universal parameterization of absorption cross sections

National Aeronautics and Space AdministrationLangley Research Center • Hampton, Virginia 23681-0001

NASA Technical Paper 3621

Universal Parameterization of AbsorptionCross SectionsR. K. TripathiSouthern Illinois University • Carbondale, Illinois

Francis A. Cucinotta and John W. WilsonLangley Research Center • Hampton, Virginia

January 1997

Printed copies available from the following:

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Abstract

This paper presents a simple universal parameterization of total reaction crosssections for any system of colliding nuclei that is valid for the entire energy rangefrom a few AMeV to a few AGeV. The universal picture presented here treats proton-nucleus collision as a special case of nucleus-nucleus collision, where the projectilehas charge and mass number of one. The parameters are associated with the physicsof the collision system. In general terms, Coulomb interaction modifies cross sectionsat lower energies, and the effects of Pauli blocking are important at higher energies.The agreement between the calculated and experimental data is better than all earlierpublished results.

Introduction

Transportation of energetic ions in bulk matter is ofdirect interest in several areas (ref. 1), including shield-ing against ions originating from either space radiationsor terrestrial accelerators, cosmic ray propagation studiesin a galactic medium, or radiobiological effects resultingfrom the workplace or clinical exposures. For carcino-genesis, terrestrial radiation therapy, and radiobiologicalresearch, knowledge of the beam composition and inter-actions is necessary to properly evaluate the effects onhuman and animal tissues. For proper assessment of radi-ation exposures, both reliable transport codes and accu-rate input parameters are needed.

One such important input is the total reactioncross section, defined as the total minus the

elastic cross sections for two colliding ions:

(1)

In view of its importance, the total reaction cross sec-tion has been extensively studied both theoretically(refs.1–14) and experimentally (refs. 15–24) for the pastfive decades. A detailed list of references is given in ref-erences 1, 13, and 16. Empirical prescriptions have beendeveloped (refs. 2–4, 10, 11, and 13) for the total reactioncross sections working in various energy ranges andcombination of interacting ions. The present modelworks in all energy ranges with uniform accuracy for anycombination of interacting ions, including proton-nucleus collisions, and is more accurate than earlierreported empirical models (ref. 10), which were accurateabove 100 AMeV but showed large errors up to 25 per-cent at lower energies.

Model Description

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

(2)

where r0 is energy-independent,δ is either an energy-independent or energy-dependent parameter, andAP andAT are the projectile and target mass numbers, respec-tively. This form of parameterization works nicely forhigher energies. However, for lower energies, Coulombinteraction becomes important and modifies reactioncross sections significantly. In addition, strong absorp-tion models suggest energy dependence of the interactionradius. Incorporating these effects, and other effects dis-cussed later in the text, we propose the following formfor the reaction cross section:

(3)

wherer0 = 1.1 fm, andEcm is the colliding system centerof mass energy in MeV. The last term in equation (3) isthe Coulomb interaction term, which modifies the crosssection at lower energies and becomes less important asthe energy increases (typically after several tens ofAMeV). In equation (3),B is the energy-dependent Cou-lomb interaction barrier (factor on right side of eq. (3))and is given by

(4)

whereZP andZT are the atomic numbers of the projectileand target, respectively, andR, the distance for evaluat-ing the Coulomb barrier height, is

(5)

whereri is equivalent sphere radius and is related to therrms,i radius by

(6)

with (i = P,T).

σR( ) σT( )σel( )

σR σT σel–=

σR πr02

AP1/3

AT1/3 δ–+

2

=

σR πr02

AP1/3

AT1/3 δE+ +

2

1 BEcm----------–

=

B1.44ZPZT

R-------------------------=

R rP rT

1.2 AP1/3

AT1/3

+

Ecm1/3

---------------------------------------+ +=

r i 1.29r rms,i=

2

The energy dependence in the reaction cross sectionat intermediate and higher energies results mainly fromtwo effects—transparency and Pauli blocking. Thisenergy dependence is taken into account inδE, which isgiven by

(7)

whereS is the mass asymmetry term given by

(8)

and is related to the volume overlap of the collision sys-tem. The last term on the right side of equation (7)accounts for the isotope dependence of the reaction crosssection. The termCE is related to both the transparencyand Pauli blocking and is given by

(9)

where the collision kinetic energyE is in units of AMeV.Here,D is related to the density dependence of the collid-ing system, scaled with respect to the density of the12C+12C colliding system:

(10)

The density of a nucleus is calculated in the hard-sphere model. Important physics is associated with con-stantD. In effect,D simulates the modifications of thereaction cross sections caused by Pauli blocking. ThePauli blocking effect, which has not been taken intoaccount in other empirical calculations, is being intro-duced here for the first time. Introduction of the Pauliblocking effect helps present a universal picture of thereaction cross sections.

At lower energies (below several tens of AMeV)where the overlap of interacting nuclei is small (andwhere the Coulomb interaction modifies the reactioncross sections significantly), the modifications of thecross sections caused by Pauli blocking are small andgradually play an increasing role as the energy increases,which leads to higher densities where Pauli blockingbecomes increasingly important. Interestingly enough,

for the proton-nucleus case, because there is not muchcompression effect, a single constant value ofD = 2.05gives very good results for all proton-nucleus collisions.For alpha-nucleus collisions, where there is a little com-pression, the best value ofD is given by

(11)

For lithium nuclei, because of the “halos” (ref. 21), com-pression is less; therefore, the Pauli blocking effect isless important. A reduced value ofD/3 gives betterresults for the reaction cross sections at the intermediateand higher energies.

There are no adjustable parameters in the modelexcept that, for proton-nucleus collisions, this method ofcalculating the Coulomb interaction barrier underesti-mates its value for the very light closed-shell nuclei ofalpha and carbon, which are very tightly bound and,therefore, compact. Consequently, for these two cases,the Coulomb barrier should be increased by a factor of 27and 3.5, respectively, for a better fit.

Results and Conclusions

Figures 1–45 show the plots of available results forproton-nucleus, alpha-nucleus, and nucleus-nucleus col-lisions. Figures 6 and 18 also show comparisons withreference10. The data set used for figures 1–5 was col-lected from references 15 and 23 and, for figures 6–14,was obtained from references 16, 17, 22, and 23. Exten-sive data available for a C + C system (fig. 18) weretaken from references 16, 17, 23, and 24. For the remain-ing figures, data were collected from the compilation ofdata sets from references 9 and 16–20. The agreementwith experiment is excellent and is better than all otherempirical models reported earlier, which is particularlyimportant in view of the fact that the agreement is excel-lent throughout the whole energy range—up to a fewAGeV. We notice, again, that at the lower energy end,the cross sections are modified by the Coulomb interac-tion, and at the intermediate and high energy end, Pauliblocking effects become increasingly important. It willbe interesting to see how the model compares with thenew experimental data as and when these becomeavailable.

NASA Langley Research CenterHampton, VA 23681-0001December 17, 1996

δE 1.85S 0.16S Ecm1/3⁄

CE–+=

0.91[+ AT 2ZT )ZP A( T⁄ AP) ]–(

SAP

1/3AT

1/3

AP1/3

AT1/3

+---------------------------=

CE D 1 E 40) ]⁄ 0.292 E 792)⁄–(exp––(exp–[=

0.229E0.453

cos×

D 1.75ρAP

ρAT+

ρACρAC

+---------------------------=

D 2.77 8.0 103–× AT

– 1.8 105–× AT

2 +=

0.8/ 1 250 E ) 75⁄–([ ]exp+{ }–

3

References

1. Wilson, John W.; Townsend, Lawrence W.; Schimmerling,Walter; Khandelwal, Govind S.; Khan, Ferdous; Nealy,John E.; Cucinotta, Francis A.; Simonsen, Lisa C.; Shinn,Judy L.; and Norbury, John W.: Transport Methods and Inter-actions for Space Radiations. NASA RP-1257, 1992.

2. Bass, Reiner:Nuclear Reactions With Heavy Ions. Springer-Verlag, 1980.

3. Bradt, H. L.; and Peters, B.: The Heavy Nuclei of the PrimaryCosmic Radiation. Phys. Rev., second ser., vol. 77, no. 1,Jan. 1, 1950, pp. 54–70.

4. Karol, Paul J.: Nucleus-Nucleus Reaction Cross Sections atHigh Energies: Soft-Spheres Model. Phys. Rev. C, vol. 11,no. 4, Apr. 1975, pp. 1203–1209.

5. Glauber, R. J.: High-Energy Collision Theory. Lectures inTheoretical Physics, Volume I, Wesley E. Brittin and LitaG. Dunham, eds., Interscience Publ., Inc., 1959, pp. 315–414.

6. Wilson, John W.: Composite Particle Reaction Theory. Ph.D.Diss., College of William and Mary, June 1975.

7. Wilson, J. W.; and Townsend, L. W.: An Optical Model forComposite Nuclear Scattering.Canadian J. Phys., vol. 59,no. 11, 1981, pp. 1569–1576.

8. Ernst, David J.: Total Reaction Cross Sections and the MatterDensity of Finite Nuclei. Phys. Rev. C, vol. 19, no. 3,Mar. 1979, pp. 896–904.

9. Gupta, S. K.; and Shukla, P.: Trajectory Modifications in theGlauber Model for Heavy Ions.Phys. Rev. C, vol. 52, no. 6,Dec. 1995, pp. 3212–3216 and references.

10. Townsend, L. W.; and Wilson, J. W.: Energy-DependentParameterization of Heavy-Ion Absorption Cross Sections.Radiat. Res., vol. 106, 1986, pp. 283–287.

11. Shen, Wen-Qing; Wang, Bing; Feng, Jun; Zhan, Wen-Long;Zhu, Yong-Tai; and Feg, En-Pu: Total Reaction Cross Sec-tion for Heavy-Ion Collisions and Its Relation to the NeutronExcess Degree of Freedom.Nucl. Phys., vol. A491, 1989,pp. 130–146.

12. Vitturi, A.; and Zardi, F.: Modified Glauber Model for theDescription of Elastic Scattering Between Heavy Ions.Phys.Rev. C, vol. 36, no. 4, Oct. 1987, pp. 1404–1407.

13. Sihver, L.; Tsao, C. H.; Silberberg, R.; Kanai, T.; andBarghouty,A. F.: Total Reaction and Partial Cross SectionCalculations in Proton-Nucleus and Nucleus-Nucleus Reactions and Phys. Rev. C, vol. 47,no. 3, Mar. 1993, pp. 1225–1236.

14. DeVries, R. M.; and Peng, J. C.: Nucleus-Nucleus Total Reac-tion Cross Sections.Phys. Rev. C, vol. 22, no. 3, Sept. 1980,pp. 1055–1064.

15. Bauhoff, W.: Tables of Reaction and Total Cross Sections forProton-Nucleus Scattering Below 1 GeV. At. Data & Nucl.Data Tables, vol. 35, 1986, pp. 429–447.

16. Kox, S.; Gamp, A.; Perrin, C.; Arvieux, J.; Bertholet, R.;Bruandet, J. F.; Buenerd, M.; Cherkaoui, R.; Cole, A. J.;El-Masri, Y.; Longequeue, N.; Menet, J.; Merchex, F.; andViano, J. B.: Trends of Total Reaction Cross Sections forHeavy Ion Collisions in the Intermediate Energy Range.Phys.Rev. C, vol. 35, no. 5, May 1987, pp. 1678–1691.

17. Barashenkov, V. S.; Gudima, K. K.; and Toneev, V. D.: CrossSections for Fast Particles and Atomic Nuclei.Prog. Phys.,vol. 17, no. 10, 1969, pp. 683–725.

18. Webber, W. R.; Kish, J. C.; and Schrier, D. A.: Formula forCalculating Partial Cross Sections for Nuclear Reactions ofNuclei With MeV/Nucleon in Hydrogen Targets.Phys. Rev. C, vol. 41, no. 2, Feb. 1990, pp. 566–571.

19. Dubar, L. V.; Eleukenov, D. Sh.; Slyusarenko, L. I.; andYurkuts, N. P.: Parametrization of Total Cross Sections ofReactions in the Intermediate Energy Region. Sov. J. Nucl.Phys., vol. 49, no. 5, May 1989, pp. 771–773.

20. Menet, J. J. H.; Gross, E. E.; Malanify, J. J.; and Zucker, A.:Total-Reaction-Cross-Section Measurements for 30-60MeVProtons and the Imaginary OpticalPotential.Phys. Rev. C,no. 4, Oct. 1971, pp. 1114–1129.

21. Riisager, K.: Nuclear Halo States.Rev. Mod. Phys., vol. 66,no. 3, July 1994, pp. 1105–1116.

22. Auce, A.; Carlson, R. F.; Cox, A. J.; Ingemarsson, A.;Johansson,R.; Renberg, P. U.; Sundberg, O.; and Tibell, G.:Reaction Cross Sections for 75–190 MeV Alpha Particles onTargets From12C to 208Pb. Phys. Rev. C, vol. 50, no. 2, Aug.1994, pp. 871–879.

23. Jaros, J.; Wagner, A.; Anderson, L.; Chamberlain, O.; Fuzesy,R. Z.; Gallup, J.; Gorn, W.; Schroeder, L.; Shannon, S.;Shapiro, G.; and Steiner, H.: Nucleus-Nucleus Total CrossSections for Light Nuclei at 1.55 and 2.89 GeV/c Per Nucleon.Phys. Rev. C, vol. 18, 1978, pp. 2273–2292.

24. Heckman, H. H.; Greiner, D. E.; Lindstrom, P. J.; and Shwe,H.: Fragmentation of4He, 12C, 14N, and 16O Nuclei inNuclear Emulsion at 2.1 GeV/Nucleon. Phys. Rev. C, vol. 17,no. 5, May 1978, pp. 1735–1747.

Zt 26≤( )(Zp Zt 26).≤

E 200≥

4

Figure 1. Reaction cross sections as a function of energy for collisions.


800

600

400

200

0100 101

Energy, A MeV

Empirical model (ours) Experiment

102 103

σR, mb

p Be94+

600

400

200

0100 101

Energy, A MeV


102 103

σR, mb

p C126+

5



1000

800

600

400

200

0100 101

Energy, A MeV


102 103

σR, mb

p Al2713+

1000

800

600

400

200

0100 101

Energy, A MeV


102 103

σR, mb

p Fenat26+

6


Figure 6. Reaction cross sections as a function of energy for collisions; dashed line is from reference 10.

150

100

50

0100 101

Energy, A MeV


102 104103

σR, mb

α H11+

1500

1000

500

0100 101

Energy, A MeV


102 104103

σR, mb

α C126+

7



1500

1000

500

0100 101

Energy, A MeV


102 103

σR, mb

α O168+

1500

1000

500

0100 101

Energy, A MeV


102 103

σR, mb

α Si2814+

8



2000

1500

1000

500100 101

Energy, A MeV


102 103

σR, mb

α Ca4020+

2000

1500

1000

500100 101

Energy, A MeV


102 103

σR, mb

α Ca4820+

9



2000

1500

1000

500100 101

Energy, A MeV


102 103

σR, mb

α N5828 i+

2000

1500

1000

500100 101

Energy, A MeV


102 103

σR, mb

α N6028 i+

10



2500

2000

1500

500

1000

0100 101

Energy, A MeV


102

σR, mb

α Sn12450+

3000

2500

2000

1500

1000

500100 101

Energy, A MeV


102 103

σR, mb

α Pb20882+

11



4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Li63 Ca40

20+

4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Li63 Zr90

40+

12


Figure 18. Reaction cross sections as a function of energy for collisions; dashed line is from reference 18.

4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Be94 Si28

14+

2500

2000

1500

1000

500

0100 101

Energy, A MeV


102 104103

σR, mb

C126 C12

6+

13



2500

2000

1500

1000

500

0100 101

Energy, A MeV


102 104103

σR, mb

C126 Al27

13+

2500

2000

1500

1000

500

0100 101

Energy, A MeV


102 104103

σR, mb

C126 Ca40

20+

14



2500

2000

1000

1500

500

0100 101

Energy, A MeV


102 104103

σR, mb

C126 Fe56

26+

2500

1500

2000

1000

500

0100 101

Energy, A MeV


102 103

σR, mb

C126 Zn64

30+

15



4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

O168 Si28

14+

4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

O168 Ca40

20+

16



4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

O168 Co59

27+

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

O168 Ni58

28+

17



5000

4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

O168 Pb208

82+

2500

2000

1500

1000

500

0100 101

Energy, A MeV


102 104103

σR, mb

Ne2010 C

126+

18



2500

2000

1500

1000

500

0100 101

Energy, A MeV


102 104103

σR, mb

Ne2010 Al

2713+

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Ne2010 Ca

4020+

19



5000

3000

4000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Ne2010 Pb

20882+

5000

3000

4000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Ne2010 U

23592+

20



5000

3000

4000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

S3216 Mg24

12+

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

S3216 Al27

13+

21



3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Cl3517 Ni58

28+

4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Cl3517 Ni62

28+

22



5000

3000

4000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Ar4018 Ag109

47+

5000

3000

4000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Ar4018 Bi209

83+

23



6000

5000

3000

4000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Ar4018 U238

92+

4000

3000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Ca4020 Ca40

20+

24



5000

3000

4000

2000

1000

0100 101

Energy, A MeV


102 103

σR, mb

Kr8436 Cu65

29+

6000

4000

2000

0100 101

Energy, A MeV


102 103

σR, mb

Kr8636 La139

57+

25



8000

6000

4000

2000

0100 101

Energy, A MeV


102 103

σR, mb

Xe13654 Bi209

83+

8000

6000

4000

2000

0100 101

Energy, A MeV


102 103

σR, mb

Kr8436 Pb208

82+

26


8000

6000

4000

2000

0100 101

Energy, A MeV


102 103

σR, mb

Kr8436 Bi209

83+

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January 1997 Technical Paper

Universal Parameterization of Absorption Cross Sections

WU 199-45-16-11

R. K. Tripathi, Francis A. Cucinotta, and John W. Wilson

L-17580

NASA TP-3621

Tripathi: Southern Illinois University, Carbondale, IL; Cucinotta and Wilson: NASA Langley Research Center,Hampton, VA.

This paper presents a simple universal parameterization of total reaction cross sections for any system of collidingnuclei that is valid for the entire energy range from a few AMeV to a few AGeV. The universal picture presentedhere treats proton-nucleus collision as a special case of nucleus-nucleus collision, where the projectile has chargeand mass number of one. The parameters are associated with the physics of the collision system. In general terms,Coulomb interaction modifies cross sections at lower energies, and the effects of Pauli blocking are important athigher energies. The agreement between the calculated and experimental data is better than all earlier publishedresults.

Nuclear reaction; Cross sections; Heavy ions27

A03

NASA Langley Research CenterHampton, VA 23681-0001

National Aeronautics and Space AdministrationWashington, DC 20546-0001

Unclassified–UnlimitedSubject Category 93Availability: NASA CASI (301) 621-0390

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universal parameterization of absorption cross sections

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