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DETERMINATION OF THE COULOMB CORRECTION AND ISOVECTOR TERMS
OF THE NUCLEON-NUCLEUS OPTICAL MODEL POTENTIAL FROM

NEUTRON ELASTIC SCATTERING AT 30.3 AND 40 MEV

BY

Raymond Peter DeVito

 

A DISSERTATION

Submitted to

Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1979

 

ABSTRACT

DETERMINATION OF THE COULOMB CORRECTION AND ISOVECTOR TERMS
OF THE NUCLEON-NUCLEUS OPTICAL MODEL POTENTIAL FROM
NEUTRON ELASTIC SCATTERING AT 30.3 AND 40 MEV
BY

Raymond Peter DeVito

Elastic scattering angular distributions(l§£fi 136%

lab;
for scattering of 30.3 and 40 MeV neutrons from targets of

12 288' 32 40 208

C, 1, 8, Ca and Pb have been measured using the

7Li(p,n)7Be

MSU beam swinger Time of Flight system. The
reaction served as a neutron source. Overall energy resolu-
tion was typically 500-1000 keV FWHM. Relative and normal-
ization errors are both typically <3%. Optical model
potentials are deduced by comparing the observed cross
sections with optical model predictions smeared to account
for the effects of multiple scattering, attenuation, and
finite angular resolution.

Comparison of deduced neutron potentials with exist-
ing proton potentials at the same incident energy for N=Z
nuclei yields directly the Coulomb correction term. The
magnitude and energy dependence of the isovector part of
the nucleon-nucleus potential is deduced by comparison of
neutron and proton potentials for N¢Z nuclei. Comparisons

are made both in terms of volume integrals and potentials

for fixed geometry.

ACKNOWLEDGEMENTS

I would like to thank my thesis advisor, Professor
Sam Austin, for suggesting this project. His encourage-
ment is deeply appreciated. His advice and assistance
were indispensable to the completion of this work.

I thank my wife, Mary Lynn for her patience and
encouragement and for typing this manuscript.

I gratefully acknowledge the assistance of Dr.
Ulrich Berg, Dr. Wim Sterrenburg and Dr. Larry Young
in taking the data.

For their assistance in construction of experimental
apparatus and in running the cyclotron, I thank the
cyclotron technical staff. In particular I thank Dr.
Peter Miller, Mr. Norval Mercer and Mr. Bill Harder.

I would also like to thank the National Science
Foundation and Michigan State University for their

financial support.

ii

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

II.

III.

IV.

INTRODUCTION
A. Nucleon-Nucleus Optical Model
B. Phenomenological Potentials

C. Present Work

NEUTRON SCATTERING APPARATUS
A. Beam and Beam Transport
B. Scattering Apparatus
1. Beam Swinger
2. Neutron Production
3. Scattering Targets
C. Detectors
l. Neutron Detectors
2. Monitor Detector
D. Electronics
1. Time of Flight Signal
2. Pulse Shape Discrimination Signal
3. Light Pulsers
DATA ACQUISITION PROCEDURE
A. Computer and Spectrum Accumulation
B. Time of Flight Spectrum

C. Normalization Procedure

D. Background

DATA REDUCTION
A. Peak Areas

B. Cross Sections

iii

vi

10

14
15
15
18
25
31
31
33
34
34

35
35

40
41
42

45

47

47

C.

D.

V. OPTI

A.

Neutron Detection Efficiency

Experimental Errors

CAL MODEL ANALYSIS

Optical Model Parameter Search Code
Center of Mass Cross Sections
Parameter Search Procedure

Volume Integrals

Coulomb Correction Term

Isovector Term

Imaginary Potential

VI. SUMMARY

APPENDIX

LIST OF

REFERENCES

iv

50

50

55
58
61
65
70
79
84

91

94

132

11.

12.

13.

14.

LIST OF TABLES

30 MeV Run Parameters

40 MeV Run Parameters

Scattering Sample Dimensions

Target Nuclei

Experimental Errors

xz/N of Global Parameter Sets
Optical Model Parameters En=30°3 MeV
Optical Model Parameters En=40.0 MeV
Volume Integrals

J/A for Protons and Neutrons on Calcium
Enucleon =30.3 MeV
J/A for Protons and Neutrons on Calcium

Enucleon =40.0 MeV

J/A for Protons and Neutrons on Lead
Enucleon =30.3 MeV
J/A for Protons and Neutrons on Lead
Enucleon =40.0 MeV

Tabulated Data

21
22
27
28
54
62
66
67

69
77
78

85

86

126

10.

11.
12.
13.
14.

15.

16.

17.

18.

19.

20.

LIST OF FIGURES

Experimental Area of MSU Cyclotron Laboratory
Navy Magnet and Beam Swinger

Lithium Target Chamber

(n,n) Apparatus

7Li(p,n) near 00

Vault Electronics

Data Room Electronics

Neutron-Gamma Pulse Shape Discrimination
Spectrum

TOF Spectra 40 MeV, Si, 600 Target In, Target
Out, Subtracted

Monitor Spectrum 40 MeV Lithium Target In, Li
Target Out

Neutron Detector Efficiency
Flow Chart for GIBSCAT
Search Procedure

Energy Dependence of Calcium Volume Integrals,
Best Fit Values

Energy Dependence of Calcium Real Potential
Strength with Fixed Geometry

Energy Dependence of Lead Real Potential
Strength with Fixed Geometry

Energy Dependence of Lead Volume Integrals,
Best Fit Values

Energy Dependence of Calcium Imaginary
Potential Strength with Fixed Geometry

Energy Dependence of Lead Imaginary Potential
Strength with Fixed Geometry

TOF Spectrum 40°, 3, 30 MeV

vi

16
17
23
24
26
37

38

39

43

44
51
57
63

73

74

81

83

89

90
95

21.
22.
23.
24.
25.
26.
27.

28.

29.
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.

40.

41.

42.

43.

TOP

TOF

TOF Spectrum

TOF

TOF

TOF

Spectrum

Spectrum

Spectrum
Spectrum

Spectrum

Si, 30 MeV

Ca, 30 MeV
Pb, 30 MeV
S, 40 MeV
Ca, 40 MeV

Pb, 40 MeV

Monitor TOF Spectrum 30 MeV

TOF Spectrum of Neutron Source, 7Li(p,n)

at 00

Laboratory
Laboratory
Laboratory
Laboratory
Laboratory
Laboratory
Laboratory
Laboratory

Laboratory

Center of Mass Cross
Center of Mass Cross

Center of Mass Cross

30 MeV

Center of Mass Cross
30 MeV

40Ca,

Center
30 MeV

Center
ZOBPb,

Cross

Cross

CI‘OSS

Cross

Cross

Cross

Cross

Cross

Cross

of Mass Cross

of Mass Cross
30 MeV

Section, 288'

325'

40

Section,

Section,

Section, 208

Section, 12

Section, 28

Section, 32

Section, 40

Section, 208

Section,
Section,

Section,

Section,

Section,

Section,

vii

ll

Ca,

C:
Si,
Sr
Ca,

Pb,

30 MeV
30 MeV

30 MeV

Pb, 30 MeV

40 MeV
40 MeV
40 MeV
40 MeV

40 MeV

288i, 30 MeV
32

S, 30 MeV

40Ca,

Best Fit
Fixed Geometry

208

Best Fit Pb,

Fixed Geometry

96
97
98
99
100
101
102

103
104
105
106
107
108
109
110
111
112
113
114

115

116

117

118

44.
45.
46.

47.

48.

49.

50.

Center of Mass

Center of Mass

Center of Mass

Center of Mass

40 MeV

Center of Mass

40Ca, 40 MeV

Center of Mass

40 MeV

of Mass
40 MeV

Center

208Pb,

Cross

Cross

Cross

Cross

Cross

Cross

Cross

Section,
Section,
Section,

Section,

Section,

Section,

Section,

viii

12C, 40 MeV

288i, 40 MeV
32$, 40 MeV

Best Fit 40C

a:

Fixed Geometry

208

Best Fit Pb,

Fixed Geometry

119
120

121

122

123

124

125

I. Introduction

A. Nuclear Optical Model

The nucleon-nucleus interaction is a complex many body
process that can not be solved exactly. In order to begin
to understand the physics of the atomic nucleus, approxima-
tions and simplifications must be employed. As more and
more information and experience is gained these simplifi-
cations and approximations will possibly lead us toward a
more accurate and complete knowledge of the nucleus. It is
the aim of this present work to add to that experience.

Nucleons incident upon an atomic nucleus may be scatter-
ed elastically, leaving the nucleus unchanged except for
some translational energy, or may react with the nucleus,
altering its internal structure in some way. Thus the in-
cident wave packet may be scattered or absorbed. In optics,
light incident on some medium may undergo refraction and
absorption. This process for light is described by the
complex index of refraction of the medium. The actual micro-
sc0pic interaction of the incident photons with the material
is very complicated. In describing the nucleon scattering,
we can think of the incident particle being scattered by a
complex potential well. The imaginary part would account
for all nonelastic reactions. By analogy to the case in optics

we call this potential the Optical Model Potential (0MP).

This idea was applied semi-classically by Fernbach et a1.
(Fe49) in 1949. They treated the scattering and absorption
of 90 MeV neutrons by a range of nuclei. The elastic and
inelastic total cross sections could be accounted for by their
process. Later, in 1952, LeLevier and Saxon (Le52) did a full
quantum mechanical calculation for 17 MeV protons on Aluminum.
In 1954 Feshbach (Fe54) showed that the energy averaged varia-
tion of low energy neutron cross sections with atomic weight
could be represented by a complex neutron-nucleus potential.
With the advent of electronic computers, wave functions could
easily be calculated from the Schroedinger equation for ar-
bitrary potentials. As the precision of the data increased the
model was refined to a point where it can account for differ-
ential and reaction cross sections as well as polarization
to a high degree of accuracy.

The study of the Nuclear Optical Model involves two
categories of work. One is phenomenological, whereby one
empirically determines the parameters of an 0MP by fitting
experimental elastic scattering data. The other is theoret-
ical in nature and involves computing the effective potential
from considerations of the many-body problem(Je77, Br77).

Aside from its intrinsic interest the study of the OMP
is motivated by the important role it plays in the interpre-
tation of many nuclear reactions (Au70). The calculated
incident and outgoing waves in a reaction undergo reflections
and absorptions due to the potential determined by elastic

scattering.

3
Consider the system comprised of A+l nucleons (Pe74),
where there is a nucleon incident on a target nucleus, de-
scribed by a wave function W. The wave function for a
state i of the target nucleus is described by ¢i(rl,...,rA)
with corresponding energy Ei' The variables rk indicate
position, spin and isospin of the nucleons. We expand using

the complete orthonormal set oi with amplitudes Xi'
W = E ¢i(rl,...,rA) xi(r0). (I-l)
The Schroedinger equation that describes this system is

‘W = E‘P (I-2)
where

34: HA(rl""’rA) + T + V(ro,rl,...,rA). (I-3)

0
HA is the Hamiltonian for the A particles of the target

nucleus, T is the kinetic energy for the incident nucleon

0
while V is the potential energy of that nucleon in the

field of the target nucleus. We note that ¢ satisfies

= €.¢. . (I-4)

Thus using the orthonormal properties of the set ¢i we

obtain a set of coupled equations for the amplitudes

(T0+Vii+€i-E) Xi = -Z Vijxj (I-S)
ii‘j
where
Vij = (¢ilv¢j)r
*
v.. = v.. . (I-6)

1] 31

We define the matrices

X1
K = X2 (I-7)
X3
and
y- :

o (V01'V02"'°) '

The matrix operator H is defined by

Eij = Toéij + vij + eiaij 1,3¢o . (I-8)

In matrix notation equation I-S becomes

Wo+%o-mm= 22
f (I-9)
(g — E)_)$ = —y_0 .

Solving formally we find

5 = 1 Yo (1-10)

where E(+)=E+in with n++0. Within the Green's function

in specifies that only outgoing waves are present in Xi
for i>0. Using equation I-10 in equation I-9 we obtain

the one body Schroedinger equation

+ _ _
To+voo+v 1 yo-E xo—O . (I 11)

_0_____.___.

E(+)-H

5

We therefore obtain the "generalized Optical Model Potential"

(I-12)

The potentia1°V is not the optical model potential,
it is the exact potential operator for elastic scattering.
The OMP is the simple effective potential that replaces
the true potential operatorov. With an appropriate choice
of replacement for‘V'the Schroedinger equation becomes more
.simply solvable. The new wave function is not exactly x0
since the replacement potential does not exactly represent
WK In elastic scattering the details of the wavefunction
are not important, but rather the asympototic behavior of
x0 is important, i.e. the potentials must be phase equivalent.
The choice of an OMP is guided by intuitive physical
ideas, but must incorporate some of the properties that can
be deduced from equation I-12. The potential operator 3/
is not Hermitian, due to the imaginary term in the Green's
function. The second term in I-12 is responsible for the
imaginary part of the OMP, but it also contributes to the
real part. This term is nonlocal and explicitly energy
dependent. The spacial nonlocality in this term arises
physically by removal of flux from the entrance channel
due to 23. This flux can propagate in reaction channels,
then some flux will reappear in the entrance channel at some
other point by the V interaction. The term V also yields

0 00
a nonlocal potential due to explicit exchange forces in the

6
two body interaction and from antisymmetrization. The spa-
tial nonlocality of the generalized OMP appears as a momen-
tum dependence if a local replacement potential is used.
It is not possible to distinguish between the explicit energy
dependence and the energy dependence due to the spatial
nonlocality of the potential operator whenOV'is replaced
by a local potential.

Recent theoretical analyses have yielded good calcul-
ations of the basic properties of the nucleon-nucleus OMP
(Je77, Ma79, Br77, Br78) starting from the nucleon-nucleon
interaction. Within the framework of Brueckners theory a
density dependent potential is derived fromma two nucleon
interaction e.g. Reid hard core (Re68) or Hamada-Johnston
(Ha62). The simple radial shape of the phenomenological OMP
suggests that it is mainly dependent on the matter density
of the nucleus. Thus it is feasible to study the OMP in a
finite nucleus by studying nuclear matter at various densi—
ties and applying a Local Density Approximation (LDA). A
simple LDA, one that assumes that the OMP at a given loca-
tion in the nucleus is equal to the same value as in a uni-
form medium with the same local density, is able to yield
semi-quantitative conclusions on the global properties of
the OMP: depth, energy dependence, non-locality, small
components and main features of the form factors. Good agree-
ment between volume integrals calculated using a sinple IDA and those
observed experimentally is achieved. However root-mean-square

(rms) radii are in general too small. An improved LDA,

7
which takes into account the finite range of the effective
interaction yields improved agreement between calculated
and phenomenological rms radii without affecting volume
integrals.

Theoretical OMP are able to render properties of the
observed average OMP with an accuracy of about 10%. Agree-
ment between the calculated and observed imaginary potentials
is worse, about 30%, due to the inability of nuclear matter

based theories to take into account shell effects.

B. Phenomenological Optical Model Potentials

There is no apriori reason to believe that the general-
ized OMP admits any equivalent simple local potentials of the
type typically used to analyze data. But, from the great body
of data analyzed over the past 25 years theneexists a simple
potential which describes very well most of the features of
elastic scattering of nucleons and other projectiles. It is
the purpose of OM analysis to determine the various terms of
this potential and to study their behavior as a function of
energy and target nucleus.

A typical phenomenological OMP is a local multi-para-

meter potential usually written as

-U(E,r) = V(E,r) + iW(E,r) (I-13)

In the present analysis, the real part of the potential

is written as 2

l CL

.- L 1 f‘Xso)
v<e,r)=vc(r)+V(n)f(xR)—vso(o-i) mnc r r

(I—l4)

8
The form factor f(xi) is taken to be.a Woods-Saxon shape

defined as

-1

f(xi)=(l+exi) ; xi=(r-riAl/3)/ai .

This shape is chosen for convenience and because nuclear
matter densities are closely described by such.florms (Ne70).
The first term of equation I-14, VC(r), is the Coulomb

potential due to a uniformly charged sphere of radius RC,

2 2 2
(Zze /2RC){3-r /RC} for r__<=RC

C (Zzez/r) for r__>;_RC (I-15)

where RC=rCA1/3, Z is the target charge and z is the projectile

charge. This term vanishes for the neutron potential since
the neutron charge is zero.

The last term in equation I-14 is the spin-orbit poten-
tial. The explicit Thomas form of the potential was chosen
by analogy to the atomic spin-orbit potential and has been
substantiated experimentally.

The central real term from equation I-14 can be written,

following the suggestion of Lane (La62), as
V(E) = V0(E) + 4V1(E) to? + AV . (I-l6)
A. C

Here V0(E) is the isoscalar part of the potential and V1(E)
the isovector part; t and T are the isospins of the incident
nucleon and target, respectively. The isovector interaction
t-T, splits the central part of the potential into diagonal

terms which are responsible for proton and neutron scattering

9
and a non-diagonal term that mediates the (p,n) or (n,p)
quasi-elastic scattering. For nucleon scattering we can

-> + _
evaluate t-T to give
V(E) = V0(E) i eVl(E) + AVC . (I-l7)

Where e=(N—Z)/A represents the nuclear asymmetry. The +
sign applies for protons and the - sign for neutrons.
The isovector strength V1(E) comes about because of the
prOperties of nucleon-nucleon interactions, Vpp=vnn#vpn.
This effect comes about because the Pauli exclusion
principle restricts states between like nucleons but not
states between unlike nucleons. The term AVC is the Coulomb

correction term, first suggested by Lane (La57), and is

usually paramerized by
AVC=BzZ/Al/3 .

In addition to the Coulomb potential (equation I-lS) the
charge of the nucleus has the effect of reducing the mean
kinetic energy of incident charged particles interacting
with the nucleus. This effect is accounted for by adding
to the proton potential the Coulomb correction term.

The imaginary part of the OMP is not expected to have
the same shape as the real part. Absorption takes place
throughout the nucleus, but especially at low energies
various factors such as the Pauli principle and surface

excitations should enhance surface contributions. As the

10

incident energy increases, both these effects should decrease
causing the absorption to be distributed more uniformly
throughout the nucleus. OM analysis confirms this . In

the present analysis a Woods-Saxon form factor together

with a derivative Woods-Saxon form factor are used. As
energy increases the strength of the surface potential de-
creases and that of the volume absorption increases. We
therefore write the imaginary part of the phenomenological
potential as

W(E,r)=WV(E)f(xV)-4WD(E)Q__f(xD) . (I-l8)
de

Just as for the real part, the imaginary part can be
parameterized by Coulomb correction, isovector and spin-orbit
terms. Jeukenne et a1. (Je77) have calculated the imaginary
Coulomb correction and isowaxbr terms starting from the
Brueckner-Hartree-Fock approximation and Reid's hard core
nucleon-nucleon interaction. The imaginary spin-orbit term
is calculated by Brieva and Rook (Br78) to be substantially
smaller than the real spin-orbit term (WSO/VSO~-0.05 for

20 MeV nucleons) and is set equal to zero in the present

analysis.

C. Present Work

A large collection of precise proton scattering data
already exists in the literature. There is a lack however,
of precision neutron data, expecially for incident neutron
energies greater than 15 MeV. The extensive neutron scatter-

ing program of Rapaport et al. at Ohio University has con-

ll
tributed good neutron data at 11, 20 and 26 MeV for a wide
range of nuclei (Ra77, Fe77, Ra78). Most of the best
proton scattering data.an3for incident proton energies
greater than 26 MeV, notably at 30.3 MeV (Ri64) and at
40 MeV (8166). The isovector strength of the nucleon-
nucleus OMP can be extracted by comparison of proton and
neutron potentials. There are two ways to compare these
potentials, atthe same energies or at energies shifted to
account for the Coulomb correction. The former method
yields the Coulomb correction from comparison of N=Z nuclei
while the latter method requires either prior knowledge of
the Coulomb correction or the measurement of angular distri-
butions over a range of energies. The Coulomb correction is
not known very precisely and due to the amount of cyclotron
time required to complete one angular distribution, measuring
several angular distributions for each nucleus was impracti-
cal. Thus we have measured neutron elastic scattering angu-
lar distributions at incident energies of 30.3 and 40 MeV

on targets of 12C (40 MeV only), 28Si, 328, 40Ca, 208Pb,

and 2098i. Comparison between N=Z nuclei yields the Coulomb
correction term and then comparison between N¢Z nuclei yields
the isovector term.

Since neutrons have no net charge and all particle
accelerators use electromagnetic forces, no direct beam of
monoenergetic neutrons exists. To produce a monoenergetic

neutron flux at our scattering target a charge exchange re-

action is used. A proton beam accelerated by the MSU cyclotron

12
strikes a target of 7Li and the reaction 7Li(p,n)7Be is
used as the neutron source. This reaction is strongly for-
ward peaked which reduces background. High energy neutrons
are produced that are well separated in energy from
neutrons produced by other reactions. To produce a neutron
flux large enough to complete an angular distribution measure-
ment in about 1-2 days an energy loss due to Li target thick-
ness of about 500 keV was used.

To achieve a large enough counting rate large scatter—
ing samples (~l mole) of cylindrical geometry are used.

Since the neutrons will not interact with the Coulomb field
within the target, a large scattering sample could be
tolerated.

To detect the scattered neutrons another nuclear inter-
action must take place in the form of (n,p) scattering within
an organic scintillator. The energy of the neutron cannot
be directly determined since directional information on the
(n,p) scattering angle is not available. The neutron velocity
can be determined however, by measuring its time-of-flight
(TOF) over a fixed flight path. Once the velocity is known
the energy can be calculated.

One advantage of the neutron scattering measurements
compared to charged particle work is that absolute cross
sections can be measured with little uncertainty. After
measuring the sequence: source reaction-scattering-detection
reaction, one can remove the scattering sample and look at

O

0 to measure the sequence: source reaction-detection reaction.

13
By comparison of these two measurements the majority of the
uncertainty in the 7Li(p,n) cross section, Li target thick-
ness, detector efficiency, neutron attenuation along the
flight path and solid angle of the detector are removed.
The beam swinger built at MSU and used in these

experiments simplified the shkfldfimg and detector position-
ing requirements. The incident proton beam is rotated to
vary the scattering angle instead of moving the neutron
detector. For long flight paths this is an important ad-

vantage.

II. Neutron Scattering Apparatus

A. Beam and Beam Transport

A beam of nearly mmxxmenxnic neutrons is produced
by bombarding a thin foil of metallic 7L1. The 7Li(p,n)7Be
(9.5.) and 7Li(p,n)7Be(0.429 MeV) reactions at zero degrees
are used to generate the neutron beam. The scattering tar-
get is located on the swinger axis thereby allowing
neutron elastic scattering angular distributions to be
measured by rotating the beam swinger.

The particle beams produced by the Michigan State
University Cyclotron are very well suited for time-of-
flight (TOF) experiments. The beam is\sharp1y bunched in
time, with a typical burst width of =300ps and a burst
interval between 50 and 67 ns depending upon particle energy.
For the experiments described herein the cyclotron was used
to produce 32 and 42 MeV proton beams. The energy resolution
AE/E of the cyclotron beam is 510.3; compared to the overall
energy resolution of this experiment, this energy spread is
negligible.

After extraction from the cyclotron, the transport of
the beam to the experimental area is controlled by a series
of bending magnets, focusing magnets (quadrupoles)and position
defining slits. The beam is defined spatially by slits in Boxes 1,
3 and 4 (see Figure 1). After being focused at slit 3 by a

quadrupole doublet Ql and 02, the beam passes through slit 4

and undeflected through M3 into the neutron TOF beam line.

14

15

In vault 5 the beam is refocused at a point just before the
Navy magnet by a quadrupole triplet located near door 5. Beam position
is checked here by using a TV monitored scintillator. The
beam is centered through the quads by requiring that there
is only focusing and no net translation when the setting of
Q7 and Q8 are changed.

The layout of the beam swinger is shown in Figure 2.
The beam is deflected through 900 by the Navy magnet prior
to entering the swinger. In addition to the focus at the
entrance to the Navy magnet, the system has a focus near

the entrance to the swinger and at the target position.

B. Scattering Apparatus
1) Beam Swinger
The swinger consists of two magnets capable of rotating
about the incident beam axis (Bh77). The beam is first
deflected -450 and then deflected +135O with the net effect
that the beam is perpendicular to its original direction.
The swinger magnets, of fully annealed 1010 steel,
are of a H design with a bending radius of 76 cm. The
poles are 10.2 cm wide with a 3.2 cm gap. A 36 minute
taper on each pole tip makes the swinger magnets double focus-
ing (n=%). The overall magnification of the swinger system
is about one. The current carrying coils are flat pancakes,
three to a pole, of 1.2 cm square hollow copper conductor
wrapped in fiberglass and vacumn potted in epoxy. Current

and power consumption at a field of 1.4 Tesla is 450 Amps

16

 

 

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18

and 22 Kilowatts. An additional power supply was connected
to the 1350 magnet to balance the two swinger magnets.

The swinger's usable angular range was from 00 to
1600 for the neutron scattering experiments. In general
the angular distributions were measured out to 130°, beyond
which point the cross sections become too small to measure
in the available time.

The shielding walls of the swinger vault were stacked
concrete blocks 1.8 meters thick. The wall, 2.15 meters
from the scattering target, provides good isolation of the
detectors from the high neutron and gamma-ray flux in the
swinger vault. A hole along the target to the detector
flight path allows transmission of the scattered neutrons.
This hole is filled with steel bars and lead bricks except
for an opening just sufficient to allow both detectors an
unobstructed view of the scattering target. A 300 kg iron
shadow bar is positioned so as to attenuate the direct flux

of neutrons to the detector from the 7Li(p,n) reaction.

2. Neutron Production

To obtain a mean energy of 30.3 and 40.0 MeV for the
neutron beam, proton beams of 32.2 and 41.9 MeV respectively
were used. In practice proton beam energies were slightly
different for each run on the cyclotron. The mean neutron
energies, along with the spread due to production target
thickness are presented in Table 1 for 30 MeV and Table 2
for 40.0 MeV. The effect on the cross sections of the energy

variation and the energy spread due to the production target

19

thickness was estimated by OM calculation to be <0.l%.

The lithium targets used in the (p,n) reaction were
made from high purity lithium enriched to 99.99% 7Li.
The lithium was pressed into disks about 1 cmin diameter
by a hydraulic press. The target thicknesses used in this
experiment were 0.64 and 0.76 mm at 32 and 42 MeV proton
energy respectively, corresponding to a total energy spread
due to energy loss in the Li target of 500 keV. This energy
spread had to be tolerated to obtain count rates sufficiently
high to make the experiment practical. The contribution
from the 7Li(p,n)7Be(0.429 MeV) reaction was included in
the neutron elastic scattering peak. The first excited state
contribution was about 25% and 30% at 32 and 42 MeV proton
energy respectively. The second excited state of 7Be is at
4.57 MeV excitation, well removed from the high energy elastic
peak. The neutron yield from three body final states (Q=
-3.24 MeV) is measured to be very small in the energy range
of interest (see Figure 28). The 7Li(p,n)7Be (g.s.) reaction
has a Q value of -l.644 MeV.

Following the Li target is a 0.127 mm thick aluminum
foil. The 27Al(p,n) reaction has a Q value of -5.592 MeV,
thus contributing no background at the elastic peak. This
aluninum foil isolates the vacumn chamber from a water
faraday cup, consisting of an aluminum chamber through

which distilled water is constantly pumped. This provides

cooling to the lithium target as well as a beam dump with
I

20
a large negative Q value (Qpn=-l6.2 MeV) for neutron
production. The high energy neutron flux from the 0.2%

18 __ 17 __ -
O (Qpn— 2.4 MeV) and 0.04% O (Qpn— 3.5 MeV) con

of
tributed negligibly to the measured spectra.
The neutron flux at the scattering target could be
adjusted by varing the distance between the Li target and
the scattering sample (see figures 3 and 4). This was done
by changing the length of the plexiglass pipe that makes up
part of the vacumn chamber. The Li target to scattering
sample distance (d) could be set at 24.4, 18.4, or 11.0
cm. This range allowed beam intensities to vary by 1:1.76:
4.9. The largest practical distance was chosen for any given
angular range since the closer is the neutron source to
the scatterer, the larger is the angle subtended by the
sample and thus the larger the finite angle correction to
be made. The angular ranges are tabulated in Tables 1 and 2.
Two proton beam collimators were machined from graphite
since 12C has a large negative Q value for neutron production
(Qpn=-18.l MeV). The high energy neutron flux from the

1.11% 13

C (Qpn=-3.0 MeV) was neglible. Beam current on
the collimators was monitored during the experiment and
was usually negligable (<1 nA). In all cases the collimator
current was kept to <0.1% of the target current.

The mean scattering geometry for the neutron scattering
is determined by the position of the Li target, the posi-

tion of the scattering sample and the position of the detec-

tor. The detector is never moved, the sample was repositioned

21

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25
to better than 1.0 mm and the Li target was rigidly held to
the swinger. Thus the mean scattering angle was well known,
and did not depend on the angle that the proton beam hit
the Li target. The cross section for neutrons scattered
in the direction of the target however does depend on the
angle the proton beam hits the Li target. The 7Li(p,n)
cross section is forward peaked with a slight flat region
around 00. The proton beam was collimated to 1.2.500 FW for
C, Si, 8 and Ca and 11.00 FW for Pb. Figure 5 shows the

7

angular distribution of Li(p,n)7Be(g.s.+0.429 MeV)

for scattering angles from 00 to 150.

3. Scattering Targets

All scattering targets used were formed in solid right
circular cylinders. The dimensions, mass, chemical purity
and isotopic enrichment of these targets are listed in Table 3.

The best shape for each scattering target was determined
by computer calculation of multiple scattering and finite
angle effects. The sample must have a symmetry axis perpen-
dicular to the beam direction. The multiple scattering effects
are reduced as the target is elongated, but then the finite
angle effects are increased As the target is made more
mdunflcal finite angle effects are reduced but multiple scat-
tering is increased. The best target shape was calculated
for the various targets. Small variations about the best
shape caused little increase in finite geometry effects.
The actual target shape was not necessarily the calculated

best shape but depended upon what materials were available.

26

 

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27

Table 3. Scattering Sample Dimensions

 

 

Dimensions Chemical

Height x diam. Purity Mass
Sample (cm) (%) (gm)
12

c 3.40 x 2.64 98.+ 33.077

288i 3.69 x 2.36 99.+ 37.777
288i 7.09 x 2.36 99.+ 72.559
325 2.86 x 3.17 99.9 42.417
40Ca 4.36 x 1.90 98.0(b) 18.697
4OCa 4.80 x 2.30 98.0(b) 27.200
208Pb 3.90 x 2.40 99.7+ 200.640
a) Borrowed from Darrell Drake,

b)

Laboratory

Isotopic
Enrichment

(%)
natural(98.89)
natural(92.2)
natural(92.2)
natural(95.0)
natural(96.94)
natural(96.94)

98.69(a)

Los Alamos Scientific

Includes oxygen impurity measured as described in

text page 29.

28

Table 4. Target Nuclei

 

 

lst excited state 2nd excited
Nuclei Energy(MeV) Spinparity state (MeV) Remarks
12c 4.44 2+ 7.66 8(2:)=0.60(a)
2851 1.78 2* 4.62 B(21)=0.40(a)
32S 2.23 2* 3.78 8(21)=0.37(a)
40Ca 3.35 0+ 3.74 =spherical
208Pb 2.61 3- 3.20 =spherical

a) reference (St65)

29

The 208Pb target was prefabricated by Los Alamos to
our specifications. The Si target was received as a cylinder
of appropriate diameter and only needed to be cut by diamond
saw to the desired length. The C and Ca targets were machin-
ed on a lathe from ingots.

The calcium targets were sealed inside thin aluminum
cans as calcium is reactive in air. The cans were fabri-
cated from 0.05 mm thick foil held together with epoxy. For
each target can an identical empty can was fabricated from
the same size andxmflght pieces of aluminum. The weight of
the target can and the empty can were the same to <1%.

An estimate of the oxygen contamination in the calcium
target was made by measuring the neutron scattering from
the sample with sufficient energy resolution to separate
the neutron groups elastically scattered from 40Ca and 16O
at a few angles around 70°. An estimate of the cross section
ratio combined with the ratio of scattered neutrons indicated

the 16

O contamination to be 2%:l%.

The sulfur target fabrication was more difficult than
the others. Molten sulfur was poured in layers into a pyrex
beaker. The layers were thin enough so the solidification
could be monitored to ensure no holes were being formed in
the target. When the desired amount of material was solidi-
fied the glass was heated just enough to melt the outer sur-
face of the sulfur target and then cooled. Best results

were achieved with fast cooling in a water bath, with care

taken to be sure no water splashed into the beaker. The

30
glass beaker usually had to be broken away from the sulfur
target. Once the technique was mastered several targets
were produced, all of which appeared to be of the same
quality. All but two were then broken open and checked for
voids. No voids were discovered. The target used in the
experiment was broken open after the experiment was complete
and no voids were discovered. The outer shell of sulfur
was hard and did not rub off. Due to these farication pro-
cedures, the sulfur target was the only target where the
diameter exceeded the height.

The scattering targets were all mounted with the sym-
metry axis along the swinger rotation axis. The targets
rested on thin aluminum trays supported by thin stainless
steel rods. The trays were made with the smallest amount
of material that still gave rigid support. The rod, tray
and target assembly was then supported from beneath by one
of four rotatable arms of the target changer. These four
arms allowed three targets to be mounted for one run. The
fourth position was taken by a blank target, i.e. a tray
and rod only. A Geneva device was used to accurately rotate
the target assembly thereby changing the scattering target.
The targets were aligned by using a survey telescope and
survey markings on the wall and swinger. Due to the magni-
fication of the telescope, alignment with the survey mark-
ings could be done to within.1.um. Accuracy of the survey
markings was checked and found to be consistant with the

swinger rotation axis. The Geneva device was rotated in

31

one direction only. The reproducability checked to within

the accuracy of the survey scope.

C. Detectors

l. Neutron detectors

Unlike charged particle detection, neutrons are not
detected directly, but rather, the recoil of a charged parti-
cle is detected if the neutron undergoes an appropriate
nuclear scattering within the detector. The charged recoil
causes the detector material to scintillate and this light
is detected by a photomultiplier.

We used two 12.7 cm diameter x 7.62 cm thick NE213
liquid organic scintillator detectors, produced by Nuclear

3 of

Enterprises. These detectors each contain 965 cm
scintillator. The liquid is encapsulated in a glass cylinder
painted with white reflective paint. Each has a teflon ex-
pansion chamber to relieve pressure caused by temperature
variations. The light is carried to the photo multiplier by
a conical light pipe 7 cm thick, one end 12.7 cm diameter
and the other 5.08 cm diameter. The light pipe is coupled
to the scintillator and the photomultiplier by Dow Corning
Optical Silicon grease.

Either a RCA 8575 or a RCA 8850 phototube was used in
an Ortec 265 phototube base. These phototubes contain 12
dynodes and the base provides signals from the 9th dynode

and the anode. The amplified signal from the 9th dynode pro-

vides a measure of the total light produced by an event.

32

The scintillator Ne213 was chosen because it allows one
to distinguish between events caused by neutrons and gamma rays.
This discrimination is possible because the recoils for neutrons
are mostly protons while the recoils for gamma rays are electrons.
The shape of the light pulse for protons and electrons is differ-
ent and can be distinguished. This Pulse—Shape Discrimination
(P80) is very important in eliminating ganma ray background.

The photomultiplier assembly for each detector was wrap-
ped in several layers of magnetic shielding. This magnetic
shielding was necessary because fringe fields from the
Superconducting Cyclotron magnet being built at MSU were
sometimes present during experimental runs.

To have a continuous check on the gain of the detectors a pul-
sed light-emitting—dynode (LED) giving a constant number of
photons was fed into the photomultiplier during data collection.
The position of this LED peak thus gave an on-line gain stabil-
ity check.

The detector and associated electronics provide a timing
pulse with a finite uncertainty. The detector thickness
provides a time spread due to the uncertainty in where the
event took place in the detector. The transit time for the
7.62 cm thick detector is 0.9 ns for 40 MeV neutrons and
1.0 ns for 30 MeV neutrons. The energy uncertainty due to
time uncertainty is given by the nonrelativistic equation

AE=(.0277)E3/2At (II-l)
d

where E is in MeV, At is in nsec, and d is in meters.

33

The energy resolution for each run is tabulated in Table l

and Table 2. The time resolution At for all runs was about

1 ns for y-rays from the production target.

2. Monitor Detector

A detector is mounted rigidly to the swinger to monitor
the neutron flux from the 7Li(p,n) reaction at a scattering
angle ranging from 210 to 240 depending on production target
to scattering target distance. A flight path of 140.0 cm
provided sufficient energy resolution to separate the
7Li(p,n)(g.s.+0.429)7Be neutrons from the neutrons produced
by 27Al(p,n) and other background sources.

The detector consisted of a cylinder of NE102 plastic
scintillator 2.54 cm diameter by 1.9 cm height coupled direct-
ly to a RCA 8575 phototube and Ortec 265 base by Dow Corning
Optical grease. The detector was wrapped in several layers
of magnetic shielding and then mounted inside a soft iron
cylinder with 1.75 cm thick walls. Since this detector was
rotated in the fringe field from the Navy magnet located 4 m
away, extra magnetic shielding was necessary.

A stability check of the monitor detector gain was made
by measuring the Compton edge for gamma rays from 228Th at
several swinger angles with full current in the Navy magnet.
The detector was stable to better than 1%.

Lead shielding 10.0 cm thick was placed between the

detector and the source. The Pb attenuates gamma rays, es-

pecially those of low energy (<1 MeV) more than high energy

34
neutrons, thus reducing the overall count rate to a manage-
able level.

The anode signal was fed into a constant fraction dis—
criminator (CFD) whose output was used for the timing signal
(see Figure 6). The dynode signal was fed into a preamp and
then to a Spectroscopy Amplifier. The NE102 does not produce
PSD information therefore a Spectroscopy Amp was used for its
convenience. The monitor detector also had an LED pulser
fed directly to the phototube to monitor possible gain shifts.

Due to the mounting position of the monitor it had to be
removed and repositioned on the opposite side of the swinger

when the swinger was rotated through 90°.

D. Electronics

1. Time of Flight Signal

The anode of the photomultiplier produces a fast nega-
tive voltage pulse when a neutron event occurs in the scin—

tillator. This pulse is fed into a CFD from which a fast

negative pulse is produced that is timed from the point where
the leading edge reaches 50% of the maximum pulse height (see
Figure 6). This method provides minimal variation of trigger-
ing time for pulses covering a wide dynamic range. The neg-
ative output of the CFD was used to start a Time to Amplitude
Converter (TAC) (see Figure 7). The stop signal originates
from the zero crossing of the Cyclotron RF, which is detected
by a Zero Crossing Discriminator. The TAC provides a voltage
pulse whose height is prOportional to the time between start

and stOp. Since we start with the event pulse and stop with

35
the cyclotron pulse we get a time spectra that gives a "normal"

spectrum with increasing energy going from left to right.

2. Pulse Shape Discrimination Signal

The decay of the light pulse from electrons, protons
or heavier charged particles is different in a way that
allows us to distinguish these events. Proton (neutron)
events have a longer decay time than electron (gamma) events.

A signal from the fast negative output of the CFD is
delayed for about lusec either by a long length of cable
or a Gate and Delay generator and then starts a TAC (see
Figure 6) - The double delay line (DDL) output from the amplifier is
fed into a Tfihfixm;8ingle Channel Analyser (TSCA) run in
the zero crossing mode. A signal timed from the zero cross-
ing is then sent to stop the TAC.

The zero crossing of the DDL output from neutron events
will be delayed longer relative to that for electrons because
of the longer decay time. A typical PSD spectrum is shown in
Figure 8. A gate can be set around the neutron events so
only neutron events are recorded in the TOP spectra. This
PSD system is based on the technique of Alexander and Goulding
(A161).

3. Light Pulsers

A temperature compensating current pump to drive an LED
was built, based on the design by Hagen and Eklund (Ha76).
This LED pulser gave a stable source of photons to act as a

gain drift.monitor. By comparison with a monoenergetic gamma

36
ray source the LED light signal was found to drift <l% over
a temperature range of :100 C from room temperature and over
many hours. The LED was mounted on the current pump circuit
board next to the thermistor that gives the device its
temperature stability. A small light pipe consisting of
a bundle of fiber optics was then used to transport the light
to the photocathode. The light pipe of the neutron detector
prevented direct access to the photocathode so the LED pulse
was directed into the detector light pipe. This caused the
signal to be greatly attenuated but it was stilltmabhein the

low light region of the spectra.

37

 

 

 

 

LED
drive

 

 

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drive

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38

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

time #1 time #2 time MON ‘
start ‘TOF start TOF start [TOE
‘TAC TAC TAC
stop #1 stop #2 stop MON
Delay
light #1
PSD #1 Octal PDP
TOP *1 ADC 11/45
computer
light #2
PSD #2
TOP #2
light MON
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Data Room Electronics

Figure 7.

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III. Data Acquisition Procedure

A. Computer and Spectra Accumulation

The TAC, light and PSD signals from both neutron detec-
tors along with TAC and light signal from the monitor detec-
tor are fed into an EG&G AD811 octal Amplitude to Digital
Converter (ADC) as shown in Figure 7. The signals of each
detector arrive at the ADC in coincidence with a strobe
signal generated by the logic output of the relevent CFD.
This Strobe activates the ADC which converts the voltage
pulse height into digital format. This ADC is linked to
a PDP 11/45 computer which reads the digitized values
of the event and then performs the defined gating and data
storage. Up to 8 spectra can be stored using the data
taking program BKNTOF, which was used in all data runs.

This program allows up to two gates on each spectrum. For
example one could record the TOP spectra for a detector
gated by PSD to require a neutron event and light to esta-
blish a known threshold. .

The strobes from each detector were ORed in a Universal
Coincidence box. The output of this Coincidence box was
sent to a 4-Fold Logic Gate where it was transformed to the
proper negative voltage to act as a strobe for the ADC.

The program records nine sealers; one internal that
counted the number of processed ADC strobes and 8 more read
from two quad scalers that recorded the total number

of strobes sent to the ADC, the charge accumulated, the total

40

41

active time of the run, and the number of LED events for each
detector.

During the time the computer is processing an event
it is not sensitive to any other events that might reach
the ADC. A measure of the computer dead time is obtained
by comparing the number of events recorded by the ADC to
the number of strobes recorded by the scaler for the same
time span. For comparison of neutron detector yield to
monitor yield the computer dead times for each detector
are the same. The dead time due to the reset time of the
electronics, with the TACs being the dominant source,
was measured to be usually <0.1%. This agrees with dead
times calculated from the reset time of the TACs. The

O

0 normalization runs had the largest dead time uncertain-

ties, introducing a normalization error of <1%.

B. Time of Flight Spectrum

For both main detectors and the monitor a TOF spectrum
is recorded. This spectrum is gated by the light signal,
accumulating only those events that produce a recoil in the
scintillator with energy above a certain threshold. These
thresholds are listed in Tables 1 and 2. In all but a few
cases the main detector TOF spectra are also gated by the
neutron part of the PSD spectrum. The monitor TOF spectrum
is never gated by PSD as no PSD information is available

for it. Figure 9 shows a typical TOF spectrum for 40 MeV

neutrons scattered from silicon at 8:650. The target-in,

42

target-out and subtracted spectra are shown. The first ex-
cited state of 288i at 1.76 MeV can be seen next to the
larger elastic peak. Additional target-in TOF spectra are
shown in Figure 37 thru Figure 43. Figure 10 shows the mon-
itor TOF spectrum for a 40 MeV run. Spectra with the lithium
target in place and without the lithium target are shown.

The monitor spectrum for a 30 MeV run with the lithium target

in place is shown in Figure 44.

C. Normalization Procedure

The absolute normalization of the neutron elastic scat-
tering cross sections was obtained by a ratio technique that
removes dependence on some of the least well known quanti-
ties in the cross section calculation. This procedure in-

7Li(p,n) reaction at 00

volved measuring the yield of the
with the same parameters as the scattering runs for each
source to scatterer distance and each monitor position.
The yield formula for the normalization run is discussed
below (IV-B).

The experimental system was designed to accomodate the
104 difference in counts/charge experienced in the normali-
zation runs compared to the angular distribution runs. The
data acquisition program generated unacceptable dead times
when the data rate above the electronic threshold was greater
than about 700/sec. The computer count rate is the sum of
the strobe rates from both detectors and the monitor. During

the angular distribution measurements, the strobe rate in

the monitor was prescaled by a factor of 10 and the detectors

COUNTS/CHANNEL

43

 

 

 

 

 

 

 

 

T T T T | T
SUBTRACTED
100..
0
TARGET OUT
100,.
0 W
TARGET IN
100-
0 1 1 ._jky 1
300 800
CHANNEL NUMBER
Figure 9. TOF Spectra 40 MeV, Si, 60° Target In,

Target Out, Subtracted

 

COUNTS PER CHANNEL

500

500

44

 

r l
WITHOUT LITHIUM TARGET

M 1

 

 

NITH LITHIUM TARGET

1)

 

 

 

 

 

 

 

.1
H00

Figure 10.

 

1 l 1
800 800
CHANNEL NUMBER

Monitor Spectrum 40 MeV Lithium Target
In, Li Target Out

45

were not prescaled. During a 0°norma1ization run, the beam
current was reduced to a few nA. The prescaler was removed
from the monitor and the strobes for the two detectors were
prescaled by a factor of 10. This procedure bridges the
count rate gap between these two measurements, allowing the
0° runs to be done in an hour or two with 1.7% statistics.

It was necessary to make a zero degree run whenever the moni-
tor detector was changed or the source target to scattering

target distance was changed.

D. Background

Upon examining the sources of background we found that
the nearby air produced about 90% of the background around
the sample elastic peak. The contributions to air scatter-
ing were mapped using a scattering target of Mylar (C10H804)°
This target was positioned at various points in the scatter-
ing area and the yields recorded, thus determining the re-
lative contributions. These measurements indicated that
the air scatter dropped off rapidly as one moves away from
the central scattering region, except for the forward direc-
tion where the fall off was slower. The holder produced the
remaining background apart from a very small residual caused
by many bounce paths.

The background causes a peak in the NTOF spectrum that
lies under the elastic peak at the smallest angles and
shifts to lower energy with increasing scattering angle.

The background yield becomes less intense, relative to target

46

yields, with increasing angle. Typically by 800 the air
peak was no longer noticed in the TOF spectrum, except for
12C where the kinematics keeps the air peak and target peak
inseparable. When the scattering sample is in place the
background due to non-sample scattering will be slightly
less than when the sample is removed because of flux
attenuation by the sample. Based on the air scattering
measurements and absorption of the samples this correction
can be estimated. There is <l% effect on the deduced
cross section for all cases except for lead at the forward
angles where there was at most a 3.5% effect on the cross
section.

A Helium bag was used during some of the early runs
in an attempt to reduce this air scattering background.
A large plastic bag that enclosed the scattering targets
was filled with helium to a pressure that expanded the
walls of the bag away from the scattering sample area.
The concept did not work well in practice. Because of

mechanical considerations and leakage problems it was

eventually discarded.

IV. Data Reduction

A. Peak Areas

For all angles and for each run both target in and
target out spectra were measured. To extract the target
yield, the target out run of each detector was normalized
by scaling it according to the ratio of monitor yields.
Then the target out spectrum is subtracted channel by channel
from the target in spectrum. This defines the elastic peak
very well, and its area is obtained with no additional
background subtracted.

The monitor yield is determined by defining specific
channels around the large peak in the TOF spectrum as seen
in Figure 10. The peak area is extracted with no back-
ground subtracted. From run to run the same limits of
integration are used and a check of the resultant centroid
is made. If the centroid varies by more than :1 channel
the limits are redefined so that the difference, centroid -

lower limit, is constant to 11 channel.

B. Cross Sections
The detector neutron yield for neutron scattering

is given by

0
Y (6)= do .(0 )TpQN x do (8)NN
n,n afiLl ;g§—A afiexp A D

6(E)A_R (IV-l)
2

where, NA= Avogadro's number

D= scatterer to detector distance

47

48

T: 7Li target thickness

Q= number of incident protons

€(E)= detector efficiency at En=E

A= area of detector

0: Li target density

d= Li to scatterer distance

N= moles of scatterer

R= finite angle correction for incident neutron
intensity.

The intensity correction R is given by

do .(1 d9
S d014< )
scatterer .

J; 991fi(0) an
detector d0

The detector yield for the 0° (p,n) reaction is

R:

 

 

given by
0 o . ,
Y (0 )=do .(0 )TQ C(E )ApN (Iv-2)
p,n afiLl 2 A
7(d+D)
where, Q'= number of incident protons

E'= energy of 00 neutrons.

Incorporating equation IV-2 into equation IV-l we find

 

O
Yn,n(°) = Yp,n(° ) (d+D)2€(E) NNA ggexp(0)R. (IV-3)
Q Q dZD2 €(E') d9

Solving for the differential cross section we find

49

Qgexp(9)=Yn n(e) Q dzDz €(E') 1661 , (IV-4)

dQ ~
0 Q' 2 NR
Yp,n(0 ) (d+D) €(E)

 

where the cross section is in mb/sr and the distances D and
d are measured in cm. In terms of monitor yields the cross

section is given by

do (0): Yn,n(6) M(0°) d2D2 €(E') 1661 , (IV-5)

—exp
d9 Yp n(0°)M(8) (d+D)2€(E) NR

 

where M(0°) is the monitor yield for the 0°(p,n) run and
M(8) the monitor yield for the neutron scattering run.
In this formulation the pairs Yn,n(°) with M(0) and Yp,n
(0°) with M(0°) can be the raw yields with no computer

dead time correction, as it is the same for Y and M. The
results of equations IV-5 and IV~4 were almost always the
same as the monitor to charge ratio was nearly always constant
to 1%. The cross section obtained from equation IV-5

is the experimentally measured cross section,

Oexp, tabulated in Table 14 as the uncorrected laboratory
cross section. It has not been corrected for finite angle,
multiple scattering or attenuation effects and is not
corrected for these effects prior to Optical model searches.
Instead the calculated OM cross sections in the Lab frame
are themselves smeared to mimic the experimentally deter-

mined cross sections by incorporating a Monte-Carlo sub-

routine in the OM search code.

50

C. Neutron Detection Efficiency

The efficiency of the neutron detectors has been
calculated using the program TOTEFF as modified by Doering
(D074). A set of calculated efficiency curves are shown
in Figure 11 for three different thresholds. The light
from the scintillator is calibrated in equivalent electron
energies. These electron energies are measured by Compton
recoil electrons due to gamma ray sources. The maximum
electron energy corresponds to the Compton edge and is the
distinguishing feature in these low Z detectors. The
228Th source (EY=2'615 MeV) was our standard and thresholds
were set in terms of the equivalent number of Th Compton
edges. The Compton edge peak and half height were measured
and the edge extracted based on a study of these quantities
in relation to the true edge by Galonsky and Doering (Ga78).

The calculated efficiencies are accurate to about 10%,
but efficiency enters the cross section calculation only
in ratios of efficiencies. The relative effeciencies depend
only weakly on the efficiency curve. For carbon some uncer-
tainty is introduced in the back angles because of the large

kinematic shift of the scattered neutrons. As the targets

get heavier this effect becomes less important.

D. Experimental Errors
The major source of error in the measured experimental
cross sections is statistics. The fractional error for the

(target in)-(target out) yield is

51

Smumcm couuomam >02

or

mm.m

mmpm anemone .hocmH0flmmm Houomuma couusmz .HH wusmfim

$3: motocm c8562

 

 

      

Eatocexr

Eatockxm\

E:_Loc._.xm\

om oN OH
I a

A

   

fl

        

 

 

CD
«a

(x) fiOUGDUJB 001108480 UOJH‘ION

52

_ l/2 _
éX-(Yin+yout) /(Yin Y )

Y out
where AY is the error in the target yield, Yin is the yield
for the target in run (this includes target yield + back-

ground) and Y is the normalized yield for the target out

out
run (see Table 5).

Compound nuclear contributions to the cross sections
are not reproduced by simple Optical Model Calculations.
Thus before fitting data with a simple OMP it is necessary
to subtract out any contributions due to compound nuclear
elastic scattering. Rapaport et a1. (Ra77) have estimated
this effect by a Hauser-Feshbach calculation for neutron
scattering at lower bombarding energies. They find the
correction to be il% for 20 and 26 MeV scattering. Since
this contribution to the elastic scattering decreases with
increasing bombarding energy, we did not repeat their
calculation but rather assumed compound nuclear elastic
scattering contributions to be negligible.

Since calcium is reactive in air, during the brief
time it was exposed to air during the canning process it
invariably absorbed some oxygen. Elastic scattering
from this absorbed contaminant would contribute at the
forward angles. Beyond about 600 the neutrons elastic-
ally scattered from oxygen will be shifted in energy away
from the calcium scattered neutrons due to the kinematics.
The oxygen contamination also caused a small uncertainty
in the absolute normalization since the samples composition

was not exactly known.

53

The Lithium target to scattering target distance (d)
is known to 31 mm causing an error ranging from 2% for
d=ll.0 cm to <1% for d=24.4 cm. The mean scattering angle
is known to about 10.50 for d=ll.0 cm, _+_0.30 for d=18.4 cm
and 320.20 for d=24.4 cm.

During each run, before the Li target to scattering
target distance was changed the yield measurement at one
or two angles was repeated. After the Li target to scatter-
ing target distance was changed, or if the monitor position
was changed the yield measurement was repeated at one or
two angles. These checks gave results consistent within

the experimental error.

54

Table 5. Experimental Errors

Relative Uncertainties(%)

 

152-90°
95 -160

Statistics in Yields: 0
Monitor Statistics

Finite Geometry

Compound Nuclear Contribution

Contaminants

Background Attenuation Due to Sample
Detector Efficiency

Incident Angle of Proton Beam

Scattering Target Position

Mean Scattering Angle

Normalization Uncertainties(%)

Statistics in Yields, 0°f1ux
Monitor Statistics, 0° flux
Dead Time Correction

Flux Anisotrophy Correction

Number of Target Nuclei

Total

l(C)

<1-3.5

<1-2
o.2°-0.s

<1
1.7

<1

<1

<1

2.6

a) Applicable only to center-of-mass cross sections

b) Applicable only to C, Si, S, and Ca

c) Ca only at forward angles

V. Data Analysis

A. Optical Model Parameter Search Code

The cross section determined by equation IV—5 is un-
corrected for multiple scattering, finite angle or attenu-
ation effects. The multiple scattering cross section de-
pends on the entire single scattering angular distribution.
The finite angle correction depends on the slope of the
cross section around the mean scattering angle. To treat
these effects we have chosen to smear the predicted Optical
Model cross sections instead of attempting to correct the
experimental data. Correcting lower energy data (Ki70)
is acceptable because the cross sections do not vary so
fast and smaller samples are used so multiple scattering
effects are not as great. From equation II-l we see that
as the energy increases, flight paths must become longer
to maintain the same energy resolution. Thus larger tar-
gets are needed to maintain a good data rate. Also, as the
energy increases, the cross section slope tends to increase,
making the finite angle correction larger.

Thus the Optical Model search code GIBELUMP (Pe66)
has been modified and this new version of the code is call-
ed GIBSCAT. The code GIBELUMP calculates the c.m. elastic
scattering cross section from a given set of OM parameters.
It established the relation between a decrease in x2 and

parameter variation where

55

56

 

N 2
2 do 6. do 6.
X =2 66°a10( 1"aOeXP( 1) (v—1)
i=1 '
do x (Bi)
d0° P

GIBSCAT differs from GIBELUMP in that prior to comparison
with experimental data the c.m. cross section is converted
into the lab cross section and transformed to include
multiple scattering, finite angle and attenuation effects
by the subroutine MULSCAT. The subroutine MULSCAT is
based on the Monte Carlo code developed by Kinney (Ki70).
This code proceeds as shown in the flow chart in
Figure 12. The initial OM parameters, geometry and experi-
mental cross sections are read in. The code calculates
the center of mass cross sections from the Optical Model
potential then converts these cross sections to the lab
frame. Then the Monte Carlo routine calculates the smear-

ed cross section 0 (6) that includes multiple scatter-

calc
ing, finite geometry and attenuation. This smeared cross

section is compared to the experimental one by equation
V—l. The gradient in x2 space is determined by first vary-
ing the OM parameters. Then a revised smeared cross section

is calculated by

new
Ocalc(°)-° (6)

— OM 0
old calc
COM (8)

(9) (V-Z)

where Ocalc(°) is the smeared cross section determined by

old
OM

calculated by the original set of Optical Model parameters,

Monte Carlo, o (9) is the laboratory cross section

57

 

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58

03;w(6) is the laboratory cross section calculated by the

' I
varied OM parameters, and 0

(0) is the revised smeared

cross section which is to be used in calculating the x2

calc

gradient. Then guided by this gradient a new set of OM
parameters is determined. The program then recalculates
Ocalc(6) from these new OM parameters until the predeter-
mined number of iterations is reached.

Due to the nonexact calculation of X2 gradient, the
procedure did not always converge. In practice the program
was only allowed to proceed a few iterations per run. Then
the best set of OM parameters was used as input for the
next run.

The vast majority of computer time in these searches
is spent in the Monte Carlo routine. The calculation of
cross sections from OM parameters for 21 data points
uses about 0.1 minutes of computer time. To correct these

data points by the Monte Carlo routine using 3000 histories

takes about 11.3 minutes.

E. Center of Mass Cross Sections

The measured cross section Oexp can be divided into

two components, the single scattering and the multiple

scattering contributions by,
I
0s

+ 0' (V-3)

Oexp(e)=OLAB(e) OLAB MS

OLAB(6)=true laboratory cross sections

0' = single scattering contribution to the
measured cross section

59

OMS: multiple scattering contribution to the
measured cross section.
The calculated smeared cross section Ocalc is also separ-

ated into single and multiple scattering compounds by

O

Ocalc (mzoom O—s— +0Ms (v-4)
OM
where COM: calculated OM laboratory cross section

Os: single scattering contribution to the
smeared calculated cross section

OMS: multiple scattering contribution to the
smeared calculated cross section

The multiple scattering contribution depends on the
entire angular distribution and not on the value at one

(6)

angle. If local fluctuations between Oexp(e) and Ocalc
have random signs and the deviations are small we can
extract OLAB(6). We assume the multiple scattering contri-
bution is determined since, if the fluctuations are random
and small, then the average cross section is well deter-
mined. Also we assume that the finite angle and attenua-

tion effects are accurately determined from the Monte

Carlo routine, i.e.

 

s 0s

0 0 ' (V-S)
LAB OM

Then the true cross section OLAB is given by
0 = (0 -0 ) 0 +0
LAB exp calc OM OM (V-6)
0
s

From this the true center of mass cross sections are

deduced by directly converting to the center of mass frame.

60

In all but one case the conditions leading to equation
V—6 are fulfilled by the final best fit OM prediction.

The deduced center of mass cross sections are tabulated
in Table 14 and shown in Figures 38 to 50.

In the case of 40 MeV scattering on lead the conditions
for equation V-6 break down. As seen in Figure 37 in the
angular range 750 to 1150 the smeared calculated cross
section Ocalc(6) is larger than oexp(6). In this region
a scale factor seems more appropriate to correct the
difference between Ocalc(e) and oexp(6). In the region

(810-1100) the true laboratory cross section is determined
by 0 .

OLAB = COM _§xp__ . (V-7)

calc

For the final determination of the true center of mass
cross section, the Monte Carlo routine with 10,000
histories was run at least twice. The results were
compared and found to be in excellent agreement. The
uncertainty in the deduced center of mass cross sections
due to the finite geometry correction is estimated to be
between 1 to 8% depending on the target. The largest
correction errors are for Pb near the first cross section
minimum. All other targets had correction errors <2%.
The corrected center of mass cross sections are tabulated

in Table 14 under the heading Corrected Center of Mass.

61

C. Parameter Search Procedure

For each angular distribution a set of OM parameters
was determined. These parameters represent the best fit
in terms of x2 minimization. The search procedure was
guided to help eliminate ambiguities in related parameters.
D’ WV and
and then V,

One search sequence was to vary first V, W

obtain the best fit then V, a W W r

R' D' v' I‘

r a W W , rI, a The other search sequence used

R' D' V I'
was to vary only uncoupled parameters such as V, rI or

and not to allow all parameters

RI

R, rI or aR, a1, 01‘ IR, WD

to vary at once, except when the fit was very good, to

r

verify a x2 minimum in all variables. Since no polariza—
tion data were available the spin-orbit term was not
varied but was fixed at the best fit value of Becchetti
and Greenlees (Be69). Only relative errors were used in
all the searches and x2 calculations.

Several global OM parameter sets were used for
starting parameters. Namely those of Becchetti and
Greenlees (Be69) (BG); Patterson, Doering and Galonsky
(Pa76) (PDG); Rapaport (Ra79) set A (RAPA) and Rapaport
(Ra79) set B (RAPB). Table 6 lists the x2/N for each of
these parameter sets.

The initial search uses the program GIBSCAT and
searches until a "good" fit to Oexp is achieved. A "good"
fit is one that meets the conditions for applying equation

V-6. From this fit the center of mass cross sections are

determined by equation V-6. GIBELUMP is used to search on

Table 6.

Nuclei

12C

2831

2851

32S

32S

40Ca

40Ca

208Pb

208Pb

xz/N of Global Parameter sets

Neutron
Energy
(MeV)

40.0
30.3
40.0
30.3
40.0
30.3
40.0
30.3
40.0

21.
71.
19.
21.
82.

73.

62

 

a) as given by equation V-1

280.
140.
200.
84.
173.
43.
111.
87.

27.

(a)

200.

21.

22.

21.
70.
42.
40.
140.
180.

120.
20.
20.
40.
22.
26.
40.

140.

180.

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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a r rfiflo mowmfiu nmgmfiqu “MEAQfi. ‘MMWV
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05.

 

65

OCM and a new set of OM parameters are determined. GIBSCAT
is run for l iteration using this new OMP. The change in
x2 is then checked compared to the previous run of GIBSCAT.
If the x2 decreases the procedure iterates as shown in
Figure 13. If the x2 has increased we go back to searching
on Oexp using GIBSCAT. The search procedure is terminated
when no further decrease in x2 is achieved and the OM
parameters remain constant. The OMP derived from this
search procedure are tabulated in Table 7 for 30 MeV
neutrons and in Table 8 for 40 MeV neutrons.

The absolute normalizations of the final results
are those values determined experimentally. For each angu-
lar distribution a search on the absolute normalization
was conducted. These searches gave either no improvement
or minimally improved x2 for normalization changes on the
order of a few per cent. For those samples that were of
natural abundance, no correction was introduced for the
small admixture of other isotOpes. A calculation assuming
the Becchettiand Greenlees Optical Model potential (Be69)

show that the correction is less than 0.1% in the potential

depths.

D. Volume Integrals

The determination of a set of potential parameters
always entails a certain amount of ambiguity. The nature
of a multiparameter search procedure, the slightly differ-
ent results for various starting values of the potential

parameters and interrelationships between parameters make

66

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67

 

 

 

 

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maco muouu0 0>flu0a0u .mcofluo0m mmouo muou0non0q A0
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0 K . 3
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m.m va.o mov.H o.o vmo.m Hmh.o mmH.H oo.Hv ammom
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68
comparison between different nuclei and energies difficult.
Feshbach in 1958 (Fe58) suggested that the volume integral

of the potential,

J=fV(r)d3r (V‘B)

is a better measure of the strength of the potential
and it is now well verified that J is determined better

than V, rR and aR separately. Another well determined

quantity is the: mean-square radius defined as

2 3

<r > = fV(r)r2d
fV(r)d3r

 

r (v-9)

For each nucleus at each energy we have determined the
volume integral per nucleon as well as the root-mean-
square radii for both the real and imaginary potentials.

For the volume terms (V , WV) in the parameterization

R

one finds that the volume integral per nucleon is

 

(approximately)
Jvol 4n Vr3 1+ "a 2 (v-10)
__ "—7— I
A — 3 rAl 3

 

 

 

while the mean—square radius is given by

2

2 1 2A2/3+7n2a ). (v-11)

rvol=§ (3r

for the derivative Woods-Saxon term the volume integral

is (approximately)

 

 

 

 

2 .
£2 = l6nr aWD 1+1 “a 2 (V-12)
A A173 3 rA1/3

69

 

 

 

Table 9. Volume Integrals and rms Radii
Neutron (J/A)real <r2>1/2 (J/A)imag <r2>l/2

. Energy 3 real 3 imag
Nuclei (MeV) (MeV-fm ) (fm) (MeV-fm ) (fm)
120 40.0 396.4 3.10 152. 3.37
2351 30.3 410.9 3.79 145.4 3.81
2851 40.0 355.6 3.78 121.6 3.96
325 30.3 427.8 3.81 125.2 4.26
32$ 40.0 383.0 3.91 125.7 4.13
4°Ca 30.3 412.7 4.17 113.2 5.02
400a a 30.3 395.3 3.99 114.0 4.67
4OCa 40.0 348.2 4.14 97.2 4.55
400a a 40.0 340.3 3.99 94.5 4.62
208Pb 30.3 320.6 5.97 67.3 7.17
20886 b 30.3 326.6 6.06 68.8 7.14
208Pb 40.0 296.4 5.00 74.6 6.88
208Pb b 40.0 303.5 6.06 77.0 6.76

a) fixed geometry(Va7l)
b) fixed geometry(Va74)

70

and the mean-square radius by

 

 

 

 

 

2 C
r2 = lZarAl/3 1+ 31——— Jvol . (V-13)
D 1/3
rA JC
D
Where Jgol and J3 are the volume integrals for the volume
and derivative form, respectively, with Vvol=VD=l.

When a combination volume and surface term is used
the total volume integral is

g; = Jvo1 + JD (v-14)

A A

 

while the :mean-square radius is

 

<r2>=<r2 >Jvol + <r2>JD . (V-15)
vol D ——
J J
. 2 1/2
The volume integrals and <r > of the present work are

tabulated in Table 9.

E. Coulomb Correction Term

The charge of the nucleus has the effect of reducing
the mean kinetic energy of incident charged particles in—
teracting with the nucleus. Because the local real poten-
tial increases with decreasing energy, the effective real
potential felt by protons is larger than that for neutrons
of the same bombarding energy. This effect is accounted
for by adding to the proton potential the Coulomb correc-
tion term, AVC(r). The real potentials of the Lane formal-

ism for proton and neutron scattering are

v‘“’(r.E)=(v0n-yE-evl(8))f(r)

71

V(p)(r,E) = (Vop-YE+€V1(E))f(r)+ AVc(r) (v-16)

where a linear energy dependence is assumed. Assuming

a charge symmetric nucleon-nucleon interaction the terms
VOnand Vop are equal. The subscripts n and p will be
left off from here on. If we compare the potentials
deduced for scattering from N=Z nuclei(€=0) at the same

energy we find

V(p)(r,E)-V(n)(r,E)=AVC(r) . (v—17)
The Coulomb correction term can now be obtained directly.
The derived potentials from scattering over a range of
energies can be fitted and these energy dependent poten-
tials determined. In Figure 15 the real well depth from
the average geometry is plotted for both neutrons and
protons (the average geometry and proton data are from

the work of Van Oers, (Va7l) for40Ca and (Va74) for 208

Pb).
Apart from a dip in the proton potentials near 20 MeV a
linear trend is clearly established. The proton potential
is (p) -

V (E)—(59.2-0.35E)MeV

and the neutron potential is

V(n)(E)=(56.5-0.35E)MeV
where the energy dependence of the neutron potential is
constrained to match that of the proton potential. Thus
from equation V-l7 the Coulomb correction for calcium is
AVC=2.710.3 MeV

where the form factor is Woods-Saxon shape with R=rOA1/3

72

1/3

=1.152A fm and ao=0.692fm. The error is estimated by

noting that slope change of 10.02 is about the maximum

allowed by the data. Using the form of Lane (La 57)
AVC=BzZ/A1/3 (V-18)

we establish the Coulomb correction for protons to be
Avc=(o.46:o.05)z/Al/3Mev .

In terms of volume integrals we need only change scale

since the geometry is fixed. Thus,

Jp/A = (494.5-2.92E)MeV fm3 ,
Jn/A = (472.0-2.92E)MeV fm3 ,
and
JA/A = (3.8610.4)Z/A1/3Mev fm3 . (V-20)

Jeukenne et al. (Je77) have calculated the Coulomb
correction in the framework of the Brueckner-Hartree-Fock

approximation. They conclude that the standard value (Pe63)

AV2t=(O.4Z/Al/3)f(r)MeV

is an underestimate. They calculate a 25% larger volume

integral than the standard value for 208Pb at 25 MeV.
Rapaport et a1. (Ra77) deduced the same value for VC as

this analysis. They compared the proton data of van Oers

to their neutron data, which is also used in the present

analysis. Their data covered an energy range of only

15 MeV. The present analysis extends that range to 29 MeV

73

 

‘1 + +
.++ **++ ‘
+ +
U)
o + A
E +
;: g
d: £30"' * * -
\ +4~*’+
.3 +
H

 

 

L10 1 1 1 1 1 1

 

 

 

 

 

20 “IO 80
EnerggWeV]
[ I f l | I
SOOr + .1
B +t€n
r2: ‘_: +4§ 0
{H003 + *1 9 4» ¢ fi
:3 ’ A + +
300r -
L 1 1 1 J L
20 '70 80
EnerggWeV]

Figure 14. Energy Dependence of Calcium Volume Integrals,
Best Fit Values

74

>Hu0Eo0w o0xflm nuw3
numc0uum H0Huc0uom H00m EdH0H0U mo 00:0pc0m0o wmumcm

H>mzammtmcm
or om
1 41 q A. q q Omw

.mH 005508

 

(AGWM

 

r 2.33 2383 232502 4
:33qu 0:60.502 0
7.30 :63 2.29.0 +

 

8 #1 1L

 

 

75
with the inclusion of higher energy data.

The determination of the Coulomb correction term from
linear fits to neutron and proton potentials over a range
of energies has certain inherent limitations. A linear
energy dependence is assumed from the very start. This
appears to be a good assumption, but as in the case for
protons on 40Ca large deviations from linearity are observ-
ed. Thegadeviations must be treated individually thus in-
troducing personal judgement errors or bias to the linear
fit. All the potentials considered are not derived from
data of comparable quality, quantity or content. Different
angular ranges are measured for the various angular distri-
butions. Some include polarization while some do not and
the experimental uncertainties of the data arerun: consis-
tant. Assigning errors to potentials based on the quality
of the data and the quality of fit is not well understood.
It is hoped however that the net effect of all fluctuations
and errors will in some average way become small.

If neutron and protons potentials are compared at
identical energies, no energy dependence needs to be assum-
ed before extraction of specific terms in the potential.
Thus we compared neutron data at 30.3 and 40.0 MeV to
existing proton data at the same lab energies. The proton
data is reanalysed restricting the angular range to match
that of the neutron data. The data of Ridley and Turner

40

(Ri64) for Ca at 30.3 MeV and the data of Blumberg et a1.

(8166) at 40.0 MeV are used. A search procedure similiar

76
to that used to search on the center of mass neutron data
is used to determine proton potentials. The proton data
are analysed in terms of the average geometry of van Oers
(Va7l) and best fit parameters. The volume integrals of
these potentials are then averaged and compared to the
same averaged potential from the appropriate neutron data.
These results are tabulated in Tables 10 and 11 for calcium.

The differences of the volume integrals for calcium

are

4% (30.3MeV)=lO.7MeV fm3

A% (40.0 MeV)=32.5MeV fm3 .

In this case AJ=JA. From the theoretical considera-

tions JA should be energy independent (Je74, Je77). Thus

we take the average value of AJ from the above to give

= (21.6:7)MeV fm3

{I’lu
>

for 40Ca. Using the form of equation V-18

£9 = (3.69il.2)Z/A1/3Mev fm3. (v—21)

3’

This value is in good agreement with the value extract-

ed from the linear fits. The average of the two determin-

ations yields

J
EA = (3.78i0.4)Z/Al/3Mev fm3 . (v—22)

77

Table 10. J/A(a) for protons

and neutrons on Calcium.

 

Enucleon=3o° 3 MeV
F' (b) -

1xed geometry Best flt Average
J
K2 real 405.1 424.2 414.7
Jn
A— real 395.3 412.7 404.0
:2 - Jn *
A A_ real 9.8 11.5 10.7
J
EB imag 104.3 103.0 103.7
Jn
Xf-imag 114.0 113.2 113.6
J _ Jn
A A_ imag -9.7 -10.2 -9.9

a) all J/A units are MeV fm3

b) rR=1.152 fm, aR=0.692 fm, rI=1.309 fm, aI=O.549 fm

78

Table 11. J/A(a) for protons

and neutrons for Calcium.

E =40. MeV
nucleon

(b)

Fixed Geometry Best fit Average

 

J
XE real 374.0 379.4 376.7
J

XE real 340.3 348.0 344.2
52-5

A X“ real 33.7 31.4 32.5
J

EB imag 102.9 102.2 102.6
J

XE imag 94.5 97.2 95.9
ig-Jn

A A_ imag 8.4 5.0 6-7

3

a) all J/A units are MeV fm

b) rR=l.152 fm, aR=0.692 fm, rI=l.309 fm, aI=0.549 fm

79

In terms of the average calcium geometry of van Oers this

term is
Avc=<.452:0.05)Z/A1/3Mev . (v-23)

F. Isovector Term
We have already assumed a charge symmetric nucleon-
nucleon interaction i.e. Vpp=vnn. However the nucleon—
nucleon interaction is not charge independent i.e.
Vppgvpn. This effect is described by the isovector
strength VI of the Lane Model potential. This term is
important not only in proton and neutron scattering in
terms of a global OMP but also in charge exchange reactions.
If we consider the case for N#Z nuclei, such as
208

Pb, we see from equation V-16 that for the same incident

energy,
V(p1r,E)-V(n)(r,E)=2€V1(E)f(r)+AVC(r). (v-24)

The nuclear asymmetry (8) is roughly Z dependent. Since
the Coulomb correction term is also Z dependent, even if
we consider a wide range of nuclei, unless we have prior
knowledge of AVG, the isospin dependence V1(E) can not be
directly extracted.

In an analysis similar to the one for 40Ca, van Oers
et al. have compiled and analysed proton scattering data
for 208Pb(Va74). Again best fit and average geometry
potentials are determined. In Figure 16 the real potential

depth with fixed geometry (ro=l.183fm, ao=0.724fm) as well as

the deduced linear fits are plotted for neutron and proton data.

80
In terms of volume integrals, the proton potential is
(Ra78)

= (485.7-2.52E)MeV fm3

{PRC-4

From a linear least squares fit to the neutron data we

find the neutron potential to be

J
X2 = (380.2-1.88E)Mev fm3 .

Writing equation V-24 in terms of volume integrals we

have

- :3 = 2: i1 + EA . (v-25)
A A

A

3,10“

From equation V—22 the volume integral of the Coulomb

208

correction term for Pb is 52.3 MeV fm3. Thus

:1 = (125.8-1.51E)MeV fm3 . (V-26)

A
In contrast to the calcium case the real volume inte-
grals of the best fit potentials do show a well defined
linear energy dependence as shown in Figure 17. A least

squares fit to the proton data yields

32 = (516.4-3.28E)MeV fm3 .
A

The neutron volume integrals, which include the present
measurements at 30.3 and 40 MeV, the data of Rapaport
et al. (Ra78) at 7, 9, 11, 20, and 26 MeV and data from

the tabulation of Perey (Pe76) are best fitted by

in = (407.5-2.85E) MeV fm3 .

A

81

hHu0Eo0U p0xwm spas
numc0uum H0fluc0uom H00m 000A mo 00:0oc0m0o mmu0cm .ma 0uomflm

_>023mtocw

 

 

  

0..) 0N

4 _ 1 cm

 

3.53 239:: 23232 4
2.3335 39:32 o
A930 ca: 2.29:. +

 

 

(AGWJA

82
This yields for the isovector volume integral

2; = (133.8-l.02E)MeV fm3. (v—27)

A
The density of neutron data points is not uniform but
rather is concentrated at low energies. To make the
neutron potential more dependent on the higher energy data,
the data of Perey is left out. Then a least squares fit

for the real neutron volume integral gives

:3 = (397.0-2.48E) MeV fm3 .

A
This neutron potential yields an isovector volume integral
of

i; = (158.6 -l.9E) MeV fm3. (V-28)

A

These three determinations of Jl/A yield different results.

As a compromise we take the average value,

<Jl/A> = (l39.4-l.48E) MeV fm3 . (v-29)

The proton data on 208Pb of Ridley and Turner (Ri64)

at 30.3 MeV and the data of Blumberg et al. (8166) at

40 MeV were reanalysed restricting the angular range to
match the neutron data as was done for calcium in section
V-E. The results of the proton searches are tabulated

in Table 12 for 30.3 MeV and in Table 13 for 40 MeV. From
this analysis we find the isovector strength at 30.3 and

40 MeV to be

83

 

 

 

 

r I + fit I ' [+
+ + + " 3 +
O) +
C "’ +
.§
4: _. ._
\ 80 A
3 ‘ A
O
O
L+0 4”, 1 if 1 1 La
20 Q0 80
EnerggIMeV]
T I r I I [

 

500 oNeufr‘ons [PeregH

 

[J/A]real
é

300

 

 

 

‘1‘0 80
EnerggIMeV)

Figure 17. Energy Dependence of Lead Volume Integrals,
Best Fit Values, symbols same as Figure 16
a) J /A=(407.5-2.85E)Mev fm3
b) J3/A=(397.0-2.48E)Mev fm

84

86.9 MeV fm3

Jl/A(30.3 MeV)

73.2 MeV fm3 .

Jl/A(40.0 MeV)

Thus if we now assume a linear energy dependence we find

that 3
Jl/A = (129.4-l.40E) MeV fm . (V-30)

This value is in good agreement with the value (equa-
tion V-29) derived from the fitted potentials. Taking the
average (V-29 and V-30) we find

Jl/A = (134.4:13)-(1.44:0.08)E MeV fm3. (v-31)

The erroris dun; due to the error in the Coulomb Correction.
In terms of the average geometry of van Oers, this isowxnnr

potential strength is

Vl = (l7.5-O.l9E) MeV . (V-32)

The present determination yields a value about 20%
smaller than the values reported by other authors (Ra79,
Pa76, 3e69, Ca75). However the energy dependence determin-
ed by Rapaport et a1. (Ra79) and by Patterson et al. (Pa76)
agrees very well with the energy dependence determined by

the present work.

G. Imaginary Potentials

The imaginary part of the OMP describes the effect of
all the non-elastic interactions of the incident particle
with the target nucleus. This potential requires a
combination of volume and surface form factors as discussed

in section I-B.

>LUC-u

>|C4
s

SwUQ 31:9 EflUQ Sfluh

a)

b)

85

Table 12. (J/A)(a) for protons
and neutrons on Lead.

Enucleon=3o’3 MeV

(b)

Fixed geometry, Best fit Average

 

real 411.4 412.6 412.0

real 326.6 320.6 323.6
X_ real 84.8 92.0 88.4

imag 110.0 104.2 107.1

imag 68.6 67.3 68.0
K— imag 41.4 36.9 39.1
all J/A units are MeV fm3

rR=l.183 fm, aR=0.724 fm, rI=1.273 fm, aI=0.699 fm

{I’lsC-a 31°C: 310;. {I’lsc-a {DEC-1

who

a)

b)

86

Table 13. J/A‘a) for protons

and neutrons on Lead.

 

Enucleon=40' MeV
. (b) .

Fixed geometry Best fit Average

real 389.9 375.2 382.6
real 303.5 296.4 300.0
X_ real 86.4 78.8 82.6
imag 107.8 104.8 106.3
imag 77.0 74.6 75.8
X— imag 30.8 30.2 30.5

all J/A units are MeV fm3

rR=1.183 fm, aR=0.724 fm, rI=1.273 fm, aI=0.699 fm

87

The total imaginary volume integrals for scattering
from 40Ca are shown in Figure 14 and the surface and vol-
ume potentials of the fixed geometry for 40Ca are shown
in Figure 18. We notice in both Figures that there is no
systematic difference between the proton and neutron
potentials.

The reanalysed proton data yield a negative value
for AJimag(AJ=Jp-Jn) at 30 MeV (Table 10) and a positive
value for AJimag at 40 MeV (Table 11). We conclude that
the imaginary Coulomb correction is very small for Ca.

The van Oers proton data was analysed using a Gaussian
form factor for the imaginary surface term while the
neutron data were analysed using a Woods-Saxon derivative
form factor. The derivative Woods Saxon is chosen to have
the same width at half maximum as the Gaussian. Results
obtained with the Gaussian surface potential replaced by
a derivative Woods-Saxon were determined by van Oers to
be very similiar (Va7l). Rapaport et a1. find the two
potentials to be not very different (Ra77 and reference
therein).

In the analysis by van Oers et al. for protons on lead
(Va74) a derivative Woods—Saxon form factor is used instead
of a Gaussian. Figure 17 shows the imaginary volume in-
tegral for protons and neutrons. The neutron volume inte-
gral is increasing approximately linearly with increasing
energy. The proton volume integrals are decreasing slightly

with energy. Jeukenne et al. (Je77) have calculated the

88

imaginary Coulomb correction for 208Pb in the energy range

up to 75 MeV. They find this term to be negative, non-
linear with its magnitude approaching zero with increasing
energy. If this term were to be subtracted from the proton
volume integral the trend would be more nearly linear and
decreasing with increasing energy. However untill the imag-
inary Coulomb correction is better known for 208Pb an accur-
ate determination of the imaginary isovector strength will
not be possible. Shown in Figure 19 are the surface and
volume components of the imaginary potential using the
fixed geometry of van Oers (Va74). Here we see that the
strength of the volume term is nearly the same for protons
and neutrons. The major difference between the proton and
neutron potentials is in the surface contribution. For
energies above 20 MeV there is a linear decrease in surface
strength with protons and neutrons having approximately
the same slope.

For protons and neutrons of the same bombarding energy
incident on 208Pb there is additional surface absorption of

the protons, perhaps due to the additional reaction mech-

anism (Coulomb excitation) available to protons.

Nv[M9V]

ND[M6V]

89

 

 

 

 

 

 

 

r I l r l
12r-
8r +
L» z + ,
)- ‘l’ +
0 l 1L_JL l 1 ll,
20 ‘1‘0 80
EnergyIMeV]
r r r I I
8" + 0" ‘+ .
o t N,
_ :3 ‘
9F. 3 ‘
0 4 1 L 1 1
20 HO 80
Energg[MeV]

Figure 18.
Potential Strength with

Energy Dependence of Calcium Imaginary
Fixed Geometry

 

 

90

 

 

 

 

 

 

 

T, r r T 4’ I
A
9 +
g .
“E; L+h- A + +'—
z I
O * _‘
'- O
0 11 l 1 l 1 l
20 80 80
Energg[MeV]
I f, * l I I I
3 + +
)2 . * ’ + . -
H '1. . . 1. -
A '0
l— A .1
0 1 l 1 l 1 l
20 L10 60
Energg[MeV]

Figure 19. Energy Dependence of Lead Imaginary
Potential Strength with Fixed Geometry

 

91

VI. Summary

Apparatus to accurately measure elastic scattering
angular distributions for 24-42 MeV neutrons is developed.
A monoenergetic neutron beam is produced using the reaction

7Li(p,n)7Be(g.s.+0.429 MeV). The neutrons are scattered

from targets of 12C, 2851, 328, 40Ca, and 208P

b. The scat-
tering angle is varied using the MSU beam swinger, thus
allowing production target and beam dump to be in a dif-
ferent room than the neutron detectors. The scattered
neutrons are detected by liquid organic scintillator detec-
tors and energy analysed by the time-of-flight technique.
Detector gain is monitored during each run by feeding a
constant photon source directly to each detector. A mon-
itor detector measures the direct neutron flux from the _
7Li(p,n)7Be reaction. Relative cross section errors range
from 2% to 5% over most of the angular range. Absolute
normalization errors are <3%.

The data are analysed using a standard Optical Model
potential. Calculated cross sections are smeared by a
Monte Carlo routine to account for multiple scattering,
finite angle and attenuation effects and then compared to

40 208

the experimental cross sections. For Ca and Pb both

best fit and fixed geometry potentials are deduced.

40 208

Already existing proton data on Ca and Pb at3043ami

92

40 MeV are reanalysed using the same procedure as was used
for the neutron data. The angular range of the proton
data was restricted to match that of the neutron data.

Comparison of proton and neutron potentials for 40Ca,
with the neutron energy dependence constrained to match
that of the proton data, yields the volume integral of the
Coulomb correction term. Comparison of the reanalysed
proton potential to the deduced neutron potentials at
30.3 MeV and at 40 MeV ynfldsani average Coulomb correction
term for 40Ca. Taking the average of these two determina-

tions of the Coulomb correction term and parameterhflng.h1

the standard way we find

JA/A = (3.78 : 0.4)Z/A1/3 MeV fm3 .

In terms of the average geometry of the proton potential

40

for Ca (rR=1.152 fm, aR=0.692 fm) we find

AVC = (0.45 i 0.05)Z/Al/3 MeV.

Both the fixed geometry and best fit volume integrals

208Pb are compared,

of the proton and neutron potentials for
each fit With an independent energy dependence. Using the
Coulomb correction term determined above, the isovector
term is deduced. Comparison of the reanalysed proton and
neutron potentials at 30.3 MeV and at 40 MeV yield an

energy dependent isovector strength. The average value

of the volume integral is deduced to be

93

Jl/A = (l34.4il3)-(1.44i0.08)E MeV fm3 .

In terms of the average geometry for lead (r =1.183 fm,

R
aR=0.724 fm) we find the isovector strength to be

V1 = (17.5il.7)-(0.1930.02)E MeV.

APPENDIX

Tabulated and plotted data

TOF spectra are target in spectra only. Experimental
cross sections are deduced using equation IV-S and are
uncorrected for multiple scattering, attenuation and finite
angle effects. These are tabulated under the heading
Uncorrected Laboratory in Table 14. Errors for the experi-
mental cross sections are relative errors only as listed in
Table 5. Center of Mass cross sections are corrected for
multiple scattering, attenuation and finite angle effects.
These are tabulated under the heading Corrected Center of
Mass in Table 14. Errors for corrected center of mass cross
sections include relative errors and unfolding errors.

C.M. cross sections for 40Ca and 208Pb are deduced from
"best fit" parameters. There is an additional normalization

error for Experimental and Center of Mass cross sections

of 2.6%.

94

95

 

 

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104

 

l V l ' l

    
  
   

E 288i[n,n]288i E
103' ELOb:30'3 MeV -
E. E
‘1(J()::' ':
E 3

dCI/dQ [mb/sr‘J

1 1 1']1111|
1_ <1 1 1 11111

1 1 1 11111|
1+1
1, 1 1 111111

 

 

 

1 l 1 A l 1 l
0'1 ‘10 80 120

6Lab
28

Figure 29. Laboratory Cross Section, Si, 30 MeV

C3

0.1

105

 

 

 

 

I I l I I :
32 E
S [n,n] 325 ,
_ ELOb:30'3 MeV '
~ I
E 5
_ q
q
_ q
.. * :
f
:- * 1
r I ‘
1 I 1 l 1 l
0 L10 80 ' 120
ech

Figure 30. Laboratory Cross Section, 328, 30 MeV

100

do/dSZ (mb/sr‘]

0.1

106

 

_
p
p
p
—

1 11'1"“ j IIIITUI

1 1 111111]

1 11 [1111'

 

' I ' l ' l

    
  
  
 

LmCCJ[r1,r1]L*OCcJ
ELODZBO‘B MeV

1 1 111111

1 11111111 111111111

11111111]

11111111

 

1 l 1 2 l 1

 

O

1
L10 80 120
6Lab

40

Figure 31. Laboratory Cross Section, Ca, 30 MeV

H
O
4:

107

 

 

 

 

.. fi ' ' 1 ' 1 :
: 208Pb1n.n)‘208Pb :
3- ELOb=30'3 MeV q
10 E' 1
”a - I
‘9 h 1
\ 100. _.
-° 5 a
E : I
o: - * 3
i 105' ':
b 5 + 1
'0 _ *
15' 1

0.1 1 I l I l l

0 ‘10 80 120

eLab

Laboratory Cross Section, 208Pb, 30 MeV

Figure 32.

0.1

108

 

P
1—
1—
p
—
r
—
1—

  
  
  
    

' llllllll I IIIIIHI I 11ll1lll

1 1111111]

 

0

Figure 33.

1 1 1 1

'10 80
9Lab

Laboratory Cross Section,

12

1111111

1 1 1111111

1 11111111

111111111

111111111

 

1
120 _

C, 40 MeV

dU/dQ [mb/sr‘]

109

 

1.3
O
4:
fi

_ 1 I r 1 I E
: 28Si[n.n)2881 3
103? ELOb:L+0'0 MeV q
E— 7:.
E i
:1()[)E? 1:
E i
10:. .5
E u E
1:7 '3
; :

E].1 1 1 1 l 1 1'1

 

 

 

O

l~10 80 120
eLab

Figure 34. Laboratory Cross Section, 28Si, 40 MeV

10“1

1—a 1—~
C) O
C) 60

do/dQ [mb/sr]

0.1

110

 

 

 

 

: 2 32 :
' 3 T.
1 S [n,n] S .
E E
r— —1
P ‘1
r— -1
" '1
E” =2
E. a
P Q
1- -1
1' '1
: J
1- d
- -1
' '1
1 l 1 J 1 l
0 '10 80 120
eLob
Figure 35. Laboratory Cross Section, 328, 40 MeV

10"r

103

10

do/dQ [mb/sr‘]

0.1

111

 

 

 

 

' 1 ' l 7 1

i
’ L10 '10 q
I C0[n,n] Co -
_ ELOsz1O'O MeV d
; :
h- -1
- 1
E E
P ‘
)II- .-
)- -1
- -1
5 5
- a
- 1
r I __
1— u-
L' 2
p— -1
b I q
1.
- q

1 1 1 l 1 1
0 L10 80 120
eLob

Figure 36. Laboratory Cross Section, 40Ca, 40 MeV

112

 

     
  
  
   

 

 

 

: r r r [ 1 f 3:
E 208Pb[n,n]208Pb E
3" EL0b=L10.O MeV ‘
10 1:— =3
,2 _ .
(r) P ‘
\ 100.. 1
E E 3
(>3) 1- q
i 103 ‘5
b E
'O _ _
P ‘1
1F

I 1

(]_1 1 1 1 1 1‘

1
90 80 120
9L06

Figure 37. Laboratory Cross Section,

D

208Pb, 40 MeV

113

 

  
   
    

 

 

 

 

q
10 1 1 1 l 1 I
28 . , 1
s.[n,n]288.
E 3
t .
H100;- ..
L. - -1
Q 3 :
E f 3
C3 _
D b -1
\
b
1’ 10:. ._
t a
1:. .:
: I ‘1 5
I 1 ‘
OJ 1 1 1 l 11 11
0 ‘10 80 120
CM ANGLE
Figure 38. Center of Mass Cross Section, 288i,

30 MeV

do/dQ [mb/er]

114

 

  
   
    
     
   

 

 

 

10‘. T I I I I I :
: 5
- q
- i
P 328 [n’n] 328 T
- -1
100:— a
10: 1
I .

1 l 1 l 1 1

0'10 1+0 80 120

CM ANGLE

Figure 39. Center of Mass Cross Section, 328, 30 MeV

dc/dQ [mb/sr)

115

 

 

 

 

10“ 1 T I r I r _‘
C I
P A
.1
" L“oCo[n,n]L+0Ccn ;
_ +

L’ I
b ..
100:. 1
; :
1— .1
)- d
_ + _
10:- 1
t: 4. :
1. —I
_ I .1
1__ 1

1 1 1 1 1 1

0'1 L10 80 120

CM ANGLE

Figure 40. Center of Mass Cross Section, Best Fit

40Ca, 30 MeV

116
10“

 

III

1 111111

LmCo[n.n]40Co
ELABZ3O'3MeV
AVERAGE GEOMETRV

l
l

103

1 1 1111q

1 1 111111

I
l

100

1 11111”
1111111d

I
9
l

dU/dQ (mb/sr)

g—o
O

1 1 1111”
+
1 1 1111“

I
J

1 1 1 1111]
1 11111d

T
J

 

 

 

OJ 1 .1 I 1 1 1
0 “0 80 120

CM ANGLE

Figure 41. Center of Mass Cross Section, Fixed Geometry
40Ca, 30 MeV

dG/dQ [mb/er]

117

10"

 

1 111111

.1 11111

20i-Q’Pb1r1.r1]‘208F’b
ELAB:30'3MeV

I
1

1 1 1111”
1.11111d

I
J

100

1 1*11r111l
1 1 111111

    
   

I
l

 

 

1 1 1111”
1,1 1 11111

I

1 11111”
1 1111111

I
l

 

 

 

1 1 1 l 1 1
0.10 I10 80 120

CM ANGLE

Figure 42. Center of Mass Cross Section, Best Fit

ZOBPb, 30 MeV

dU/dQ [mb/ 61"]

118

 

 

 

 

 

 

10” 1’ 1* V 1 ' 1 4
F 208 8 1
)— ._.
. Pb[n,n]20 Pb q
103.. .1
E AVERAGE GEOMETRY 3
- 1 -
100_ ..
: I :
c 1 . 1
b 1 4
~ 4
f 1
1— + q
1:- 1
1— -1
0J| 1 11 1 1 .1 1
0 -+0 80 120

CM ANGLE

Figure 43. Center of Mass Cross Section, Fixed Geometry
208Pb, 30 MeV

do/dQ [rub/er)

10“: I r I I T I :1
: 12c[n,n]12c :
3 ELAqu’OMeV
10 _
E :
L _
look— 3
E z
)— —4
I 1
10? .3
1:” '1
C 1
0J_ 1 l 1 1 1 1
0 I110 80 120
CM ANGLE

119

 

    
   
 
   
 
   
 
     

 

 

 

Figure 44.

Center of Mass Cross Section,

120, 40 MeV

dc/dQ [mb/sr)

120

 

  
   
  
   
   
    
    

 

 

 

10“: T I T I I i _‘
2881(n,n]2881 q
ELAB=%0Mev

103:. .1
E 3
f .
.. "1
100:. .1
I; I
1 4
S .
10: 1
r -
1:. .1
Z 3

1 1 l 1 ,1

OJ *J‘ 40 80 120

CM ANGLE

Figure 45. Center of Mass Cross Section, 288i, 40 MeV

dc/dQ [rub/er]

121

 

 

 

 

10*: T’ r l l I r 4
: 3
b 1
F -+
_ 328 [n,n] 328 q
3 ELAqu0MeV
10.—
: 3
:- J
+- '1
- A
100-— _
E i
‘0: _
: + i
F q
+— —1
F- .1
r— -l
1: -:
r- --1
OJ' 1 l 1 l 1 l
0 40 80 120

CM ANGLE

Figure 46. Center of Mass Cross Section, 328, 40 MeV

122

 

     
  
  
  
   

 

 

 

 

T T I 7 l E
: L*OCQ[n,n]L+0Co I
3_ ELob=L+0.0 MeV ‘
10 5' '5:
,Z:, _ A
(D ,— d
100_

3 a
g I: :
O} , ’ "
z 105- :-
b E -'-‘
'0 _ 1
1.-.- 1
E E
C -

[).1 1 l 1 i l 1 +

1
"1‘0 80 120

earn.

Figure 47. Center of Mass Cross Section, Best Fit

40Ca, 40 MeV

C3

dU/dQ [mb/sr]

123

 

I I I l

H
C)
(A)

I I IIIIII]

H
O
O

I I IITIIII

1...:
O

'7 ITITTT]

I I I lllll'

 

OJ 4

l l I 1’ I

LWCOInJflLmCo
ELob='-+0.O MeV

FIXED GEOMETR

1
. .1 ill

1 1 1 [11

Y

1 1 111111] 1 #1 11111111

i1JJ11uI

 

, 111111111

 

(I)

Figure 48.

l+0 80 120
ec.m.
Center of Mass Cross Section, Fixed
40

Geometry Ca, 40 MeV

124

 

  
  
   

 

 

I ' l I r :

I 208Pb[n,n]208Pb E

3r ELob='-+0.0 MeV 4

10 E' '2

E '3

,2 __ -
(0 P n
\ 100: 1
D E 5
E I :
\ 1°:
b : :
'C - q

 

 

 

1 41 4 l 1 l
0'10 1+0 80 120
9m.

Figure 49. Center of Mass Cross Section, Best Fit
208Pb, 40 MeV

OJ

125

 

  
  
 

 

 

_ I I 1* I I l E
E \ 208Pb[n.n]208Pb ?
_ EL0b=L+0.0 MeV "
f FIXED GEOMETRY:

I I IIIIIII
c1 1 1|111fl

  

I I VIII!"

-
q
.1
—I
_
q
q

 

 

1 l L l 1 l

L+0 80 120
€)c>.rYI.

Figure 50. Center of Mass Cross Section, Fixed

Geometry 208Pb, 40 MeV

 

CD

RELA11VE
ERROR
(NB/SR)

(MB/SR)

DIFFERENTIAL
CROSS-SECTIUN

CLNTER-OF-MASS
(DEG)

CORRECTED
ALGLE

126

ERROR
(MB/SR)

Measured Differential Cross Section
RELATIVE

30.3 MEV
DIFFERENIIAL
CROSS-SECTION

(MB/SR)

Table 14.
2551(R,R)
bNCOthCTED
LABORATORY

ANGLE
(DEG)

»

L

00000
000000000000000000

‘

.9665 02
.5075 0
g
u
8
In
8
6
Z
7
9
B
9
5
2

3.4212125135232111
o o o

u~33.32n.£.£2222111111000
00000000000000000000

”7093630u1771283
67212..“ 91600u501876050
.0u200u0765321u70665;
1031556u3212675232

O O O

53062713.“.306028000
5790232789001111086

O 0 O O O O I 0 O O O O O O O O O O O O
50516161616272722211
122%”5566778890123

22111111110100000000
00000000000000000000

EEEEEEEEEEEEEEEEEEEE
90262001u01206u255
50139882352“ 351572
959“.“112125135232111

“33222222121111110
00000000000000000000

1731056u321817763

23
3122691897u306 50702
117

.

00000000000000000000
cocooooooooooooooooo

50505050505050500000
19.-2%.“.5566778090123

HELPIIVt
EHhUH
(Mb/SR)

DIFFERERIIAL
CROSS-SECTION
Mb/SR)

CORRECTED
CENTER-OE-MASS

ANGLE
(DEG)

RELATIVE
ERROR
(MB/SR)

0.3 MEV
MB/SH

<

'-
of

DIFFERENTIAL
CRO?S-SEC§ION

325(N,R)
bLCOthCTED
LAEUHATORY
ARGLE

(DEG)

fiL21|1|1|11l1l1|1l100000011l1
00000000000000000anw0.
LEEEEEEEEEEEEEEEEEEE
02.23707068119.“ 09010.0
90911 665.001; 1531.40
m416“6n¢22111uuu221769

O O .......

n“. 3332 2322?.22211111000
00000000000000000000

095.48 310006u227812u

706333210 16631691033929

“”005502025108210907

11(17919521118‘062763
I

8.401370231410150 .1106 0000
”$319023“ 56678 bBBBRuHOhnm
O O I 0 O I O O O O O 0 O O O O O
50C).0,04I61|61I616 1611111-
12233.45566775890123

111111

21111111111100
0

000000000000

“33223222222111.1110
00000000000000000000

0 0392052021.“. 70000000
7922003HB1 58 95722
3293000928211 91373110
17219110521118862113

505050505050500000

1111

127

Table 14. (cont'd)
MOCA(N,N) 30.3 MEV

OROORR8018O CORR80180
LABORATORY CENTER-OF-MASS
ANGLE DIFFERENTIAL RELATIVE ANGLE DIFFERENTIAL RELATIVE
(08G) CROSS-SECTION ERROR (DEG) CROSS-$807168 ERROR
(MB/SR) (MB/SR) (Mb/SR) (HE/SE)
15.0 .16888 08 .1158 03 15.39 .18058 08 .1168 03
20.0 .53038 03 .3188 02 20.51 .53618 03 .3238 02_
25.0 .1 8 8 0 .1208 02 25.6 .17028 03 .1228 02
0.0 .92768 0 . 708 01 30.; .80028 02 . 778 01
5.0 .13008 03 . 378 01 35. .12638 03 . 518 01
0.0 .15868 03 .8768 01 0.96 .163 8 03 .5008 01
85.0 .53Z18 O .3378 01 86.03 .181 8 0 .3818 01
50.0 . 68 0 . 18 01 51.1 .888 8 0 . 988 01
55.0 . 2 8 02 .1288 01 56.22 88228 02 .138E 01
60.0 .2 z 8 02 .1368 01 61.29 .23568 02 .1398 01
65.0 .2 8 02 .8208 00 66.35 .22278 02 .8568 00
70.0 .2 8 02 .gg18 00 71. 0 .20738 02 .7798 00
$5.0 .1 018 02 . 08 00 36.88 .1 118 02 .7018 00
0.0 .1 058 02 .5888 00 1.8g .1 158 02 .5658 00
85.0 .81 08 01 .2858 00 86.8 1818 01 .2998 00
90.0 .53 08 01 .1858 00 91.83 8608 01 .1598 00
95.0 .37 08 01 .2898 00 96.8 .31878 01 .2928 00
100.0 .30 08 01 .1368 00 101.50 .28758 01 .1398 00
110.0 .1 008 01 .1228 00 111.80 .19908 01 .12 8 00
120.0 .1 808 O1 .2088 00 121.30 .17SZE 01 .20 8 00
130.0 .69008 00 .1668 00 131.10 .663 8 00 .16 8 00
1 5.0 .60008 00 .1208 00
160.0 .35008 01 .8208 00

RELATIVE
ERROR
(ME/SR)

MB/SR)

DIFFERENTIAL
CROES-SECTION

CORRECTED
CENTER-OF-MASS
ANGLE

(080)

128

RELATIVE
ERROR
(MB/SR)

(cont'd)

30.3 MEV
DIFFERENTIAL
CRO?S-SEC§IUN

MB/SR

Table 14.
208PB(N,N)
UNCORRECTED
LABORATORY
ANGLE
(DEG)

0000000000000000000

00

2211111111

u 222222222222222222222

0000000000000000000000000000000000
EEEEEEEFREELEEEBEEEEEEEEEEEEEEEEEE
552 7.4 90 9526 90 51 58u235720379u11uu 2a.".
780293781002282869.9320052202768
66718 939037716059212072295167370715
2368 6 2235222122111072233

O O O O C O

0011111.11!11222222222222222223332
O

0
O O O O O O O O O O O O O O O ....... C O O I O O I O O O O
5

r3811“. 700 923 .Iu—Ioggz 581" 70505050
1| 122 23533.15.“ 55266677788890.100112 23

22211111111111111100000

00000000000000000000000

OO
00

83l86u322211111111755555uu

n.332222222222222222222222211111111
0000000000000000000000000000000000
EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE
.46“. 372333175037050 3112282068537
10133000619“.02181u01u0852257685953
3‘”29523802210362u201‘2875299
272196330876u353322222
O O ....... I O O O

000000000000000000000000000000000
..................................

RELATIVE
ERROR
(ME/SR)

DIFFERENTIAL
CROSS-SECTION
(ME/SR)

CENTER-CF-NASS

CORRECTED
ANGLE

(DEG)

129

RELATIVE
ERROR
(NE/SR)

(cont'd)

“0.0 MEV
DIFFERENTIAL
CROSS-SECTION

(ME/SR)

Table 14.
12C(N,A)
DRCORRLCTED
LALCRATORI
ANGLE

(DEG)

2211110000000111

0000000000000nw0.n.v

EEEEEEEEEEEEEEEE

32.5129715u152387

902203066331052

11.5321RUCJBB111866
0

3333222211111110
0000000000000000

337.01.“ 0 2881627361
”b 31u73255055322
29052“ 7u80198706
CJrfiw31rO211896u2116

........... O O O O

011072u8333280
37122313578990

................
61727356323395
12%‘33u556677889

21.111100000001111
EEEEEEEEEEEEEEEEE
29180373610u1029
721.38 2.4“.“171305312
18fl21175331117661

O ........ O O O I I O O O

333.52222211111101
00000000000000000.
EEEEEEEEEEEEEEEEE
73302uu2u0000000
526230277u671060
091786 3708 16“.?“57
9.5317321197532193

0 O O O O O O O ..... O O O 0

000000 00000000000
505050 5050 5050505
12,632.".526778891

1

RELATIVE
ERROR
(Mb/SR)

DIFFERENTIAL
CROSS-SECTION
(MB/SR)

CORRECTED
CENTER-OE-MAss
ANGLE

(DEG)

RELATIVE
ERROR
(MB/SR)

No.0 MEV
DIFFERENTIAL
CRO?S-SECTION

MB/SR)

28SI(N,N)
UNCORRECTED
LAtORATOR!
ANGLE

(DEG)

22111111000000011
3191153usn/L8207212
60022808” 57135282
627321218 52221199

0 O O O O O O O O O I O O O I O O
u3322222221111110
00000000000000000
680.40.“.896 9921910

32 376 2 1123771007

508.". 91660 17631940

182956 532 16533112
. O

£251;l280otbfironxc731AGXU
57902356789001119

0 O O O O C O O O O O O O O O O O
50516161616272726

1

9.2111111000000011
8&15CJ150u 5522832
“82,817802u92u122
61622111852121199

0 O O O C O O C C O O O O O O 0
“332222221111110
00000000000000000
EEEEEEEEEEEEEEEEE
367.“.920880531500
7o 08 563191822101.“
“01115782.“ 115081“
183176532197uu221

O O

00000000000000000
0 O O O O ..... O O O C O O C
50 505050505050505
12233u5566778891
1

RELATIVE
ERROR
(MB/SH)

LIEFLHLLTIAL
CROSS-SECTION
(ME/SR)

CORRECTED
CENTER-OF-MASS

ALGLE
(DEG)

130

RELATIVE
ERROR
(MB/SR)

(cont'd)
no.0 MEV
DIFPERENTIAL
CROSS-SECTION
(MB/SR)

Table 14.
328(N,N)
UNCORRECTED
LABORATORY
ANGLE

(DEG)

22211111000000010
0000000 0000000040
EEEEEEEEEEEEEEEEE
34138205596nu8u063
1503“ 62u7871731u1
721..“3221952211181
O O O O O O O I O O O O O .
“33222952221111110
00000000000000000
buug9032213812
5972782625.“. 535290
61.102715220863239
1829786u217753213
O O O O .

39.4513uu30715780
“67902327788887
00000 O O O O O O I O O O O 0
50.506161616161616
12233.4” 526778891

1

95211111100
0000000 000

00
E 00
E 00
E 00
E 00
E-01
E 00

“3.4432222221111110
00000000000000000

EEEEEEEEEEEEEEEEE

1831086u21975321

O O O O O O O O O O O O O O O

SOSOSOSOSOCJOSOSOS
1.2.2 33““ 52677-8891
1

RELATIVE
ERECR
(LL/SH)

CBCSS-SLCTIOL
(MB/Sh)

CENTER-OF-MASS
DIFFERENTIAL

ANGLE

CORRECTED
(DEG)

RELATIVE
ERROR
(MB/SR)

“0.0 MEV
DIFbERENTIAL
CROSS-SECTION

(EB/SR)

uOCA(N,N)
UNCOHEECTED
LABORATORY
ANGLE

(DEG)

2211111110000000000

0000000000000000000

EEEEEEEEEEEEMBE.EEEEE

1325:0696025044t C28u1l7

26955991801526.571h428
2

0037.“ 332112

“3331.32rc22211111000
0000000000000000000

EEEEEEEEEEEEEFE REEF“
1850 5501.5 72857u6c.,203
50179053 07135.4 76 270
72231295n28 7.7.0.43“.
19r/h11l18311196311u1u.

91in! 567658615600000
3:67890123uur5533

O I O O O O O 0 C ..... O O O
5050506161616161111
122339u55§b77559012

221111110000
000000000000

“33332222211111100
0000000000000000000

0

80
626.“?1095252 O.
1921118u11 96

.1

O I O C O O O O C O O
505050505050 5050000
12233.“ FJ267768 9011/.

RELATIVE
ERROR
(ML/Sh)

DIFFERENTIAL
ChOSS-SECTICN
(ME/SR)

CORRECTED
CENTER-DR—MAES
ANGLE
(DEG)

131

RELATIVE
ERROR
(ME/SR)

(cont'd)
”0.0 MEV

DIFFERENTIAL
GROSS-SECTION
(MB/SR)

Table 14.
ZObPE(N,N)
UNCGRRECIED
LABORATGR!
ANGLE
(DEG)

211111111111110000

00
00000000000000000000

2
0
EEEEIDEEEEEELDEEEEEEEEEEE
920.81u0918027131033961u76537318
01C/8109hfi 92.05.“ n£0293613737un¢9016
al.1fl755u32ncré11111976555u322111987

211111111111

2
0000000000000
E

@“1790503133
111656u3211111

0.0.1I.1I.1111112.62222222222222222533322

000000000000 O O C O O O O O O O
21.“.703092 581“ 70129258L5m5nw5lu5050
11222333””5556666677788990011223

2222111111111111000000000000001111

0000000000000000000000000000004w4w

EEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

867970”599u351u83u6bou10512u536004

820037986171uu310329u2801626310695

211175”332221111987555uu3221119877
O O O

333§3§3222222222222221111111111
000ooooooooooooooooooo000000000000
BEEEEEEBEELEEEEEEEEEEEEEEEEEEEEEEE
67u8155780909106662508310000000000
606 37.727919560120010“ 0866671.“.91 185
Run/70090032107869.9252075070153293
62211111196555“ .222111876u321
. C .

00000.00000000000000000000000000000
581“",030995561u701292315050505050
11229.3333”“555666667758990011223

1111111

Al

Au

Be

Bh

Bl

Br
Br

Ca

DO

Fe

Fe

Fe

Fe

Fe
Fe
Fe
Ga

Ha

61

7O

69

77

66

77
78

75

74

49

76

77

54

58a
58b
62
78

76

132
LIST OF REFERENCES
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N. Austern, "Direct Nuclear Reaction Theories",
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F.D. Becchetti,;hu &(LE. Greenlees, Phys. Rev.,
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R.K. Bhowmik, R.R. Doring, L.E. Young, Sam M. Austin,
A. Galonsky & S.D. Schery, Nucl. Inst. Meth., 143
(1977)63.

L.N. Blumberg, E.E. Gross, A. van der Woude, &
A. Zucker, Phys. Rev., 147(1966)812.

F.A. Brieva & J.R. Rook, Nucl. Phys. A291(1977)3l7.
F.A. Brieva & J.R. Rook, Nucl. Phys. A297(l978)206.

J.D. Carlson, C.D. Zafiratos, & D.A. Lind, Nucl.
Phys. A249(l975)29.

R.R. Doering, Ph.D. Thesis, MSU, 1974.

S. Fernback, R. Serber & T.B. Taylor, Phys. Rev.
Z§(l949)l352.

J.C. Ferrer, et al., Phys. Lett. 62B(l976)399.

J.C. Ferrer, J.D. Carlson and J. Rapaport, Nucl.
Phys. A275(l977)325.

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