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ABSTRACT
THE EFFECT OF INTERNAL NUCLEON

MOTION AND NUCLEAR BINDING
ON ELASTIC PION-NUCLEUS SCATTERING

By
Kenneth E. Gilbert

This thesis is a theoretical study of elastic pion-
nucleus scattering at energies near the pion-nucleon (3,3)
resonance ( EiTT = 100:300 MeV). Specifically, the work
investigates the effect of internal nucleon motion and nuclear
binding on resonant elastic pion-nucleus scattering. The nuclei
considered are uHe, 120, and 160.

The basic approach is first to calculate the effects
using exact numerical methods and then to develop analytic
approximations which take into account the main features of
the numerical results. In all cases, an optical potential is
obtained which contains no adjustable parameters.

Nucleon motion effects are calculated exactly using
an independent particle model for*uHe and the free pion-nucleon
t-matrix. This exact model, which is numerical, is compared
with three existing models for pdonénucleus elastic scattering.
The existing models all treat nucleon motion in an ad hog way
and none of them give satisfactory agreement with the exact

numerical model. By systematically approximating the multiple

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Kenneth E. Gilbert

integrals required in the exact model, an improved analytic model
is developed. The improved analytic model gives excellent agree-
ment with the exact numerical model and, in addition, allows a
transparent understanding of the main effects of nucleon motion.

The nucleon motion model for'uHe is generalized for the
cases of 12C and 16O and a comparison is made with experimental
data fOr all three nuclei. Both the exact numerical model and
its analytic equivalent give significantly improved agreement
with the 1’He data and good agreement with the 12c and 160 data.

We conclude that nucleon motion effects are significant
and Should be accurately accounted for since §9.222 treatments can
lead to spurious results.

Nuclear binding effects are calculated using a 3-body
model for'uHe. The target nucleon is bound to the rest of the
“He nucleus by an s-wave separable potential which has a single
bound state at ~20 MeV. In this model, binding effects are
relatively small and can be approximated with a simple analytic
formula. By extending the formula to include an infinite number
of bound states, an upper limit on binding effects is estimated.
In this unphysical limiting case, the effects of nuclear binding
are about twice as large as in the single-state case.

After generalizing the uHe 3-body model for the cases
of 120 and 16O, the results are compared with experiment. The
relatively small binding effect in the singleestate model is

found to be compatible with the experimental data. However,

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Kenneth E. Gilbert

the unphysical model with an infinite number of bound states gives
poor agreement with experiment.
We conclude that nuclear binding is a relatively small

effect in resonant elastic pion-nucleus scattering.

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THE EFFECT OF INTERNAL NUCLEON
MOTION AND NUCLEAR BINDING

ON ELASTIC PION—NUCLEUS SCATTERING

By
., I)
~d$‘

d-
Kenneth E. Gilbert

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Physics

1976

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ACKNOWLEDGMENTS

I would like to thank my advisor, Professor J. H.
Hetherington, for suggesting this problem and for helping me
through some difficult periods in the work. I am grateful
also to Professors J. Borysowicz and H. McManus for many
helpful suggestions.

I gratefully acknowledge the use of the excellent
computing facilities of the Michigan State University Cyclotron
Laboratory and the able assistance of the computer staff.

I also thank Drs. R. R. Doering and W. F. Steele for generous
help with computer codes.

The Center for Naval Analyses is very much thanked
for printing the thesis.

Most of all I want to thank my wife, Lynn, for her

patience and understanding. This thesis is dedicated to her.

ii

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v

TABLE OF CONTENTS

Page
ACMWI‘EDGM'ENTS I I O I I C O O I I O D O I 0 ii
LIST OF TABLES o o o o u o o o o o o o o o o Vii
LIST OF FIGURES o o o o o o o o o o o o o o o Viii
Chapter
I. INTRODUCTION . . . . . . . . . . . . l
l. The Pi-Meson As a Nuclear Probe . . . . l
2. Elastic Scattering of Pions from Nuclei . 2
3. The Present Experimental Situation . . . 3
h. Previous Calculations . . . . . . . 3
5. This Calculation . . . . . . . . . 4
II. THE OPTICAL POTENTIAL FORMALISM AND
PION-NUCLEUS ELASTIC SCATTERING . . . . . . ll
1. Formal Theory . . . . . . . . . . ll
2. Qualitative Remarks . . . . . . . . 18
III. CALCULATION OF THE ‘TT'-uHe OPTICAL POTENTIAL:
‘ INTEGRATION OVER THE TARGET NUCLEON MOMENTA . . 2O
1. An Independent Particle Model for 4He . . 20
2. Some Numerical Considerations . . . . . 2
3. Derivation of the TT'-uHe Optical
Potential Using Vector Brackets . . . . 22
IV. THE FACTORED FORM OF THE 1T -uHe OPTICAL
POTEINTIAL: Two E2. Egg- MODES I O O O O O O 32
1. Definition of the Factored Form . . . . 32
2. The Ad E22 Models . . . . . . . . . 33
3. Angle Transformations . . . . . . . 35
h. The Simple Static Approximation . . . . 36
5. The Modified Static Approximation . . . 37
6. The Effective Collision Energy . . . . 39

iii

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Chapter

V.

VI.

VII.

VIII.

IX.

THE FACTORED FORM OF THE 11’ -”Ha OPTICAL
POTENTIAL: A RECENT 52 Egg MODEL . . . . .

THE FACTORED FORM OF THE 1? - He OPTICAL POTENTIAL:

L.

THE EFFECTIVE PION-NUCLEON IMPULSE INTERACTION.

1.

2.

3.

Definition of the Effective Impulse

Interaction . . .

The Fully-Integrated Impulse Interaction
Compared to the Simple Static Approximation

and the Modified Static Approximation .

Comparison of the Fully-Integrated Impulse
Interaction with the Kujawski-Miller—Landau

MOdel I I I I I I I I I I I I

A SYSTEMATIC APPROXIMATION FOR THE EFFECTIVE
PION-NUCLEON IMPULSE INTERACTION . . . . .

l.
2.

3.

A Linear Approximation . . . . . .
A Quadratic Approximation . . .

Qualitative Effects of Nucleon Motion in.
the Effective Impulse Interaction . . .

CORRECTIONS TO THE IMPULSE APPROXIMATION . .

1. A 3-B0dy MOdel o o o o o o I o o
2. Numerical Results for Tm, . . . . .
. A Single State Approximation for
Binding Effects . . . . . . . . .
14'. A Closure Limit 0 I o o o o o o o
CALCULATION OF THE ELASTIC SCATTERING
CROSS SECTIONS . . . . . . . . . . .
1. Relation of the Cross Section to

GUI-[3‘ DON

the PionéNucleus t-Matrix . . . . .
Calculation of the Pion-Nucleus t-Matrix
Partial—Wave Decomposition of the
Optical Potential . . . . . . .
The Form Factors . . . . . . .
Treatment of the Non-Resonant Channels
Coulomb Effects . . . . . . . .

COMPARISON OF THE CALCULATED CROSS SECTIONS
WITH EXPERIMENT O ‘ o I o o o o o o o

1.

Impulse Approximation Results for “He .

a. The First Minimum in a e . . .
b. The Second Minimum in He . . . .

iv

Page

41

A6

46

48

53

58
61

68
71
72

80
90

100

100
102

103
107
109
109

110

110

110
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Chapter
X. c. The Discrepancy at Forward Angles . . .
d. Relation of the “He Cross Sections to
the Angle Transformation . . . . .
e. The Quadratic Approximation and the
Kujawski-Miller—Landau Approximation . .
2. Impulse-plus-Binding Correction Results for “He
a. The Single State Approximation for
Binding Effects .
b. The Closure Approximation for Binding Effects
3. Impulse Approximation Results for 120 . . .
a. Nature of the Minima in 12C . . . .
b. The Fully-Integrated Impulse Approximation
Results for 120 . . . . .
0. Comparison of the Modified Static Approxi-
mation (MSA) and Simple Static Approximation
(SSA) to the Fully-Integrated Impulse
Approximation (FIA) o o o o o o o o
d. The Quadratic Approximation and the
Kujiwski-Miller-Landau (KML) Approximation
u. Impulse-plus-Binding Correction Results for 120
5. Impulse Approximation Results for 16O . . .
a. The Fully—IntegratigO Impulse Approximation
(FIA) Results for . . . . .
b. Compigison of the MSA and SSA to the FIA
for I I I I I I I I I I I
c. The Quadratic Approximation and the
Kujigski-Miller-Landau (XML) Approximation
6. Impulse-plus-Binding Correction Results for 16O
XI. Summary and Conclusions . . . . . . . . . .

BIBLI WEAR-[Y I I I I I I I I I I I I I I I I I
WI cm I I I I I I I I I I I I I I I I I
A. REFERENCE FRAMES AND KINEMATICS . . . . . . .

B. PARAMETERIZATION OF THE OFF-SHELL PION-NUCLEON
t-MATRIX c o o o o o o o I o o o o o O

122
125

127
127

128

134

135
146

146

152

152

165
165
171
175
179
180

188

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APPENDICES Page

Go NOTATION I o I o o o I o I o 0 I c o 193
D. VECTOR BRACKETS . . . . . . . . . . . . 201
E. THEORETICAL JUSTIFICATION OF THE 3-BODY MODEL

IN PION-NUCLEUS SCATTERING I c I o o I O O 210
F. CALCULATION OF BINDING CORRECTIONS USING

VECTOR BRACKETS . . . . . . . . . . . . 217
G. SOLUTION OF THE LIPPMANN-SCHWINGER EQUATION . . . 230

vi

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Table

5-1

LIST OF TABLES

A comparison of the angle transformations used
in the simple static approximation (SSA), the
modified static approximation (MSA), and the
Kujawski-Miller—Landau approximation (KML). .

vii

Page

. . 43

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LIST OF FIGURES

Figure Page
3.1 The relationship between the pion and nucleon
momenta in the 'fl'DCM frame and the total
momentum 9c“ and relative momentum &" . . . . . . 30
~ ~
5.1 Graphical representation of the angle transfor-

mations used in the simple static approximation
(SSA), modified static approximation (MSA), and
the Kujawski-Miller-Landau approximatifn (KML)

for‘S‘ \P'\= E;=l.5fm........44

6.1 The real and imaginary parts of the effective
pion-nucleon impulse interaction as a function
of the cosine of the pion-nucleus C.M. scattering
angle calculated by three methods: (a) (solid line)
the effective interaction resultin from full integra-
tion of the impulse approximation %FIA); (b) (short-
dash line) the sim 1e static approximation (SSA);
(C) (dot— dash line the modified static approximation
MSA). The laboratory kinetic energy of the pion
is 110 MeV. . . . . . . . . . . . . . . . . . . . . . . 60

6.2 . As in Figure 6.1 except at 180 MeV. Note the
strong down bending at backward angles in the real
part of the fully—integrated impulse interaction (FIA). 51

6.3 As in Figure 6.1 exoept at 260 MeV. . . . . . . . . . . 52

6.4 The real and imaginary parts of the effective pion-
nucleon impulse interaction as a function of the
cosine of the pion-nucleus C. M. scattering angle
calculated by two methods: (a) (solid line) the
effective interaction resultin from full integration
of the impulse approximation %FM) (b) (large dots)
the Kujawski-Miller—Landau approximation (KML) to
case (a). The laboratory kinetic energy of the pion
is 11-0 neVI I I I I I I I I I I I I I I I I I I I I I I 55

6.5 As in Figure 6.4 except at 180 MeV. . . . . . . . . . . 56

6.6 As in Figure 6.4 except at 260 MeV. . . . . . . . . . . 57

viii

. . : a. i L. .. I...) ... a .3 3 v .
2. H. C :. 3 i 3 3 I. .r“ E a” K... .L
e X E :. ...,-.\... 1: X ..
. . .1 8 E .. 74\ .. .A E a. E C .
3 t; E e n 1.. .1
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."1....v..r. v..: 2 I. 5 .... v..." 5 a“
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‘4‘

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“Us

7.1

7.2

7-3
7.4

8.1
8.2

803

Page

The real and imaginary parts of the effective

pion-nucleon impulse interaction as a function

of the cosine of the pion-nucleus C. M. scattering

angle calculated by two methods: (a) (solid line)

the effective interaction resulting from full

inte tion of the impulse approximation (FIA3%

(b) large dots) the linear approximation (LA

case (a). The laboratory kinetic energy of the

pion is 110 MeV. . . . . . . . . . . . . . . . . . . . 63

As in Figure 7.1 except at 180 MeV. The down bending

at backward angles in the real part is due to the

higher average two-body scattering energy for large

angle scattering, which for this case actually raises

the energy from below to above resonance. . . . . . . . 64

As in Figure 7.1 except at 260 MeV. . . . . . . . . . . 65

The real and imaginary parts of alga as a function

of the pion-nucleon C. M. kinetic energy. The

quantity' OJ¢~\is the total relativistic pion—nucleon

C. M. energy so that uh,“- (mwi' (““303

is the corresponding kinetic energy. . . . . . . . . . 66

Diagrammatic representation of equation (8.10) . . . . 76

An Argand diagram of the L = 1 component of

(P‘ \ <0 “:1: fl\Q>\P PE) (impulse result).

and

(P1: \(O Y“ 1"“ \ 0>\P1> (illlpulse plus

binding effects).

The solid line is the impulse result and the dashed

line is the impulse result plus the binding correction.

At a given energy, the arrow connecting the solid line

and the dashed line is the binding correction itself. . 79

The real and imaginary parts of the binding correction
shown in Figure 8.2 (i.e., the arrow in Figure 8.2).
These curves result from an exact numerical evaluation

°f < E. ‘ (“1533859 Ru ‘30P?“ °7\ E! 5

Note that the binding correction is only 10-20% of the
impflsetemoooooouooooooocoo-coo. 8].

ix

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calculated two ways: (a) (solid line) the exact

result using vector brackets; (b) (dashed line)

the single state approximation from equation (8.33).

The solid line is the same as in Figure 8.3. . . . . . . 88

Three different versions of the effective pion-

nucleon interaction: (a) (solid line) the fully-
integrated impulse result FIA); (b) dashed line)

the single state approximation for binding effects

(SSAB); (c) (dot-dash line) the closure limit for

binding effects (CAB). The laboratory kinetic energy
OfthepionisllOMeV...o............. 97

As in Figure 8.5 except at 180 MeV. . . . . . . . . . . 98
As in Figure 8.5 except at 260 MeV. . . . . . . . . . . 99

The elastic 17 -uHe differential cross sections
resulting from three different effective pion—
nucleon impulse interactions; (a) (solid line)

the fully-integrated impulse result (FIA);

Eb) (short dash line) the simple static approximation
BSA); (0) (dot-dash line) the modified static
approximation (MSA). The energies shown are the

. laboratory kinetic energy of the pion. . . . . . . . . .112

The elastic TT-uHe differential cross sections

calculated by two methods: (a) (solid line) the
fully-integrated impulse result FIA); (b) dashed line)

the quadratic approximation (QA) to case (a . The

energies shown are the laboratory kinetic energy of

the pion. . . . . . . . . . . . . . . . . . . . . . . . 119

The elastic 1T ~uHe differential cross sections

calculated by two methods: (a; (solid line) the fully—
integrated impulse result (FIA ; (b) (dashed line) the
Kujawski-Millethandau (KML) approximation. The energies
shown are the laboratory kinetic energy of the pion. . .121

The elastic 1T J‘LHe differential cross sections

resulting from three different effective pion-nucleon
interactions: (a) (solid line) the fully-integrated
impulse result FIA): (b) (short-dash line) the single
state approximation for binding effects (SSAB); (c) dot-
dash line) the closure limit for binding effects (CAB).

The energies shown are the laboratory kinetic energy of
the pion. . . . . . . . . . . . . . . . . . . . . . . . 124

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10.6
10.7
10.8

10.9

10.10
10.11
10.12

10.13

10.14
10.15
10.16
10.17

The elastic'Tr -12C differential cross sections
resulting from three different effective pion-
nucleon impulse interactions: (a) (solid line)
the fully-integrated impulse result FIA); (b)
(short dash line) the simple static approximation
SSA): (c) (dot-dash line) the modified static
approximation (MSA). The energies shown are the
laboratory kinetic energy of the pion . . . . .

AS in mm 10. 5 o o o o o o o o o o o o o o c a .
AS in Figure 100 5 o o o a o o o o o o o o o o o o 0
AS in Figure 10.5 o o o o o o u o o o o o o o o o o

The elastic Tl -12C differential cross sections
calculated by two methods: (a) (solid line) the
fully-integrated impulse result (FIA); (b)

(dashed line) the quadratic approximation (QA)

to case (a). The energies shown are the laboratory
kinetic energy of the pion. . . . . . . . . . . . .

As in Figure 10.9 . . . . . . . . . . . . . . . . .
As in Figure 10.9 . . . . . . . . . . . . . . . . .
AS in Fim 10I9 I I I I I I I I I I I I I I I I I

The elastic 1T ~12C differential cross sections
calculated by two methods: (a) (solid line) the
fullyeintegrated impulse result (FIA); (b)

(dashed line) the Kujawski-Miller—Landau (KML)
approximation. The energies shown are the labora-
tory kinetic enerw 0f the Pionc c o o o o o o o o 0

AS in Figlre 10I13I I I I I I I I I I I I I I I I I
As in Figure 10.13. . . . . . . . . . . . . . . . .
AS in Figure 10.130 0 o o o o c o o o o o o o o o o

The elastic‘TT ~12C differential cross sections
resulting from three different effective pion-
nucleon interactions: (a) (solid line) the
fully-integrated impulse result (FIA); (b)

(short dash line) the sin e state approximation
for binding effects (SSAB§% (c) (dot-dash line)
the closure limit for binding effects (CAB). The
energies shown are the laboratory kinetic energy
Ofthepionooooooocoocoo-0000000

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AsinFigure10.17................ 149
AsinFigure10.l7................ 150
AsinFiguIeIO.l7................ 151

The elastic TT -160 differential cross sections
resulting from three different effective pion-
nucleon impulse interactions: (a) (solid line;
the fully-integrated impulse result (FIA); (b
(short dash line) the simple static approximation
SSA); (c) (dot-dash line) the modified static
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AsinHSUI‘BlO.21..............o. 155

AS in Fim 10.21 I I I I I I I I I I I I I I I I l56
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The elastic TY - O differential cross sections
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tory kinetic energy of the pion. . . . . . . . . . 158
ASinFigureloozu'ooooooooooooooon 159
AsinFigurelO.24................ 160

The elastic TI -l60 differential cross sections
calculated by two methods: (a) (solid line)

'the fu1ly-integrated impulse result (FIA): (b)
(dashed line) the Kujawski-Miller—Landau (KML)
approximation.‘ The energies shown are the labora-

‘tory kinetic energy of the pion. . . . . . . . . . 162

ASinFigurel-09270000000000000000 163

ASinFigllrel-Ooz'ZQcoo-00000000000 16+
16

The elastic‘l'l’ - O differential cross sections

resulting from three different effective pion-

nucleon interactions: (a) (solid line) the fully-

inte ted impulse result (FIA); (b) (short dash
linegrihe sin 6 state approximation for binding

effects (SSAB : (c) (dot-dash line) the closure

limit for binding effects (CAB). The energies

shown are the laboratory kinetic energy of the pion. 167

xii

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xiii

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CHAPTER I

INTRODUCTION

1. The Pi-Meson As a Nuclear Probe
It is certainly reasonable to ask why pi-mesons (pions)

should be used to probe the nucleus. After all, it is much easier

(and cheaper) to use more conventional projectiles such as protons
The answer is that pions have some unique properties

and electrons.
First, pions

Which make them particularly useful as nuclear probes.

have three charge states (17* , 11", 1r°) so that double charge
Second, pions are bosons and therefore can

exchange is possible.
(Absorption

be absorbed by "clusters" of nucleons in the nucleus.
by a single free nucleon cannot occur.) Hence, pion absorption

experiments may yield new information about correlations among

nucleons inside the nucleus. Third, and perhaps most important,

the pion-nucleon interaction is dominated in the 100-300 MeV region

by the well-known (3,3) resonance;L The (3,3) resonance makes it

Pessible to vary the strength of the pion-nucleus interaction by

¥

e quantum numbers associated with the (3,3) resonance

SCI-re I (isospin) = 3/2, J (total angular momentum = 3 2, and L
eorbital angular momentum) = 1. At resonance, the laboratory kinetic
nergy 01‘ the incident pion is about 190 I‘19),-

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more than an order of magnitude simply by varying the energy of

the incident pion beam. This convenient "knob" on the interaction

strength is not present with conventional probes such as protons and
electrons. Fourth, the (3,3) resonance has total isospin 3/2 so that

positive pions interact mainly with protons and negative pions
interact mainly with neutrons. Pions should therefore be useful

in determining proton and neutron distributions. Finally, the zero

spin and relatively small mass of the pion considerably simplify

theoretical analysis of pion-nucleus scattering.

2. Elastic Scattering of Pions from Nuclei
When an energetic particle such as a 100-300 MeV pion

COllides with a nucleus it usually scatters elastically, so in

many cases,inelastic scattering can be treated as a small perturbation

On the elastic wave function. In order to use a perturbative method

f 01' inelastic scattering, it is necessary to have an accurate elastic

Wave function as input. A careful experimental and theoretical

Study Of elastic scattering is therefore the first step toward a

COmplete description of pion-nucleus scattering. Hence, in recent

Years, elastic pion-nucleus scattering has received increased attention

by both experimentalists and theorists.

,, lOne such perturbative approach is the well-Imown
distorted-wave" Born approximation (DWBA) .

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3, The Present Experimental Situation
The present experimental situation in pion-nucleus physics

is a bit strange. Several "meson factories" have recently become

operational and should produce high quality pion-nucleus scattering

data in the very near future. But, so far, no results from the

meson factories have been published and the only data presently

available come from older accelerators which were not designed for

pion-nucleus scattering. While these data are not of the quality

eXpected from the meson factories, some fairly good elastic scattering

measurements have been made for light nuclei. Binon et. al. have

measured differential cross sections for elastic 1T -1 C scattering

at 120, 150, 180, 200, 230, 260 and 280 MeV (Bin70). The same group

has reported preliminary results for elastic 'W-uHe scattering at

110; 150, 180, 220 and 260 MeV (Bin7l). Bercaw et. al. have measured

elastic differential cross sections for 11' -l60 scattering at 160,

170: 220. 230 and 240 MeV (Ber72). Other data have been reported 1

but the “He, 12C and 160 data cited here are the most systematic

data available at present in the (3,3) resonance region (100-300 MeV).

LP . Previous Calculations
Many theoretical calculations have been done to explain the

12
T- C and ‘W-16O data2 and a few have been done to explain the

1;
1r- He data (Mac73, Ger73, Fra74). Most of the calculations use

‘1

1For a listing of experimental results see K0170, Cer74.

re 2It is impractical to give a complete list here. Some more
cent results can be found in Lan73, Pha73, Kis74, Kuj74.

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either the Glauber theory (Gla67) or an optical potential model
derived from Watson's multiple scattering theory (Wat53, Ker59).
These calculations describe the ‘W-lZC and 'W—l60 data fairly well,

regardless of the model, but are in poor agreement with the Tr-uHe

data. 1

A survey of the pion-nucleus scattering data indicated

120 and 16O,

to us that while multiple scattering is important in

Single scattering dominates in “He. Our preliminary calculations

and multiple scattering calculations by Charlton and Eisenberg (Cha7l)
also showed strong single scattering in He. The dominance of single
scattering in this case explains why most "standard" models give

120 and 160 data, but poor agreement

fairly good agreement with the
with the “He data. Apparently, the gross features of multiple

Scattering do not depend much on the details of the model so that

maranally accurate models which give poor results for He can still
$1"€3'3f'&:1.:r:ly good agreement with the 12C and 160 data. Hence: it

appeared that a more careful calculation of the single scattering

ten Was needed for a good description of the 'W-uHe elastic

chamelting data.
5 ‘ % Calculation

The objective of the present investigation to explain the

‘fl' L;
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ms°nance region, using a theoretical optical potential which contains

\
However, R. Mach obtains good agreement with the 1f fiHe

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no adjustable parameters. Since the existing 1 -12C and IT -160

data can be fit fairly well with almost any model, most of our effort

is directed toward a more accurate optical potential for He. We then

generalize our TV -uHe optical potential to obtain optical potentials

for 1ac and 16o.

In the optical potential formalism used in this calculation,

the single scattering term is given by the momentum-space Optical

potential. Hence, a more careful evaluation of single scattering

L,
in He is equivalent to a better calculation of the TV -uHe optical
In this work we make a careful investigation of two

Potential.
1" —uHe optical potential:

factors which we expected to affect the
1) internal motion of the target nucleons; 2) nuclear "binding"

effects. Both of these effects were expected to be important because

of the strong energy dependence of the pion-nucleon interaction in

the 100—300 MeV region.
The motion of the target nucleons inside the nucleus

causes the pion-nucleon center-of-mass energy to be "smeared" over

a. range of about 100 MeV. This C.M. energy distribution gives rise

‘to a distribution of pion-nucleon interaction strengths due to the

Plasence of the (3,3) resonance. In most previous calculations the

effect of nucleon motion in pion-nucleus scattering has been either
neglected entirely or else treated in some arbitrary way. In this
28‘:'-<3‘\12l.a.tion we make an exact numerical treatment of nucleon motion

using an independent particle model for ”He. We find considerable

3‘fffiér‘ence between the predictions of our model and typical "standard"

In
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our exact numerical result with a simple analytic model. Both our

exact numerical model and its analytic approximation give significantly
1.2C

improved agreement with the 4He data and good agreement with the

and 160 data.

We investigate nuclear binding effects using a simple

3-body model. In our model a single target nucleon is bound in a
The nucleon thus interacts with the incident pion

potential well.
We expected binding effects to

and also with a binding potential.

be large in the (3,3) resonance region because the pion-nucleon

However, we find that binding effects

interaction is strong there.
Our model predicts

are fairly small even quite near the resonance.
that binding effects weaken the optical potential by 10 - 20 % in

the resonance region and that the effect is considerably smaller

away from the resonance region.
In order to provide a more detailed outline of the approach

taken in this work, we now discuss the main points contained in each

Chapter.
In Chapter II we outline the Watson optical potential

formalism and give the major approximations used in our calculation.
We also discuss the motivation for the calculation in light of the

LPHe 12 16

o C and 0 data and the qualitative features of the Watson

optical potential formalism. Both the data and the formalism suggest
L;

that multiple scattering effects are significantly smaller in He

than in 120 and 160. (Numerical calculations confirm this conjecture.)

Hence, we argue that an accurate evaluation of the single scattering

”berm (Le. , the optical potential) is necessary for a good descrip-

‘t‘ioh of Tr-uHe scattering.

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F-

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vss: 5,; ans for

itr‘»
Mes untir “a

Us. -

.2333“!
. .4. 03‘." ; ac‘ 01-3

:5.
"ivdf

ion for the .

 

 

ti”-
we 1 ¢
T- 991.8 f

i
M.u9f"er$
A. L, sema
K“
4...??54!
c._“°

 

In Chapter III we discuss our calculation of the 11‘ flHe

Optical potential. We use an independent particle model for He and

a simple separable parameterization of the free pion-nucleon t-matrix.
(In most of this work we take the basic pion-nucleon interaction to

be the free pion-nucleon t-matrix, i.e. , we use the impulse approxi—
mation. ) Our objective is to properly treat the motion of the target
nucleons so that the essential features of the free pion-nucleon
t-matrix are accurately represented in the pion— He optical potential.

In calculating the pion-“He optical potential we use the method of
"vector brackets" (see Appendix D) to take advantage of the separability

0f the pion-nucleon t-matrix. (Vector brackets are the same as

Moshinsky or oscillator brackets except the coordinate-space radial
Wave functions are spherical Bessel functions instead of harmonic
Oscillator wave functions.) The vector bracket method is shown to

considerably reduce the computational effort required in the calculation.

However, even with the vector bracket method, it is necessary to

Perform non-trivial multiple integrals.
Chapter IV is devoted to a discussion of the so-called

“ factored form" of the optical potential and to two typical ad 103

Prescriptions for obtaining a factored form. The factored form

involves writing the optical potential as the product of a pion-nucleon
interaction factor and a nuclear ground state form factor. The
motivation for the factored form is twofold. First, one avoids the

1it‘ll-time integrals discussed in Chapter III and, second, one obtains

Convenient separation of nuclear structure effects from pion-nucleon

1
I“terraction effects. However, most "standard" versions of the

o “ "
::oo“vi“ {:3 m3. r'"

.‘I ‘V.

swig: features We

c"fl5"9€l if, 228

.clo-il'“"
.

i.“ a: .‘Fbe 0“. fl" .‘

"a. y‘ f
.v *- va¢ v -
.

T,m. .

, p -
4... v.1..yd'. 1
-

I'pu 7

‘ g . A“; I .
tau. kl“ a: thp f
v

‘3”, .
.‘xgte: ‘A
v.
W a:

 

factored form treat nucleon motion in an arbitrary way so that the
essential features of the pion-nucleon interaction are not accurately

represented in the optical potential.
In Chapter V we discuss a recently published impulse model

which is a significant improvement over the models discussed in

However, the derivation of this more recent model is

Chapter IV.
This recent

just as arbitrary'as the two models in Chapter IV.
E 1122 model is included in this study for the sake of completeness.

In Chapter VI we take a more systematic approach to the factored

form of the optical potential.
In Chapter VI we make a definition for an "effective"

P1 (In-nucleon impulse interaction in terms of the fully integrated
1+
-“ - He optical potential which is calculated in Chapter III.

0111‘ definition allows us to have a factored form for the optical

Potential and at the same time to make an exact treatment of nucleon

“touch. We compare our "exact" or fully—integrated effective impulse

interaction with the "standard" prescriptions discussed in Chapters

IV and v and find there are significant differences.
In Chapter VII we develop a simple but accurate approxi-

mation for the exact or fully-integrated impulse interaction of
chapter VI. By comparing our approximation with the exact result

at SeVera]. different energies, we show that our approximation gives

a. Very good representation of the exact result. While no more com-
131fleated to apply than the "standard" approximationSv Our a'I’Pl‘m‘in‘a"Ci 0“
allows us to easily understand the differences between the exact
e:t‘fGCB‘Izive impulse interaction and the standard prescriptions.

ii 7: '3‘

.«w’nm

.1 Pr
.-Von

" yu-A‘ 0"" ‘
. .“v.vvoo l

.p‘ol‘ ” a
n v .VCV

P‘ ‘3‘ V 0'.

I“ era-.— .‘3. "

A I It I ' In a 1
0:...va a: ‘"-"e n I

0'“
to c-

F“
UQA

‘
“"5
any-

cut. 0“
~- In»!

2

u.

‘ .0 ‘
2‘. Q. .”=. ‘h
u a.

«any

..."e

I.- q : «A
.. HQ I: F .0 A o
A. “n" ”to 9.-.:C
V

7 ‘7' fl“. ,.
""2.“ G hie v: v-“ -
”’“th I

‘ -v-

omn‘vb a inrle a:

22:52. "1.71?”

...e ex;

 

4:: ‘.
e...“ .0 estizate -.
.‘n
3“” i- 1
0‘ I. 9‘
‘og‘du‘v‘ar t a
(,1 .u I
u‘ \ §
”haves (a Sara-”,2.
. ‘1
he ‘\ .,
‘. fl ‘7‘
«chi hie effEnon
v
'0
.43.
y\ H‘ l .
I“ u?;:."’&
Vw
“- II
o. ‘
I.| & a. .
d~~ '12:.1 w. a
Q... ‘ c N Q.
~-
'5 {1‘9 F
v V I“e $3.“:
~h‘
:N!
N.‘ I“
ur a 1' -._,_
I" “1‘.

 

Using our approximation as a guide we give a qualitative discussion

of the role of nucleon motion in the effective impulse interaction.
In Chapter VIII we consider corrections to the impulse

In our model the pion scatters

approximation using a 3-body model.
The pion interacts

from a nucleon that is bound in a potential well.
with just the nucleon and not with the potential well itself. We

use an 8 -wave separable potential to represent the potential well

and find that the "binding" effects of the potential well are relatively

small. The main effect in our model is a 10 - 20 % reduction in the
We also

strength of the optical potential near the (3,3) resonance.
construct a simple analytic approximation which gives fairly good
a'El'eelnent with the exact numerical calculation for binding effects.
In order to estimate an upper limit on the binding effects we extend
O‘lu approximation to a potential well with an infinite number of
bound states (a separable potential can P13064108 only one bound state)

and find the effects are about twice as large as with a single bound

State a

In Chapter IX we discuss the details of the calculation

120 and 16o.

of the elastic differential cross sections for He,
He Sj-Ve the ground state form factors and indicate how we obtain the

input for a Lippmann-Schwinger scattering equation.
The calculated elastic differential cross sections for

“He: 12(2 and 16o are compared to the data in Chapter x. For each
nucleus, the discussion is divided into two parts: 1) impulse inter-
a’Q‘l‘dxan results and 2) impulse-plus—binding correction results.

We find that the fully-integrated impulse model (and good approximations

t9
it) gives satisfactory agreement for all three nuclei, and that

 

. a ‘| 3'.
""9. :DLEOS :0 r.V d .
qw'.

~- w “it wire :1

"(let-n ‘06 ~

. ‘0‘]...-
""-e rad.-.

o".

“ .l‘lvu" AF no‘ Av-
e.)- o-fio‘ir- vvoo u a- v.-
..

"" ' 3:“ ~......~ State

.o-u . A.“ .

"n' a“ q, ' '

- r

9.5.. - fl- _
O

 

10

16

120 and o the data do not distinguish

other models do not. In
between the various models.
The relatively small binding correction given by the single
state model is found to be compatible with the experimental data.
The larger correction given by an (unrealistic) model with an infinite
number of bound states overestimates the binding effects.

Chapter XI contains a summary and our conclusions.

FO— f‘:
.o... L.

vb. 7n‘t
U o‘».‘

1. 32:12.1 Them
W
Watson e‘..a‘.
ieszrlbe the scatteri;
13:5 of a two-5013. c;
:cvaztial' cinema: (3
esseztial elements of

of“ \ .
N: 33’ Ker-nan, £333."

We assure 3

ins:
37 opera“! and

 

(2-2) V ::

“P

C,.

is
the Sum of We b0-

EJCIQOHS. r

 

CHAPTER II

THE OPTICAL POTENTIAL FORMALISM
AND PION-NUCLEUS ELASTIC SCATTERING

1. Formal Theory

Watson et.al. have shown that it is possible to formally
describe the scattering of a projectile from a nucleus exactly in
terms of a two-body operator which is commonly called the "optical"
potential" operator (Wat53). In this section, we present the
essential elements of the Watson optical potential formalism as
given by Herman, McManus, and Thaler (Ker59).

We assume a Hamiltonian of the form

(2.1) Hm: = H, + \<1r *Vw

where H» is the nuclear Hamiltonian, K“. is the pion kinetic

energy operator and

A
(2.2) Vt» "‘ 3%..”

is the sum of two body interactions between the pion and the target

nucleons .

ll

«e icno' '

‘.
et“
.b. ‘

i 3
{.732-

.-' ‘
=“J/

TVLE) =

ters- E is the k‘.
1
5:2“; state energr.‘
TN“) batwe
irate ( M!

ii) and \m.‘

and '
~~J 9.929;:

irritant of the 1

 

 

12

The pion-nucleus scattering operator T (E) 18
1"”
Given by

-|
(2.3) 12“,“) = V.” + VW(E.-H,,- Kwi-Le} TWLE)

Where E. is the kinetic energy of the pion plus the nuclear
ground state energy.1 In calculating the matrix elements of

I-‘ruca 3 between antisymmetrized nuclear states we need to

calcmate < I“: \ :- “SINILB‘ \ M > where
Lin.) and \M: > denote the nuclear states. Since the
c .
matrix element (M: \ 61;"(3) \NL» is

independent of the label J . we can write equation (2-3) as

(2-4) twat): AWN + Afiflece)Tw<c-3.

Here A is the number of target nucleons, and G (,E 3 =
. '-|
QaLE‘Hv-Kw+tpe\ ,where O" is

a. Projection operator for completely antisymmetric nuclear states.

we now define the t-matrix 1‘ (e) for scattering
1m

0:?
a pion from a bound nucleon;

\

1 I
“1‘1 Throughout this work, we use E. s to denote energies
1;] ch do not include rest mass energ, and £035 for energies

Ni; do. For example, E 1? means the kinetic energy of the pi on

‘AJ‘, 18 ' E“ + We?! o

 

r--.:Ap *fi-a :me '1 -

ng-vu. .JV 6. 5‘
0

hi, t‘“Le-\ :

 

.
.‘~

l.‘; 3
~ .‘ V, ‘

‘V‘i‘d g c’ . NY “ ‘c‘

a ,V. P‘ u '

r— .r

.
I. ‘
'1‘. I
-.., r "r
. £¢ Ul‘l T "
‘I ',

13

(2.5) Tn“) -.-. ’\S;m- drums) fiwceu.

The operator 1;“ is a many-body operator and therefore
extremely difficult to evaluate, so we will later approximate
’Vm‘ with the t-matrix for scattering of a pion from a free

nucleon. The free pion-nucleon t-matrix is given by

(2' 6) t““LEX = «the +43an 0‘09 ('2’) tree)

-\
Where %°Le\ '= (Q'Kfi‘K-w +Le\ . We can relate
:1“. to t‘“ by using equation (2.6) to get 4);” in terms
01‘ t‘“ and substituting the result into equation (2.5). After

Some operator algebra we obtain

(2.7) Thus 3 = twat» + twfitox [G £63~ %.Lefl’\‘mt€\ .

Note 2. does not necessarily equal E and that in fact e.
is an arbitrary parameter. We shall return to this point in
QI‘lapter VIII. For now we shall assume fig E .

We continue the formal development by using equation (2.5)

t0 get “11;“ in terms of 1;“ so that we can write the

c .. fi:"a‘
J-v W"“ 9 C SV’V'v‘
o—‘v-v‘~

-"‘

l . Q
«- 'gmzoo - c“""
I: ..:..~a - ~-::~

- ' :
-.: :wve 3""2" "" '
‘53”, .uJ—ovoVOO ‘
0

i.e'
'.“h’
{J
. o
C)
H

i" ‘ n’\
\~'.¥j

Us...”
J‘wu.‘
““4 o... I.
Q
a. F r-
.“V‘ J:

 

in

pion-nucleus scattering operator as

(2.8) T179 = ATV.“ + (Pt-“1“”, 6(va .

(We hereafter suppress the variable E in the scattering operators.)

The above equation for T1“, can be put in a more symmetric form

0
by defining an auxillary scattering operator T1?!)

(2.9) Tr.» -._._-.-. [Us-n he] Tin) .

I
The operator T is given by the more symmetric integral

e<11—‘lation

(2.10) T“ =.- (ppm/o 4.. (page 61",
(W TN TN 10"

I
The auxillary scattering Operator ‘1“) can be thought of as

a. “multiple scattering" operator. For example, if there were just
one target nucleon (A -- 1) then there would be no multiple scattering
I
6.1-1d we would have T = 0 and T = ’\\' .
10’ TV 11)

At this point, equation (2.10) is in the form of an optical

I><>~tential equation where the optical potential is taken to be
N

(A - l) l 11’“ . However, the propagator in equation (2.10)

a:Lllows excited intermediate nuclear states so that the equation

-' "~21 pc“"
pen “a ' i-..—
‘ if» '

U'U-
.

- 1"“. U a:
- o;V‘. “p

.25.”; UW =- L

. . . ‘

“a. on» o

. . fir , 9 five-

“‘ “' V --<:-e 4e Ky .-
O

 

15

would be extremely difficult (if not impossible) to solve. The
formal remedy is to put the excited nuclear states into the definition
0f the optical potential. We define a "first order" optical potential

9
Operator UV” as ,

(2.11) U12.” :1 LA-W’I‘jw”

With the complete optical potential operator being given by

(2.12) U1“, = U119 + U1; Li- to><o\\ (:1 U1“,

where ‘07 represents the nuclear ground state. The scattering

l
oPentator T- I written in terms of the complete optical potential,

 

1r
U1“, , is
(2-13) T. = U + U \°><°‘ '
1W 1w 1” E-KV-Kw+LeTWV '

Hel‘e E- is the kinetic energy of the pion plus the kinetic energy
due to motion of the nucleus as a whole (for example, in the pion-
111-‘~<:1eus C.M. frame) and the operator Ky is the kinetic energy
Qberator for translation of the nucleus as a whole. With the internal
Illlclear degrees of freedom removed, we have a two-body equation which

Qalt). be easily solved.

16

For elastic scattering we want to calculate
. l
h k s
<?1r\<o\-‘:I\'p ‘0) \?1r>,wherete et if!)
and ~‘P1;> describe respectively the initial and final pion

~
States in the pion-nucleus C.M. frame. Writing

<P§K°\T1;p\°)\?g7 as ELLE’PE" PI)
and <?1;‘<0‘ Utv‘°7|P1r7 as an“); Pu)

We have

Equation (2.14) involves only the pion coordinates and gives an

exact description of elastic pion-nucleus scattering. However, it

13 clearly just as difficult to calculate the complete optical
UWV as it is to solve the original equation for T .

Th “7’
Q value of the optical potential formalism is that it allows us

‘t:
O relate the pioné-nucleus scattering operator 1:”) to the pion-

A
nucleon scattering operators ‘1“) and t1”) in a systematic way.

.
~-- :'. .uv-mAN ; -

aw . 1”-.JJ_...
.n—o'..“: a.

 

 

Tfid-“o -

 

hpg-‘l
A
A”?“’
“h‘-.-
.-
Av- _
5—4' ,°
V>-‘;
, .
,
‘..,“
.a_‘-
i.
. I .
\vq.
. 4 ‘ I! -
"" _-."‘:v-~-
‘1 ‘
vv..‘-~ ‘
-s
I-\
'l
b

I

 

 

I
.
\‘--;
.w. ’.
‘1 '
n J $- A”
fl. r4 5
'a..4
.. .
n .,
~. u‘a.
V.
"~ .-_‘_N
4 v
_'-' A-
--
.
~.
v" :
‘\.‘
-.
" «. 52 A-
‘v "‘ ‘V-
.._.
Q.‘
..
C.
.- rn-"
‘ ...n.
v‘- a
"s,_

 

 

17

In order to actually calculate ‘1“) we make the
following approximations :
a) The Coherent Approximation:
We approximate the complete pion-nucleus
optical potential U“, with the first
order optical potential KlKV . The coherent
approximation is equivalent to neglecting excited

intermediate nuclear states in equation (2.10).

b) The Impulse Approximation:
The scattering operator ’T\ for
I?!)
scattering of a pion from a bound nucleon is

approximated by the free pion-nucleon scattering
o rator ~tkur .
pe 9 “

IJESi—Ilg the above approximations, we obtain for the pion-nucleus

Optical potential operator

(2.15) UT?” 2. LR~\3‘t‘WN .

:Iirl Imost of our calculations we use equation (2.15) for the pion-
1”‘l‘LIQ‘leus optical potential. In Chapter VIII we investigate corrections
—t;<:’ 'the impulse approximation in a simple 3—body model.

We remark finally that the optical potential formalism
3:223E33ented here is by no means unique. There are many other ways of

tiefining Operators and grouping terms. The reason for using the

. ' 3‘.
H-
. ~‘FA
”-2 ‘l ‘::u--uv‘
IUO’.
‘ 1
PA.“ V
:vc’ .2.,.-pv-A C... -u
— “iv.
“v- :."“ .
‘ u-o

.0. n'."' A a" Q

,.
'- .‘_‘”v-AU v: ‘

fl. .-v.o
.

_ V u “

“r...“ :0 " ‘ “ A.

-.t . . “u. do

¢—.o.' -4
I

9
- .o- 4
.5; f!" ‘3
... I.‘""‘.
., .

A

A P Q, I ’

~- .--- u-o "L"’
, .

.. !#0. .‘v o- ' v n -

Y .n

p O»: f‘ ~.'
‘ ..
0“ u-‘y ~ -
‘ n

on; u. .. A '
- fl d-v: A o
5-! "VA. .q‘v-~‘~ v - u

 

a...

-

10“ to .4.

O‘ ”y H-‘.
a

’65 a ".

7‘“

fl.
'/

 

 

.
:rmrj ‘.
u‘ A.
t. v \‘4u-y Ar
f
$.-
at ~
J... or I
V‘. >-
w
u. w

 

18

formalism described here is to put the resonant character of the
free pion—nucleon interaction directly into the approximation for
the pion-l-nucleus optical potential. If we could solve the problem
exactly it would of course be irrelevant how the various terms

were grouped.

2. Qualitative Emarks
In this section we discuss some qualitative features of
the pion-nucleus optical potential and pion-nucleus scattering.

We indicate how these features motivated the approach taken in this

 

work.
If we write the auxillary pion—nucleus scattering operator,
1
‘1“) , as a power series in U‘WV , the first few terms are
‘ 10><o\
2. z:
( 16) 1;.” U + U 4. . . .

Ta-IQng U‘WV to be (A-‘St‘fi‘fl , we have

 

(2.17) T‘}; = LA-W‘bwn + LPN-St?” |o><o\ tm

E'- K1‘Ky+£’e
+ .. . .

I {e 1 ‘
h e the Born a IOXimati on f or i

YA ”.48 DTP “:1 A tWN , i.e., the single scattering

a':DII~Z>‘...r‘oximation. For a small nucleus such as He it is plausible that

the Born or single scattering term is more dominant than in larger

12 16

nuclei such as C or 0 where multiple scattering is stronger.

F .1 “l 5*,
“’35 we .-€:~ 9X?"

" ‘fl .x .Am

:R:’°=‘J ‘6 VV u---
,p.r“o“

q/
9‘

113-"

A -
«r V

u r _
a 4 00’ 9 "F fl”-.. V"
:‘r. .3... Vane V" ‘f" ‘

In ”I

l
' n nan
L" up 2" 1‘.
a. J. a..‘ -uo
‘) Of I ‘ w.
a.” v“. u‘ .....n
‘0‘“! at“- n-A t -l' u
u v

v . -
' .3313: avg” a031, A.
‘ ‘ V WMJ-v JV- '.

.-
30zou A':‘I .3 r1 ...- . V‘
- hung-“J ~.-¢-.~-

It: :’ ;| K ”v. f‘
- I
, on fine - 0—9 v

“H .3 ’, ‘.

E r 5
' Ht‘.4'u ..

:':‘._' " A \
u...34. ‘ L A y- n

.
"C. '
ngax“
.l...;._‘_4r‘ :m a“-
as. . .
O A

v

I
..‘:- u
I. :- V‘ I: “ u
‘fi' '3 I!“
-u “8 due - W

: {DETC ‘1 r2-..
‘ ' h.-

h

..".e

'3 ' -‘

I 4 u

.‘fik,~ xca"‘ -‘\
‘ vv“'-k

O.

 

 

.‘ ~
‘v E»! .
v“ ‘F V,
M‘ ea {A N...“
g I
«G‘

 

19

Hence we might expect the characteristic features of pion—nucleon
scattering to be more prominent in ‘W -U’He scattering than in
'W 4-1261 or 'W —l60 scattering. Indeed, this appears to be the
case when one compares the elastic differential cross sections for
aHe, L20, and 16O in the 100—300 MeV region. (See, for example,
Figure 10.1 and Figures 10.5 #- lO.8 in Chapter X.) The strongly
p -wave character of the pion-nucleon interaction causes a nearly
stationary minimum in the 'W -4He elastic cross section at about
75° in the 'W -’+He C.M. frame. In 120 and 16O, on the other hand,
the minima appear to be diffractive, moving inward with increasing
energy. In 120 and 16O the details of the pion-nucleon p -wave
interaction are apparently masked by multiple scattering effects,
"hi-1e in “He the Born term dominates, producing a characteristic

P ”—wave minima.

The fact that optical potential models based on the impulse

approximation give reasonable agreement for 12C and 160, but poor

a'St‘eement for “’He, coupled with the fact that the "IT —uHe cross
Sections have a definite p ~wave minimum led us to conclude that
a. I“ore accurate treatment of the single scattefing term (i.e. , the
o:D‘tical potential) was needed in the case of 'W ~4He scattering.
It was expected that an improved “He optical, when appropriately
€61'leralized for larger nuclei, would also give better agreement with

the 12c and 160 data.

H ' m7 0-Way f5:
Ju-‘a‘~.-~..\
"v—Tfl: I ”'7 P‘v

I‘d. .u'J‘QI. . - L.‘

\

It h"‘15 show-

'.-' , _
« “In“se an" vxiflc '

a
a u
N “ie f. 0 vi A
' rQJnQ'. -
:-

 

 

' “
L s‘:~l=
‘ ~

 

“‘ this ..
V
“'3 W I'9 I

 

 

CHAPTER III

CALCULATION OF THE 1T -4He OPTICAL POTENTIAL:
INTEGRATION OVER THE TARGET NUCLEON MOMENTA

It was shown in Chapter II that the coherent approximation
and. impulse approximation give a pion-nucleus Optical potential Of

the form

(3 -l) Um, K 9". P1,) '3 (A")< §\<¢qs \‘HJ‘QQWO

h

where the kets ‘ PE) and \ Pi> describe respectively the
P1011 states in the pion-nucleus C.M. frame (TDCM frame) and

\ WQ§> describes the nuclear ground state. The Operator t1".
is ~the free pion-nucleon t-matrix in the TUCM frame. In this
cit‘l-‘aqfier we discuss the explicit evaluation of equation (3.1) for

1“ J‘He scattering.

l ‘ An Independent Particle Model for ”He

we want to take the main effects of nucleon motion into
a’<-‘-C=<Junt in a simple way so we assume an independent particle model
for “He. In this model equation (3.1) gives a very simple form for

the 1V J’He Optical potential

20

I

?

w 1
PM '

‘
'- 'i‘-
c

1-..“. \IT.‘ "

| ‘
“'v gu,‘

. _ .
--o‘-~
3")- o_ _'

5v“ "‘va‘. U . ‘- -
.

*.

‘1': 9H,“.‘flr .5 -

c .¢.¢\r.. ‘u- .

Hahn?" "Gate! /
"~r‘“v."' 'Ugvv. \

-
A >—-
-‘..‘ ‘
/
I u
I
“‘ «'1: "V 0w“? ‘
t‘. ‘-‘ v v.

0.. I
""5 ".‘A‘G'°w‘ I,“
J M'““~-..~.A

 

-
4‘ -.

 

 

I
I
\' a‘
..
v-‘
I. ‘
.“A\. ~'_
1“ 3.1-. 4;:- .
A
A.“

"“3“? .f‘ :-
~' ' -
“Q ~
~.; ‘A
1 *1 ‘V._e to A,

 

21

(3.2)

U L?

W w

241’.

m- -u W ”a; “3% Mytgméjg
LN 3

In equation (3.2), tw“ is the spin—isospin averaged free

Pi Orr-nucleon t—matrix and K" is a single particle ground state

The vector 7a; is defined as the seven

’0: ,9}: )where we“ , DE,

wa-Ve function for uHe.

Conlponent vector ( a)“ ,

a, ‘ .

11d & are respectively the collision energy, initial pion
A;

momentum, and final pion momentum,all in the pion-nucleon C M frame
is the nucleon momentum

( TNCM frame). The vector P“

A!

in the C. M. frame of the ”He nucleus alone and the vector %/ is

the 3-momentum transfer, Pn- "' PW, . The reference frames and

N

I’i‘irlematics involved here are discussed in Appendix A.

2 - Some Numerical Considerations

Although equation (3.2) is very simple in form, it is

quite tedious to evaluate numerically. First, a 3-dimensional integral

InIlst be done to Obtain WC?“ , %\ . Then another
11Thegral is required to Obtain“ a l’partial wave decomposition Of

UV ”W 1‘, 9 3 for use in a scattering equation.

o o ‘
.. . o
.. .u o' "
a... da -.
U. ‘ .

t1)

 

 

. a- _:
v _ -
.n... easy one 9 .4
l ‘ I
(par... owl. in" ‘.'v ,
" .. . ;
.on.vu,:o:c MV
.. n...4.vu "' “
.1”. q. 0"; fl‘-“ 0;.
h ..-..v ..A' 'v. . A ./-.~ .
.
' ' n . - . .
no... - . .. .
‘ ‘-o o‘cqr' a to ~‘_
”".-' in-uu‘c‘v -‘
.
~ Q .. a
a.
t 'u ‘2 fl" 2'." ”v- P.
' "" v~y“a.-... a-
.' l I
u _ c , -
.. '-":“v; \y--.-,
§I-~..“v¢.~ F‘. ~-

 

.._.:A' /A‘ H\

.-v.. \’.4/‘ “3“..--

“ca
o

 

v .
-‘.. o

. .- a... ’v-

4 a--- _ - A.

-.. -‘

'o- I ‘
' ‘9‘. " .
‘UI' — A!

-\

‘cn

“-.\.

 

 

 

.
a.
A "R
~-..4
\. .
.' n‘iu., .“
i
s‘)‘ a, A.."
H .
r.
u
.
1.
-' 'vn v
‘.
" "“ “v- 4
- . Q
‘-_. ‘
I
h. - v.
N“ .‘ ‘
. .
as“: fir:‘._
‘- V-.,.\ “VI-
A dd.
\.

 

 

.
~~' u.“
.‘V
‘I
'I
'E. 9,-
v n'.
.g“ e
i
a _\
4
14' ‘

 

 

22

0
Hence for each value Of \P-w\ and \ P“. \ . a u-dimensional

integral must be done, so that one is faced with calculating a large
nunber of multiple integrals. In Appendix B we give a parameteri-
Zation for the free pion-nucleon t—matrix which is separable in the
Coordinates ’g’ and 23:. . We can exploit this separability
to reduce the computational task considerably. Using the "vector
bracket" technique described in Appendix D, we reduce the problem
to the calculation Of many 2-dimensional integrals rather than many
4-dimensional integrals. Obtaining the final algebraic form Of
eclllaiion (3.2) using vector brackets is tedious although the final

form itself is rather simple. A derivation is given in the next

Sec-"'t:’1.on.

3 ' Derivation of the TI —uHe Optical
Potential Using Vector Brackets

In this section we give a derivation of the T —uHe

OD"tical potential using the vector bracket method described in
AIDIDGndix D. Our derivation serves as an example Of the use Of vector

b:E‘Ei-czkets and also shows how the separability of the pion—nucleon

t‘matrix allows a considerable reduction in computational effort.
The derivation itself is not essential for an understanding of the
lehaining chapters so this section can be skipped over by the casual

: eElder.

We want to calculate the matrix element

(3.3) u a P.\= (A-q<a.‘\<o\'tw\o>m> .

l
wv EN

_ u v
v. n--’ v
v
a I
Ago ‘ ' f
. we-..

 

 

23

~

Inserting a complete set Of nucleon momentum states on both sides of tin!

We Obtain

(3.4)

U1“) &?1\‘_: PI.\ =

 

 

KA‘\)SS A311» SS P4.” q)" C? ~26)? (P404)
(”33 Gm)” "’

X<"t' \<M-:E\1m) KOWB

where WLPA) and “2* (24.33 are <3; do} and
<0‘P Pfiifi> ~respective1y.

make the following definitions (see Appendix c for details con-

QeI‘l'iing the notation) =

<3 - a A) can a mint: to

(3-6) Hi“); 2 Y: (W2H)\WL1MF>

Lwa

l
with a similar definition for \ P > .

1"

~

(b

.q -,

- _. 4- oooooo

I. 2 A? -—

.c- a u. ‘o... ......
. I’ \

. 1

.~' .3 " ‘V x 4 A

.~ ...—.-y.. \/.H '

 

 

y
I I
“ "r-;
o. Av,
.l. N .A
:¢§- ‘V-
-\
s
:v.‘
. ..
\.1, Q”; .
' "‘v ‘n‘ 7“!
's-
‘s

 

 

24

Also, we define
(3.7) \P H») _LZM :{f} M240 \‘umd LN MM>
MN

I
With a similar definition for \ P > .

Now equation (3.4) can be written

em ka'fié as = we“ P11: AP; aim

(aw? QNTP

 

 

Mice, Lm \ Z Y (PJHY L6)

LWMW Lu“ W
Hr MK

. _
x <21L;H;(K\>4L LLzo HM'=o\tm\ P L =0 MN=°>12W LT Mb

The unit operator in terms of the WM QM coordinates (or equiva-

lthly, the relative and center-of—mass coordinates) is given by

(3.%

_ u u $322." $31 I, u "
A“ " “’i 7‘7‘» 78 mg as“ W“

25

or in terms of "coupled" angular momenta

(3.10)

1:

Q" N \| i! n 88—- &" WAK'I 40;, u u u
\ a: { M2,.” 12 )3" Tm? (a ——S-3<&‘7£ wm‘ 'La 12,31.

2.... 9-1"

Inserting the unit vector of equation (3.10) on both sides of t1?»

we have (The "coupled state" \ .kuxui "‘M" flan £1“

defined in Appendix C.)

(3.11) U U’t‘P £3= Lee-oi: Y n‘iflrf‘i‘

“N {W «M1
L; M' imm‘"
_;__AE“S?: Agni “8L" Awflgkul—fl-fl 8L“; 4%“ 8% 1,31%“
:2.—
Us, (in: (NY 33 0:: 3"" (’1 W33 (2103 @‘W S3

e edifices <P; e1, MW Le Lew \w w (”MW 1e"
x<&"e<"'£"’iv1“1£ £9: \E“ \ may" {"Im "' 14" 2e”)

‘ <‘e‘moKm10utmmfi’m £¢rm\ P“. szf‘M L1 L”: O>

I.“

 

 

26

where the states \?'w K24“ i‘YA L‘W Ln: 0 >

and H’t‘ 9.1. 1' ’M' L's L‘s =0 5

are-5‘15" ‘ MLWNV‘ Luszn ~0>

and \ Q“! L“ M‘?\ P4.» L7 .0 M '30) respectively.

Note that i = i. — L1?- — L“ since LN- - L;
~ ~ a~ IV "‘
We now need the matrix elements of £1!» between the TVNCM

kets and also the overlaps between the WNCM kets and the

“Dc-M kets. (These overlaps are the so-called "vector brackets".)

The matrix elements of 41““ are given by

(3 .12)

093%" 1' ma" 1*» 52 «\t' \24"'x"'i'in"'i&"'1

.. on 851:: ”so mm “1”)Wu" mswm

 

x ZUZI‘t‘W‘t‘S t: L 3,3004)k' 9t 3
G I: 2.1%" 4" k"
Where
(3 -13)

 

06%
t1 L chM,&!, 2‘): XtILkwr,hg’h3 fitxfi‘hu%ueifih
E“! %t (he)

H.

'~~."...s
--

x A ‘
\EL’k)
«m

27

and

(3.110 “$353149."

.. 2 e, -—\
(Wm) Lark-3: 8W “0 [$3146. 1°

1.; w ”a meet,

 

 

A more complete discussion of the parameterization of H» is
given in Appendix B. The vector bracket expressions are of the form
(See Appendix D for a detailed discussion of the various factors in

equation (3.15).)

(3 - 15)

<&I\|.Km 1m mm 9“". 1%»: ‘ PW ?4-N im LT? L“ >

..._. Law\3(4w5‘£U"'i \ sw'mscw 8. as

X

 

 

mam W.)

woe-.1“ “MK

W xiii] [? W w\"’"">Y“’“Y‘f*:]'

LVN-n.

$r- "A‘5

'V
; UVOU' du ‘.U.
0 v

-__ 1‘. A. av:
..
..«

51.

C .
:4. ’0' " c {Le U u
.9...Vu~ I ‘

V ' R,
up‘“"'"" 0! I
.‘ \ U a y. A v-
1V”.'. '

I
P

n. ‘2": L
c a“. r

:.X ‘ ‘ - A
_ V‘s‘cvflcp. A.
u

- ou¢~-..,-___v v.
.

I

‘-Q .-
3 ‘v‘ ‘
V‘-‘-—..‘A

.‘Q‘: n“: .
‘§-»-.... \ *3 $.-
V‘...— v-
v
‘0‘. -‘
h h " V.
-m.

 

F.
‘1.

~u
.
“.1: a.
V .“.‘.~V‘&
. 1 -
.'
.
‘M V
"“1? :1“ .
‘ “‘35:! ‘u.‘ \

 

 

28

The factors to note in the vector bracket expression are the delta

functions. These delta functions, together with the delta functions

in t1”! (equation (3.12) ), allow us to simplify things

‘ I
considerably. For example, since L“. = f and L“: t ,
~ ~ ~

( u ~
we have Lw = L‘. ‘-= 1 : f.“ . Hence, in equation (3.11),
A. ~ ~

~

II m m
we can eliminate the sums over ( f" , m ), ( i ,m ), and

( L19: M; )0 Also, we have Rafi": 2km and LI" Zfi‘km -

The sum over M." can be done trivially since the final result
is independent of (M , and W?- MT .
The kinetic energy conserving delta functions SON) and SCW')

allow us to eliminate two of the momentum integrations. (We

eliminate P4” and P4; .) The integral over (Km ' is
eliminated using the delta function S (”Kn-km) in equation (3.12),
leaving us with integrals over just CK" , k“ , and km .
From equation (A.10) in Appendix A, we obtain the relationships

<£’>‘=(a¢ - 93' M" and

w’ A!
I" 2,.- o u 2/
L&)- (PW‘ .L‘?) . Wethenwrite
N we N n H
a
the integrals over &" and «9L as integrals over X

m u A, u "'_ A An
and. X where X '2: P‘K o ‘2 and X -" P17“ OK .
(This eliminates the need for the 9 functions.) A straight-
forward substitution of the vector bracket expressions and the

Pi On-nucleon t-matrix eXpression into equation (3.11) now gives

1W

W")

x “(“41" -

 

 

 

29

(3.16) Um) (19" P3!) 7-

 

 

 

 

(arenas-Jr \) $5";
U’V")JCZ (ZL“+\)L2.2&"+\) ML“. V)
L‘Qx“
I’E’Rk"
1’ 9e M“
t §fcffl 0’ o ' "L Q n (? (Kuhn-£1")
‘ ,Kofl-A‘Ku : 4&7. EQRY 1r 3:)? 1'
(fig)
L 13’1ch "‘
where
(3.17) Lear“ L115.“ =
131*." \
/2.

\ u u
u u l wwLfi ) wfigh )]
SA‘ 0503i ) Raw} 0.0%) meta!)
-\ 13.9..."

>k

X if {[ 249x" it" ”1%) “WOW: ( €")Yc&u)]

i! “*- Rx" Mo"

x [15%) Yo} $43113 .

=. "-r Mk‘k

 

Viv

' r‘o ' a».
3( ), the v c

0;“ Y :
7.

I. ‘n‘

. \l
”"71 X ~
-
.fi‘.
it» ::."
. u~~.-:I' ey‘m

 

. 5 file
"‘53“. k» C M 4'
"Q, in ‘

 

 

 

30

The square root factor in )1: 9*" comes from the X
factor in equation (3.13). (See equation (13.5) in Appendix B for

a discussion of the \6 factor.) In the integrals in equation (3.16)

H
and (3.17), the vectors & , P: and in; are related to
~

~

0
v1- and ‘K’ via the equations
~

~

1213“ ‘2' - @i/wTE' > CE." = ‘1‘" 92:

~
I
\
and r, = ?4; ~ T P; (see Appendix A). For example,
~ ~ ~

(3.18) Kk: )1 : .
“3' Y’ *W»~>‘w§>‘- ems/neg W \

where

A
(3.19) X“ = “I ' CK"

, 2. I 2'
With similar expressions for (PH) and ($110.) . The
~ I
9

1 and 96

N

l
orientation of the vectors R“- . 9h,
"" N

can be drawn as shown in Figure 3.1.

H

(K /

 

~ I
PN
‘ku + ~

Figure 3.1 - The relationship between the pion and nucleon momenta

in the Tr I
c f d h t ai t ' d it
mom en t 1’? .rame an t e ot momen um 26’ an re a ive
~

Tr $‘n V "5'
-.. '., no~..-

l G
. I 9 " -
5..., ..- Ar 5 i", ...‘:
'.;‘“'.l - .
~’.

.
II?
7.1": .‘ G

Ign'v “IV 9“V :‘V..

..'.." i c we ‘3

~':.:.-.' ..V.'

i.ek;{’21"mcf‘v':*‘
.0.« k1,!) ‘ V"

 

' 1
e... .
."“f\ a“? ‘ w:. 1
Int“ ”‘VOVU.. -. V
n d ofly‘-4

a. ‘4‘ A“ O
Q... ‘9." ‘V 5a..»
vv‘ - 41
a. v-c .gg.
.
i
I-..
. V " “WI: ‘
no".. 3 .‘L 2 ":~'
‘ Q 4.“ fl ‘
0"
—: AA-‘.‘..: A‘ Q
.1...‘ 3"” p.
O “5 “A...
v-
..
o
._ 3"‘:$3 p, \
M .v“- A ‘ A
l JL ' ’

 

 

31

In the numerical calculation we assume the pion-nucleon
interaction is in the (3 ,3) channel only ( 1: '5/2- , 3:272, , 1 "7" ).
and we take the pion-nucleon t-matrix to be the same off-shell as
on-shell, i.e., we take 5 l . We have found the results

1112*:
in the (3,3) resonance region to e rather insensitive to the choice
or 0
km...

The advantage of using vector brackets should now be clear.
Vector brackets allow us to take advantage of the separability of the
pion-nucleon t-matrix, so we just have to calculate the function

2.
u and integrate it with 't, / %
3131* 133k" 131k“
asirflicated in equation (3.16). In addition, we automatically
obuun.a partial wave decomposition of the optical potential. Hence,
the computational effort is considerably less than in the original

form (equation (3.2) ) where h-dimensional integrals are required.

a“ V 0'
'~‘ 'cA a- I
U

 

1

 

Hi

.1,

.-.
it---
“A
.fi
-u‘
‘
:2"-

v-. R
.. .
.'+‘* 22-“ ....-_
a..- *~—._~,._ _-
-
.r';v-:-‘
iv.‘ A
‘ “-‘-.
Q
.. A;
d:
V ‘
.L“ o- .
.‘ ‘-..‘_
‘
-‘ H‘< I
' -
4“. §< r‘
v "o a. *n.‘ .
s‘." H
~— ngv .
C-. :
‘ D I
a... rfl-‘~ ‘ a
§- _ _ ‘ ‘
'“V‘ .
‘ ‘ v

 

‘.=‘ ‘u
's ’3
u‘.~
—
a.

r‘;
r

CHAPTER IV

THE FACTOBED FORM OF THE 1T -uHe OPTICAL
POTENTIAL: Two 513 Egg MODELS

In this chapter we review some of the conventional ad £3.99.
models for the factored pion-nucleus optical potential that are
currently popular and show the shortcomings inherent in these models.
First, however, we need to consider the factored form in general and
show the assumptions which are necessary to obtain a factored optical

potential .

1. Definition of the Factored Form
In order to avoid. the multiple integrals involved in

calculating the fully integrated optical potential (equation (3.2) )
the "factored form" of the optical potential is often used (Lan73,
Kuj7u, Mil7u). In the factored form, the pion-nucleon t-matrix

"' I

t‘fl'fl( Jl.\ (recall (3:: (wCM)k )k” is evaluated at
some "effective" value of .g and taken outside of the integral
over the nucleon momenta. Denoting the effective value of .2;
as I} and using the factored form for the ”He optical potential

we have

1,, 31,29:- (A fink-(1)506)

32

72:9 (,9 :5 if
.2; (1):

‘ I
Q.
.‘gA "":V.‘ .
a

I

‘4

hour <o-De. v- .
o

M

o A .. p «Q a
- V'
. IA. W. * w

a..- o 1“-..

 

d-‘..~- u.

 

 

! _\ L
1
Q
|.//
l ‘ I ‘ ¢nw .
.P :us—wr‘:
nu H...‘ M- .v-.| ‘
.
. A '
“2? g a" ’;»\¢-~.
MI“ ‘- AV-.. .‘*u “V-
.
a..‘R.I A,
‘~“"*VL.'
l
I‘ ..
I 1; "' ”an Vr,
~I any u‘ Hwy ‘.k‘
' ~
.
r A y
«.6 a. _
~ -
.
n 'l |
' .5.
c- a '3 “52“‘0 I.
use ya..._ ‘V ;.
.
~
1 .
.v .N; .
K‘ 4.4. .
s.
o
‘ “-r ~4-
Q:U‘d... .-
‘u
"h a ‘ ‘
:‘a-CfiE a“, ”A .
~ eo“ ‘U
I
b“
h I; A
CI‘ Vha‘
O
‘:,-. M
“~ ~ ‘
.» 5«he A‘

p
V‘Jl~

 

33

where (09%) is given by

 

(4.2) (“1)?- “V (P J+A2%)W(Pw)fi

~ (1;)

The quantity (09%) is the ground state form factor and is just

the Fourier transform of the ground state density Fc‘g) :

' .V‘
e% ~

(t3) («$32 (Mfg Agr .

An empirical form factor is usually used in equation (4.1) rather
than a form factor obtained from some assumed ground state wave

function.

2. The Ad Hoc Models

 

The ad hgg versions of the factored form which we consider
in this chapter are typical of the models found in the literature
in that .£E is obtained by some plausible but rather arbitrary
prescription. These models are presented solely for pedigodical
purposes and do not give good agreement with the fully-integrated
'1‘.-uHe optical potential of Chapter III. Our objective is to

make some connection with the work of other workers and not to give

\vo
with; a“? .
:Y'Lv:"i' V“
..-

r:

. .vov "O ”I.
rue-35‘ ‘LL Iv E”.

Jogl‘ I".
.

. tug-0"
‘C .‘3- on. 2 2 v:
.‘ 5
3-8: “" o
. . "'

.
. A-A 0"
I"! :‘V ' y" .-

“fl ... -9.“ .. w-
.

'9' “v 4’:
‘. u a.

. . A
w "w: ..
u- d.v.v $-54

 

 

 

 

o q I . '
:0:‘0Am‘ Fl'. A;
'O‘vtdgv- V: wow“ ‘0
-
" r
€- .... -
‘ -
3 re ,
Aetv ‘
Ill. ‘
A
C '~
...v
f
EFF)
"p;
. . \ .
I
‘:‘ 2 ‘C ‘r -
..""~ ‘I -~ “‘
.
. I
“v A .. "_
; ‘- tc°~l
u-vv ‘~.‘“ -
Q
~
- A
5-. A r‘CT A.
59““; V- ‘-
I-u. ' _
. M ._
"on " "in u‘ ~—‘!
" at”, .
. ‘ ‘4‘-

.-
“:M"Afi\‘ I.":
"h..g,‘ \A A C
s-~ ~
‘.‘:. L. ~ .
A D
v \ Q ~
~w-:.,,€_-, ran,
:“’\-‘
“‘v
r...
\J , 2,.
b. .
O‘. Na g" -
N. ‘_ V
*‘ fige ‘AA--
‘.
_ d.
y,.‘- ‘
q
«‘I ‘V..
”a.” §-
a s..1~:‘ _ ‘.
y“d

 

34

an exhaustive account of the various (incorrect) choices for .{3 .
In Chapter VII we give our own version of the factored form which
results from a systematic approximation to the integrals over the
nucleon momenta.

In our discussion we consider just the choice of .EE ,
but there are actually three distinct problems involved in generating
a factored optical potential from first principles:

a) The off-shell parameterization of tw”(.q:\

b) The choice of .£1
~

0) The 2‘, -space form of the Optical potential
(assuming the calculation is done in ’1; —space)

PrOblem (a) is in practice often closely connected to problem (c)

since useful 1: 4space optical potentials are obtained only for a

special class of unphysical divergent off-shell parameterizations

such as the Kisslinger and "local" parameterizations (Lan73, Lan74b).

We use a simple non-divergent zero-range separable parameterization

throughout this study and work entirely in momentum space so that

we isolate problem (b), the choice of 33. .1
Our criteria for a "good" choice of 1g}. is one for

which the factored form is in good agreement with the exact or

fully integrated optical potential of Chapter III. As will be shown,

most typical ad hgg models involve "bad" choices for I]: . How-
ever, the prescription we obtain in Chapter VII by systematically
approximating;§he integrals over the nucleon momenta gives a "good"
choice for J}; .

 

1Our calculations and the work of Landau and Tabakin
(Lan74b) indicate that a non-divergent zero-range t-matrix gives
essentially the same results as a short-range t-matrix.

A a-<
H on“.
we“? .

. a ’0‘”

 

"FD w-¥ ' ;.'
.eov “-v~-v
.

'33. w 5-13 Ak'fi: «a "

V \.
.V-.. v“ “GU V“V-V

. ~ "
0";‘O .5, CPD VIIIAy-r
"v'" .4. Unnv ‘-..
.
-

I - I I
"L “3, MI!» .,_
‘~‘. .U- «A.

g‘ x v .
*3". " -""\" ”‘20:
“a--- u. "-‘w- ‘4 "’-

.
""-£‘~:< nu; yr-
"--uv...v‘ -U‘. ‘.:
O W'W.
0....

~‘-~
.

A q. I
‘-"‘ '" "M mun“!
'1 *~ A
.' .‘~P-.
.
o. '

u
5‘. ‘
h a.
"un,. OR
\UV‘I‘ a
e
"““‘-v "_§‘ A P
. J‘vl. ‘

 

35

3. Angle Transformations

The problem of choosing a value for -£3 in equation (4.1)
has received considerable attention in recent years (Ded7l, Fal72,
Lan73, Kuj7u, M11714). For the most part, however, the emphasis has
been on the choice of the so-called "angle transformation" which is
closely connected to the choice of effective values for the 1TNCM
momenta a and g, (which we denote as E and E I respectively).
Except for the work of Landau et.al. (Lan73), the question of a good
value for {San (the effective collision energy) has not been dis-
cussed in much detail in the literature. Hence, we shall mainly be
concerned with the various angle transformations currectly in vogue.

The purpose of the angle transformation is to take into
account the "mapping" of the scattering angle from the 1tNCM frame
to the TENChK frame. In the early work of Kisslinger (Kis55), a
One—to-one mapping was used so that C0591!" '3'— CO"; 9“," .1
Dedonder (Ded7l) and Faldt(Fal72) were the first to point out that
a transformation of angles is important in pion¥nucleus scattering
and they proposed a linear angle transformation of the form
C0591“: Q,+ b COS any .‘2 An angle transformation similar

¥

1The relation between the cosines and the associated
momentum vectors is

C°59...= E‘-E/\‘§,\\E\ , c059,“; E'PE/IEIWH'

2However, to our knowledge, the "local"optical potential
of Lee and McManus (Lee7l) was the first model to actually take the
angle transformation into account. The angle transformation implied
by Lee and McManus is the same on-shell as the one proposed by
Dedonder and Faldt.

‘ -1 - ’1
‘ A n»-
": ‘- w- __ O I
v “flu—-..h ’
' .O v
. ~ .
.oqopgw--- - ‘
1 .— p‘
_\u—--u-..'v.' ‘
' O
.70 1: wt: -2
‘~ -"-'~~.u--
- N
a - . .
a. “q. .
’P‘ A. -c* A a .:
'. -"~-.-~‘ c-- -
a
~
" ’u: \‘ 1.- :..
““ ~--"—-v b

‘1 '
~‘;‘-: "v-a:
o—av-.~ a- ‘
.
:w .. .
Q, i
o._ *;e H;-_
3
~
‘A I~‘
‘ k -
‘
\
.
V.~\
«)

q
e
‘V
‘ I: ..
'V “‘5“-
mg: s‘c
~.,
.—
.
._‘
~v-
\._ .,‘ ‘
‘ I— “a
- ‘.t
‘~,C.
v
l a \
~ it -~
.-‘

l
T x

36

to that of Dedonder and Faldt was later used in the momentum space

calculation of Landau, Phatak, and Tabakin. Hence, for purposes

of discussion we consider two models: (a) a "simple static approxi-

mation" '- this model uses a one-to-one transformation ( C0361”:-
Q0581“, ) and is representative of the early Kisslinger-type

angle mappings; (b) a "modified static approximation" - this model

incorporates a linear mapping of the form CO”; 6““:- 0;? b C05 6”

and is representative of models incorporating the angle transformation

of Dedonder and Faldt.

1%. The Simple Static Agaproximation
Our quasiJ-relativistic kinematics relate the WUCM
momenta and ‘WNCM momenta by (see equation A.13 in Appendix A

for the details of the kinematics) by
.. _ a
(ban) 15- %[\-(&A__‘.) “ft/hf] -(UOW/wo) Em

(4.5) .951: Pfl'IE‘ — %)w;/wa - Luz/”0) P4,“ '

If we assume the struck nucleon remains "frozen" in the “He nucleus
I

then we have P4“ '-"- 4“: O and the effective values of

I ~ A.

k and k are

N N

(4.6) E : %[\-(E—E)w;/wj

' ‘ ‘ I '1 '0: A
1- . ' 2
” .I" m a I-.. (-b

u an or-” '0

a u 1 V . -5
-::- "'0 be c '-
..v~ do v4 chow ~~-:--

use“, .

h
I:‘ ‘ a ‘h a
q
Mug," ‘ ‘nn‘ua ‘ ’
- \
~

-— \
[1

‘1)

\_ .J

'l
u 9
/,,.,I

“W C0

 

 

 

37

(in?) «E; = E’[\-— as)”; Adj

0
In equations (4.4) -'- (4.7), b)“. [00. is essentially the ratio
of the relativistic pion mass to the nucleon mass. It should be

clear that the simple static approximation implies C05 en.“—

5. The Modified Static Approximation

The modified static approximation is the same as the
simple static approximation except the 'ITNCM angles are related
to the “TQM angles via a transformation of the form COS 611R "’
O.+b C0591,” . The relations between the magnitudes of the
momenta are taken to be the same as in the simple static approximation.

Hence, we have for the modified static approximation

(”-8) [Eh- \ffl [\-— (6.3)w;/wfl
(m9) HQ”: HEWEI - (A- \w'fi :V/wil

= ‘0 0 .
(4.10) Case‘m CL-t- C 89“”

~
at." :1 .5.
Via-.0 a!

on. Q
.U- "d

3
.r.
c.

by

‘5'-“‘11
--"-_.J~ a.

'2.-

sbe.

' V '3
3 q“ :0 FF
p“--- o""

nq‘
.'~/

m-
'-

Cost

[m w

t -.
*.'I.L

 

 

 

‘
y;
‘H

'5 a fin
‘cU

v.l

a -
3a..-

38

Numerous prescriptions have been given in the literature
for choosing the "a" and "b" in equation (4.10) (Ded7l, Fal72, Lan73).
We shall use here a prescription that is typical of those in the
literature. We shall not be concerned with the details of previous
calculations because,as we shall see later, the proper relationship
between (.059 and C059 is, in fact, nonlinear. A linear

1TI| 1L9
relationship is obtained only if the recoil of the struck nucleon

is neglected. For example, our quasi-relativistic kinematics give

W

~ ~

on) (Jag—g)": (P ~33“

or

(4.12) c.0591”):
I
[Ui‘t— Qz‘z— 9;:- 9"?) AM + (Pris Aggy cosew
where k denotes Ig‘ , etc. .

Due to the recoil of the struck nucleon, the quantities l (E \
and ‘Q“ actually depend on C059 so equation (4.12)

~ 1W
represents a nonlinear angle transformation. However, as equation
(4.12) is often used (Fal72, Lan73), l-Q \ and ‘jSI\ are taken
to be independent of C05 9“,” and a linear angle transformation
is obtained. For instance, Landau, Phatak, and Tabakin (Lan73)

suggest taking the momenta on-shell and calculating \k‘ and ‘ tl‘

1 AV

CIA
:2311’3 3 I1"- ""
“u

that. '~'

.1!
9
tn
’ c
(D
—
m

lt\=\

:..-.,_..'u..;1 ,. fizz:
U‘ul— o- Hunt-v-0 d ivy-

yrs-31A, .' .R o.
”-5.98. ".44 “‘3

GS) COSE

M >
v-g_

i=(‘if/tj.

3:5“:
.. ’10 Emeritus
a

:91: ,
May-1’ ‘F. 9;.h'ta“ A_y-

1
“A H:

4L --
a. Otnew

lcosemA to

 

 

39

assuming a nucleon which remains frozen in the nucleus so ”3' ‘ 2".

[&'\E k. . Using the ansatz of Landau, Phatak, and Tabakin

wetake |Pfll 2'. |fi;Ԥ F1: sothat
N ~

Hal: Ms“: D - “iflwé/wfle; =3 96.

Substituting these values for the magnitudes of the momenta in

equation (4.12) we get the transformation
(a 13) (.056 " \- (9° [k 37' ( o /& icos 9
w ' Wu ' " ° + 2“ A w.

Hence, in equatio; (4.10), we have (L: \ -- ( El: /&°\2. and

b =(?1: / 4&0) . Equation (4.13) is valid for forward scattering
where no momentum is transferred to the struck nucleon but is
incorrect at other angles. For example, equation (4.13) allows

‘ C056“”\ to exceed 1 at backward angles. These unphysical
values of £0561". introduce serious discrepancies into the optical
potential. As we shall see later, the agreement between the modified
static approximation and the fully integrated optical potential

is very poor at backward angles.

6. The Effective Collision Energy

So far, nothing has been said about the choice of we“
in the g hoc models we have discussed. In general, most workers

use a. value of men that is plausible for forward scattering (Lan73 ,

Kis 74). Hence, for our study we use the value of we.“ obtained

in Chapter VII with our systematic approximation, but evaluated for

H)
t)
(n

_ . M, .3“:1
«:1 an».

n: é’u-p " ”f:

I.
. 0". V .‘v‘
fins—(e V‘ *'

v.4 1:

.
.,_,- I -'

'v 3‘" “P L Q.
cut/13:9“: ~'-~ ,

2.
avenge of Pg" <

N

.

I ‘ '

.: One m :o’ovs‘ no: a
c. an .v--§v-".: v-'.

n
a

r “2‘5 $‘
L A... Le 929:3“:

M.
w: EY‘VOQQ’ rm 4",

fiy -
.§v~~,‘.- -

 

 

"iere 7.
PK

 

(e91.

 

4O

forward on-shell scattering. This choice is intended to be repre-
sentative of the models found in the literature.1 The explicit

result is.2

(4.14) 6‘”: (,3°- (‘K1CT/Zwo)[ ( “673'? ‘2? * < 94:71'

In equation (4.14), the quantity (P P2“) is the ground state
average of P‘N (we use (9: “7: ~l fm 2). The variable P1:
is the relativistic momentum of the incident pion in the “TQM frame.

We take the energy parameter 00° to be

a...) w°= m; me + (*‘°"/2m.&)[<"2.> + tat/S] -

This expression for (10° is just the relativistic energy of the
pion plus the ground state average of the relativistic nucleon

0
energy (neglecting the potential energy). That is, (.0 is

0 V;
(has) w=(v\‘ep°"+m;e‘) +mfic} + (N- é/zm c501?)

‘

when. ( eel) = < 3:“) + (P; A?

 

lThe difference between equation (4.14) and typical choices
is, at most, a few MeV. (See, for example, (Lan73) ).

zThe result 6€ven here is obtained from the "linear approxi-
mation" (equation (7.6 of Chapter VII) for [P \-.= (9‘: he“ 9' and

£059wa .-

'IQ
I'D“

.5v..

III" '7 \ " Q a

._ I , '-
‘~i ’1’ ‘m‘¢le‘

;~:“

-
‘EFQ Inn:- A
~uauu.\,w I

l
00. V0.

‘s
:57- rat
“‘

I ~..
_ “"" LSCJD

R ' o
’ '1’“. ‘.
a. *“'¥er ‘I I
fi
7"‘3‘1
a

‘5: . .
3.1311?“ 0:5.
12519;;
:‘~‘ ,
0' liq- . ‘
. amt“ ha... Ca
‘—

(5.1) .
t- P.

 

S

 

CHAPTER V

THE FACTORED FORM OF THE 11' fine OPTICAL
POTENTIAL: A RECENT 5p p99 MODEL

When our work was nearing completion, Kujawski and Miller
(Kuj74), Miller (M1174), and Landau (Lan74) proposed angle trans-
formations which take into account the recoil of the nucleon. We
shall not discuss these recent developments in great detail since

in Chapter VII we present a more systematic choice for all of the

variables (wgm ’ h... ‘ ‘E‘ 3 : _a~ which is just as

N
easy to use as any of these more recent ad _h_g_c_:_ prescriptions.

However, for completeness, we shall give the general form of these
prescriptions.
.. l .—
Instead of assuming ‘% \ — \k \: a. as in the
angle transformation used in the modified static approximation,
we could have calculated \h\ and ‘h'\ by assuming some
~ A:

value for P in the quasi—relativistic relationsl

~

(5-1) Eli'- ‘3- Kw%/w°)‘)£ > 23 = Pu: (“g/c.3635,

 

lSee Appendix A for a discussion of kinematics.

41

1..
”- an 1.1 e
1 l v
rq:vu0 "

"I... a: 33;? Cf

'. A. v- use“-

e‘r’w getwee"

v..
.hou-vn

-
‘---.olv‘! 5"" a“ .
.v;~ “"C' U:v-v ‘ -

u
I I ‘
- Y“"="'¢“-1 c.
u... -a.uv¥.auv .
C v
.:,Q_:.. Y ”v Y
‘6 and. v‘ -I . -..
a
‘ -3 ..
O‘.:.A”.F.y A“ -'.':‘
wka-o i- a- V.. v. . ..
n—‘VI W‘ tog-.9." f‘ ,1 ‘w‘
v~ “""""-.4‘V a-"
- -

.r.:'.g :Q
:"““-~h..

~- '
2 .‘H :‘u : .
. ‘
"v "-'"4y 2“
‘. ‘~o~a u‘

.92: 4
E -
w- \8 8 p
""‘Sdslfizaa
“*OTQ _
." I‘d:
f... .
\ 21":- s.’
” V
w; .

 

42

where GEE :: “ES. :; ‘E;'«+-‘i‘ :3 “;.*.i%: for a free
nucleon. The value for ~Pfl used bymli‘lille’rui is P" = 0
an

and the value used by Landau is E = — ii (93 _ 93' 3 .
With either of these choices, equation (5.1) leads to a non-linear
relation between Cosg‘n‘ and QO$9vy and the
resulting Optical potential is in much better agreement with the
fully integrated optical potential than the two Older ad hgg models
Of Chapter IV. In fact, the choice made by Landau gives an angle
transformation that is almost identical to the one we Obtain in
our systematic approximation to the fully integrated optical
potential.

A convenient way to compare the transformation of Kujawski,
Miller, and Landau (KML) with the transformations of the simple
static approximation (SSA) and the modified static approximation
(MSA) is to set [93 \-:. \ P35 '\- .. (9° (1. e., on—shell)
and expand the transformation formulas in powers of b); I (,3... \l 5
keeping terms up to the first power in w“- I my . With this
procedure, we get the results shown in Table 5.1.2 In Figure (5.1)
we compare these transformations graphically for P1: :7- |.5 ¥M.
(5“,: (88 MeV) . At Ova-0° , all three
transformations have the same value. The MSA and KLM approximations

0
also have the same slope at 9 7- O . At 6 = 180° the
1w 1”;

 

1The form given by Miller is equivalent on-shell to the
form given by Kujawski and Miller.

2To order ”31/93. , the transformations of Kujawski,
Miller, and Landau are the same for ‘9‘, -_-, ‘Qé\= 9
~

”7" —.\.“# ’-

. Z ' f‘ A
:— l‘-“. CUHF--.‘
--J AQ~A. - ‘1' A,“
$1.: ‘ .gfi 2

"' ”Hv-v..
..

~.<.'A..-' 1 ‘

 

 

3:: CDSBR‘ -_

I: [0% 6U“ -

\

 

 

43

TABLE 5.l--A comparison of the angle transformations used in the simple

static approximation (SSA), the modified static approximation (msa) , and
the Kujawski-Miller-Landau approximation (KML).

 

 

SSA: cote“: case“,
0 a O
as... (.0561... = “ZUw/w" + (\ + awe/w 3 Case”

0 2.
XML: C05 6“.“ -: -m§/m° + C069“, 4- (ww[w°)COS 91W

 

 

 

 

 

MSA

I

/

I

FIGURE 5.1--Graphical representation of the angle transformations
used in the simple static approximation (SSA), modified static
approximation (MSA), and the Kujafski-Miller-Landau approximation
(1011.) for "3‘ =\|3\= e“; 1.5 fm

u.‘ ~
....,.
Lina-1-. -..... «-
.

.‘ I 1 1 n
‘4 V" ": v- no r“
5.5- .oom‘v- A. U- ‘.

,u -
‘D’HA.. "
3.. doag‘;.‘

\v

(D
u.
DJ

(D

 

 

45

SSA and ma approximation both give cos 9.“: -‘\_ while
the MSA gives COS 91m?! -2. . As discussed in previous
chapters, the unphysical backward angle values for 005 91"“
obtained with the MSA leads to large discrepancies in the optical
potential. The ICLM transformation has a C051 9““, term
which cancels the constant term for 91W... 0. or 150° 50
that C0591“ varies nonlinearly between + l and - l .
The inclusion of nucleon recoil via the KML angle transformation
apparently remedies most of the problems associated with the simple
static approximation and modified static approximation.

Although the more recent prescriptions which include
nucleon recoil are certainly an improvement over the prescriptions
represented by the simple static approximation and the modified
static approximation, they are no less arbitrary. Our systematic
approach to the factored form in Chapter VII eliminates this

arbitrariness by relating the prescription for' .aE: to the
fully integrated result and perhaps more importantly, our prescrip—
tion allows some insight into the physics of elastic pion-nucleus
scattering.

In Chapters VI and X we show results using the angle
transformation of XML. The general agreement with the fully
integrated result is surprisingly good although some important
features of the fully-integrated result (which are associated with

the choice of to“ ) are not given by the XML ansatz.

.0- - “
-... IL” ;,~..
.0.- “‘U. .
"‘V'v.
":2- ‘I- .. .
a...

 

‘1': “I -

.‘i‘ , f “ I h

“N :“'-“..ac-t

0‘... 6‘ I

*4, n A ‘
W “ ‘ -
a —-J..-.‘uc-t

'N .. ‘
.1. 31131.11 Cora-
A. ”61"
a.
‘5!

 

’ X
‘
I
: ‘ q
I A.
k ‘8: ces‘f‘fl “
N -r-e."
b
‘fld, a
m-«be 1“

CHAPTER VI

THE FACTORED FORM OF THE 1'" -uHe OPTICAL POTENTIAL:
THE EFFECTIVE PION-NUCLEON IMPULSE INTERACTION

1. Definition of the Effective Impulse Interaction

Writing the ‘fi' -uI-Ie Optical potential in a factored
form allows a convenient separation Of nuclear structure effects
from the pion-nucleon interaction effects; however, we must realize
that the pion-nucleon interaction factor which multiplies the ground
state form factor is actually an effective pion-nucleon interaction.
The interaction factor is a function of just the pion coordinates
but should contain the effects of internal nucleon motion. It is
not clear.§‘pgigri that the simple static approximation or modified
static approximation properly represent the nucleon motion effects
(a pgsteriori they do notl) so we define an effective pion-nucleon
impulse interaction by writing the fully-integrated 11' -He optical

potential in the form

(.1, U». a e.,»: A M 06%)

 

1However, the more recent KML approximation gives much
better agreement with the fullyeintegrated result than either the
simple static approximation or the modified static approximation.

46

. . \
“:12; t

. —- a-

‘f‘:'61 9" «“4 a}. A. . ;
"vi... a

 

47

0r

-\
(6.2) <t1m> :[( A-t) ‘0 ($31 Um!) E“: P“)
~ ~
where (tfiflv is the effective pion-nucleon impulse interaction
and U1", is the fully-integrated 1‘ -4He optical potential
calculated as described in Chapter III. The above definition for
(tVfl allows us to have a factored form for the Optical
potential without making any arbitrary choice for £2. . We can
now focus our attention on (th> rather than the Optical
potential itself.
It should be noted that (t‘tu) is well defined only

in cases where P($3 has no zeros or where U1“, L P‘: PW\
xv av

can be written explicitly in factored form. Since we expect that

(an) does not depend strongly on the details of the

nuclear wave function, we take LP L P4N\ to be of the form
3, 2, ~

No EXVQ'HkR. 9“» with R0... (,5 {.m . This

choice results in a gaussian form factor so that <t“fl>
' l
is well defined. We calculate U using the
W L ‘3 Pl! 3
gaussian wave function and then obtain the fully-integrated result
for (Hg) via the definition in equation (5.2). In the
next section we compare the exact or fully-integrated result for
t with the result given by the simple static approxi-
1!“

mation and the modified static approximation.

9 two

‘54

W

 

afactezed for-.1 <35

.o'

-
A‘D"
\

v. I

‘U

'2

angers“

tau-"

fie u-.—~:'-

.1 ‘
.rizer t

§

 

Ii “" e
in 8115
veracti

1m

{
d

“33
on

..
99:1 in

HF.
can

a
v1 '0‘

“U‘

'V.‘
ԤU-
H
9-“
”g-
h.“
“‘in.

u
-A

Dove form

‘

 

48
2. The Fully-Integrated I Julse Interaction Compared to the
Simple Static Approximation and the Modified Static Approximation

The two .a_.d hgg approximations used in Chapter IV to obtain
a factored form of the 1" -I+He Opti cal potential can now be re-
garded as approximations for the exact or fully-integrated effective
impulse interaction, i.e., (hfl'flw 9.: tfifl Q .9: 3
where .95 is given by the prescriptions in Chapter IV. We will
consider the simple static approximation and the modified static
approximation to be approximations for (tfiflv in the
remainder of our discussions.

In Figures 6.1 - 6.3 we show on-shell <t“N»
the following cases: (a) the fully-integrated result Obtained
from a numerical evaluation of U‘WV k g“; a: 3 and the
definition in equation 6.2; (b) the result Obtained from the simple
static approximation; (c) the result obtained from the modified
static approximation. In all three cases we have assumed the pion-
nucleon interaction is in the (3,3) channel only and we have used

the following zero-range form for tm(£1.\:

A A
(63’ t,,,L4L\= Yak “tub \QA'AIA

The above form is just the parameterization described in Appendix B
with I: 3’2)I= 3/2, , and L3‘ . The factor d

is hence (| ’3“) tle .
It is apparent from Figures 6.1 - 6.3 that there are

33»

significant differences between the fully-integrated result for

49

FIGURE 6.1—- The real and imaginary parts of the effective pion—
nucleon impulse interaction as a function of the cosine of the
pion-nucleus C.M. scattering angle calculated by three methods:
(a) (solid line) the effective interaction resulting from full
inte ation of the impulse approximation (FIA); (b) (short dash
line§rthe simple static approximation (SSA); (c) (dot—dash line)
the modified static approximation (MSA). The laboratory kinetic
energy of the pion is 110 MeV.

 

50

‘ RE <t1rN>
- IOOO(Mev-fm3)

IIO MeV

 

 

 

 

 

------ SSA - -Iooo

IM <t1TN>
- 800 (MeV -fm3)

 

 

 

 

" “800

FIGURE 6.1.

//

‘
-‘
“
u.“ U. “A‘S i
u ‘4: -
é‘c‘lrlg at
\a
3‘2; 4 ‘ ‘
‘ ‘ ~— ‘ :‘
.a ,9». A..::n\

51

 

 

RE <t1m>
\‘ - 200(MeV-fm3)
\\
\
~\ I80 MeV
\\ . L :00
\\ \‘
\\\ \
\ .
II Jr, 4;>‘:" ‘\x f\; ‘3' e? é)
‘\ ~ 1T 7T
\ \ \‘
‘\\
\\\\
--Ioo \\
-'200
IM<t1rN>

~ 2000 (MeV-(m3)

 

 

 

—— FIA
----- SSA ‘\
----- MSA

 

~‘2000

FIGURE 6.2--As in Figure 6.1 except at 180 MeV. Note the strong
down bending at backward angles in the real part Of the fully-
integrated impulse interaction (FIA).

[4/
(' .7\

LA,

I
II-
! I)

26C

52

RE <t1m>
260 MeV ’ 800 (MeV-M3)

 

 

 

 

” L-soo
\\
‘\ r 800 (MeV -fm3)

\

\
\
\

\
-|
l J
— FIA
----- SSA
----- MSA

 

 

" ‘800
FIGURE 6.3——As in Figure 6.1 except at 260 MeV.-

Q
.- ‘x 9" P4
' .pn - A H
:n‘ a 5"----'“
A’- "H "’
- "F‘ A».
- 0 *4...
"a: 4’ G rave-
M-.- ”-OV-
" '
wr‘ Q
R“ ._...
v-O: ‘ .“~V- "F -
‘n F \
-*a “-dub ‘._. -

T" “H:_
”Vi-
-

-. A .
A. “9-?“ CA“ A‘
I " ""‘-~b-. V‘
- V
«-1 *P‘o 2'. .-_
.'-o a..- ..g_ _-
g- ._ _
‘.\ a <
V t-v_.
." ' .
'9. F4 .‘
."-:--.: ‘9‘; To.
. " Uta- ..VI
~‘:- 4
v‘v: AA- 0
u ‘ val ’ a ‘
‘-v

 

av

5' l
fiw .‘

y‘.‘ a ‘N":
~_ ] —
--*vo_“

c
..-’
- v
‘~¢:: _‘A,:
~“'=“. m -‘
A‘
\",~~ ,
u‘ 0-1 5‘
'--- s a»! c.-
-K‘ , .‘_
‘ ““. 4"“
V“--.
V
i‘ O
‘
u 5,
\
.
.
Q ~
:'\ {3...- ,

v. “4“ A“ n—

v' ‘
s

 

53

()5an and the results given by the simple static approximation
and the modified static approximation. In trying to understand
these differences we have arrived at a simple prescription for
(tun) which is in good agreement with the exact result and
which allows us to understand in a transparent way the main effects

of nucleon motion in elastic pion-nucleus scattering. We present

our prescription for (“kw“) in Chapter VII.

3. Comparison of the Fully-Integrated Impulse Interaction
with the Kujawski-Miller—Landau Model

 

As mentioned in Chapter V, when our study was nearly
finished, the newer models which take nucleon recoil into account
began to come into use. For the sake of completeness, we have
calculated <th> using the KML angle transformation in
Chapter V. The results are shown in Figures 6.4 - 6.6. Clearly,
the KML approximation is a much better representation Of the fully-
integrated result for (tun) , although there are still
some significant discrepancies.

It is important to note that at each energy the KML
approximation for <t‘RN) (as well as the other two ad hog
approximations) is just a complex constant times the function for

(.05 0“" given by the associated angle transformation
in Table 5.1. The surprisingly good agreement between the KML
approximation and the fully-integrated result thus indicates that

nucleon recoil is an important effect in (twu> .

FIGURE 6.4--The real and imaginary parts of the effective pion-
nucleon impulse interaction as a function of the cosine Of the
pion-nucleus C.M. scattering angle calculated by two methods:
(a) (solid line) the effective interaction resulting from full
integration of the impulse approximation (FIA); (b) (large dots)
the Kujawski-Miller—Landau approximation KML to case (a). The
laboratory kinetic energy of the pion is 110 MeV.

IIO

 

FIGURE 6.4.

55

RE<f1r~>
IIOMeV PIOOO (MeV-fm3)

 

 

 

. . OKML b'lOOO

IM (LVN)
F 800 (MeV-fm3)

 

 

 

 

“”800

56

 

 

RE <t1, >
- 20b (Mev-fm3)
l8 MeV
. . . ”'00 O
O
I
. O
- 0 e
1' l 11””1.‘g fix. 1' av .f;
0 1r 1r
0
O
O
O
e-Ioo '
O
L"200
FIA

 

o o o KML IM<t1rN>
-2OOO(Mev-fm3)

 

- 'IOOO

 

"2000
FIGURE 6.5--As in Figure 6.4 except at 180 MeV.

260

 

iti/
." "

$1 ‘\

s
at“ a
-'

57

 

 

 

RE “MP
260 MeV P 300 (Mev- fms)
- 4oo ,
- H
I 1 l J 6!”. $7
r-4OO
FIA "800
C C C KML
IM <:,,N>

- eoo (MeV-fmz’)

 

1 Elk, A
P1r'P1r

 

--4oo

 

h-BOO
FIGURE 6.6--As in Figure 6.4 except at 260 MeV.

~
~.
- d
.i 6 ~ 0
-
|v -V‘
‘0. d.-
II
m-~-- ~ a...“
. 1 .— A- f" r. W
'u.'—.-~‘U-v .— -" _
~ - -
3-;- ‘:
_w_fi- A
-“'--‘v J‘V...
Q . -
" 22v ‘ ..
-o .. .5"_,__‘ ..

q a- .
-.‘ - ‘.~-‘

a.-. qr
0‘ -..‘.v‘ o

...'.‘W: A
- -“ F~
"~- . - ‘ A*
‘— -v‘
I
1* ~\
.- ,‘
‘\ 0s}!
\“ ‘
~¢=- .-
‘ ‘h
s \‘ ‘
- ‘ ‘
‘~ 4
‘1
‘ «‘5

CHAPTER VII

A SYSTEMATIC APPROXIMATION FOR THE EFFECTIVE
PION-NUCLEON IMPULSE INTERACTION

In this chapter we outline a method for obtaining a
systematic approximation for the effective pion-nucleon impulse

interaction.

1. A Linear Approximation

We assume the fluctuations in t“”L-I}:’ \ due to

nucleon motion are small and make a linear expansion of 11“” (.0. \

obtaining for <£WN7 1
(\I "‘ #r
(7'1) <twu> = f) (.33 SW‘fiufi' at?“ ‘3

’43:“ (‘99 +(V; t 1h); (xv-n A3? ‘34»
an

N

 

1It turns out a similar approach has previously been
used in nucleon—nucleus scattering (Kow63, Ade72). We thank
R. Landau for bringing these calculations to our attention.

A
.-
h.“

.o
-n

u
do. V

-—¢,"¢

2

AA
3 .‘
vs:

1
“I-
I. «0'

I.

I
i t
X
A.

A.

‘..

.
:2

 

59

where the expansion point _fl. is arbitrary at this time. Clearly
~

if we choose _a. to be

~

(72)}: (“$3 S‘ffiwzib, +F_\_" 59““? “CA?“
~ (210

 

the linear term vanishes so that we obtain the simple prescription

(7.3) (twp a Em (:9

where :é is defined in equation (7.2). We call the above
prescription the "linear approximation" for <t“"> .

In order to obtain an explicit expression for (t‘u>
we use the gaussian form for “P(? 4N} given in Chapter VI and

make a change of variable P: “a 12%“ L" so we have
2A

(7.4) 3; 2 €(%)-\ex9(- R7; ‘B‘I‘A A: 4.
P

x811. N: eva‘ 2R0 P /3)E;‘:33

The form factor associated with our gaussian wave function is

efo- “tit/L) so we have finally

(7.9%: S41. M Dex? (~23. P “M ‘33 9*
' (2.1V)

C .r- "‘:
.: “t." n: H“
o ”V '0 -

‘
-
. u a
“I .12.
up... --

 

"- ~\ —

\Iv'i

' an“-

N -‘ _-: R '—
. U-v..:

.--

~
{
Lru‘
\- _
-‘k
c
-
5
‘ ‘-»
v— -
k‘.‘ 0.:—
~— . ,‘
‘~V-‘ ELV- A
V n.
-g ..
.
“
, h\
H .:‘ ‘
J

‘3: ‘
\\\ ‘
\ a
t; g
‘

 

60

We use the quasi-relativistic kinematics given in Appendix A and

obtain for .2. 1
~

(7.6) 13m = (5-- -——-.‘:[L .L‘AiUEmgia-(véil

(7.7) E '-"‘- E‘Q’D‘; /u>° %3(II*?}£»
(7.8) 1: = flaw;/m°)(%i\(vg ‘43 ‘31?) '

~

I:

In equations (7. 6) - (7. 8), E and P}; are respectively
the initial and final momentum of the pion in the WVCM frame.
The quantity' 000 is the relativistic energy of the incident pion
(Chin. -(‘fiC PW; + mWC >72; and (P41) is the
usual ground state average of Pat: . The energy parameter
0 0 ~
(ID is 03.“. plus the ground state average of the relativistic

nucleon energy (neglecting the potential energy), i.e.,

(7.9) (0° 2 w; + Mun?" + “1‘1/2M“0)<?:7

 

1We could use fully relativistic kinematics just as
easily. For purposes of comparison we consistently use the quasi-
relativistic kinematics of Appendix A.

(9:7 =<

-1. ‘1 ~
'.o :. O'-‘ 8-- '
.25. .U’I .0 -ao
.

AV. “‘1 A'-

n

..-O‘ c- J.-
p - a v-

‘._.' g.v~v- .

. A F: ‘
w "a 6"" e (if...

,.v woo-or-

.. r:-~
a ’A—

‘0 - ‘7-
~—

 

23: (tun)

‘ 9‘»..-
- m: 93:22.21
-

h.
‘I .
“g s
\ FW-
. A
~.. 4-..:1‘ .
-“

>—
‘-—.V’ A: 3"“
' a- A's ‘
ubwv'q «I‘LI v
I...-

.--I

 

-S‘Cv- ‘ .

\ 'r
‘Oy .
R‘.

.
.._ '
_ ..
-H
~“~..;Q‘ .
" ".2 e.
V 5
‘~
9.“-
.‘:'\
-1 _ A
. . ‘5‘ F_
s '-
~

 

61

where (P: 7" —<::“7 + LP“? hot. It should be noted

that for on-shell values of ?W and PE, and 61' p: 00 , the
above prescription for (tun) reduces to the 1'prescriptions
in the simple and modified static approximations.

In Figures 7.1 '- 7.3 we compare the linear approximation

for (t'fi “7 with the fully-integrated result.

2. A QuadraticA Jproximation

A smoothing effect is mis. ing in our linear approximation
for <t““> that can be included by using energy averaged
input for E““ or by including higher order terms in the

expansion of ti?» . We consider here a quadratic approximation
1
to in the ero-ran e t-matrix of e uation 6. .
0'.“th _ z 5 <1 ( 3)
We thus expand the to second order in we». but only to
I
first order in k and l . This procedure provides a simple
~ ~
way of accounting for the main part of the nonlinear energy depen-
dence in t“
In Figure 7.4 we show 0‘33 k wa\ in the energy
7- 1..
region UM“ \- M “A C. ’50 (mvemfiyc - 400 MeV .
In regions where d3; L wa\ shows strong nonlinear
behavior, we want our quadratic approximation to produce an averaging
effect. For example, the peak in IM L433 3 at
.— 7- 2.’
00m (MW-rm“); .. ISO MeV should be lowered

somewhat with a quadratic approximation.

 

1The parameterization in equation (6.3) is

tmg 4.3 = Y 0&3} L “3.3 43.3. .

62

FIGURE 7.l-—The real and imaginary parts of the effective pion—
nucleon impulse interaction as a function of the cosine of the
pion-nucleus C.M. scattering angle calculated by two methods:

(a) (solid line) the effective interaction resulting from full
integration of the impulse a proximation (FIA); (b) (large dots)
the linear approximation (LA§ to case (a). The laboratory kinetic
energy of the pion is 110 MeV.

 

 

 

63

RE <t,,~>
~Iooo (Mev-fm3)

_500 HO MeV

3| A, A
Pr'P-n-

L-

 

 

-'500

.
-———'FIA wflOOO
. ' . LA IM <17TN>

r 800 (MeV-me)

h 400

 

 

 

-’800

FIGURE 7.1.

6+

 

 

RE <t,,~>
|£3() “AE3\/ F'IBCND (“AG“V"ffi03)
- |00
\ A’ A
0—H P1743”

 

~‘l00

 

 

Q b'1ZC)C)

In“ <:I1rh‘:’
- zooo (MeV-fms)

 

 

 

+| A, A
l J PW'P'rr
FIA
o o a LA -'|000 .

 

*- " 2000

. FIGURE 7.2--As in Figure 7.1 except at 180 MeV. The down bending
at backward angles in the real part is due to the higher average
two-body scattering energy for large angle scattering, which for
this case actually raises the energy from below to above resonance.

26

65

 

 

 

RE <t1rN>
260 Me v ' 800 (Me V°fm3)
- 400
“I *I A, A
l 1 1 P77. P.”
b'400
FIA ”800
o o 0 LA
IM (LVN)

- eoo (MeV-fm3)

 

 

 

* ’800

FIGURE 7.3-——As in Figure 7.1 except at 260 MeV.

 

L003 1

66

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00m -
000...
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67

We choose 0(3‘3kw0M3 to be of the form

— - 2.

where “3‘“ is given in equation (7.6). We require that
ti . h ° 0)
equa on (7 9) be exact at t e pOlnts CM 3 WHAT. 6.0
where 6" 1' is the avera value of
no se ( wcm— "2'93

calculated from equation (7.5). The explicit result for 6:0 is1

(7.10) 23(ch
a

2;.) [0:9 <59
we». 3:) < 2.)]

(Note 6202' depends on the scattering angle, being larger~
at forward angles.) Using the above quadratic form for 0‘33

and proceeding as with the linear case, we obtain

(7'11) (twp : é-[Emf It,“ Hi“ (:1- 3]

 

lWe us; equation (7.5) with .{L -) ( wcn" acmx
to define G.” . For R. - L5 v§wx we have

(P;“)’” i *m-z. and ($.73 5/3 tn .

6239'.“ is about 30 MeV near the (3, 3) resonance.

 

where

-1- — - -,
(7.12)_(L :(w +6- 32:32 .

C.M— 0.)) N

The difference between the quadratic approximation and the linear
approximation is fairly small, but there is a slight improvement
so we use the quadratic approximation in the actual calculation of
elastic scattering cross sections.
3. Qualitative Effects of Nucleon Motion in the

Effective Impulse Interaction

The linear approximation for (t-“Q> contains the

main features of the fully-integrated result (see Figures 7.1 - 7.3)
so we can use the simple form in equation (7.3) to understand how
the effects of internal nucleon motion enter into the effective
pion-nucleon impulse interaction. First, however, we need to under—
stand the properties of _{L . The dependence of .22., on

P“-

~ ~

fact that the average value of P ,4 is "' L "'1" \
it 2A ’3?

and P“. shown in equations (7.6) - (7.8) is due to the

and not zero (we define "average“ here by using equation (7.5)

with .[L -" ? ). This result for ? is valid for any
~ 2.” f8“

single particle wave function of definite parity. Due simply to

momentum conservation the average value of the final QNCM nucleon

momentum is “9410+ A ..:.\%_ "' E:— . Thus for

E4“ 3 TA
elastic scattering (viewed in the hNCM frame) where the nucleon

"absorbs" momentum Eff but no energy, the nucleon wave

function favors collisions where the nucleon momentum vector is

 

69

reversed without changing magnitude. The effect of this momentum
— - "1
"selection rule" is to make “an , ‘91 and k all larger for
«I ~
large angle scattering than for small angle scattering. (We assume
‘ ‘ .’
\?£\ _ ‘9“ ‘ here for purposes of discussion.)
A. _
Let us now examine how the properties of .{L are reflected
- ~
in the behavior of t1”) L4}. 3 . We consider only the
parameterization of equation (6.3), but the main effects should be
present in any parameterization. First, in the angle factor, our
I
prescription for “E. and ‘93: generates an effective angle trans-
formation from the WNCM frame to the “VCM frame. This angle
transformation is essentially the same as the Kujawski-Miller—Landau
(KML) angle transformation given in Table 5.1 and shown in Figure 5.1
in Chapter V. In the limit £347 ‘1, our transformation is identical
with Landau's. (The (A- \)]A factor arises in our transformation
because we take into account the (small) effect of the recoil of the
nucleus as well as the recoil of the nucleon.) Hence, in our trans-
: . ' — —l
formation c.0561.”3 _ «92 £3 /‘.e£\\k\ A
varies nonlinearly between + l and - l as Cos a“ 2 fit. 9",

v
varies between + l and - l . It is interesting to note that for

\ fl: \: \P“.\ , Miller's choice( 9,, = O ) gives
—' ~ .I- ~ 1 La ' ~ .6 - - l. L '
‘g ‘ 1; ‘£\ .1. whiAe\ ndau 5 choice 2 - 2. 9“,-wa
.. ‘ a "’ "
arid— our resu-l-t ( Pt? .. .5? L 93 .. PI 3 ) both give
‘k' ‘ z: ‘t \ . Hence, elastic scattering in the WVCM
frame corresponds (on the average) to elastic scattering also in

the WNCM frame when nucleon motion is taken into account correctly.1

 

lThere is actually a different “NC-M frame for each
component of the nucleon momentum. We really mean the "average"
“NW frame when we speak of "the" WNCM frame in elastic
pion-nucleus scattering.

 

70

Next, the factor X in equation (7.3) (see Appendix B

for an explicit form) has a nonneglible dependence on P‘- and

~

I
at . This factor has previously been treated as a constant
~
for each scattering energy. (Lan73, Kuj74). At backward angles
X is approximately 1, whereas at forward angles X is about

.8 or .9 for the energies we consider.

Third, the most interesting feature of the effective
impulse interaction is the dependence of 0‘33 c we“) on

fir

~
scattering amplitude (He or Im part) is increasing with energy, the

l
and F“. . In energy regions where the free pion—nucleon

factor 433 me‘x (Be or Im part) is larger at backward angles

because u) is larger there. The reverse is true in energy

CM
regions where the free scattering amplitude is decreasing with
increasing energy. This effect is quite evident in the real part
of <t1tfl7 at a pion laboratory energy of 180 Mev (see
Figure 7.2). For this case the forward scattered pions have an
average WNCM collision energy that is below the resonance
energy while backward scattered pions have an average collision
energy that is above the resonance energy. The real part of the
free scattering amplitude changes sign at the resonance energy,
causing the real part of (£11137 to be strongly distorted
at backward angles. At other laboratory energies the effect is
much less dramatic, but is still present.

I Finally, it is interesting to note that although our
parameterization of £1!» in equation (6.3) is a zero-range
form, the effective impulse interaction (tun has a finite
range (which changes with the WDCM scattering energy (0. ) due

_ '
to the dependence of (Om on P“, and ES. .

~

 

CHAPTER VIII

CORRECTIONS TO THE IMPULSE APPROXIMATION

Up to this point we have assumed the t-matrix for
scattering of a pion from a bound nucleon is the same as the free
pion-nucleon t—matrix, i.e.. we have used the impulse approximation.
We now consider corrections to the impulse approximation using a
3-body model.

The calculation described in this chapter was motivated
by estimates given by Goldberger and Watson which indicated that
binding effects in pion-nucleus scattering are large in the (3,3)
resonance region (Gol6h). Our calculations indicate that the effects
are, in fact, not very large. Moreover, since the calculation of
binding effects is very tedious even in a simple 3—body model, we
have developed a fairly good approximation to our exact calculation.

Most of the work presented in this chapter was done some
time before the calculation presented in previous chapters and,
consequently, there are some minor differences between the model
used here and the one used in previous chapters. In this chapter
we assume an infinitely heavy nucleus and use a finite-range
off-shell parameterization of the free pion-nucleon t-matrix.

Our experience indicates that these two factors do not significantly
affect the results. Since the binding correction itself turned out

to be fairly small, it was not necessary to modify this early model.

71

72

l. A33gBody;Model

Instead of treating the complete problem of a pion and
A target nucleons, we consider the interaction of a pion with a
single nucleon which is bound in a fixed potential well.1 We
shall assume in this chapter that our 3-body model is a reasonable
approximation for the (A + l)—body problem. In Appendix E we give
a theoretical justification for the model.

In our model the incident pion interacts only with the
nucleon and not with the potential well itself, so the model

Hamiltonian is

(8.1) Hsz+HN+V“N.

Here K“ is the relativistic pion kinetic energ, H" is the
Hamiltonian for the bound nucleon and VWN is the potential
between the pion and the target nucleon. The nucleon Hamiltonian

H” is taken to be

(53-2) H“: KN +UM

 

1We note that since the potential well is fixed in space,
nuclear recoil effects are neglected and the “FUR coordinates
are identical to the laboratory coordinates.

73

where K“ is the nucleon kinetic energy and U” is a single
particle binding potential. Our 3-body Hamiltonian is formally
similar to the original (A + 1)5-body problem in that the nuclear
Hamiltonian H, is replaced with H" .

By replacing H, with H“ in the formalism of
Chapter II, we can easily obtain the t-matrix for scattering of
a pion from a bound nucleon in terms of the free pion-nucleon

tz-matrix. 1

(8.3) Tum) sqfle'l *tcke"[°8“” gill-4)] Tmtey

Here t1“. is the free pion-nucleon t-matrix and °t and 0x0

are given by

. '4
(804) *Le\ = (E-“fi'K”- UN +L€\

' 3 . -\
(8.5) arcs): LE-Kw-K“+cg\ .

 

1We thus replace the nuclear degrees of freedom with
the degrees of freedom associated with a single bound nucleon.

74

The propagator %-(,E\ can be written as

(8.6) 08m): 08.63 *- caAnku ‘3’“)

where RN is defined as

(8.7) RN = UN + UN cEOLE3 RN °

That is, RN is the t-matrix associated with the potential U
’t“
“’N

(8.8) if“: E\ -.-. inf) + GNU?) [050(3) - (34.5)
+’ %OKE')R“ $0C‘X] 4:".(23

In Chapter II it was stated that the energy E. in

N 0

We can now write as

tn“ ‘1.) is an arbitrary parameter. Several authors have
suggested that E, can be chosen so as to minimize the second
term in equation (8.8) (Sch72, Lan73). We intend to calculate
the second term explicitly so we simply take E}?- E. . Setting

EL:- E. in equation (8.8) we obtain for 1““

75

(8.9) Thu) = tugs) + tut?“ (30(6) RN 08002) flute.) .

Rather than attempting to solve the full integral equation for
1;.“ , we calculate the first two terms in the series. It
turns out that the second term is small compared to the first
term, so our procedure is reasonable. We also assume that t1!“
acts only in the (3,3) channel (i.e., “tn“: +333 ), so we

want to calculate

(8.10) first“ = tum) tint?) 33.5)?“ ‘3.£€\t3§F-3 .

The separable parameterization of Appendix B is used for tag .
The off-shell factor fibfi) used in the separable parameterization
(see Appendix B) was taken to be k /(\+ kt ‘3‘) 000%}
rather than unityl. The t—matrix RN is assumed to have a single
:5- wave bound state at EN = - 20 MeV.
Equation (8.10) can be thought of in terms of multiple

scattering and represented diagrammatically as in Figure 8.1.

1In previous chapters a zero-range force was used so that

the off-shell factor was one. The off-shell factor given here was

taken from the work of Charlton and Eisyeznberg (Cha71). The factor
2.

(out) is (ta-tut“ + m“. a) and r.

was taken to be .5 fm. We have found the elastic cross sections

to be relatively insensitive to the form of the off-shell factor.

76

“- 1r
\ 1 4
\\ I \ -~
‘ I, \ /’ \ I
\ \ / \ /
T .. \ I + ‘ \’
Km ' 'H‘ N

Figure 8.1 - Diagrammatic representation of equation (8.10).

The first diagram is the impulse term and the second is a correction
to the impulse term. In the second diagram the nucleon is scattered
by the potential well (via RN ) between scatterings with the pion.
The interaction of the nucleon with the potential well causes the
nucleon wave function to be "distorted" in the intermediate states.
Thus, our 3-body model takes into account multiple interaction

of the pion with the same nucleon and the distortion of the nucleon's
wave function by the binding potential.

In preliminary calculations using a square well for UN ,
the main contribution from RN appeared to be coming from the
bound state pole. It is well known that near a bound state pole
a one-term separable potential gives a good approximation for a

local potential (Lov6h). Also, a separable potential for U con—

N
siderably simplifies the calculation,so the binding potential was

taken to be an s —wave separable potential of the form

(es-11> UNLP; 9&3 -.:. XMLP43uflch

77

where unififi = (dz+?:\‘) 9“: 3A“ {'M-‘
and I 7" 38‘35 MeV' £M.‘ . These values for OK
and X give a bound state at -20 MeV with an EMS radius of
1.61 fm. (A one-term separable potential can produce only one
bound state.) The t-matrix associated with the above separable

potential is

(8.12) RuLE3P;)P“\‘-=XKNLPJ)VNLE\1JL.CP“\ .

The energy factor Y‘NLE} is given by

km» \ "
91:11." 24 (a + up]

 

 

(8.13) V“ LE3=[\ + (

7
‘where in; =: (EZ.VV\..EL /h${“)‘z'°

The matrix elements for the binding correction term were
calculated using the vector bracket method as described in Chapter III
for the impulse term. The binding correction term considered here is
considerably more complicated than the impulse term, so instead of
giving a detailed derivation we indicate schematically the procedure.
A more complete development is given in Appendix F.

We want to calculate

78

(8.14) A‘EW“E (P11-K°\t33°30?‘w“5ot33\°7)Pg> °

Denoting a complete set of two—body states for the pion—nucleon

C.M. frame as ‘WNCM)S(‘WNQN\\ and a complete set
for the pion-nucleus C.M. frame as \ WVLM) S <‘“‘ng\

we calculate

(8.15) at.” -.-. < 9130 \wvcu7g<www\wwcw7(<wwcw\

x t33\lTNCM>S<TNtM\TWQM) SOWLM
" %0R“ %0 \WVCM7S (WCM\ “N CM>S<1TNCM\
X £33 \WNQM)S<'WNCM\ “ch784'WVCM\ O)‘ ?fl'> .

2. Numerical Results for T1311

We have calculated exactly the L = O and L = 1 components

I
of < V: ‘(o \Tfl'fl ‘0>\PE> for energies between 31 and
373 MeV (P. = .5 tn“ +0 2,5 {.M“ ) . In Figure 8.2
the result for L = l is plotted on an argand diagram. The solid

line is the impulse result and the dashed line is the impulse result

79

 

 

 
   

(3|) ’

l 1 l l 1 J L l l l l I

1 1 1 1 RE
' 500 0 -500 -|000 -|500
(AM/4m” '

FIGURE 8.2--An Argand diagram of the L = 1 component of

< E“: \(O \t‘HJ 0) \ PE) (impulse result)

and

< PI: ‘< D‘TII’N\ O > ‘ P‘s) (impulse plus binding effects).

The solid line is the impulse result and the dashed line is the
impulse result plus the binding correction. At a given energy, the
arrow connecting the solid line and the dashed line is the binding
correction itself.

 

 

 

80

plus the binding correction. At a particular energy, the arrow
connecting the solid line and the dashed line is the binding
correction itself. The real and imaginary part of the binding
correction (i.e., the arrow) is shown as a function of energy in
Figure 8.3.

One immediately apparent feature in Figure 8.2 is that
the binding correction term (i.e.,the arrow) is small and "rotates"
about twice as fast as the impulse term. A more careful study
indicates that the binding correction term varies approximately
as ( -i ) times the square of the impulse term. The simple relation-
ship between the binding correction term and the impulse term
prompted us to look for some simple approximation which we could
justify theoretically. In the next section we derive an approxi-

mation for the binding correction term.

3. A Single State Approximation for Binding Effects

As we mentioned in the previous section, the main
contribution from the t—matrix RN comes in vicinity of the
bound state pole, i.e., RN(e) is the largest for 3% EB .
We want to consider now a simple model which takes into account
just the bound state pole contribution.

The one-body t-matrix associated with a potential V can

be written

(8.16) T = v + v (E- K+Le)“T

81

)\ I .53 magma“ 93 Mo fiomloa ago ma cogomfihoo mega? on» page. 0902
Auafio_nflpbwiflififlan—OV—u Vmo cofigao decreases: 3.88 cm eohm edema mogo omega .Amd 85mg ca
son-He 93 :93 N5 chem?" a“ 555m :ofioonhoo 336:3 one mo 38mm ngefi one down unkind $505

«>35 hm nub 0mm mm. NO. _m 2.0 mmN mm. NO. _m

\\

 

 

.Eshnn. . can . ow . a... . o... . mwl mm . cam . n... . o... s m.
: cowl
Eaa 55532. Eaa 43m .
co... .
> i
. O ..
oo. .
oo~.

 

 

 

A.
c.

 

 

 

82

where K is the kinetic energy operator. We can also write T

as

(8-17) T :. V + V Z M0<MA

m .
E'EM‘t-Le

whence the kets \M.) are the eigenstates of K + V and include

"W

the continuum. Near a bound state with energy E the L—

i
tens-In dominates so we can approximate the t-matrix there by

(8....) Ta V “N“ ‘V
E- Evie

 

T1113 :form, which is discussed in some detail by Lovelace (Lov64),
is the form we will take for RN . However, in our problem RN is
giVen by equation (8.7) which includes the pion kinetic energ,r

0P8rator in the propagator. Writing out the propagator % o
exPli citly, we have for RN

, -\
(8.19) R“: UfirU“(E“KV-K“+L63 RN .

An ec[trivalent eQuation: analogous to equation (8'17)’ is

(8°20) RN=UN+UNZ \;?<b\ r U
“ -KT-ELt-Le

 

83

where \b) are the eigenstates of KN + UN , including the

We assume the potential UN has only one bound

state and use the form in equation (8.18) to obtain an approximate
form for RN near this single pole.

continuum states.

(8-21) Rfig UN ‘°><O\ .U“
EEKW-EBH‘B

Hence EB is the energy of the bound state ‘0) . Now the energy
E is E. = 56* E.“- where E.“ is the kinetic

energy of the pion. The denominator in equation (8.21) hence is

Thus, in our single pole approximation, we have for RN
IO)<o\
(8.22) R = U“ . V»
N En‘Kwfle

use, we take for the pion-nucleon t-matrix the spin-isospin
a-Ve:I:'aged (3,3) channel component of tag which we denote as

t3; . Then using equation (8.22) for RN we have for the
bil’lding correction term

_. ' - UN‘°><°\U '-
<8.23) at“-<3\<o\t33%. {aw-Kn; getalmpg) -

 

84

Inserting a complete set of pion states on both sides of

% R“ ck.) and remembering that %o and RN are diagonal

with respect to the pion momentum we obtain

‘3 ll
- ) at s S A P - ..
(8 21» we in? < PI: \(O\‘t33\ 2.5 >

U“ \o>(0\ U“

x°ho , afio<§l¥rflio>‘%>'

 

I .
The denominator E17- K‘. +Le can be written

 

(8.25) LET‘K:+LES‘Z K“ - LWSLEW‘K;\ .

II

I
’1.
2
Since K“. 2 {tic} P13 + M;C.~)" MEG} we can

Write the S -function in terms of the momentum

 

 

_\ . o
(8.26) (ET- " + L6) = W n - urea... MP} P‘s)
Flt-K“, ‘Wc? F1:

1. - "I" 0 ‘2-
Where E“. = (ktct e; + fiQA‘) _ mtg?- E on“... MWC’

 

85

3 u u" u
Remembering that A P.“ means ?'W A?“ (3.1?“

'W
and keeping just the S -function contribution from RN we have

ll
a’:jE'—«t;ez‘ the integration over ‘3‘.

(8 -27) At‘WN: 3,3119; (a; \<O\:E33 \ ?1:">

m )3 ~

X(- L“ (Q; ?:)%OU” \°)<°\U” C30
'W’c.‘

X<?‘:“\:€3‘5\0>\PI> °

 

H
In equation (8.27) the notation ‘ P“. > means a state where
N 0 ~ 0
\ ? | = 9 but the direction of P is still an
W ‘- 1"
~ ~
1ritegiltaxion variable. We can further simplify Atw“
I
noting the form that (to now has. Before we took the LS

Part of RN , ‘30 was given by

by

-\
(8.28) 06": (EB+E“-Kw-\<N+Le) .

. " 0

Taking the LS part of RN puts IP“.\: P so Kw
~ 1"

and %0 becomes

(8.29) ”16°:(EB-KN4r LEY‘ .

86

-\
Now by definition (EB- KN+ 3.6:) UN ‘0) '3 ‘0)

so we can write equation (8.27) as

(8-30) Stun: ;R:’R;Séa9 «<9; \M‘ %‘0>\?1:u>

X <P“°“\<o\:€:n ‘°>\?w> ‘

—- on
The quantity < P“: \(O \f33 \ o)‘ P“ > is the

same matrix element that was calculated for the optical potential
in our independent particle model for “He, 1.8.. we wrote in

earlier chapters (for just the (3,3) channel):

(8 31
> o“ ”(P

2:1-

, 91:") s (A-\\< v1," \(o \‘Esgl o>\ VETS

so We can write equation (8.22) as

 

(8.32) At“ : ' “L”; P;
" LIA-t)" “W‘s"

‘ .u o"
xgdha‘uUmki BE)U“§P,. PB) .

N

 

87

W113 a partial-wave expansion of U‘WV we obtain:L

 

3) At "' ‘ (—-—"‘-’-i' in; Pa
(8 ’ 3 1‘" " (rt-n" 81F we
2._L__-H

L

x m -——U1' U4? 9;)U;vLP;PW\PLLz;-$W\.

In Figure 8.1+ we compare the approximation for At-"N given

in equation (8.33) with the exact result for L = 1 . Our approxi-

mation is fairly accurate and since the binding correction is small

we felt that the approximate form was adequate for our purposes.

Our approximation allows us to understand the main features

of At

“I in a simple way. The fact that the bound state pole

0f RN contributes strongly means that the factor %0Rn *0

is largely imaginary and the pion momentum is on-shell so that
< 9.; \<o \tficbmuex. tag \o) \ 2.)
varies approximately as - L[<P PK‘<O‘:€33‘°> ‘ va-lz’.

This relationship between the phase of the impulse term and the
phase Of the binding correction causes the impulse + binding

term to lie "inside" the impulse term on a argand diagram (see

Figure 8.2). In the resonance region the binding correction term

is approximately 1800 out of phase with the impulse term so that

—-—\‘

1In equation (8. 33) the partifl—wave components
are defined by the expansion

”with” z—“me' WE: it.»

-.P

 

88

.m.m mhswfim 3H we mean on» ma mafia daHow one .Amm.mv soapmswo eonm coavmeonHmmm opmvm mawcfim
onp wocfia conmmcv ADV “mvoxomhn Hovoo> mcams padmon pomxm map Aoqfia caaowv Adv .mhmz 03¢ dmpmadoamo
Hug How K. fin o 1 0 An 0 m. mo mg renewal—h.“ can flag one}. . gum
A AiAo, a. so who u; vrev em

92): km mum mom mm. NO. _n mbm 0mm m9 NO. .0

 

     

“75.1%. .. hum . ohm . m. . m. . 0:. mm . era. . h... . m. . m.
08. .
54.. 542642. \ / EE 44mm .
\ 007 ..

 

 

  

~
~
m
_
_
.
_
_
~

(

 

 

 

89

the imaginary part of the pion-nucleon interaction strength is
reduced by 10 - 20 % . Above and below the resonance, the phase
1. 5 such that the real part is decreased.

The bound t—matrix ‘1‘“ is now obtained using our
a,,13:92::‘cbximation for At““ . (We also write the impulse term

Ln terms of U‘w.)

(8-34) <91;\<°\’:m\°>\?}!>

.. \ ZLH ‘- a
PM 2' 41! UMP?“ PW} +

-i. an“ .
A-\ Bw":3c.‘- UWW'P‘: “2“th 915:}
x PLL 9.. - P“\

 

If We extend our approximation to include all terms in the series

implied by equation (8.9), we obtain

(8.35) < a; \<°\1:m\o>\?£> :-

l 2__L+\
gA-QL _41;_UH
UL g b L o
_ “L919... junta. Pa
“3;?- BVC.) an]

(A-\\( ‘—
HT '48;‘-;\‘c)1w U U)“;

a: 2.»

 

x [10:43“)

90

We shall refer to the above approximation as the "single state

a,1119:]:‘03‘21111331510n" for the t—matrix T'WN since we have assumed a single

bound state for the nucleon. Equation (8.35) is used in the actual
calculations of the cross sections.

The numerical differences
between "equations (8.314) and (8.35) are very small.

1;, - A Closure Limit
In order to estimate an upper limit on binding effects

of the sort we discussed in the previous sections, we now consider
a

model where there are an infinite number of bound states (as in

a hamonic oscillator potential). We want to emphasize that our

aim is only to estimate an upper bound for binding effects and not
to represent any physical system.

We generalize equation (8.22) to represent an infinite

number of bound states by taking RN to be of the form

03-36) R LE) = “ .
“ z; E- EM-t- Le.

5° “that equation (8.30) for At'KN becomes

(a...) Atmeggfigz inpu<P£\<O\t,3\m)\P;'>

1“.

K < 91;"‘<M‘€33\ °>‘?lr> '

 

91

Using the factored form as was done in the optical potential

calculation (see Chapter VI) we can write equation (8.37) as

(8.38)

X (P'

.Mt-i33fmct*-vo<a.u§x29>

N N

/here no? -03" +4.1 W—E 8met 8?

the flation; =( m?- (.4. + mfi'c} L PM31)VL. The vector
p“. has fixed “magnitude

is defined by

“‘3‘? but the direction is variable.

0
Since the energy ml’ is much larger than £0“ EM
we ’

M M. o
at to and to obtainin
8 mm “31‘ at: P“ g

(8. :39)

mm =( gfighgm <2. 39>

(ML?

 

‘3»th W'- 41?”)? L95— 4,3.

92

Now the form factors are given by
M0

‘8‘” Emits“ Sfiquotgéii

am"

so that using closure we have

(8.41) '_ o . = l_ .
E (Don‘t: EXPMOLPAS-?I\ (000( P: 912‘

|
The Quantity foo L E“.- wa is just what we have
~ I0

veen calling ()(%3 so equation (8.39) can be written

(8A2)
4. P1? m;

At“: ( Evita“) ‘0 Cm

"941953333”;egl><£w§30 .

“an mg a partial-wave expansion of (t3‘5 L P". 9‘: 3) and
~ ~

< {33 (PI? Pw\> we obtain the result

4 .‘fl'

 

93

(8A3)

Atfiuz (2:2:Wt:t) (0 ($3

K 23—“:<1E33L?‘P°3><:E33LP; PM) PU" P'X'

 

@tending our result to include all terms implied by equation (8.9)

/e have

(8.111») 0;“)
"L" __1r__

New»: ext—T—K‘c" 1»? 30"

2— 2m (tncewv. ))._<t33(91. m) P LA

at 3.» -
I. 4‘“ LOOK?"
l *’ __“.
8‘? RC} (twéw “ 9" )>L

Using this approximation we obtain for the closure limit form

f ‘
° 1!»

 

(8A5)

 

P“!K°V‘\m\°>\fiv w>= (“5192” ZL-H x

 

F- .

P A A
< t33(er %‘>L- -( 31:21; <t33(:f 3» its“ 1&5}. fawn?»
L ‘ + 37.1%; 1<t33QP1§PW°>>L_

 

 

’lhe 01-0 sure limit varies with energy in roughly the same way as

the single state approximation, but the effect is about twice as

large. For example, near the (3,3) resonance the single state

approximation reduces the imaginary part of the impulse term by

about 10 - 20 % in the forward direction while the closure approxi-
m§tions reduces it by about 30 - 1&0 %
The results we obtain for AtWN

in the closure

limit are in good agreement with results given by Goldberger and
Watson (G016+) who estimate the binding correction term to be

about 2/3 of the impulse term near the (3,3) resonance. Goldberger

311k Watson do not sum the binding correction series so their result

comesponds to the result we obtain in equation (8 1+3) Our equation
0
(8.4-3) gives the binding correction at 9 = 0 to be about 60 %

of the impulse term near the (3,3) resonance, but after the series

is S‘ummed in equation (8.144) the correction is only about 30 - 1+0 76
of the impulse term.

 

‘ c
w.
I

 

95

In Figures 8. 5 - 8.7 we show the effective impulse
interactions given by the single state approximation and the closure
limit.

 

96

FIGURE 8.5-~Three different versions of the effective pion-nucleon
interaction: (a) (solid line) the fully-integrated impulse result
(FIA); (b) (dashed line) the single state approximation for binding
effects €SSAB); (c) (dot-dash line) the closure limit for binding
effects CAB). The laboratory kinetic energy of the pion is 110 MeV.

97

RE <hm>
IIOMeV - IOOO (MeV-fm3)

 

 

 

 

—— FIA "I000
----- SSAB
_.-... CAB IM <1’1rN>

- 800(MoV-fm3)

 

 

 

- 400
N‘ *| A, A
l _l
.\ Pvr' P7
\ S
\\\
\\\\
l"4’00 \
\
\
-"'800

mm 8.5.

 

FIGURE

98

RE <t1m>
I80 MeV ~200 (Mev-fm3)

- IOO

 

 

/ --200 \
I M (tn-N)
- 2000 (MeV-fins)

 

 

 

 

L--.'a000

8.6--As in Figure 8.5 except at 180 MeV.

 

 

V
‘s

Q: I!

99

 

 

RE (tn-N) 3
'260 MeV :- 800 (MeV-fm )
- 400
I”
“I I”: ’.’;I A A
. . ,: Ply-Pt
P’4OO
-— FIA b-goo
----- SSAB
----- CAB
IM.<t1rn>

- 800 (MeV-m”)

 

 

 

 

. . "9.00 .
FIGURE 8.7--As in Figure 8.5 except at 260 Mev.

 

CHAPTER IX
CALCULATION OF THE ELASTIC SCATTERING CRwS SECTIONS

In previous chapters the calculation of the pion—nucleus

optical potential was discussed; in this chapter we discuss the

calculation of elastic cross sections from the pion-nucleus

optical potential .

l. Elation 0f the Cross Section to the Pion-Nucleus t-Matrix

The elastic differential cross section is related to the

elastic scattering amplitude {“9 9‘,\ by

(9.1) Aw/sz. :. \,§ng\ \2.

and the elastic scattering amplitude in related to the pion-nucleus

t_matnx Tm

 

0 O
(9.2) _. __ ‘ W “3»

100

101

where |P; ‘1:- ‘Y‘; \-{> , P1: being the incident

pion momentum in the mug frame. The energies (.0; and
b): are given by hf}... _ (M1g*+‘w'c7’ P413».
and “012- (“43¢ W+MIP°2)7~
Defining the partial wave components of 17‘”(P“ PW}

as

I
L A, A A
(9.3) T1‘.” (9‘: PW} : ZWSTRV(?L: P3\ R} Pu. P1X Ac 9%?»

~\

so that

(9.4) TV 9“)“- PE=\ @(ZL;‘»T 17120)“? \? (Pw'afl

 

we easily obtain for the total elastic cross section,

(9_5) *‘S‘Q‘LB‘? \\1A-¢

PH b); V5___v\ ZZL-H
7.1V w°+ to?)

 

USing the Optical Theorem, we have for the total cross section,

am]

(9.6) C)" : 4'“.
TOT

 

       

102

The reaction cross section is then given by

R
Q 6’ = ‘ E
(9 7) TOT 0"?" <3; 01' .

2. Calculation of the Pion-Nucleus t—Matrix

I

The auxillary pion-nucleus t-matrix ‘YRTV was defined

in Chapter II using a relativistic Lippmann-Schwinger equation

of the form

(9.8)

RM?“ ‘30 3 Unit?" P39
u I N
w.- tomb +Le (210”

with the actual pion-nucleus t—matrix being given by

(9.9)

I
RW‘V‘Qz (A/A-\)TW(P1;F\.

» N fl

103

I
In order to actually solve for -r;ry we must first reduce

equation (9.8) to a one-dimensional integral equation. We define

I
the partial wave components of 1;.” and U‘WV using the same
definition as for -‘;EV (see equation (9.3) ) and obtain a one-

L.
I
dimensional Lippmann—Schwinger equation for' -T;EV

(9.10)
IL , L c
T (P o“): Um’kPIPfi
00 , " IL 4 "2'
Ufivwwqrfiwui‘ 9‘) PW do:
(0,-(ok9‘h + Le. er)’

 

L.
l
The above integral equation for -T;r is solved numerically
P

using matrix inversion techniques. The method is discussed in

Appendix G.

3. Partial-Wave Decomposition of the Optical Potential
As discussed in the previous section, a partial wave
decomposition of the 3-dimensional optical potential is needed
for the solution of the scattering equation. In this section we
discuss the procedure for obtaining such a partial wave decomposition.
We always assume a 3-dimensional pion-nucleus optical

potential of the form

104

(9.11) UWL P; gr) : (NWCCMD {90%)

~~

where A is the number of target nucleons and (0(3) is an
empirical ground state form factor.

When we have an explicit 3-dimensional form for (tWN>
(as in the quadratic approximation, for example), it is simple to

obtain the partial wave components of UT”) . We first write

W as

(9-12) (0%): EL, (0L ( h: P“) R} :1 gr) ’

Then we write (t1...) as

(9.13) (imp -= 2;. (2:;‘)<‘bm L R} 5’. SW) .

'W

 

So we have

(9.11»)

 

 

105

Multiplying equation (9.14) by PL‘ P“, AI'P‘.\ and

 

5' A
integrating over F“ . P‘. we obtain
(9-15)
L P:
0“ W.)
(fi-WLZ M<L"Il"oo\\.o;<t ( ‘
m 2L+ t “flvfl' EL." PW 9'\ .

When we obtain (“00) numerically by calculating the

“He optical potential using a gaussian ground state wave function

L.
and the definition in equation (6.2),we calculate Um) in a

slightly different way. The definition in equation (6.2) gives us

(9.16) (tun) : ":5— U (a: PE)/®(%D

where U‘ (9"? P) is the numerically computed gaussian “He
optical potential~ and F ($3 is the associated gaussian form
factor. The numerical calculation of Uo‘ automatically gives us
the partial wave components U; so it is convenient to use

these components directly in the calculation of the partial wave

com nents of U . The general optical potential U is
P° 1W 1W
given by

106

(9.17)

Um ( P1." Pr) = (9“) (in) {0 (517)

= (““3 [‘97 ”PE'EV (19.3] (“9.)

(9.18)

o (PufP\=

“V

 

(NOB‘Z 2(2L¥)U;(P P'P P) PH“; Jfl’gy/[Nggy

We now define a ""reduced form factor ‘0(’%)/P(%) and make

a partial wave decomposition of this reduced form? factor.

(9.19)

fly?- (Np/@933 2.? any.» $61.39

107

Then following the same procedure as in equations (9.19) and (9.15),
we obtain for the partial wave components of the pion-nucleus
optical potential

 

(9.20)
((1:11 P,‘ P,» =
(351)2J :LL:\‘)<L.....O oo‘LO>U°: (91:9 ‘3‘?”on

The above method thus allows us to calculate optical potentials
with realistic form factors directly from the fully integrated
aussian ”He optical potential. We note that even for ”He this
method is necessary since the empirical “He form factor is not

purely gaussian (Fr067) .

1+. The Fog-Ln Factors

In calculating elastic cross sections for uI-Ie, 12C, and 16O

the form factor in the relation U‘fl'y( p‘P EV (A-\)<t‘N » P(%)

is always a ground state form factor compatible with electron scatter-
ing experiments.

The ground state form factor is defined as

(9.21) (OLgX ‘5'- [OQC %)/€P(z\

108

where (O is the nuclear charge form factor obtained from electron
scatterifg and F is the charge form factor for a proton. The
division of f Pby f is intended to remove the effect of the

0. P
finite proton size s0 {2 I F? is the form factor for an

ensemble of point particles.1 We take (0 to be

P

(9.22) to (5p .-. exp (- r: f/ a.)
P -

where r? = .76 fm. (Ehr59).
For “He we take Fa from the work of Frosch et.al.
(Fr067), so using equation (9.21), we have for the “He ground state

form factor

(9.23) (9(1): L | — («fag-)5] ex? (- 511:.)

where Q: .316 fm and b= .606 fm .

For 120 and 16O we use the parameterization of Ehrenberg

et. al. (Ehr59) for (O . The resulting ground state form factor is
Q

 

1The finite nucleon size is already contained in the
pion-nucleon t—matrix so it should not be included in the nuclear
form factor.

109
(9.21»)
(“51.) =

Cl - o(( ‘d‘"‘(z/z, (at-343} ex? L— ($QCRf/‘fl

where QC”: 1.66 fm, Q

Ch: 1.59 fm, a( = (A—u)/6.

5 . Treatment of the Non-Resonant Channels

 

In calculating nucleon motion effects and binding effects
we a5811111861. the pion-nucleon interaction is in the (3,3) channel only.

In the actual calculation of elastic cross sections we include the

non-resonant parts of the optical potential via the simple static
a-I’P1‘OX2‘-Ina.ti0n. Consequently, whenever nucleon motion or binding

effects are taken into account the optical potential is the sum of
two parts. A "resonant" part which includes nucleon motion and/ or

binding effects and a "nonresonant" part obtained from the simple

static approximation. This procedure is adequate since the non-

resonant part of the optical potential is relatively small at the

energies we consider.

6° Conlomb Effects

Finally, the coulomb potential is neglected in this calcu-

lation. The work of Lee and McManus (Lee71) indicates that coulomb

effects are unimportant in the (3,3) resonance region.

CHAPTER X

COMPARISON OF THE CALCULATED
CROSS SECTIONS WITH EXPERIMENT

1- IEPulse Approximation Results for “He

The results for'nHe given by the fully-integrated impulse
aPPI‘OXimation (FIA), the simple static approximation (SSA), and
‘the mOdfiafied static approximation (MSA) are compared to the
experimental data in Figure 10.1. The results for the quadratic
approximation are shown in Figure 10.2 and for the Kujawski-Miller—
Landau (KML) approximation in Figure 10.3.

We shall be mainly concerned with the predictions of the
ETA (01?:1ts equivalent) since this model is preferred on theore-
tical encounds. However, in order to show the consequences of using
§§;§22_Inodels which are poor approximations to the FIA optical

potentigil, we also discuss the predictions of the SSA and MSA.

a. The First Minimum in ”He:

The first minimum in “He is apparently due
to the p -wave pion-nucleon resonance and is
approximately stationary at ~ 770 in the WUCM
system. The correct prediction of this minimum
is an important test of any 11' —uHe optical

potential.

110

111

FIGURE lO.l--The elastic 17 -uHe differential cross sections
resulting from three different effective pion-nucleon impulse
interactions: gag Esolid line) the fully-integrated impulse

result EFIA); b short dash line) the simple static approximation
(SSA); c) (dot-dash line) the modified static approximation (MSA).
The energies shown are the laboratory kinetic enerat of the pion.

 

 
 
 

IIO MeV .

’o-O—.-."-

     

 

 

 

 

 

(Ia/do. (Mb/9')

 

 

 

l- -— FIA -‘ I:
0000 SSA . .. . ' o
.. MSA \ . “ $§
.on - . t -
' 260ro

 

L l l l l

0 so so 90 IZO I50
PION-NUCLEUS C.M. ANGLE (DEG)

 

F
IGURE 10.1.

113

As can be seen in Figure 10.1, the FIA
correctly predicts the location and relative
depth of the first minimum at all energies
except 260 MeV where the experimental minimum
is somewhat weak. At 260 Mev the predicted
minimum is too far forward by 5° and too
deep.

The SSA and MSA do not fit the first
minimum in “He nearly as well as the FIA.

The MSA gives the first minimum too far

forward at all energies and the SSA gives the
first minimum at too large an angle at all
energies except 150 MeV. For most energies,

the relative depth is too large with the MSA.

The SSA generally gives a more accurate depth

for the first minimum, but fails badly at 180 MeV.

The first minimum becomes considerably
weaker at 260 MeV. All three models indicate
a decreasing depth for energies above 220 MeV,
but none of the models have the proper degree
of weakening at 260 MeV. The FIA and MSA minima
are too deep, and the minimum vanishes completely
with the SSA.

The predictions of the various models for
the first minima in “He are directly related to

the angle transformations involved (i.e., the

11h

relationshi between 0 0 and 059 .
p C. 5 1m C w )
These angle transformations will be discussed

in detail in a later section.

The Second Minimum in “He:

At 180 MeV, a second minimum appears and
is quite prominent at 220 and 260 MeV, moving
toward smaller angles with increasing incident
pion energy. This second minimum is consistent

with the experimentally observed zero in the

”He form factor (Fr067). Although we use form

factors compatible with electron scattering in
all our models, we still fail to predict the
second minimum. Our form factors are obtained
by dividing the electron scattering nuclear
charge form factor by the proton charge form
factor. This procedure is intended to remove
the effect of the finite proton size so that
we are left with the nuclear form factor for
an ensemble of point nucleons. For an inde—
pendent particle model and harmonic oscillator
wave functions, our procedure is valid (Ehr59).
However, for ”He it is an approximation of
questionable validity since the nucleons are
highly correlated and hence only approximately

described by single particle wave functions.

115

It may be that a more accurate treatment
of nucleon size effects is necessary for

a correct prediction of the second minimum.

The Discrepancy at Forward Angles:

At all energies, except 260 MeV, the
predicted cross section is too large for all
the impulse models. The binding correction,
which we discuss later on, weakens the pion-
nucleus optical potential somewhat and thus
gives improvement in the forward direction.

At 180 MeV, the FIA is too large at
forward angles and also at backward angles,
although the overall shape is good. The
weakening effect of the binding correction
improves the agreement with experiment at
both forward and backward angles.

Relation of the ”He Cross Sections
to the Angle Transformation:

Single scattering dominates in ”He so
that the cross sections vary with angle

2.
approximately as \ \J \ . Hence, the

‘TV
features of the cross sections are closely
connected to the features of the effective

interaction <t1ffl> used in each model.

Since most of the larger effects in

(t KN» can be related to the angle

116

transformation, it is simpler to discuss the

features of dr/AQ, in relation to the

features of the various angle transformations.

The angle transformations shown in Figure 5.1

can be used to understand most of the effects.

The KML curve in Figure 5.1 can be taken to

represent the angle transformations of the

FIA and quadratic approximation since they all

are the same (on-shell) to order (W; Iw'jz": V25 .1
For a given model, the predicted position

of the first minimum in “He is directly related to

the zero in the corresponding C0$9“» V‘ (.05 91W

curve in Figure 5.1. For example, the MSA

minimum in Acr/AJZ. is at the smallest angle

and SSA is at the largest angle with the FIA

in between the MSA and SSA. Looking at

Figure 5.1, we see the same relation exists

for the zeros of the various angle transformations.

The features of the three angle transformations

(SSA, MSA, KML) are also clearly seen in the

backward scattering. The FIA and SSA are not

too different at backward angles,with the FIA

generally being slightly higher. At backward

 

1There is no explicit analytic form for the FIA

angle transformation since the FIA result is
obtained numerically.

117

angles, the MSA cross sections are larger
than the FIA and SSA cross sections (and also
the data) by a factor of 4 to 6. This large
discrepancy in the MSA arises from the MSA
angle transformation which allows \ C05 EMA
to exceed one at backward angles.

The Quadratic Approximation and the
Kujawski-MillereLandau Approximation:

The “He results for the quadratic approxi-
mation are shown in Figure 10.2. The quadratic
approximation results from a systematic approxi-
mation to the fully-integrated impulse model
(see Chapter VII). It is evident that the
quadratic approximation.gives a very good
representation of the fully-integrated impulse
approximation (FIA) .

In Figure 10.3 we show the results for’uHe
using the Kujawski-Millethandau (KML) model.
The agreement with the FIA results is essen-
tially the same as that obtained with the
quadratic approximation. Hence, we see that
a proper angle transformation is essential for

a good representation of the FIAl. However,

 

lAgain, recall that the angle transformation
in the KML model is essentially the same as the
one in the FIA.

118

FIGURE 10.2--The elastic "TY-“He differential cross sections cal-
culated by two methods: (a) (solid line) the fully-integrated
impulse result (FIA); (b) (dashed line) the quadratic approximation

(QA) to case (a). The energies shown are the laboratory kinetic
energy of the pion.

 

 
    

IIO MeV
1.1.

L

 

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B

:ns cal- E

mfiif suv
Pro '1 LC:

kinetic %

\

b

'0

l l l l l l l l l 1 l L l l

0 30 60 90 I20 I50
PION—NUCLEUS C.M. ANGLE (DEG)

 

 

 

FIGURE 10.2.

120

FIGURE 10.3--The elastic"fl' -uHe differential cross sections cal-
culated by two methods: (a) (solid line) the fully-integrated
impulse result (FIA); (b) (dashed line) the Kujawski—Miller—Landau

(KML) approximation. The energies shown are the laboratory kinetic
energy of the pion.

121

 

IOO

IO

 

IOO

IO

 

IOO

IO

dc/dfl (mb/sr)

 

.Ol

 

l l J l j l l l l l l J l l

o 30 60 90 I20 I50
PION-NUCLEUS C.M. ANGLE (DEG)

 

 

FIGURE 10.3.

122

we should point out that at lower energies
the angle transformation effects and collision
energy effects (i.e., won effects) are com-
parable although neither is a dominant effect.
For example, at 110 and 150 MeV, the simple
static approximation (which contains no angle
transformation or collision energy effects)
gives cross sections which are not too
different from the FIA result. Angle
transformation effects, which vary as

w; I we , become increasingly important

at higher energies while collision energy
effects remain about the same as at lower
energies. Hence, the angle transformation

gives the dominant effect for the energy

region above the (3,3) resonance.

2. Impulse-plus—Binding Correction Results for “He

We now consider the results obtained for ”He with the
following models: 1) the fully-integrated impulse approximation
(FIA); 2) the single state approximation for binding effects (SSAB);
3) the closure approximation for binding effects (CAB). The results
for all three cases are shown in Figure 10.4.

a. The Single State Approximation for

Binding Effects:
Due to the weakening of the optical

potential produced with SSAB, the SSAB cross

123

FIGURE lO.U--The elastic 1T -uHe differential cross sections
resulting from three different effective pion-nucleon interactions:
(a; (solid line) the fully-integrated impulse result (FIA);

b short dash line) the single state approximation for binding
effects (SSAB); (c) (dot-dash line) the closure limit for binding

effects CAB). The energies shown are the laboratory kinetic
energy of the pion.

.0

‘ Al
O‘ f‘
aaaaaa

a I
u lv-’4 '...-

.....

da-lda (mb/sr)

IOO

IOO

IOO

IO

IOO

IO

IOO

IO

.OI

 

 

 

 
 

 

IIO MN .I

 

 

 

 

 

 

 

" ~-— El: \ ¢ 260 MeV I
--- AB 11’
— —- CAB I: - 1......»
- 3‘ I *irro-.-:
iI’
0 30 60 90 I20 I50

FIGURE 10.14.

PION-NUCLEUS C.M. ANGLE (DEG)

b.

125

sections are lower than those given by the
FIA. At 110 and 150 MeV the reduction is
fairly small, giving a slight improvement
at forward angles and a slight worsening
at backward angles. At 180 and 220 MeV
the forward angle cross section is improved
some and the backward angle cross section
is improved significantly. At 260 MeV

the forward angle cross section is not
improved (or perhaps is slightly worsened),
but the backward angle cross section is
improved noticeably. At 110, 150, and

180 MeV the position of the first minimum
is not changed by the binding correction,
but at 220 and 260 MeV the minimum is
shifted f0rward by a few degrees. Thus,
the overall result of the SSAB is to
noticeably improve the agreement with the

uHe data.

The Closure Approximation for Binding Effects:
The CAB affects the “He cross section

a good deal more than does the SSAB, with

the effect being greatest at 150 and 180 MeV.

Generally, the CAB overestimates the binding

correction. At 110 MeV the forward angle

126

result of the CAB is as much too low as the
SSAB result is too high. The fit at the
minimum is considerably worsened, but the
backward angle result is left unchanged.

At 150 MeV the CAB gives very poor results,
being too low at forward angles and too
high at backward.angles. The first minimum
at 150 MeV is too far forward and too shallow.
The forward angle result at 180 MeV is as
good as the SSAB result, but the first
minimum is too far forward and the cross
section at backward angles is too high.

At 220 MeV the CAB result goes through the
one data point at forward angles and is
comparable to the SSAB at backward angles.
The 260 MeV result is somewhat too low at
forward angles but comparable to the SSAB
result at backward angles. Also, the CAB
minimum at 260 MeV is far too deep.

The overall effect of the CAB is to
worsen the agreement with experiment. This
result is not surprising since the CAB is
only intended to give an estimate of an
upper limit on the binding correction and

is not intended to be a realistic model.

127

12

3. Impulse Approximation Results for C

We now consider the impulse approximation cross sections
for 120 using the same models as used for'uHe. The results are
shown in Figures 10.5 - 10.8. Again, we are mainly interested
in the FIA result (and good approximations to the FIA), but we
discuss the other models in order to show the sensitivity of the

elastic cross sections to the features of the optical potential.

a. Nature of the Minima in 12C:

In “He, single scattering dominated so we
could identify the first minima as coming from
the pion-nucleon p-wave resonance and the
second minima as being diffractive in character.
In 12C, such an identification is not possible
because multiple scattering strongly affects
the shape of the elastic cross section and
masks the features of the single scattering
term. For example, one of the models studied
by Lee and McManus (Lee7l) is a simple local
optical potential which completely neglects
the p-wave nature of the pion—nucleon interaction.
Although this crude model does not give a satisfac-
tory fit to the data, it nevertheless is capable
of reproducing the gross features of pion-120
data (i.e., two minima). Hence,it is pointless
to try to relate the features of d 6" lAIL

directly to the details of the optical potential.

b.

128

However, the main differences between the
various optical potentials are usually
reflected in some way in the elastic cross

sections, even though for 12

C the general
shape of 36" (All. is independent of the
details of Ilwy .

The Fully—Inte ted Impulse 12
Approximation FIA) Results for C:

The FIA results for 12

C are shown in
Figures 10.5 - 10.8. At forward angles the
agreement of the FIA with the data is good.
In particular, the first minimum and first
maximum are described fairly accurately up
to 260 MeV. The agreement at the second
minimum and second maximum is only fair,

with the cross section at backward angles
being underestimated at all energies. The
tendency to underestimate the large angle
cross sections in 120 seems common to many
models (Lan73, Lee7l, Kuj74) including models
with the optical potential parameters adjusted
for a best fit (Ste7o). Lee and McManus
(Lee7l) have suggested that ground state
deformation effects are possibly important

in 120 so it may be that a more sophisticated

treatment is required for 120. Nevertheless,
the fit obtained with the FIA is as good or

better than any so far published.

129

FIGURE 10.5--The elastic TT' ~12C differential cross sections

resulting from three different effective pion-nucleon impulse
interactions: Ea) (solid line) the fully-integrated impulse

result (FIA); b short dash line) the simple static approximation
(SSA); c) (dot-dash line) the modified static approximation (MSA).
The energies shown are the laboratory kinetic energy of the pion.

130

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133

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rather

sparse
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model

the ME
minima
Whole
Parab
diffs
some

too h

Co

13LI

Comparison of the Modified Static Approximation
(MSA) and Simple Static Approximation (SSA) to
the FullyaIntegrated Impulse Approximation (FIA):

12C the differences in the models are

For
not as clearecut as in.uHe. As already mentioned,
the gross shape of the 12C cross section does not
depend much on the details of the model. A
rather model-independent shape, together with
sparse backward angle data, makes it difficult
to reach any definite conclusions as to which
model is to be preferred.

Except for the 260 and 280 MeV results,
the MSA does not fit the 12C data near the first
minima and maxima as well as the FIA. On the
whole, the MSA fit at backward angles is com-
parable to the FIA even though the MSA and FIA
differ by as much as an order of magnitude at
some energies. The MSA is generally as much
too high at backward'angles as the FIA is too low.

At 120 MeV the SSA is actually better than
the FIA and at 150 MeV the fit is comparable.

At 180 and 200 MeV the SSA is too large at the
first minima and maxima, but is as good as the
FIA at backward angles. Beyond 200 MeV the SSA
continues to be too large at the first minimum
having only a "shoulder" where the first minimum
should be. However, at 280 MeV the FIA fit

has degenerated to the point that it is no

better than the SSA.

d.

135

Although we cannot clearly distinguish
between the FIA, SSA, and MSA 0n the basis of

120, it is evident that the calculated 120

the

cross sections are still somewhat dependent on

the model, even though the general shape of
Ar/An. is the same for all the models.

For example, the depth and position of the

second minima is quite sensitive to the model

and at backward angles the differences in the

different models are quite large. In 12

CI

LI
as in He, the unphysical values of c.0565",
in the MSA show up rather'dramatically in the
large angle cross section. Moreover, the
difference between the FIA and SSA at higher
energies and backward angles is even greater

in 12C than in.uHe.

The Quadratic Approximation and the Kpgawski-
MillerbLandau (KML) Approximation in C:

The quadratic approximation (discussed in
Chapter VII) is compared to the fully—integrated
impulse approximation (FIA) in Figmres 10.9 - 10.12.
The quadratic approximation obviously provides
an excellent approximation to the FIA, although
there are a few small differences at backward

angles at some energies.

136

I50 MeV

FIGURE lO.9—-The elastic‘TI’ 320 differential cross sections cal-
culated by two methods: (a) (solid line) the fullyaintegrated
impulse result (FIA); (b) (dashed line) the quadratic approximation
(QA) to case (a). The energies shown are the laboratory kinetic
energy of the pion.

7r—‘2c

137

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138

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10-3 -—FIA

--- QA fig:
lM0 ‘ 810 l' 1210 I 150

PlON-NUCLEUS c.m. ANGLE (deg)

FIGURE 10.12--As in Figure 10.9.

lhl

FIGURE 10.13-¥The elastic WT -12C differential cross sections cal-
culated by two methods: (a) (solid line) the fully-integrated
impulse result (FIA); (b) (dashed line) the Kujawski-MillerbLandau

(KML) approximation. The energies shown are the laboratory kinetic
energy of the pion.

142

MA .02“ SUE

303 unoz< .Ed mam-632-205

 

 

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--- KML [\—
1o‘*_ - l .L .4 i .J
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PION-NUCLEUS c.m. ANGLE (deg)

FIGURE lO.l6—-As in Figure 10.13.

146

The KML approximation for 120 is shown
in Figures 10.13 - 10.16. At lower energies,
the KML approximation is not as close to the
FIA as is the quadratic approximation because
of the neglect of collision energy effects in
the KML model. At higher energies, where angle
transformation effects dominate, the KML model

is just as good as the quadratic model.

12C

4. Impulseeplus-Bindinggcorrection Results for

In Figures 10.17 - 10.20, we show the 120 results obtained
using the single state approximation for binding (SSAB) along with
the closure approximation for binding (CAB).

In.uHe, the SSAB gave noticeably improved agreement with
experiment, but in 12C the SSAB leaves the (fairly good) FIA result
essentially unchanged.

In 12C, the CAB causes some rather drastic changes in the
cross section. The largest effect occurs at 150 MeV where the CAB
result at 9: 0° is too low by a factor of 3. Even at 280 MeV,
where the CAB binding correction is smallest, the CAB result under-
estimates the first maximum by a factor of 3 or'4. In 12C, as in

”He, the CAB model evidently gives much too large an effect.

16

5. Impulse Approximation Results for 0

The results for 160 using the fully-integrated impulse
approximation (FIA), simple static approximation (SSA), and modified

static approximation (MSA) are shown in Figures 10.21 - 10.23.

147

FIGURE lO.l7--The elastic 11' -12C differential cross sections
resulting from three different effective pion-nucleon interactions:
Ea) (solid line) the fully-integrated impulse result (FIA);

b short dash line) the single state approximation for binding
effects (SSAB); (c) (dot-dash line) the closure limit for binding
effects CAB). The energies shown are the laboratory kinetic
energy of the pion.

 

148

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151

 

7T —'20
103
100 1 280 MeV
.73 10
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b I
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'-
10"3 I-
"T 1 4| 1 .. J T-~4‘
1° 0 so 120 180

 

PlON-NUCLEUS c.m. ANGLE (deg)

FIGURE 10.20--As in Figure 10.17.

152

The quadratic approximation is shown in Figures 10.2u - 10.26,

and the Kujawski-MillereLandau (KML) approximation is shown in

Figures 10.27 — 10.29.

a.

b.

The Fully-Inte ted Impulse 16
Approximation FIA) Results for

In 160, as in 12C, the FIA gives a good

0:

description of the data. Although the FIA cross

sections are quite similar in 16O and 120,

there is one interesting difference. In 120
the FIA generally was too low starting at about

600 or 700 and got worse toward larger angles.
The 160 results do not show the same tendency to
underestimate the data. Since 160 is a spherical
nucleus, these results lend some support to the
suggestion of Lee and McManus (Lee7l) that the
backward angle discrepancy in 120 is a result

of ground state deformation effects. However,
since the 160 data only extends to about 75°,
definite conclusions on this point must await

further large angle data.

Comparison of the MSA and SSA to the FIA forléo:

The FIA gives a somewhat better account of

the region around the first minimum in 160 than

the SSA or'MSA. This feature is the same as in

12 16

C, although in 0 there seems to be less

dependence on the model. At 230 MeV, for example,

153

FIGURE 10.21é--The elastic.“ -160 differential cross 39 0,5102:
resulting from three different effective pion-nucleon 1111 so
interactions: (a) (solid line) the fully-integrated imp“; ximation
result EFIA); (b) (short dash line) the simple static 3,29% (MSA)-
(SSA); c) (dot-dash line) the modified static approxima he pion-
The energies shown are the laboratory kinetic energy 0 f ’5

 

154

 

 

 

 

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155

 

 

 

 

 

<92 ....I

73 (mm 000 was
....... 4H“. 0.
x), «-2 3. ~32 D
I. lllllll \. I. d ..... \ x /
/ -l;| ".3 .. n?

. 3 .. 3
+ .. . U
. a + a w
,.... q
..i . . t /
S S S
U\

>22 0mm 2: >22 omm _ as

no" mos

GEAR

156

 

 

 

 

7r-'60

103

mo . 240 MeV
E 10 o
.o *9 a
E 1 FIA
C,’ ' 0 0 SSA
E 0.1\.\ X... .:::.T.SA
8 10’2 ' )3.

10"3 -

A} J ; J l 1 J
10 ‘0 so 120 180

PlON-NUCLEUS c.m. ANGLE (deg)

FIGURE lO.23--As in Figure 10.21.

157

ions 031‘
FIGURE lO.2’+--The elastic 1T -l60 differential cross 58 0-1:;

ted
culated by two methods: (a) (solid line) the fully-in'be filtration
impulse result (FIA); (b) (dashed line) the quadratic 359:9 memo
(QA) to case (a). The energies shown are the laboratory
energy of the pion.

 

158

:33 3924 .Ed 8382-20:

0 a a a O o
W . oqw . OtIOH OW J OW q um . flog

so --.. m
-3 .3
m (Hm I m
MID." NIOu
\// I! l/ .
/\ 9. #5 /t.. “.0
w H H
, K
O ‘
3 n 3
>22 Gt 2: >22 om. c3
«.2 . m2

om. lkh.

3N .OH EDUHh

(IS/aw) vp/np

159

God
W

 

.smdd assess 5. 2:3.2 ESE

BoE mrczq .80 8382-20...

 

4 cm” . mm . 9.13 cm” . ow“ o_w . 0.13
<0 ..--

.-s S... ..l .-e

+ .3 # do

a i r r
e

3 .,3

>22 0mm as >22 0mm 2:
was so“

 

 

09E. _

(19/ am) tip/op

160

 

 

7T- '50
m3
1% - 240 MeV
35 m o
.o
E 1 .
€504 f
\b — FIA
‘0 ”'2 ---‘ GA
10'3 i l
10-..- ‘ l fiL 1 J
a so 120 180

PlON-NUCLEUS c.m. ANGLE (deg)

FIGURE 10.26--As in Figure 10.21».

161

FIGURE 10.27--The elastic 1‘ -160 differential crc) 5 5 5901510118 cal-

culated by two methods: (a) (solid Line) the fill-1r“ : Jimdau

fitory kinetic

impulse result (FIA); (b) (dashed line) the Kujawsicjl—
(KML) approximation. The energies shown are the lbbo
energy of the pion.

 

162

Bog 3oz< .Ed 8382-205

 

cm”
..I

 

3

OS

as

am. mm s. 2: our 8 c,
1. q q .4 m. ’IOH fit a fit a q .
a 42x ......
‘ .. 4H”. I
i :3
_ \)// 'I’
a o .
/IL\ 0 any / \
/\
. _
..\ . 1x
.u
>22 ON. ..3 >22 00.
c2

AVmw_.erho

mm .3” "55lo

(JS/qw) UP/DP

163

.smda 83E as urimmda gas

303 m..cz< .Ed mamdnz - zo_d

emu ON"
1.

ow
4H

9 cm“

 

 

9N"
..1

nmwx
.

o

 

. J! u . -flcn 1 4 d u -..IQA
42x ......
.-s ..1 «H... II ..2
..... - IOH luallul ”ha
"6 # do
+ .a
.H cu
>32 0mm 2: >22 0mm as
no” nos

(‘S/Q‘“) UP/DP

16h

 

 

 

7r-‘50
103 '
100 . 240 MeV
E 1| 0
9
.n
E 1 ...
3 0.1 f
B —— FIA
'O 10-2 J "'""“'" KML
10'3 ~
"" n . 1 L J
1° 0 60 Ti?) 180

PION-NUCLEUS c.m. ANGLE (deg)

FIGURE lO.29--As in Figure 10.27.

165

a comparison of the results near the first

minimum shows the models in noticeably closer

agreement in 160 than in 12C. The backward

angle results, however, are just as sensitive

to the details of the models.

c. The Quadratic Approximation and the Kpgawski-
Miller—Landau (KML) Approximation in o:

The quadratic approximation gives an accurate

representation of the FIA in 160 just as in “He

and 120. There are again some small differences

at backward angles.

e 16

Th 0 results for the KML approximation

are also similar to the KML results in 12c. At

lower energies the neglect of collision energy
effects causes some significant differences
between the KML approximation and the FIA.
However, at higher energies the differences are

quite small .
6. Impulse—plus—Binding Correction Results for 160

The 160 results for the single state approximation for

binding (SSAB) and closure approximation for binding (CAB) are shown

in Figures 10.30 - 10.32.

160 the small change in dt/Aa. given by the SSAB

is entirely compatible with the data. (In 120 the SSAB worsened

In

the fit by'a small amount.) The SSAB is hence compatible with the

experimental results for all the nuclei we have studied.

166

ti
FIGURE lo.3o——The elastic 1T 360 differential cro 55 Sigtegzfionsg
resulting from three different effective pion—nucle o I (FIA);

(a) (solid line) the fully—integrated impulse result f or binding
(b) (short dash line) the single state approximati. 01:14; for binding
effects (SSAB); (c) (dot-dash line) the closure 1:1. kinetic
effects CAB). The energies shown are the laboratory

energy of the pion.

 

167

 

om.oa mmDUHa

BoB ”.3024 .55 8582-205

 

 

 

 

o o H ems 3 o
..a . a. . s . ..-s a . . . . i ...-s
9.8 l.l
mnoH mqwm o o o 79
<H...._ . p
.....1 -2 .. -2
It. 5., N .. “a . N D
. c/ .. .r n ..... . .fl 0 W
a ..\ ~ o lo . .
... . fl o . \ a o
/o|\ A s /o/..\. .l. \ U
all. /s )
. a . .... a w
. .. m
OH r. ca mm
. (
>22 on. as >22 or. .7 as
no” .e moH

168

.omda osstoHa 5.” neiaméa ESE

Ems uaoz< .Ed 8382-205

 

 

 

 

om” oNH cm 0.
0qu a CW“ 4 . mm a ornoa . . a . 4 1:0“
._
ooooo / e e m. ”IDA MIC."
.......... u/.\ I _
/ ... ~33 Nan
i “.0 .. ad
,  b N . .:.
urns a _ .1 a
. A . ...
. . 3 V
S o."
>22 0mm 2: >22 CNN 2:
no“ mo."

om_lnF

(IS/Gm) UP/DP

169

 

 

 

 

‘17-'60
103
mo 5 240 MeV
33 10 '
\ J...
.0 .
E 1 \./".\
g 0'1 ‘ FIA
B . - .. SSAB
.0 10'2 l. _._ CAB
10-3 ! '\ .-':.-\....;,:::;
1°-th i so . 150 l 150

PlON-NUCLEUS c.m. ANGLE (deg)

FIGURE lO.32——As in Figure 10.30.

170

The CAB results for 160 are the same as in 120; the

CAB overestimates the binding correction by a considerable amount.

 

CHAPTER XI

SUMMARY AND CONCLUSIONS

We have studied elastic scattering of pions from He,

12 160 in the 100-300 MeV region. We began our study by

C and
calculating an optical potential for “He which was based on the
impulse approximation and a harmonic oscillator shell model.

The integration over the nucleon momenta was carried out exactly.

We defined an effective pion—nucleon impulse interaction as the

fully-integrated 1T -uHe optical potential divided by the associated
harmonic oscillator form factor. Because we calculated the

scattering in momentum space, it was not necessary to put the

effective interaction or the associated optical potential into a

coordinate space form.
We next studied three 951 hoc models for the effective

impulse interaction, two of which gave poor agreement with the
The third if} .1392 model, which took into

fully-integrated version.
account nucleon recoil, gave much better agreement with the fully-

integrated results, although there were still serious discrepancies.
By systematically approximating the integrals over the

nucleon momenta, we developed our own simple but accurate approxi-
Our systematic

mation to the fully-integrated impulse interaction.
approximation provided some insight into the effects of nucleon

motion in the effective impulse interaction.

171

 

172

We found that the optical potentials based on the fully-
integrated impulse approximation (FIA), or its equivalent, gave a
satisfactory account of the data for “He, 12C and 16O. The 29.222
models which were not good approximations to the FIA (such as the
modified static approximation) gave poor agreement with the “He

12C and 160

data. No definite conclusions could be reached for
due to the lack of sufficient backward angle data.

On the basis of the “He results, and on theoretical grounds,
we conclude that it is important to accurately approximate the
integrals over the nucleon momenta. We further conclude that an
accurate approximation can be made by evaluating the free pion-
nucleon t-matrix at certain "effective" values of the pion-nucleon
C.M. momenta and energy. The prescription we give in the "quadratic"
approximation (Chapter VII) provides an accurate representation
of the fully-integrated impulse interaction and is as easy to use
in momentum-space calculations as any of the conventional ngQQQ
models.

Corrections to the impulse approximation (which we call
"binding" corrections) were studied using a 3-body model where the
pion scatters from a single target nucleon which is bound in a
potential well. Using a separable s-wave binding potential, we
exactly calculated the first correction to the impulse approximation
and found it to be relatively small. A simple approximation (called
the single state approximation) was developed which gave fairly good

agreement with the exact result. In order to estimate an upper

limit on binding corrections, the single state approximation was

 

173

extended.to include an infinite number of bound states. This extended
(approximation, (called the closure approximation) was found to give
zi'binding correction which is approximately twice as large as in

the single state approximation. We found for ”He that the single
state approximation gave noticeable improvement but that the closure
.approximation (which was not intended to be physically realistic)

rmmie the agreement with experiment somewhat worse. For 120 and 16O,
'the single state approximation was essentially the same as the impulse
.approximation, but the closure approximation overestimated the binding
correction by a large amount. We thus found the ”He, 12C and 160 data
'to be compatible with the relatively small binding correction of the
single state approximation but not compatible with the larger correction
of the closure approximation.

We conclude, therefore, that binding effects are relatively
small in the (3,3) resonance region. We expect other effects
(neglected here) such as physical absorption of the pion and excitation
of virtual nuclear states (dispersive effects) to be as important as
the binding effect considered here.

Representing the pion—nucleus optical potential in momentum
space and solving a momentum—space Lippmann—Schwinger equation for
the elastic scattering proved to be a useful and convenient approach.
This method enabled us to easily study the sensitivity of the cross
sections to various kinematical and dynamical factors. For example,
we easily verified by direct calculation that in the (3,3) resonance
region the elastic cross sections are not very sensitive to the

details of the off-shell parameterization. As a result, we were

able to use a simple zero-range t-matrix with confidence.

174

We feel that this work provides a useful approach for
obtaining an accurate impulse—type pion-nucleus optical potential,

thus making it possible to obtain meaningful estimates of higher

order corrections from a comparison of impulse approximation results

with experiment. We also feel that our single state estimate of

binding effects is reasonable and shows that such effects do not

greatly change the predictions of the impulse approximation in the

(3 ,3) resonance region.

 

BIBLI OGRAPHY

175

 

 

 

 

Ade72

Ba169

Ber72

Bin70

Bin7l

Cer7llr

Cha71

Ded71

Doe73

Ehr59

Fa172

Fra74

Fro67

Ger73

Gla67

BIBLIOGRAPHY

M. L. Adelberg and A. M. Saperstein, Phys. Rev. C5
(1972) 1180. “‘

R. Balian and E. Brézin, Nuovo Cimento pl,(l9o9) 403.
R. W. Bercaw, J. S. Vincent, E. T. Boschitz, M. Blecher,

K. Gotow, D. K. Anderson, R. Kerns, R. Minehart, K. Ziock,
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F. Binon, et. a1., Nucl. Phys. 317 (1970) 168-188.

F. Binon, et. a1., preprint CERN, submitted to the 4th
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CERN 74—8, 10 April 1974: Proceedings from "Topical
Meeting on Intermediate Energy Physics," Lyceum Alpinum
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L. A. Charlton and J. M. Eisenberg, Ann. Phys. (N.Y.)
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J. P. Dedonder, Nucl. Phys. Alzu (1971) 251-272.

R. R. Doering, A. I. Calonsky and R. A. Hinrichs,
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H. F. Ehrenberg, R. Hofstadter, U. Meyer—Borkhout,
D. G. Ravenhall and S. E. Sobottka, Phys. Rev. 112
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Goran Faldt. Phys. Rev g5_(1972) 400.

Victor Franco, Phys. Rev. Ci (1974) 1690.

R. F. Frosch, J. S. McCarthy, R. E. Rand and M. R. Yearian,
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J. F. Germond and J. P. Amiet, Nucl. Phys. A216 (1973) 157.
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EC. Alexander, ed.), p. 311, John Wiley and Sons, New York
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176

 

 

Haf70

Hagflt

Her70

Het65

Gol64

Ker59

Kis55
K1570

K0170

Kow63
Kuj74
Lan72

Lan73

Lan74a

Lan7hb

Lee7l
Lov6b
Mac73
M1174
M0559

Pha73

Rod67

177

M. I. Haftel and F. Tabakin, Nucl. Phys. A158 (1970) l.

R. Hagedorn, Relativistic Kinematics, (W. A. Benjamin, Inc.,
New York, 1964).

D. H. Herndon, A. Barbaro-Caltieri, and A. H. Rosenfeld,
LRL Report No. UCRL—20030 (1970), (unpublished).

J. H. Hetherington and L. H. Schick, Phys. Rev. 122 (1965)
B935-

M. L. Goldberger and K. M. Watson, Collision Theory,
(Wiley, New York, 1964).

A. K. Kerman, H. McManus and R. M. Thaler, Ann. Phys.
(N. Y.) § (1959) 551.

L. s. Kisslinger, Phys. Rev. 2Q (1955) 761.
L. S. Kisslinger and F. Tabakin, Phys. Rev. 02 (1974) 188.

D. S. Kolton, "Interaction of Pions with Nuclei," Advances
in Nuclear Physics, Vol. 3, (Academic Press, New York, 1970).

 

K. L. Kowalski and D. Feldman, Phys. Rev. 129 (1963) 276.
E. Kujawski and G. A. Miller, Phys. Rev. 09 (1974) 1205.
R. H. Landau and F. Tabakin, Phys. Rev. 25 (1972) 2746.

R. H. Landau, S. C. Phatak and F. Tabakin, Ann. Phys.
(N. Y. ) 7_8 (1973) 299.

R. H. Landau, Private Communication

R. H. Landau and F. Tabakin, Nucl. Phys A221 (1974)445-454.
H. K. Lee and H. McManus, Nucl. Phys. Alé2,(l97l) 257.

C. Lovelace, Phys. Rev. 5 (1964) B1225.

R. Mach, Nucl. Phys. 5295 (1973) 56—72.

C. A. Miller, Phys. Rev. £19 (1974) 1242.

M. Moshinsky, Nucl. Phys. 13 (1959) 104.

S. C. Phatak, F. Tabakin and R. H. Landau, Phys. Rev. CZ
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Leonard S. Rodberg and R. M. Thaler, Introduction to the

 

Quantumgflheory of Scattering, (Academic Press, New York, 1967).

 

Sch72

Ste70

Wat53

Won72

178

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M. M. Sternheim and E. H. Auerbach, Phys. Rev. Let. 25
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K. M. Watson, Phys. Rev. §2_g1953§ 575; N. Francis and

K. M. Watson, Phys. Rev. 92 1953 291- W. B. Riesenfeld
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C. W. Wong and D. M. Clement, Nucl. Phys. A183 (1972) 210.

 

 

APPENDICES

179

 

 

APPENDIX A

REFERENCE FRAMES AND KINEMATICS

180

181

At a laboratory kinetic energy of 100 - 300 MeV, a pion
is quite relativistic, so at first glance it seems necessary to
use fully relativistic kinematics. However, due to the fact that
m“, /m‘?.‘ 1/7, it is possible to use essentially nonrelativistic
kinematics except for the pion energy.

The primary reason for developing "quasi—relativistic"
kinematics is to allow a straightforward application of the vector
bracket method which is discussed in Appendix D. The vector bracket
method was originally constructed for nonrelativistic problems where
the various momenta are related via linear equations. Fully
relativistic kinematics involve nonlinear relations; but as shown
in this appendix, we can linearize the relations at the energies
we consider. A secondary reason for quasi—relativistic kinematics
is to make contact with the nonrelativistic forms so we can use
our physical intuition more effectively.

Experimental pion-nucleon phase shifts give us information.
about pion—nucleon scattering in the pion-nucleon C.M. frame
( TTNCM frame); however, the pion—nucleus scattering calculation
is usually done in the pion—nucleus C.M. frame ( ITVCN\ frame),
so we must make a transformation between the two frames. Even
though the pion is relativistic in both the TTNCM frame and the

1EPCN\ flame, the transformation between the two frames is not

very relativistic because the WINCH frame is moving at a

182

nonrelativistic velocity with respect to the TWCM frame.l
First order relativistic effects can be taken into account without
losing the simplicity of nonrelativistic kinematics. We present
both the relativistic and nonrelativistic equations and then show
how to construct a simple formalism that is nonrelativistic in
form but that agrees to within a few percent with the relativistic
formalism.2

We first define the momentum and energy variables that

we will deal with.

: pion momentum in the “TOM frame.

nucleon momentum in the “TQM frame.

total pion-nucleon momentum in the
“VCM frame; K: “4.?" .

IX li‘zfl‘o

pion energy in the TIDCM frame.

28 as

nucleon energy in the WVCM frame.

0

8

scattering energy in the WNCM frame;
0

pion momentum in the WNCH frame
(the nucleon momentum is —ak, ).

~

is.

(Jon: scattering energy in the WNCM frame.

(Our "momenta" here are actually wavenumbers in fm-l.)

 

I J'Due to the fairly small ratio in“. mM ,the velocity
of the “nuts frame relative to the 10mm me is less than
~ .Sc. at the energies we consider.

2The transformation between laboratory and 1TWCBQ
coordinates presents no problems, so it is not discussed here.
The lab-1rvcan transformations used in this calculation are
fully relativistic.

183

The nonrelativistic kinematics are given by (Rod67):

(A-l) 00“. '-'- mug." + ”k" (it/2m“.

(A.2) U“ = mnc} t +3 (fit/2m“

"‘w

“‘fl'mn

 

(L3) {a =(mqu - m“Pay/(Wilma) = at-

a 1‘.
(AA) (0cm 3 (OW-t o0“ '- ‘kz‘Kz /2. (mqu

and the relativistic kinematics are given by (Hag6u):

a. 4- ‘7- " V2.
(1.5) 00,," (“‘19 “‘ T“ Er)
(A.6) h)”: (mug * C. E

-\
(1.7) “E, =- E‘tfie‘twmtmmvdfl Lefqihg-(m‘lw‘b‘é

‘/
1L
~ 2.

(A.8) wen: L (0° ..‘kzd-‘KZJ ,

184
Since the target nucleons move at nonrelativistic velocities we
can write equation (A.6) as

(A.9) 0oN ’-‘-.1 mug} +t‘9: /zmM

Also, in the 100 — 300 MeV region we have \P ‘3' \- 2. tmd ,
15—0-3 4m"), to“?! 300 MeV, and oo’... (.0 s (200 MeV.
Using these values in equation (A.7), we find the leading term is
about 1.5 fm_1, the second term is about .05 fm—l, and the third

term is about .5 fm-l. Hence, the second term, which is complicated,
can be dropped without any significant loss in accuracy. Further,
since 'RC— 1X7'L‘ (5.2. and 0c M— 03 we can write equation

(A.7) and (A.8) as

(A.10) & g R, " (wt/0301C,

2
“Rd"

‘3
5.
7).

(A.11) LDC“ -

The above equations for 4k. and ODCF\ can be obtained from the
nonrelativistic equations (A.3) and (A.4) simply by making the
an t.
replacements mw-t h)“ /c , and Mu" wulc where
Q“, is calculated from equation (A. 5). We will use this pre-

scription unless stated otherwise.

185

Sometimes it will be useful to call 4%; the relative

pion-nucleon momentum and 9C the center-of—mass momentum.

~
Then our transformation can be regarded as a transformation from
“”04 coordinates to relative and center-of-mass coordinates.

The momentum and energy transformations we have given
thus far are all that is needed if the initial and final states
of the nucleon are free particle states. If the nucleon is bound
in a nucleus we need to consider a third frame, the C.M. frame of
the nucleus alone, since we use single particle ground state wave
functions which depend on the nucleon momentum in the nuclear center-
of-mass frame. We shall refer to the nuclear C.M. frame as the
4NQM frame since we mainly consider W -1+He scattering in
our derivations. However, we shall leave the number of target
nucleons, A, arbitrary in our equations in order to obtain a more
general result.

The initial and final momentum of the “He nucleus in the
frame are respectively "‘ P‘, and - 9‘: . We show the situation
schematically in Figure A.1:v In Figure A.~l, the symbol ® denotes
the He center-of-mass. Using P4“ to indicate the nucleon

~

momentum in the 4NCM frame, we have for the nucleon momentum

in the ‘IUCM frame

(A.12)

:P

.F

N _ (M‘/MHD Pu
.. *0/90 P1,. .

‘3

ill
1’ 2

2»

186

(Before collision)

Q a
W",

 

 

(After collision)

E

 

Figure A.l - Schematic representatio of an elastic 'IT-u'He
scattering. The symbol 8 denotes the He center of mass.

187

We can now write the TTVCW\ frame variables in terms of the

variables and 41“.“ variables.

A-l w u.)
“'13) ‘1‘." "w["(T)'z§6 " 7;:- P...

2

fi-i
(A.14) (K. 3 (T)P“ + 19*“

It should be noted that there are actually two 4NCM frames
involved (the initial and final) since the nuclear center-of-mass
itself is deflected by the collision with the pion. We ignore
this point in our discussions and speak of "the" 4NCM frame
as if there were only one.

Finally, in pion-nucleus scattering, it is necessary to
distinguish between on-shell and off-shell kinematical variables.
On-shell variables are fixed whenever the 'ITUCM scattering energy
is fixed so we write them with an "o". For example, we write the

on—shell value of 03-“ as Q;- .

APPENDIX B

PARAMETERIZATION OF THE OFF-SHELL
PION—NUCLEON t~MATRIX

188

189

In pion-nucleus scattering we need the entire (on-shell
and off-shell) pion-nucleon t-matrix in the pion-nucleus C.M. frame
(“DCM frame). The experimental pion-nucleon phase shifts give
us the pion-nucleon t-matrix only on the energy shell and in the
pion-nucleon C.M. frame (TTNCM frame) so that we must first
make some parameterization for the off-shell part of the
t-matrix and then transform to the WPCM frame.

The on-shell pion—nucleon t-matrix in the WHOM frame

2"ng
~8W‘K’C‘ L _g
(B-l) tI‘JLiwcmikMkv): [me

/“x 2th.

where 1J3, L denote respectively the pion—nucleon eigenchannel

is related to the pion-nucleon phase shifts by

 

 

isospin, total angular momentum, and orbital angular momentum.

The quantity 5:31 is the eigenchannel phase shift and
7L I 3L is the eigenchannel absorption parameter. The

on-shell momentum in the “WWW frame is 4R. , and /'(R is

the relativistic reduced mass, which is defined as

0) (9e 3 00 V2,
(B. 2) ”R = 1f 9 fit.) ]
00,, («2.1+ the.»

190

/2.
where D.“ (k a» = (M: C" + t16¢kt )

‘I
and touch.) = (mic + ‘k‘e 9e?) 2

The phase shifts and absorption parameters were taken from the

CERN theoretical fit as tabulated in Her70.

There are several popular off-shell parameterizations
for the pion-nucleon t-matrix (Lan73), but the one best suited to
our purpose is the separable parameterization used by Landau and

Tabakin (Lan72).

ct
<B-3>te..tw...t'.t)=tm utmmfijgm .... )

71*.)

Landau and Tabakin obtain the g's in equation (B. 3) by solving

an inver e att ri roble 0 th t % i C3:
8 so e ng P m S a 13')... I3

defines a separable potential which reproduces the on-shell pion-
nucleon t-matrix. We have found our pion-nucleus scattering results
to be relatively insensitive to the choice of the g's (for smooth
nondivergent forms) so we take the g's to be unity.
In part of our pion-nucleus scattering calculation

we need the spin-isospin averaged form of the pion-nucleon t—matrix.
Using the parameterization of equation (B.3) and averaging over spin
and isospin we obtain for the 3-dimensional t—matrix in the “MGM

frame

191

(3.4) is“ (“emit . 99:) =-

"To"; 2 (1+ mgr. v.3 tnLLoom We) ficfififi) .
m *

In order to insure Lorentz invariance of probability
we must make a transformation of the pion—nucleon scattering operator
itself. The scattering operator in the 'WVCM frame is related

to the scattering operator in the “NC.“ frame by

(13.5) (tmywm‘fi 7" 10%.. mucus

where Y is given by

Y2

 

(13.6) 7 -.-. MW) one) once) one)
0149;) 0.59.13 (ow um from

In equation (B.6) the unprimed momenta and primed momenta refer

respectively to momenta before and after the collision. The 'I
.. 1. 4- 1 z. 3
energies are defined as LOVLM - (MTG +‘k C. X1)

192

V7.

and (3.)“le 1:: (MJCA- + kid; X13 .

The transformation in equation (B.5) is a purely relativistic effect
and should not be confused with the momentum and energy transformations

discussed in Appendix A. In a completely nonrelativistic formalism

Tm)1w¢M :3 (t'WNBTtNCM

even though momentum and energy transformations would still be necessary.

we would have (t

There is a minor point we have so far glossed over. When
equation (B.5) is used for the pion-nucleon t—matrix in the
frame, the 1‘ factor gives rise to a form that is a function

I
of the WV CM nucleon momentum Pu and P.) and the “TC-M
' ~ ~

I
pion momentum P“ and P“- as well as 00‘" , k and IL .
~ ~ ~ ~

However, since the kinetic energy of the nucleon is always much

I
less than its rest mass energy, the dependence of ){ on “ and F:
4-!

~

I
is very weak. Further, P“. and P“. are fixed in the integral

~ ~
over nucleon momentum (see equation (3.2)) so that for convenience

we write the “VCM t-matrix as a function of just won , k

«I
I
and -85 just as we would in a completely nonrelativistic theory.
In the actual calculations the dependence of the TU’CM pion—
(
nucleon t-matrix on P“- , P“ , 1”,.) and I“ is taken
A. ~

~ «1
into account properly.

APPENDIX C

N OTATI ON

193

194

We denote the state vector of a particle with definite

momentum as i a) . The normalization is taken to be

< I“ P) :- LZWIS (P... P\ . The coordinate representation
of ‘37 is A,

'F.r A h- ,. .L,
(0.1) (13$): 3" ”'4 4W2YLMLY)YLMLP3L.)LC 9r)
LM

where we define

(0.2) <33) 2.) E Z YL:($)<£ \ PLM§
LM

so that

* A
(0.3) \E') =2. 2 )(Mcei\e1~(>
LM

The state ‘PLM> is a state of definite angular momentum and

kinetic energy. The normalization is < P‘L' M. i P L M» :

(WW)3 SLLSL) $003M) SLP.‘ P) /P7’

195

A two particle state for particles of definite momentum is

written \P.)\ 9:» 2 \9‘: ‘32) or

* A A
(0.4) \Q 927 '-‘- Z YLM (VAYL ML Pt»‘?.L.M.>‘V1 HMO
LIM\ t I .
L2“;

We often need to couple the angular momentum of particles

1 and 2 . We write such a "coupled" state as

(c.5) \ 9‘ 91-31% L\\-r>=- Z. <L.L1M.M‘\1’WL§

mm -

x \?‘L.M\>\P1L1Mz7 .

If one of the particles has spin, say particle l, we can couple

the spin to LI and write (we assume a spin of 1/2 so
A:

(0-6) \R 3‘ *Sti-L‘» = Z <L\S‘ M\S\%\ 3\ 3\%>

W 5h}

K \V. L. VH>\SI%> .

196

Then this state can be coupled to the angular momentum of particle 22

to obtain

(0.7) \9? “*3 LL>= 2 <3 L13.%Mz\%%e>
diam

x \R mom L2M1> .

If more convenient, we could of course use a different order for
the coupling.

Since the states ‘P‘>\P‘L> are complete we have
~ ~
that

((1.8) \?¢7\91§S:3:;38A_j_:z (‘1ka \ -_-_-

Similarly we can write

(0.9) “fl 911M L.L1>Z 889‘ ‘9‘? Sig-SS! (9 P1 {M L LA: 1
4cm (1“): L1“

L\L1.

197

and

 

(0.10) \e,?,\\3%3\L,L,>% 2:31;; 1:36 P 95%th \=1.
'1‘

AxLiLL

Suppose now 9‘ and P7, are related to two other
~

~
vectors /k and ‘17,, by the relations
~

~

(0.11) Y’, 2 (1,22, +‘ofli
((2.12) \31 2: and): + \oic‘fl.

where we assume the Jacobian “1451‘ “‘sz is equal unity.

For example, ‘9: might be the relative momentum and (K the

~
center-of-mass momentum (see Appendix A). The state \&> ‘33»
'th '~1 tt t” stt P ,wa
is en equiva en 0 he a e \ A)‘ P2» b can now

define as before

(0.13) wag,»— 7:. Y9: (M w: JZ MD

ghmk

198
if A
(0.11») mg» 2 Z YQ‘JECK) \‘K ’Q‘KwW-w
980‘“): 0‘
(0'15) \‘k‘Kfi'WLRk 1.3K) :-

‘Zijb‘<:5L%15{q<"figt)~\Q<.\jgfwyli;>\4k.Q¢:WN§:j7\%C15"“q£:> .
"‘4: “'K

The spin can be included as before
“'16) ‘3‘ 31k 5(a- Q43 2

z <‘QQA/W‘k AW“ 5&&&%§ \‘kflkmk>\ ”4%) .

MkAa

And as before, we can further couple the two single particle states

((3.17) ”(OK- (3'39: 5k 11% Rex? :2

Z <541R'K39a: W‘fl %%%> ”i Sma- £fl>\kflkmk>,

){a “k

199

The states ‘%;§\°&> also form a complete set of

states so we can write

(c.18) mmwjrx 8&1 («KM =— 1

0M3

01‘

NM. Lac
(0.19) \k'K (m H RX>EK 88*— (7:33 Q“), 1<‘&.K {m 1‘4. 29C\

:1

01‘

(0.20) “£50“qu Sklhflpg 88—“— ‘86. X it): (“x t»? szged
1‘

 

on cm

35.16.. ’Q'k

II
P

200

The completeness relations given in this appendix are
useful when we want the matrix elements of a two-body operator in
terms of some "unnatural" coordinates and the operator has a simple
known form in terms of "natural" coordinates. In such a case the
unit operators in terms of the natural states can be inserted in
the proper places so that the operator is "sandwiched" between
natural states. Then we need only the overlaps between the natural
and unnatural states. This procedure is discussed in detail in

Appendix D.

APPENDIX D

VECTOR BRACKETS

201

202

"Vector brackets" can be used in scattering problems in
the same way Moshinsky or oscillator brackets (M0559) are used in
bound state problems. The purpose of both types of brackets is to
provide a more convenient and systematic method for evaluation of
certain matrix elements. Oscillator brackets are generally used
to calculate state "overlaps" for two independent particles moving
in a harmonic oscillator well so that the coordinate-space radial
wave functions involved are proportional to EX P(- QTY?) X
(Laguerre polynomial). Vector brackets, on the other hand, are
related to overlaps between free particle states so that the
associated radial wave functions are proportional to spherical
Bessel functions. Except for the different associated radial wave
functions, oscillator brackets and vector brackets are identical.
However, it is possible to obtain an explicit expression for vector
brackets while oscillator brackets usually must be tabulated in
numerical form or evaluated on a computer. We give a general
expression for the vector bracket in this appendix.

Vector brackets have been used in various forms in 3-body
theory for quite some time. We thank.Dr. Nancy Larson for intro-
ducing us to these overlaps and for providing notes which made it
possible to derive results appropriate for Z-body states. Balian
and Brezin (Bal69) give a derivation which leads to a more useful
form than we derived originally. Recently, Wong and Clement (Won72)

generalized the overlap and introduced the name "vector bracket" into

203

the literature. The general expression obtained by Wong and Clement

is given here. Suppose we want to calculate

(13.1) (PcPlf'm‘ 1:1; ‘ Q \ 9. him Q‘Q‘>

U C ’
where ‘2H|*'IL1L'=’ jfi’ and 9L"*'jlt. ::' ‘iz

~

(See Appendix C for notation).

Also, suppose we know Q in terms of the k , K coordinates
‘3 ~
(for example, Q and ‘K; might be the relative and center-of—mass
~ ~

momenta). That is, we know

(13.2) (fi'X'i'M‘fl&12¥l\Q\& (K i M Ea at».

We can evaluate the expression in equation (D.l) by inserting the

unit Operator

(v.3)

'& X” i m“ 2*” 9 10>"ZBS%2A:WZEW%<&H ‘X f" 9n." 3*" flxul
Ifl’W (2 W3)
ԣԣWQ?<AI

=1

201+

on both sides of Q in equation (D.l) and performing the sums and

integrals,provided we have an expression for overlaps of the form

(M) (m) = < P.‘ F: i' m‘ 2.‘ 21 Wk" (”"1" 1e" fix") '

The expression in equation (D.4) is the so—called "vector bracket."

I I I
When ‘9‘. and K are related to P\ and F‘ by
~ N ~ 41

(13.5) 8:“ = 0L,?,' + 0., v,‘
~ A!

N

_ I I
(13.6) x "' b. P‘ + b ?
~ 1. 1.
~ A!
Wong and Clement show that the expression in equation (D.4) is given
by
(13.7)

(I m) : M416”; (1'1"); Um‘ M“ ) g ( w}

1 | ‘ __
“ Budam“ M mu ?.‘?£%¢"K" AIflI

205
where

(D.8) N2 = a normalization factor (we use N = (211’ )3 )

(D.9) w = b.5{kua‘ Q,‘ QI'KM.
., (...... mum v:"- m. w‘)

(”'10) X, = (9'3“: at? 9:7" 11?: ) /( 2- % 0We: ‘92-.)
I Z- 2
(13.11) x2: UK" - In} ?,'L- 52?: ) /(2_ b.b,?,'e;) .

The variables X, and X2. are eXpressions for the cosine of the

angle between P, and P: . The functions which are defined as
A. ~

(13.12) 9 (XL) =

(.... ~23

just insure that \X‘ \ and \X1\ are less than or equal 1 .

The symbol A is
1,11

206

(DJ-3) “I,“ =

x' *-
2 “:2 <1,Q; Wmdfm>Y. ( $3 Y: {61:}
cm 3 ." man, 91 M, 12‘“:
[2a. «2 mm: m>YW>Y<$Q )
Zhlluh °
Mi 'K
The vectors ‘1‘ , Yt’ £3 and ‘X' all lie in a plane and

the relative angles between the vectors are fixed by the magnitudes
of the vectors. The orientation of the plane itself is arbitrary.

For computational purposes it is useful to take the vectors? to lie

 

in the X-Y plane so that YLM( 9 :‘W/z JQ (ewn Ywhere
(Marya. y
(D 1L») (A ) [3+2 (1+M)‘(L-M)flz
. __ 1+M_____! {-M ’ ’

40v (l-m)evcn
0 JF-ov Ll-M) 04A

The factor 5(w) expresses conservation of kinetic

 

emery. For example, for the familiar transformation

=P(m.,, —n\ 9.: \/(M Ma.) ,‘Xu

P." + R."

~ A!

we obtain the result

207

(13.15) 30,4) = UIZ/A)S(&.Z/~+ Tut/gm

.. P,“ 2m, - ‘37sz

where fl: M‘M‘ /(N‘*M1) and (m: M‘+ M1. 0

Finally, we note that parity conservation requires
(9.. +11+2fiu+1exn even
The inclusion of spin is straight forward. We shall list some
results for the case where one of the particles has spin 1/2.

First, some useful expansions:

N

(13.16) for 3" = 1".» v2 , El: :3: + 1'1!
A» ’V’ ’~' ‘fi’

and

Sw‘tli'tl}; ’33:;3-FJ‘L

~

”595%”351211147
’7- 71+LI+QL+§

"1 1:5: . .
= i. {1' (3‘ 1'} [(3.1 H)(2j.+t\ Q")

x We} %‘%;i'£.'11>

208
(D.17) for & ”2 2k" + 71. , 3" '3 11h" + 1%”
«g: ': 2£u+gffl ’8“: 1::4’ ’1
u (c 0| '| .
\k x % 8% 3k" 1*." ’Q‘XU»
- 2 YA 1“" 54:“ ya

u ‘24’1 n + I I!
a" ’Q'k" 3“ {II3E21+I)(23&0+fl(-I)I -& W4"?

x WW” y" $3; {"1“ 2k»)

(D.18)

For the simple case

<42. ,Ku ‘25“ g; 1"th 1,64?" V: %I3; ill: 1...,»
= gw‘msq; warm
X (fin Xui'w'=o£&“ £564 9"F111'1h':0 1:21)

The simple case in (D.18) together with the expansions in equations
(p.16) and (D.17) can be used to obtain

209

(p.19)

3€.=lg.+‘lz’ %:J,+‘/2.

1:! ':, 5L:‘+-JL:. ft: 1: lt‘ 4r ”5L

A: N ~

(HX (39% ’81." NJ’Q "‘9 P ‘Xhi 1 Ql>
-..-. 3”,“?! \8L%;%%\{: :1: Jaguzi'uflzdh n+0]

V7." 2*" +.Q¢x" +%

h

fiLf‘) <‘fiu 'Xui'm':° £h|'£,xfl‘?" ?,:1L'W'=O,Q:Q;>
(D. 20)
. 2 V1. I! ‘
J41. “a + ~ )1—225"+()A42"
J": 0;”: ,%= 0,4. .3:
<48 u u

'K‘b (3% 3,..0. Mulf'?” ”LY 0' .0 '5
- no” WW \
V1119: 3k ’1. A .a i - -0 IA-
x 2&2 2x" 8' &' H111 i'dat' (21“)E2’0'f'xzdu“)l

102,1. +2-K"+£' +9.
xL-n («'an 0m- ="o£,{,"1ax IP' ? “110m .0‘0')

APPENDIX E

THEORETICAL JUSTIFICATION OF THE 3-BODY

MODEL IN PION-NUCLEUS SCATTERING

210

211

In Chapter VIII we obtained the t-matrix for scattering
of a pion from a single nucleon which is bound in a potential well.
This 3-body t-matrix was later used in place of the free pion-nucleon
t-matrix in the calculation of elastic pion scattering from ”He,
12C, and 160. In this appendix we give a theoretical justification
for the use of the 3—body t—matrix in pion-nucleus scattering.

In the development of the optical potential formalism

in Chapter II, we eliminated the pion—nucleon potential, ‘45“ ,

17M

by writing “J. in terms of the t—matrix I?» where we defined

WM 1!»

A
E.l 4‘ = ’0-
( ) 1m WN+’\J'.;mC-, ‘u‘
The propagator (I was given as
-\

(m) C; = (L (E-Ev—Kwuq

where Q; is a projection operator for completely antisymmetric
states. The elimination of 1‘.“ using equation (E.l) is clearly

an arbitrary step. Any operator equation containing KEEN could
have been used to eliminate “3;“ . However, the resulting equations
for the optical potential do not have conventional forms for some

choices.

212

Kerman, McManus, and Thaler (Ker59) give a systematic
way for eliminating VII“ in their Appendix IV. Their approach
is completely general and shows the conditions necessary to end up
with a conventional Schroedinger (orVLippmann-Schwinger) scattering
equation. We first present their approach and then we apply the
result to our 3-body model.

Kerman, McManus, and Thaler (KMT) point out that we can

define a whole series of scattering operators '136.

(m) tor: «gm MR,“ (gr-to.

which can be used to eliminate ’01:“ . The propagator Gd” is
completely arbitrary; however, as we shall see, there is only a
certain class which lead to a final scattering equation of conventional
form. Our objective here is to show that we can make a consistent
definition of the pion—nucleus optical potential when 60" is
the propagator used in our 3—body model.

If we use equation (13.3) to eliminate fig”; , we obtain

for any choice of (ia_

__ l
(EA) TTW ‘- {A /LA-\) ‘1”

U INJ
E. I '= A- b 1+ "I!
( 5) 1U) L ‘) crL Gr ‘1”,\

213

(E's) ao- : E k ILA-"311:6!" Gr/A‘] '

Equations (E.u) — (E.6) are the same results we obtained in Chapter II,

N
except now we have 60" instead of G and td' instead of
/\
l1m . Defining the first order optical potential as

0
(13.7) U1” : LIX—”fir

and the complete optical potential as

(M) U1»: U; + U13, (l- \O><6\\21J¢ Um)

I
the equation for ‘12:“, becomes

...! ~ I
(12.9) ‘w = OW + Um, l07<o\(a,.\o><o\‘[jw

Further, the t-matrix t6. can be written in terms of the free

pi on—nucleon t-matrix

(mo) to. = "CTN-k- in.“ C66: 3.\£o-

where, as before, 't:

is 'ven b

(an) {31“ = 417“ + ’U'T'm (3651”

In order for equation (E.9) to be the conventional one—body

Lippmann-Schwinger equation we must have 1

(E.12) <O\Q r\‘=07 (E‘w -KK+L£:\

or equivalently,

(E.13) <O\G6_\0V: <0\ G \ 07

-|
(Recall that (No.10): (ETI- Kwt-(é)

 

1We assume an infinitely heavy nucleus here so the nuclear
kinetic energy terms are not present.

-)

215

The Hamiltonian for the bound nucleon in our 3-body problem
was HN= Kid" UN . If we use an independent particle model
for the nucleus and single out one of the nucleons (call it nucleon #1),
we obtain the one—particle Hamiltonian H , '-'- K, + U , . We

therefore define a t-matrix t (its; in terms of H, and ”I!“

(“4) tab: “Raw 1m G'ftso

. -‘
where 6‘: (E‘+E.“."K“U“ KW+L6\

The t—matrix tab is the same t-matrix that was calculated in
Chapter VIII. (In Chapter VIII we denote the 3-body t—matrix as
If. H th tati t t id nf i

“0‘ . ere we use e no on 35 o avo co us on

with the t-matrix in equati n (E. l) .) Hence, we now want to see

f<°létlo>= <0‘G\O)= ~LE1r-KW+L&).|

Since G1, is a one-nucleon operator and '0) is an antisymmetric

nuclear (many-body) state, we have

é \

|
A d:\ E.“‘ Kw + CEV'EAQ'EG

 

(E.15) <0\G,\o§ '-

W‘here ck indicates the occupied single—particle states and E q is

the single-particle energy. In an independent-particle picture of

the ”He ground state, all the occupied single particle states have

216

the same energy' fig , so for that case we have the required result:

< o \ e,\o>m= (E..— Kwt tey’ ,
10

Hence, for He, we end up with the usual Lippmann-Schwinger equation

for Tr." when we use the 3-body t—matrix, tsfi , to obtain
12

the (first order) optical potential. For c and 160, we have that

-\
~ 0
(OHM ‘07 =(E1r‘K1r *“e
since we consider energies where E“. )7 CE" E‘B .
Further, in equation (E.6) the 6'6- term is multiplied by l/A .

Hence, for 120 and 16O, we obtain an ordinary Lippmann-Schwinger

equation to a very good approximation.

Thus we have shown that it is possible to use the 3—body
t-matrix in the definition of the optical potential, but we have not
shown that this procedure makes the first order optical potential
(which is what we use for U1.” ) a better approximation to the
complete optical potential. To show that there is an improvement
would require showing that the higher order terms are smaller for

U1?” 2 Q,-\ \ £33 than for q” '2'. (A- ‘)tTTN .
We therefore take a "try and see" attitude in this work, and require

that the final justification come from a comparision with experiment.

APPENDIX F

CALCULATION OF BINDING CORRECTIONS

USING VECTOR BRACKETS

217

218

The first binding correction term is given in our

3-body model by

(F.l) Atwnr-tggaogngotg} -

In Chapter VIII we showed in a schematic way how this term is
calculated. In this appendix we discuss the details of the
calculation.

We want to calculate the matrix elements of AtITN
and average over the z-components of spin and isospin of

target nucleon. Hence, we want to evaluate (we use <1 [5‘3“ ‘:>

to denote the matrix elements of AtTVN

(F 2) < AtvN>- '-

 

 

(2": H3 (23' “3:“:

LtA 3.»

Mi <2: AZMNMSJQW

In equation (F.2), \* L1. Al> E \W)‘ Ll>\A’”z»

where \W» denotes the wavefunction for the bound (s-wave)

N 'N
nucleon; A}: and L, ‘2 are the z—components of the nucleon

219

spin and isospin respectively. The initial and final pion momenta
l .
are I, and ‘1‘ respectively and L};
r h . 1r .
o t e pion isospin; also we define ‘51,." L; >2... \ ?E» h... >

Since all the operators in 6%: are "diagonal with respect

is the z-component

o‘ . L:
to the total isospin I = L +L~ , we can easily sum on .

~
We first write the isospin state 335‘ L,” LN E»%

as

(as) \ L“ L“%)\ L” Li)

7, (mg LIL” if", Q) III.)

11:,

Then we calculate

(FW)<A't-“» 2““ H i+\219 Z

11% (v.3 A"?!

 

(11%“ w LfiLZLfit><IX%\<:t\-KWAQ M‘At \+A%>\?“>\112> ,

The matrix elements of At?“ are independent of I; and

220

 

 

 

. . u“ . 2 _ QI+|
(R5) 2 (J (N Li L; \II;> " .1r
...) lb +\
1%L%
So taking 1%:0 we have
(F.6)
y 11+\ 2 ‘
<15th ‘ r (12,“ “((10 ((3 N in.“ H
41-

x <11§0K§KWA:\Atm\\\!a:>\§>\llz‘°>

We assume the pion-nucleon interaction is only in the (3,3) channel

so I: yz . Hence, we want

(F.7)

(came/e») QHZG’I’L W ”32‘ ARM “N ‘39
1%

The sum over the nucleon spin can also be easily done, but it is

 

not quite as straightforward as for the isospin. First, we write

the state \ W0 A'N%>\?£» as

221

(F.8)

N A- EWW E); <%%%L1TA M,r AN>
"‘ %
RNA;
8"» “‘° '
‘ RN} 9'” ‘1)“ I»: % 3&1 LT Ln”)

(aw )3

 

In equation (F.8), i:- L“ since LN=O . (We use this
redundant notation in order to be consistent with the notation
in Appendix D.) The function RL?A\ is defined by

l __ t " '
W93) «(mung RC9.)
is the "radial" part of the momentum space wavefunction. Using

the expansion in equation (F.8) we have

\
(m) (aim =33- (‘2

 

a... M.- :mDQgskLW/a M. “D

”DY (0.
x YL'w an») S?” AP“\°”‘\Q'° an'fim)
Ln)” 0“")

x < 91; 9; 959,233.13? L'ueoptm We?» mg L,L,=.>

 

 

222

Now conservation of total angular momentum and conservation of
l l
parity require (‘1? '3 L“: i 3 1 (Recall the nucleon
wavern tion is s-wa e. Also ‘tHT i 1 de ndent of
c v ) , 4<:£§ .{j7' s n pe

so using the fact that
29

(MO) 2 (Ln-A? ”WA-3, \%’%’2—$ = 1L“

A: %% lLrlq
(p.11) 3 (3' \= gilt} 9(516'
ZMWYHHWWM: 4-“ L" K 0
\ -——\ “| "
om: HEALTWwW‘LW-i’a

 

 

‘7- ' 2
x g?» “f R?“ ““3 R00”; \Roo(efi\

2 CN \3 2 ON 33

K < 9‘" a: 95953-1 L1; Ln“ ‘AthTrRs $931L1TLH=Z>

223

At this point we take advantage of the separable form for RN

(see Chapter VIII) by writing %0 a.) 630

(ms)

 

“somo 73" g‘i— 9‘ " ”5"» “49;"

8.3% I (2W33 L2“\3 f0“. 33
1" L3

xH> “9" y'gfl‘l; $.93 (e 9,“,P.¢")u~(e.1')

x1Y~L5~H(P'S\KuN (9N0!) %I(E, 91:311.“)
x<EKu§§ "%9: 1 Ln Ln ’0‘

(Note: u 2 || sin e u 2’. 0 )
a L“ c LN

Inserting the above form for (kofincao we obtain
H
(Again, by conservation of angular momentum and parity, L11- : Luv .)

(F.1u)

<mo=3~ Z (zwgbm M‘ 02- mm
W‘BLT 01W)

x 3 m‘ a" ”6 MEL P—«Pfls L0

 

224

where

@19ch P " (25 Lxfl- .—

 

 

0234“ ag'flAP" R P‘ L " 1:
° Lam?" Ego (2“):- C ‘08 ’ ‘r’ N )uuu’»)

" <Pn‘ R; “121: L“ L~:°\t33 “>1: P: Mei L'L: z°>

II
The expression for B L P“. fl. %’Lw 3 is quite similar to

the impulse term calculated in Chapter III. Instead of

R(?J)R(efl“3 we ha::R(? “308° (E)? ‘PNIM)

and we have t3} instead of t33 . As with the 1Timpulse term
in Chapter III, we insert a complete set of “NC“ states

on each side of tB'S , i.e. , we insert the unit Operator

(F.16)
\42"°""‘)5"‘b; «1,. .. ZSV'flW on" “K"
A” Nag.» cm" m )3
’B‘hu’Qhuktkn

K @"cx" wfiw 2M“ =- :L

 

 

225

where X‘:‘ +,Quand"8 :Il '/
’B’ku 'k, 4?." {L + ’7;

W ~ ~
The matrix elements of ”€33 for the above states are

(F.17)
< w w y 54' w 2“. 2%.. \t” W" 1'" y" 3;" in.» My»
: w)3§‘£°ru_oxm )

 

()(N‘L S ( ‘bn %") g(%; 3: Bgmr’n‘m) Maths/2‘)
x 3024;," 94533 (210." A S“ y" ,quch xt33(wm 42"4'")

where from Appendix B we have

 

(F.18)

t33( we.“ w w" x = X ’63-,me MA (333C w macaw)
and %:3( Q.)
(F.19)

tntwcm 42.9“): ‘8‘W2t‘CZ-Z 7233 e' "' 1

NR 1;. (a. ‘

 

226

(See Appendix B for the details. The off-shell factor ‘3.” should
not be confused with the free particle propagator 3° .) Using

the separable form in equation (F.18) for t33 we obtain for B

(F.20)

Ewe": (3 HA: RX"?- y‘: in ( we“ )9“ 92°)
(2W? <33” ( 0.203 I

x 31(P1‘1“%Lw\'52 (1)131“ ELI-3

where

(F.21)

31“} (Kn '3' L“\:

2 "Rt" UV“ ‘5‘? at? much")

m9 (3“ 8‘

 

 

x (PWPN' $85.01 LwLnto H551" Ni- ’86. 492 an :1 LX>

227

(F.22) 3.“ P1“ ‘1“ % LW\
2 \Xz"“~\&_'_'(£:§ Act“

(3“ )3 (3“- “~33

 

1.49m (2 P"i>.1')gn(h”'5

x <23?" ?‘%%%1 L.“- L: 1'— O \Q"‘ “K“ 35% zit'VzLQ-"11k’>

The vector bracket expression in equation (F.21) and (F.22) is
given in equation (D.19) of Appendix D. The quantities E31. and
are evaluated in the same wa as the :E}‘ factor
B2. y 131*
of the impulse term (see section 3 of Chapter III). Proceeding as

in the impulse term we obtain for Bi and B

.Q + ‘/
(F.23) BL(?‘R CK (3 LT\: -L4W§ZL_1)%+ ‘k 2.

V1 x. 3!; 1 .Jz,
X‘Qx (3 LT} 8}”?th WWWJM

 

 

 

 

‘1 1mm ammo (9&1

*—

L'l‘
x Z “1%fux “udWD 13: W223:
fiz-Lw A
M {WWW}

228

Where -F|L is KL?N\%3(‘&\ for L21 and
uu(?n\%o (59 {Pu\’\n% (st) for (a: “'2. The

variable is o
X PW‘K
Using the results from equations (F.14), (F.20) and (F.23)

and regrouping some of the factors, we have finally for (At‘fi'flw

 

_. 31 29m
(1.321») (As-t1» "' fiz‘. (XLQL'nHBL 1?Lr(81(" '? “.3

1:: (EV
V1 1 3,1 1 V7, 1 3,2. “12' H
x i 9F A?“ \YLE’KI’LPJIfD

O
x (50’s ?1? L'W ledé L P; (’1: Lwiqfl
The function fi is given by
(F.25)
ll CD «7' fl
(3 t it ?t Ln m = gin w tacos“ t, 023
° tied.)

x Y11(?‘l\”b’<" LTI’ QQ v7! ?1:°K"L11 (ex)

 

229

 

(F.26)

P (P 1" 1

L w Lwlx\=§Ax-¥L[U1:Lh\wp(k\1
“>me (P,,\

x Z“: (i Rxmkux\L,‘Yn§Y (4: \Y (‘X qr“; mY ( 9,1}

As before, 4-1:. R(e “3%33(h\ and
4-,; u («mob (a? m ntu- Th
U 0

integration variable x is at ,

In order to avoid the singularities on the real axis,
the integrals over PF” and x are taken along complex
contours by making the substitution P: —> 91:69 (P
and ‘k"—§ °Kn 6L ? . The validity. of this procedure

is discussed by Hetherington and Schick (Het65).

APPENDIX G

SOLUTION OF THE

LIPPMANN-SCHWINGER EQUATION

230

231

Haftel and Tabakin have developed a simple matrix
method fOr solving the Lippman-Schwinger equation in momentum
space (Haf70). We present here a modified version of their method.
Our method does not involve "subtracting out" the singularity in
the free two-particle green function and produces the transition
matrix itself rather than the phase shifts.

The Lippmann-Schwinger equation for two spinless particles

interacting via an arbitrary potential \/ is given by

(G-l) TU»..:' 33):: V(‘f§\+ V(§?."3T(wo)?."f.\ 43—9..
w,- may)» +‘te Q“):

I
where ?> and ‘E: are respectively the initial and final

~

momenta1 in the two particle center—of—mass frame. The energy

variable 090’") is given by

2 ’ 2. Ii"
(gm) QL€3=M\Q}+m1c3+-‘VEE« /2.W‘t «$sz lam...

for the nonrelativistic case and by

 

1As in previous sections, the momenta here are actually
wave numbers.

232
u 'I. 4. 1c} u" Y7, 1 q 2.; Ht)’;
(G.3) COL?“ )-=va. c. +‘Vs E; ) *- (mtg t-‘btg P

in the relativistic case. The energy parameter w, is related

to the on-shell momentum Po by
N

(6.1») 9).: (ALE)

where QCVQ) is given by equation (G.2) for the nonrelativistic
case and by equation (G.3) for the relativistic case.
For a central potential, we have after a partial wave

expansion

(6.5)
II

v (Y'P")TL(°J. .P" P) f "A?
0 (or w(P")+Le (1“)3

 

TL(Q.,P'P3=VL(?'P) +800

where

(es) VLP‘ 113?- 2&2qu (P' P)? (P'- P3

(G.7) TLQ” ‘2' 3.3: Z {ng‘BTLLwPIP'ISRLW'n
L

233

The momentum variables without ,N, in equations (G.5) - (G.7)

denote the magnitudes of the vector momenta.
By rationalizing the denominator of the integral in

equation (0.5) we obtain

(at) itafle'm =an'e) *-
(I) 1 n "
+ g E(P.P”)V(PP MT (wc,P"P) HAP"
where O P: - PT. + (a e (3.1"):

 

(G09) é (PO e") : XM‘M‘L _L
I‘d-m1 'th

for the nonrelativistic case and

(G.lO) c? 9“: (“3“?o\*’wi(?"3\(wz(?03*wzc?")) _|_
F) 0|(?c\+w\(.?')+w1(?c\+wi(P") ‘d'

for the relativistic case.
In equation (G.lO) the energies LO‘ and UL are

given by

(G.ll) L.) (P): LN: c, ”Ii-Vac} Phi) L“ “1).;

234

Using the identity

Ck,‘ E>f> CL)!

(0.12) 1:

. + (Jr 80““)
X”‘)(."‘ué: XL" X15

 

we can write equation (G.8) as

(G 13)

710.3,)? P)- " VL (? P) -LL?o )3“? MW (? V. ‘T Ltwo)?o P)
agar?

 

m ‘
9’8 P“’o?“W..<?'P P‘)TL.(°)¢)?" *‘3 P 4?"
o. i?.ut" ‘?;F' (ngr)3

We now obtain a finite integral by making the substitution
I!
P = ? LLX . Also, we are interested only in

O \_ x
the half-off-shell t-matrix, T C (40° ) ? )0? B , so we consider

the equation

(c.1u)

T (e.,? 93-: v u 93+...— 0““ Mpcve.)V(va Lana. ...)
1m)

 

__ WK @U’ ?")v (P 'P")T may P'o) LL
29., X m?

—-l

235

Let us now digress for a moment and consider just the
principal-value integral in equation (G.14). In a Gaussian formulation

of numerical integration we have the quadrature rule

3 u
(G-15) _amtofix 9-" Z WLIV (XL)
A in

where RX) is an arbitrary function to be integrated and .(L(X')
is a preassigned weight function. The weights WL and points
XL are found by requiring the moments ‘MM.
3 N
M Z M
L —
. _ M
A L 4

be exact for M equal zero to 1N-\ so that equation ((3.16)

is exact if +Q‘\ is a polynomial of order 2 2N‘\ . For

Gauss-Legendre integration we would take _flLX)=\ and E A) 81:
[-X) 1] . In our principal—value integral in

equation (G.11+) we can choose the weight function to be |/X .

With this weight function and N even, we obtain the result:L

VV QAuss
«2.17) X; = X;

 

1For N odd, the "center" {L is zero, so the
center weight is not defined.

236

N 691055 (Moss
(c.18) We 37- WU / b

where "PV" refers to m): VX and "Gauss" refers to M): ‘ .

Thus, an Nepoint Gauss—Legendre evaluation of integrals of the form

3'; DQAX/x is exact if ‘9“) is a polynomial of order

.2 ZN... (N even), provided the points are taken symmetrically

about x = O .

For convenience, we now make the following definitions:

((2.19) TL. L U30 , PL F.) '5 TL
(c.2o) Vt. ( Pi. ‘33 E VLJ

(0.21) FL?¢,?L\E€D

Where 1);: ?0L \+ Xffl’wss 3/Q - XEIAUSS) .

We call ?0 the N + 1 point, and write equation ((3.14) as

237

 

(9.22)
,._..\I . _FV'W? ..
Tk ”NH :L11::\ 9"“ VHN“ \RH
Gauss
3:.‘_E( PJ/(W?)PJV “J TJWJ /X6240$S
““4 on

We further define weights

ff" (39,055 a, .
K... ‘ __:!_3 FAQ’S /lewss) 4:5»

93““ L11")

 

_—_

 

(02.23)“:
_. FTW PM,”

1 Law )3

 

Po $:N+\

so that equation (G.22) can be written

M-H

((524) T1: VL,N+\ + i-W-J VLJT;
A“

238

Or equivalently,
NH

(G35) 2 (SL3 —-W-J- VL\T3 = VL,N+\ .
A“

Equation (G.25) defines a system of complex equations which we
solve by Gauss elimination.

In order to check the accuracy of our method we calculated
differential elastic cross sections for several different local
optical potentials using our method and the optical potential code
GIBELUMP (Doe73). We found that 16 points gave results that agreed
with the GIBELUMP results to 3 digits. In addition, the pion-nucleus
.differential cross sections calculated with 16 and 24 points agreed
with each other to three digits. Hence, we feel that our code

produces reliable results for the cases calculated in this work.