Accurate universal parameterization of absorption cross sections III – light systems

Accurate universal parameterization of absorption cross sectionsIII ± light systems

R.K. Tripathi a,b,*, F.A. Cucinotta c, J.W. Wilson a

a NASA Langley Research Center, Mail Stop 188B, Hampton, VA 23681, USAb National Research Council, Washington, DC, USA

c Lyndon B. Johnson Space Center, Houston, TX, USA

Received 30 March 1999; received in revised form 11 May 1999

Abstract

Our prior nuclear absorption cross sections model [R.K. Tripathi, F.A. Cucinotta, J.W. Wilson, Nucl. Instr. and

Meth. B 117 (1996) 347; R.K. Tripathi, J.W. Wilson, F.A. Cucinotta, Nucl. Instr. and Meth. B 129 (1997) 11] is ex-

tended for light systems (A 6 4) where either both projectile and target are light particles or one is light particle and the

other is medium or heavy nucleus. The agreement with experiment is excellent for these cases as well. Present work in

combination with our original model provides a comprehensive picture of absorption cross sections for light, medium

and heavy systems. As a result the extended model can reliably be used in all studies where there is a need for absorption

cross sections. Ó 1999 Published by Elsevier Science B.V. All rights reserved.

1. Introduction

The transportation of energetic ions in bulkmatter is of direct interest in several areas [1,2]including shielding against ions originating fromeither space radiations or terrestrial accelerators,cosmic ray propagation studies in galactic mediumor radio biological e�ects resulting from the workplace or clinical exposures. For carcinogenesis,terrestrial radiation therapy, and radio biologicalresearch, knowledge of the beam composition andinteractions is necessary to properly evaluate the

e�ects on human and animal tissues. For theproper assessment to radiation exposures, bothreliable transport codes and accurate input pa-rameters are needed.

One such important input is the total reactioncross section (rR), de®ned as the total (rT), mi-nus the elastic cross sections (rel), for two collidingions:

rR � rT ÿ rel: �1�We have developed a model for absorption

cross sections [3±6] which gives very reliable resultsfor the entire energy range from a few A MeV to afew A GeV. It is gratifying to note that severalagencies and institutions have adopted the modeland are using it with success in their programs.

Nuclear Instruments and Methods in Physics Research B 155 (1999) 349±356

www.elsevier.nl/locate/nimb

* Corresponding author. Tel. +757-864-1467; fax: +757-864-

8094; e-mail: [email protected]

0168-583X/99/$ ± see front matter Ó 1999 Published by Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 5 8 3 X ( 9 9 ) 0 0 4 7 9 - 6

The present work extends the model to lightersystems, where either both projectile and target arelight particles or one is light particle and the otheris medium or heavy nucleus. The details of ourprevious model are discussed elsewhere [3±6]. Wereproduce here the main features of the formalismfor completeness and to put the light systems inproper context.

2. Model description

Most of the empirical models approximate totalreaction cross section of Bradt-Peters form:

rR � pr20�A1=3

P � A1=3T ÿ d�2; �2�

where r0 is a constant related to the radius of acolliding ion, d is either a constant or an energydependent parameter and Ap and AT are the pro-jectile and target mass numbers, respectively. Thisform of parameterization works nicely for higherenergies. However, at lower energies for chargedions Coulomb interaction becomes important andmodi®es reaction cross sections signi®cantly. Forthe neutron-nucleus collisions there is no Coulombinteraction, but the total reaction cross section forthese collisions is modi®ed by the strength of theimaginary part of the optical potential at the sur-face, which was incorporated by introducing a lowenergy multiplier Xm that accounts for the strength

of the optical model interaction. Because we usedthe same form of parameterization for the neu-tron-nucleus case as well [4,6] ± which helped toprovide a uni®ed, consistent, and accurate pictureof the total reaction cross sections for any systemof colliding nuclei for the entire energy range- weincorporate the absorption cross sections for lightsystems in the same formalism also. We note thatstrong absorption models suggest energy depen-dence of the interaction radius. Incorporatingthese e�ects, and other e�ects discussed latter inthe text, we use, as before, the following form forthe reaction cross section:

rR � pr20�A1=3

P � A1=3T � dE�2 1

�ÿ Rc

BEcm

�Xm; �3�

where r0� 1.1 fm, and Ecm is colliding systemcenter of mass energy in A MeV. The second tolast term on the right-hand side is the Coulombinteraction term which modi®es the cross sectionat lower energies and becomes less important asthe energy increases (typically after several tens ofA MeV). The Coulomb multiplier Rc is needed inorder to have the same formalism for the absorp-tion cross sections for light, medium and heavysystems and for reasons discussed later in the text.In Eq. (3), B is the energy dependent Coulombinteraction barrier (right hand factor in Eq. (3)),and is given by

B � 1:44ZPZT

R; �4�

where ZP (ZT) is atomic number of the projectile(target) and R, the radius for evaluating the Cou-lomb barrier height is

R � rP � rT � 1:2�A1=3P � A1=3

T �E1=3

cm

; �5�

where ri is equivalent sphere radius and is relatedto the rrms;i radius by

Table 2

Coulomb multiplier for light systems

Systems p + d p + 3He p + 4He p + Li d + d d + 4He d + C 4He + Ta 4He + Au

Rc 13.5 21 27 2.2 13.5 13.5 6.0 0.6 0.6

Table 1

Parameters for a + X systems

Systems T1 G

General settings 40 75

a + a 40 300

a + Be 25 300

a + N 40 500

a + Al 25 300

a + Fe 40 300

350 R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 155 (1999) 349±356

ri � 1:29rrms;i �6�with (i�P,T). Our computer routine to calculatethe radius of a nucleus is given in reference [7].

There is energy dependence in the reactioncross section at intermediate and higher energiesmainly due to two e�ects±transparency andPauli blocking. This is taken into account in dE,which is

dE � 1:85S � 0:16S=E1=3cm ÿ CE

� 0:91�AT ÿ 2ZT�ZP=�ATAP�; �7�

where S is the mass asymmetry term and is de®nedas

S � A1=3P A1=3

T

A1=3P � A1=3

T

�8�

and is related to the volume overlap of the colli-sion system. The last term on the right-hand sideof Eq. (7) accounts for the isotope dependence ofthe reaction cross section. The term CE is related tothe transparency and Pauli blocking and is givenby

Fig. 1. Reaction cross sections as a function of energy for n + d

collisions.

Fig. 2. Reaction cross sections as a function of energy for

n + alpha collisions.

Fig. 3. Reaction cross sections as a function of energy for p + d

collisions.


p + helium 3 collisions.

R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 155 (1999) 349±356 351

CE � D�1ÿ exp�ÿE=T1�� ÿ 0:292exp�ÿE=792�� cos�0:229E0:453�; �9�

where the collision kinetic energy E is in units of AMeV. Here D is related to the density dependenceof the colliding system and can be nicely related tothe densities of the colliding systems for mediumand heavier systems [3±6]. This in e�ect simulatesthe modi®cations of the reaction cross sections dueto Pauli blocking. Eqs. (1)±(9) summarize ouroriginal model. For systems discussed in our pre-vious work T1� 40 in Eq. (9) gave very good

results. For light systems studied here, there is asigni®cant amount of surface in both the projectileand target nuclei and each case behaves somewhatdi�erent from the other. The best values of pa-rameter D and T1 in Eq. (9) for the cases studiedhere are as follows:

n (p) + X systems

T1 � 18 �23�

D � 1:85� 0:16

1� exp��500ÿ E�=200� : �10�


p + alpha collisions.


p + lithium 6 collisions.


p + lithium 7 collisions.

Fig. 8. Reaction cross sections as a function of energy for d + d

collisions.


d + X systems

T1 � 23

D � 1:65� 0:1

1� exp��500ÿ E�=200� : �11�

3He + X systems

T1 � 40

D � 1:55 �12�4He + X systems

D � 2:77ÿ 8:0� 10ÿ3AT � 1:8� 10ÿ5A2T

ÿ 0:8=�1� exp��250ÿ E�=G��: �13�Table 1 gives the values of the parameters T1 andG for alpha-nucleus systems.

For medium and heavy systems D can be ex-pressed in a very simple way [3±6] in terms of thedensities of the colliding nuclei. There is interestingphysics associated with constant D. The parameterD in e�ect simulates the modi®cations of the re-action cross sections due to Pauli blocking. Thise�ect is new and has not been taken into account


d + alpha collisions.Fig. 11. Reaction cross sections as a function of energy for

helium 3 + beryllium collisions.


d + carbon collisions.


helium 3 + carbon collisions.


in other empirical calculations. The introductionof the parameter D and its association with phys-ical phenomenon of Pauli blocking helps present auniversal picture of the reaction cross sections. Atlower energies (below several tens of A MeV)where the overlap of interacting nuclei is small(and where Coulomb interaction and imaginarypart of the optical potential modify the reactioncross sections signi®cantly) the modi®cations ofthe cross sections due to Pauli blocking are small,and gradually play an increasing role as the energyincreases, since this leads to higher densities wherePauli blocking gets increasingly important.

This method of calculation of Coulomb energydoes provide a uni®ed picture of reaction crosssections for any system of colliding nuclei. Wenote that for light systems Eq. (5) over-estimatesthe interaction distance and consequently Eq. (4)underestimates the Coulomb energy e�ect. In or-der to compensate for this e�ect and still maintainthe same formalism for light, medium and heavysystems, there was a need to introduce a Coulombmultiplier parameter Rc, in Eq. (3). The defaultvalue of Rc� 1 was used in all the cases studiedhere (including 3He + X systems) except for thecases mentioned in Table 2. For p + C system [3]


helium 3 + aluminum collisions.


alpha + alpha collisions.


alpha + beryllium collisions.


alpha + nitrogen collisions.


Rc� 3.5 gives good agreement with experiments.Although, there are indications for some generaltrends in light systems as well, but due to paucityof data it is premature to crystallize them untilmore data are available. Till such detailed analysisis done, it will be good to take the default (generalsetting) values for the parameters T1� 40, G� 75and Rc� 1 in Tables 1 and 2 for the cases notdiscussed here unless data demand otherwise. Theoptical model multiplier as introduced in refer-ences [4,6] is given by,

Xm � 1ÿ X1eÿE=X1� SL �14�with

X1 � 2:83ÿ 3:1� 10ÿ2AT � 1:7� 10ÿ4A2T: �15�

For the n + 4He system X1� 5.2 gives betteragreement with experiment. The function SL forlight systems as used here is,

SL � 1:2� 1:6�1ÿ exp�ÿE=15��: �16�A comment is in order regarding the use of the

model at very low energies, which was graciously


alpha + aluminum collisions.


alpha + iron collisions.


alpha + tantalum collisions.


alpha + gold collisions.


pointed out to us by the referee. For a givensystem as we keep lowering energy, at some energyeither the Coulomb factor (1ÿRc B/Ecm) or theoptical model multiplier Xm in Eq. (3) wouldbecome negative. This would be the lowest energyto which the model should be used for the systemto maintain the positivity of the cross sections (thecross section can never be a negative quantity). Wehave built in a check in our code where a valueof zero is returned for the cross section in suchcases.

3. Results

Figs. 1±20 show the plots of available resultsfor neutron-nucleus, proton-nucleus, deuteron-nucleus, helium 3-nucleus and alpha-nucleus sys-tems. The data in Figs. 1 and 2 are from references[8]; and for Fig. 3 have been taken from references[8,9]. For Fig. 4 data has been taken from refer-ences [9±12]. Extensive data set exists for p + 4Hecollisions (Fig. 5) and have been taken from ref-erences ([8,9], [11±17]). Data for Figs. 6 and 7 havemainly been collected from the compilation of Ref.[9] and that of Figs. 8±10 are from reference [13].Not much data are available for 3He±nucleuscollisions and have been taken from reference [18]for Figs. 11±13. For 4He + 4He (Fig. 14) data havebeen taken from references ([13,14,18]). For Fig.15 data have been taken from reference [18]. ForFigs. 16±18 data have been taken from reference[19] and that of Figs. 19 and 20 have been takenfrom reference [20].

4. Conclusions

The agreement of our results with experimentsfor light systems is excellent for the entire energyrange from a few A MeV to a few A GeV and is ofthe same quality as that of our previous work [3±6].

Present work in combination with our originalmodel provides a comprehensive picture of ab-sorption cross sections for light, medium and heavysystems. We are not aware of any published or re-ported model which gives as good agreement forabsorption cross sections for light, medium andheavy systems for the entire energy range as foundhere.

Acknowledgements

One of the authors (RKT) acknowledges NRC-LaRC fellowship.

References

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Accurate universal parameterization of absorption cross sections III – light systems