental rates as they become available.Such acombination of theoretical and experimental rates isprovided, e.g., in theREACLIB compilation. Further detailsonREACLIB, theNON-SMOKERcode,andthecalculationsare presented at http://www.nucastro.org/reaclib.html. Ratesbased on additional mass models can also be obtained fromtheauthors on request or directly at the latter URL.

Acknowledgments

This work was supported in part by the Swiss NationalScience Foundation (Grant 2000-053798.98) and theAustrianAcademy of Sciences(APART).T.R.acknowledgesa PROFIL professorship from the Swiss National ScienceFoundation (Grant 2124-055832.98).

53

References

1. T. Rauscher and F.-K. Thielemann, ATOMIC DATA AND

NUCLEAR DATA TABLES 75, 1 (2000), doi:10.1006/adnd.2000.0834.

2. P. Moller, J. R. Nix, W. D. Myers, and W. J. Swiatecki,ATOMIC DATA AND NUCLEAR DATA TABLES 59, 185 (1995),doi:10.1006/adnd.1995.1002.

3. J. M. Pearson, R. C. Nayak, and S. Goriely, Phys. Lett.B 387, 455 (1996).

4. P. Mohr et al., Phys. Lett. B 488, 127 (2000).

5. K. Vogt et al., Phys. Rev. C 63, 055802 (2001).

6. T. Rauscher, in preparation.

7. R. B. Firestone, in Table of Isotopes, 8th ed., edited byV. S. Shirley (Wiley, New York, 1996).

8. T. Rauscher, F.-K. Thielemann, and K.-L. Kratz, Phys.Rev. C 56, 1613 (1997).

9. P. Moller, J. R. Nix, and K.-L. Kratz, ATOMIC

DATA AND NUCLEAR DATA TABLES 66, 131 (1997),doi:10.1006/adnd.1997.0746.

10. T. Rauscher et al., Phys. Rev. C 57, 2031 (1998).


http://www.nucastro.org/reaclib.html


EXPLA NATIO N OF TABLES

TABL E I. Sampleof Cross Section Tables

Sampleexcerpt of theASCII filecs_frdm.asc containing crosssectionsfor reactionson theground statecalculated using input data from theFRDM mass model.

Z Chargenumber of targetA Mass number of targetp Projectile: 0 for neutron, 1 for proton, 2 for α particleL Number of calculated energies and cross sectionsb Index b, which assumesvaluesof 0 or 1 depending on whether crosssectionsarelisted (b = 1)

for thefirst particleexit channel (defined by column f)c Index c, which assumesvaluesof 0 or 1 depending on whether crosssectionsarelisted (c = 1)

for thesecond particleexit channel (defined by column h)Zc Chargenumber of thecompound nucleusAc Mass number of thecompound nucleusSN Neutron separation energy in thecompound nucleus, in MeVSP Proton separation energy in thecompound nucleus, in MeVSA α-particleseparation energy in thecompound nucleus, in MeVT Target written as combination of element nameand mass numberx Projectile: +n for neutron,+p for proton,+a for α particlee, f, h Exit channels:−g for γ ray (capturereaction),−n for neutron,−p for proton,−afor α particle.

In thistable, crosssectionsfor thecapturechannel arealwayslistedandcolumneisalways−g; in the table of reactions with Q < 0 (the cs frdm endo.asc and cs etfsiq endo.ascfiles), no capture reactions aregiven and column eis omitted.

E1. . .EL Center-of-mass energies in MeV for which the nuclear reaction cross sections have been cal-culated. There are L values with up to eight entries per line. The energies are only givenonceas they are thesame for thesamecombination of target and projectile.

G1. . .GL Capturecrosssectionsin barn; L valuesaregiven with up to eight entriesper line. Valueslowerthan 10−30 barn areset equal to zero.

(X1. . .XL) Nuclear cross sections for the first particle exit channel (specified by column f ) in barn. Lvalues are given with up to eight entries per line. They are listed only if b = 1. Valueslower than 10−30 barn areset equal to zero.

(Y1. . .YL) Nuclear cross sections for the second particle exit channel (specified by column h) in barn; Lvalues are given with up to eight entries per line. They are listed only if c = 1. Valueslower than 10−30 barn areset equal to zero.

TABL E II. Sampleof RateTables

Sampleof theASCII file rates_frdm.asc of laboratory rates NA 〈σv〉lab calculated with theground statecross sections of Table I and stellar rates NA 〈σv〉∗ calculated from stellar cross sections with a thermallyexcited target, given in cm3 s−1 mol−1. Values lower than 10−30 cm3 s−1 mol−1 areset equal to zero.

Z Chargenumber of targetA Mass number of targetp Projectile: 0 for neutron, 1 for proton, 2 for α particleb Index b, which assumes values of 0 or 1 depending on whether rates are listed (b = 1) for the

first particleexit channel (defined by column f )c Index c, which assumes values of 0 or 1 depending on whether rates are listed (c = 1) for the

second particleexit channel (defined by column h)T Target written as combination of element nameand mass numberx Projectile: +n for neutron,+p for proton,+a for α particle





EXPLA NATIO N OF TABLE S Continued

e, f, h Exit channels: −g for γ ray (capture reaction), −n for neutron, −p for proton, −a for αparticle. Rates for thecapturechannel arealways listed and column eis always−g.

G1. . .G24 Laboratory capture rates NA 〈σv〉lab (see Section 2.2). Eight entries each in three lines aregiven, corresponding to the grid of 24 temperatures T9 = 0.1, 0.15, 0.2, 0.3, 0.4, 0.5,0.6, 0.7, 0.8, 0.9, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0 (T9 isgiven in 109 K).

H1. . .H24 Stellar capture rates NA 〈σv〉∗ (see Eq. (10) in RATH), given as eight entries each in threelines for the 24-temperature grid. Only the stellar rate should be used in astrophysicalapplications and when applying detailed balance in order to derive the reverse rate.

(R1. . .R24) Laboratory ratesfor thefirst particleexit channel (specified by column f ) at 24 temperatures.They are listed only if b = 1.

(U1. . .U24) Stellar ratesfor thefirst particlechannel (specifiedby columnf ). They arelistedonly if b = 1.Only the stellar rate should be used in astrophysical applications and when applyingdetailed balance in order to derive the reverse rate.

(R1. . .R24) Laboratory rates for the second particle exit channel (specified by column h) at 24 tempera-tures. They are listed only if c = 1.

(U1. . .U24) Stellar rates for the second particle channel (specified by column h). They are listed onlyif c = 1. Only the stellar rate should be used in astrophysical applications and whenapplying detailed balance in order to derive the reverse rate.

TABL E III. Sampleof Nuclear Level Table

Upto20experimentally knownnuclear stateswereused in thecalculation (seeSection2.4). Anexcerptfrom the full on-line ASCII file levels.asc is given here as an example. Only the experimental values usedin thecalculation are listed. The theoretical ground statespins werealready given in RATH.

Z Chargenumber of the isotopeA Mass number of the isotopeL Number of given energy levels (including theground state)T Isotopewritten as acombination of element nameand mass numberE1. . .EL Excitation energy of the level, given in MeVp1. . .pL Parity of the levelJ1. . . JL Spin of the level




TABLE I. Sample of Cross Section TablesSee page 54 for Explanation of Tables

56 Atomic Data and Nuclear Data Tables, Vol. 79, No. 1, September 2001


TABLE II. Sample of Rate TablesSee page 54 for Explanation of Tables



TABLE III. Sample of Nuclear Level TableSee page 55 for Explanation of Tables


tables of nuclear cross sections and reaction rates: an addendum to the paper “astrophysical...

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