Nuclear Instruments and Methods in Physics Research B 129 (1997) 1 I - 15 $.I& __ l!iiJ mm

NOM B Beam lnterections

with Materials 6 Atoms

ELSEVIER

Accurate universal parameterization of absorption cross sections II - neutron absorption cross sections

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

a Hampton University, Hampton, VA 23669, USA ’ NASA Langley Research Center, Hampton. VA 23681, USA

Received 25 October 1996; revised form received 3 December 1996

Abstract A recent parameterization (here after referred as paper I, Ref. [4]> of absorption cross sections for any system of charged

ions collisions including proton -nucleus collisions, is extended for neutron-nucleus collisions valid from u 1 MeV to a

few GeV, thus providing a comprehensive picture of absorption cross sections for any system of collision pair (charged and/or uncharged). The parameters are associated with the physics of the problem. At lower energies, the optical potential at the surface is important and the Pauli operator plays an increasingly important role at intermediate energies. The agreement between the calculated and experimental data is better than earlier published results.

1. Introduction

The transport of neutrons in matter is of direct interest in several technologically important and scientific areas [l-3] including space radiations, cosmic ray propagation studies in galactic medium, nuclear power plants, radiobio- logical effects impacting on industrial and public health. For the proper assessment to radiation exposures both reliable transport codes and accurate input data are needed.

An important ingredient of the input data is the total absorption (reaction) cross section, defined as the total minus the elastic cross sections for two colliding ions:

UR = ur - (&, . (1)

Recently, we have developed a simple accurate formalism (paper I and Ref. [5]) for the total absorption cross sections for any system of charged colliding ions including protons.

The present work extends the formalism to neutron-nucleus collisions, thus presenting a simple accurate comprehen- sive formalism for the absorption cross sections for any combination of colliding ions (charged and/or uncharged) valid for the entire energy range. We have retained the high energy features of the model [6] developed here at NASA Langley Research Center (LaRC) where it is accu- rate, and improved on its behavior at lower energies within

the universal formalism recently developed (paper I and Ref. [5]). The other commonly used model [7] gives good

* Corresponding author. Fax: + l-804-864-8094; email:

rkt@hesbl .larc.nasa.gov

results for higher energies but shows increasingly larger deviations from experiment for lower energies.

2. Model description

Most of the empirical models approximate total reac- tion cross section for charged ion collisions of Bradt-Peters form [8], but have very different form for the neutron- nucleus collisions. We have maintained the uniform con- sistent picture for all the systems proton-nucleus, nu- cleus-nucleus, and now for the neutron-nucleus colli- sions, and have written total reaction cross sections for neutron-nucleus collisions also of the Bradt-Peters form:

where rO is energy independent and 6 is either energy-in-

dependent or dependent parameter, and A, and A, are the projectile and target mass numbers respectively.

It is well known that this type of parameterization is

suited for higher energies and is modified through Coulomb interaction at lower energies (paper I and Ref. [5]) for the charged ion collisions. For the neutron-nucleus collisions there is no Coulomb interaction, but the total reaction cross section, in this case, is modified by the strength of the imaginary part of the optical potential at the surface. Since we are using this form of parameterization for the neu- tron-nucleus case for the first time (which helps to pro- vide the unified consistent and accurate picture for the total reaction cross sections for any system of colliding nuclei for the entire energy range), we introduce a low

0168-583X/97/$17.00 0 1997 Elsevier Science B.V. All rights reserved

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12 R.K. Tripathi et aI./ Nucl. Instr. und Meth. in Phys. Res. B 129 (1997) I I - 15

energy multiplier (X,) accounting for the strength of the optical model interaction. In addition, the effects of the transparency and Pauli blocking are taken into account

through the energy dependent parameter, 6,, defined latter in the text. We, therefore, propose the following similar

form for the reaction cross sections for the neutron-

nucleus:

us = ~TT;( Ab/’ +Ay3 + 6,)‘X,, (3)

where r0 = 1. I fm. The low energy multiplier X, is given

by (i) For A, < 200

E X,= I-X,exp -~

i i X,Sl_ ’

where S,=0.6for A,< 12, S,= 1.6for A,= 12, S,= 1

otherwise with

X, = 2.83 - 3.1 x IO-*A, + 1.7 x W4Az,. (5)

For the low energy, shell effects play an important role in light nuclei and the optical potential at the surface varies differently with these effects showing variation in S,.

(ii) For A, 2 200

x,,= [I -03exp( -q)][t -exp(O.9-E)]. (6)

The effects of the energy dependence at intermediate and higher energies due to Pauli blocking and transparency are taken into account in 6, , which is defined as,

6, = 1.8.5s + o.l6s/E;c

- C, + 0.91( A, - 2Z,)Z,/( A, A,), (7)

where S is the mass asymmetry term and is given by

s = A;“A;”

Al/? + A!/” P r

(8)

and is related to the volume overlap of the collision system. For the results reported here A, = I. The last term on the right hand side of Eq. (7) accounts for the isotope dependence of the reaction cross section. The term C, is related to the transparency and Pauli blocking and is given

by

C, = D[ I - exp( -E/T,)] - 0.292exp( -E/792)

x ~os(0.229E’.~~“), (9)

where T, = 40 (except for 11 I A, < 40 a value of T, = 30 is recommended for a better fit). Here D is related to the density dependence of the colliding system and the global value is given by

0.538 Dr - ~

PA,, + PA, (‘0)

The density of a nucleus is calculated in the hard sphere model, and for a nucleus Ai is given by

(11)

where ri is the equivalent sphere radius and is related to

the rrms,i radius by

r, = 1 .29r,,,,i, (12)

where rrms., values are taken from the experiment [9].

There is interesting physics associated with the constant

D. This in effect simulates the modifications of the reac- tion cross sections due to Pauli blocking. This effect is

new and was introduced in our charged particles reaction

cross section work (paper I), and is being used for the first time in neutron-nucleus work. This is an important physi- cal ingredient of the model and helps present a unified picture of the reaction cross sections for any system of

colliding particles. Let us elaborate on the choice of D in Eq. (9). This has

contributions from the global characteristics of nuclei and also takes into account the specific characteristics related to their stability. The density of a nucleus is a very useful global property. We, therefore, associate the global value

De to the nuclear densities (Eq. ( 1 I)) of the colliding system. Consequently. for most systems, we have:

D=Da> (13)

where D, has been defined in Eq. (IO). This value of D is modified in certain groups of nuclei to account for their special features in the following way:

The shell structure plays an important role in the size of a light nucleus and hence its cross sections. These effects are more prevalent upto double closed sd shell nuclei (A I 40) and manifest themselves as a tightly bound nucleus and also as isotopic effects. Hence for this group of nuclei, by taking into account these special character- istics, the best value of D is given by:

For A, I 40

D = D, - I .5( A, - 2Z,)/A,

+ 0.25/{ I + exp[( E - 170)/100]). (14)

The second term on the RHS of Eq. (I 4) accounts for the isotopic effects and the last energy dependent term accounts for the shell effects upto doubly closed sd shell nuclei. For still heavier nuclei shell effects start becoming less important but the isotopic effects persist on, We, therefore, found better agreement using the following ex- pression:

For 40 < A, < 60

D = Dg - I .5( A, - 2Z,)/A,. (15)

For even higher nuclei ( A 2 60) special characteristics do not dominate the stability of a nucleus and the global value of D defined in Eq. (13) works well. However, for

R.K. Trip&hi et (11. /Mud. Instr. and Meth. in Phys. Res. B 129 (19971 1 I- /5 13

s 600

r b” 400

200

ot 1.’ ..I 8 .... I . . ..A 1 10 100 1000 10 000

Energy (A MeV)

Fig. 1. Total absorption cross sections for n + :Be as a function of

neutron energy.

very heavy systems the Coulomb force becomes an impor- tant effect and manifests itself as neutron excess (or proton

deficiency) for the stability of a nucleus. We found that it is necessary to modify the global value of D (Eq. 13) for nuclei heavier than lead to account for this effect. We determine that the best value of D for, Z, > 82 is given

b)

D=D,-&/&-Z-r). (16)

For any regions not covered by these special effects the global value of D (Eq. (13)) gives good results.

3. Results/ conclusions

Figs. l-12 show the plot of the total absorption cross sections for the neutron-nucleus collisions. We observe that cross sections are never negative and are always positive quantities. For a particular system the energy at which the cross section drops down to zero is the lowest energy to which our model should be used. The data for

n+ 126C lOOOr . . ..“I . . ..‘.I .“....I .““.I

800 i

+ + Experiment - LaRC !

-1 10 100 1000 10 000

Energy (A MeV)

Fig. 2. Total absorption cross sections for n + A” C as a function of

neutron energy.

n + 2713Al

+ + Experiment - LaRC

i --.-. LaRC (Prior) _ ‘...‘..” Letaw et al. _

01 1. “‘_..I “.....I “..,,,I ” . ..A 1 10 100 1000 10 000

Energy (A MeV)

Fig. 3. Total absorption cross sections for n + :: Al as a function of

neutron energy.

higher energies (379- I73 1 MeV) has been taken from Ref. [lo], and for lower energies has been taken from the compilations of ENDF/B-VI, and Refs. [I 1,12].The agree- ment with experiment is excellent for all nuclei for the

“+ =26Fe 2000~ ...I I ..... ( ‘-q

‘+

i

1 10 100 1000 IO 000

Energy (A MeV)

Fig. 4. Total absorption cross sections for n + $ Fe as a function of

neutron energy.

n + 5g28Ni

2ooo -

1 10 100 1000 10 000

Energy (AMeV)

Fig. 5. Total absorption Cross sections for n + z:Ni as a function of

neutron energy.

R.K. Tripathi et al./Nucl. Instr. and Meth. in Phys. Res. B 129 (1997) 11-15

+ + Experiment

..-.. LaRC (Prior) - ,.. Letaw et al.

0;

1 10 100 1000 10 000

Energy (A MeV)

Fig. 6. Total absorption cross sections for n + !GCu as a function

of neutron energy.

entire energy range. We have also compared (Figs. 2, 3, 6,

8, 10) present results with the two popular models: model developed by Wilson et al. [6] here at the NASA Langley Research Center (LaRC (Prior)), and that of Letaw et al.

10 100 1000 10 000

Energy (A MeV)

OF . ..... 0 0 8 j

1 10 100 1000 10 000

Energy (A MeV)

Fig. 7. Total absorption cross sections for n +$zZn as a function

of neutron energy. Fig. 10. Total absorption cross sections for n + i?Pb as a function

of neutron energy.

2500[ n + 1 074’Ag

. . . . . . . . . . . . . . .““l + + Experiment - LaRC ..... LaRC (Prior) ‘....‘.‘. Letaw et al.

0’ I 1 10 100 1000 10 000

Energy (A MeV)

Fig. 8. Total absorption cross sections for n + i?Ag as a function

of neutron energy.

2500- n + 11g50Sn

. . . . . ..x . . . ..I . . . . . ..T + + Experiment - LaRC

0’ “““’ ......a ......I I 1 10 100 1000 10 000

Energy (A MeV)

Fig. 9. Total absorption cross sections for n+$,‘Sn as a function

of neutron energy.

[7] at the Naval Research Laboratory (NRL). We see from

the plots that both these models agree very well with experiment at higher energies, however, there are depar-

4000, n + 20882Pb

..I ..I ,.I . ..l + + Experiment - LaRC

3000

i

----- LaRC (Prior) ,... ““‘._

i-------- . ..__ p______‘y . ...,

f *ooo “~~~~~~

4000 L n + *Oge3Bi

. . . . . . . . . . . . .-

3000

z E. 2000

b”

1000

O10 1 100 1000 10 000

Energy (A MeV)

Fig. 1 1. Total absorption cross sections for n + it’Si as a function

of neutron energy.

R.K. Tripathi et al./Nucl. Instr. und Meth. in Phys. Res. B 129 (1997) 11-15 I5

4000, n + 238 .,,.( . . . . . 9?c1 . . . 4

+ + Experiment - LaRC

1 10 100 1000 10 000

Energy (A MeV)

Fig. 12. Total absorption cross sections for n + zi”U as a function

of neutron energy.

tures at lower energies. In general LaRC gives better results than NRL model at lower energies.

Clearly present model (LaRC) improves on earlier models and gives accurate results to much lower energies, and is a welcome improvement for the transport model data base. Neutrons are the dominant component of the

radiation environment for the High Speed Civil Transport (HSCT) mission where the data base proposed here is of immense iinportance.

We have successfully presented simple accurate and unified model for the total absorption cross sections for any system of charged and/or uncharged collision systems valid for the entire energy range, which are important ingredients for several data bases including transport. The model will also be useful where analytical forms of the cross sections are needed.

In view of its utility the computer program calculating the comprehensive accurate absorption cross sections of paper I and present paper will be placed in the COSMIC program library for the public use.

Note added in proof

There is an error in Eq. (12) of paper I (Ref. [4]). A

pair of parentheses is missing. The correct equation should

read as follows:

D = 2.77 - 8.0 X 10-3A1A, + 1.8 x 10-5A2,

- 0.8/( 1 + exp( (250 - Q/75)).

Acknowledgements

(12)

One of the authors (RKT) acknowledges NASA Lang- ley Research Center for the support.

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