8- AND P - WAVE EFFECTNE RANGE ‘
PARAMETERS FOR THE NUELEUN - NUCLEQN {NTERACTION

Thesis far the Degree of Ph. D.
MCHIGAN STATE UNEVERSITY
MICHAEL S. SHER
1989

 

rum. a; 2r 47% 6r v V
MICHIGAN STATE u~v

mm M w mu'uuuummtumum L. 9 mi;

293 00628 1277

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

University

 

This is to certify that the
thesis entitled

8- AND P-WAVE EFFECTIVE RANGE
PARAMETERS FOR THE NUCLEON-

NUCLEON INTERACTION
presented by

Michael S. Sher

has been accepted towards fulfillment
of the requirements for

Ph.D. degree in PhYSiCS

 

 

Date Januarv 14, 196

 

0-169

 

ABSTRACT

8- AND P-WAVE EFFECTIVE RANGE PARAMETERS FOR THE
NUCLEON-NUCLEON INTERACTION

BY
Michael S. Sher

A two parameter description for each of the low energy
proton-proton 3P phase shifts is justified and used to perform
the first comprehensive analysis of all the proton-proton scat-
tering data below 27.6 MeV. Effective range parameters for the
150 and 3P states are determined. The analysis reveals several
inconsistencies among the data below 10 MeV. we point out the
importance of the central, tensor, and spin-orbit P-wave para-
meters in any low energy analyses, and determine the ranges of
these parameters.

Low energy magnetic and vacuum polarization effects
in the proton-proton interaction are investigated. The magnetic
amplitudes are found to contribute negligibly at these energies
and the vacuum polarization effect is shown to be present and
with a strength 1.053 1 .097 times its theoretical value.

Finite core, hard core, and soft core potentials for

the ls

0 state have been found, each of which has a smooth

radial dependence which avoids the large discontinuities of
presently pOpular potentials. Our nuclear potentials each

fit equally the proton—proton scattering length and the same set

of 130 phase shifts, and produce very similar neutron—neutron

Michael S. Sher

scattering lengths. Finite structure effects of the nucleon
charge and magnetic moment distributions are examined and are
shown, under the assumption of charge-symmetry, to produce a

change in the neutron-neutron scattering length of + .44 F.

8- AND P-WAVE EFFECTIVE RANGE PARAMETERS FOR THE
NUCLEON-NUCLEON INTERACTION

by

v ‘2-
\"

,.:\’
Michael Si Sher

A THESIS

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

DOCTOR OF PHILOSOPHY
Department of Physics

1969

ACKNOWLEDGEMENTS

I am most grateful to Professor Peter S. Signell for
providing support and guidance throughout the time encompassed
by these calculations, and for his invaluable insights into
many of the problems encountered.

Special thanks goes to Dr. Leon Heller who, together
with Professor Signell, introduced me to the problems of the low
energy two-nucleon interaction, and provided continuing discus-
sions concerning the electromagnetic corrections to the Coulomb
interaction. I am also grateful for the hospitality shown to
me by Dr. Heller and other members of the staff of the theoreti-
cal division of the Los Alamos Scientific Laboratory during the
summer of 1967.

Drs. N. R. Yoder, J. Brolley, M. Gursky, and J. Holdeman
kindly provided discussions and computer codes which greatly
facilitated the completion of this project. Discussion with
M. Miller Concerning statistical analysis and Dr. D. J. Knecht
concerning the Wisconsin experiments are gratefully acknowledged.
I should also like to thank L. Turner for his assistance in
programing portions of the final computer code used in the
analysis.

For their efficiency, cooperation, and patience, I
extend my thanks to the management and staff of the Computer
Center of Michigan State University. In particular, I wish to

thank Geremy R. Coon and.Charles J. Gillengerten.

ii

My thanks to the Cyclotron Staff of Michigan State
University for allowing me to use the Sigma 7 computer
in the early stages of this work.

For her speed and endurance, I thank Marie E. Ross
for typing this manuscript.

Finally, I wish to thank my wife and daughter for

their patience and companionship.

iii

Chapter
I.

II.

III.

IV.

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . .
SUMMARY OF PREVIOUS THEORETICAL WORK .
A. Vacuum Polarization Potential . .

1. Electric Potential . . . . .

Page

. . . . . 6

2. Electric Plus Nuclear Potentials . . . 7

3. Utility of Electric Wave Functions . . 8

B. Magnetic Potentials O O O O O O O

. . . . . 9

1. Reduction of a Relativistic Hamiltonian

to a Non-Relativistic Potential

2. Finite Structure Effects . .
C. Nuclear Potentials . . . . . . .
D. Scattering Amplitudes . . . . . .
E. Effective Range Theory . . . . . .

8- AND P-WAVE ENERGY DEPENDENT PHASE
PARAMETERIZATIONS AT LOW ENERGIES . .

Proton-ProtonlS Nuclear Potentials

A. 0

B. P-Wave Phase Shift Parameterizations . . . .

1. 39 Phase Shifts . . . . . .

2. Central, Spin-orbit, and Tensor

Parameters . . . . . . . . .

LOW ENERGY DATA ANALYSES . . . . . . . .

A. Single Energy Analyses of the Data
10 Mev O O I O O I O O O O O O O
B. Multi-Energy Analyses . . . . . .

1. Data Below 11.5 MeV . . . . .

2. Data from 0.3 - 27.6 MeV . .

iv

Operator 10
16
18
20

23

SHIFT

. . . . . 26
26
31

31

36

39

4O
43
47

50

Chapter
V. ELECTROMAGNETIC CORRECTIONS TO THE LOW ENERGY
NUCLEON-NUCLEON INTERACTION AND IMPLICATIONS
OF CHARGE-SYMMETRY . . . . . . . . . . . . .
A. Vacuum Polarization Interaction . . . .
B. Magnetic Amplitudes . . . . . . . . . .
C. Implications of Charge-Symmetry . . . .
1. Point Nucleons . . . . . . . . . . .
2. Finite Structure Effects . . . . .
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . .

Appendix A.

Appendix B.“

Appendix c.

Appendix D.

AEDI->endix E .

TABLE OF CONTENTS

Numerov Method of Obtaining a Numerical Solution

to the Schroedinger Equation .

Numerical Procedures and Calculations

1. Calculation of the Vacuum Polarization
Phase Shifts

2. Calculation of the Electric Wave Functions

81 and T2 .

3. Calculation of A j

Potentials Due to the Finite Structure of the

Nucleon . . . .

1. Finite Charge Distribution . .

2. Finite Magnetic Moment Distribution
First Order Corrections to

1. The Breit Interaction

2. Garren's Amplitude .

Search Program .

the

Coulomb Potential

Page

66
66
66
67
67
68

71

75

78

78

80

82

84
84
84
89
89
91

94

Table

II.

III.

IV.

VI.
VII.

VIII.

LIST OF TABLES

Comparison of phase shifts in single-energy

analyses . . . . . . . . . . .

25 MeV Phase Shifts . . . . .

Comparison of multi-energy analyses usinglthe
effective range parameterization for the

phase shift . . . . . . . . .

Comparison of multi—energy analyses with

potential parameterization . .
(p,p) data from 1 to 27.6 MeV

References for Table V . . .

Search Parameters for 1S0 and 3P phase shifts

Numerical values of the vacuum polarization phase

shifts and the Alj phase shifts

vi

S

0

Page

44,45

48

49

54
55,56
57

58

81

Figure

1

LIST OF FIGURES

Page

Hard and "soft" core potentials which provide
equal fits to the 0-330 MeV lS0 phase shifts.
Also shown is the Ramada-Johnston lSO potential . 27

Finite core potentials which provide equal fits
to the 0-330 MeV 1S phase shifts. Also shown is

the Bressel-Kerman S0 potential . . . . . . . . 28

Hard core, "soft" core, and finite core potentials
which provide equal fits to the 0-330 MeV lS0 phase

Shifts O O I I O I O O C O I O O C O O O I O O . 29

Energy dependence of the 3P phase shifts exhibited
by different descriptions of the two nucleon
interaCtionooo o o o o o o o o o o o c o o o o 33

Energy dependence of both the Hamada-Johnston 3P2
phase shift, 62%, and the difference between this
phase shift and the one-pion-exchange 3P2 phase

. Opec
Shlft' 612 o o o o o o o o o o o o o o o o o o 34

Energy dependence of the difference between the
3P2 phase shift for the two-nucleon models of
Bryan-Arndt(BA), Hamada-Johnston(HJ), and Lomon-

Feshbach<LF) . . . . . . . . . . . . . . . . . . 35

Central P-wave parameters from the single-energy
analyses of Table I... . . . . . . . . . . . . 46

1S0 effective range function from multi-energy

analyses 6, 7, 10, and 11 of Table III... . . . 51
Departure of the Wisconsin 3.037 MeV and the
Berkeley 9.918 MeV differential cross-sections from
the preferred phase shift solution using the data
set of Table V . . . . . . . . . . . . . . . . 59

vii

Figure

10

ll

12

l3

14

15

16

17

LIST OF FIGURES

Page
Central P-wave parameters from 0-25 MeV from
single- and multi-energy analyses... . . . . 60
Central P-wave parameters from 0-10 MeV (an
enlarged version of the low-energy region of
Figure 10)... o o o o o o o o o 0 l O o o o o 0 61
Central P-wave parameters from 0-50 MeV from
analyses and models... . . . . . . . . . . . . 62
Tensor P-wave parameters from 0-50 MeV from
analyses and models... . . . . . . . . . . . . 63
Ratio of the tensor P-wave parameters and the
one-pion-exchange (OPEC) tensor P-wave parameter
from 0-50 MeV from analyses and models... . . . 64
Ratio of spin-orbit to tensor P-wave parameters
from 0-50 MeV from analyses and models... . . . 65

Nucleon-nucleon electromagnetic potentials
derived by using the Hofstader-Wilson dipole form

factor... . . . . . . . . . . . . . . . . . . . 87

Proton‘proton electromagnetic potentials... . . 88

viii

I . INTRODUCTION

Until recent months, no energy dependent phase
shift analysis had been carried out which concentrates on
the proton-proton (pp) scattering data in the energy
range 0 to 30 MeV. This is because electromagnetic cor-
rections to Coulomb scattering, primarily vacuum polari-
zation effects, play a significant role in analyzing the
extremely accurate data below 10 MeV and are rather dif-
fiCult to prOperly incorporate in computer codes used for
such analyses.

The low energy region is interesting for a number
of reasons. The accurate pp data aLknvprecise determina-
tions of the singlet pp scattering length and effective
range, quantities which are crucial in the question of
charge symmetry of the strong interaction, a hypothesis

that states that the proton-proton interaction in the absence

 

of the electromagnetic interaction is exactly the same as
the neutron-neutron interaction. Accurate analysis may also
provide an experimental check on long-known theoretical cor—
rections to the Coulomb interaction such as vacuum polari-
zation, magnetic moment interactions, and finite charge and

magnetic moment distributions of the nucleon.

Recent analysesl’2

of the low energy data have used
somewhat questionable forms for parameterizing the P-waves,
which contribute significantly at low energies because of the
accuracy of the data. There has been no detailed examination
of P-wave parameterization in this energy range, aside from
extrapolations of values from phenomenological models which
themselves were found by fitting the high energy data.

The S-wave phase shift has been parameterized at
low energies by effective range theory and over the energy
range 10-350 MeV by polynomials of functions which exhibit
the prOper energy dependence; e.g., associated Legendre
functions of the second kind are used by Livermore3. No
use has been made of potential parameterization of the S-
wave to fit the low energy data, although this is a natural
method which has unique advantages. The one limitation in doing
so is the limited theoretical understanding of the 8 state
potential, exCept for the long range one-pion—exchange
potential (OPEP) contribution4. Hence, if a phenomenological
potential is used, it should be examined to see if the result-
ing analysis is form limited.

The principal advantage of using the potential
parameterization of the S-wave is that it allows one to directly
separate the nuclear from the electromagnetic effects in the

5

two-nucleon interaction . It has been suggested by Heller6

that "to compare proton-proton scattering in a given state

with . . . neutron-neutron scattering, in connection with
the study . . . of charge symmetry of the nuclear force,

it is desirable to isolate as many electromagnetic effects
as possible". Heller pointed out further that a "clean
separation is not possible at small distances. One example
. . . is a possible alteration of the nucleons' anomalous
magnetic moments when they are at small separations, and
the resultant change in their electromagnetic interaction

at small distances. Another possibility is that the charge

 

and magnetic structure of the nucleons is altered when they
are in strong interaction".
However, despite these difficulties, one can make a

fairly clean separation of the long and short range electro-

 

magnetic effects from the total potential. Although both
effects are weak compared to the Coulomb potential, the long
range effects such as vacuum polarization produce scattering
in many angular momentum states. Thus, although any single
partial_wave may be small, the scattering amplitude may be
significantly effected by the infinite sum of these partial
waves. The short range effects, such as finite charge and
magnetic moment distributions, although small and easily
absorbed into the phenomenology of the nuclear S-wave potential,
should be separated and their effect estimated, if possible,
in order to make clearer the use of the nuclear part of the

potential in the question of charge symmetry.

In summary, the purpose of this thesis is to com-
bine the available theoretical information on the electro-
magnetic and long range nuclear interactions with a judiciously
chosen phenomenological parameterization of the two-nucleon
non-OPEC phase shifts so as to simultaneously analyze the
pp data in the energy region below 30 MeV and provide the
best separation of the electromagnetic and nuclear parts of
the pp singlet S-wave potential in order to examine the im-
plications of charge symmetry in the nucleon-nucleon inter-

action.

II. SUMMARY OF PREVIOUS THEORETICAL WORK

In the introduction. effects such as vacuum
polarization, magnetic moment interactions, and finite
charge and magnetic moment distributions were mentioned.
These are all first order corrections in the fine
structure constant, a, to the Coulomb potential eZ/r, and
are believed to be present on the basis of the well-
studied interaction between electrons and nucleons. It
is the purpose of this section to briefly review what
is known about these effects and how this knowledge has
been used.

A. Vacuum Polarization Potential

Quantum electrodynamics predicts7 interactions be-
tween charged particles in addition to the Coulomb effect
which, although small, are significant in the description
of sufficiently accurate eXperiments. The largest effect,

arising from the following diagram,

 

is due to vacuum polarization (v.p.). The effective v.p. potential

6
to first order in a.was first derived by'Uehling8 and is
given by 2

_ e
(l) va(r) _ A _f'I(r)

 

 

°° _ 2 1/2
With I(r) = [ e 2Kr€ (1+ 12) (E -1) I d5 I
"’l 25. £2
1 = 2—‘3; = 1.549x10'3 ,
a -1
and (2<) =’5/2mc = 193.1 F

is the Compton wavelength of the-electron-positron pair
divided by 2“.
The limiting forms of I(r) are (252<r)

I(r) = -y-5/6-ln(Kr)+e(z) , z<<l

(2)
3(2101/2 e-2Kr

I(r) = 4 377 (1+e(l/z)) , z>>l

 

(2Kr)
where y is Euler's constant = 0.5772 . . . .

This interaction was found to-contribute an important part

of the total Lamb shift between the electronic energy levels
2 2 - t 9

25 Sl/Zand 2p P1/2 in the hydrogen atom (See reference 10

for a bibliography on the history and derivation of the

vacuum polarization potential).

1. Electric Potential
The electric potential, defined as the sum of the

Coulomb plus v.p. potentials,

2
VE(r) = E-r--(l+XI(r))

7
satisfies a radial wave equation

2
d ”2(r’ 2 1 2(2+1)
—'§—+“‘ “"
dr
There are two linearly independent solutions to the

_ AI(r)
Rr ‘j‘z—luz‘r’ ‘17—‘12“) -

equation. SR, the regular solution, vanishes at the

origin and we choose, for the asymtotic form, following

the notation of Heller,11
(3) S£(kr) 2:;331 F£(kr) + tan 12 G£(kr)

where Pg and G£ are the regular and irregular Coulomb
wave functions respectively and 11, the v.p. phase shift,

satisfies the equation

F IS

(4) 9. i

 

tan 1 = -2nA I dr
2 o

where
n = eZ/fiv .
The other Solution TR is taken to have the asymtotic

form

(5) T£(kr) ZFFFEI G£(kr) - tan 1 F£(kr) .

l

The numerical calculation of the electric wave functions

is described in Appendix B.
2. Electric Plus Nuclear Potentials

Considering the total pp potential to be the sum of
the nuclear and electric potentials, the radial wave
functions becomes, at distances large compared to the range

r0 of the nuclear force,

 

E
(6) R£j(kr) cos %,

, E
res-1:0 j S£(kr) + Sin 613' T (r)

Q

with GEj the nuclear phase shift in the electric potential.

Using equations (3), (4), and (5) this becomes

 

(7) R1j(kr)2:??$1 cos K . F2(kr) + sin K . G£(kr)

1]

with
K

III
09
+
a

lj ij 2

One can relate GEj to the nuclear phase shift éij,

 

which is the phase shift the same nuclear potential would

produce ifkthere were no v:p.. If W£j(kr) is the radial

 

wave function in the same nuclear plus Coulomb potentials,

but with no v.p. potential, then Gij is defined by

c . c
(8) W£j(kr) cos Sij F£(kr) + Sln sz G£(kr)

r>>r
0

Using perturbation theory, one can show that

_ E _ c
(9) sz — azj + rz — agj + Azj
°° WEI
with A.=_2nxjdr_1_
£3 r
o

3. Utility of Electric Wave Functions

In solving for the nuclear phase shift due to a
particular nuclear potential, one solves the Schroedinger
equation containing the total potential (nuclear + electric)
and matches the wave function in the "asymptotic" region to
the solution of the Schroedinger equation without the nuclear

potential. The question of where the "asymptotic" region

9
begins depends upon whether sz of equation (6) or equation
(7) is used, i.e., does one directly calculate GEj or Kij?
In order to use equation (6), the asymtotic region is
near 15-20 F.’ For equation (7), the asymptotic form
isn't reached until many hundreds of fermi since (2K)-l=l93.l F.
Thus, if one has to solve the Schroedinger equation many times,
it is advantageous to calculate the electric wave functions and

phase shifts and make use of equation (6).

B. Magnetic Potentials

 

At this time, an exact relativistic theory for the
two-body electromagnetic interaction is not known in closed

12'13'14'15 used in the

form. There are several methods
past to find a two-body Hamiltonian that represents the
electromagnetic charge-charge, charge-moment, and moment-
moment interactions. Each of these methods uses questionable
assumptions but all lead to essentially the same potential

to first order in a, the fine structure constant, and

v2/c2. The method preferred by the author is that of Barker
and Gloverlz, who start from the Bethe-Salpeter equation

and use the Foldy-Wouthuysen transformation to reduce a
sixteen-component two body Hamiltonian in the non-relativistic
limit of order vz/c2 to a four component potential Operator.

A short discussion of this method follows, while descriptions
of other methods giving comparable results are presented

in Appendix D. Also described is a method to account for

the finite structure of the charge and magnetic moment

distributions of the nucleon.

10

1. Reduction of a Relativistic Hamiltonian
to a Non-Relativistic Potential Operator

A relativistically covariant description of the two-

16 who derived

body interaction was given by Salpeter and Bethe
the integral-differential equation

' a a ' b b
(ua P11 in + pu Yu ma)(ub Pu Yu pu Yu mb) w(pu)

(10) _ _ . -1 4
— (2N1) [a k G(ku) Mpu + k“)

which holds in the center of mass system for two particles

a and b where Pu is the fixed energy-momentum four—vector

of the center of mass (Pu = (O,E)). E is then the eigen-
value of this equation.‘ pH is the relative energy-momentum
four—vector while y: and y: are Dirac operators in the spaces

I
of a and b respectively. The dimensionless coefficients “a

I I
and “b are given by ”a

I
=m5/(ma+mb) and “b =mb/(ma+mb) while
the wave function w(pu) is a sixteen-component spinor in
the direct product space of particles a and b. G(ku) is the
interaction function corresponding to a simple exchange of

one quantum. For the case of electrodynamics, two alterna-

tive expressionsl7 may be used. In momentum space, they are

 

 

 

2 YaYb
(11) G(k ) = e2 —3§3
u 2n k
V
2 Yayb YaYb
and G(k ) = - 32 ( 424 + 121
“ 2n E. isl ku

where the second expression consists of two parts: the
first part represents the Coulomb interaction, instantane-

ous in a particular frame of reference: the second part,

11
called the transverse part,has the sum over i=1,2 involving
a summation over two mutually perpendicular directions of
photon polarization, both perpendicular-to k. This second
expression is more convenient to use and its lack of manifest
Lorentz-invariant form is compensated by the fact that a
non-relativistic treatment involving the first part
already furnishes a fairly good approximation.

Salpeterl7 considered the slightly mysterious
"instantaneous interaction", by which he meant replacing
G(ku) by G(k), where the Fourier transform of G(k) is an
instantaneous potential in coordinate space. Such an
interaction function, as noted by Salpeter, is not Lorentz
invariant, but leads to a three dimensional Bethe-Salpeter

equation written in momentum space as

[E - Ha(p) - Hb(p_)]¢(_p_) wimp/12(3) -Ai‘<p)A’3<g>1

(12’ *I d3k G'<g>¢<2+r>
where a

Ha(p) = maB + 2.0

Hbe) " mbBb - 2 ob

G'qg = vjvfi’ em

and the sixteen component three dimensional wave function
¢(p) is given by

¢(E) =J dp4 ¢(2,P4)
The Casimir prejection operators A: are defined as usual

by

 

A+(B) = 3(2) i H(E).
- E (p)

12

where E(E) = (m2 + 22)l/2

Four wave functions representing positive and negative
energy states are defined by
_ a b
¢++(p) - A+<e> A+<2> ¢<2>
a b
¢+_(p) A+(p) A_(p) 49(3). etc.

For the scattering problem, we are only interested in

¢.++(p).
The postulate of an instantaneous interaction is

not well understood except that it is known18

that ¢(p,p4)
is peaked at p4=0 for p2<<m2, thereby making the replace—
ment p4=0 in G'(pp) a good approximation in the non-
relativistic limit. The instantaneous interaction also

allows comparison with other two-body descriptions such

as the Breit equation (see Appendix D) which is of the

 

form r _ 3
.(E - Ha(p) -Hb(p)) ¢<2> = [a k G'(E) ¢<2+t>
where 2 aa ab
2 _"_‘
G'(]£) =E..Z(l:2-_ Z 121)
211 15 i=1 1;

In the non-relativistic limit, with both particles in
positive energy states, only ¢++(p) is important and the
two equations are identical.

Thus, if we choose to set p4=0 in the second
expression for G(ku)' the three-dimensional Bethe-Salpeter
equation may be written in Hermetian form in coordinate

space for two equal mass particles of charge e as

13

 

l a b a b
(13) Hwy — (Ha+Hb+ §[{A+A+—A_A_}.UB(3:_)1+)1Mg)
2 1k.r a b a ‘ b “
where UB(r) = 9—2 I " dE-(l - E E ‘ E E E E)
_' 2n :2
e2 a-ab oa-f ab-r
= E_ (l - — — _ — — — — )
2 2

The above equation describes the interaction
between two fermions without anomalous magnetic moments.
Although a quantum electrodynamic explanation exists for
the electron's intrinsic magnetic moment, no similar funda-
mental explanation exists to explain the nucleon‘s intrin-
sic magnetic moment. However, the anomalous moment inter-
actions may be accounted for by adding to UB(£) terms
which represent (a) the energy of each charge in the mag-
netic moment of the other; (b) the mutual potential energy
of the additional magnetic moments; and (c) the energy
which the electric dipole of each nucleon, associated with
the anomalous magnetic moment, possesses in the electric

field of the other. In classical terms, these would cor-

respond to terms of the form19
(a) e gé°Ap
where A? = m? x £/|£J3 , m? = uqu , and “b is the anomolous

magnetic moment of particle b.

 

 

(b) _§a Eb
where Eb = 3iffifflp)‘fl§ 8n
|£|3 +3‘T5E)
(c) i Hag Eh
where _b = e 3 E

14

This was first done in a relativistically covariant manner
by Pauli for the Dirac equation representing a single

particle in an external field20

, and has been applied

to the two—nucleon interaction by“Schwinger21 and others.
For two nucleons of charge e, terms of the form (a), (b),
and (c), each multiplied by the appropriate Dirac
matrices, in analogy to Pauli's single particle treatment,

are combined to give an anomalous magnetic moment inter—

action term UAM(£) of the form

 

 

 

 

r r
(14) UAM(£) = -e [uiBaOa-g§ x — 3 - ugebgb-ga x _ 3]

' ’ Isl iii
a b a b
o '0 _ 3(o 'f)(o -‘f_)

+ Bash paub [ _ '1 ’ " ’ A - 31 0a ob<S(r)]
a a 3 3 3 — — —

|r| Isl

. a 3 b
+ 18 [uzeag 'g/lgl -uaBbgp'£/|£l]
1.79276 no for the proton
where u =
-1.9128 “0 for the neutron
and no = efi/Zm is the nuclear magneton.

AM(£)’ Barker and

Glover12 used the Foldy-Wouthuysen method to reduce equation

With UB(£) replaced by UB(£)+U

(13) in the non-relativistic limit (keeping only terms of
fi2
m2r
Operator for two nucleons of mass m and charge e in positive

 

order EZ/m2 and ) to the following Pauli four-component

energy states:

22 e2 4
(15) Hfree + Hem = fi_ + 2m + F” - __3
4m
e2 2
+ ——2 [E . f E + ——§ (r-p) ]
2m r
2e e”- 21
"-a[“rr"s]£'§ + uT:3-812
eh eh 8n 2 a b
[8"7saua+4"§sauo+T“T°-i 15””
where =
u ua+uo

m
U‘
.9!

Hence, the relativistic treatment of Coulomb-scattering
results in the appearance Of spin—dependent forces between
the protons, similar in origin to the spin-orbit forces
characteristic Of the Dirac theory Of an electron in a
central field.

This agrees with the formulations of Breitl3 and
Garrenl4 described in Appendix D except for differences in
the momentum dependent terms. These differences are believed
to arise from the difference in the methods used to reduce the
sixteen-component Hamiltonian Operator to a non-relativistic
four-component potential Operator, i.e., the method of large
components vs. the Foldy-Wouthuysen transformation (see discus-
sion in Appendix D).

An amplitude due to the Breit momentum-dependent

terms from the reduction by large components has been calculated

by Tubis22 and found to be small compared to the v.p. amplitude,

16

the ratio Of the two being less than 0.2 for e < 20° at 2 MeV
and decreasing rapidly with increasing angle. Although the
effect would increase with energy, the experiments themselves
become less accurate. Rather than calculating a similar ampli-
tude for the relativistic terms of our equation, the approach
taken here is to replace n=;§: in all calculations of amplitudes
and phase shifts by its relativistic value. This is believed

to simulate the relativistic correction terms for the accuracy

desired, and is consistent with previous treatmentslo'23.

2. Finite Structure Effects
The terms in equation (15) arise from having assumed
the two nucleons to be point charges. However, we know from

24 that the nucleons have finite

electron-nucleon scattering
structure in their charge and magnetic moment distributions.
This finite structure will presumably be observable only in
the S state, being a short range effect, and hence only the effect
on the Coulomb term and magnetic contact terms in equation (15)
need be considered.

The electric GE and magnetic GM
by Hand et al25 are associated with the charge e and total mag-

form factors given

netic moment “T of the nucleon and are related to the Dirac
form factor Fl' associated with the magnetic moment no, and the
Pauli form factor F2, associated with the anomalous magnetic

moment pa, by the relations

l7

2
GE = F1 ‘ f L? 11an
m
(16)
Gm = Fl + pan

Electromagnetic potentials for use in the Schroedinger

equation may be calculated by Writing2

 

<17) v(5) = 1 3 J eiS'E V(g) d9
(2w)
where
V(q) = fl(q) f2(g) V(g)

1
where fl(q) and f2(q) are the Fourier transforms of the

two interacting charge or magnetic moment distributions

and v(q) is the Fourier transform Of the interaction between
two point sources. The interpretation of F(q) and G(q)

as Fourier transforms of the spacial distribution Of their
associated charge or magnetic moment is a reasonable one

27,28

at low energies , and they are interpreted as such here.

GE and GM are generally easier to measure than

F1 and F2. Although several theoretical forms have been

propbsed, none is entirely adequate. It has been found29’30

that the Hofstader-Wilson dipole formula,

 

_ 1 2
(18) GEP _ 2 ,
1 +
.71(§?C—‘1)2

in conjunctions with the Rosenbluth formula, fits the
electron-proton elastic scattering cross-sections to within
15% over the entire range for which measurements presently

exist. Because these form factors are for free nucleons

18

and are to be applied to account for effects of strongly
interacting nucleons, and since the largest deviations from
dipole form factor occur for large values of q2, this is be-
lieved to be a sufficiently accurate description for our purposes.
It is well known, e-g., see reference 31, that the proton

and neutron form factors are related by the formulas

G G
_ M _ Mn
(19) GEp " —‘§§7‘e'2.7 ‘ We '
GEn = o.

The above relations are used in Appendix C to cal-
culate a potential representing the finite structure effects

of the two-nucleon interaction.

C. Nuclear Potentials

A description of the nuclear interaction by a
potential model has been attempted for many years. Since
we have little theoretical understanding of such a potential,
various phenomenological forms have been used with various
degrees of success. Existing phenomenological nuclear
potentials have recently been reviewed by Signell.32

The only criteria for the acceptance of any phe-
nomenological nuclear potential is its x2 fit to the data.
Any xz-based model analysis of a set of data Dn seeks to

minimize the parameter
2 Pn — Dn 2
X=5‘“‘”€}T”

19
where the Pn are the model-predicted data values and the
en are the experimental uncertainties associated with each
Dn’ If (1) theDn are distributed-normally about the the-
oretical Pn' (2) the En are good estimates of the standard

deviation of the data D , and (3) the model parameterization
n

accurately represents the true functional form, then the
expected value of x2 is the number of degrees of freedom;
i.e., the number of experimental data minus the number of
search parameters.33 A goodness-of—fit criteria is there-
fore represented by what is known as the chi-square ratio

XZR, where
2

XII: xz/x2(expected) .

If all three above conditions are met, then on statistical

grounds we would expect a minimization of x2 to yield

X =1. Failure Of any one of them will effect XZR to some

extent.

Two prominent existing potential models, of the

hard core and finite core variety, are the Hamada-Johnston34

35 2

(HJ) and Bressel-Kerman R

(BK) potentials, which yield x
of 2.98 and 2.13 for the existing 9-330 MeV pp data.36
The BK potential is essentially the same as the HJ potential
for radii greater than 0.7 fm but has a finite core inside
this radius rather than the infinite hard core of HJ. Both

these potentials were formulated by fitting the pp data above

9 MeV and therefore give questionable quantitative information

20

at lower energies as indicated by their very different pp 150

scattering lengths of -7.738 F and -8.163 F, although they
each fit the data above 9 MeV rather well. Another drawback
of these potentials is their unrealistic shapes--the S-wave HJ
potential has a hard core at 0.5 F followed by a 1.5 BeV
attraction while the BK potential has a l BeV discontinuity
at 0.7 F.

We will be interested in a potential for the 180
state that fits the data well, especially at low energies,

and if possible avoids the pathological behavior of existing

potential models.

D. Scattering Amplitudes

 

Having assembled an electromagnetic potential
approximately correct to first order in a, we now examine
the method of combining this potential with a short range
phenomenological nuclear potential whose formulation will
be discussed in a later seCtion. The procedure followed
is that of Heller11 who solved the problem of the Coulomb
plus vacuum polarization potentials.

The complete unsymmetrized singlet scattering
amplitude f(6) may be decomposed into partial waves and
written as
216T
f(6) = ——— Z (22 +l)(e 2“

l
Zik

£ -1) P£(cos 6)

. T _ em E .
We may write all - a2 + 12 + 611 + 5km where o£ is the Coulomb

21

phase shift, I is the vacuum polarization phase shift,

1

5:? is the phase shift due to other electromagnetic cor-
rections to the Coulomb potential, and 6M is the nuclear

phase shift in the "electric potential" defined in Section

II.A.2. We then write the identity

0 T I o o ' I cm
216 210 210 211 21(0 +T ) 216
e “2-1 = (e 2-1) + e z (e “-1) + e 2 g (e M—1)
. E
2i(o£ +1 ”+6 m) 216
+ e £1 (e £g-l)

Then, f(6) may be written as
f(6) = fc(e) + fvp(e) + f€m(6) + fn(e)

The Coulomb amplitude, fc(e), can be written in closed

form as

 

fc(9) = ,nz e-21n ln(s1ne/2)
2ksin 0/2

with n calculated relativistically.

The vacuum polarization amplitude, fvp(e), can be

calculated to first order in a by using the electric poten-

tial of Section II.A.l,
e2
VE(r) = E— (l + .001549 I(r)) .

10

Durand has given the v.p. amplitude, exact in the Coulomb

distortion effects, in an integral representation which may

be calculated numerically.37

Arising from the following terms of equation (15),

(20) V€m(r) = " --3(IJ ‘ — “T :3 312 I

the amplitude fem(e) may be written as

f m(6) = fLS (6) + fT(6)

22

In the past, these terms have been used to calculate Born
amplitudes.38 Breit et ails have found ng(e) to have a much
smaller effect on polarization data at 150 MeV than f§2(e),
and therefore neglect fT(6) in their calculations in this
energy range. However, the Coulomb distortion effects may
be significant as in the case of v.p.lo, thereby requiring
a distorted wave Born approximation amplitude to be used
for both. This will be discussed more fully in Section V.

We now examine fn(9), and explain why the amplitude

was divided as it was. At low energies, GE is appreciably

23'
E em

different from zero only for i=0 or 1, while Ktj = 6£j+T2+5£j

is non-zero for very many 2 due to the long range character
of the vacuum polarization potential. For energies below
30 MeV, all phases for 2>l are known to have essentially

their OPEC value.39

If we neglect Coulomb distortion of
the OPEC amplitude minus the S-and P-wave contributions--
believed to be a good approximation since the higher phase
shifts are less sensitive to Coulomb effects due to the

centrifugal barrier effects--we may write for the unsym-

metrized singlet amplitude

1 ziaggec
s _ s _ _
fn(6) - fopec(e) 21? (e l)
1 2i(ro+53$) 21630
+ -—-;—-21k(e ) (e ’1)

+

fc(6) + fvp(e) + f€m(6)

23

The symmetrized singlet amplitude is

s _ s s _
fn Sym(9) — fn(8) + fr1 (TT 8) .

The triplet amplitude is treated in an analogous manner, but
symmetrized by

ft

_t_t_
n Sym(e) — fn(e) fn (n e) .

E. Effective Range Theory

Low energy pp scattering experiments done in the
1930's were the first experiments relating directly to
the nucleon-nucleon interaction. This first data was of
limited accuracy and could be fit by almost any potential
well with two adjustable parameters. In the late 1940's
effective range theory explained why only two parameters
were needed. Due to the short range of the nuclear force,
scattering below 5 MeV takes place almost entirely in the
S-state and due to the strength Of the nuclear force the
energy dependence of the S-wave phase shift is adequately
described by only two parameters.

For a short range nuclear force, the nuclear phase
40

shifts dzj satisfy the relation

lim k2£+1cot 6 .= constant .

k+0 £3
If, in addition to the nuclear force, the Coulomb force

is present, then the relation becomes

24

2 2
. g 21 2 c h(n) i
(21) 1:20 ll (1 + 37-) k (CO k cot 6£j+ -§f—J— constant
' m=l

where

W

2 l

2wn
C=-—f————. ,h(n)=nZ————2—-1nn-Y
° -~e "”-1 i=1 £(£+ n )

R-’f12_ __1__
I 2 - 2nk
me

These are known as effective range functions, and can be
shown to be analytic, even functions of k out to the nearest
singularity produced by the nuclear potential, e.g., k=iiu/2fic

for a Yukawa potential of range u-l.6

This means that since
the long range part of the nuclear force is believed to be OPEP
with u=l35.0 MeV for the pp interaction, that the effective
range function can be represented by a power series in k2 which

will converge for lab kinetic energies below 10 MeV. The series

expansion is commonly written as

2 3 4 5 6
r k - P ro k + Q ro k + ...

2 _ - i
(22) X(k ) - a + o

NH—J

where the constants a and no are known as the scattering length
and effective range, and P and Q are called the shape parameters.
For energies up to about 5 MeV, experimental techniques
have developed such that cross-sections may be measured to an
accuracy of 0.1 to 0.3%, the higher errors being associated with
small angle measurements, ecms20°. This high precision causes

analyses Of the data to be highly sensitive to the effects Of the

25

v.p. amplitude and v.p. Smand P wave phase shifts as well as the
split P-wave nuclear phase shifts.
The S-wave effective range expansion which accounts

explicitly for v.p. + Coulomb is given by

2

C k 2 (n)
(23) o E _ h(n) o _ 2

fio[(l 4' X0) COt 500 tan To] + -——-R + ——R - X(k)
where 60, x0, and £0 are functions of energy defined by
Heller.ll

In the case of equation (21) for 2 = 0, the constant

will be referred to as the pp "Coulomb" scattering length,

agp, defined by
-1 . 2 c
24 ac = 11m -C k cot 6 .
< ) pp k+o < o 00 >

The scattering length of equation (23) will be referred to

as the "electric" scattering length, asp, defined byll
_ l + x
(25) aE 1 = lim (-c2 k 0 cot sE ) .
pp k+0 O l _ ¢ 00
O

The Coulomb scattering length is the one normally calculated
from nuclear potential models with the Coulomb potential
present, while the electric scattering length is normally

the 1So pp scattering length quoted in the literatureG.

III. S- AND P-WAVE ENERGY DEPENDENT PHASE SHIFT
PARAMETERIZATIONS AT LOW ENERGIES

A. Proton-Proton lSn Nuclear Potentials

For the present problem, we are interested in parameter—

izing the nuclear potential only for the 1S0 state. We have

 

found that sums of Yukawas with various hard core radii will

fit the most recent set of energy independent S-wave phase

shifts from 9 to 330 MeV and the pp scattering length. Two

such potentials, with radii of 0.1 F and 0.4 F, are shown in
Figure 1 along with the Hamada-Johnston lS0 potentia134. The
shapes found were rather unique beyond 1 F. although one could
produce any shape between them by varying the hard core radius
between 0.1 F and 0.4 F. The 0.1 F hard core potential is refer-
red to as a "soft" core potential because of the Yukawa repulsion
at smaller radii - a somewhat questionable terminology, but common
in the literature.

We have also found a class of finite core potentials
with heights ranging from 400 to 2000 MeV which fit well the
same set Of S-wave phase shifts and pp scattering length. Two
Of these, Of core heights 400 and 1000 MeV, are shown with the
Bressel-Kerman lSO potential35 in Figure 2.

In develOping these potentials, an effort was made to
keep the potential as smooth as possible, avoiding the large
discontinuities present in other potential models. This smooth-
ness, besides being more physically appealing, has the pragamatic

26

2]

 

 

 

 

 

 

 

 

 

T I I
500b_ a
/O.l F HARD CORE
‘9 0.4 F HARDCORE
- g o I l l l l l l l l L l J
z I I I I l [I f r I I I I I
\p
-d
S,
'5
h:
k.
E \HAMADA-JOHNSTON
g -500 'f' ..
£0
-|OOO F' -(
' -| 500 - .1
1 1 1 1 1 1 1 1 1 1 1 1
(15 . L0
ROdIUS (F)
Figure 1. Hard and "soft" core potentials which provide

equal fits to the 0-330 MeV 1S0

is the Hamada—Johnston

phase shifts. Also shown

SO potential.

28

 

 

 

 

 

 

 

 

 

 

 

I I I I T I I I I l u T I
I000 ‘
g /l000 MeV FINITE CORE
0
I! .a//
" 500 q
2‘
1‘ BRESSEL-KERMAN
s /
.‘3
C)
Q.
o /400 MeV FINITE CORE
£0
0 i i i i i .7
_500 J l I l l l l l l I l l
' (15 L0
Radius (F)
Figure 2. Finite core potentials which provide equal fits

to the 0-330 MeV 1S phase shifts.
Bressel-Kerman

lSO Opotential.

Also shown is the

29

 

 

   
 

 

 

 
  

 

 

 

 

(000- _
‘ 3.?
3.
d
S,
.5 0 I. F HARD CORE
8 500- -_
~mo \ HARD CORE
/400'Mev FINITE CORE
0 i 5 3 5 u u u i u i i i i
1 1 1 1 l 1 1 1 1 I 1 1 1
,0 (15 . - L0
Radlus (F)

Figure 3.

Hard core, "soft" core, and finite core potentials

which provide equal fits to the 0-330 MeV lSO phase shifts.

30

value of being easier to use in calculations involving numerical
integrations. Each potential becomes equal to OPEP beyond about

4 F. The 0.1 F hard core potential, the 0.4 F hard core potential,
and the 400 MeV finite core potential have the following forms:

(a) Soft Core 1S Potential

0
_ _ -4x -7x 2
V — VOpep 1411.3 e /x + 5281.9 e /x + e /r ,r > 0.1 F
= w r < 0.1 F
(b) Hard Core lS0 Potential
v = v - 6.5525 e'zx/x - 1054.7 e'4x/x
Opep
+ 6727.5 '9x/x + eZ/r r > 0.4 F
:00 r<0.4F
(c) Finite Core lS0 Potential
8
v = (v - 3.0235 e-ZX/x - 1097.8 e'4x/x)*(1 - e'(X/Xc) )
Opep 8
+ 400 e-(X/xc) + ez/r
where xC = .78589, and the potentials are in units of MeV.
x E m r/fic, V = -f2 m e-x/x, f2 = 0.08, and m = 135.0 MeV.
w Opep n n

We note the very different character of these potentials
in the region less than 1 F and the fact that they are all nearly
equal beyond 1.2 F as shown in Figure 3. Each was fitted to a pp
scattering length of —7.819 t .004 F, showing that the pp scatter-
ing length depends primarily on the potential beyond ml F -- the
potential shape inside 1 F consists of varying degrees of attraction
and repulsion that, as far as the scattering length is concerned,
cancel the effect of one another. The nn scattering lengths calcu-

lated after removing the Coulomb potential from the total pp potential

31

are -17.29 F, -l7.17 F, and -l7.26 F for the potentials of (a),
(b), and (c) above, respectively. If, in addition to removing
the Coulomb potential, the mass in the Schroedinger equation is
changed from the proton to the neutron rest mass, the nn scatter-
ing lengths become -l7.60 F, —l7.57 F, and -l7.55 F respectively.
It was originally intended to combine each of these
potential forms with the prOper electromagnetic contributions
to fit the pp data directly and to investigate the resulting nn
scattering length in each case when charge symmetry was imposed.
However, the above discussion indicates that there will probably
be little difference in the nn scattering lengths of each model.
We therefore choose to use only the 400 MeV finite core potential

form for the 180 potential in our investigation.

B. P-Wave Phase Shift Parameterizations

1. 3P Phase Shifts

The 3P phase shifts of the Hamada-Johnston34(HJ) poten-
tial, the Lomon-Feshbach4l(LF) boundary condition model, and the
Bryan-Arndt42(BA) one-boson-exchange-contribution model exhibit
the low energy behavior shown in Figure 4, where the P-wave effective
range function of equation (21) is plotted vs lab kinetic energy
(in the case of the BA model, k3 cot GIj would be plotted since the
Coulomb interaction is not present). Note that the 3P0 and the
3P1 phase shifts may be parameterized by a scattering length and

effective range up to about 30 MeV, while the 3P2 effective range

32

function exhibits large curvature below 10 MeV (It should be

noted that all of these are phase shifts of the type 5Ij defined

in equation (8)).

3 5

It is known that the P OPEC phase shift has k

2
energy dependence below 10 MeV, rather than the usual k3
dependence Of the other P-wave phase shifts. Figure 5 shows

the effective range function for the HJ 3P2 phase shift and

the difference between the HJ 3P2 phase shift and the OPEC 3P2
phase shift. It is seen that by using this difference, the
P-wave effective range function is well described below 30 MeV
by only two parameters. This behavior is exhibited by the LF
and BA models as well, as illustrated in Figure 6.

Since each of the three models HJ, LF, and BA represent
a very different kind of description of the two nucleon inter-
action, and each exhibits a common energy dependence in the 3P

states, we feel justified in using these parameterizations for

our analysis in the energy region 0-27.6 MeV. Thus, we represent
3 3

 

the P0 and P1 phase shifts by two parameters each in the
formula

2 2 2 c h(n) _ _ l 1 2
(26) (l + n ) k (CO k cot Glj + R ) — 3:; + 2 rlj k

where n, Cg, h(n), and R are defined in equation (21). For j = 0

or 1, GIj is the 3Pj nuclear-bar43 phase shift with respect to the

Coulomb wave functions if no v.p. potential is present, and alj

and rlj are the P-wave scattering lengths and effective ranges re-

spectively. For the 3P2 nuclear-bar phase shift, we use the relation

33

 

 

 

 

l j l l l
5— 3 ..
P2
4P\ -(
.q
'1’
k
‘m .31- ‘
l\ 3
a P0
8
c
+ .2- -
Uhs
66‘
'0.
o
c; I _ _
cf
0
U
\l
a. 0 i 1 e 1r I
a:
‘K
N
G>
1.
\ .-
\.
1 1 1 1 1

 

 

 

0 IO 20 30 4O 50

EHJAB(A06\/l

Figure 4. Energy dependence of the 3P phase shifts exhibited
by different descriptions of the two nucleon interaction (see
discussion in Section III.B.1).

3H

 

ELAB(MeV)

 

 

 

 

 

 

I u I 8
-0
V‘
_<3
«I
_O
«I
b —
I- _9
1 1 1 1 1
I) Q- «I «I -— cf)
( J) ” +"‘e:oa 1,09 In 4+1)
" (IN . a a 2
Figures . Energy dependence of both the Hamada-Johnston 3P2
phase shift, and the difference between this phaseopec
6 0

shift and the zone—pion-exchange 3P2 phase shift, 12,

35

 

.Amqv nomnnmmmlcoqu 6cm “Away CowmanOh
Imnmemm .A<mv ucsn<s mmhm mo mHmUoE comaosclozp map pom nuanm mmmnm

mmm mwcmnomeCOfim
Imco map cam phflnm mmmsa mm map cmmzpmn moCmAmMMfiu mg» no mocmvcmamc mwhmsm

.m mhzwfim
A>m2v 044w

on.
_ _

on ov 0m 0. o
_ e _ q _ i _

_ o

 

 

F..—
L)
UJ
at
C)

l
QN
[O
J
N

 

J
V

a

1

CD
(c.dl[zz‘7’7+lmi’e-§’9) m #5:) jawzau)

 

 

36

2

2 2 k .

2 c _ Opec
) k (CO k cot (612 6

12

 

h(n) l
)=-———+
R a12

Having two parameters describing each 6ij’ we must now

1
’+ 7 r12

(27) (l + n

recognize that the phase shift Klj defined in equation (17) is the
phase shift actually used in the nuclear amplitude. In order to

find Klj from Gij’ we must calculate (see Appendix 3.3) the phase

shifts Alj given by equation (9) and use the relation

The Klj phase shifts enter the triplet nuclear amplitude in the

form
Zial 2iK

lj
21k e (e

-1) cos 6

and the triplet v.p. amplitude must then enter the calculation

11
as

figmmgferized - Egfi-(eZlTl-l) cos 6 .

As a final check of the freedom of our parameterization,
we tested its compatability with the Livermore energy dependent
parameterizationl by fitting their energy dependent 3P phase shifts
from 1 to 25 MeV. We found that we could reproduce their phase

shifts with the following parameters in units of F: alo = -4.54,

= 5.53. = 2.68, r = -8.57, = -.392, and r = 9.54.

r10 a11 11 a12 12

2. Central, Spin-orbit, and Tensor P-Wave Parameters

At energies below 20 MeV, the more useful P-wave parameters,

which are linear combinations of the 3P phase shifts, are commonly

37

E

referred to as the central parameter AB LS’

C’ the spin-orbit parameter A

and the tensor parameter A5, where

E _ 1 E c E
AC - 2 (610 + 3611 + 5612)
E_1_E_E E
(28) ALS— I5‘ 2510 3511 + 5512)
E _ 5 _ E E _ E
AT ' 75‘ 2610 + 3611 512)

It has been shown44 that if the Born approximation
holds separately for each component of the nuclear P-wave potential
and if only central, spin-orbit, and tensor forces were present,
then the A? are given by
A? = - 2%; J” Vi(r) S
o
for i = C, LS, and T, where k is the c.m. momentum of either

2

l (k,r) dr

particle, Sl(k,r) is the regular electric wave function for L = l,

and the total nuclear potential is the sum? of V L-§_V and

C"- LS'

S V

12 T'

If all phase shifts for L > 1 are zero and the Coulomb

and v.p. interactions are present, then the differential cross-

section, app’ for small P—wave phase shifts, may be written as

(29) k? a = k2 a (nuc)
PP PP

2 . 2
+ k 3 int + k Coul + v. .
p( ) opp( p )

P

 

T The P-wave tensor matrix elements 812 are -4, 2, and -2/5 and

the Spin orbit matrix elements E-S are -2, -4, and l for J = 0, 1,

and 2 respectively.

38

E
2 A
2 _ . 2 E E 36 LS 2
where k :3pp(nuc) — Sln 600 + 18 AT [jg-+(ZE—) ]
T
2 2 12 “is 2
2 E E
+ 9 C08 6 [3AC + ZAT ( 33 “ (ZE—) )] ,
. . E T
21(3 + T ) 216
2 . l o o oo 5*
k int)= — Im e e -l f
5pp( 2 I ( ) 1
2i(3 + T ) *
+ 9 cos 9 Re [e l 1 ft ] A2 I
2 l s 2 3 t 2
and k {Jpp (Coul + v.p.) = I If I + Z If | ,
s _ _ _
where f — fc(e) + fc(n e) + fvp(e) + fvp(n 6)
ft:

fc(8) - fc(n-6) + fvp(6) - fvp(n-6)

and the Coulomb and v.p. amplitudes f

and f are described in
C Vp

Section II.D.

For energies less than 10 MeV, the dominant contribu-

tions to opp(nuc) comes from 650 while the only P-wave contribu-

tion to 0 (int) is due to AE Thus, since MES/Ag)2 is generally

pp 0'
less than 0.3 (see Figure 15) the low energy differential cross-

. O O O E
sections are insenSitive to A

LS’ slightly sensitive to AE, and most

sensitive to A5.

IV. LOW ENERGY DATA ANALYSES

In a statistical analysis of a set of experimental data,
the chi-square ratio, xi, will be near the value of 1 if the three
conditions discussed in Section II.C hold, i.e., if (1) the experi—
mental measurements exhibit a Gausian (normal) distribution about
the true values, if (2) the eXperimental errors are good estimates
of the true standard deviations, and if (3) the parameterization
is adequate, i.e., the model used to describe the data is not
limited by its form in doing so.

We have found in both the single energy and multi-

energy analysis that X: is abnormally low for most of the data
below 30 MeV. There is no question of form restriction in a single-
energy analysis and an inadequate parameterization in a multi-
energy analysis would produce a high rather than low value of x2.
We therefore feel that the responsibility of the low x2 lies in an
over-estimation of experimental errors. This will be particularly
apparent in our discussion of the Wisconsin45 data, and is true to
some degree for most of the other data in our analysis.

This presents a problem to the analyst, who must rely
on statistical arguments in making judgements as to the compatability
of data sets with one another and in determining standard deviations
of parameters found in fitting the data. We will be interested in
determining the standard deviations of such quantities as phase shifts
and effective range parameters. The procedure for doing so is well

known if x: a l, but problems arise if the data errors are over-

39

 

 
 

40

estimated, producing x: << 1 as in the present case. We follow

33

the procedures described by Cziffra and Moravcsik who assume

th

that the true standard deviation” oi, of the i datum is related

to its eXperimental estimate, 5i, in approximately the same manner
for all data in a given data set, i.e., the ratio gi/gi is approxi-
mately constant for all the data. This is a rather questionable
assertion, but we feel the simplest and most reasonable that can

be made with the information available about the experimental
errors. We therefore proceed with a word of caution to the reader
that all reported standard deviations of phenomenological parameters
would have a stronger foundation if the experimental errors had

been reported more fully.

A. Single Energy Analyses of the Data Below 10 MeV

There are currently four sets of data below 10 MeV. The
380 keV LASL46 data has been analyzed by Heller and Gursky47, and
our analysis adds nothing to their conclusions. This data is at
such a low energy as to be not at all sensitive to the 3P phase
shifts: it is fully describable by two S wave parameters.

45

The 1-3 MeV Wisconsin data, together with the LASL data,

has been the primary source for information on the pp scattering
length and effective range6. There are two major problems with
these data. 'First, the experimentalist encountered difficulties48

in treating different sources of errors in this data set. Systematic

41

errors which were angle-dependent were combined with random
errors. The proper treatment would be to report the angular-
dependance of these systematic errors and not combine them with
the random errors, allowing the analyst to treat the systematic
errors prOperly as angle-dependent normalization errors. Secondly,
and perhaps not independently, the chi-square ratio for this data
is unusually low. The single energy analyses of these data yield*
a x: of .37, .08, .21, and .39 for the four energies 1.397, 1.855,
2.425, and 3.037 MeV. The only probable explanation for this is
that some or all of the errors were grossly overestimated. A
possible solution to both of these problems is that the angle-
dependent systematic errors were overestimated. Since these

represent the principal source of error45’48

in the experiment,
separating them from the relative errors could easily raise the
x2 contribution at each angle to a more reasonable value. The
overestimation of error would then be centered in a single normal-
ization datum at each energy, and would be of negligible effect in
the analysis.

The presence of other systematic errors in the Wisconsin

data was pointed out by Noyes50

who found that the vacuum polar-
ization strength, when treated as a free parameter in single
energy analyses of the Wisconsin data, systematically decreases

with energy. We also find this to be true. If we assume

 

*
On the basis of Chauvenet's criterion49, the 90° datum was removed

from the 1.855 MeV data in all our analyses.

42

that ALS = 0 and AT = Agpec’ and search on the 150 phase shifts
and the E phase parameter along with the strength factor of

C
the vacuum polarization, AVp' we find for the Wisconsin energies

1.397, 1.855, 2.425, and 3.037 MeV that XVp = 1.34 i .08,
1.07 t .07, 1.04 i .12, and 0.63 i .15 respectively. Whether
this is related to the problems we discuss above is unknown.

The Berkeley data51 was reported as two sets of data
at each of the three energies 6.141, 8.097, and 9.918 MeV. These
were referred to as BGS and D data, the difference being in the
treatment of the subtraction of background events. The Berkeley
experimentalists believed the D data to be more consistent with
other data sets, but the BGS data to be the more accurate. We
will use the BGS data in our analyses, having found the D data
to be less compatible with other data sets (this will be shown
in our discussion of multi-energy analyses). In addition, one
of the Wisconsin authors48 has expressed the Opinion that the
BGS data should, in principle, be more consistent with the
Wisconsin data than the D data.

52

An analysis of the Minnesota data at 9.68 MeV in

combination with the Saclay53 data at 11.4 MeV has been reported

by Noyes and Lipinsky54.

As discussed in Section III.B.2, the data below 10 MeV
E
T!
In order to perform single energy analyses

E
T

are rather insensitive to AES/Ag’ slightly sensitive to A and

very sensitive to AE.

on this data we have used the ranges of IAES/Ag' and A

43

indicated by two of our multi-energy analyses using (1) a

data set containing only higher energy data and (2) our final

data set containing all data below 27.6 MeV except for the

3.037 MeV Wisconsin and 9.918 MeV Berkeley data (see Figures 13, l4:

and 15). The results are shown in Table I. Note that the only

 

phase parameter unambiguously determined throughout this energy 'r*-
range is Ag. This is because 180 and A3 are tightly coupled in
cpp(nuc) of equation (29).

The single energy A: values are shown in Figure 7. We
see that the 3.037 MeV Ag appears to be incompatible with all the ng
other Ag, and that the 9.918 MeV A2 is incompatible with the
9.68 MeV Ag.

B. Multi-Energy Analyses

 

We perform two types of multi-energy analyses. They

l

differ in the energy parameterization of the S0 phase shift

and the range of energies analyzed. In one case, we use the four-
parameter effective range function of equation (23), while in the
second case we use a 400 MeV finite core potential of the form

discussed in Section III.A, i.e.,

-2x -4x -(X/xc)8
(30) V = (V + a e /x + b e /x) (1 - e )
Opep 8

-(X/xc)

+ c e + VE

‘Mhere a, b, and xc are free parameters and V the electric

El

jpotential, is the sum of the Coulomb and vacuum polarization

44

 

 

Table I. Comparison of phase shifts in-single—energy analysesa
mine E E E E 2
of 1 A -A IA /A | x
ELab(MeV) Data So C T 4 LS T
1.397 110 39.3169¢.0090 -.004li.0034 .05 .0 3.302
39.3168:.0090 -.0041¢.0034 .05 .4 3.301
39.3051i.0091 -.0033i.0034 .15 .4 3.309
39.3062i.0091 -.0033i.0034 .15 .0 3.315
1.855 120 44.3481i.0044 -.0035i.0016 .08 .0 .794
44.3478i.0044 -.0035i.0016 .08 .4 .796
44.3174i00044 -00021i00017 .25 .4 0781
2.425 140 48.3648i.0082 -.0071i.0031 .15 .0 2.691
48.3639i.0083 -.0073i.0031 .15, .4 2.704
48.3283i.0082 -.0064i.0031 .30 .4 2.671
48.3322:.0081 -.0059i.0031 .30 .0 2.626
3.037 130 51.038i.015 .0053i.0052 .2 .0 4.240
50.973i.015 .0056i.0051 .4 .4 4.215
50.979i.014 .00651.0051 .4 .0 4.128
6.141 170 55.52.91.050 -.095:.017’ .4 .o 6.176
55.533i.050 -.096t.017 .4 .4 6.206
55.267i.051 -.104i.018 .8 .4 6.290
55.292i.050 -.1001.017 .8 .0 6.166
8.097 160' 55.694i.082 4.076i.024 .5 .0 10.14
55.685i0082 “.077i.024 05 04 10014
55.403t0082 ‘0086-‘150024 09 00 10915

45
Table I (continued)

 

 

N°ITYPe 1 E E E E 2
of s A -A |A /A | x
ELab (MeV) Data 0 C T LS T

9.918 170 54.777:.090 -.l46i.028 .8 .0 11.02‘
54.756t.089 -.l48:.028 .8 .4 10.91
54.321:.088 -.164:.027 1.2 .4 10.71
54.366:.088 -.158¢.028 1.2 .0 10.89
9.68 27. 55.69 ..12 .022:.020 .8 ’.0 11.20
55.68 1.12 .018:.020 .8 .4 11.17
55.24 4.12 -.001..020 1.2 .4 11.18
55.28 1.12 .007:.020 1.2 .0 11.17

 

E

aThe range of values of AT

by the multi-energy analyses of Figures 14 and 15.

and phase parameters are given in degrees.

and AES/Ag are taken from those indicated

The phase shifts

 

.116

 

 

 

 

 

 

 

I I I I I I I I I I I
1- , -
1
A
55
lu
°‘ 0 1 1 1 1 1 1 1 1 1 1
8 4L 1
o
\p
mo
Q
-.11. _
,_ 2 .1 9'1 1, 1 1 1 1 1 1 11 1
'o 5 I0
ELAB(MEV)

Figure 7. Central P-wave parameters from the single-energy
analyses of Table I. (A) = Wisconsin data, (0) = Berkeley
data, (0) = Minnesota data.

47

potentials. The parameter c is fixed at 400 MeV, and a and b

have the dimensions MeV. The effective range parameterization

 

is here used in detecting inconsistencies in the data below 10

MeV, while the potential parameterization is used in analyses of

 

data over the entire range 0-27.6 MeV. The parameterization of
the 3P phase shifts is that described in Section III.B.l. The
1D2 and 3F2 phase shifts are put at their OPEC values while the

£2 coupling parameter is represented by €3pec, (1-.00814 ELab)’
with ELab in units of MeV. This fits the single-energy £2 at

50 MeV and approaches the OPEC value at lower energies. The
effect of all higher phase shifts is represented by OPEC ampli-
tude contributions.

In order to search on the effective range S- and P-wave
parameters, we must be able to specify the P-waves more completely
than the data below 10 MeV alone are capable of doing. As our
first analysis, we therefore use the potential parameterization

for the 1S phase shift to fit the 380 keV LASL data plus all

0
data between 10 and 27.6 MeV in order to get the 3P phase shifts
at 25 MeV. The results are shown in Table II along with the
Livermorel’3 phase shifts from single-energy and multi-energy

analyses.

1. Data Below 11.5 MeV
Various combinations of data below 11.5 MeV are analyzed
and the results shown in Table III. All eleven analyses contain

the LASL 380 keV data, the 11.4 MeV Saclay Axx/Ayy datum, and the

48

Table II. 25 MeV Phase Shifts

 

 

Our Livermore (MAW-VII) Livermore (MAW-X)
Multi-Energy Single-Energy Multi-Energy
Analysesa’d Analysesb'd Analysesc’d
150 48.60 . .16 48.61 . .26 48.81 t .07
3P0 8.40 1 .44 8.53 . .45 8.21 . .11
391 -4.88 . .12 -5.01 . .21 -5.08 t .03
392 2.41 . .09 2.43 . .16 2.59 t .04

 

a 380 keV LASL data and data from 11—28 MeV
b Data from 23.4-28.2 MeV
c Data from 1-400 MeV

d The Saclay data used in the Livermore analyses were treated
with absolute normalization furnished by H. P. Noyes, Bull.
A.P.S. 11, 845 (1966). They should all refer to a common

normalization factor as was done in all our analyses.

 

25 MeV 3P phase shifts discussed above. There are ten free
parameters in each run - four S wave effective range parameters
and two effective range parameters for each of the three 3P
phase shifts as discussed in Section III.B.l.

In examining Table III, we see from analyses 1-7
that the 3.037 MeV data is clearly incompatible with any and
all of the rest of the data above 2 MeV. We also see from
analyses 9-11 that the 9.918 MeV and 9.69 MeV data are completely

incompatible, while from analyses 8 and 10 it is apparent that

the 6.141 MeV and 9.69 MeV data show some incompatibility. These

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49

 

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50

conclusions fully agree with the inconsistencies exhibited

among the AE phase parameters of Figure 7.

C
The values of 1S effective range parameters

0
found in some of these analyses are of interest and are also
given in Table III. The effective range function for each of
these is plotted in Figure 8. In the multi-energy analyses to
be discussed in the next section in which a potential parameteri-
zation is used for the 1SO phase shift, we find that we can
duplicate the fits in all analyses Of Table III except for those
containing the 9.918 MeV data set. In the latter, the potential
parameterization always gives a much higher value of x2 for the
Berkeley data than the effective range parameterization. The
reason for this is apparently the large values of the 180 effec-
tive range parameters P and Q in analysis 11 of Table III re-

quired by the 9.918 MeV data which the potential model can not

duplicate. If a potential model were found that did dUplicate

the low energy behavior shown by curve 11 in Figure 8, it surely

would not fit the 25 MeV datum, the triangle in Figure 8.

2. Data from 0.3 - 27.6 MeV
Our multi-energy analyses including the data up to
27.6 MeV uses the three-parameter finite-core potential of
equation (30) for the 1S0 phase shift. The results are shown
in Table IV. Comparing analyses 3 and 4 with 5 and 6, we see
that, as mentioned earlier, the Berkeley D data is less compati-

ble with the other data than is the BGS data. We therefore use

51

 

ALL BERKELEY DATA (ll)

9.9I8 M.v BERKELEY DATN

.6 _ REPLACED av 9.69 MeV
- MINNESOTA DATA (10)

(7)
.5 '- / §

.4- ' \ 4

NO BERKELEY DATA (6)

— .1 + 1 r.-k‘-Pr31‘+0r,’k‘(F'€7

 

 

 

/
.3- -
' /

.21— -
| L C
o J. 1 1 1 14

0 IO 20' 3O
ELAB(MCV)

Figure 8. The 180 effective range function from multi-

energy analyses 6, 7, 10, and 11 of Table III. All analyses
contain the three Wisconsin energies 1.397, 1.855, and 2.425
MeV. Analysis 7 contains no Berkeley data but includes the
3.037 MeV Wisconsin data. The dashed straight line is from
the first two terms of the effective range function. The
triangle is the single-energy analysis value at 25 MeV.

52

the BGS data in the rest of the analyses. Analysis 1 shows
the incompatibilities discussed in the previous section.
Analyses 2-4 show that the Minnesota data is to be favored
over the 9.918 MeV Berkeley data.

We therefore reject the 3.037 MeV Wisconsin data
and the 9.918 MeV Berkeley data. The results of our final multi-
energy analysis are given in Table V which shows the final data set.
Comparisons between the experimental and best fit predicted values
at 3.037 and 9.918 MeV are shown in Figure 9.

The 3P central, tensor, and spin-orbit parameters, AE,
from two multi-energy analyses, together with single-energy
analysis values and values from the Hamada-Johnston, Lomon-
Feshbach, and Bryan-Arndt models are shown in Figure 10
through 15. The multi-energy analyses shown are (1) from our

final data set and (2) from a data set excluding all Wisconsin

 

and Berkeley data. In addition, points below 10 MeV from a
third analysis containing no Berkeley data are shown in
Figures 14 and 15. These strongly suggest an incompatibility
between the Berkeley data and the Wisconsin and Minnesota
data. The values of the search parameters from these three
analyses are given in Table VII.

The 25.6 MeV and 50 MeV single-energy analysis values

of Ag shown in Figure 12 have the values 0.648 1 .0270 and
1.904 t .0460 respectively. These errors range from 3 to 10

3

times smaller than the errors of the P phase shifts(see Table II

 

53

and reference 1). The central P-wave parameter Ag therefore
reflects much more clearly the ability of a model to fit the
scattering data than does any one of the 3P phase shifts. Any
future potential model should, we suggest, be fitted in this
energy range to the Ai parameters in the 3P state rather than
to the 3P phase shifts themselves.

We see that the questionable compatabilities of the

 

6 and 8 MeV Berkeley data with both the lower and higher energy
data, as well as the over-estimation of errors in the Wisonsin*
data, does not permit us to confidently arive at final values

3

for the 1S and P effective range parameters. This is unfortun-

0
ate because these parameters offer a means of comparing the low
energy limits of theoretical models of the two nucleon interac-
tion. These parameters also offer simple phenomenological
descriptions of the interactions in these states. We therefore
conclude that it would be most advisable to remeasure all of

the data below 10 MeV to at least the accuracy reported by the

presently available experiments.

*If the over-estimation of errors in the Wisconsin data should
be primarily in the angle-dependent systematic errors, it may
be possible to analyze this data in a better manner. This

question is presently under investigation.

511

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55

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56

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57

 

Table VI. References for Table V.
Berkeley (1968)A BGS data of reference 51
(1968)B D data of reference 51
Los Alamos (1960) See references 46 and 47
(1967) N. Jarmie, J. E. Brolley, H. Kruse,
H. C. Bryant, and R. Smythe, Phys.
Rev. 155, 1438 (1967)
Minnesota (1959)A See reference 52
(1959)B See reference 52
(1960) J. H. Jeong, L. H. Johnston,
D. E. Young, and C. N. Waddell,
Phys. Rev. 118, 1080 (1960)
Princeton (1954) J. L. Yntema and M. G. White,
Phys. Rev. 95, 1226 (1954)
(1959) W. A. Blanpied, Phys. Rev. 116,
738 (1959)
Rutherford (1964) C. J. Batty, G. H. Stafford, and
R. S. Gilmore, Nuc. Phys. 51, 225
(1964)
(1965) A. Ashmore, B. W. Davies, M. Desire,
S. J. Hoey, J. Litt, and M. E. Shepherd,
Nuc. Phys. 13, 256 (1965)
Saclay (1966) See reference 53
(1968) P. Catillon, J. Sura, and A. Tarrats,t

Phys. Rev. Letters 29, 602 (1968)

 

58

Table VII. Search parameters for 1SO and 3P phase shifts

 

 

 

Phase Phase Shift Analysesb’c
Shift Parametersa l 2 3
a (MeV) ~14.5*1.3 -12.811.2 -7.6t2.7

lS0 b (MeV) —1007.7t7.9 -1017.4t6.9 -1049.tl6.

XC .78059t.00031 —.78052 (fixed).78071 (fixed)
a (F) -15.7*5.0 -7.l*3.l -2.6*2.0

3P 10

0 rlo(F) 6.04*.20 5.611.44 4.312.0
a (F) 4.42t.95 2.721.86 2.8t1.3

3P ll

1 rll(F) -9.49*.38 -8.69t.77 —9.0tl.0
a (F) -.187t.038 -.227i.045 -.45t.28

3P 12

2 r12(F) -5.2t6.6 .915.6 15.t10.
x: .57 .47 .54

a The 1S0 parameters are defined in equation (30), while the 3P
parameters are defined in equations (26) and (27).

b Analysis 1 uses the final data set described in this section,
analysis 2 uses a data set containing no Berkeley data, and
analysis 3 uses a data set containing neither Berkeley nor
Wisconsin data. The x: for each analysis is presented.

c

The parameter values and standard deviations change negligibly
‘when the xc parameter is free as in analysis 1 or fixed as in

analyses 2 and 3.

59

 

 

 

 

 

 

 

 

. .- I'H' I I I I I I I I I
T T 3.037 MOV
.4 '- T ..
‘T
2 .. I; 7 _
fl
1 I T
4 ‘5 1
O V l I
1 l

 

 

 

 

-.21- _
‘2
c 1 1 1 1 1 1 1 1 1 1
0
c:
s
g.
Q.
\- 6 I I r* I r* I I I I I
9.9I8 Mov 1
4- ‘—

 

 

«I»
d-
«1-
GP
~1-

DEPA RTURE 0F MEA sun ED VA LUE FROM CA L CULA TED moss SECTION

 

 

1 1 1 1 1 1 1 1

1

O 20 4O . 60 80 I00
CENTER-OF-MASS SCATTERING ANGLE (Degrees)

Figure 9. Departure of the Wisconsin 3.037 MeV and the

Berkeley 9.918 MeV differential cross-sections from the
preferred phase shift solution using the data set of Table V.

 

60

 

 

.1

3;.

“U

ha

0:

E3 _

C3

\-

“Mb

‘3 _
-.| A

 

l ' l I l

 

 

 

'0 5' '0 I5 20 25
ELA3(MOV)

Figure 10. Central P-wave parameters from 0-25 MeV from
single- and multi-energy analyses. The single-energy analysis
values shown are (A) = Wisconsin data, (C) 8 Berkeley data,
(D) = Minnesota data, (A) = 25 MeV data. The lined region
represents the error corridor from the analysis using the data
set of Table V. The cross-hatched region represents the error

corridor from the analysis which excludes all Wisconsin and
Berkeley data.

 

61

 

 

.‘

' Q'o’o'o '

Q

0 O
3.0

     

O
.0
9'.

o

O

 

...,

   

I O

o.

on

o/owéououo
.000... . .

    

   

    

 

                     

db

1
IO

 

 

.061-

.04-

.021-
O
-u02-

ammmmeme ms

-.04 -

-J06*-

-.08 )-

-.|O*-

-.|20

The

The lined region repre-

(A) = Wisconsin data, (o)=

(D) = Minnesota data.
sents the error corridor from the analysis using the data set

of Table V.

ELAB (MOV)
Central P—wave parameters from 0-10 MeV (an en—

larged version of the low-energy region of Figure 10).
The cross-hatched region represents the error

corridor from the analysis which excludes all Wisconsin and

single-energy analysis values are from those data sets used
Berkeley data.

in the final data set of Table V.

Berkeley data,

Figure 11.

62

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323 23w
00 cc oc on on mm, ON 0. o. n
. d _ d fl _ _ _ q d
i. .. x i .1 x x x to

 

I q.
V.

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1/
nu
.1
w

1N.. N“
c.
[I

4o;

 

 

I ooN

 

 

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Hmcm
zmhmcmlfipase u on .> magma mo pom spmc map mcfins mammamcm zmhmcmnfipass u ADV .zm museum
lamp Mo mammamcm mmumcolmawcfim u :uv .mfimmamcm ammocmlmawsfim ago I A<V .m opswfim on coHpQMo
mcp ca omoapommc mamcoe cam .wmnaamcm Scam >mz omlo 80pm mammempmo m>m31m somcoe .ma mpzwfim

 

 

 

_ 32,: 93m . I
on . O¢ On . ON 0. O
a . . a . . o o
o\
\\O
0"
l I.
. x
\
\
\
\
ox _
3
6 \\ W
I \\. .IN 1/
AV . nu
. A\\\ .1
a. \ 9
\ w...
.x\. .J
21 \\ ma.
\\\
l: \\
(a \\
\X\
\\\ Ilka—
\\\
‘\\\\
fl. . _. . . .it

 

64

 

 

 

 

 

 

 

 

C)
I I I K)
)— 1 —4
<1
_ _,CD
I m V
I
_ I .a
I
,' 3‘
I E
)— :. —I m
1 3
.. / Lu
\/' O
b u_ I 2: -‘0I
..1 I .3
I It
_ '/ ..
I
_ I ..o
' ..
"! o
.. ’I a
- -i. o
gl
1 l I“-"' 1 l
0. In. 0. In. 0
N ... .—

aodo lV/ 17

Figure 14. Ratio of the tensor P-wave parameters and the one-
pion-exchange (OPEC) tensor P-wave parameter from 0- 50 MeV from
analyses, and 2models described in the caption to Figure 6.

f = Co (1 + n 2), which simulates the Coulomb effect in the OPEC
parameter. (0) = multi-energy analysis using the data set of
Table V, (A) = multi-energy analysis excluding all Berkeley data,
(0) 8 multi-energy analysis excluding all Wisconsin and Berkeley
data. Error bars not shown are smaller than dots.

65

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lavage u on .mpmc mmamxpom Ham wcHBSHoxm mammamcm mmnwchHpHsE u A<v .> magma no new some
on» wcfims mammamcm awhmchHpHSE u AOV .m opswam on COHpQMo map CH confipommc mfimooe com
.nmmzfimcm Song >62 omIo Edam mmmmednmo m>m31m pomcmp on cannoncflom mo capmm .ma mpzwam

$2); 93m

00 0? On ON _ O. a
q _ . . _ A, . . _ _ . ¢..

, , r..-
. , «mm WW

"II

 

 

 

 

 

 

 

 

 

 

V. ELECTROMAGNETIC CORRECTIONS TO THE LOWNENERGY
NUCLEON-NUCLEON INTERACTION AND IMPLICATIONS

OF CHARGE-SYMMETRY
A. Vacuum Polarization Interaction

The v. p. interaction does not measurably effect

3

the data above 11.5 MeV. Using the P phase shifts at 25 MeV

and the data below 11.5 MeV, but excluding the 3.037 and 9.918

MeV data, we use the effective range parameterization of the

1

S state in an eleven parameter fit, one parameter being the

0
v. p. strength factor, AVp' a multiple of all V. p. amplitudes
and v. p. phase shifts present in the analysis. The result

is Avp = 1.053 t .097.

B. Magnetic Amplitudes

We now investigate the questions raised in Section II.D
concerning the effects of the magnetic interaction terms of

equation (20) upon the analysis of the low energy data. If the

DWBA
LS '

Q .
added to the total amplitudes', the fit to the data remains un-

symmetrized spin-orbit distorted-wave-Born amplitude, f is

changed, i.e., the change in x2 is miniscule. For example, the

DWBA

difference in differential cross-section with or without fLs

present is 51 part in 50,000 for E s 25 MeV. The effect in

Lab
polarization measurements is about 20% at 20 MeV, but present

data do not approach this accuracy. The effect of ngBA appears

66

67

to increase with increasing energy and decreasing angle, and
this amplitude may have some measurable effect at higher energies.

Although we did not calculate the tensor distorted-wave-Born amp-

litude, fgWBA, there is no reason to expect it to have a sig-
nificantly larger effect than does fggBA (see earlier discussion

in Section II.D.).

C. Implications of Charge-Symmetry

Due to the lack of intense neutron beams, the measure-
ment of the nn scattering length has depended upon the use of

many-body interactions. These methods have been reviewed by

55 56

Slaus , and discussed by Van Oers in whose Opinion the most

reliable eXperimental value for the nn scattering length is

53

at present based on the study of the reaction D(w-, y)2n, which

gives the result
ann = -18.43il.53 F.

The quoted error includes a theoretical uncertainty of 1 1.0 F.

1. Point Nucleons

Using the potential parameters of analysis 1 in

Table VII and removing the v. p. potential, we calculate the pp

Coulomb scattering length of equation (24) to be

aC = -7.814 F.

PP

This compares with the pp electric scattering length of equation (25)

68

from analysis 10 of Table III,

aE = -7.8205 i .0039 F.

PP

If the Coulomb potential is now removed and the proton
rest mass replaced by the neutron rest mass, the resulting nn

scattering length, if charge-symmetry holds, is

a = -l7.37 F.
nn

2. Finite Structure Effects

If the potential terms V3;

and V229 of equations (C-1)
and (C-2) due to the charge-charge, charge-moment, and moment-
moment interaction of two protons with finite charge and magnetic
moment distributions are combined with the v.p. potential and
the phenomenological finite core strong-interaction terms of
equation (30), a fit to our final data set produces the identi-
cal pp scattering length agp described above, but with new
potential parameters having the values a = —l3.2il.l MeV,
b = -1014.7i6.7 MeV, and xC = .78055 i .0037 (these should be
compared to the values of analysis 1 of Table VII).

Combining this strong interaction potential with the

potential Vflig of equation (C-3) and allowing for the neutron-

proton mass difference produces a nn scattering length

ann = ~16.93 F.

Comparing this with the result of the previous section, we see

 

69

that the effect of finite structure of the nucleons is to

change the nn scattering length by +.44 F.

This result is quite different from that of Schneider

and Thaler26

who calculated a change in arm of -.02 F. Their
calculation differs from ours in three respects. First, they
used the now outdated form factors of deVries, Hofstadter, and
Herman58 which give a non-zero electric potential for the nn
interaction. Secondly, for the pp interaction they use the
moment-moment magnetic term but neither of the charge-moment
terms of equation (C-2). Third, they use a two parameter hard
core Yukawa potential, with no one-pion tail and a depth of
about 400 MeV at a core radius of 0.388 F. They adjust the
two parameters to fit a pp scattering length and effective
range, and they find nn scattering lengths of -l6.46 F and
-l6.44 F with and without the finite structure potentials
present.

As discussed in Section III.A., our finite core,
hard~core, and soft core potentials give very similar nn
scattering lengths when no electromagnetic interaction other
than the Coulomb potential is present. This is rather surpris-
ing considering the very different nature of the potentials
in the region 0.5 F to 0.8 F (see Figure 3). Since the finite
structure potentials are no larger than the Coulomb potential

throughout this region, we believe there would be little dif-

ference in the nn scattering lengths produced by the hard core

70

and soft core potentials with finite structure effects con-

sidered. However, this point remains for future investigation.

10.
ll.
12.
13.
14.
15.
16.
17.

18.

19.

BIBLIOGRAPHY

M. H. MacGregor, R. A. Arndt, and R. M. Wright, University
of California Radiation Laboratory Report UCRL-70075

(Part X).

R. J. Slobodrian, Phys. Rev. Letters gi, 438 (1968).

M. H. MacGregor, R. A. Arndt, and R. M. Wright, Phys. Rev.
169, 1128 (1968).

H. P. Noyes, Phys. Rev. Letters 19, 171 (1964).

L. Heller, P. Signell, and N. R. Yoder, Phys. Rev.
Letters 99, 577(1964).

L. Heller, Rev. Mod. Phys. 99, 584 (1967).

R. P. Feynman, Quantum Electrodynamics, W. A. Benjamin
(New York), 1961.

 

E. A. Uehling, Phys. Rev. 99, 55 (1935).

S. Triebwasser, E. Dayhoff, and W. E. Lamb, Phys. Rev. 99,
98 (1953).

L. Durand III, Phys. Rev. 999, 1597 (1957).

L. Heller, Phys. Rev. 999, 627 (1960).

W. A. Barker and F. N. Glover, Phys. Rev. 99, 317 (1955).
G. Breit, Phys. Rev. 99, 1581 (1955).

A. Garren, Phys. Rev. £99, 419 (1956).

G Breit and H. M. Ruppel, Phys. Rev. 991, 2123 (1962).

E. E. Salpeter and H. A. Bethe, Phys. Rev. 99, 1232 (1951).
E. E. Salpeter, Phys. Rev. 91, 328 (1952).

J. Connell, Ph.D. Thesis, University of Washington, 1967
(unpublished).

J. D. Jackson, Classical Electrodynamics, John Wiley and Sons,
Inc. (New York), 1962. ‘

 

71

20.

21.

22.

23.

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

35.

36.

BIBLIOGRAPHY (continued)

A. S. Davydov, Quantum Mechanics, Addison-Wesley Publishing
Company, Inc. (Reading, Mass.Y, 1965, pp. 247-50.

 

J. Schwinger, Phys. Rev. 19, 135 (1950).

A. Tubis, Ph.D. Thesis, Massachusetts Institute of
Technology, 1959 (unpublished).

G. Breit, Rev. Mod. Phys. 99, 766 (1962).

D. R. Yennie, M. M. Levy, and D. G. Ravenhall, Rev. Mod.
Phys. 99, 144 (1957).

L. N. Hand, D. G. Miller, and R. Wilson, Rev. Mod. Phys.
99, 335 (1963).

R. E. Schneider and R. M. Thaler, Phys. Rev. 137, B874
(1965).

F. J. Ernst, R. G. Sachs, and K. C. Wali, Phys. Rev. 119,
1105 (1960).

R. G. Sachs, Phys. Rev. 126, 2256 (1962).

W. Albrecht, H. J. Behrend, F. W. Brasse, W. Flauger,
H. Hultschig, and K. G. Steffen, Phys. Rev. Letters 91,
1192 (1966).

M. Goitein, J. R. Dunning Jr., and R. Wilson, Phys. Rev.
Letters 99, 1018 (1967).

E. B. Hughes, T. A. Griffy, M. R. Yearian, and R. Hofstadter,
Phys. Rev. 139, B458, (1965).

P. Signell, The Nuclear Potential (a review), in Advances
in Nuclear Ph sics, M. Baranger and E. Vogt, ed., Vol. 2,
PIenum Publishing Corp. (New York), in press.

P. Cziffra and M. Moravcaik, University of California
Radiation Laboratory Report UCRL-8523 Rev.

T. Hamada and I. D. Johnston, Nucl. Phys. 94, 382 (1962).
C. B. Bressel and A. K. Kerman, Quoted in: P. C. Bhargara
and D. W. L. Sprung, Annals of Physics (New York) 99, 222
(1967).

M. Miller and P. Signell, (private communication).

72

37.

38.
39.

40.

41.

42.
43.

44.

45.

46.

47.

48.
49.

50.

51.

52.

53.

BIBLIOGRAPHY (continued)

J. E. Brolley, Los Alamos Scientific Laboratory Preprint
LA-DC-9535. The author is indebted to Dr. Brolley for
furnishing a c0py of the computer code used to calculate
the vacuum polerization amplitude.

G. Breit and M. H. Hull, Jr., Phys. Rev. 7, 1047 (1955).

P. Signell and J. Durso, Rev. Mod. Phys. 3 , 635 (1967).

J. D. Jackson and J. M. Blatt, Rev. Mod. Phys. 99, 77
(1950).

E. L. Loman and H. Feshbach, Annals of Physics (New York) 9_
94 (1968).

R. A. Bryan and R. A. Arndt, Phys. Rev. 150, 1299 (1966).

H. P. Stapp, T. J. Ypsilantis, and N. Metropolis, Phys. Rev.
105, 302 (1957).

J. L. Gammel and R. M. Thaler, Progr. Cosmic Ray Phys. 9,
99 (1960).

D. J. Knecht, P. F. Dahl, and S. Messelt, Phys. Rev. 148,
1031 (1966).

J. E. Brolley, J. P. Seagrave, and J. G. Beery, Phys. Rev.
135, 31119 (1964). The data from this experiment is analyzed
1,__

in Reference 47.

M. Gursky and L. Heller, Phys. Rev. 136, B1693 (1964).

D. J. Knecht, private communications.

L. G. Parratt, Probability and Experimental Errors in Science,
John Wiley and Sons, Inc., New York (1961), esp. p. 176. I"

The results of Noyes' analyses are unpublished, but were dis-
cussed by L. Heller (Bull. Am. Phys. Soc. 9, 154(T) (1964)).

R. J. Slobodrian, H. E. Conzett, E. Shield, and W. F. Tivol,
Phys. Rev. 174, 1122 (1968).

L. H. Johnston and D. E. Young, Phys. Rev. 116, 989 (1959)
and L. H. Johnston and Y. S. Tsai, Phys. Rev. 115, 1793 (1959).

P. Catillon, M. Chapellier, and D. Garreta, Nucl. Phys. B2,
93 (1967)

73

54.
55.
56.

57.

58.

59.

60.

61.

62.

63.

64.

65.

66.
67.
68.

BIBLIOGRAPHY (continued)

H. P. Noyes and H. M. Lipinski, Phys. Rev. 162, 884 (1967).
T. Slaus, Rev. Mod. Phys. 99, 575 (1967).

W. T. H. Van Oers, University of Manitoba, Winnipeg, Canada,
preprint.

R. P. Haddock, R. M. Salter, Jr., M. Zeller, J. B. Czirr,

and D. R. Nygreen, Phys. Rev. Letters 14, 318 (1965);

R. M. Salter, Jr., Ph. D. Thesis, University of California,
Los Angeles, 1965 (unpublished); D. R. Nygreen, Ph. D. Thesis,
University of Washington, 1968 (unpublished).

C. deVries, R. Hofstader, and R. Hermen, Phys. Rev. Letters 9,
381 (1962).

The author is indebted Dr. J. Holdeman for furnishing the
computer code used to calculate the distorted-wave-Born
approximation spin-orbit amplitude.

J. Raynal in Methods in Computational Physics, Vol. 6, B. Alder,
S. Fernbach, and M. Rotenberg, ed., Academic Press, New York
(1966).

P. M. Morse and H. Feshbach, Methods of Theoretical Ph sics,
Part II, McGraw-Hill Book Company, Inc., New York (195%)
p. 1670.

The author wishes to thank M. Gursky for furnishing computer
codes which calculate the Coulomb wave functions and the vacuum
polarization potential.

G. Breit, Phys. Rev. 99, 553 (1929).

M. E. Rose, Relativistic Electron Theory, John Wiley and Sons,
Inc., New York (1961).

S. DeBenedetti, Nuclear Interactions, John Wiley and Sons, Inc.,
New York (1964), esp. chapter 6.

Z. V. Chraplyvy, Phys. Rev. 99, 1310 (1953).

L. Heller, private communications.

P. Signell, N. R. Yoder, and J. E. Matos, Phys. Rev. 135,
B1128 (1964)

74

APPENDICES

APPENDIX A

 

Numerov Method of Obtaining a Numerical Solution

to the Schroedinger-Equation-

 

The Schroedinger equation for the 2th partial

wave may be written»

u£(r) = f£(r) u£(r)
where
_ will ai _ 2
f£(r) — 2 +$2V(r) k
with boundary condition u£(0) = 0.

Defining rm 5 nA ,

u

n u£(rn) ’

and f
n

where A is the step size, we find by Taylor series expansion

of u(riA) the difference equation

A2 A2
un+1 (l - I2 fn+1) + un-l (l - I2 fn-l)
2
= 2n (1 - A— f ) + A2 f u
n 2 n n n
with a local truncation error of (A6/720) uVI(r). Thus,

1f un-l and un_2

Since we are dealing with a second order equation,

are known, un can be found.

two boundary conditions (b.c.) must be used to uniquely
determine the wave function. One b.c. is u£(0) = 0. The
second b.c. is u£(A) = constant. This constant only

effects the normalization of the wave function. Thus,

75

76

this procedure gives the unnormalized wave function at

any radius r. An error analysis of this method is given

by Raynal60, but tends in practice to depend somewhat upon
the particular potential being used. The step size must be

small for high accuracy, but not too small or the difference
1 - %; fn will loose significant figures.

As a test of the accuracy of this method for our
particular needs, use was made of the exponential potential

V = V e-r/a ,
o

The i=0 wave function and phase shift are known analytical-

ly61. With a step size of .05 F, the wave function was

calculated out to 2500 F and found accurate to 7 significant
figures on the CDC 3600. The S-wave phase shift was found
by steping out far enough (dependent upon the value of a)

to match the exponential wave function to the solution for

zero potential, i.e., using the equation

 

F' - y F
O 0 0
(A-l) tan 6 = - , _
00 GO YOGO
I
where _ E9
YO — 110

F0 = Sin kr

and G0 = cos kr

00 to six significant figures, it was found
1

for V0 = 0.1 MeV, a step size of 0.05 F, and k=0-1-0-3 F- ,

In order to get 6

77

that one must calculate 600 at distances greater than 300,
800, and 1600 F for d = 20, 50, and 100 F respectively

(notice that the normalization of the wave function calculated
using the Numerov method is irrelevant for the calculation of

600 since only Y0 enters equation (A-l)).

 

APPFND IX B

 

Numerical Procedures and Calculations

 

1. Calculation of the Vacuum Polarization Phase Shifts.

The v.p. phase shift was defined in Section II.A.l

by equation (4) as

(4) tan 12 = -2n1

 

°° FIS
Jar 2 1

r
0

where F1 is the regular Coulomb wave function, and S2 is

the regular electric wave function defined in equation (3)

to have the asymptotic-form

(3) S£(kr) 2::Tfl F£(kr) + tan TR G£(kr)

Hellerll numerically calculated T2 by Born approxi-
mation, i.e., by replacing S2 by F2 in the integral of
equation (4). Since 1 is so small, one would expect this

to be a good approximation, but no estimate has been made of
its accuracy.

We have calculated TR exactly for i=0 and l. The
Coulomb wave functions can be calculated to an accuracy of
about five significant figures over the entire integration
range and the v.p. potential I(z)/z has been approximated by a
POlynomial in z with an accuracy of 1 part in 10,00062. We used
a Simpson's 3/8 integration routine which, for the class of

integrand encountered here, is accurate to 1 part in 2000.

The difficulty of calculating Tfl arises in the calculation of

78

 

79

S We use the Numerov method of integrating the Schroedinger

2.
equation containing the electric potential (see Appendix A).
The v.p. potential has an asymptotic form given in

equation (2) as

1/2 -2
(2) I(z) = 19%— ET (1 + 0(1/z)) , z>>1
z /2 i
where z = 2Kr and (21<)-l = 193.1 F. Although I(z) is weak,

due to the long range we must integrate out to large distances
before the asymptotic form of equation (3) is reached. One
steps r to the asymptotic region and at each step adds the
contribution to the integral for tan T2. At regular intervals,
one can normalize the integral by iteration, i.e., in Fortran

notation,

F1

unnormalized
2

TANT£= Norm * TANTR

Fl + TANT2 G£
unnormaliZed

1

 

Norm =
S

 

Norm =
S

TANT£= Norm * TANTR

etc., until there is no change in the value of Norm. When
tin: value of Norm over many intervals stabalizes, then the
asymptotic region has been reached. It was found that the

ratio

(Fg + TANT£ G1)

unnormalizedfi
R

 

Norm
S

 

80

is stable to within 4 parts in 1000 from 400 to 2000 F for
energies below 10 MeV.

The values of T2 calculated in this way agree
with the Born T£(TE) to within 1%. Because present ex-
periments determine the nuclear phase shifts with an error
larger than 10% of the v.p. phase shift, T2 is more than
sufficient for our purposes. Table VIII shows the values of

To and T1 used in our analysis. (For energies below 4.2 MeV,

To may be calculated using a formula given by Hellerll).

2. Calculation of the Electric Wave Functions SR and T .

 

 

2
The integral equations satisfied by Si and T2 are11
m” GQISR
R l r r
w F2151
+ G£(tan1£ + 2nA f dr' )
r r'
and m F£IT£
_ I
T2 — G£(l + 2n1 f dr r' )
r
00 GRITQ
_ I
F£(tanr2 + 2n1 i dr ——ET— )

Using perturbation theory, we may keep only the first order

correction terms which are obtained by replacing Si and T1

wherever they occur in the above integrals by F£ and G1

by T2, S and T may be

respectively. Also, replacing T i g

I

 

Table VIII. Numerical values of the vacuum polarization phase
shifts and the A

"T

-A

j phase shiftsa

-A

-A

 

ELabmeV) o 41 10 11 12
1.397 .0851 .0490 .0496 .0490 .0493
1.855 .0800 .0468 .0475 .0468 .0471
2.425 .0753 .0446 .0455 .0446 .0449
3.037 .0712 .0427 .0437 .0426 .0431
6.141 .0593 .0371 .0384 .0364 .0373
8.097 .0550 °0350 .0364 .0341 .0352
9.68(9) .0523 .0334 .0351 .0325 .0338
9.918 .0520 .0332 .0350 .0324 .0337

10. .0519 °0332 .0347 .0321 .0333

11.4 .0504 00321 .0340 .0313 .0326

 

aAll phase shifts are given in degrees.

is abOut 1%.

The error in the phase shifts

 

82

calculated to first order with an accuracy better than 1 part
in 50,000, the above integrals being computed in the same
manner as the integral for Tl described in the previous
section.

These electric wave functions can then be used to
find the electric phase shift 650 from a potential V = Vn + VE
by using the Numerov procedure to step out to a range where
Vn is negligibly small and matching the wave functions, due
to V, to the electric wave functions. In practice, for five
significant figures in égo,this distance is between 15-20 F-
Our calculations used S2 and TR and their derivatives at 20 F-

3. Calculation of Alj'

As discussed in Section III.B.1, we parameterize

the 3P phase shifts de but use the phase shifts Klj in the
nuclear amplitude, the two being related by Klj= éIj + Alj'
The use of
' °°21

=— T—
(9) Alj 2n1 é “zj r dr

depends upon knowing the nuclear wave function le which can
be calculated only by uSing one of the existing two-nucleon
potential models. Fortunately, the Alj are almost totally
model independent at low energies, most of the contribution

to the integral of equation (9) coming from outside the region

where all potential models deviate from OPEP. Table VIII shows

83

the values of A1. used in this analysis, calculated with
the use of the Hamada-Johnston potential. The Alj receive
less than 2% of their total value from r54 F for Elab s 5

MeV and less than 6% of their total value from r54 F for

Elab < 10 MeV.

 

APPENDIX C

 

Potentials Due to the Finite Structure of the Nucleon
1. Finite Charge Distribution
Using the results of Section II.B.2, we calculate here

the potentials due to the charge structure of two interacting

 

nucleons. v(q) is then the Fourier transform of ez/r, or
4ne2
v(g) = 2
9
while fl(q) = f2(q) = GEp for two protons
and fl(q) = f2(q) = O for two neutrons.

Using equation (17) and the Hofstader-Wilson dipole form factor

of equation (18) for GEp' we find

 

 

 

. 2
(C-1) v::(£) = 1 3 J eli’i l 2 4 4flze dq

(2“) <1 4. g \ g-

.71 BeV 2/
<-———-C 1
e2 —4 27r 3 2
_ E‘ [ 1 - e ° (1.62 r + 3.42 r + 2.94 r + 1) I

and Vii(g) = 0-

Equation (C-l) is displayed in Figure 16 together with the

2
reference curve e /r.

2. Finite Magnetic Moment Distribution
For the potentials due to the magnetic moment structure

of the nucleons, we use the contact terms v(£) of equation (15)

84

85

to find the v(q), i.e., the Fourier transforms of the three

magnetic contact terms for the Spin-singlet case

eh
Voa(£) = - 80 555 “a 5(£) :
e‘h
VOO(£) = - 411 mg no 5(£)
8 2 b 2
and VTT(£) = - —% uT gé-g_ 6(5) = 80 0T 6(5)
are
at
Voa(g) = - 8‘” ch “a r
eh
Voo(g) = - 4‘” 2mC uC) ’
and v ( ) = 8n 2
TT 9 “T

Associating the prOper form factors with each u as

described in Section II.B.Z gives the proton—proton magnetic

 

 

 

potential*
_ a9 = 1 i922
(C 2) Vgp (5) (2w)3 [e [Voa(q)Fl(g)F2(g)
2
+ Voo(g)Fl(g)
+ v (g)G2 (g)] dg
2 TT Mp .
l + u if
where Tp 4Mp
Fi‘fi) = GEp $12
1 + 2
4M
P

 

* The terms containing voa and VCD arise from the charge-

moment interaction, while the vTT term arises from the moment-

moment interaction.

 

 

and F2(q) = GEp l
l + —3§
4M
P
. e1 . uu en
The potential V , together With the components V and V
PP PP PP
defined by
(C-3) uu 1 ig-r
v = e — v ( )F ( )F ( )
pp (2“)3 J [ ca 9 1 3 2 9 r-
2
+ voa(g)Fl(g)] dg ,
and
(C-4) Ve“ = l J eigt'E v (9)G2 (9) d9 I
pp (2")3 TT Mp‘ -
4;

 

are shown in Figure 17. For the neutron-neutron interaction,

there being no charge-moment interaction,

_ ag = l ifl'E 2
(C 5) Vin (E) '73:;3—I e VTT(q) GMn (g) dq
where GMp and GMn are given by equation (19) and are to be
normalized to unity at 92 = 0 for use in the above equations.

The integrals of equations (C-2) through (C-S) are
most easily treated numerically, and the resulting potentials

are shown in Figures 16 and 17.

 

 

 

 

 

87

3

2 2
E.
is
o;

g I
In
\I
S
'5

:30
o
o.

-I

'52

I I I I I I I I J I I I I

 

 

0.5 . 1.0
RADIUS (F)

_Figure 16. Nucleon-nucleon electromagnetic potentials
derived by using the Hofstader-Wilson dipole form factor.
Shown are the proton-proton electric, magnetic, and total
electromagnetic potentials as well as the neutron-neutron
magnetic potential. The dashed line is the Coulomb point
interaction.

 

88

 

 

 

 

PRO TON-PROTON FORM FACTOR POTENTIA 1.3 (m V)

 

 

Figure 17. Proton
charge—charge (Veg
(V n) interactiog

 

RADIUS (F)

-proton electromagnetic potentials--the
), charge-moment (Ve ), and moment-moment
potentials derived y using the Hofstader-

Wi on dipole form factor.

 

APPENDIX D

 

First Order Corrections to the Coulomb Potential

 

l. The Breit Interaction.

Breit63 studied the interaction of two electrons
via the exchange of a single photon by applying classical
electrodynamics. The question asked is: what operator has
matrix elements between eigenstates of the Coulomb inter-
action that gives the correct interaction energy in second
order perturbation theory? The equation of motion for a

stationary state is

2
e -
(1)-1) (I7 - Ha - Hb - f—) I) — B111
ab
W is the total energy of the system. Ha = mafia + Ea-ga
where Ia = Re - eéext(£a) in general. ConSidering the
exchange of transverse photons,
0 '.
A(r ) = a e1E £a
_. —a _
and 15(Eb) = a E e’¥£3£b ,
where E-k = E-k' = 0. B is an approximately relativistic

current-current interaction deduced from the mutual emission
and absorption of transverse quanta between the two electrons.
To determine B, the zero-order solutions of equation (D—l)
with B=0 are used in second order perturbations treatment,
the first non-vanishing approximation. This gives an inter-

action energy

89

 

90

 

'k'or -ik-r
2 3 . <o|a el— —a |n><n|d e
(D-2) AW =‘_ e 2 Z I dkk { a1 b)
4n 1 n k' + W - W
n o
ik'- -ik-r
+ <oldbx e — —b|n><n|dax e — —a|o> }

 

k' + w - w
n O

This corresponds to the emission of a photon with wave
number k', polarization 1 by particle b and its absorption
by particle a and the same process with a and b interchanged.
The index n designates an intermediate state in which the
energy is k' + Wn'

The Breit interaction results from neglecting

64

retardation effects , i.e., assuming IWn-WO|<< k'. This

allows the completeness relation Z|n><n|= l to be used

 

n
giving
AW’= <o|B|o>
2 3 .
. e d k' 1k'°R

With B = - —— Z I e — — a a

2TT2 A k,2 a1 b1
where B = 5b — Ea’ and |o> is the eigenstate of equation (D-l)

Iaith B=0. The sum over polarization state A is

 

91

giving the final result

.3) .

Mm
5|»

B = - (9a.gb + 2Esta-95b
Thus, the complete Breit two-electron interaction Operator

2
—§_
Hab-R +B

is the correct interaction energy to first order in vz/cz.
Breit has used the method of large components to
reduce this Operator to two component form65. This procedure,
in effect, mixes positive and negative energy states, leading
to somewhat questionable results and difficulties in inter-
preting the prOperties of certain Operators in this lowest
order approximation of order v2/c2. The Foldy-Wouthuysen
method66 appears to be free of such ambiguities, and clearly
separates the positive and negative energy states for both
relativistic and non-relativistic regions.
Barker and Glover12 used the Foldy-Wouthuysen transfor-
mation to reduce the Breit Hamiltonian and arrived at a result

differing from Breits' only in the momentum dependent terms

(See Section II.B).

2. Garren's Amplitude.

Garrenl4 solved the problem of finding the Born

approximation amplitude for two Dirac particles with static

 

92

anomolous magnetic moments. He started with a Hamiltonian

of the form

. l . a
D—3 H = - A + — i O F
( ) 1eBYu u 4 u 8 uv “v
where A is the four—vector potential, F = 8 A - 8 A ,
U UV UV V11
= — =‘ a' '
o1W Yqu Yvyu , no efi/ch, and u is the nucleon s

anomolous magnetic moment, i.e., (00 + pa) 3 is the total
magnetic moment. The first term of (D-3) is the usual inter-
action between the charge and the electromagnetic field,

and the second is the Pauli term describing the interaction
of a static anomolous moment with the field. The Born
approximation transition matrix is found by calculating four

diagrams of the type

Pi Pi

pl ]92

where one or the other of the two terms in H are used at

the two vertices to emit or absorb the photon. The resultant
scattering amplitude, properly symmetrized, is then given

as matrix elements in the spin-space of the two protons.

One may ask what potential, in Born approximation,

will give the Born amplitude of Garren? The Garren amplitude
may be reduced to a four component operator in momentum space

by the method of large components, yielding a Hamiltonian

 

93

operator of the form

M = 5:5; H(g_IEfI Bi’ ga’ Eb) ga, 512)

where ga and Eb are two-component Pauli spinors for
particles a and b and q = Bf - pi. The operator, V(£), that

gives this amplitude in Born approximation, i.e.,

-iE-_r_- is-r .
M = -—— f e f V(£)e 1 dr

may then be determined by inspection. This procedure has
been followed67 and the resulting potential operator agrees
with equation (15) except for the momentum dependent terms.
We believe this difference to be due to the reduction method

used.

 

APPENDIX E

 

Search Program

 

Our analysis was performed using a computer code
develOped by P. S. Signell and N. R. Yoder, known as EDPS,
and modified by the present author. The modified version
will be referred to as EDPSM.

EDPS is basically a search routine which contains
energy dependent parameterizations of the two-nucleon
phase shifts. The program performs a least-square search
on a chosen set of these parameters in order to fit the
scattering data by minimizing the statistical quantity
x2 (see Section II.C). The common procedure used to search
for the x2 minimum was first used in phase shift analyses
by Signell et al68 and has been adopted by others.

EDPSM contains modifications which permit para-

meterization of the 1

S0 phase shift by either a potential
or by the effective range expansion of equation (23) and
parameterizations of the 3P phase shifts by either the
effective range expansions of equations (26) and (27) or

by the Ai parameters of equation (28). Vacuum polarization
phase shifts and amplitudes were added to effect all data
below 12 MeV. Also added was the ability to predict un-
certainties in quantities which are functions of combina-

tions of search parameters, e.g., the uncertainty in a phase

shift which is a function of more than one search parameter.

94

 

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