NASATechnicalPaper3324

July 1993

Geometric Model FromMicroscopic Theory forNuclear Absorption

Sarah John,Lawrence W. Townsend,John W. Wilson,and Ram K. Tripathi

NASATechnicalPaper3324

1993

Geometric Model FromMicroscopic Theory forNuclear Absorption

Sarah JohnViGYAN, Inc.Hampton, Virginia

Lawrence W. Townsendand John W. WilsonLangley Research CenterHampton, Virginia

Ram K. TripathiViGYAN, Inc.Hampton, Virginia

Abstract

A parameter-free geometric model for nuclear ab-sorption is derived herein from microscopic theory.The expression for the absorption cross section inthe eikonal approximation, taken in integral form, isseparated into a geometric contribution that is de-scribed by an energy-dependent e�ective radius andtwo surface terms that cancel in an asymptotic seriesexpansion. For collisions of light nuclei, an expressionfor the e�ective radius is derived from harmonic os-cillator nuclear density functions. A direct extensionto heavy nuclei with Woods-Saxon densities is madeby identifying the equivalent half-density radius forthe harmonic oscillator functions. Coulomb correc-tions are incorporated, and a simpli�ed geometricform of the Bradt-Peters type is obtained. Resultsspanning the energy range from 1 MeV/nucleon to1 GeV/nucleon are presented. Good agreement withexperimental results is obtained.

Introduction

The nuclear absorption cross section is a mea-sure of the sum of all the reaction processes inducedduring nuclear collisions. This metric is appliedwidely in such diverse �elds as fundamental nu-clear physics, the study of radiation e�ects on livingcells, and radiation-shielding design for future space-exploration vehicles (ref. 1).

In recent years, considerable advancements inboth the theory (refs. 2{19) and experimental meth-ods (refs. 20{26) of this fundamental observable havebeen made. Most signi�cantly, the energy varia-tions of the absorption cross sections for all nuclei inthe intermediate-energy range from 10 MeV/nucleonto 1 GeV/nucleon are qualitatively similar andcan be related to the underlying two-nucleon inter-actions. At energies above 1 GeV/nucleon, the ab-sorption cross section is essentially constant and ap-proximately equals the total geometric cross-sectionalarea of the projectile-target system; this consistencyallows an extension of the geometric model and itsparameterized forms (refs. 13{17, 25, and 26) to theentire intermediate- and low-energy range. The geo-metric formula provides a quick method for the cal-culation of the absorption cross section and is highlydesirable for inclusion in radiation-shielding trans-port codes.

Microscopic formulations are based on the quan-tum collision theory of many-body composite sys-tems. The solution of this complex problem can beapproached with the methods of Glauber (ref. 2),Watson (ref. 3), Feshbach (ref. 3), and the KMT ap-proach (ref. 4). Wilson (ref. 5) extended the Watson

formalism of nucleon-nucleus collisions to nucleus-nucleus collisions, from which the cross sections inthe eikonal approximation can be obtained. Thesesolutions, which are generally solved by numericalmethods (refs. 5 and 6), are given in integral form.

Closed-form solutions for the microscopic theoryhave been obtained for the simplest case of Gaussiannuclear density distribution (refs. 7 and 8). Thesolution applicability to the more realistic densitydistribution of the Woods-Saxon form is based onmatching equivalent Gaussian forms to the surfacedensities of heavier nuclei. This approach is suc-cessful because the energy dependence for the trans-parency is largely a function of the low-density sur-face region with a saturated, absorptive inner corethat remains essentially energy independent.

The purpose of this paper is to derive a geomet-ric formula, valid for both light and heavy nuclei, forthe nuclear absorption cross section from the micro-scopic formulation. This e�ort will be accomplishedby separating the integral expression for the absorp-tion cross section in the eikonal approximation intoan e�ective geometric radius and two surface com-ponents that approximately cancel in the asymptoticexpansion.

The concept of the e�ective radius in microscopicformulations is not new (refs. 8 and 9). However, inall previous works the contributions beyond a cuto�radius are negligible, whether the e�ective radiusis de�ned at half-absorption density (ref. 8) or ata distance of one mean free path from the surface(ref. 9). In this work, however, the e�ective radius athalf-absorption density has a precise de�nition suchthat the transparency within this radius cancels theabsorptive contribution from the outer region.

A summary of the format for this paper is pre-sented. In \Theory," the results from the micro-scopic theory are reviewed and developed into a geo-metric formalism. In \Application to Light Nuclei,"a method is illustrated for harmonic oscillator mat-ter densities that are applicable to light nuclei. Ex-pressions for the eikonal phase shift and the totalabsorption cross section are derived for the nucleon-nucleus as well as for the nucleus-nucleus inter-actions. An exact formula for the e�ective radiusat half absorption is derived, and its �rst approxima-tion is used to evaluate explicitly the geometric termand the surface contributions. Results comparing ex-periments for energy values from 50 MeV/nucleonto 1 GeV/nucleon are presented. Also, a simpli-�ed version of the expression for the e�ective radius,which is used in the rest of the sections, is given.In \Extension to Woods-Saxon Densities," a direct

extension of the expression mentioned above is madefor the heavier nuclei described byWoods-Saxon den-sity distributions. In \Coulomb Interactions," thee�ects of Coulomb de ections are incorporated, andresults for light- and heavy-ion projectile-target sys-tems spanning the energy range from 1 MeV/nucleonto 1 GeV/nucleon are presented. In \Bradt-PetersForm," the expressions are further simpli�ed and theresults compared with those of the previous section.In \Nuclear-Medium E�ects," comments are madeon nuclear-medium e�ects. \Concluding Remarks"summarizes the suggestions for improvement submit-ted herein.

Theory

Microscopic Theory

The nuclear absorption cross section in the eikonalapproximation is given by

�abs = 2�

Z1

0

f1� exp[�2Im �(b; e)]gb db (1)

where Im denotes the imaginary part of the complexphase shift �(b; e), which is a function of the impactparameter b and energy e, and can be derived froma multiple-scattering series expansion for the opticalpotential (refs. 5 and 6). Within the Watson formof impulse approximation, when the propagator G isreplaced by the free propagator Go (ref. 5), the phaseshift is given by

�(b; e) =A2

PA2

T

k(AP + AT)

Zdz

Zd3��T(�)

�

Zd3y�

P(b+ z + y + �)t(e;y) (2)

where k is the wave number of the projectile, APand AT are the mass numbers, and �P and �T arethe projectile and matter densities ; t(e;y) signi�esthe two-body transition amplitude for free nucleonsas follows:

t(e;y) = �� e

m

�1=2��(e) [�(e) + i] [2� B(e)]�3=2

� exp

"�

y2

2B(e)

#(3)

Here, e is the two-nucleon kinetic energy in thecenter-of-mass (cm) frame of reference, m is thenucleon rest mass, ��(e) is the isospin-averaged totalnucleon-nucleon cross section, B(e) is the averagedslope parameter, and �(e) is the averaged ratio of

the real to imaginary components of the forward-scattering amplitudes.

Geometric Model

A geometric formula for the absorption cross sec-tion of high-energy nuclear collisions is given by theBradt-Peters form (ref. 2),

�abs= � (RP + RT � �o)2 (4)

where RP and RT are the radii of the projectile andtarget nuclei and �o is a constant. If �o is small com-pared with the radii, the formula shown above impliesan annular region of transparency for peripheral colli-sions. To show that this form may indeed be derivedfrom microscopic theory, the integral given in equa-tion (1) is separated at an impact parameter Re asfollows:

�abs= 2�

Z Re

0

f1� exp [�2Im�(b; e)]g b db

+ 2�

Z1

Re

f1� exp [�2Im�(b; e)]g b db

= �R2e � �1 + �2 (5)

such that �1 and �2 are the small surface contribu-tions given by

�1 = 2�

Z Re

oexp [�2Im�(b; e)] b db (6)

and

�2 = 2�

Z1

Re

f1� exp [�2Im�(b; e)]g b db (7)

Appendix A includes the asymptotic seriesexpansions,

�1 � 2�Reexp[��1(Re; e)]

@�1(b; e)=@bjRe(8)

and

�2 � 2�Ref1� exp [��1(Re; e)]g

2

@�1(b; e)=@bjReexp[�1(Re; e)]

(9)

where�1(b; e) = 2Im�(b; e) (10)

Note that �2 can also be expanded into a conver-gent Taylor series, provided that �

1(b; e) < 1 beyond

2

Re and that, in the �rst approximation, �2 is givenby

�2 �

Z1

Re

�1(b; e)b db (11)

From equations (8) and (9),

�2 � �1 = 2�Re�(Re; e) (12)

where

�(Re; e) =exp[�1(Re; e)]

@�1(b; e)=@bjRe

f1� 2 exp[��1(Re; e)]g

(13)Hence,

�abs = �R2

e � 2�Re�(Re; e)

� �[Re � �(Re; e)]2

(14)

If the integration limit Re is the sum of the pro jec-tile and target radii, the required geometric formulashown in equation (4) is obtained. However, for con-venience, Re can be chosen such that � is identicallyzero. When this condition is valid, equation (13)shows

exp[��1(Re; e)] = 0:5

or�1(Re; e) = ln 2 (15)

That is, Re is chosen at half transparency or, equiv-alently, at half absorption. (See �g. 1.) The trans-parency function T is

T = exp[��1(b; e)] (16)

and the absorption function A = 1� T . With � � 0,

�abs= �R2

e (17)

where the e�ective radius Re is now a function ofenergy. This equation gives the desired geometricformula for the nuclear absorption; Re is derivedfrom equation (15). However, a more appropriateformula for Re can be derived in terms of the classicalgeometric radius Rg, which is de�ned as Re!1.Note that at the high-energy limit, as e ! 1, thewavelength of the projectile nucleons is su�cientlysmall compared with the nuclear dimensions, and thee�ective radius assumes the classical geometric radiusRg, given by

Rg = ro

�A1=3P + A

1=3T

�(18)

where ro is a variable determined by the condition

�1(Rg;1) = ln 2 (19)

such that � = 0 is satis�ed. The e�ective radius Re

at an arbitrary energy e can then be related to thegeometric radius Rg by equating

�1(Re; e) = �

1(Rg;1) = ln 2 (20)

This relation is useful only if an algebraic expres-sion for �(b; e) is known. For Gaussian and harmonicoscillator density functions, �(b; e) is readily obtainedby the methods in \Application to Light Nuclei."

Application to Light Nuclei

In this section, the phase-shift function �(b; e) isderived from the harmonic oscillator density func-tion, which is applicable to light nuclei for A � 16:Also, an exact expression for the e�ective radius isobtained. The nucleon-nucleus and nucleus-nucleuscollisions are treated separately.

For the nucleon-nucleus collision, the nucleondensity is simply

�(x� x0) = �(x� x0) (21)

because the nucleon wave function has been e�ec-tively incorporated into the two-body transition am-plitude t(e; y), and the nucleus charge density isgiven by

�c(x) = �o

1 +

x2

a2

!exp

�x2

a2

!(22)

where and a fm (1 fermi = 1 � 10�15 m) areconstants taken from reference 27. The expression for�(b; e) is derived in appendix B and has the followingform for the nucleon-nucleus system:

�1(b; e) = f1(e)

�a1 + a2b

2

�exp[�P (e)b2] (23a)

and for the nucleus-nucleus system:

�1(b; e) = f1(e)

�a1+ a2b

2+ a3b4

�exp[�P (e)b2]

(23b)

where explicit expressions for the functions f1(e) andP (e) and for the constants a1, a2, and a3 in each caseare given in appendix B.

Figure 2 displays the absorption function

A = 1� exp[��1(b; e)] (24)

as a function of the impact parameter b, for 4He{4Heand 16O{16O systems, at an energy of 2 GeV/nucleon,which is taken as the geometric limit. The position ofthe e�ective radius Re that is de�ned at T = A = 0:5

3

is also indicated. Note that even within the geometriclimit, considerable transparency for peripheral colli-sions occurs.

The e�ective radius Re for three di�erent energyvalues is shown in �gure 3, where the absorptionis plotted at 50 MeV/nucleon, 250 MeV/nucleon,and 1 GeV/nucleon for the 16O{16O system. Asshown in the �gure, Re passes through a minimumand then increases as the energy decreases. Thestrong absorption radius Rs is reached in the low-energy limit. The underlying basis for this energydependence can be traced to the nucleon-nucleontotal cross section, which exhibits a similar trend. Anexpression for the e�ective radius Re in terms of itsgeometric value Rg is obtained from equations (23)and the condition

�1(Re; e) = �

1(Rg;1)

For nucleus-nucleus collisions,

R2

e =P (1)

P (e)R2

g +1

P (e)

(ln

�f1(e)

f1(1)

�

+ ln

a1 + a2R

2e + a3R

4e

a1 + a2R2g + a3R

4g

!)(25)

with a similar expression for the nucleon-nucleus casefor a3 = 0. In the equation above, the geometricvalue Rg is given by

Rg = rTA1=3T (26a)

for the nucleon-nucleus system and

Rg = rTA1=3T + rPA

1=3P (26b)

for the nucleus-nucleus system. The value r is deter-mined from the condition

�(Rg; 2A GeV) = ln 2

which, for the nucleus-nucleus system, is given by

ln f1(2A GeV) + ln

�a1+ a2R

2

g + a3R4

g

�� P (2A GeV)R2

g = ln(ln 2) (27)

Substituting for Rg from equation (26), equa-tion (27) is solved iteratively with an initial valueof ro = 1:0 fm. For the nucleus-nucleus system, iden-tical target-projectile systems are considered withrT = rP = ro. The resulting values for ro are plotted

Table I. Collision Values for ro

ro, fm

Atomic mass

number Nucleon-nucleus Nucleus-nucleus

2 0.50 0.47

4 1.02 0.85

7 1.14 1.13

9 1.19 1.17

10 1.19 1.16

11 1.18 1.14

12 1.19 1.13

14 1.20 1.16

16 1.23 1.21

>16 1.25 1.25

in �gure 4 and listed in table I. From equation (20),this determination is clearly su�cient to ensure thatthe condition � = 0 is also satis�ed by the value ofthe e�ective radius given by equation (25).

For the heavier nuclei within the harmonic oscil-lator category, ro � 1:2 fm; this length approximatesthe generally accepted value that determines the nu-clear radius. Clearly, the values of ro for the lightestnuclei such as deuterium and helium are considerablysmaller than those for the heavier nuclei; this di�er-ence can be explained by examining the convolutionintegral for the phase shift. In the case of Gaussiandensity distributions, the resultant width increases

by the factor�a2P + a2T

�1=2

, whereas the total geo-

metric width is given by the sum aP + aT , as shownin equation (26), where aP and aT are the widthsof the target and projectile, respectively. Hence, forthe lightest nuclei, which have near-Gaussian distri-butions, ro must be lowered to account for this dis-crepancy. The convolution widths for the Woods-Saxon forms, however, are expected to be close tothe geometric value.

The advantage of writing Re in terms of Rg, asgiven in equation (25), is that in the �rst approxima-tion the second logarithmic term may be neglectedwhen Re � Rg, with the result

R2

e �P (2A GeV)

P (e)R2

g+1

P (e)ln

�f1(e)

f1(2A GeV)

�(28)

Although this approximation o�sets the condition� = 0, the integration limit in equation (5) is su�-ciently de�ned to evaluate �1 and �2 explicitly; theabsorption cross section is given accurately by

�abs = �R2

e � �1(Re; e) + �2(Re; e) (29)

4

Figures 5{9 display the absorption cross sectionusing equation (29) for light-ion projectile-target sys-tems that agree well with experimental data. Theseresults are also in close agreement with the numericalevaluation of the integral given by equation (1). (Seerefs. 5 and 6.)

Hereafter, the expression for Re from equa-tion (28) will be used with the assumption that(�2 � �1) is negligible. In equation (28), the expres-sion P (e) for the nucleon-nucleus collision is equiva-lent to equation (B10),

P (e) =

�a2T �

2

3r2p + 2B(e)

��1

(30a)

and P (e) for nucleus-nucleus collision is equivalent toequation (B17),

P (e) =

�a2T + a2P �

4

3r2p + 2B(e)

��1

(30b)

where rp = 0:86 fm is the radius of the proton.Because the slope parameter B(e) is approximatelya constant equal to 0.4 fm2 within the energy rangeconsidered, P (2A GeV) � P (e); equation (28), withRg given by equation (26), then simpli�es to

R2

e =�rTA

1=3T

�2+�a2T + 0:35

�g(e) (31a)

for the nucleon-nucleus collision, and

R2

e =�rPA

1=3P + rTA

1=3T

�2+�a2T + a2P

�g(e) (31b)

for the nucleus-nucleus collision, where

g(e) = ln

�f1(e)

f1 (2A GeV)

�

� ln

���(e)

��(2A GeV)

�(32)

The terms AP and AT are the charge density widthparameters for the harmonic oscillator function givenin reference 27, rP and rT are the respective valuesfor ro given in table I, and the parameterized formfor ��(e) is given in appendix C.

Extension to Woods-Saxon Densities

The Woods-Saxon density does not yield a simpleexpression for the phase shift �(b; e). Therefore, asimple modi�cation of equations (31) is given. The

two-parameter Woods-Saxon charge density is givenby

�c(r) =�o

1 + exp�r � R

c

�where R is the radius at half density, c is relatedto the nuclear skin thickness t by the expressionc = t=4:4, and �o is a normalization constant. Fromreference 27, the parameter c is a constant approx-imately equal to 0.55 fm for all nuclei. Hence, weassume that only one parameter R describes theWoods-Saxon form.

The equivalent half-density distance R for theharmonic oscillator function satis�es

1

2=

1 +

R2

a2

!exp

�R2

a2

!(33)

which can be rewritten as

k = ln 2 + ln(1 + k) (34)

where

k =R2

a2(35)

Equation (34) is solved iteratively for k withan initial value of k = ln 2. The parameter k isplotted in �gure 10 as a function of atomic massnumber. For the heavier nuclei with the harmonicoscillator density form, k � 2 or R �

p2a. Also,

for these nuclei an equally valid description would bethe Woods-Saxon form with the half-density radiusgiven by R =

p2a. This equivalence suggests that

the substitution of

a =Rp2

into equations (31) can extend its range of applicabil-ity to the Woods-Saxon densities of heavy ions withA > 16. This substitution yields for the nucleon-nucleus system,

R2

e =�rTA

1=3T

�2+

R2

T

2+ 0:35

!g(e) (36a)

and for the nucleus-nucleus system,

R2

e =�rP A1=3

P+ rTA

1=3T

�2

+1

2

�R2

T+R2

P

�g(e)

(36b)

where RP and RT are the half-density values for theWoods-Saxon forms given in reference 27.

5

Coulomb Interactions

The derivation of the eikonal approximation as-sumes straight-line trajectories for the projectile andan impact parameter at the collision site that isequal to its asymptotic value. For heavy ions atlow energy, the Coulomb force de ects the straight-line trajectories signi�cantly, thereby decreasing thecross-sectional area of incident particles that actuallyundergo nuclear forces. The eikonal approximationmay still be used (refs. 8, 10, and 18) if the e�ectiveimpact parameter at the collision site is the distanceof closest approach d (refs. 8 and 18), where b and d

are related by

b2 = d

2

�1�

2a

d

�(37)

with

a =ZTZPe

2o

Ecm

where ZT and ZP are the charges of the target andprojectile, respectively, eo is the electron charge, andEcm is the total center-of-mass energy.

Hence, the asymptotic impact parameter be, cor-responding to the radial distance Re, is given by

b2e = R

2e

�1�

2a

Re

�(38)

where be now de�nes the integration limit for equa-tion (5). Therefore, the Coulomb-modi�ed geometricformula is readily obtained as

�abs = �R2e

�1�

2a

Re

�(39)

In �gures 11{21, the absorption cross sectionfor several light- and heavy-ion projectile-target nu-clei is plotted with and without the Coulomb cor-rections for energy values from 1 MeV/nucleon to2 GeV/nucleon. The results with Coulomb correc-tions agree well with experiment. The following val-ues for ro (see table I) were used for nucleus-nucleuscollisions:

A = 4; ro = 0:56 fm

7 < A < 16; ro = 1:20 fm

A > 16; ro = 1:25 fm

Figures 22{26 show the cross sections for spe-ci�c projectiles on various targets with and withoutCoulomb interaction. The linear relationship withatomic number is clearly evident. As expected, the

Coulomb corrections are more signi�cant for heavyions.

Bradt-Peters Form

In the present formalism, the geometric limit forthe nuclear absorption cross section is given by

�abs= �r2o

�A1=3P + A

1=3T

�2compared with the Bradt-Peters form at high energy,

�abs= �r2o

�A1=3P +A

1=3T � �o

�2The di�erence comes from the choice of ro. Here,

ro satis�es the geometric criteria that are valid athigh energies and small wavelengths such that

Rg = ro

�A1=3P + A

1=3T

�(T = 0:5; e!1)

An equally valid description is to de�ne the strongabsorption radius Rs at low energy as the referencevalue such that

Rs = rs

�A1=3P + A

1=3T

�(T = 0:5; e! 0)

The strong absorption parameter rs is greaterthan ro because the absorptive content of the surfaceregion increases at lower energies due to s-wave scat-tering. This low-energy description results in �o 6= 0at high energy, a state that conforms to the Bradt-Peters formula.

However, since the eikonal approximation is validonly at high energies, the low-energy descriptioncannot be implemented. Also, the corrections thatshow the Coulomb and Pauli correlation, signi�cantcontributors at low energies, were not incorporatedinto the phase-shift function. Hence, the choice ofthe geometric limit as the reference point is justi�ed.

The more general equations (36) for the e�ectiveradius Re at intermediate energy will now be cast inthe Bradt-Peters form. The simplest form is obtainedfor heavy ions when rT = rP = ro. Using theparameterized form for half-density charge radius R,given by

R = c1A1=3� c2

withc1 = 1:18

c2 = 0:48

)(40)

6

in equations (36) and simplifying, the Bradt-Petersform is obtained for the nucleon-nucleus system

R2

e = r2(e)hA1=3T � �o(e)

i2(41a)

where

r2(e) = r2o +c21g(e)

2

�o(e) �g(e)

2r2(e)c1c2

and for the nucleus-nucleus system as

R2

e = r2(e)hA1=3P + A

1=3T � �o(AP ; AT ; e)

i2(41b)

with r as shown before and

�o(AP ; AT ; e) �g(e)

2r2(e)

0@c1c2+ c21

A1=3T A

1=3P

A1=3T +A

1=3P

1A

where ro = 1:25 fm.

In deriving the formula above, �o is assumed to be

much smaller than A1=3P and A

1=3T . Note that r is now

energy dependent, which supports the observationmade by Kox et al. (ref. 25).

After the above equation is substituted into equa-tion (39) to account for Coulomb correction, the re-sults are compared with those of the previous sectionin �gures 27{30. The agreement is good for the heav-ier as well as for the proton projectiles. The discrep-ancy for the alpha projectiles in the nucleus-nucleuscase results from the assumption rP = rT = 1:25 fmfor all nuclei in the Bradt-Peters form.

Nuclear-Medium E�ects

In the microscopic formulation, the two-nucleoninteraction is de�ned between free nucleons. Thisde�nition excludes nuclear-medium e�ects fromFermi motion and Pauli blocking. The exclusionof these e�ects can be justi�ed in both the high-and low-energy (below 20A MeV) regimes but not

in the intermediate-energy region. At high energy,many scattering states are available; hence, the ef-fects of the occupied states are comparatively negli-gible. At low energy, peripheral collisions dominatewhen the nuclear overlap densities are low and whenthe number of occupied states is small. Peripheralcollisions dominate at low energies for the followingreasons: the nuclear absorptive power increases dueto s-wave scattering between nucleons, and Coulombforces de ect the trajectories into the peripheral re-gions. Thus, reasonably good results are obtainedwith the Coulomb corrections alone. The nuclear-medium e�ects should actually increase the meanfree paths (ref. 19); hence, more nuclear transparencyresults. For uncharged projectiles such as the neu-tron, these e�ects can be considerable, as shown in�gure 14(b).

Concluding Remarks

A parameter-free geometric model for nuclear ab-sorption was derived herein from microscopic theory.The expression for the absorption cross section in theeikonal approximation, taken in integral form, wasseparated into a geometric contribution that was de-scribed by an energy-dependent e�ective radius andtwo surface terms that cancelled in an asymptoticseries expansion.

For collisions of light nuclei, an expression for thee�ective radius was derived from harmonic oscilla-tor nuclear density functions. A direct extension toheavy nuclei with Woods-Saxon densities was madeby identifying the equivalent half-density radius forthe harmonic oscillator functions. Coulomb correc-tions were incorporated, and a simpli�ed geometricform of the Bradt-Peters type was obtained. Re-sults spanning the energy range from 1 MeV/nucleonto 1 GeV/nucleon were presented. Good agreementwith experimental results was obtained.

NASA Langley Research Center

Hampton, VA 23681-0001

April 23, 1993

7

Appendix A

Surface Contributions to Total Absorption Cross Section

The surface contributions to the total absorption cross section, given by �1 and �2, are evaluated using the

Laplace method for asymptotic series (ref. 27)

�1 = 2�

ZRe

o

exp[��1(b; e)]b db

= 2�

ZRe

o

exp

��

��1(Re; e) +

@�1(b; e)

@b(b�Re) + : : :

��b db

� 2�Re

exp[��1(Re; e)]

@�1(b; e)=@b��Re

(A1)

Similarly,

�2 = 2�

Z1

Re

f1� exp[��1(b; e)]gb db

= 2�

Z1

Re

exp�ln f1� exp[��

1(b; e)]g�b db

= 2�

Z1

Re

exp

�lnf1� exp[��

1(Re; e)]g+@

@b(lnf1� exp[��

1(b; e)]g) (b�Re) + : : :

�b db

� 2�Re

f1� exp[��1(Re; e)]g

2

exp[�1(Re; e)][@�1(b; e)=@b]��Re

(A2)

8

Appendix B

Derivation for Phase Shift �(b;e)

Expressions for the phase shift �(b; e) are derived from the harmonic oscillator density functions. The

charge density is

�c(x) = �o

1 +

x2

a2

!exp

�

x2

a2

!(B1)

where and a fm are taken from reference 27 and �ois a normalization factor.

The matter density is derived from the charge density (ref. 29) as

�m(x) = �o(a0 + b0x2) exp(�px2) (B2)

where

a0 =a3

8s3

1 +

3

2 �

3

8 a2

s2

!(B3)

b0 = a5

128s7(B4)

s2 =a2

4�

r2p

6

p =1

4s2

where rp = 0:86 fm is the radius of the proton. Nucleon-nucleus and nucleus-nucleus collisions are considered

separately.

Nucleon-Nucleus

The matter density function for the nucleon is given by

�P(x� x0) = �(x� x0) (B5)

and for the target nucleus by

�T(x) = �TO(a0

T + b0Tx2) exp(�px2) (B6)

The phase shift is given by equation (2),

�(b; e) =A2

P A2

T

k(AP +AT )

Zdz

Zd3��T(�)

Zd3y�P(b+ z + y+ �)t(e;y)

Substituting for �P , �T , and t(e;y) from equation (3), the integral is easily performed, obtaining

�1(b; e) = 2Im�(b; e) = f1(e)f2(b; e) (B7)

where

f1(e) = AT�TO��(e)

��

P (e)

�1=2 �

2B(e)(p+ p0)��3=2

(B8)

f2(b; e) = (a1+ a2b2) exp[�P (e)b2] (B9)

9

with

a1 = a0 + 1:5b0

p+ p0+

a2

2P (e)

a2 =p

02

(p+ p0)2b0

p0 =1

2B(e)

p =1

4s2T

P (e) =1

4s2T + 2B(e)(B10)

Also an expression for �2 in the Taylor expansion is derived as

�2 = 2

��

P (e)

�3=2

f1(e)

�a1+ 1:5

a2

P (e)+ a2R

2

e

�exp

h�P (e)R2

e

i(B11)

Nucleus-Nucleus

The matter density function for the projectile is

�P (x) = �PO

�a0P + b0P x2

�exp(�px2) (B12)

and for the target is

�T (x) = �TO(a0

T + b0T x2) exp(�qx2) (B13)

which gives�1(b; e) = f1(e)f2(b; e) (B14)

where

f1(e) = APAT ��(e)�TO�PO�2�B(e)[p+ p0(e)][q + q0(e)]

�3=2

(B15)

and

f2(b; e) =�a1 + a2 b2 + a3 b4

�exp[�P (e)b2] (B16)

with

p =1

4s2P

q =1

4s2T

q0 =1

2B(e)

p0(e) =h4s2T + 2B(e)

i�1

P (e) =h4�s2P + s2T

�+ 2B(e)

i�1

9>>>>>>>>>>>>>>>>>>>>=>>>>>>>>>>>>>>>>>>>>;

(B17)

10

a1 = a0Pa00

T +1:5

p+ p0

�a00Tb

0

P + a0P b0

T

�+

15

4

b0T b0

P

(p+ p0)2

a2 = a00T b0

P

p02

(p+ p0)2+ a0Pb

0

T

p2

(p+ p0)2� b0T b

0

P2pp0

(p+ p0)3+ b0Tb

0

P

3

2

p2+ p02

(p+ p0)3

a3 = b0Tb0

P

p2p02

(p+ p0)4

where a0P and b0P are the constants that relate to the projectile as given by equations (B3) and (B4) and

a0T =a3T

8s3T

" 1 +

3

2 T�

3 Ta2T

8s2T

!+

Ta2T

16s4T

1:5

(q + q0)

#

b0T =a3T

8s3T

Ta2T

16s4T

"q

02

(q + q0)2

#

a00T = a0T + b0T1:5

q + q0

relate to the targets.

An expression for �2 in the Taylor series expansion is derived as

�2 �f1(e)

2P (e)

r�

P (e)

(a1+ a2

�1:5

P (e)+R2

e

�+ a3

"15

4P (e)2+

3R2e

P (e)+ 4R4

e

#)exp

h�P (e)R2

e

i(B18)

The following integral relations were used.

I =

Z1

�1

exp[�qr2� p(r� y)2] dr =

��

q + p

�3=2

exp

��

pq

p+ q

�y2 (B19)

Zr2 exp[�qr2 � p(r� y)2] dr = �

@

@qI (B20)

Z(r� y)2 exp[�qr2 � p(r� y)2] dr = �

@

@pI (B21)

Zr2(r� y)2 exp[�qr2� p(r� y)2] dr =

��

@

@p

���

@

@q

�I (B22)

An alternate partial evaluation is given in reference 30.

11

Appendix C

Parameterization for Experimental Two-Nucleon Total Cross Section

For completeness, the isospin-averaged nucleon-nucleon cross section taken from reference 1 is given. A

parameterization for the experimental two-nucleon total cross section in millibarns (1 barn = 1� 10�28 m2) is

given for the proton-proton interaction by

�pp(e) =

�1 +

5

e

�n40 + 109 cos

�0:199

pe�exp

h�0:451(e� 25)0:258

io(C1)

for e � 25 MeV and for the lower energies by

�pp(e) = exp

�6:51 exp

��� e

134

�0:7��

(C2)

For proton-neutron interaction at e � 0:1 MeV,

�np(e) = 38 + 12 500 exph�1:187(e� 0:1)0:35

i(C3)

and at lower energies

�np(e) = 26 000 exp

��� e

0:282

�0:3�

(C4)

The isospin average is calculated as follows:

��(e) =�(NPNT + ZPZT )�pp+ (NPZT + ZPNT )�np

�=APAT (C5)

where A, N , and Z are the atomic, neutron, and proton numbers, respectively.

The nucleon-nucleon slope parameter is parameterized (ref. 31) as

B(e) = 10 + 0:5 ln

�s0

so

�(C6)

where s0 is the square of the nucleon-nucleon center-of-mass energy and so = (1 GeV=c)�2.

12

References

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Nucleus Reaction Cross Sections. Canadian J. Phys.,

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15. Townsend, L. W.; and Wilson, J. W.: Energy-Dependent

Parameterization of Heavy-Ion Absorption Cross Sec-

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16. Townsend, L. W.; and Wilson, J. W.: Comment on

\Trends of Total Reaction Cross Sections for Heavy Ion

Collisions in the Intermediate Energy Range." Phys.Review C , vol. 37, no. 2, Feb. 1988, pp. 892{893.

17. Shen, Wen-Qing; Wang, Bing; Feng, Jun; Zhan,

Wen-Long; Zhu, Yong-Tai; and Feng, En-Pu: Total Re-

action Cross Section for Heavy-Ion Collisions and Its Re-

lation to the Neutron Excess Degree of Freedom. Nucl.Phys., vol. A491, 1989, pp. 130{146.

18. Vitturi, A.; and Zardi, F.: Modi�ed Glauber Model for

the Description of Elastic ScatteringBetweenHeavy Ions.

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19. Hussein, M. S.; Rego, R. A.; and Bertulani, C. A.:

Microscopic Theory of the Total Reaction Cross Section

and Application to Stable and Exotic Nuclei. Phys. Lett.,vol. 201, no. 5, 1991, pp. 279{334.

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A.: Total-Reaction-Cross-SectionMeasurements for 30{

60-MeV Protons and the Imaginary Optical Potential.

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21. Barashenkov, V. S.; Gudima, K. K.; and Toneev, V. D.:

Cross Sections for Fast Particlesand AtomicNuclei. Prog.

Phys., vol. 17, no. 10, 1969, pp. 683{725.

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P.; Favier, B.; and Richard-Serre, C.: Reaction Cross

Sections for Protons in the Energy Range 220{570 MeV.

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23. Perrin, C.; Kox, S.; Longequeue, N.; Viano, J. B.;

Buenerd, M.; Cherkaoui, R.; Cole, A. J.; Gamp, A.;

Menet, J.; Ost, R.; Bertholet, R.; Guet, C.; and Pinston,

J.: Direct Measurement of the 12C + 12C Reaction Cross

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Lett., vol. 49, no. 26, Dec. 27, 1982, pp. 1905{1909.

24. Buenerd, M.; Lounis, A.; Chauvin, J.; Lebrun, D.;

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El-Masri,Y.; Longequeue,N.; Menet, J.; Merchez,F.; and

Viano, J. B.: Trends of Total Reaction Cross Sections for

Heavy Ion Collisions in the Intermediate Energy Range.

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26. Dubar, L. V.; Eleukenov, D. Sh.; Slyusarenko, L. I.; and

Yurkuts, N. P.: Parametrization of Total Cross Sections

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13

28. Bender, Carl M.; and Orszag, Steven A.: Advanced Math-

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pp. 185{188.

14

1.2

1.0

.8

.6

.4

.2

0 2 4 6 8 10b, fm

Abs

orpt

ion

δ1

δ2

Figure 1. Absorption as a function of impact parameter for nuclear collisions. At half absorption, shaded areas�1 � �2.

1.2

1.0

.8

.6

.4

.2

04 5 6 7 8

b, fm

Abs

orpt

ion

1 4He–4He

2 16O–16O

R1 R2

3

Figure 2. Absorption as a function of impact parameter for pro jectile-target systems of 4He{4He and 16O{16Oat 2 GeV/nucleon. Terms R1 and R2 denote geometric radii for systems.

15

1.2

1.0

.8

.6

.4

.2

04 6 8 10

b, fm

Abs

orpt

ion

1 50 MeV/nucleon

2 250 MeV/nucleon

3 1000 MeV/nucleon

R2 R3 R1

Figure 3. Absorption as a function of impact parameter for three energy values: 50 and 250 MeV/nucleon and1 GeV/nucleon. Corresponding e�ective radii R1, R2, and R3 are indicated.

1.4

1.2

1.0

.8

.4

.2

0 5 15 20A

r o, f

m

Nucleon-nucleus

Nucleus-nucleus.6

10

Figure 4. Variation of ro with atomic mass for harmonic oscillator density functions for nucleon -nucleus andidentical nucleus-nucleus collisions.

16

.50

.40

.30

.10

0100 10 000

EL, MeV/nucleon

σ abs

, b

p–12C

p–4He

.20

1 00010

Figure 5. Variation of absorption cross section with energy for proton-nucleus collisions.

.50

.40

.30

.10

0100 10 000

EL, MeV/nucleon

σ abs

, b

p–16O

p–9Be

.20

1 00010

Figure 6. Variation of absorption cross section with energy for proton-nucleus collisions.

17

.8

.6

.4

.2

0100 10 000

EL, MeV/nucleon

σ abs

, b

d–12C

d–4He

1 00010

Figure 7. Variation of absorption cross section with energy for deuteron-nucleus collisions.

1.5

1.0

.5

0100 10 000

EL, MeV/nucleon

σ abs

, b

16O–16O

16O–p

1 00010

Figure 8. Variation of absorption cross section with energy for oxygen-nucleus systems.

18

1.5

1.0

.5

0100 10 000

EL, MeV/nucleon

σ abs

, b

C–He

C–p

C–C

1 00010

Figure 9. Variation of absorption cross section with energy for carbon-nucleus systems.

2.5

1.0

.5

018

A

k

168

1.5

2.0

141210

Figure 10. Variation of k = (R=a)2 for harmonic oscillator functions with atomic mass number , where R ishalf-density radius and a fm is harmonic oscillator parameter.

19

.8

.6

.2

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1

.4

100 1 000

p–C

Figure 11. Absorption cross section as a function of energy for proton on carbon with and without Coulombe�ects.

1.5

1.0

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1

.5

100 1 000

p–Al

Figure 12. Absorption cross section as a function of energy for proton on aluminum with and without Coulombe�ects.

20

1.5

1.0

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1

.5

100 1 000

p–Cu

Figure 13. Absorption cross section as a function of energy for proton on copper with and without Coulombe�ects.

21

5

4

010 10 000

EL, MeV/nucleon

σ abs

, bExperiment

Coulomb

w/o Coulomb

1

1

100 1 000

Proton on Pb

3

2

(a) Proton on lead.

5

4

010 10 000

EL, MeV/nucleon

σ abs

, b

1

1

100 1 000

Neutron on Pb

3

2

(b) Neutron on lead.

Figure 14. Absorption cross section for proton or neutron on lead as a function of energy with and withoutCoulomb e�ects. Without Coulomb e�ects, curve corresponds to neutron projectiles.

22

2.0

1.5

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1

.5

100 1 000

C–C

1.0

Figure 15. Absorption cross section as a function of energy for carbon on carbon with and without Coulombe�ects.

3

2

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1 100 1 000

C–Al

1

Figure 16. Absorption cross section as a function of energy for carbon on aluminum with and without Coulombe�ects.

23

4

2

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1 100 1 000

O–Ca

Figure 17. Absorption cross section as a function of energy for oxygen on calcium with and without Coulombe�ects.

8

4

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1 100 1 000

O–Pb

6

2

Figure 18. Absorption cross section as a function of energy for oxygen on lead with and without Coulombe�ects.

24

3.0

1.5

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1 100 1 000

O–Si

2.5

1.0

2.0

.5

Figure 19. Absorption cross section as a function of energy for oxygen on silicon with and without Coulombe�ects.

5

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1 100 1 000

Ca–Ca

4

2

3

1

Figure 20. Absorption cross section as a function of energy for calcium on calcium with and without Coulombe�ects.

25

8

010 10 000

EL, MeV/nucleon

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1 100 1 000

Ar–Bi

6

4

2

Figure 21. Absorption cross section as a function of energy for argon on bismuth with and without Coulombe�ects.

3.0

010 10 000

AT2 /3

σ abs

, b

Experiment

Coulomb

w/o Coulomb

1 100 1 000

Proton–X 40 MeV/nucleon

2.5

1.5

.5

2.0

1.0

Figure 22. Absorption cross section as a function of�A2=3T

�for protons on targets at 40 MeV/nucleon with

and without Coulomb e�ects, where AT is target mass number.

26

3.0

0 40

AT2 /3

σ abs

, b

Experiment

Coulomb

w/o Coulomb

20

p–X 60.8 MeV/nucleon

2.5

1.5

.5

2.0

1.0

Figure 23. Absorption cross section as a function of�A1=3T

�2for protons on targets at 60.8 MeV/nucleon with

and without Coulomb e�ects, where AT is target mass number.

5

070

(AT1 /3 + AP

1 /3)2

σ abs

, b

Experiment

Coulomb

w/o Coulomb

10

Alpha–X 25 MeV/nucleon

4

2

1

3

6050403020

Figure 24. Absorption cross section as a function of�A1=3T + A

1=3P

�2for alpha particles on targets at

25 MeV/nucleon with and without Coulomb e�ects, where AP is mass number of alpha particles andAT is target mass number.

27

5

060

(AT1 /3 + AP

1 /3)2

σ abs

, b

Experiment (30 MeV/nucleon)

Experiment (300 MeV/nucleon)

Coulomb

w/o Coulomb

10

C–X

4

2

1

3

50403020

Figure 25. Absorption cross section as a function of�A1=3T +A

1=3P

�2for carbon on targets at 30 MeV/nucleon

and 300 MeV/nucleon with and without Coulomb e�ects.

5

080

(AT1 /3 + AP

1 /3)2

σ abs

, b

Experiment

Coulomb

w/o Coulomb

20

Ne–X 30 MeV/nucleon

4

2

1

3

70504030 60

Figure 26. Absorption cross section as a function of�A1=3T + A

1=3P

�2for neon on targets at 30 MeV/nucleon

with and without Coulomb e�ects.

28

2.5

0 40

(AT1 /3)2

σ abs

, b

Experiment

Bradt-Peters form

Original

Proton–X 60.8 MeV/nucleon

2.0

1.0

.5

1.5

20

Figure 27. Comparison of Bradt-Peters form with that for absorption cross section of proton on targets at60.8 MeV/nucleon.

4

060

(AT1 /3 + AP

1 /3)2

σ abs

, b

Experiment

Bradt-Peters form

Original

Alpha–X 25 MeV/nucleon

2

2010 30 40 50

Figure 28. Comparison of Bradt-Peters form with that for alpha on targets at 25 MeV/nucleon.

29

5

0 80

(AT1 /3 + AP

1 /3)2

σ abs

, b

Experiment 30 MeV/nucleonExperiment 300 MeV/nucleonBradt-Peters formOriginal

C–X

2

20 40 60

1

3

4

Figure 29. Comparison of Bradt-Peters form with that for carbon on targets at 30 MeV/nucleon and300 MeV/nucleon.

5

080

(AT1 /3 + AP

1 /3)2

σ abs

, b

Experiment

Bradt-Peters form

Original

Ne–X 30 MeV/nucleon

4

20 30 40 50

3

2

1

60 70

Figure 30. Comparison of Bradt-Peters form with that for neon on targets at 30 MeV/nucleon.

30

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1. AGENCY USE ONLY(Leave blank) 2. REPORT DATE 3. REPORT TYPE AND DATES COVERED

July 1993 Technical Paper

4. TITLE AND SUBTITLE

Geometric Model FromMicroscopic Theory for Nuclear Absorption

6. AUTHOR(S)

Sarah John, Lawrence W. Townsend, John W. Wilson,

and Ram K. Tripathi

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

NASA Langley Research Center

Hampton, VA 23681-0001

9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES)

National Aeronautics and Space Administration

Washington, DC 20546-0001

5. FUNDING NUMBERS

WU 593-42-21-01

8. PERFORMING ORGANIZATION

REPORT NUMBER

L-17187

10. SPONSORING/MONITORING

AGENCY REPORT NUMBER

NASA TP-3324

11. SUPPLEMENTARY NOTES

John and Tripathi: ViGYAN, Inc., Hampton VA; Townsend andWilson: Langley Research Center, Hampton,VA.

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Subject Category 73

13. ABSTRACT (Maximum 200 words)

A parameter-free geometric model for nuclear absorption is derived herein from microscopic theory. Theexpression for the absorption cross section in the eikonal approximation, taken in integral form, is separatedinto a geometric contribution that is described by an energy-dependent e�ective radius and two surface termsthat cancel in an asymptotic series expansion. For collisions of light nuclei, an expression for the e�ectiveradius is derived from harmonic oscillator nuclear density functions. A direct extension to heavy nuclei withWoods-Saxon densities is made by identifying the equivalent half-density radius for the harmonic oscillatorfunctions. Coulomb corrections are incorporated, and a simpli�ed geometric form of the Bradt-Peters typeis obtained. Results spanning the energy range from 1 MeV/nucleon to 1 GeV/nucleon are presented. Goodagreement with experimental results is obtained.

14. SUBJECT TERMS 15. NUMBER OF PAGES

Reaction cross section; Nuclear scattering; Bradt-Peters formula;

Spacecraft shielding31

16. PRICE CODE

A0317. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION

OF REPORT OF THIS PAGE OF ABSTRACT OF ABSTRACT

Unclassi�ed Unclassi�ed

NSN 7540-01-280-5500 Standard Form 298(Rev. 2-89)Prescribed by ANSI Std. Z39-18298-102