A simple analytic method for calculating nucleon-nucleus optical potentials using a single folding of a Gaussian two body interaction with an arbitrary nuclear distribution is presented. When applied to proton-lead elastic scattering, the predicted real part of the Woods-Saxon potential is in substantial agreement with the experimentally determined phenomenological potential, although there are no adjustable parameters. In addition, the volume integrals of both real potentials are nearly identical

Bidassaria, H. B.

Townsend, L. W.

English

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iUhc_ft-Tn_ - J_/Lo,II
NASA-TM-84611 19830013665
NASA Technical Memorandum 84611
ANALYTICDETERMINATIONSOFSINGLE-FOLDING
OPTICALPOTENTIALS
L. W. TownsendandH. B. Bidasaria
February1983
BIANA
National Aeronautics and
SpaceAdministration
Langley Research Center
Hampton, Virginia 23665
https://ntrs.nasa.gov/search.jsp?R=19830013665 2020-03-21T03:50:38+00:00Z

Table of Contents
NOMENCLATURE
INTRODUCTION.
"'"''''""''"''"'""'"''"''"----'--------.--...,.. 2
RESULTSFOR PROTON-LEADELASTICSCATTERING.............................. 6
REFERENCES
FIGURE
NOMENCLATURE
A nuclear mass number
B(e) average slope parameter of nucleon-nucleon scattering amplitude,fm
c Woods-Saxon surface diffuseness, fm
e two-nucleon kinetic energy in their center of mass frame, GeV
J defined in Eq. (21)
m nucleon mass, kg
R Woods-Saxon half-density radius, fm
.
r position vector, fm
r u radius of equivalent uniform distribution, fm
t average two-nucleon transition amplitude, MeV
t o defined in Eq. (3)
Vo defined in Eq. (16)
Vopt optical potential (defined in Eq. (15)), MeV
x relative position vector of projectile, fm
y two-nucleon relative position vector, fm
z position vector of projectile in beam direction, fm
_(e) average ratio of real part to imaginary part of nucleon-nucleon
scattering amplitude
B defined in Eq. (6)
6(_) Dirac delta function
p nuclear density,fm-3
Po normalizationconstant in Eq. (20), fm"3
_(e) averagenucleon-nucleontotal cross section,mb
ii
Subscripts :
P projectile
T target
; Arrows over symbols indicate vectors.
iii

ANALYTICDETERMINATIONSOF SINGLE-FOLDINGOPTICAL POTENTIALS
Lawrence W. Townsend
Langley Research Center
and
Hari B. Bidasaria
Old Dominion University
SUMMARY
A simple analytic method for calculating nucleon-nucleus optical
potentialsusing a single-foldingof a Gaussian two-body interactionwith
an arbitrary nuclear distribution is presented. When applied to
proton-leadelastic scattering,the predictedreal part of the Woods-Saxon
potential is in substantialagreement with the experimentally-determined
phenomenologicalpotential, although there are no adjustable parameters.
In addition, the volume integrals of both real potentials are nearly
identical.
INTRODUCTION
In order to properly assess the optimum shield requirements for future
manned space efforts, especially missions of long duration, accurate
methods for predicting the interactions of cosmic rays with spacecraft
materials and inhabitants are required. Although theories capable of
accurately predicting high-energy nucleon-nucleus and nucleus-nucleus cross
sections exist (refs. 1 through 6), disagreements between theory and
experiment are evident at low energies. These are probably due, in large
part, to the inapplicability of the eikonal approximation at low energies
(refs. i and 2). In a recent work (ref. 7), substantial improvement in the
agreement, at low energies, for light and medium nuclei was noted when
partial wave analyses, utilizing complex WKB (Wentzel-Kramers-Brillouin)
phase shifts, were performed. The method outlined in reference 7, however,
appears to be suitable only for collisions involving light and medium
nuclei whose matter densities are Gaussian or harmonic well distributions,
since their nuclear optical potentials can be analytically determined
(ref. 8). For heavier nuclei whose density distributions are neither
Gaussian or harmonic well shapes, their optical potentials are not readily
reduced to analytic forms (ref. i). In this work, a method is presented
for obtaining an approximate analytic expression for the nucleon-nucleus
optical potential, involving the single-folding of an arbitrary target
density distribution with a Gaussian two-nucleon interaction. As such it
represents an initial effort toward discovery of approximate analytic
methods for eventual use in evaluating double-folding optical potentials
involving arbitrary nuclear density distributions. When applied to i GeV
proton-lead collisions, this method yields a predicted real part of the
nuclear potential which is in excellent agreement with the available
phenomenological results obtained from elastic scattering experiments
(ref. 9).
THEORETICALANALYSIS
In previous work (ref. 2) it was shown that the optical potential
approximation to the exact nucleus-nucleus multiple-scattering series is
V(_) = ApAT I d3_ PT(_) I d3_ Pp(_ + _ + _) t(e,_) (1)
where the two-body transition amplitude, averaged over constituent types
(ref. 1) is
t(e,_) = to exp[-y2/2B(e)] (2)
with
1 3
to = -(e/m)_ a(e)[a(e)+ i] [2_B(e)]-_ (3)
and PT and pp are the target and projectile single-particle number
(matter densities). In equation (3), e is the two-nucleonkineticenergy
in their center of mass frame, and _(e), e(e), and B(e) are the usual
nucleon-nucleonscatteringparameters(refs. 1 and 2).
For nucleon-nucleus scattering involving a target nucleus of mass
number AT, the projectilesingle-particledensity is
pp(_ + _ + _) : a(_ + _ + _) (4)
with Ap = 1. Equation (1) then reduces, for nucleon-nucleusscattering,
to
Vopt(_) = AT f d3_ pT(_) t(e, _ + _) (5)
Insertingequation (2) into (5) and letting
B2 = [2B(e)]-i (6)
, yields
• Vopt(_): 2_toAT f_z2dz pT(_) exp[-B2 (x2 + z2)]
o
(7)
_o exp[-2xz B2 cos G] sin GdO
The angularintegrationis of the form
_exp(-q cos e) sin Ode : [exp(q) - exp(-q)]/q (8)
0
which, upon collectingexponents,yields
Vopt(_) = (_toAT/{32x) [=zdz pT(Z) {exp[-82(z - x)2]0
(9)
-exp[-8 2 (z + x)2]}
Evaluatingthe integralsin equation(9) gives
f=zdz pT(z)exp[-8 2 (z - x) 2] =
o
(io1
8-I [= (x + 8-1 S)PT(X+ (3-1 s)exp(-s2)ds
-8x
and
_=zdz pT(z)exp[-82(x + z)2] :
o
(ii)
8-1 _ (8-1 s - x) pT(8-1 s - x)exp(-s2)ds8x
Incorporatingthe results from equations (10) and (11) into (9), and
assuming that the density distributionis sphericallysymmetric yields,
after some algebra,
Vopt(_)= _toATB-3 {(8 x)-I _=sds exp(-s2) [PT(X+ 8-I s)0
- PT(X - 8-1 s)] (12)
+ f= ds exp(-s2) PT(X + 8-1 s)}
The integrals in equation (12) are of the same form as those in
equation (86) of reference 1, where a similar method was utilized to
extractmatter densitiesfrom nuclearcharge densities.
The first integral, is generally smaller than the second since it
contributesonly when x is near the nuclearedge. For the extremecase,
where PT is a finite uniform distribution, the ratio of the first
integralto the second,at the uniformradius,ru, is
: 1
Ratio = ru'l[2B(e)/_] (13)
Noting that this ratio is a maximum for large B(e) and small ru, which is
the situationfor light target nuclei and high energies,an estimateof the
ratio is made. Choosing ru - 3 fm (a nominalvalue for a lithiumnucleus
from reference 10) and B = [5 fm2 (ref. 11) gives a maximum ratio of 19
percent. It is expectedthat the error resultingfrom neglectof the first
integral, for realistic densities, will be substantiallyless than this
value.
Neglecting the first term in equation (12) yields an approximate
nuclearoptical potential
Vopt(X) = _toAT_-3 f_ ds exp(-s2) PT(X + B-I s) (14)
For an arbitrarydensitydistribution,the integralcan be approximatedby
a two-pointGauss-Hermitequadratureformula (ref. 12) to yield an analytic
nuclearoptical potential
Vopt(_) = (VoAT/2)[PT(X+ vrB)+ PT(X - V_B)] (15)
with
1
Vo : -(e/m) -Z _(e)[a(e) + i] (16)
Should the need arise to includethe first integral in equation (12), it
can be similarly approximated by changing variables such that
I_sds exp(-s2)[PT(X+ B-I s) - PT(X - B"1 s)]
o
i
= _if_dp exp(-p)[PT(X+ (p/B2)- _) (17)
• 0
i
_ PT(X _ (p/B2)--Z)]
This allowsthe integralto be easily approximatedby a Laguerrequadrature
formula (ref. 12).
5
RESULTSFORPROTON-LEADELASTIC SCATTERING
To illustrate the application of equation (15), the real part of the
nuclear optical potential for proton-lead elastic scattering at an incident
kinetic energy of 1.04 GeV is calculated. Thus, taking the real part of
equation (15) gives
i
Vopt(_) = -[(e/m) _ _(e) _(e) aT/2][PT(X + _FB) + PT(X _ vrB)] (18)
which, for 1.04 GeV protons colliding with lead yields
Vopt(_) : 9041.5 [PT(X + .49) + PT(X - .49)] (19)
For PT given in fm-3 (normalized to unity) . ", Vopt(x ) In equation (19) is
given in MeV. Choosing PT of lead to be a Woods-Saxon charge density
using the methods in references 1 and 2 yields.
pT(_) = Po{1 + exp[(r - R)/c]} -I (20)
where R = 6.624 fln, c = 0.860 fm, and p(o2 i 7.059 x 10-4 fm-3. Theresults obtained from equations (19) and (] are plotted in figure i.
Also plotted in figure 1 are the phenomenological Woods-Saxon potential
results obtained from elastic scattering experiments (ref. 9). The
agreement between theory and experiment is impressive particularly since
there are no arbitrarily adjustable parameters in the theory. In addition,
the volume integrals, J , of both real potentials
J = 4_ I _ r 2 V(r) dr (21)
0
agree te within 0.02 ,percent.
Finally, we note from equation (13) that the maximum expected error
due to the neglect of the first integral in equation (12) is less than 6
percent. The actual error, for the density used in the calculations
(eq. (20)), was found by numerical methods to be less than 5 percent
_ (typically 2-3 percent) for values of x up to 12 fm.
CONCLUDINGREMARKS
In this work a simple analytic method for approximating nuclear
optical potential integrals involving the single-foldingfor a Gaussian
interactionwith arbitrarynuclear distributionswas presented. Applying
the method to 1.04 GeV proton-lead collisions, the real part of the
predictedpotentialwas found to be in remarkablygood agreementwith the
phenomenologicalresultsobtainedfrom elastic scatteringexperiments. The
resulting Woods-Saxon radial shape, for the potential, obtained when a
Woods-Saxontarget density was utilized also suggests that there may be
some credibilityin the usual assumptionthat the spatialdependenceof the
nuclear potential should follow that of the associated nuclear density
distribution. It is anticipatedthat the analyticmethodsdescribedherein
will be useful for determining complex WKB solutions to low energy
scattering problems and for other nuclear potential calculations where
purely numericalmethodsmay not be appropriate.
REFERENCES
1. Wilson, J. W.; and Costner, Christopher M.: Nucleon and Heavy-lon
Total and Absorption Cross Section for Selected Nuclei. NASA TN
D-8107, 1975.
2. Wilson, J. W.; and Townsend, L. W.: An Optical Model for Composite
Nuclear Scattering. Canadian J. Phys, vol. 59, no. ii,
November 1981, pp. 1569-1576.
3. Townsend, L. W.: Optical-Model Abrasion Cross Sections for High-
Energy Heavy lons. NASATP-1893, 1981.
4. Townsend, L. W.: Harmonic well Niatter Densities and Pauli Correlation
Effects in Heavy-lon Collisions. NASATP-2003, 1982.
5. Townsend, L. W.; Wilson, J. W.; and Bidasaria, H. B.: On the Geo-
metric Nature of High Energy Nucleus-Nucleus Reaction Cross
Sections. Canadian J. Phys., vol. bO, no. I0, October 1982,
pp. 1514-1518.
6. Townsend, L. W.: Abrasion Cross Sections for 2°Ne Projectile at
2.1 GeV/Nucleon. Canadian J. Phys., vol. 61, no. i, January 1983,
pp. 93-98.
7. Bidasaria, H. 8.; Townsend, L. W.; and Wilson, J. W.: Theory of
Carbon-Carbon Scattering from 200 to 290 MeV. J. Phys. G.: Nucl.
Phys., vol. 9, no. I, January 1983, pp. LI7-L20.
8. Bidasaria, H. B.; and Townsend, L. W.: Analytic Optical Potentials
for Nucleon-Nucleus and Nucleus-Nucleus Collisions Involving Light
and Medium Nuclei. NASATM-83224, 1982.
9. van Oers, W. T. H.; Haw, H.; Davidson, N. E.; Ingemarsson, A.;
Fagerstrom, B.; and Tibell, G.: Optical-Model Analysis of p+2°sPb
Elastic Scattering from 15-1000 MeV. Phys. Rev., ser. C, vol. i0,
no. i, July 1974, pp. 307-319.
10. Hofstadter, R.; and Collard, H. R.: Nuclear Radii Determined by Elec-
tron Scattering. Landolt-Bornstein Numerical Data and Functional
Relationships in Science and Technology, Group I, vol. 2,
H. Schopper(Ed.), Springer-Verlag, 1967, pp. 21-52.
ii. Hellwege, K.H. (Ed.): Elastische und Ladungsaustausch-Streuung von
Elementarteilchen. Landolt-Bornstein Numerical Data and Functional
Relationships in Science and Technology, Group I, vol. 7,
Springer-Verlag, 1973.
12 Abramowitz, M.; and Stegun, Io A., (Ed.): Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Tables. National
bureau of Standards, AMS-55, June 1964.
T20
-_" ---- Theory
>=
i; *°
o
€'_ 0 _ ! -"-
0 3 6 9 12
x, fm
Figure1. -Real part of the proton -lead opticalpotentialat 1.04GeVincident
kinetic energy.
I. Report No. 2. Government AccessionNo, 3. Recipient'sCatalogNo.
NASA TM-84611
"4. Titleand Subtitle 5, Report Date
ANALYTIC DETERMINATIONS OF SINGLE-FOLDING February 1983
OPTICAL POTENTIALS 6.PerformingOrganizationC de
199-20-76-01
7. Author(s) 8. PerformingOrgan;zation Report No.
"-'LawrenceW. Townsend and
**Hari B. Bidasaria 10.WorkUnitNo.
9. Performing Organization Name and Address
Langley Research Center 11. Contractor Grant No.
Hampton, VA 23665
13. Type of Report and Period Covered
12. Sponsoring Agency Name and Address Technical Memorandum
National Aeronautics and Space Administration 14 S_,onsnringAgencyCode
Washington, DC 20546
15. S_pplementary Notes
*NASA Langley Research Center, Hampton, Virginia
**Old Dominion University, Norfolk, Virginia
16. Abstract
A simple analytic method for calc_Jlating nucleon-nucleus optical
potentials using a single-folding of a Gaussian two-body interaction witil
an arbitrary nuclear distribution is presented. When applied to
proton-lead elastic scattering, the predicted real part of the Woods-Saxon
potential is in substantial agreement with the experimentally-determined
phenomenological potential, although there are no adjustable parameters.
In addition, the volume integrals of both real potentials are nearly
identical.
17. Key Words (Suggested by Author(s)) 18. Distribution Statement
Single-Folding Potential Unclassified - Unlimited
Optical Model Potential Subject Category - 73
19. Security Classif. (of this report] 20. Security Classif. (of this page) 21. No. of Pages 22. PTice
Unc lass ified Unc inssified 13 A02
,-305 Forsale bytheNationalTechnic;_lInformationService,Springfield.V,rgmla22161




Analytic determinations of single-folding optical potentials

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