Cabibbo, Nicola

Volant, C

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TOTAL REACTION CROSS SECTIONS IN DEUTERON ANDPROTON INDUCED REACTIONSM. Cabibbo and C. VolantDAPNIA/SPhN, CEA/Saclay, F-91191 Gif-sur-Yvette CEDEX, FRANCE.(June 28, 2000)AbstractTotal reaction cross sections for deuteron and proton induced reactions areparameterised through models whose parameters are adjusted to the existingdata from low energies to the domain of around GeV per nucleon.I. INTRODUCTIONThe total reaction cross section is one of the rst basic measurements needed to un-derstand the nuclear interactions in ion-ion collisions. Its parameterisation is also of greatinterest and many attempts have been already done [1{8]. Studies of the spallation residueproduction by means of the reverse kinematics using heavy ions of energies around 1GeV/nucleon bombarding hydrogen targets (hydrogen and deuterium) are currently per-formed at GSI Darmstadt (Germany) [9]. The target used is relatively thick and secondaryreactions are quite probable. One way to correct the data for them is to rely on reactioncross section parameterisations. This has been done for proton targets but for deuteron tar-gets such work had to be performed. That is the aim of the present report. To be completewe will also applied this formulation to proton induced reactions.ext-2000-18428/06/2000II. USED FORMULATIONWe performed a simple routine which permits to calculate the total reaction cross sectionof deuteron and proton-induced reactions on heavy nuclei. It is based on two models:For energies per nucleon greater than 100 MeV we use the Karol microscopic model[1]. In this model the reaction probability is calculated assuming the interaction due tosingle nucleon-nucleon collisions in the region of the overlap between projectile and target.Therefore in this model the choice of the density distribution function plays a crucial rule. Inthe case of light nuclei (deuteron or proton in our case) coherently with the Karol descriptionthis function is assumed as a Gaussian:(r) =A(ap)3exp( (ra)2) (1)where a is related to the root mean square radius bya = (r:m:s: radius)  1:5 1=2(2)In litterature one can nd as r.m.s. charge radius of deuteron 2.1 fm [10], but in ourroutine we x this value equal 2.8 fm to have a better reproduction of experimental data[2,11]. This last number which corresponds to twice the Compton wave length of the pion isused as the square potential well radius in the simplest model for the deuteron system [12].In the case of p-induced reaction we x the correspondent r.m.s. proton radius equal to1.034 fm. In fact also one can nd in litterature [10,13] a r.m.s. proton radius equal to 0.8fm, but to have a better reproduction of experimental data (see the next section) we choose1.034 fm which is the radius of equivalent uniform model [13].For heavier nuclei (A > 40 target nuclei) the density distribution function is obtained inKarol model using the so-called \surface-normalized" Gaussian density distribution. With-out going into details a Gaussian form is used(r) = (0) exp( (ra)2) (3)2where both (0) and a are parameterised following Karol to t the Gaussian distributionto the surface of a Fermi distribution. This method permits an analytical solution for thereaction cross section.Another fundamental ingredient of this model is the average nucleon-nucleon collisioncross section <  > which is a function of the incident energy per nucleon (see ref. [1]).This cross section is a weighted average over the proton and neutron numbers of the targetand the projectile, between the neutron-neutron (proton-proton) total cross section iiandthe neutron-proton total cross section ij. The parameterisation which we use to take intoaccount the energy dependence of these cross sections is a compromise between a faithfulreproduction of the existent experimental data and simple analytical forms which allow fastreconstruction of the cross sections (see ref. [14]).At low incident energies the Karol model is no more valid because it neglects manytypical low energy eects (Pauli blocking, Fermi motion, Coulomb barrier). Therefore forenergies lower than 100 MeV per nucleon, the total reaction cross section is calculated, inour routine, using the Kox et al. geometrical formula [3]. This semiempirical approach startsfrom a simple geometrical formula corrected by the Coulomb barrier BCR= Rint2[1 BCEcm] BC=ZpZte2rc(A1=3p+ A1=3t)(4)where Ris the total reaction cross section, Ap(t)and Zp(t)are, respectively, the mass andproton number of projectile (target), and Ecmis the kinetic energy in the centre of masse.Kox et al. propose to divide the interaction radius Rintinto a volume term Rv=r0(A1=3p+ A1=3t) = Rp+ Rtand in a surface term Rs:Rint= Rv+ Rs(5)Introducing this surface term, it is possible to include in a simple geometrical descriptionthose part of the total reaction cross section due to the interaction between the projectileand the target in the region of the overlap. In Kox formula the surface term is given by:Rs= r0[aA1=3pA1=3tA1=3p+ A1=3t  ck] + D (6)3The rst term takes into account the mass asymmetry and is related to the volumeoverlap between projectile and target, ckis an energy dependent parameter which takescare of increasing surface transparency as the projectile energy increases. These two termsrepresent the main physical eects introduced by Rs.The last term is the neutron excess term DD =5(At  2Zt)ZpApAt(7)It takes into account the fact that at low incident energy the total cross section forneutron-proton (n-p) collisions is greater than those for n-n and p-p collisions. Thereforethis term is very important for heaviest or neutron-rich targets specially in the case of protonprojectile.Our routine uses an adjusted Kox formula. In the case of deuteron the projectile volumeradius Rpis xed (in Rv= Rp+Rtand in BC) to be equal to 1.9 fm. Remark that a r.m.s.value of 1.95 fm for deuteron is also quoted in ref. [15]. With this choice, which permits ageneral agreement with the experimental data (see next section), we take into account theloose structure of the deuteron. In this way we will have for Rvand BCRv= r0A1=3t+ 1:9 BC=ZpZte2(rcA1=3t+ 1:9)(8)where r0was x equal to 1.1 fm as in standard Kox parameterisation. For rcwe usevalues between rc=1.1 and rc=1.3 fm (see the pictures in sec.III); the last one being thevalue used in standard Kox parameterisation.Remark we do not modify the asymmetry term in eq.(II.6) which could be felt as aninconsistency because we change Rpfor deuteron from 1:1  21=3= 1:39 to 1.9 fm. In factfrom a microscopical point of view the interaction probability in the overlap region dependsnot only upon the overlap volume but also from the density in this region. In deuteroncase there is an increase of the overlap volume due to its large diusivity but on the otherhand the external projectile density is lower. In the parameterisation of Kox et al. thedensity prole is not taken into account. So the mentioned eect cannot be included on theasymmetry term without including new unknown parameters.4In the case of p-induced reaction calculations we x Rp=1.fm (in Rv= Rp+ Rtand inBC) and rc=r0=1.1 fm.In our routine the transparency terms of Rs(ckin eq. (II.6)) depends on the incidentenergy per nucleon in the following way:ck= 2:2704  log10(En)  2:6173 (9)This behavior was obtained by tting the ck-values obtained by Kox et al. for lowerenergies [3].III. COMPARISON TO EXPERIMENTAL DATAA. DeuteronsWe present some comparisons between results from our routine and experimental data.We calculate (see g.1-4) the total reaction cross section of d + Pb; Zr;Ni;Ca collisions;using the parameterisation explained in the previous section. In particular for each reactionwe show the calculation of Rusing both mentioned values for the rcparameter (1.1 and1.3 fm for incident energy lower than 100 MeV/nucleon).One can see a general agreement with experimental data; the small discontinuity at 100MeV per nucleon is due to the change of formula. For the lowest energies it is dicult to saywhich value for the rcparameter gives better results. One can note that using the standardvalue rc=1.3 fm we have a better agrement for the two heaviest target nuclei (Pb and Zr),but at the same time we have a worst continuity at 100 MeV with respect to rc=1.1 fm.In general at low energy the choice of rc=1.3 fm could be preferred because these resultsare obtained by changing in the Kox parameterisation only one parameter: the projectilevolume radius Rp.5For the highest energies, in g.1 and 3a) results obtained with two dierent values ofdeuteron r.m.s. radius: 2.8 and 2.1 fm are shown to illustrate the sensitivity of Karol formulaon this parameter. The good general agreement obtained xing this parameter equal to 2.8fm is very encouraging. Considering the scarse data available at the highest bombardingenergies, this justies our choice for this parameter in Karol formula.B. ProtonsIn g.5 is shown a comparison of our calculation with experimental data [16] in the caseof p +207:2Pb reaction. As can be seen in this case we obtain a very good continuity at100 MeV using the present parameterisation. The agreement with the data is also quitesatisfactory specially in the high energy regime. Note that we adapted Kox parameters tohave a good continuity with the Karol calculations. See also on the gure the low energycalculation using the standard Kox parameter (rc=1.3 instead of r0=1.1 fm).In g.6 is shown a standard Kox calculation for the full energy range. Note the discon-tinuity at 200 MeV because above this energy the neutron excess term D is put equal tozero. The authors of ref. [3] recommand to do so above 200 MeV/nucleon because nnandppare no more so dierent from np. This however is not completely true (see next sectionand g.10). In principle to solve this problem, an energy dependence of the D term shouldbe considered. This is yet out of the scope of this report.C. Comparison with other formulationsIn litterature dierent semiempirical models are available; see for example Tripathi et al.[4,5] and Prael and Chadwick [6] which tend to be universal.To be complete we show also the results obtained using other existing semiempiricalmodels in their published standard versions. In g.7 the standard Tripathi et al. model [4,5]and the Wellisch [6,7] formulation is applied to the p +207:2Pb reaction. As can be seenfor the lowest energies the corrected Wellisch [6,7] formulation reproduces the experimental6data better than our adjusted Kox formula. However our parameterisation works morenicely at the highest energies. The published Tripathi et al. parameterisation [4,5] givesworst results but the nice agreement shown in ref. [8] with the same model has to be claried.In g.8 and g.9 are shown the standard Tripathi calculations for the reactions d + Pband d + Ni respectively. The two curves (full and dashed) in each picture are obtainedusing two dierent values of the deuteron r.m.s. radius (2.8 and 2.1 fm respectively). Forthe targets, the r.m.s. radii are taken from [10] as recommanded in [4,5]. The agreementwith the experimental data is worst than the one obtained in the present work. One canalso notice the weaker sensitivity of the Tripathi et al. calculations comparative to ourformulation (see g.1 and 3a) on this parameter.IV. ENERGY DEPENDENCE OF RFig.10 shows the p-p and n-p total reaction cross section as function of the bombardingenergy calculated with the parameterisation of J. Cugnon et al. [14] used here. Clearlythe maximum transparency (minimum in R) seen around 300 MeV/nucleon in g. (1-9)arises from the variation of the nucleon-nucleon cross section as already pointed out in ref.[2]. Above this minimum Rstarts again to increase because of the opening of the pionproduction.Note a discontinuity at 80 MeV due to the change of parameterisation formula at thisenergy in [14]. In the present work only energy above 100 MeV are considered. Also, inintranuclear calculations these low energy collisions are of low importance.7V. CONCLUSIONThe present parameterisation can permit to perform the needed corrections mentionedin the introduction. The discrepancy observed at low energy between our parameterisationand experimental data (see g.1) is reasonable because in our formulation we don't takeinto account dierent physics eects releated to: projectile wave length, energy dependenceof Coulomb barrier, overlap volume (for deuteron case), energy dependence of D term. Onthe other hand the good agreement for the high energy range is very encouraging since thatis mainly in that region that the cross-section evaluation are needed. The main ingredientis the deuteron r.m.s. radius which has been adjusted to the data but the obtained phe-nomenological value can be reasonably well justied on a theoretical ground. Data above 400MeV/nucleon for deuteron total reaction cross section are not available for targets heavierthan carbon [18]. This makes dicult a complete check of the present predictions.8REFERENCES[1] Paul J. Karol, Phys. Rev. C 11 (1975) 1203 .[2] R. M. DeVries and J. C. Peng, Phys. Rev. C 22 (1980) 1055.[3] S. Kox et al., Phys. Rev. C 35 (1977) 1678.[4] R. K. Tripathi, F. A. Cucinotta, J. W. Wilson, Nucl. Instr. and Meth. B 117 (1996)347-349.[5] R. K. Tripathi, F. A. Cucinotta, J. W. Wilson, \Universal Parameterization of Absorp-tion Cross Sections" NASA Technical Paper 3621 (January 1997).[6] R. E. Prael and M. B. Chadwick, \Applications of Evaluated Nuclear Data in LAHETTMCode", Proceedings of the International Conference on Nuclear Data for Science andTechnology, May 19-24, 1997 Trieste, Italy (1997).[7] H. P. Wellisch and D. Axen, Phys. Rev. C 54 (1996) 1329.[8] R. E. Prael et al., \Plots Supplemental to: Comparison of Nucleon Cross Section Param-eterization Methods for Medium and high Energies", Los Alamos National Laboratoryreport LA-UR-98-5843 (December 1998).http://www-xdiv.lanl.gov/XCI/PEOPLE/rep/plist.htlm[9] W. Wlaz lo et al., Phys. Rev. Lett. 84 (2000) 5736 .F. Rejmund et al., submitted to Nucl. Phys. A.J. Benlliure et al., submitted to Nucl. Phys. A.T. Enqvist et al., submitted to Nucl. Phys. A.[10] H. de Vries, C.W. de Jager and C. de Vries, Atom. Data and Nucl. Data Tables 36,495-536 (1987).[11] Nguyen Van Sen et al., Phys. Lett. 156B (1985) 187.9[12] E. Segre, Nuclei and particles, W. A. Benjamin, Inc. New York 1965 Amsterdam.[13] H. R. Collard, L. R. B. Elton, and R. Hofstadter, in NuclearRadii, edited by H. Schop-per (Springer-Verlag, Berlin, 1967), Landolt-Bornstein, New Series, Vol.2.[14] J. Cugnon, D. L'Ho^te, J. Vandermeulen, Nucl. Instr. and Meth. B 111 (1996) 215.[15] J. Arvieux and J. M. Cameron Adv. in Nucl. Phys., Vol. 18. Edited by J. W. Negeleand Erich Vogt (Plenum Publishing corporation, 1987).[16] B.C. Barashenkov, Cross sections of Interactions of Particle and Nuclei with Nuclei,JINR, Dubna, 1993.[17] A. Auce et al., Phys. Rev. C 53 (1996) 2921.[18] J.Jaros et al., Phys. Rev. C 18 (1978) 2273.10FIGURESFIG. 1. (a)Total reaction cross sections in mb as function of incident energy per nucleon forthe system d +208Pb. The full line is our adjusted Kox-Karol calculation using rc=1.3 fm forE=n < 100 MeV (see text), the experimental data (open symbols) are from ref. [2,17]. (b) Thesame of (a) but using rc=1.1 fm. The dashed line in (a) is a Karol calculation obtained using ar.m.s. deuteron radius equal 2.1 instead of 2.8 fm (full line).FIG. 2. (a)Total reaction cross sections in mb as function of incident energy per nucleon for thesystem d+90Zr. The full line is our adjusted Kox-Karol calculation using rc=1.3 fm for E=n < 100MeV (see text), the experimental data (open symbols) are from ref. [2,17]. (b) The same of (a)but using rc=1.1 fm.FIG. 3. The same as in g.1 but for the system d +58Ni. In this case the experimental data(open symbols) are taken from ref. [11,17].FIG. 4. The same as in g.2 but for the system d +40Ca. In this case the experimental data(open symbols) are taken from ref. [2,11,17].FIG. 5. Total reaction cross sections in mb as function of incident energy for the systemp +207:2Pb. The dashed line is a standard Kox calculation. The full line is our calculation usingthe parameterisation presented in section II. The experimental data (open symbols) are from ref.[16].FIG. 6. The same as in Fig.5 but only with a standard Kox calculation for the full energy (fullline). The experimental data (open symbols) are from ref. [16].FIG. 7. Total reaction cross sections in mb as function of incident energy for the systemp +207:2Pb. The full line corresponds to the formulation of Prael and Chadwick and the dashedline is a Tripathi et al. calculation (see text). The experimental data (open symbols) are from ref.[16].11FIG. 8. Tripathi et al. calculations for the total reaction cross section (mb) for the reactiond +208Pb. The two calculations was obtained using a r.m.s. deuteron radius equal 2.8 fm. (fullline) and 2.1 fm. (dashed line). The experimental data (open symbols) are from ref. [2,17]FIG. 9. The same of g.8 but for the system d+58Ni. In this case the experimental data (opensymbols) are taken from ref. [11,17].FIG. 10. Proton-proton and neutron-proton total reaction cross sections as function of thebombarding energy calculated with the parameterisation of J. 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Total reaction cross sections in deuteron and proton induced reactions

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