Evaluation of fusion-evaporation cross-section calculations
aa r X i v : . [ nu c l - e x ] J u l Evaluation of fusion-evaporation cross-section calculations
B. Blank a , b 1 , G. Canchel a , F. Seis a 2 , P. Delahaye c a Centre d’Etudes Nucl´eaires de Bordeaux Gradignan, 19 Chemin du Solarium, CS10120, F-33175 Gradignan Cedex, France b ISOLDE/CERN, EP Department, CH-1211 Geneve 23, Switzerland c Grand Acc´el´erateur National d’Ions Lourds, Bd Henri Becquerel, BP 55027,14076 CAEN Cedex 05, France
Abstract
Calculated fusion-evaporation cross sections from five different codes are compared to experimental data. The present comparisonextents over a large range of nuclei and isotopic chains to investigate the evolution of experimental and calculated cross sections.All models more or less overestimate the experimental cross sections. We found reasonable agreement by using the geometricalaverage of the five model calculations and dividing the average by a factor of 11.2. More refined analyses are made for example forthe
Sn region.
Key words: fusion-evaporation reactions, comparison experiment - calculations
1. Introduction
On Earth, 255 stable nuclides are available for nuclearphysics studies. In addition, 31 quasi stable nuclides hav-ing a half-life comparable to or longer than the age of theEarth exist. All other nuclei must be created in order to beusable for experimental studies. Different types of nuclearreactions exist to produce these unstable and radioactivenuclei.Two methods can be used to create basically all boundor quasi bound (i.e. bound for a short laps of time) nu-clei: spallation or fragmentation. Spallation reactions areusually induced by light particles (protons or neutrons) onheavier stable nuclei. In these spallation reactions, the in-cident light projectile ejects nucleons from the target nu-cleus by nucleon-nucleon collisions and the excited frag-ment (often called pre-fragment) evaporates light particles(protons, neutrons, α particles) to get rid of excitation en-ergy. With e.g. incident proton energies of a few hundredMeV up to 1 or 2 GeV, basically all nuclei, bound or quasibound, but lighter than the target nucleus itself, can be pro-duced. However, as these spallation reactions are basicallyalways ”thick-target” reactions, the reaction products haveto diffuse out of the target to become useful. As this takessome time and depends very sensitively on the chemistry of ∗ e-mail address: [email protected] † Summer student at CENBG the element of interest, short-lived nuclides of condensableelements are very difficult to produce by this means.Fragmentation reactions employ heavy-ion induced re-actions on different heavy-ion targets. Therefore, target aswell as projectile fragmentation can be used. Target frag-mentation suffers from the same problem as spallation re-actions: the products have to diffuse from the target itself.Therefore, this process is again limited to relatively volatileisotopes with sufficiently long half-lives. In projectile frag-mentation reactions, one can use ”thin targets” which al-lows the products to recoil out of the target due to the in-cident projectile energy. This approach is basically univer-sal and allows all nuclides to be produced. However, thereare at least two drawbacks of projectile fragmentation: i) itneeds high-energy heavy-ion accelerators and ii) the beamquality of these fragment beams is rather bad.In deep-inelastic or transfer reactions, two heavy nucleiinteract with each other at energies around the Fermi en-ergy (typically 20-60 MeV/A) and nucleons are transferredfrom one nucleus to the other producing thus more or lessneutron-rich or neutron-deficient isotopes. However, as thenumber of nucleons transferred is limited, only nuclei rela-tively close to stability can be produced.In nuclear fission, a very heavy nucleus, e.g.
U or
Cf,fissions by creating two medium-mass nuclides. This fissionprocess can be induced (e.g. by proton, neutron or γ -rayimpact) or spontaneous. Due to the curvature of the nuclearvalley of stability, the heavy fissioning nuclei have always Preprint submitted to Elsevier 9 April 2018 n excess of neutrons compared to lighter nuclei. Therefore,nuclear fission always produces neutron-rich isotopes in themass range of A ≈
50 - 170.Finally, neutron-deficient nuclides can be produced byfusing two lighter nuclei. In this case, the situation is re-versed compared to fission. The light stable nuclei that in-teract are proton-rich compared to the heavier nuclei in thevalley of stability. For example, the reaction of a stable Canucleus with a stable Ni nucleus produces as the com-pound nucleus, i.e. the sum of all nucleons, Cd, a nucleuswhich is 8 neutrons more neutron-deficient than the mostneutron-deficient stable isotope of the element cadmium.From this list of possible reactions, it is evident thatthe experimenter has some choice to use the reaction bestsuited for the production of the nucleus of interest. How-ever, evidently this choice depends also strongly on the ac-celerator available, the separation possibilities and muchmore. For each type of reaction, parameters like the reac-tion partners and the incident energy have to be optimizedin order to achieve the highest production rates of the iso-tope of interest. For spallation, fragmentation, and deep-inelastic reactions, it is most often advantageous to use astable nucleus close to the desired final nucleus to enhancethe production rate. This choice basically does not existof fission because only a few quasi stable fissioning nucleiexist. For these reactions, analytical codes have been de-veloped which have a rather good predictive power for thereaction cross sections. Let us mention the EPAX code [1–3] for projectile fragmentation, the SPACS code [4,5] forspallation reactions or the GRAZING model [6,7] for deep-inelastic reactions. The ABRABLA [8,9] code deals withfission, fragmentation, and spallation.For fusion-evaporation, the situation is different inthe sense that all nuclei can be produced with differentcombinations of projectile, target and incident energy.Therefore, an optimization of these three parameters isneeded for any nucleus to be produced. To do so, differentcodes are available, some of them being analytical, oth-ers being of the Monte-Carlo type. In the present work,we have used five codes to calculate fusion-evaporationcross sections: CASCADE [10,11], HIVAP [12], POT-FUS+ABLA called CNABLA [13,14,8], PACE [15], andPOTFUS+GEMINI++ [14,16]. All codes have advantagesand draw-backs and we could not decide a priori whichcode would perform better over a wide range of nuclei.The original reason for the present work was to determineproduction rates for SPIRAL2 where fusion-evaporation re-actions were foreseen as a tool to produce neutron-deficientisotopes from mass 20 or so to the heaviest nuclei in thesuper-heavy element region by means of a target - ion-source ensemble in the production building. However, dueto financial constraints, the construction of the SPIRAL2production building was put on hold. The same work wasused in the mean time to predict production rates for theS separator [17] or at other facilities.For this purpose, we performed a literature research of allfusion-evaporation reactions used to produce proton-rich nuclei. Using the projectile-target combination and the en-ergy given in the literature, cross sections were calculatedwith the five codes. To predict SPIRAL2 production rates,the in-target yields were determined using the predictedprimary-beam intensities and the extracted yields were ob-tained by means of release functions found in the literature.In this way production rates could be predicted for morethan 700 proton-rich nuclei.In order to evaluate the performance of the fusion-evaporation codes and the quality of the production ratepredictions, we have performed calculations with all fivecodes and compared the results to either fusion-evaporationcross sections found in the literature or to production ratesfrom the GSI on-line separator [18]. The former values con-stitute a more direct comparison, however, in most casesthe authors had to use transmissions of their separatorswhich contain quite some uncertainties. For the seconddata, release efficiencies are needed in order to determinein-target production rates and thus production cross sec-tions. To compare our cross-section calculations with thesevalues we will use release data collected in the frame workof the SPIRAL2 facility [19] where these release functionsare needed for fusion as well as fission products.The purpose of the present paper is to describe the resultsof this comparison between calculated cross sections or pro-duction rates and experimental data for fusion-evaporationreactions. The general outcome is that the different codesoverestimate the experimental data by about a factor of 10.Therefore, for e.g. planning an experiment using a fusion-evaporation reaction, the predictions deduced from calcu-lations using fusion-evaporation codes should be reducedby this factor in order to obtain a realistic estimate of theproduction rates to be expected.
2. Experimental data
In this section, we summarize the experimental data usedfor the comparison with the theoretical predictions. Table 1gives the experimental cross sections used in the presentwork ‡ .In general, relatively few fusion-evaporation cross sec-tions are found in the literature and those found have oftenlarge error bars or, even worse, no uncertainties at all. Thisis to a large part due to the fact that the cross sectionsare often determined at ISOL facilities where informationof effusion and diffusion is scarce and induce large uncer-tainties. Other cross sections are determined by means ofmass separators or velocity filters where the transmissionsare not well known.Another problem with a comparison of experimentalcross sections and calculated values is that it is often notclear whether the beam energy given is the one at theentrance or in the center of the target. We always use the ‡ The authors are eager to increase the present data base of exper-imental cross sections and encourage readers to communicate otherexperimental fusion-evaporation cross sections to us. able 1Experimental cross sections from literature. Given are the mass andcharge number of the nuclei of interest, the mass and charge numberof the projectile and target nuclei, respectively, the incident beamenergy, the experimental cross section and its error if available, andthe reference.A Z A p Z p A t Z t E cross error Ref.(MeV) section (mb)(mb)light N ≈ Z nuclei:64 30 12 6 54 26 37 1.60E+02 7.00E+00 [20]64 31 54 26 12 6 150 7.90E+01 [21]64 32 40 20 27 13 102 4.00E-01 6.00E-02 [22]64 32 54 26 12 6 165 6.40E-01 7.00E-02 [20]64 32 54 26 12 6 150 3.40E-01 9.00E-02 [21]64 32 54 26 12 6 165 5.00E-01 3.00E-01 [23]64 32 12 6 58 28 40 2.00E-01 5.00E-02 [24]68 34 58 28 12 6 175 3.80E-02 1.60E-02 [23]68 34 58 28 12 6 220 2.00E-01 5.00E-02 [25]72 36 16 8 58 28 55 1.00E-01 3.00E-02 [24]72 36 58 28 16 8 170 6.00E-02 2.50E-02 [26]76 38 54 26 24 12 175 1.00E-02 5.00E-03 [23]80 40 58 28 24 12 190 1.00E-02 5.00E-03 [27]80 39 58 28 24 12 190 2.00E+00 1.00E+00 [27]80 38 58 28 24 12 190 4.40E+01 4.00E+00 [27]
Sn region:95 45 58 28 50 24 250 1.10E+00 4.00E-01 [28]97 45 58 28 50 24 250 3.40E+00 2.00E-01 [28]98 46 58 28 50 24 250 2.20E+01 2.00E+00 [28]98 47 58 28 50 24 250 3.00E-01 6.00E-02 [28]99 47 58 28 50 24 250 3.60E+00 4.00E-01 [28]99 48 58 28 50 24 249 3.20E-02 2.00E-02 [29]99 48 58 28 50 24 249 3.20E-02 2.00E-02 [29]99 48 50 24 58 28 225 2.50E-02 8.00E-03 [29]99 48 58 28 58 28 348 1.10E-02 8.00E-03 [29]99 48 58 28 58 28 371 2.80E-02 2.10E-02 [29]99 48 58 28 58 28 394 3.10E-02 2.00E-02 [29]100 47 58 28 50 24 250 3.90E+00 2.00E-01 [28]100 47 50 24 58 28 225 3.90E+00 [28]100 48 50 24 58 28 225 1.00E+00 [30]100 49 50 24 58 28 225 1.00E-03 [30]100 49 58 28 50 24 319 2.60E-03 [29]100 49 58 28 58 28 325 8.00E-04 [29]100 49 58 28 58 28 348 1.70E-03 [29]100 49 58 28 58 28 371 1.70E-03 [29]100 49 58 28 58 28 394 1.60E-03 [29]100 50 50 24 58 28 225 4.00E-05 [30]101 47 58 28 50 24 250 4.70E+01 3.00E+00 [28]101 48 58 28 50 24 250 1.80E+01 2.00E+00 [28]101 50 58 28 50 24 249 1.60E-05 4.00E-06 [29]101 50 58 28 50 24 250 1.00E-05 [29]101 50 58 28 58 28 325 9.00E-06 4.00E-06 [29]101 50 58 28 58 28 348 1.30E-05 3.00E-06 [29]101 50 58 28 58 28 371 2.80E-05 1.00E-05 [29]101 50 58 28 58 28 394 7.00E-06 4.00E-06 [29]102 48 58 28 50 24 250 6.30E+01 1.90E+01 [28]102 49 58 28 50 24 249 9.00E-01 5.00E-01 [29]102 49 58 28 50 24 249 1.30E+00 7.00E-01 [29]102 49 58 28 50 24 348 1.10E+00 6.00E-01 [29]102 49 58 28 58 28 325 1.20E+00 6.00E-01 [29]102 49 58 28 58 28 348 1.20E+00 6.00E-01 [29]102 49 58 28 58 28 348 7.00E-01 3.00E-01 [29]102 49 58 28 58 28 371 1.00E+00 5.00E-01 [29]102 49 58 28 58 28 394 9.00E-01 4.00E-01 [29]102 50 58 28 52 24 225 2.00E-03 [31]103 47 58 28 50 24 250 3.60E+00 4.00E-01 [28]103 48 58 28 50 24 250 2.70E+01 2.00E+00 [28]103 49 58 28 50 24 250 6.40E+00 8.00E-01 [28]104 48 58 28 50 24 250 1.79E+02 7.00E+00 [28]104 49 58 28 50 24 250 5.80E+01 1.60E+01 [28]104 50 58 28 50 24 250 1.80E+00 2.00E-01 [28] A Z A p Z p A t Z t E cross error Ref.(MeV) section (mb)(mb)105 49 58 28 50 24 250 1.16E+02 6.00E+00 [28]105 50 58 28 50 24 250 1.00E+01 2.00E+00 [28]Ba nuclei:114 56 58 28 58 28 222-248 2.00E-04 1.00E-04 [32]114 56 58 28 58 28 203-244 2.00E-04 1.00E-04 [33]116 56 58 28 60 28 209-249 3.00E-03 1.00E-03 [33]116 56 58 28 63 29 249-284 8.00E-04 4.00E-04 [33]117 56 58 28 63 29 249-284 5.50E-02 2.00E-02 [33]118 56 58 28 63 29 249-284 1.90E-02 6.00E-03 [33]heavier nuclei:171 79 78 36 96 44 361 1.10E-03 [34]171 79 78 36 96 44 359 2.00E-03 [34]171 79 78 36 96 44 363 6.00E-04 [34]170 79 78 36 96 44 386 9.00E-05 [34]173 80 78 36 102 46 384 4.00E-06 [34]172 80 78 36 96 44 361 4.00E-06 [34]171 80 78 36 96 44 361 2.00E-06 [34]176 81 78 36 102 46 384 3.00E-06 [34]172 80 78 36 96 44 375 9.00E-06 [35]173 80 80 36 96 44 400 1.50E-05 [35]174 80 80 36 96 44 375 3.30E-04 [35]proton emitter: pn channel:185 83 92 42 95 42 410 1.00E-04 [36]185 83 92 42 95 42 420 6.00E-05 [37]p2n channel:109 53 58 28 54 26 195 1.00E-02 [38]109 53 58 28 54 26 220 1.60E-02 4.00E-03 [39]109 53 58 28 54 26 240 3.00E-03 [40]109 53 58 28 54 26 229 5.00E-02 [41]109 53 58 28 54 26 250 4.00E+01 +4 . E +01 − . E +01 [42]109 53 58 28 58 28 250 3.00E+01 +3 . E +01 − . E +01 [42]113 55 58 28 58 28 250 3.00E+01 [42]147 69 58 28 92 42 260 1.80E-02 [43]151 71 58 28 96 44 266 7.00E-02 1.00E-02 [44]161 75 58 28 106 48 270 6.30E-03 1.80E-03 [45]167 77 78 36 92 42 357 1.10E-01 [46]171 79 78 36 96 44 389 2.00E-03 [46]171 79 78 36 96 44 370 6.00E-04 [47]171 79 78 36 96 44 361 1.10E-03 [34]171 79 78 36 96 44 359 2.00E-03 [34]171 79 78 36 96 44 363 6.00E-04 [34]177 81 78 36 102 46 370 3.00E-05 [48]p3n channel:108 53 58 28 54 26 240-255 5.00E-04 [49]112 55 58 28 58 28 259 5.00E-04 [50]146 69 58 28 92 42 287 1.00E-03 [51]150 71 58 28 96 44 297 2.56E-03 [52]150 71 58 28 96 44 292 3.05E-03 [53]160 75 58 28 106 48 300 1.00E-03 [54]166 77 78 36 92 42 384 6.30E-03 [46]176 81 78 36 102 46 384 3.00E-06 [34]p4n channel:117 57 58 28 64 30 310 2.00E-04 [55]117 57 58 28 64 30 295,310 2.40E-04 +2 . E − − . E − [56]131 63 40 20 96 44 222 9.00E-05 [57]141 67 54 26 92 42 285,305 2.50E-04 [57]141 67 54 26 92 42 315 3.00E-05 [58]145 69 58 28 92 42 315 5.00E-04 [59]145 69 92 42 58 28 512 2.00E-04 [60]155 73 58 28 102 46 315,320 6.00E-05 [61]165 77 78 36 92 42 384 2.00E-04 [46]p5n channel:130 63 78 36 58 28 432 9.00E-06 +9 . E − − . E − [62]140 67 54 26 92 42 315 3.00E-06 [58]p6n channel:121 59 36 18 92 42 240 3.00E-07 +3 . E − − . (0 E − [63]135 65 50 24 92 42 310 3.00E-06 [64] able 2Experimental production rates from literature. Given are the massand charge number of the nuclei of interest and the reference.A Z Ref. A Z Ref.60 31 [32] 105 49 [66]61 31 [67] 106 49 [68,69]62 31 [70] 107 49 [66]94 47 [71,72] 101 50 [29]95 47 [73] 102 50 [31]96 47 [74] 103 50 [75]97 47 [76] 104 50 [28]98 47 [77] 105 50 [28]100 49 [29] 114 56 [33,78]102 49 [29,79] 116 56 [33]103 49 [80] 117 56 [33]104 49 [66] 118 56 [33] energy given in the paper for the calculations. If the energyis the one at the target entrance and thus too high com-pared to the energy in the center of the target, we believethis in not a problem. The maximum of the cross sectionsis reached at a certain incident energy. At higher energies,the cross sections fall off slowly, whereas at lower energiesthere is a threshold effect to overcome the Coulomb repul-sion which makes that the cross sections fall off much fasteron the low-energy side. Therefore, taking in some cases aslightly higher beam energy is somehow on the ”safe” side.Experimental production rates can be found in a numberof publications from the former GSI on-line separator [65].They are summarized in table 2.
3. Simulation codes
In this section, we give a short overview of the fusion-evaporation codes used to calculate the theoretical crosssections. In total, five codes were used: i) CASCADE, ii)HIVAP, iii) CNABLA, iv) PACE, and v) GEMINI++.These codes use a two-step scenario for the reaction: pro-jectile and target nuclei completely fuse and then decayaccording to a statistical model approach of compoundnucleus reactions. They take into account competition be-tween different decay channels like proton, neutron, and α emission as well as γ decay and fission. All codes give a va-riety of decay information like the particles emitted, theirenergy and angular distribution etc. In the present work,we only use the production cross section of the isotope ofinterest. All programs used the Atomic Mass Evaluationdata base from 2012 [81].3.1. The CASCADE code
The program CASCADE was originally written by F.P¨uhlhofer [10]. The original version of the program wasmodified by different persons (e.g. E.F. Garman, F. Zwartsand M.N. Harakeh) to perform calculations for specialstates of good spin and parity, to include isospin and par-ity properly in the statistical decay as well as to includethe electric quadrupole decay. CASCADE is an analytic program which is quite fast andthus convenient to optimize projectile-target combinationsand the beam energy. In the present work, we use a versionof CASCADE provided by D. R. Chakrabarty [11].3.2.
The HIVAP code
HIVAP is a statistical evaporation code written byW. Reisdorf [12]. Several improvements were introducedlater [82,83]. We used a version provided to us by F.Hessberger [84]. Like CASCADE, HIVAP is an analyticalprogram being thus very fast.3.3.
The CNABLA code
CNABLA is a program which combines the POTFUSfusion code [14] for the first step of the reaction with theABLA part from the ABRABLA code [8] for the evapora-tion. POTFUS is a quite successfully used fusion code andallows us to prepare an input file with a predefined numberof events with four parameters: the mass and the charge ofthe complete-fusion product, its excitation energy and itsspin. These events are then used with a special version ofABRABLA [9] to perform the evaporation part by meansof a Monte-Carlo technique.3.4.
The PACE code
PACE is probably the most widely used fusion-evaporation code. It was originally written by A. Gavron [15].This Projection Angular-momentum Coupled Evaporation(PACE) code is again based on the statistical model anduses the Monte-Carlo approach for the de-excitation of thecompound nucleus. Only the equilibrium part of the decayis treated, no pre-equilibrium emission is considered.3.5.
The GEMINI++ code
The GEMINI++ code [16] is the C++ version of theoriginal GEMINI code [85,86] written by R.J. Charity. Inaddition to light particle emission and symmetric fission, itallows for all binary decays to occur. This new version curesproblems with heavier systems in the original code. Thecomplete fusion compound nuclei are again produced bythe POTFUS code [14] and read into GEMINI++ where aMonte-Carlo procedure is used to perform the de-excitationstep.3.6.
Averages from calculations
In order to compare the experimental results to the the-oretical predictions from the five codes, some averaging ofthe calculations is needed. This task is not so easy becausethe calculations can differ by one or two orders of magni-tude from one code to another. A standard average would4 ig. 1. Comparison of experimental cross sections taken from Table 1 and calculated cross sections with the five different models. Each figuregives the scale factor by which the calculated cross sections had to be divided to match the experimental cross sections. The deviation givesthe average difference factor between the experimental cross sections and the scaled calculated cross sections (see text). The left column(a-d) compares the experimental cross sections to all fives model calculations, for all data (a) and for different mass ranges (b-d). The rightcolumn compares all experimental data with the different models (e-i). ig. 2. Comparison of experimental and calculated cross sections (with the adopted scaling factor of 11.2) for selected elements. favor the larger cross sections (e.g. the average of 1 mb and100 mb being about 50 mb). Therefore, we decided to usethe geometrical average yielding for the example above anaverage of 10 mb. As the uncertainty range we used themaximum and minimum value from all codes.In general, not all codes give results for all isotopes orprojectile, target, and energy combinations. The average istherefore made with the results available.
4. Results and discussion
Comparison with experimental cross sections
Figure 1 gives an overview of all experimental data com-pared with the results of the individual codes and the aver-ages of these calculations as explained above. As indicatedon the figure 1a to get the best match between the averageof the simulations and the experimental data, we had to re-duce the results of the calculations by a scale factor of 11.2.The parameter called deviation is a measure for the scatterof the calculated cross sections, after scaling, around the experimental ones. Again due to large differences betweenthe calculated values from different models, we used a log-arithmic difference defined as: deviation = 10 ∗∗ " /n X n abs (cid:18) log (cid:18) σ cal /sfσ exp (cid:19)(cid:19) where n is the number of data points, σ cal and σ exp arethe calculated and the experimental cross sections, respec-tively, and sf is the scale factor mentioned above. There-fore, this deviation is the average factor by which the cal-culations deviate from the experimental value: the smallerthis value is, the better the model calculation, once scaledby a constant factor, agrees with experimental data.From figure 1a, we conclude that the average of the fivemodel calculations corrected by a scale factor of 11.2 devi-ate on average by a factor of close to 5 for individual value.As can be seen from the left-hand side of the figure 1, thescatter between the models and the experimental data ismuch better for lighter nuclei and gets worse when movingto heavier nuclei.The right-hand side of figure 1 gives an analysis of theresults as a function of the model used to calculate the cross6 ig. 3. Comparison of experimental and calculated production rates with a scale factor of 7.3 for the mass A=60 region (a) and of 4.6 forthe mass A=100 region (b-e). sections. From a first glance, it seems that PACE is thebest model, because the scale factor is the smallest of all.However, the scatter of the data is the largest of all models.Overall we believe that the GEMINI++ model coupled tothe POTFUS fusion program gives the most convincinganswer for fusion-evaporation cross sections. As in the othercases, the agreement is better for the low-mass region (A <
90) with a scale factor of 3.4 and a deviation parameter of1.9 and for the medium mass region (90 < A < Sn region, a lot of experiments have been per-formed and experimental cross sections determined, no-tably at the former GSI on-line separator [65]. Therefore,this region allows for a detailed comparison of experimentaldata and calculations. If we use the overall scale factor of
Table 3Scale factors for calculations and deviations between calculated val-ues and experimental data for the five different models in the
Snregion. model scale factor deviationCASCADE 6.3 1.9HIVAP 1.9 3.5CNABLA 22.8 7.0PACE 2.1 8.9GEMINI++ 2.4 1.9
Sn region. If we comparethe large body of experimental data from A=94 to A=117to the different models, we get scale factors and devia-tions as given in Table 3. In this region, HIVAP and POT-FUS+GEMINI++ give the best results with small scalefactors and small deviations.4.2.
Comparison with on-line production rates
Another possibility to compare predictions and experi-mental rates is to use production rates achieved in exper-iments and compare them to calculated rates. This com-parison is possible with production rates published fromthe former GSI on-line separator (see Table 2). However,in such a comparison the uncertainties are expected to beeven larger because, in order to calculate these rates, onehas to make assumptions about release and ionization effi-ciencies. This is a rather difficult task, because it involves alot of chemistry and the on-line rates are known to fluctu-ate from one run to the other due to often apparently mi-nor differences of the experimental conditions of differentexperiments.Nevertheless, we have attempted to predict production7 ig. 4. Comparison of experimental excitation functions for nuclei in the mass A=200 region [87] with predictions from the five modelsused. The energy range is unfortunately too small to draw conclusions about the accordance of the maximum of the distributions betweenexperimental data and models. rates for the future SPIRAL2 facility at GANIL, be it forneutron-induced fission of
U or fusion-evaporation reac-tions for proton-rich nuclei [88]. For this purpose, we havecollected experimental parameters of two types: (i) empiri-cal parameterizations of the release fractions based on mea-sured data at different facilities or (ii) parameters from dif-fusion and effusion laws which then allow the determinationof the total release efficiency, as was established by Kirch-ner et al. studying the performances of the UNILAC targetion source systems [89]. The latter approach has been usedfor the present study, where the diffusion and effusion coef-ficients were mostly obtained from measurements at UNI-LAC [89,90], CERN and Dubna [91]. Because of a lack ofdata in the case of Ga and In, we used diffusion coefficientsof the neighboring Ge and Sn elements, respectively. TheFEBIAD ionization efficiencies were estimated from effi-ciencies measured at ISOLDE for rare gases [92]. For themetallic elements of interest, an interpolation in mass givesresults which are compatible with the order of magnitudeof the efficiencies quoted by Kirchner for UNILAC (30 - 50% [93]).Figure 3 shows the results of this comparison. The in-target yields were estimated from the cross-section averagesas described in the previous section. The extracted yieldsare in-target yields multiplied by the diffusion, effusion andionization efficiencies, and have to be compared to the ex-perimental production rates measured at UNILAC. As inthe case of the production cross sections, the productionrates also scatter a lot. However, with the cross section scalefactor for the low-mass region of 7.3 (figure 3a) and 4.6 forthe mass A=100 region (figures 3b-e), we reach a reasonable agreement which seems to indicate that a reduction of thecalculated cross section is also needed for this comparison.We note that some of the less exotic isotopes have notbeen produced in ideal conditions, but experimenters settheir apparatus for a short while on these nuclei to starttheir experiment. As for these nuclei the beam energy wastherefore certainly not optimized, the simulation codes mayhave even larger deficiencies.4.3.
Excitation function of fusion-evaporation crosssections
As mentioned above the body of experimental data forproduction cross sections is quite scarce. This is even worsein terms of excitation functions where the production crosssections are measured as a function of the energy of theincident beam. We have found one example where sufficientdata are available to make a meaningful comparison. In theBi - Po region [87], a few cross sections have been measuredas a function of the incident beam energy, however, onlyover a short range. In figure 4, we compare this excitationfunction to the different models used in the present work.Interestingly, if we exclude the CNABLA model for thetwo A=200 nuclei, the maximum of the calculated valuesis rather close for the different models. It is difficult tosay whether the experimental trend is reproduced by themodel predictions. For such a statement, more data over awider range of energies would be needed. The figure alsoevidences that, in case of doubt, a slightly higher energy ismore convenient to move away from the threshold effect at8ow energies.
5. Summary
We have performed a detailed study of fusion-evaporationcross sections and production rates. Our first finding wasthat there is a rather limited number of experimentaldata available in the literature. In addition, these data aremost likely subject to large uncertainties keeping in mindthat for most of these data no experimental error bars aregiven in the literature. Therefore, in order to improve thebasis for this kind of studies, experimenters need to makeefforts to extract cross sections or production rates withexperimental uncertainties.We found that all codes that we tested over-estimate theexperimental production cross sections or rates with factorsof 4 or more. The most reliable code is maybe the GEM-INI++ evaporation code coupled with the POTFUS fusioncode, where a relatively small scale factor is needed and arelatively small scatter of the different rates or cross sec-tions is observed once the simulated data are scaled down.The overall overestimation of the cross sections seems toincrease towards the heaviest elements. A general recom-mendation is to divide predicted cross sections or rates bya factor of 5 - 10 to obtain experimental production ratesin reasonable agreement with ”experimental reality”.
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