Differential cross sections for 200 MeV protons elastically scattered from C-12 were analyzed utilizing a local, complex, spin-dependent optical potential with a harmonic well radial dependence. Analyses were performed using the WKB and eikonal approximations. For the latter, first-order corrections to he phase shifts were incorporated to account for the spin-orbit contribution. Large disagreement between theory and experiment was observed when the usual Thomas form for the spin-orbit potential was utilized. Substantial improvement was obtained by allowing the parameters in the central and spin-orbit potential terms to vary independently

Bidasaria, H. B.

Townsend, L. W.

English

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I :
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National Aeronautics and
Space Administration
tomy Research tenter
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ALIT l og2
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AC4. Nib,
NASA Technical MemormWum 84498
PIENONIENOLOGI CAL OPTI CAL POTENTIAL ANALYS I S OF
PROTON-CARBON ELASTIC SCATTER I NG AT 20O MeV
N. B. Bidasaria and L W. Townsend
J  ne 1982
(NASA-TH-84498) PHENOMENUGGICAL OPTICAL
POTENTIAL ANALYSIS OF PEO'PUN-CARBON ELASTIC
SCATTERING AT 200 Me g
 (NASA) 21 p
HC A02 u   A
N82-30066
14
01	 CSCL 20H
	 Unclas
63/73 28523
NOMENCLATURE	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 . ii
SUMMARY .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 1
INTRODUCTION	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 2
ANALYTIC METHODS
	
.	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 3
Optical	 Potential	 Parameterization .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 3
Scattering	 Formalism .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 4
Eikonal
	 Phase Shifts .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 6
WK,B	 Phase	 Shifts .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 7
RESULTS .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .	 g
CONCLUDING	 REMARKS	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 g
REFERENCES	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	
.10
TABLES	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .12
FIGURES .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 .	 . .	 .	 .	 .	 .	 .	 .	 .14
1
NOMENCLATURE
Symbols:
Ao	 harmonic well optical potential parameter (central term),
dimensionless
A1 ,A2
	harmonic well optical potential parameter (central term), fm-2
Bo	 harmonic well optical potential parameter (spin -orbit term),
dimensionless
B1 ,B2	harmonic well optical potential parameter (spin-orbit term), fm-2
b	 impact parameter vector, fm
E	 projectile kinetic energy in center of mass system, MeV
e	 proton charge, esu
F(0)	 defined in equation (7)
f(0)	 spin non-flip scattering amplitude, fm
9(0)	 spin-flip scattering amplitude, fm
11	 Planck's constant, joule-second
J	 total angular momentum voctor, dimensionless
k	 center of mass momentum wave number, fm-1
L	 orbital angular momentum vector, dimensionless
G1	 nucleon mass, kq
A
	
unit vector normal to scattering plane, dimensionless
PL	 Legendre polynomial of order L, dimensionless
Pl i	 associated Legendre function of first kind and order L,
dimensionless
Pi	 incident beam polarizatino %lector, dimensionless
r	 relative position vector, fm
r± L	 WKB turning point, fm
rC	 coulomb turning point, fm
s	 spin angular momentum vector, dimensionless
ii
:ORIGINAL PAGE IS
OF POOR QUALITY
U+	defined in equation (22)
Uc	 defined in equation (21)
Vc	 central potential term, MeV
Vs	 spin-orbit potential term, MeV
Veff	 defined in equation (23)
v	 relative velocity in center of mass system, m/sec
ZT	 target nuclear charge number, dimensionless
a	 defined in equation (11)
r	 gamma function of complex argument, dimensionless
a L±	 defined in equations (16) and (20)
nL	 defined in equation (13)
a	 twice the spin angular momentum, dimensionless
a(0)	 differential cross section, mb per steradian
0	 scattering angle, radians
X	 eikonal phase shift, dimensionless
Subscripts:
C Coulomb
cm center of mass
C central
s spin-orbit
T target
iii
ORIGINAL PAGE IS
OF POOR QUALITY
PHENOMENOLOGICAL OPTICAL POTENTIAL ANALYSIS OF
PROTON-CARBON ELASTIC SCATTERING AT 200 Mev
Hari B. Bidasaria
Old Dominion University
Norfolk, Virginia
and
Lawrence W. Townsend
Langley Research Center
Hampton, Virginia
SUMMARY
gIfferential cross sections for 200 MeV protons elastically scattered
from C are analyzed utilizing a local, complex, spin-dependent optical
potential with a harmonic well radial dependence. Analyses were performed
using the WKB and eikonal approximations. For the latter, first-order
corrections to the phase shifts were incorporated to account for the
spin-orbit cintribution. Large disagreeme,vt between theory and experiment
was observed when the usual Thomas form for the spin-orbit potential was
utilized. Substantial improvement was obtained by allowing the parameters
in the central and spin-orbit potential terms to vary independently.
• .r
ORIGINAL PAGE 15
OF POOR QUALITY
INTRODUCTION
The development of an accurate heavy ion (HZE) transport theory is
required to support cosmic ray shielding analyses for future space appli-
cations. Transport theory development, in turn, requires a simple, yet
accurate, HZE reaction theory as input. For the past several years, such a
theory has been under development for incident projectile energies above
several hundred MeV/amu (refs. 1-7). At present, no satisfactory theory
exists for energies below (approximately) 200 MeV/amu where the eikonal
(high energy) approximation appears to become inadequate (refs. 3 and 6).
At lower incident kinetic energies, contributions from low order partial
waves (orbital angular momentum states) increase appreciably. In addition,
the incident ion's trajectory deviates significantly from its initial
straight line path such that the eikonal approximation may no longer yield
accurate predictions. At these lower energies, a better approximation is
the Wentzel-Kramers-Brillouin (WKB) method (ref. 8). This work describes
osir initial investigation into utilizing this approximation method to
analyze elastic scattering. Proton-carbon scattering was chosen for
analytic convenience due to its simplicity (elementary rather than compos-
ite projectile) and because it is the subject of much current theoretical
rind experimental interest (refs. 9-11). Additionally, a harmonic well
potential parameterization rather than the usual Woods-Saxon form (refs. 9
and 10) was chosen since it more accurately represents the actual carbon
nsiclear distribution (refs. 7 and 12) and should therefore yield agreement
with experimental data for fewer free parameters than the 24 required by
the double-Woods-Saxon potential analysis of reference 9.
i
l
ORIGINAL PAGE 13
OF POOR QUALITY
ANALYTIC METHODS
Optical Potential Parameterization
The optical potential chosen for this analysis consisted of a central,
Vc, and spin-orbit, Vs, contribution (ref. 8)
V (r) - Vc ( r ) + Q	 ^ Vs (r)	 (1)
where a' is twice the spin angular momentum, s, and L the orbital
angular momentum vector for the incident proton. The scalar product o • L
is
o	 t =	 L	 , parallel	 (2)
- (L + 1) , antiparallel
where	 parallel / antiparallel	 refers to the relative orientation of a"
with respect to the total angular momentum vector
J = t + (1/2) a .	 (3)
From reference 13, the spin-independent central potentiN for p - 12C
scattering, using a harmonic well charge distribution for C, ' s of the
form
Vc (r) = A  (1 + A 1 r 2 ) exp ( -A2r2 )	 (4)
where Ao	 is complex and A l 	and A2 are real. The usual Thomas-
Fermi form for the spin-orbit contribution is (ref. 11)
V (r) = 1 d V (r)	 (5)
s	 r ar c
which can also be parameterized as
V s
 {r) = Bo
 (1 + 8 1 r z ) exp (-B2r2)(6)
with Bo complex and B 1 and B2 real. Using equations (4) and (5) it
can be shown that the Bi	 (i = 0 -2) are functions of the Ai	 and
that B2 = A2 . In this work, tnese parameters are treated as completely
free.	 Hence, we find that the restrictions imposed on the 	 Bi	 by
equation (5) do not hold and in particular, B2 is A 2 . (see RESULTS).
IF
I
3
zISORIGINAL QUALITY
t?F
Scattering Formalism
From reference 8, the scattering amplitude is
F (0) - f (0) + a e
 n g (0)	 (7)	 -
where n is the unit vector normal to the scattering plane and
	 -
f (0) = f  (0) + (2ik) -1 I exp (2in L )	 (8)
L=0
x J(L + 1) [exp (2 '6L+) - 11
+ L [exp (2i6 L - ) - 1] } P L (cos 0)
g (0) = (2k)-1LI1 [exp (2i6 L+ ) - exp (2i6 1 - )] P1 (cos 0)	 (9)
where fC(0), the coulomb scattering amplitude is
fC (0) = - [ 0/2k sin 2 ( 0/2)] exp [lino - i8 In sin 2 (0/2)]	 (10)
In equations (8) through (10) above, k is the center of mass wave number,
the
	
6	 are the phase shifts (to be determined in the following
sections, the PL(cos 0) are Legendre polynomials, and . PL I (cos e)
are associated Legendre functions of the first kind (see ref. 14).
Additionally, for the coulomb contrihution, we use
6 = Z Te2/hv	 (11)
n o = Arg r (1 + i8)	 (12)
and
n L = nL-1 + tan -1 (s/L)	 (13)
where v is the relativistic relative velocity between the projectile and
target.
The elastic differential cross section is then given by
do - a (0) = i F (0) I 2	 (14)
4
5
E
___ __
ORIGINAL PACE IS
OF POOR QUALITY
Eikonal Phase Shifts
In terms of the eikonal phases, x, the partial wave phase shifts, with
first order corrections (rrf. 11) to account for the spin-orbit
contribution, are
6 Lt
	(1/2) Ix  (b) t (kh ; 1/2) x  (^)}	 (16)
where
kb = L + 1/2
	 (17)
and
x c
	-(m/ k )
 j Vc (r) dz	 (18)
_W
x S (^) = -(m/k) j VS (r) dz	 (19)
6
ORIGINAL PAGE I$
OF POOR QUALITY
W R Phase Shifts
To first order, the WKB phase shifts for outgoing scatt+
given by (ref. 15).
dt = f k [1 - U^(r,L)]1/2 dr
L
r 
s
- j k [1 - U  (r,L)11/2 dr
rC
where
L(L + 1) + ZTe2
UC (r,L)
 
_	 7— ;rE —
	2mr E
	 Er
and
U-
 (r,L) = Veff/E
with
+ = L(L + 1) + I Vc + LVs	 + ZTe2
Veff ^	 V  - (L + 1) Vs
	
r
In the above expressions, 	 E	 is the projectile k!netic; energy ?n the
center of mass system.	 The complex turning points rLt 	and rC are
those values of r for which
1 - U± (r,L) = 0	 (24)
and
1 - U C (r,L) = 0	 (25)
respectively. These were located using a computer search routine based
upon the Newton-Raphson method in the zomplex plane (ref. 16). The inte-
grals for the complex phase shifts in equation ( 20) were then evaluated
numerically.
7
//'^fi n %c ',N .	 ,-_- "'lli ,I e1^I1 :^ E' ,tw L- t^
4F PWR QUALITY
RESULTS
Figure 1 displays theoretical crass sections obtained using
equation (15) with eikonal phase shifts ohtained from equation (16). For
comparison, the experimental results from reference 9 are also displayed.
The polarization of the incident heam was approximately 70 percent
(ref. 9). The opticml potential paremeter values utilized for the anal-
ysis, listed in Table I, were obtained using a gradient -search least-
squares fit to minimize the chi-squared deviation between the theoretical
predictions and experimental results (ref. 17). As displayed in Figure 1,
the agreement between theory and experiment is excellent except for small
angles (®cm < 10').	 This disagreement is not unexpected when using an
eikonal forma
-
Tism (ref. 18). Table II displays the calculated parameters
for the Thomas-Fermi spin-orbit form from equation (5). Comparing Tables I
and II shows the calculated parameter values to be quite different than the
parameter values ohtalaed with the computer search.
Figures 2 and 3 display results utilizing the WKS approximation.
Complex turning points obtained from equation (24) are shown in Fi ure 2.
For these calculations the potential parameters used in equations ^4) and
(5) are those listed in Table I.	 Note that all turning points lie on or
near the real axis. Figure 3 displays the cross sections obtained using
WO phase shifts.
	
Improved anri-ment hetween theory and experiment is
noted for small angles.	 There is, however, significant disagreement for
large angles (05m >55'). This is likel y
 due to the use of the potential
parameters obtained from the nikonal phase shift search. Improvement
should be obtained when the parameter search using WKB phase shifts is con-
ducted, since the previously mentioned eikonr.l shortcoming for small angles
should be corrected.	 A pArem^`er search utilizing WKB phase shifts
will, however, be very t.irn crnsirnind since any change in a parameter
requires that all turning	 airl phas , shifts be re-determined. Upon
completing the parameter sca--r li wi " h the 14 1"o method, the conclusion con-
e^rning the use of the Thomas- r errii form ;or the spin-orbit contribution
should he rechecked since `h^ pr( uOntls conclusion may no longer be valid.
8
CONCLUnING REMARKS
This work has demonstrated that reasonably good agreement between
theory ,tad experiment can be nhtained with only 8 parameters in the nuclear
potential. The use 'of potential parameters, obtained from an eikonal
formalism, for a WKB analysis, is probably not adequte but may serve as a
useful starting point for a WKB parameter search. -Other improvements to
 the theory could be made by discarding the point-charge Coulomb potential
in favor of a more realist;i. extended charge distribution. Coulomb spin-
,	 orbit effects and Pauli blocking effects should also be included. 	 The
latter, however, will increase the number of free parameters from 8 to 18. 	 00
9
ORIGINAL- QUAOF POOR QLITY
REFEREKCES
1.	 Wilson, J. W.: Multiple Scattering of Heavy Ions, Glauber Theory, and
Optical Model. Physics Letters, vol. B52, no. 2, September 1974,
pp. 149-152.
1.	 Wilson, J. W.: Composite Particle Reaction Theory. Ph.D. disserta-
tion, The College of William and Mary in Virginia, June 1975.
3. Wilson, J. W.; and Costner, C. M.: Nucleon and Heavy-Ion Total and
Absorption Cross Section for Selected Nuclei. NASA TN D-8107, 1975.
4. Wilson, J. W.; and Costner, C. M.: High-Energy Heavy Ion Absorption
Cross Section. NASA TM X-73929, 1976.
5. Townsend, L. W.: Optical Model Abrasion Cross Sections for High -
Energy Heavy Ions. NASA TP-1893, 1981.
6. Wilson, J. W.; and Townsend, L. W.: An Optical Model for Composite
Nuclear Scattering. Canadian Journal of Physics, vol. 59, no. 11,
November 1981, pp. 1569-1576.
7. Townsend, L. W.: Harmonic Well Matter Densities and Pauli Correlation
Effects in Heavy-Ion Collisions. NASA TP-2003, 1982.
8. Joachain, Charles J.: Quantum Collision Theory. American Elsevier
Publishing Company, Inc., 1975.
9. Meyer, li. 0.; Schwandt, P.; Moake, G. L.; and Singh ^P. P.: Elastic
Scattering of 200 MeV Polarized Protons from
	 C and the Optical
Potential. Physical Review C, vol. 23, no. 2, February 1981,
pp. 616-622.
10. Blanpied, G. S.; Hoffmann, G. W.; Barlett, M. L.; McGill, J. A.;
Greene, S. J.; Ray, L.; Van Dyck, 0. B.; Amann, J.; and jhiessen,
li. A.: Large Angle Scattering of 0.8 GeV Protons from i C.
Physical Review C, vol. 23, no. 6, June 1981, pp. 2599-2605.
11. Waxman, D.; Wilkin, C.; Germond, J. F.; and Lombard, R. J.: Eikonal
Corrections for Spin-Orbit Potentials. Physical Review C, vol. 24,
no. 2, August 1981, po. 578-587.
12. De Jager, C. W.; De Vries, H.; and De Vries, C.: Nuclear Charge and
rlagnetization-Density-Distribution Parameters from Elastic Electron
Scattering. At. Data P, Nucl. Data Tables, vol. 14, no. 5/6,
Nov/Dec 1974, pp. 479-508.
13. Bidasaria, H. B.; and Townsend, L. W.: Analytic Optical Potentials
for Nucleon-Nucleus and Nucleus-Nucleus Collisions Involving Light
and Medium Nuclei. NASA Tri-83224 , 1.982.
14. Gradshteyn, I. S.; and Rvzhik, 1. N.: Table of Integrals, Series, and
Products. Academic Press, 1955.
10
ki
15. Chan, C. K.; Suebka, P.; and Lu, P: Higher-Order WKB Phase Shifts for
the Heavy-Ion Optical Potential. Physical Review C, vol. 24, no. 5,
November 1981, pp. 2035-2031.
16. James, M. L.; Smith, G. M.; and Wolford, J. C.: Applied Numerical
Methods for Digital Computation with Fortran and CSMP, Second -
Edition. Harper & Row, 1977:
17. Bevington, P. R.: Data Reduction and Error Analysis for the Physical
Sciences. McGraw-Hill Book Co., 1969.
18. Glauber, R.: Asymptotic Diffraction Theory. Proceedings of the
Fifth High Energy Heavy Ion Study, Berkeley, CA, May 18-22, 1981,
Lawrence Berkeley Laboratory Report No. LBL-12652, p. 396.
11
Table I. - Optical Potential Parameters from the Eikonal Formalism Search
Ao = -4.95 - i 48.01 MeV
A l = 0.380 fm-2
A2
 = 0.495 fm -2
Bo = 8.52 + i 1.81 MeV
B 1 = 0.431 fm-2n
B2 = 0.688 fm-2
m
12
13
Table II. - Calculated Thomas-Fen
IBo = 1.1,
E^ 1 =	 1.6
B2 = 0.4
ORIGINAL PAGE
OF POOR QUALITY
•
•
•	 — Theory
• Experiment (ref. 9)
101
100
•
70	 90
10	 30	 50
ecm, degrees
Figure 1. 200 MeV proton-carbon elastic scattering
differential cross sections. The theoretical curve
was obtained using eikonal phase shifts.
a
L
N
L
d
N
CL
LV
E
CD
v
Co
105
103
101
14
Ew
JL
O
Ad
ivO
♦i
ORIGINAL PAGE IS
OF POOR QUALITY
Ull ` li jo lied Aieui6ewl
a^L
CO
C
ev ^,.
A E
C C
cn 'r+
C. C_ cvC
a.. Q
m ♦'' C
Tt 	ti..d
aE 4A
p ^ C
LL.
U O
cat eCv ^°
O ^ CL v O
O 0.
15
}ORIGINAL F IY^..t, f 
OF POOR Q'IMATY
1
I^
105
C
ev
ccL
H
L
d
N	
lU
CL
OL
V
i
	 E
•
•	 --- Theory
• Experiment (ref. 9)
iol
io° .:
CDb	 70	 90 1
101
10	 30	 50
ecm , degrees
Figure 3. 200 MeV proton-carbon elastic scattering
differential cross sections. The theoretical curve
was obtained using WKB phase shifts.
16


Phenomenological optical potential analysis of proton-carbon elastic scattering at 200 MeV

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