Medium modified nucleon–nucleon cross sections in a nucleus

Medium modi®ed nucleon±nucleon cross sections in a nucleus

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

a Hampton University, Hampton VA, USAb National Research Council, Hampton, Virginia, Washington DC, USA

c Lyndon B. Johnson Space Center, Houston, USAd NASA Langley Research Center, Hampton VA, USA

Received 24 November 1998; received in revised form 26 January 1999

Abstract

A simple reliable formalism is presented for obtaining nucleon±nucleon cross sections within a nucleus in nuclear

collisions for a given projectile and target nucleus combination at a given energy for use in transport, Monte Carlo and

other calculations. The method relies on extraction of these values from experiments and has been tested for absorption

experiments to give excellent results. Ó 1999 Elsevier Science B.V. All rights reserved.

1. Introduction

A signi®cant amount of information aboutnucleon±nucleon (NN) collisions is obtained fromexperiments performed for free NN collisions.This information, when used for nucleon±nucleusand nucleus±nucleus collisions, has to be modi®ed,mainly due to presence of other nucleons in anucleus [1,2]. However, there is no simple reliablerecipe for this modi®cation directly obtained fromexperiments. The present work ®lls in this void andgives reliable values of nucleon±nucleon crosssections in a nucleus for a given projectile targetsystem at a given energy. This method will beuseful for transport, Monte Carlo and many other

calculations where medium modi®ed nuclear crosssections are needed.

2. Method

Following the coupled channel approach [3±8]developed here at NASA Langley Research Cen-ter, recently [9,10], we have demonstrated that thevalues of free nucleon±nucleon cross sections needto be modi®ed in a nucleus to explain availableexperimental absorption cross sections data. Sincein-medium nuclear cross sections are needed inmany studies (e.g. use in intranuclear cascademodels and heavy ion reaction models), and it isnot so obvious as to how to extract their valuesfrom Refs. [9,10], we decided to make this clearhere, so that it can be used by anyone not wantingto go into the details of the coupled channel ap-proach [3±8]. The detailed steps and expressions

Nuclear Instruments and Methods in Physics Research B 152 (1999) 425±431

www.elsevier.nl/locate/nimb

* Corresponding author. Address: NASA Langley Research

Center, Mail Stop 188B, Hampton VA 23681, USA. Tel.: 757-

864-1467; fax: 757-864-8097; e-mail: [email protected]

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

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

used for the formalism are given elsewhere [9,10],however, we sketch here some of the steps forcompleteness.

The absorption cross section in this formalismis given by

rabs � 2pZ 1

0

b dbf1ÿ eÿ2Im�v�g; �1�

where b is the projectile impact parameter and vthe eikonal phase matrix (see Refs. [3±8] for de-tails) is given by,

v�b� � vdir�b� ÿ vex�b�: �2�The direct and exchange terms are calculated usingthe following expressions [3±6],

vdir�b�� APAT

2pkNN

Zd2q exp�iq � b�F �1��ÿq�G�1��q�fNN�q�

�3�and

vex�b� �APAT

2pkNN

Zd2q exp�iq � b�F �1��ÿq�G�1��q�

� 1

�2p�2Z

d2q0 exp�iq0 � b�fNN�q� q0�C�q0�;

�4�where F�1� and G�1� are projectile and targetground-state one-body form factors, respectively,and C is the correlation function [6]. The massnumber of projectile and target nuclei are repre-sented by AP and AT, respectively. The two-bodyamplitude, fNN, is parameterized as

fNN � r�a� i�4p

kNN exp

�ÿ Bq2

2

�; �5�

where kNN is the relative wave number in the two-body center of mass system, r the two-body crosssection, B the slope parameter, and a the ratio ofthe real part to imaginary part of the forward,two-body amplitude.

Notice the absorption cross sections in Eq. (1)depend on the imaginary part of the eikonal phasematrix. This leads us to write the two-body am-plitude in the medium, fNN;m, as

fNN;m � fmfNN; �6�

where fNN is the free NN amplitude and fm thesystem and energy dependent medium multiplierfunction [9,10]. It follows that the nucleon±nucle-on cross sections in the medium (rm) can be writ-ten as

rNN;m � fmrNN; �7�

where rNN is the nucleon±nucleon cross section infree space, and the medium multiplier is given by

fm � 0:1 exp�ÿE=12�

� 1

�ÿ qav

0:14

� �1=3

exp

�ÿ E

D

��; �8�

where, for AT < 56,

D � 46:72� 2:21AT ÿ 2:25� 10ÿ2A2T �9�

and for AT P 56,

D � 100: �10�

In Eq. (8), qav refers to the average density of thecolliding system,

qav ÿ1

2qAP

ÿ � qAT

�; �11�

where the density of a nucleus Ai (i�P, T) is cal-culated in the hard sphere model, and is given by,

qAi� Ai

�4p=3�r3i; �12�

where the radius of the nucleus ri is de®ned by

ri � 1:29�ri�rms: �13�

The root-mean-square radius, (ri)rms is obtaineddirectly from the experimental charge radius[11,12]. For the bene®t of users our subroutine tocalculate the radius of a nucleus is listed in Ap-pendix A.

Use of Eqs. (7)±(10) modi®es free neutron±proton (rnp), and proton±proton (rpp) cross sec-tions which in turn can be used to calculate nu-cleon±nucleon (NN) cross sections for any systemof colliding nuclei by taking the isospin averagevalue as,

426 R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 425±431

rNN � ZPZT � NPNT

APAT

rpp

� NPZT � ZPNT

APAT

rnp; �14�

where, ZP, NP and AP (ZT, NT and AT) are thecharge, neutron and mass number of the projectile(target) nucleus. Note that in Eq. (14) we use theusual assumption that the nuclear pp cross sectionis equal to the nn cross section.

3. Procedure/discussion

For convenience, we ®t free neutron±proton(np) and proton±proton (pp) cross section data[12±14] at available energies by following smoothfunctions of energy for ease of interpolation andfor their availability at a given energy,

r � A exp

�ÿ E

25

�� B

1� exp��400ÿ E�=75�� C

1� exp��E ÿ 300�=75�� 28 exp

�ÿ E

125

�cos�0:26E0:453�; �15�

where for pp system A� 300, B� 44 and C� 15;and for np systems A� 922, B� 40 and C� 50.We make the usual assumption that the free nncross section is equal to the free pp cross section(baring Coulomb interaction).

Figs. 1 and 2 show the present representation(Eq. (15)) of free np and pp cross sections com-pared with the available data [12±14].

The procedure for obtaining in-medium crosssections is as follows. For a given projectile col-liding with target at laboratory energy (A MeV),the medium multipliers are calculated using Eqs.(8)±(13). Having calculated the medium multiplier,Eq. (7) is used to calculate in medium modi®ed npand pp cross sections, and then in-medium NNcross sections are calculated using Eq. (14).

4. Results

Figs. 3±6 show results of in-medium cross sec-tions in proton±nucleus collisions, for a set of

representative nuclei. For any other particularcase, it is good to use our formalism to get speci®cvalues. It is interesting to note that at lower en-ergies (less than 200 A MeV), there is system de-pendence of the cross sections and for energiesbetween 200 and 600 A MeV the system depen-dence is less important. It is also interesting to notethat at energies greater than 600 A MeV the me-dium modi®cations almost disappear and the freenucleon±nucleon cross sections appear adequate.

Fig. 1. Free neutron±proton cross sections. Stars represent

values from compilation [12±14]. The solid continuous line our

representation Eq. (15).

Fig. 2. Free proton±proton cross sections. Stars represent val-

ues from compilation [12±14]. The solid continuous line our

representation Eq. (15).

R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 152 (1999) 425±431 427

Figs. 7±10 give medium modi®ed cross sectionsfor neutron±nucleus collisions for various systems.An important comment is in order for this case,which was graciously pointed out to us by thereferee.

Since we do not know free neutron±neutroncross sections, we have assumed (rnn)free� (rpp)free

and used Eq. (15) for this also. Although, we havea fairly good idea of the neutron±neutron crosssections in the medium, the exact separation ofmedium and free space contribution, in this case, is

di�cult to infer from the present study. In fact, itis non-trivial, outstanding and as yet unsolvedproblem as remarked by the referee. Clearly, thisremains a challenge to scienti®c community today.

To make the comparison with the proton±nu-cleus systems, we have kept the same set of nucleifor these collisions as well. The general features ofproton±nucleus collisions are seen here as well.However, it is very interesting to compare themedium modi®ed cross sections for the two cases.We note that the medium modi®ed cross sections

Fig. 4. Modi®ed neutron±proton cross sections for various

proton±nucleus collisions.

Fig. 6. Same as Fig. 5. The energy range is extended to 2 GeV

to demonstrate the variation of nucleon±nucleon cross sections

with energy. See text for details.

Fig. 3. Modi®ed proton±proton cross sections for various


Fig. 5. Modi®ed nucleon±nucleon cross sections for various



needed for the proton±nucleus system are verydi�erent from the neutron±nucleus systems. Ob-viously, Coulomb interaction plays a role in themedium modi®cation of nucleon±nucleon crosssections. This is an interesting observation and toour knowledge has not been pointed out in theliterature earlier, and will signi®cantly impact theapplications.

Figs. 11±14 give results for nucleus±nucleuscollisions. To make our points we have takencarbon as the projectile. However, we would liketo emphasize again that for a speci®c case of in-terest it will be good to generate system modi®ed

cross sections using our formalism for that par-ticular system rather than taking the values fromother systems, since the values are system depen-dent as shown here. This is a really interestingobservation. We note here again that for nucleus±nucleus collisions the system modi®ed cross sec-tions are signi®cantly di�erent from the proton±nucleus and neutron±nucleus collisions. Our cal-culation seems to indicate that not only the nuclearforce, but also the Coulomb force plays a role inmodifying the cross sections in a nucleus, an e�ectwhich has been ignored by others.

Fig. 7. Modi®ed neutron±neutron cross sections for various

neutron±nucleus collisions.





Fig. 10. Same as Fig. 9. The energy range is extended to 2 GeV

to demonstrate the variation of nucleon±nucleon cross sections

with energy. See text for details.


5. Conclusions

We have shown very interesting features for thesystem dependence of the cross sections for vari-ous systems. We found that the system dependentcross sections depend both on the kinetics and thedynamics of the system. These e�ects are impor-tant. Our formalism includes them in a simple andreliable way. Microscopic methods [1,2,15,16] aretime consuming and do not extract these valuesfrom experimental observations as done here. Inaddition, our approach points out several inter-esting features which were not recognized by oth-

ers. Our method is easy to use and gives excellentagreement with experiments [9,10]. The usefulnessof our method for various applications ± trans-port, Monte Carlo, radiation protection, radiationtherapy ± is noted.

Acknowledgements

One of the authors (RKT) acknowledgesNASA Langley Research Center under grantNCC1-242 and National Research Council for thesupport.

Fig. 14. Same as Fig. 13. The energy range is extended to 2

GeV to demonstrate the variation of nucleon±nucleon cross

sections with energy. See text for details.


nucleus±nucleus collisions.

Fig. 11. Modi®ed proton±proton cross sections for various





Appendix A

CC*******************FUNCTION RADIUS*****************CFUNCTION RADIUS (A)DIMENSION NA(23),RMS(23)DATA NA/1,2,3,4,6,7,9,10,11,12,13,14,15,16,17,18,19,20,22,*23,24,25,26/DATA RMS/0.85,2.095,1.976,1.671,2.57,2.41,2.519,2.45,2.42,*2.471,2.440,2.58,2.611,2.730,2.662,2.727,2.900,3.040,2.969,2.94,*3.075,3.11,3.06/FACT� SQRT (5./3.)IA�A+0.4RADIUS�FACT * (0.84* A**(1./3.)+ 0.55)DO 1 I� 1,23IF (IA .EQ. NA(I)) GO TO 2GO TO 1

2 RADIUS�FACT*RMS(I)1 CONTINUERETURNEND

CC**************************************************************C

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Medium modified nucleon–nucleon cross sections in a nucleus