A method for calculating proton–nucleus elastic cross-sections

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

a NASA Langley Research Center, Mail Stop 188B, 23681-0001 Hampton, VA, USAb Lyndon B. Johnson Space Center, Houston, TX, USA

Received 14 June 2001; received in revised form 16 January 2002

Abstract

Recently [Nucl. Instr. and Meth. B 145 (1998) 277; Extraction of in-medium nucleon–nucleon amplitude from ex-

periment, NASA-TP, 1998], we developed a method of extracting nucleon–nucleon (N–N) cross-sections in the medium

directly from experiment. The in-medium N–N cross-sections form the basic ingredients of several heavy-ion scattering

approaches including the coupled-channel approach developed at the NASA Langley Research Center. We investigated

[Proton–nucleus total cross-sections in coupled-channel approach, NASA/TP, 2000; Nucl. Instr. and Meth. B 173–174

(2001) 391] the ratio of real to imaginary part of the two body scattering amplitude in the medium. These ratios are used

in combination with the in-medium N–N cross-sections to calculate proton–nucleus elastic cross-sections. The agree-

ment is excellent with the available experimental data. These cross-sections are needed for the radiation risk assessment

of space missions.

� 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

The transportation of energetic ions in bulkmatter is of direct interest in several areas includ-ing shielding against ions originating from eitherspace radiations or terrestrial accelerators, cosmicray propagation studies in galactic medium orradiobiological effects resulting from the workplace or clinical exposures. For carcinogenesis,terrestrial radiation therapy, and radiobiologicalresearch, knowledge of beam composition andinteractions is necessary to properly evaluate theeffects on human and animal tissues. For the

proper assessment of radiation exposures bothreliable transport codes and accurate input pa-rameters are needed. One such important input iselastic cross-sections. The motivation of the workis to develop a method for calculating accurateelastic cross-sections. These are needed in trans-port methods both deterministic and Monte Carlo.

Nucleon–nucleon (N–N) cross-sections are thebasic ingredients of many approaches [1–10] toheavy ion scattering problem. Most of the infor-mation about these N–N cross-sections comesfrom the free two-body scattering. These cross-sections are significantly modified in a nucleus, dueto presence of other nucleons, which is affected [11]through the Pauli exclusion principle and modi-fication of meson field coupling constants. Thein-medium corrections are very important effectsand nuclear-physics literature is quite rich on this

Nuclear Instruments and Methods in Physics Research B 194 (2002) 229–236

www.elsevier.com/locate/nimb

*Corresponding author. Tel.: +1-757-864-1467; fax: +1-757-

864-8094.

E-mail address: r.k.tripathi@larc.nasa.gov (R.K. Tripathi).

0168-583X/02/$ - see front matter � 2002 Elsevier Science B.V. All rights reserved.

PII: S0168-583X(02 )00690-0

subject (for example, see [12–19]). Most of thesetheoretical methods either use nonrelativisticmany-body [12–14] or relativistic many-body [15–19] approaches. Unfortunately, there is no agree-ment between different approaches. As a result,there is no consensus on this issue. The essence ofthe problem is to answer a simple question as tohow a nucleon behaves and interacts inside a nu-cleus. The theoretical approaches incorporate therenormalization effects to include the effects of themedium in the chosen approach. The goal, ofcourse is, to be able to explain the experimentalresults. Our work [20–23] is technology based. As aresult, we approach the problem from the otherend. And ask ourselves the question as to whatexperiments tell us about the in-medium effects.Consequently, our results may also provide themeeting ground for various approaches on thisissue and on the Coulomb effects discussed later inthe text. Our theoretical approach is based on thecoupled channel method [1–6] used at the NASALangley Research Center. This method solves theSchrodinger equation with an eikonal approxi-mation. The method needs modifications at lowand medium energies. In an earlier work [20,21],we developed a unique method of extracting me-dium modified N–N cross-sections from experi-ments and found that the renormalization of thefree N–N cross-sections is significant [20,21] atlower and medium energies. These modified in-medium N–N cross-sections, in combination withthe newly developed ratio of the real to imaginarypart of the two-body scattering amplitudes in themedium, were used to calculate the total cross-sections for proton–nucleus collisions [22,23]. Wedemonstrated that the blend of the renormalizedN–N cross-sections, the in-medium ratio of thereal to imaginary part of the two-body amplitudeand the coupled-channel method gave reliableapproach to the total cross-sections. The purposeof the current paper is three folds: (i) To put inplace a reliable method for calculating elasticcross-sections for collisions of protons with ions.(ii) To use our previously developed N–N cross-sections in the medium and modified two-bodyamplitudes to calculate elastic cross-sections forproton–nucleus collisions. (iii) To validate andcompare the calculated results with the available

experimental data. And provide theoretical resultswhere data are not available (due to nonexistenceof experimental facilities and/or difficulty in ex-perimental data analysis).

2. Method

We briefly sketch here the essentials of thecoupled-channel method for completeness (see [1–6] for details). In this approach the matrix forelastic scattering amplitude is given by

f ðqÞ ¼ � ik2p

Zd2b exp ð�iq � bÞ

� fexp ½ivðbÞ � 1g; ð1Þ

where f and v represent matrices, k is the projectilemomentum relative to the center of mass, b is theprojectile impact parameter vector, q is the mo-mentum transfer and vðbÞ is the eikonal phasematrix.

The total cross-section is found from the elasticscattering amplitude by using the optical theoremas follows:

rtot ¼4pkIm f ðq ¼ 0Þ: ð2Þ

Eqs. (1) and (2) give

rtot ¼ 4pZ 1

0

dbbf1� e�ImðvÞ cosðReðvÞÞg: ð3Þ

And the absorption cross-sections (rabs) is given by[20,21]

rabs ¼ 2pZ 1

0

dbbf1� e�2ImðvÞg: ð4Þ

Having calculated the total and absorption cross-sections, for many nuclei, elastic cross-section isthe difference of these quantities,

rel ¼ rtot � rabs: ð5Þ

The eikonal phase matrix v (see [1–6] for details) isgiven by

vðbÞ ¼ vdirðbÞ � vexðbÞ: ð6ÞThe direct and exchange terms are calculated usingthe following expressions [1–6]:

230 R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 194 (2002) 229–236

vdirðbÞ ¼APAT

2pkNN

Zd2q exp ðiq � bÞF ð1Þð�qÞ

� Gð1ÞðqÞfNNðqÞ ð7Þ

and

vexðbÞ ¼APAT

2pkNN

Zd2q exp ðiq � bÞF ð1Þð�qÞGð1ÞðqÞ

� 1

ð2pÞ2Z

d2q0 exp ðiq0 � bÞfNNðqþ q0ÞCðq0Þ;

ð8Þ

where F ð1Þ and Gð1Þ are projectile and targetground-state one-body form factors, respectively,kNN is the relative wave number in the two-bodycenter of mass system, and C is the correlationfunction [6]. The mass numbers of projectile andtarget nuclei are represented by AP and AT, re-spectively. The two-body amplitude, fNN, is para-meterized as

fNN ¼ rða þ iÞ4p

kNN exp

�� Bq2

2

�; ð9Þ

where r is the two-body cross-section, B is theslope parameter, and a is the ratio of the real partto imaginary part of the forward, two-body am-plitude.

It is well known that the absorption cross-section depends on the imaginary part of the eik-onal phase matrix. This led us to write [20,21] thetwo-body amplitude in the medium, fNN;m, as

fNN;m ¼ fm fNN; ð10Þwhere fNN is the free NN amplitude and fm is thesystem and energy dependent medium multiplierfunction [20,21]. It follows that the nucleon–nucleon cross-sections in the medium (rNN;m) canbe written as

rNN;m ¼ fm rNN; ð11Þwhere rNN is the nucleon–nucleon cross-section infree space, and the medium multiplier is given by

fm ¼ 0:1 exp ð�E=12Þ

þ 1

�� qav

0:14

� � 13ð Þexp

�� ED

��; ð12Þ

where E is the laboratory energy in units ofAMeV. D is a parameter, in units of MeV, as

defined below. The numbers 12 and 0.14 are inunits of MeV and fm�3 respectively. For AT 6 56(mass number for iron ion representing heavyelements considered in our transport phenomena),

D ¼ 46:72þ 2:21AT � 2:25� 10�2A2T ð13Þ

and, for AT > 57,

D ¼ 100 MeV: ð14ÞIn Eq. (12), qav refers to the average density of thecolliding system,

qav ¼1

2qAP

�þ qAT

; ð15Þ

where the density of a nucleus Ai (i ¼ P ; T ) is cal-culated in the hard sphere model, and is given by

qAi¼ Ai

4p3r3i; ð16Þ

where the radius of the nucleus ri is defined by

ri ¼ 1:29ðriÞrms: ð17ÞThe root-mean-square radius, ðriÞrms is obtaineddirectly from experiment [24] after ‘‘subtraction’’of the nucleon charge form factor [2].

We also note from Eq. (3), that total cross-section depends on real component of eikonalphase matrix and hence (see Eqs. (5)–(7)) on theproduct of ra in two-body amplitude. Since wehave determined and tested thoroughly [20,21] themodification of the cross-sections in the medium,we study here the modification of a-ratio of real toimaginary part of the two-body amplitude – in themedium in order to calculate the total and elasticcross-sections. Some data for total and elasticcross-sections are available for a few systems athigh energies. Unfortunately, no data are availablefor total and elastic cross-sections at low andmedium energy range (there is some data forpþ Pb in the 100 A MeV range). Therefore, valuesof the medium modified a have been tested forhigher energies. At low and medium energies, ourtheoretical results, which incorporate the in-medium two-body amplitudes, can be validated, ifand when experimental data becomes available.

A best estimate of medium modified a ac-counting the enhancement of the cross-sections[33] and stability is given by

R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 194 (2002) 229–236 231

am ¼ 3 exp�� E�

� 13A1=32=5000

�

þ K1þ exp 10�E

75

� ; ð18Þ

where

K ¼ 0:35þ 0:65 exp

�� 2

3ðN � ZÞ

�; ð19Þ

with N being the neutron number of the nucleusand Z its charge number.

We have also modified Eq. (3), to account forthe Coulomb force in the proton–nucleus cross-sections. Coulomb effects are very interesting andhave been extensively worked on in the literature(see, for example Refs. [25–30]). There is generalagreement that there is a need to modify the freenucleon–nucleon coulomb interaction for the nu-cleus–nucleus collisions. There is model depen-dence on the modifications of the used theoreticalapproach. Here, again, we approach this fromtechnology perspective and ask ourselves what theexperiments tell us about this effect. Secondly, tokeep the accepted form of the effect uniform weinvestigated [20–23] this using the available ex-perimental data and found what medications areneeded to explain the experimental results. In viewof the importance of the problem and its wide use,in many different areas, in physics there will re-main an active interest in this area. Our work mayprovide a useful bridge between various theoreticalmodels and experiments. Our work indicated thatthis has significant effects at low energies and be-comes less important as the energy increases andpractically disappears for energies around 50AMeV and higher. In the present work we use thisformalism and recap the essentials features forcompleteness.

For nucleus–nucleus collisions the Coulombenergy is given by

VB ¼ 1:44ZPZT

R; ð20Þ

where the constant 1.44 is in units of MeV fm, ZP

and ZT are charge numbers for the projectile andtarget respectively and R, the radial distance be-tween their centers is given by

R ¼ rP þ rT þ 1:2 Að1=3ÞP

�þ Að1=3Þ

T

�=Eð1=3Þ

CM : ð21Þ

The number 1.2 in Eq. (19) is in units offmMeVð1=3Þ. In our earlier work [20,21], these ex-pressions were used also for the proton–nucleuscollisions in order to have a unified picture of anycolliding system. However, as shown in [20,21],Eq. (21) over estimates the radial distance betweenproton–nucleus collisions and hence Eq. (20) un-der estimates the Coulomb energy between them.To compensate for this, we multiplied Eq. (20) bythe following factor in these cases (see [20,21] fordetails), which gives the Coulomb multiplier to Eq.(3),

XCoul ¼ ð1þ C1=ECMÞð1� C2VB=ECMÞ: ð22Þ

For AT 6 56 (mass number for iron),

C1 ¼ 6:81� 0:17AT þ 1:88� 10�3A2T;

C2 ¼ 6:57� 0:30AT þ 3:6� 10�3A2T:

ð23Þ

The constant C1 is in units of MeV. For AT > 57,

C1 ¼ 3:0 MeV;

C2 ¼ 0:8:ð24Þ

For the nucleus–nucleus collisions, C1 ¼ 0 MeVand C2 ¼ 1. We have found that this form of Cou-lomb energy works well for the proton–nucleusabsorption cross-sections [20]. Eq. (5) is the mainequation. The total cross-section (Eq. (3)) and ab-sorption cross-section (Eq. (4)) section are multi-plied by Eq. (22) to get the total and absorptioncross-sections in the medium and then these areused in Eq. (5) to get the results shown in Figs.1–6. For clarity we mention below the final ex-pressions used for calculating the total and ab-sorption cross-sections in the medium

rtot ¼ 4pð1þ C1=ECMÞð1� C2VB=ECMÞ

�Z 1

0

dbbf1� e�ImðvmÞ cosðReðvmÞÞg ð25Þ

232 R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 194 (2002) 229–236

and

rabs ¼ 2pð1þ C1=ECMÞð1� C2VB=ECMÞ

�Z 1

0

dbbf1� e�2ImðvmÞg: ð26Þ

The constants C1 and C2 are given by Eqs. (21) and(22) and the in-medium eikonal phase matrix vm

takes the form

vmðbÞ ¼ vm;dirðbÞ � vm;exðbÞ: ð27Þ

The direct and exchange terms in the medium arecalculated using the following expressions:

vm;dirðbÞ ¼APAT

2pkNN

Zd2q exp ðiq � bÞF ð1Þð�qÞ

� Gð1ÞðqÞfNN;mðqÞ ð28Þ

Fig. 1. Elastic cross-sections for proton–beryllium collision as a

function of energy. The experimental points are taken from

[31,32]. The solid line includes the modifications discussed in the

present work. The dotted line is without the corrections.

Fig. 2. Elastic cross-sections for proton–carbon collision as a

function of energy. The experimental points are taken from

[31,32]. The solid line includes the modifications discussed in the

present work. The dotted line is without the corrections.

Fig. 3. Elastic cross-sections for proton–aluminum collision as

a function of energy. The experimental points are taken from

[31,32]. The solid line includes the modifications discussed in the

present work. The dotted line is without the corrections.

Fig. 4. Elastic cross-sections for proton–iron collision as a

function of energy. The solid line includes the modifications

discussed in the present work. The dotted line is without the

corrections.

R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 194 (2002) 229–236 233

and

vm;exðbÞ ¼APAT

2pkNN

Zd2q exp ðiq � bÞF ð1Þð�qÞGð1ÞðqÞ

� 1

ð2pÞ2Z

d2q0 exp ðiq0 � bÞ

� fNN;mðqþ q0ÞCðq0Þ: ð29ÞThe two-body amplitude in the medium, fNN;m;

includes the in-medium nucleon–nucleon cross-sections (rm),

fNN;mðqÞ ¼rmðam þ iÞ

4pkNN exp

�� Bq2

2

�: ð30Þ

Eqs. (25)–(30) are used to calculate the elasticcross-sections shown in Figs. 1–6. The procedure isas follows: using the in-medium NN cross-sectionscalculate the two-body amplitude in the mediumby Eq. (30). Then use Eqs. (28) and (29) to cal-culate direct and exchange part of the eikonalphase matrix and Eq. (27) for the full eikonalmatrix, which in turn is used in Eqs. (25) and (26)to calculate the total and absorption cross-sectionsin the medium. And finally, Eq. (5) then gives theresults shown in Figs. 1–6.

3. Results and conclusions

Figs. 1–6 show the results of our calculationsfor the elastic cross-sections for proton on beryl-lium, carbon, aluminum, iron, lead and uraniumtargets respectively. The solid line includes themodifications discussed in the present work. Thedotted line is without the corrections. The experi-mental data have been taken from the compilationof [31,32]. There is paucity of data at lower andintermediate energies, where the medium modifi-cations play a significant role. For the energyranges considered, where the data is unavailable,our results should provide good values of elasticcross-sections, since many renormalization effectsdue to medium, which play important role incross-sections, have been built in the formalism.The reason for the enhancement in elastic cross-sections in the intermediate energy range is mainlydue to the fact that real part of the two-bodyscattering amplitude is more dominant comparedto the imaginary of the two-body scattering am-plitude in this energy range. Fig. 7 shows thein medium nucleon–nucleon cross-sections as afunction of energy for various systems, and Fig. 8shows the in-medium ratio of real to imaginarypart of the two-body amplitudes. It is good to notethat these values are needed to explain the avail-able experimental results and may also be usefulfor comparison with detailed theoretical calcula-tions.

Fig. 5. Elastic cross-sections for proton–lead collision as a

function of energy. The solid line includes the modifications

discussed in the present work. The dotted line is without the

corrections.

Fig. 6. Elastic cross-sections for proton–uranium collision as a

function of energy. The solid line includes the modifications

discussed in the present work. The dotted line is without the

corrections.

234 R.K. Tripathi et al. / Nucl. Instr. and Meth. in Phys. Res. B 194 (2002) 229–236

We find very good agreement with the experi-mental results for all the systems at higher energieswhere some data is available. We note that the in-medium cross-sections derived earlier in combi-nation with the modified ratio of real to imaginarypart of the amplitude discussed here provide goodresults for the proton–nucleus elastic cross-sec-tions. It is gratifying to note that the presentmethod gives a consistent approach for the totalreaction and the total cross-sections, hence, asdiscussed in the present work for the elastic cross-sections for the entire energy range for all thesystems studied here. The in-medium two body

modifications developed in the present approachcan be used with ease in other nuclear processes.

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