Nuclear absorption cross sections using medium modi®ed nucleon±nucleon amplitudes

R.K. Tripathi a,*, John W. Wilson b, Francis A. Cucinotta b

a Hampton University, Hampton, VA 23668, USAb NASA Langley Research Center, Hampton, VA 23681, USA

Received 8 May 1998; received in revised form 25 June 1998

Abstract

We extract the in-medium nucleon±nucleon amplitudes from the available proton-nucleus total reaction cross sec-

tions data. The retrieval of the information from the experiment makes these estimates very reliable. Simple expressions

are then given for the in-medium nucleon±nucleon amplitudes for any system of colliding nuclei as a function of energy.

Excellent agreement is demonstrated in ion-nucleus absorption cross sections. Ó 1998 Elsevier Science B.V. All rights

reserved.

1. Introduction

Reliable estimates of nuclear absorptive prop-erties of various materials are required for radia-tion therapy, shield design, and radiationinteraction processes important to system re-sponse. Theoretical models have been derivedfrom multiple scattering expansions of the n-bodyamplitudes in terms of two-body scattering pro-cesses. Two-body scattering is signi®cantly modi-®ed due to the presence of other nucleons in anucleus. This is mainly a�ected by the Paulisuppression in the intermediate stages whichbrings in the density dependence [1] of the inter-

action in a very natural way. The detailed mi-croscopic reaction matrix calculations [1±3] aretime consuming. It is, therefore, desirable to ex-tract the renormalization of free nucleon±nucleonamplitudes in the medium directly from experi-ment. This method, although not completelymicroscopic in nature (in that it extracts renor-malization from the experiments), does includeseveral renormalization e�ects, which are di�cultto calculate in a microscopic theory. In addition,it has the advantage that the modi®cations aredirectly linked to the experimental observable,and hence can be used with ease (and with morecon®dence) in explaining data. Given the modi®edtwo-body amplitudes and the associated n-bodyformalism, the cross sections for the reaction ofarbitrary nuclear system can be evaluated accu-rately and e�ectively.

Nuclear Instruments and Methods in Physics Research B 145 (1998) 277±282

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

8094; e-mail: rkt@hesb1.larc.nasa.gov

0168-583X/98/$ ± see front matter Ó 1998 Elsevier Science B.V. All rights reserved.

PII: S 0 1 6 8 - 5 8 3 X ( 9 8 ) 0 0 4 2 0 - 0

2. Method

We follow the coupled-channel approach de-veloped at the NASA Langley Research Center. Inthis method, the Schrodinger equation for heavyion scattering can be solved in the eikonal ap-proximation [4±9] resulting in the following matrixof scattering for elastic amplitudes,

f�q� � ÿ ik2p

Zexp�ÿi~q:~b�fexp �iv�~b�� ÿ 1gd2b;

�1�where the bold face quantities (f and v) representmatrices, k is the projectile momentum relative tothe center of mass, b is the projectile impact pa-rameter vector, q is the momentum transfer, andv(b) is the eikonal phase matrix and is given by(see Ref. [7] for detailed expressions and morecomplete discussion),

v�b� � vdir�b� ÿ vex�b�: �2�The direct and exchange terms are calculated usingthe following expressions [7]

vdir�b� �APAT

2pkNN

Zd2q exp�iq:b�F �1��ÿq�G�1��q�

� fNN�q�; �3�

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 [10]. 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, again, the relative wave number inthe two-body center of mass system, r is the two-body cross section, B is the slope parameter, and a

is the ratio of the real part to imaginary part of theforward, two-body amplitude.

The total cross section is found from the elasticamplitude by using the optical theorem as follows

rtot � 4pk

Im�f �q � 0��: �6�

Eqs. (1)±(6) give,

rtot � 4pZ 1

0

bdbf1ÿ eÿIm�v� cos�Re�v��g: �7�

The total absorption cross section is found asfollows. Integrating Eq. (1) for the elastic channelby using dX�d2q/k2 and the relation

rtot � rabs � rel: �8�Further reduction using Eqs. (7) and (8) yields [7],

rabs � 2pZ 1

0

bdbf1ÿ eÿ2Im�v�g: �9�

Eq. (5), which is used in the calculation of thephase v, is the two-body amplitude for the freenucleon±nucleon interaction and needs to bemodi®ed for the nucleus±nucleus collisions. In thepresent work, we look into the modi®cation of thefree nucleon±nucleon cross section (r) in the me-dium. We, calculate the total absorption crosssection for a wide range of proton±nucleus colli-sions using Eq. (9) and modify the nucleon±nu-cleon (NN) cross section in Eq. (5) until goodagreement is found with the experiment. We write

fNN;m � fmfNN; �10�where fNN;m is the nucleon±nucleon amplitude inthe medium (nucleus) and fNN is the free NNamplitude from experiments and fm are the systemand energy dependent modifying functions de®nedlater in the text.

The renormalization of the amplitudes, thus, isextracted directly from the experiment and is areliable measure of the medium modi®cations. Themedium multipliers de®ned in Eq. (10) for protonenergy tlab in A MeV, are given by

fm � 0:1 exp�ÿtlab=12�

� 1ÿ dens

0:14

� ��1=3�exp ÿ tlab

tden

� �" #; �11�

where, for AT <56,

278 R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 277±282

tden � 46:72� 2:21AT ÿ 2:25� 10ÿ2A2T �12�

and for AT P 56,

tden � 100: �13�In Eq. (11), dens refers to the average density ofthe colliding system,

dens � 1

2�qAP� qAT

�; �14�

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

qAi� Ai

4p3

r3i; �15�

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

ri � 1:29 �ri�rms: �16�

The root-mean-square radius, (ri)rms is obtaineddirectly from the experiment [11] after ``subtrac-tion'' of the nucleon charge form factor [4].

We have also modi®ed Eq. (9), to account forthe Coulomb force in the proton±nucleus absorp-tion cross sections. This has signi®cant e�ects atlow energies and is less important as the energyincreases and practically disappears for energiesabove 50 A MeV.

For nucleus±nucleus collisions the Coulombbarrier energy [12] is given by,

Bc � 1:44ZPZT

R: �17�

With,

R � rP � rT � 1:2�A1=3P � A1=3

T �=E1=3cm ; �18�

where ZP and ZT are charge numbers for theprojectile and target, respectively, and R is theradial distance between their centers. We have keptthese expressions also for the proton±nucleus col-lisions in order to have a uni®ed picture of anycolliding system. However, we know that Eq. (18)over estimates the radial distance between proton±nucleus collisions and hence Eq. (17) under esti-mates the Coulomb energy between them. Tocompensate for this, we need to multiply Eq. (17)by the following factor in these cases, which givesthe Coulomb multiplier to Eq. (9);

XCoul � �1� C1=Ecm��1ÿ C2Bc=Ecm�: �19�For AT <56

C1 � 6:81ÿ 0:17AT � 1:88� 10ÿ3A2T

C2 � 6:57ÿ 0:30AT � 3:6� 10ÿ3A2T

�20�

and for AT P 56,

C1 � 3:0; C2 � 0:8: �21�The above formalism is also used for nucleus±

nucleus collisions. We have to be careful in ac-counting for the surface and coulomb e�ects of theprojectile. The surface e�ects (skin thickness) aretaken into account by the parameter tden in Eq.(11). For alpha particle and carbon projectiles,where skin thickness is prominent, we used thevalues of 300 and 100, respectively, for tden. Forrest of the nuclei considered, where there is notmuch density variation at the surface, a value oftden of 200 has been used.

Consequently, the Coulomb energy is takeninto account by multiplying Eq. (9) by the Cou-lomb factor Eq. (19) with parameters C1 and C2

given by Eqs. (20) and (21) for proton±nucleuscollisions. For nucleus±nucleus collisions parame-ter C1� 0 and the values of the parameter C2 forvarious systems are as follows:

C2 � 0:625 for a� C

� 0:3 for a� Pb

� 0:35 for C� C

� 0:5 for all other systems:

3. Discussion/results

Figs. 1±4 show the results of our calculationsfor reaction cross sections for proton on carbon,aluminium, iron, and lead targets respectively. Theexperimental data have been taken from thecompilation of Ref. [13]. The formalism presentedhere gives very good agreement with experimentfor all the systems for the entire energy range. Themedium amplitude multiplier in Eq. (11) is shownin Fig. 5 and shows that as the energy increases themultiplier tends to a value of unity, as expected,

R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 277±282 279

since the Pauli e�ects responsible for the much ofthe modi®cation of the amplitudes in the mediumbecome less important. We also note that as thetarget gets heavier, the multiplier gets smaller in-dicating that here the Pauli e�ect plays a biggerrole and gives greater reduction of the amplitudein the medium as expected.

Figs. 6±8 show the results for reaction crosssections for nucleus±nucleus systems. The data fora+C system (Fig. 6) has been taken from Refs.[14±17], and for a+Pb were obtained from Refs.[14,15,18±20]. Extensive data available for C+Csystem (Fig. 8) were taken from Refs. [14,15,17,21].

The agreement with the experiments is very good.To test the validity and usefulness of the methodproposed here, it has been used for heavier systems[22]. The agreement of the results obtained by thismodel with experimental observations for theheavier systems is also as good as the results dis-cussed here. It is satisfying to know that the in-medium nucleon±nucleon amplitudes derived heregive good results for all nucleus±nucleus collisions.

Our medium multiplier Eq. (11) is simple to useand gives reliable results for the nucleon±nucleonamplitudes in the medium down to very low en-ergies.

Fig. 1. Absorption cross section for proton±carbon collision as

a function of energy. The experimental points are taken from

Ref. [13]. The solid line represents our results.

Fig. 4. Absorption cross section for proton±lead collision as a

function of energy. The experimental points are taken from Ref.

[13]. The solid line represents our results.

Fig. 2. Absorption cross section for proton±aluminum collision

as a function of energy. The experimental points are taken from

Ref. [13]. The solid line represents our results.

Fig. 3. Absorption cross section for proton±iron collision as a

function of energy. The experimental points are taken from Ref.

[13]. The solid line represents our results.

280 R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 145 (1998) 277±282

Acknowledgements

One of the authors (RKT) acknowledgesNASA Langley Research Center under grantNCCI-242 for the support.

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