SDME ASPfiCTS OF THE 95.33%
TfiEORY OF. NUCLEAR FORCES

Thesis £0: ms bum a? M. S.
Mifii‘iiéflfl STATE COLLEGi‘
Riafimw K. Sahara
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LIBRARY
Michigan State
Universlty

 

 

 

  

 

This is to certify that the

thesis entitled

SOLE ASPECTS OF m @503 THEORY 01' NUCLEAR FORCES.

presented by

Richard Kent Osborn

has been accepted towards fulfillment

of the requirements for

M. 8. degree in Pm1¢l

 

 

Major professfi

Date Hay 26, 1919

 

 

 

 

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
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SOME ASPECTS OF THE MESON THEORY OF NUCLEAR FORCES
by

Richard K. Osborn

A Thesis

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Departmant of Physics

1949

ACKNO Warm

I wish to express my sincere appreciation to
Dr. Chihiro Kikuchi for suggesting this problem and
for his patient, stimulating guidance throughout the

conduct of this iork.

I fleas? WW

filWfifl-l.

I.
II.

III .

VII.

VIII .

X.

XII.

XIII .

TABLE OF CONTENTS

General Discussion and Purpose . . . . . . . .
The Field Concept and Its Mathematical Formalism . .
The Equations for the Vector Meson Field . . . . .
The Nucleon Equations . . . . . . . . . . .
The Derivation of the Equations of Motion . . . .
TheHamiltonian . . . . . . . . . . . . .
Reduction to S-Vector Notation . . . . . . . .
Integral Representation of the Field Quantities . .
Calculation of the Exchange Dipole Moment
ortheDeuteron . . . . . . . . . . . . .
Calculation of the Potential Function for the Nuclear
Particles in the Meson Field . . . . . . . . .
Contribution to the Electric Quadrupole Moment of the
Deuteron by Mesic Space Charge . . . . . . . .
Indication as to How the Calculation of the Exchange
Moment of the Triton Could Be Accomplished . . . .

Discussion Q o e e e o e e e o e e e 0

Bibliography 0 e e e o o e e e o e o o o

16

20

22

24

50

47

54

5'7

67

SOME ASPECTS OF THE MESON THEORY

OF NUCLEAR FORCES
I. General Discussion and Purpgse

The main purpose of this paper is to illustrate some of the

 

elementary, general principles of field theory and its application
to nuclear problems. This object will be realized by selecting

a specific field postulate, by applying the general principles to
this particular case to obtain the field equations, and by applya
ing these equations to the solution of a particular problem for
which experimental data are available.‘

The field postulate that will be selected will be that of
the charged.meson theory in which the field quantities will be
'specified by two complex conjugate 4-vectors, U and'U*. After
having obtained the equations of motion for these fields (the
electromagnetic field will, of course, also play an important part
in the following development), these equations will be applied to
the problem of the calculation of the exchange magnetic mement of
the deuteron. The equations will also be used to obtain an
expression for the potential function employed to describe the
nature of the interaction between two nuclear particles.

These last two objectives are to some extent mutually exclusive,
as the charged-meson-field-postulate does not allow, to the order

of approximation employed in this paper, for interactions between

 

I I III. IIIII'hill-Iv all valslll .1 I, ll if r I; I, flu El! ELL uni I‘ilicfi ll.- u',ll' r..h.]!!~\ EVE! .lbui‘Wb ”’5‘ 1 or; .. I

1.. o. N: I....-.l.1 .

nucleons of the same charge; whereas charged mesons must be assumed
if the phenomenon of exchange currents between nuclear particles

is to be predicted by the theory. A.more general field postulate,
the pseudcscalar field, is available to remedy this particular
defect but will not be considered in this work.

There are three reasons for the choice of the exchange
magnetic moment problem as a test case for a particular field
theory in its application to nuclear physics. 1) There is
considerable reliable experimental data on nuclear.magnetic
moments with which to compare the quantitative predictions of the
theory. 2) some of these data are apparently not explainable on
the basis of "classical" assumptions, that is, new assumptions
concerning the nature of intra-nuclear forces must be made--
perhaps even new assumptions concerning the nature of the nucleus
itself. 3) The problem.of carrying through the calculations
necessary for a quantitative prediction is a comparatively

‘ straightforward process.
II. The Field ConceE and Its Mathematical Formalism

.A field can be defined as a region in space in which at every
point there is associated, in a unique manner, one or more quanti-'
ties. The field equations are then the analytical representations
of the dependence of these quantities upon their position in
space and time. These equations are therefore the field analogues
of the classical equations of motion of particle mechanics. In

fact, as pointed out by Schiffl, the concept of wave fields as

 

employed in quantum field theory is actually a sort of generaliza-
tion-to-the-continuum of classical many-body mechanics.

As in particle mechanics, the task of setting up the equations
of motion of the system may proceed from several starting points.
For the purposes of this development, we shall start from the
basic assumption that the correct equations of motion can be
derived from.the variational principle when applied to a proper
Lagrangian

5 L dt = o. (2.1)
t' .
(For a more general discussion of field theories and their
properties, see Pauliz.)

The first problem is therefore to construct the proper
Lagrangian function to describe the system. To ensure Lorentz
invariance of the resultant equations of motion, the Lagrangian
shall consist only of quantities that are relativistically invar-
iant. Since we shall be dealing with nucleons (protons and
neutrons) and their interactions with an electromagnetic and a
meson field, the Lagrangian will, in general, consist of five
parts.

L a L' L“ L" L‘" L“r
where
LP: proton Lagrangian
L": neutron Lagrangian
L": meson Lagrangian
L“: electromagnetic field Lagrangian
Lt: interaction Lagrangian.

It should be noted here that fundamental to the field theory is the

 

-4-

postulate that there is no particle-particle interaction, only
interactions between particles and fields. The specific form of
the component parts of the above function will depend, of course,
upon the nature of the wave equation assumed for the nucleons and
the type of meson field chosen for study. The general procedure
for construction of the Lagrangian for the nucleons and the
'electromagnetic field are well known and will not be entered into
here. However, the interacting terms will first be written into
the particle Lagrangians to illustrate the manner in which the

field assumptions affect the nuclear particles.
III. The Equations for the Vector Meson Field

Yukaw'a:5 was the first to recognize that the range properties
of nuclear forces could be at least qualitatively accounted for by
the assumption that there is associated with nucleons a field

which, if described by the scalar quantity3 d), satisfies the

equation
9’ _.
CI CI) “K (I) “ O (3.1)
"‘ ~«If-4:
where [j :: X7‘2—— 2%: 35?}! 3 (i):: It I

Irk - r1] being the distance between the 1th and kth nucleons. This
may be immediately generalized in analogy with the electromagnetic
system of equations (the above corresponding to Laplace's equation
for the electrostatic field) by designating 14) - U4, the fourth,

or time, component of a 4-vectcr, and‘Ui,‘Uz, 03 as the space

-5-

components of the same vector and obtain the set of four equations
“'9’ 3-D
DU-KU8O. (3.2)

If we compare this with the Klein-Gordon equation for a free

particle .
4
s?- = p2+uzc4 (3.5)
which becomes
44262 Z O (304-)
n \I .52. r
after making the operator substitutions
$
E -> 111 2 , p->-ih grad,
at

we see that we” may identify a heavy "quantum” associated with the
U field whose rest mass is related to the range constant, K by

K we. M-Ea-
‘E‘ c

The relationship between K and the rest mass of the particle
associated with the field equation
:1 I! - K3 «I» =4 0
may be interestingly brought out in another manner.
Separate the variables and designate the separation constant

by 002 2
I-
W ::. (A39. (3.5)
we

e—I 9‘ (t) .‘2 a.
m; % __ I< zw (3.6)

_ Solve the time uation b
°q ’ ° ”mite um t
LI) (t) = H 6
thus indicating the nature of the time dependence of the solutions.

To this point we have regarded the equation as a field equation.

-6-

Now let us regard it as the equation of motion of a particle and
solve for the energy eigenvalues of this system.

E‘I’ = (f 9545

(few) 2.411;): 1:420) wt)

Choosing the negative exponediial,'we have

' E = in: me
But if 7:?- 0, the eigenvalues of the momentum operator are zero;
the particle is at rest. Since

w-o,E=Kne=Mc2

we have M = K £0

'6'—

This quantum, or meson, plays an entirely analogous role in
nuclear interactions to that of the photon in electromagnetic
quantum theory of radiation. In the former case, the nucleons
emit and absorb mesons in the same fashion that charged particles
emit and absorb photons. Hence in formulating the problem.math-
ematically it would seem reasonable to be guided to some extent
by the formalism already developed for electromagnetic radiation
theory. For instance, a charged particle in an electrostatic
field experiences potential energy due to its position in that
field given by

PT 8 e 4).
Similarly a nuclear particle in a mason field will experience
potential energy given by

PE-eU4
where g is a strength constant characteristic of the particle,

- 9
and has'the same dimensions as charge, cmab‘secl gm ‘. Since in

.7 ..

the electromagnetic case the Lagrangian for the field is known

 

to be
an F 31
L- we dt (3.7)
1+
1:
where
= 2%.. 123“
F“? ‘QX. 9X55 (3‘8)

the CI)» '3 beingthe components of the electromagnetic 4-potential,

the proper Lagrangian for the meson field alone is assumed to be
3 2
(*1
L-[IGPVI +_ KRIMFI }<lr (3.9)
2
I‘

where

as: ... as» (3.10,

However, the mesons in this theory are assumed to be
charged and hence will be acted upon by electromagnetic fields.
In general, the momentum operator, 3%,. , for a particle in the
absence of an external field transforms in the presence of a

9 ...). .2. _. £3- (PF
37? 9X? C34;

Hence the tensor, GF’ , will be transformed in the presence of

field as follows:

an electromagnetic field

2.. $3. > (,4 - 2. _. i? > u
GP, —> (9x? c.4- (PF P 9X, C72” (I)? F (5.11)
and its complex conjugate
7‘ .2 (e I :2 (e W I‘
...) + __ u __ __—_ + __ .
G5” (9)9. C“? CPP y C 9X), of 9P” j a”
Two sets of field quantities are necessary: one set to be associ-

ated with positive mesons and the other with negative mesons. These

equations were first set up by Proca4 as a device for linear-
ization of the wave equations for the electron.

Hence in this theory the complete Lagrangian for the mesons,
including their interaction with the electromagnetic field, is

assumed to be

.’ * r ..
M ., . . ’3
I. z ‘{ G’giC-f” + K2 ”PIP )cn (3.12)
It .4

(For a discussion of the mathematical formulation of the many other
types of meson theory, see Frohlich, Heitler, and Kemmer5, and
Kemmersg and for a detailed development of the theory of charged

mesons, see Bhabha7.)

IV. The Nucleon Equations

For lack of a better one, Dirac's equation will be chosen to
describe the motions of the nucleons. To put it in 4-vector‘
notation we proceed as follows: The proton equation in the

electromagnetic fi old is

E-<34> = C(x-p3-e(a-H) +(3mc1:1 _ (4.1)
:/’O 0— _. I O‘\
0‘ KIT 0) (3— (o —|)
Q o I _ o -( = I 0
V( 0) WC 0) °”* (o «I

The quantities at and p are Dirac's 4 x 4 matrices, and the I)" 's

where

are Pauli's spin matrices. Rearranging and going over to the

operator equation where

E-a>ih p—>-ihV

.9...
9t
we have

"I
r- I

dig)? .. 3.: a.) — ergo-0007 — g; H) firs-smegma,» (4.2,
4*

Multiply through by (3' and define a new set of matrices as follows:

4 ..
FEE-((3%? a"=€ b” =-L;3€-=F>

6 being +311. It is to be noted that throughout these calcula-
tions 62 is not regarded as a complex number, that is, if

‘A z a -F£;b, AI f a - E b,
.A, a, and b being real quantities. Employing the above definitions,

we obtain
# P 69 ——- r1 0 .
I I” (ma-“- 4W J“ 3% It?” ”-3)

And fer the neutron

 

 

4 P2. Mto‘ __
3 Ir 2X? + t It)" - o (4.4)
where
:0? =.1_ .
on

If now we assume that a nucleon experiences potential energy
due to its position in the meson field generated by nearby nucleons,
then additional terms of the form 311 and gU* must be added to the
.above equations. And this brings us to what is perhaps the most
important concept in the whole field theory: the concept of a
mechanism whereby forces of interaction between particles are

generated.

We designate the meson fields due to the presence of nucleons

-10-

by U and 11*. Let us assume that the proton interacts with the
U*-field, the energy accruing to the proton due to the interaction
being gu*. Furthermore the proton is assumed to absorb a negative
mesOn and become a neutron. Likewise neutrons in the U-field
would have potential energy equal to go and would absorb a
positive meson and become a proton. Symbolically this exchange
process that is postulated as the mechanism of nucleon interac-
tions may be represented as follows:

P tiM' —efiN

tld-ld*-e>fh
Or conversely

P +M+ + N

N+M‘+P.

To incorporate the new energy terms and the exchange process
into the nucleon equations, we first postulate that the proton and
neutron states of a nucleon are merely the different states of the
same particle. We introduce a new coordinate into the description
of the system. This coordinate is usually referred to as the
isotopic spin variable. New define an eight-rowed, one column
eigenfunction, 3E) , which we may refer to as the nucleon eigen-
function.

awe = (3:)

Define the following 8 x 8 matrices

t ... 11:59. J. 531.1 HPLI ,
P" a —2 ()0 (,‘OJ

-11-

-7Li§-=i. JZLL —( —O’:(— 0‘0
tNP‘ 3 3- 1:0 :

The following equations are readily verifiable

an a (“2") m»? =0?)

The t 's are the charge exchange operators.
We may now write the two nucleon eouations as a single

equation

W(% " Erma (fir) " NP gaff “(Pu jar)

a” rig;— mm Err—chm me

All matrices in the equation are, of course, 8 x 8

P W" O _
X :— ~O~t ‘8’}, ,etc.

It has been shown experimentally by Rabi8 and his co-workers
that the deuteron behaves in an applied electromagnetic field as
though it possesses'a non-vanishing electric quadrupole moment.
That this experimental fact suggests the necessity of postulating
a tensor interaction term in the potential function of the
Lagrangian may be illustrated as follows: We consider the
interaction between elements of charge P2 C1 (6 in the

electron cloud distribution about the nucleus and the elements

of the charge ‘Oncic in the distributed nuclear charge.

-12-

.KdTe

 

 

-—->
r... l
e -—->
. e R l
9"“; )w '
4a . a
\‘(J/ \ \ \
V\ y“ .I-c-xdrfl\ ‘ J
4’" \ I
\ I
x \I
Figure 4.1

The interaction energy is given by

v= { “3'57“? (4.6,
where u e
I u h. X EQ(COSu.9) ’
W: {(-2 J "T h; < kg (4.7)

From the addition formula for Legendre polynomialeg, we have

‘ l “—M M r \
[99(C05u9> -: %.l (R - 53)”) I: (Cosge)f} (0056,0003 “(Cpe‘luj

Inserting this in the equation for'V and examining the terms for

1 = 8,1e have

:0“ 5,... (W -v~ meow->9 ><

<9 M)! [ Padre
r;

Z'e

f R

N

by gyms 9") cos “(<22 “91) c1?“ (4.8)

-13-

If now we assume that the nuclear charge distribution is axially
symmetric (that (0,, does not depend upon (PH ), then the only

non-vanishing contribution will be for M I 0. Thus

v, = f “if,“ Bees 93ij n," a (case) cm. (4.9»
'C 'Cu

e
It is the quantity

9C2 = [6v h2(3CogDe~ «1) CH)
C.

that Rabi detected experimentally. This integral indicates that

 

the charge distribution in the nucleus exhibits angle dependence.
But the charge distribution is a function of the forces between
particles, hence it is reasonable to assume that these forces must
also be angle dependent, that is, non-central. In view of these
considerations, we add to the potential function of the nuclear

equation the tensor interaction terms
59» GT? . ‘*
55 T X G 3’ Z‘ r c
“—— 1‘
2 K PH P? 2 K NP /°
The nucleon equation is now

4?? __~--~ *_
2‘ [i9 i53§9‘LEE”Z;W’C¥19 £3, (”F’Lfio 23' 7:97 L1f=

__ [3? 6,3){~ '7' it \ M (I

This ecuation is still objectionable as it predicts an integral

dipole moment (in units of nuclear magnetons) for the proton

and no dipole moment for the neutron. This situation could be

-14-

somewhat improved by adding additional terms of the form
u~U~°H + TIN/P (HP—MG>° H

where M” and “P are the observed dipole moments of the
neutron and proton respectively; This paper, however, is con-
cerned primarily with the exchange properties of the charged
.meson theory, and consequently the incorporation of these
terms would be merely an unnecessary complication.

To put the nucleon equation into the form of a Lagrangian

we note that if’H is any Hermitian operator, the requirement that

g§{£]3/4423T3t=o
t2:

leads to

[(363) H 42 + J3 H<242D14I<H = o

T

t k___'__\
j [(gqj)}{¢ +§$>H+Jdtdi=~o
f L——-\.

HQJ2O , HSZ/ZO

I‘
Thus we obtain two equations, one the Hermitian conjugate of the
other. Hence we obtain the desired form for the nucleon equation

by defining

.—

eLPB“: E

and writing

§[yto fir.) ..qu (70.0 _2( TNP (”Ca ‘ (31???” MP

__ C993 MPG [VINO
2K EVE "Gm +INP GP v})+ I” 3; +TP”;§- @(4.11)

for the final.form.of the Lagrangian density of the nucleons in
the electromagnetic and meson fields.
For convenience in calculation, we break it down into

three parts:

“—'P r“ L C? 2) L
L 2 2 4)? [Y 5—); ”3:1994- $42449 (4.12)
J4 [2:9 .. (L
It}; “ 9X”? SD” (4.13)
——-I "‘ ”“ ‘ <
L :1 El? f((mo U1: + UPI: up)
3 6’? ._ . iv __ 4 \_
—'23t_< 3’ CNP‘ GP? + (p... 60.)] 95 (4.14)

The bar indicates that these are Lagrangian densities. To

summarize, we have also

 

L " - + K P (‘7 (4.15)
2
and __
E E” 2 (r9? F33,

% (4.16)

-16...

fer the meson and electromagnetic fields. The summations are
over i protons and J neutrons, thus theoretically extending the

validity of the results to cases of many particles.
V. The Derivation 9_I_‘_ the Equations of Motion

Now that the Lagrangians for the system under consideration
have been set up, the process of deriving the equations of
motion is, in principle at least, a simple one. We proceed as
in classical particle mechanics and require that

5 J;(I.r + I.“ + I-m'+ L + LE ) dt I 0,
or, as in the previous notation,
g£[£ [(LP+LN+LH +i‘+i“‘ )Jr]c(t =0 (5.1)

_To illustrate the procedure, we will carry through the '
variation with respect to. Lye in detail. The other equations
resulting from.variations with respect to L4: and (fé'will then
be simply written down.

As the Lagrangians have been defined, it is seen that the

coordinate Lfl° does not appear in-LP , L , or'L . Hence
I n
B Lcifl: (5L +8L><JJZ=O (5.2)
Consider first

8 LMAJQ. = [S GHQ—g0» +K2Up“; C(JZ (5.3)
.11 . "Ll

G6,: a ’)
:5 g(%—X: :4) “1M :-,L (Pydp)+KaI/(:§ifl('s’

' I

J1 _)

‘x

 

«r Os.)a (Xv),
:[ggl’(§::z J5, -- [:fflfiup) 632
3’ XPI) 3:. <

The integrals over $1 and $2 vanish because of the requirement
that S up (Xe), = Eufi (X4); =— O at the boundaries.
Remembering that repeated indices are to be summed and that

Gf’v : — va

111. (5.3) becomes

96 96— * /, 4x
éflgazo fi‘“ +84“ 36:“: +91TG-(ov CR, 3%

+ K”; M): SMF]CLQ

‘V .
[97352 “PIC—56?); + K0? [1;] S“? 4‘0“ (5'4)

[0.

-18..

Consider now
EfLIc-(Jl [2 EFCSDUOTPNJ 5%)
.1). 4C).!

-23 )9)? T .1 ng‘p]CLD—‘

9K PM

\I

(5.5)

Considering the second term in the integrand by itself

Z _—;—:[f km) (P:[<9%) —I(Pp EU?)

”(92%) ”IQ? 3%)]9043-

(5.6)
(XM)
gZEEEWI‘; 92 SM La:
(3).
“9(9):
53,-:(3? 3"”? g 2%] ((3
(x),
*;>: 39[ [Suki ..- (EEPVTPJ E)
‘1“: as x?
<5 (0’? ('EQBCJyK-Ipyg .1
"' /(°ng (E b ('F’N _ (13) f E

...qu’ 5643;; (pyfflj EjgfL

-19-

6_ we -
Neting that 8' v “\E and recalling the remain-

der of the integrand in Eq. (5.5), we obtain

i; (-3,; as m> ape-We]

(5.7)

Hence

affd‘azffiifi? ("‘I‘PXGe: 2:55.; P 3” P~ 9)

..Zgfl? UPC: g + Muff] BLIP An = 0

Thus the equations of motion resulting from the variation with

respect to U’ are

(5-)) +71“ (83(57): fléfi Te): 93:)

~ 2:: 5’ J
jig 2r 7;. :75 —+ (<qu -—— o
,J
(5.8)
The conjugate equations are

“1%)(Gw “- 2—371?“ W22; a)
"'5'er— X’O'QPLE + («QM/a = O

(5.9)

-20..

The equations that result from the variation with respect to 4,.

are

9a»

-————‘

F" (a ( _ 1" . '1‘
9X.) '— .114}, I} 4/, +41% G?” "u? at»)
(.

'37:: pl) __..1 *_( _
+ 2: -‘K‘ ’27-: b] MV(PN ‘0’? (HP 515 :O-(5.1o)
‘JJ

These last equations are of particular importance, as they
yield the expression for current and charge density due to protons

and charged mesons. We rewrite it as follows

91%» -1=.J.' 5:.”
QXY c c c
where
P "" Q,
%= Z “Pg—[Ii]??? (5'11)
L
and

N :0) ii...- ( ~ ‘l
(— :2: E :8 ((1,, (‘NP " (I), (PN)£, (5.12)

VI. The Hamiltonian

 

Since we will not be concerned with the specific form.of the
nucleon Hamiltonians, we will indicate them byHN anleP for
neutrons and protons respectively. The Hamiltonian density for the

remainder of the system.may be obtained in the usual way

-21..

-_ " ° 4: o ___ — (‘1 — EH ‘ I
KEEN/“’75:”? “Lima“ L ‘L ‘L
The quantities W are the momenta canonically conjugate to the

us

field coordinates Ur . They may be obtained from the defining

relations
_ (9 T=_.£.2. H =——-——
«”r‘"cg2up,.’“'/P- CQMPJJ?P C9<Pm

where the notation LB .4 indicates 239°. The momenta are
9X.

“Tr = 1“ [67:4 “*ZKEE X94 Z—Ffl EVE] (6.1)

’° 9Q

é—(QLGM‘B EFF 7W Tm» £13] (5.2)

‘1.
.34
H

QT; 2' 52C FF+ (5'3)

The summation over nucleons has been neglected since we shall
be concerned only with the potential energy terms contributed to

the Hamiltonian by the interaction between two nucleons. Hence
we have QM
:PJFH H 9Q G634 7; E; B PHI-LP

#32“ P4 JQHJ ( 94%
fish "PET my“) 9t +2c5 9t

-22-

:7!

+6; m < (40> <5):

VII. Reduction 2". B-Vector Notation

@[38 04(0th a; q.) «3‘16

 

For purposes of further calculation it is desirable to rewrite
the equations of motion in ordinary B-vector notation. We make use
of the following defining relations between the tensors and 4-vectors

employed previously, and the S-vector that will be employed here-

after.
-G3 . 0 G1 -iF2
G p =
‘9 G gel 0 -1r

3
131 mg 135 0

plus conjugate relations.

”H3 0 H1 -iEz
FF, ..
Hz “Hi 0 ”“33
ml 1 32 1E3 0

<3)T> :. (Egipvrmg) LEEPXC"%>
(Mm) = (miffed?) 3i? CONE)

and the conjugate vectors 8*, T*, M* and the scalar n* involving
the converse exchange operators, T;..

Employing the above definitions, it is easily verifiable that
the equations of motion become

(V-IIA) x (s+s*) - (g 9 - HA4)(F+T*) -m*+ Kan ,_. o (7.1)
c 91
(V -m) - (F +T*) - N"‘+ Kan4 - o (7.2)

plus conjugate equations. Equation (5.12) becomes

l"- [[5 x (G*+ 5)]- flfi“ x (c + sq] -flfi4*F -U4Fj7
O

-]I[fi4*'r* -U4§7 (7.3)
for the current density and
m .
J4=elTfi°F - U*- 37 -e1[fi-T - U*oT_’f7 (7.4)

for the charge density.

Again it is to be noted that the summation over the nucleons
has been neglected. It would be entirely feasible to bring the
summation through to the above eouations, and in fact necessary,
if one were to apply these equations to the problem of the Triton
for example. This paper, however, is in the main restricted to the
two particle problem, 30‘we shall dispenae'with the many particle

notation henceforth.

-24-

The Hamiltonian in vector notation is

—s‘-'—N . 2g. .25 T .25.! +
H H+H +%_;[(F* 91")HF 9t)” )

i .
(T* . 23 )+(E - %§fl+(m .U)+(M*- U*)

9t
(NU4+N*U4*) + (T - F) + (T* - F*) - (G - s) - (G* - 3*)
- K3(U - U* +U4u,*) - (G - G*) + (F . F*) + 9.:(E2 - Hz).

The defining relations for G and F are
G: (VXU)+.I[(UXA)
G’" . (VXU*) -]I(U*XA)

r . -124 -VU -I[A4U+-]IU4A
'6' t 4

..—

*
F* s --1_ 9H - VU4* +1IA4U* - IIU4*A
c 2}t

VIII. Integral Representation 9_f_ the Field Quantities

It is desirable at this point to express the field quantities
in an analytical form that lends itself to the application to
specific problems. The method employed for this purpose is
identical in principle to that developed bvadller and Rosenfield10
and by Ma and run. The actual technioue differs slightly.

We restrict ourselves at the outset to problems that depend
only on the stationary states of the system, hence all time deriva-
tives may be neglected. We further assume that the motion of the
nucleons in a given system is such that non-relativistic wave
equations will adequately describe them. Thus all quantities

involving the operator a will be. small and may be neglected. The

(7.5)

(7.6a)
(7.6b)

(7.7a)

(7.71:)

latter approximation depends upon the interesting significance of O<

-25-

as a velocity operator.

The operator to be associated with the time-derivative of a

coordinate x is defined in quantum mechanics by

dI-Vx-i'EI-I-Ig.

dt
The Hamiltonian that appears in the commutator for this case is
given by Eq. (4.1). Since the quantities depending upon the
field coordinates commute with the quantities depending upon the
space coordinates, it is only necessary to evaluate

[-C(M°V)x + XQ(D(°V)]LP

—-cx(a-v<}/) —Q4/(0<°Vxx)+ xQOchp)
—Cwu.

hence the operator to be associated with v: is

H

vx~>ca‘ , or v—>co<.

Consider
COUP=ASD

'where the 1% 's are the eigenvalues of the velocity of the system;

a :3.

(2(4) (L L%:4

waw=gwe
so that for velocities appreciably less than the velocity of light,
quantities involving (X will be small and can reasonably be
neglected. A.non-relativistic apnroximation for the meson cannot
be made.

A.further simplification of the eouations results from.the

-25-

fact that the electromagnetic field ouantities need not be con-
sidered in the cases proposed herein. In fact, one of the reasons
for restricting the actual computational problem to systems of at
most one proton is to facilitate this particular simplification.
Implicit in this approximation is the assumption that the electro-
magnetic interaction between a charged meson and a proton is small
compared to protonemeson-field interaction. or course this
approximation also neglects the meson-meson interaction through
the intenmediation of the electromagnetic fields resulting from
mesic charge.
As a consequence of these approximations, the field equations

now become

V X(G+S"‘)+ ((28.0 (8.1)
V x(G*+s)+ K2U*uo

V-r-N* + (<2U4-0 (8.2)

V°F*-N + 1(2U4*:0

G-VXU, G*-VXU* (8.3)
Fs-VU4, F*I‘VU4*O (8.4)
And the equation for the current density due to exchange of

charged mesons between nuclear particles becomes

in: I[ fix (G*+ 317 4116* x (9 +8*)] qyfigr - U433? (8.5)

c

whereas the charge density is
H
J4 ueflfi-fl-W-fl (8.6)
From the field equations we readily deduce that
9 :2
VU'KU:VXS* (8.7a)

vaw - K"1 (1* =V x s (8.71:)

-27-

2

V

9 3

:2
U4 ' K U4 3 'N* (8.7s)

NOW, in general fl
((4/79): - amber = (Owe— swym : o
0"
as the surface recedes to infinity. A130
7391) = KN}— Sfl- r)
so

{{‘PVQCP " TD??? — $0-t’)]}c(‘i = o

“C

(WICPSO' FM? = - [(Vggb— («9494mm
7 t

m 2 gm Help...

(8.8)

Having defined 8 (r-r') by the relation
VOL/J '* KDQD . - 8(r-r')

for regions of no nucleons, we prove that
{go-mar -- H—

as follows

—— {8 (r- (Oar = {07942 — (842%)? ={[Vo(vq»)-K7+j]dt
T I C

=- {Va/2.60— — @qu4:
r

U'

-28-

New ‘kai

-Ky ~KF
76 .. _§_ g Ive “>
r " 3 r

so we have v
. e—Kr —K KP
EM)“: <73 + 5'65 ”V K)? "
V’

and Lin SS“- ”((1

 

~—)<r «r 9 2
1m [4W6 + W‘Hve — 4TH K]

 

f») r->O 3
= 411'.
It follows immediately from Eq. (8.7) and Eq. (8.8) that
U(r')a- ( [(VXS*)?(r-r')d;, (8.9)
447
and r
‘U (r') u l ‘r’ N*’ (r ~ r') d:, (8.10)
. ....“ . 4»
F (r') = - ' {W‘V‘lph' - r') 3;, (8.11)
1+1T r '
G (r') n - V Ifl'V X 3*) ¢ (r-r'_)_7 dr, (8.12)
Tn
where
(Pu-fl) = O‘KIr - r'l .
Ir - r7)

Equation (8.12) may be put in somewhat more convenient' term as

follows:

V XJVXS*)LP(r-r'j_7
=VcP(r-rv)x(vxs*) =-vP(r-rv)X(va*)
:-{‘V xfip(sz*_)7-47B7X(VX 8‘17}.

-29-

30 (8.12) becomes
G(r'):__)___ {V X[6P(VXS*_)7d;
‘ITT y
.. 1 f (P [VX( sz*_)7dr. (8.13)
WT )

The first integral may be transformed to a surface integrallg

which will vanish upon integration. Therefore

c(r').- I {(PfVJHszfiflu—r>
LHT r
gt) [f' V7 V7. 8* - )7 o‘V’ sf7’&:’

...L.

‘f7r r _9
.) pvovs*a'§-n cPVV-S*dr
«+17 r 4V V

The first integral may be partially integrated, the integrated
part vanishing on the boundaries as the boundaries recede. ‘We

have finally

G(r').-.__I__f 8* Vchszf-l [(PVV-yg:
477' r

471 r
=___L__ j 9*[K2(P - 8(r-r'l7d:

‘7’ r

-) @Vchd-r)
LHT , a
-. -4>

= K7 I 3* din-t [(9 V Vos*dr-S*.

4 r 47

)— (8.14)

-30-

IX. Calculation.2£_the Exchange Dipgle_

Moment 2£_the Deuteron

To calculate an exchange mag-

’ netic moment, it is necessary to

Figure 9.1)

 

 

 

obtain an expression for the.mag-
netic moment of a volume distribu-
tion of current of density, 313.

The components of the electromag-

netic vector potential A (see

are related to the

volume distribution of current

1‘ (9.1)
(21‘

C'

Figure 901

where Z:

t (119 yl: 31); :19 yl’ 21

being the coordinates of rl-space, a region whose linear dimen-

sions are small'with respect to r2 and R.

New
—> --> ->
f (1.2) I f (R " r1)

IOXpZ—I—

-Y

5’29

by Taylor's expansion. Hence

12
1 exp { -y —— 7 (1).
1‘2 . x1? 9X yl 9:41-22 (R)

.2192]: (R)

-31..

80 Eq. (9.1) becomes

A(x, y, z) -- .T( 11, yl, zl) expErl-VJ( l ) d'C .

l
3. (R (I: Y: 25—)

I:
(9.2)
The first term is

11-1 _J_d‘(.
c R

T
New I am dal is a current element along. X. So

4 -v --e
Idrl .IJ) daldrl . 181‘.

Hence the volume integral reduces to a line integral along the
path of a current filament. All filaments are presumed to close

within the region, 7.- , hence

..5
A 'Ll... drl a 0. (9.3)
cR k

The second term in the expansion is

A, = i [E 1'1 ' V (29] Jdt In .2'f51 0 V (3)] $1. (9.4)
c (R) c (R)
‘C )1
We employ the identities

a {51- v g] .,} - 5.- v (1.3.7 ..., + [al- v 3.1)] :1.
R R

and

(r Xdr)XV(_l_)=r-V(_l_) dr-dr-V(_l_) r
1 1 (R) [1 (12)] 1 1 m] 1

-32-

to reduce (9.4) to
-I d{_/_I-‘ °.V(_1_) r} -}_f r Xdr I VQ).
TEE-g 3‘ (11)] 1 2c X l 1 (R)'

The first integral vanishes around a closed filament. Thus

(R)

The dipole moment of the current filament is by definition

A,= -__:L_ I rlX drlx VLL). (9.5)
2c 8'

[My - _I_ r1 X drl =_l_._ r1 X Idrl.
2e 2c
8 a X

If we sum over all the filaments, that is, over the total current

distribution, we obtain the dipole moment of the distribution

[u =1;- (r1 1: flat, ’ (9.6)
C ‘ _

'Z
since

Idrl : J'd'C .-
.Thus.the dipole moment of a system such as the deuteron may

now be calculated from the following expression (see Eq. 8.5)

M ;.]_I_{rx{fix (G*+s)_7-[J*X (G+S*)_7_
2
. r

-,
- 54*3‘ _ {1415?} dr. (9.7)
we evaluate two integrals that will be of assistance in
later calculations
-K'r'Rll " K’r-Rzl ~—>

l e dr, (9.8)
1517’ (r -Rll (r-Rzl

and

-35-

 

 

_> ~KIr-Rll -k)r-R2(
1 re 8 dr. (9.9)
[677? )r‘RlI )r'Rzl
P
N
\
l\
l \
\ w.
) \R R-R -R
f; l \ - l 2
9' I . ..
Ea) \ P-R1+R2
/ : if“ \ I'l =~'1‘ ' R1
{I i // F: r R
23"" 2
(3 /
‘ 1‘ 31‘ 4—H ..R g —R
/—§ 1 2 2 l I‘2

 

 

Figure 9.2

The second integral.may then be written

)6 317’

’-

7.:0<+)<P(b)d)/ = --' ”JG +€)<‘3

Ir.) (Mr?!

‘K ”.9-
Hf.) c:

)3 Iblel

( _.
+/éTT

thl

r2;r1+R1-R3=r1+R

—)<(),+R(

 

(bl

#_—

RI—KIHI

A)

1.

: ...!— (E‘L’Qp) @(FJCPCI‘a) C)"
P

 

luv"
) 4 —)<(),( 4(1),).581
+ 277)) 6T7)" em C“
+ ...L. E €—!<H./ <3 ~)<l)r.~/?)J
léIT’ t. W 7b- (W h.

Let the variable of integration in each of the last two integrals

be X. They then may be combined as follows:

 

—)< (X( -)< ( XV?) -!</X— m \
t [-13 C {e + €—_. }<(21.

75W; )1) (X +8! 11—- r?)

The integrand is clearly odd and hence will integrate to an even
function which will vanish when evaluated over space. 80 Eq.
(9.9) becomes

_:G_ (90‘!) LPCA)<))‘,

327’ ,

Write the last integral as follows:

-35..

e-KI) -)<R)-— (m9
pee <4).
3;" ()2) ))' ‘N‘

 

 

’/
I).-f\’l = (52+ V12"? R0055) D

CT)“: =- )3? Smeclbde CH)

 

 

Figure 9.3

When these substitutions are made, the integrations over a? and 9

may be'performed directly. We obtain

... (<09 K? )fic )9“ IT
- , + -;> 9 086
I67fR’K
)zo 9=°

3

00 -K()'2+R)-K')‘a
Se

 

{are ...,
r.=o

+f) R —K(W“('a)‘K/"a

IQWK 6 9“:
(“f-0

00 ”KOVR)" (V);
+ if. e a
Ib‘fl'KK (a.

).= R

-35-

When the integrations are performed, we have as a final result

_> ~K’R
1...?)er IVWDQO‘JJd)’ = «Ci—576:3?
’5"? )~ (9.10)

Let us consider first the contribution to the exchange moment by

the term

5%: F x [uJ‘F —u. F70!)

I“
{r x[{$fi@fi,gp(a)d@};-i FILM VcPChHRflJF
fig; g)X[{T;r§N.*cP(h>dIQ}{-' W21 wwdgfld.

’3

NH

Making use of the relation V ((1) (b) 2 4) de + ('6 V4.)

we have

= “331574) 490.3490.) ( xgwfmer/(flclfldfiid)
)’ Q,

J-

-37 _

- R
F x[w,*gn,—N.Vfl.”je N ((67.49...

H
)
Lu
24’ (H
7C
50"“
83"“

Hencefo rward

-#((§7 -FT/?’

:1 ,) S§;_ 1: (F?:>.
6 LP R7 C?

NOW’

LR: X (VJNJWJCJ’QQ: N2[W2WX(Q2]C(‘Q2

J

by partial integration. In the same way

{A}: x (EN.)LJ)d@ -_. "'{Nirfi XVDSULHQQ.
pa.

4%

Hence the integral becomes

3.7),.) Q [N ’Y F Was/2) [N7 M? Wflfism.

IQ)?

Since Vawz—KRCP(?)
Vac/u 1075490?)

so we have

= £117? ( N. (FAQMMR’JAQM.
m?

-58..

,1“ N Na (p X?) Mew/8.118.,

(9.11)

 

:léJTC Iv

Consider now the remainder of the expression for the exchange

%—£1 xflu x (63+ 3)] -[u*x (5.1 angel)

= -IIK:()>4>(1.)LP<8)X{((V, 219*) xf Sagdmfiich
3977 r 8‘ )Q

J

+ -> ‘ , 0
315739 <90.) M1.) X{ f (V, X Sf) XIQVJ Z°§JPdRHm<H

9,
+1... jfi m1><P<1>x( ( (v.11) 21 (37}11211211
3'7" )1 R’

.1 ’9:

 

 

 

;@(1D(>C*1)X§§(Vg VH3) XJVZVm °§}c()Q,J/€dl
9.1

32/7” 1 ’Q'
__ {W “D “'
— fig ) P XRV) X 31 J X SJNJCW'C‘VQ (9.12.)

...:(1111111 111111 a

-39-

+1 {(5 x [(v: x Sf“) >< ($7.17a 61)]94343
69.1? (9.1212)

WWW. x 3.) x (v. 1.33)]1115’143
R

3 (9.12d)

To illustrate the method whereby these integrals were
evaluated, we shall carry through the process in detail for
Eq. (9.120). The others may be evaluated similarly.

Consider first

Sfé’x [mm x (1111-3331119.
R.

Let X . v; V: e 32. We have, after partial integration

over R1

=§<srx14>q1dr< + Z’flegfimfxwl
R. 1 a

#32141 (shunt). + 3,14,. (fl. XVN’)»

1141\zx41(s.*m.>x]ae 11’flm1x

R’

+Qz/4Q(S,* x V3409 + S'Jflg (Q X V1409

~40-

it :> . ’4‘
1 143% MS. 11233111). 1 1g“;
+ «9,3 [Q3 )<S,* XV: (P); 4' 342 Ha (p: X VA”);

+141V.142(S.*><1<<)2]a1€.

This becomes, upon rearranging and collecting terms,

3(37‘XFO4/dp. + (“9.11057 x v.41) 1W.
19‘. 1?.

+ LIHI:R ° <V. ‘P X Sikflclfi

= g (37mm), 1 jgxpmwsfflae.
R 9.

Employing the identity

V X (fab) = V43 7%! + $(VXU)

we have

$07. 16.34118 - {(sane
1. 1).

so that

-41-

31.

=§§ Q, KB! x (11,111 3*):{dfl336

19.?)

{a x[<v,1s.*>m]11w@
A?

3

Hence

Hf? xlm x 2:.) 1 (v2 1.133)]11419963
IQ?

'3

._._ $1.815: x (v. may 1159,1113

1%)? WVB 17,1 3.) x (W x Sfflewa

R.

We now note that

VeVNP: V,v,1p=§

so the first integral becomes, after two partial integrations

over R2

-—H[(391§2) x 3"]31990.

R. F;

whereas the second one becomes

~42-

“9 x[(V. $5.12,, 1S,*](v.4»)arad@
R

I?
W) x[(v.v,1 a.) 1w] 3.241211).
9. R

a .

p X [(372 V2 ° 33) ° 3'11] V'LP
:[(v,v,1g,) 1 3*][09 11:1)101. x VIM
=[(V3V)° 3)) . S'*l[((\’,x\7,&(J) —- (W; XVafil’B]

and

re, not!» = R, Xvi-.191 == fiat—(awe),

Hence the term (9.12c) becomes

“SS U31°§>X Vii/8.1g

R19;
...SRIJOSVPXZ:<VQV 3M))(\7¢2]§,*JPJIQ3

(9 13)

In a similar manner (9.12d) becomes

__ SS U332?) x $314641?)

Rt ”3

-43-

_ (g px[(v:\z13,*)oncpjsaamlg.
(a

R: (9.14)

Now

g8 P XUVNVSQMH 2241,41,

R; “3

becomes, after two partial integrations,

6% (111.11%(1.111111%

RR};

1 3 g {m x( 1, - 1,1. 4131+[631’x30671'va «We
R, (9;

and

((12 x[:(V, V. 133*) 4.411% 1111.411,
KR,

= ..fl (P11395747. (WV. 4034534?»

a p,

‘ + x ,’ ,1, cWJ,,
+£5R§[3,x(s. 1mm] [(Y. 3‘ )(VW)]} P

We now have for the sum of the integrals (9.12s) and (9.12d)

igugpor) x inflame

-44-

“L“(FXS M31? °V(V Mqu’HdIVJPJ
“([1 x51 V V= 9’9] [(3 XS.)(V mflppm
M1

+S§[(3.*°§)XSJ]JRJA§

R;

(CF X 31>[31’°V. (V. 1V. 41)] cm. 4/9,

”3

1 (g [3, x (S.*-V.\7.4»)]+[(3.x3,*) (V, ~t1fl}amg
R

.1?

2%

U

=.e°§(Px§*)[3 V.(V,1ch)]14ram

P
—33.*l(((>><33[3 0V,(V,o «40139.41?

1,;2S3(3*x3,)(v v-«mecIP. (9.15)
RJ?

1

In a similar manner it can be shown that the contribution from

(9.12s) and (9.12b) is

~2H(31*><S.)LP 1W. we,
are,

-45 ..

- gmmrm 1) x Salaam
W;

+ Up x [(3, x v, 41) x 3,711.11;

’7'”: (9.16)

Hence the total contribution to M of terms involv-

ing P is

M (terms involving (0 ) =

 

“3 ." (re cm, are;
levcm‘tig ((9 x R) N N’ (P >

 

+(€ it 3 °V V3°V9¢ 3/949:
32V¢K#£(Fxg')[a 9( D]
R. a

— (Q
Barrow}

(F. x 3.35?“ .V, (V. «V.«P)]<£R.ag
R1

(p x[(§," x v.41) x 321M. 41?.
p

t 2.

 

R

F
-LeK g

 

397m J5?

 

K 11
+3(2:Te% U Q X[(52 X79 9'") X 3' JAR (W‘-
R.(9 (9.17)

1

-46-

The tenms involving f) explicitly can all be made to vanish
by locating the origin of our coordinate system midway between the

two nucleons. The total contribution of terms independent of (7
is

11 (independent of f7 )

mm “if”: XSN’CWCW
—I ”(so SQW-VWCW M1
mm e.

 

 

4 0K? 1£<S' X3007 V+KDNJCWCWJ'
xrr

Now ‘2 -—kffi?’ -—Fffi?
'V7JDCL’ =' #7 E? -— lec’E?
g?
and
V" == in.§f
(1

so we have

 

_kfi?
M: "- WQDM)I(S XS°W —%)€ asap

Rafi?

(9.18)
This result agrees, except in sign, with that obtained by
M8. and Yull o

It is obvious that for the deuteron ‘1
spin vectors 31*

= O, as the
and 82 must be either parallel or anti-parallel.

X. Calculation 2£_the Potential Function

for the Nuclear Particles in the Meson Field

To determine the potential function of the nuclear particles
in the meson field, we must examine those terms contributed to the

Hamiltonian by the particle-meson field interaction. These terms

-- _ _ 94 ' QUE
V1 “20(Gg4 2E5?) ENE)???
_ "‘ P‘i Do!‘t

+ 2c (91+ 1,; Q 7 ”61192) gr-

._ g, E (Vi/1,1111 + 11,} TM, ) J:
632?}? E (”aw TM “L GP: INJE-

(10.1)
If we consider only those terms independent of the time and

make the usual non-relativistic approximation for the wave func-

tions for the nucleons, Eq. (10.1) reduces to (in S-vector nota-

tion)
1.: N*U4* + 11114 + s*~ (1* 4- so. (10.2)

we may evaluate the potential energy from the integral

VIAT
If.

We consider first the contribution from the terms

(Ya/(4* + Na.)

2 a ~—'—N M20011?)
”Hf/1401) 1111,11 <f

APR

(Referring again to Figure 9.2)

“1'” H 4111,331241-4—5—T3‘frx N (PUHKL
”(a R, (10.3)

Next , the terms

LHT R7
+39 Eij 4’0’06? ”'LfoO'DYIV “(SC)“1_ 3!]
11’,
_(_-K-2 30 S)cf(r1)cilx7 +£_:[(3 .g‘gkpofir)?
WT;
,1

_ ...L 1001963111; 1118311121 “EJCPUXSVV 215711121:
WT}? ,7
J l

“ 131(3):) ° 31> . (10.4)

.49-

New

$19 VV1 8) Ar = [[V((() V13) ~(V-S)VGD:(JZ

z —f(V-S)V<PC” {31va at

so Eq. (10.4) becomes

ifl <s.*1 110011111 1 fife Emma?
L{
“a

1— 11' (5,313,137, )3 ()(ojaa — fiflSySflV V43<11)]cli?.
. K.

—;Z(Sl*° 32>. (10.5)

If new we choose the origin at particle 1 (the proton) and
evaluate the potential energy function due to meson interaction

at the other particle, we have

? = -? 49(1) = 4901) = 190?),
V, V, 49(1,)-V1P(1.)= (717ch)

so [Including Eq. (10.3)]V

(V: = 7117”)” IV OWN)? 4- [firLMUV NCPUV)&W,
V1

-5 0..

LITI-

:2 V - 4? b ,,
+1: (3,2 3;)MMJCM; + 2,57% (J1 SQMXHM’,
f8; "7'

_ .1 [33152 1 VVCPWDFE) - #flioi 1VV<PCFOJ<MZ
a

w

I?)

—2(Sl*° 3;) . (10.6)
Now
9 -K? «I? 4:? .9 n>
VV<P<W = 115-3“ 63 + 35.63 +9: )QVQ
fl? ’93

I 4“? ~KF

.M@0w: +§% )lf- 50—@)) (:54

and where I is the idem factor, 80

 

(3.53.9) 11 (VVchfl = 3gg1fi><sg1re><£2

R79 3
we _ '2
+£L+| 3 ~V323)F i. k“
’9 K” r? ’ 3’ 7?” K79 F6
- 30%)
_ .7
Let {0?} — (.3’: + _g +JQ2> , then

_. ‘1 1:
V '2 “LI [if/Y @(Wd’fl + M H" 21796“?
I Q (’J 47].. J 3 7—HT ( d() I

A?
1 M

-51-

Q
+3115 3: S, VPWDCMZ, + £3.16; 2° S: ((90069.
WT R 4/7 5?
3 l

-33. 3N(SJ°’Q> —SJ])[(’?> (0(5))0‘62

———— 3
471 ’9

fl

— 2i - (3595;?) _ 371%?) @CWDCW,
W)” K p9

+ games - 2(35 39).

a (10.7)

The above expression is written as shown in an effort to
bring out more clearly the precise nature of the integration
indicated. The first integral, for example, expresses the poten-
tial energy of the neutron in the field of the proton; the
second integral expresses the potential energy of the same proton
in the field of the same neutron. Thus the potential energy of
the system is actually. Just one-half the sum of the two space
integrals (ignoring the spin dependent terms, though the same
argument applies for these also). But the system possesses a
further degree of freedomp-the isotopic spin variable that
characterizes the charge state of the nucleons. The integration
should therefore be extended over the spin coordinates also,
as indicated by the summation over i and J. After integrating

and summing, we have

-52-

KR

__. l m _’__ (2) (u) (2) :-
\41 :3! (Tree Li"! +-.7}~ 27;” —) fi?

—hrR’
'f' é: 332<T~:) Tm?) + rm?) (N?) > <67 a 0—3) 7:?

_+ igé? (171”?) 779:»,‘+'7T%:)-z}:?):>[k:07 °(E;:>

3C0PWXVS°W gig-W.
”733‘“ (H?) )9

(10.8)

To consider the significance of the operator

.... a) (a) __ (0 (a)
L NP TPN + (- Pu Twp
(2

we recall the original definitions

 

‘-

—- _ c,—<z;

(Np -

and

-‘(+(L
‘17P" '“ ‘ a
2

 

and expand as follows: ) CD) ) c) ()
<0 (2) 0 —— U) —- C) ‘ 3
7:NP ZZWV 'F‘Z;W’ YTHF’ (I (I -F'7r; .(3 _ H

9 fl . 31 '
The eigenfunctions of the isotopic variables may be written

explicitly

1‘ -——— (€363)

Z“ VHCETWLJ + (2)00]
7L3= {mm}

1‘2 Hem) 42x10]

-55-

where the character operator (goperating on the state func-
tions 3 and It yields 14
T3 .5 = § T3 (’L = - (4 ’
Hence 5 designates the neutron state and K the proton state of
the nucleon. Making use of the following relationsg,
T‘§=¢( T,S=~(b( T3§=§
1', v1 = S T, 4

it is easily verifiable that

P“Z'=o P"Z"=Z’ P7330 1372-24-

I!

-('§ T3‘4“"\

Since the functions 1' and ZErepresent two-nucleon systems
of two neutrons or two protons, it is seen that the charged meson
theory indicates no interaction between like particles to the
order of approximation employed herein. In the case of unlike
particles there is an interchange of charge states between the
particles due to the operator P" . It is to be noted that the
form of the potential function derived from the charged meson
hypothesis is identical in form with that obtained in the
neutral theory15, with the exception of the operator P“. The
neutral theory indicates interactions only between unlike

particleS.

-54-

XI. The Contribution to the Electric Quadrupole

Moment of the Deuteron inesic Space Charge

In Section IV we derived the expression for the electric

quadrupole moment

EC? '1)?" (712(3C059‘914‘DJTM
Y:

N
From Section VIII Eq. (8.6) we obtain

Hence we have

a 1(—Z’—%((V‘ x 3,?) (P (0% ffiflmwefl Ki]
K. (V:

-11“ [714% x 3;) 4) (1911?. 1 7%)fo 174905146].

1? A, (11.1)

U

a a E [[H {[vfipog x Vipofléj NQJEJEZ
ILTTQ
pg

%g (D719 0,)” 71%)] Sle, was
“.81?

:31 L); (”33* m, 3) [veep xwaflama
7

(11.2)

Now V. 00'!) = [K@(}') + @]<:“ é?)
‘VVQ 2 = J (a Q—(Jz) —>“ —_'>
Mr) [W *> + a ](r K421).

Let [KQOJ + CBS—0]“:- A.

‘>

V.<P<r1)= MF- 1‘3)

$4001 = ’42 (he)
=fll(r+f:’:)

Henc e

 

vacpm X V430.)
= ,4,A2[(r~/<J.)x(*+@]
... 4. ,4;[(+xr1’/L)+(WQ1>]
=2I4./42 (H1471).

Also

 

 

V, (P (n) = )4, (H191)

Figure 11.1

(See Figure 4.1)

so V2 (002) X V, (DUI)

,4, ,4; [0142?) mm]
-24 H1 (”331).

-56...

Thus
219V122L<11a>~<111mj
2/4, 1:1,.0 K #9)

where R 3 R1 - R2, (the internuclear distance vector).

ll

22 [V9 490.) x V. 4900]

H

Hence

‘ 1Z7}?

p" 1T. [flmjh‘nM33)1/4,H;(HR)]J;<J,J&;.
«4/11

(11.3)

Thus

(“f 0/: '13: ”[3: (N; S +/Y.;.*5>° HA (*XIQ)P%(“01_~(JJerjvJ

(11. 4)
where the prime indicates contribution from exchange processes

only.

Making use of the fact that
2 ([1003? 3:46)? + (k SMCP 31.18) T + (fCOSG) a:

«5
and R g Bk, B being the magnitude of the internuclear distance,

we have

(v M?) = (r8349 3:11p)? -<+63146C03¢P\)T

so that we have now

(Q Qj‘ — IZI/‘ETJ[{[{ <NQS, 4' N: Sg>° H. H [O'BSMG 344?):

HY} (8}

. \ $\ 2 1:”, 'i‘ J i '. ..f/j‘ , I f)
— (kw/JHQCOiiJ J]? ("“30” C) ‘0}JV1J/‘~¢ 0/1.. .
(11.5)

-57-

(6 Q); /ZIT79)({'4' fl, 6 13(3 Corie - ORA/,3”

ilfllflfi

+N.* 3:133:16 Sump *2 Sue ((1-ch giddy. 4N;

3 *
_%2ff £2,qu 5 I» (300339 -1>[N, 3.3
mus,

m," 3,3) Sue Coscp )3 m. 41454ch «7, an,

(11.5)

_>
where dr has been replaced by

‘f‘
r2 Smechde d+‘.
It is clear that the integration over {>‘will vanish for both

integrals, hence

(€3CQ:)’== C3 .

HI. Indication _a_s_ 33 How the Calculation 2; the

Exchange Moment 2!; the Triton Gould 3353 Accomplished

Making use of the general development carried out in Section V,
and particularly equations (5.8), (5.9), and (5.11), it is
entirely feasible to carry through the calculations for magnetic
and electric quadrupole moments for systems of more than two
particles. However, it is obvious that the problem.would be
extremely complicated if the system contained more than one
charged particle, since then it would be necessary to consider

the coulombian interactions that appear explicitly in the

-58-

expression for the current density. Consequently the triton,
a system of two neutrons and one proton, is the most reasonable
test case fer the extension of the development herein. There
follows a brief development of the expression for the exchange
magnetic moment, analogous to the previous development for the

deuteron.

CD Proton
® Neutron

@ Neutron

are.
b=”‘ R3

f"F?3

 

H

*3

 

 

I —>
VM==§EJkFXJ)3f
_ ?

Figure 12.1

3... 4131 X {(31131} -1_T[u* x (c + 2: SW]

-IL-[aff - 41., V] (12.1)

r
‘3‘}[690‘4 )VV 23 (d?
y (

We also have

X «It? 3!
7&1): mowed) - (mg);

* léz‘rK‘

(12.3)

~60-

Va, -/v m 1 ~ZHJ (12.4)

For the triton

m _ K2 :2
J” —_U_<L( X[IF£S,<PO.)A@ 4.3% $3<PUQCM§

; 3

714490967.VaoSJCIK—z‘JLV£<P<*1)(EV1'3154(§()

1* 2 .
.1164 Xfigrfg' 43. 0154?, 117:;[cp(+.)(m 13’)<WD
1?.

f9.

“1(“:[;}TT/£Nl * V, CPO.) JR] " “(ZELWLND )7}, Cp("a)JRz

+11% M, 73 We) 4&0

:y 1=._

-51-

.0 VP mom: x f 3.1 420,319,
'9:

       

+[<Vl }( SfDCQUJJR Xf33 CPO?) 4/93]
p, "9.?

+

 

I [fl V: X 3}") 4900M, >407; ”v; 13;)41‘094g
1;

Ié-TT
Ki

+((V, 1 §.*)c0(r.)c1/€ x(<vav3183)m>ag]

IQ, R3

 

a _
... If; [f(V, ”pupae, x swamp,
1?, K).

+ ((V, x $4303)ch fo-fipmm]
R’,

E, Iflw mew/<3 2< R(VV,1 3 meme.

+{(V3 XSBJCPOJd/g Xf<V, V, °S'*) CPOJJQ']
g 6’.

/173[{N1‘p(’)3fl [3N WVDJPHN, wow?

+1“ IQ[[NI*C( ((1) dfi/I( N2 )7; ((509)553 +£H3 V3 CfOQC/QJ
éTT p ft’ / ‘

g

(12.6)

Consider the contribution from the last two terms

“72% [f ,(pOQN V<P((~.,)c(l© CW
KR

1031; V0914,“ 2410):»? cm
{€72,413}?

 

LUM Va,» szqpflgwfldfl
6:27?-
1?;

3):!

WV)!” 4) (1) H3 V ((309353 A?
(7.29

H

$4

/é

2m (2111) Vowmcmcw
m,

 

N (VN “"MDGWCVHWQ
((1773

K. g

. (mwmwemm

a
:4
\J

(
1 a“; {M (V3N3)<P<+>4>03Hg 4g
6

(12.7)

-63...

Sotheterm
“3%)? XLU4F*”UJF]3*
y.

becomes

IL ‘> .1 _ , .1
fi1£&(r.)c(?<e) ( XRIRDY, (V, (Y, ) N, (v, [dag/,2;

llfcpowagi xm‘w (m)- Him/dam

(12. 8)

.-.—-. $57me 112.) xlmmm - N."(V1N1)]1W@

+327TK££(F)‘ H93) X(:N3 (V' N > N (V3 N3 flci/ficlfi?

Kl-qfifi/

.. gm N [<11 W1 V].

WAD-E)

+léTTK[[N‘ Vim HQ)" (’9 VJ]? 3,? 4?;

-64-

(9 H2) MIR—Kt) = «QW. we)

so we have finally

{.«A? m '
§—:K££N.* N (W xfi’)e . (NRA?

(12. 9)

In exactly similar fashion the other terms can be shown to
yield contributions to the exchange moment of precisely the same
form as those obtained for the deuteron, with the addition of a
complete set of identical terms in R1, R5. Given assumptions

concerning the relationships between the vectors R R and R

l’ 2’ 3’
it would be possible to carry through the indicated integrations.
waever, there would then remain the problem of calculating the
expectation values for the dipole operator, taking into consider-
ation all permissible combinations of space, spin, and charge
wave functions that accord with the exclusion principle. Such

a task appears feasible theoretically, but almost prohibitive from
the computational standpoint, especially in view of the apparent

inadequacy of a purely charged meson theory of nuclear forces.

XIII. Discussion

 

The purely charged meson theory is evidently unable to pro-
vide a mechanism to account for the anomalous magnetic moment of the
deuteron or the electricquadrupole moment of the deuteron in
terms of exchange pheonomena. Since the magnetic moment of the
deuteron is not the sumV of the moments of its constituent
particles*, a contribution of the right order of magnitude and of
the right sign to this moment by the charge exchange process would
have been very encouraging. Also it might have been reasoned
that the distributed mesic space charge due to exchange would
contribute to the electric quadrupole moment**. The calculations
based on the charged meson hypothesis, however, reveal that the
exchange process contributes nothing to these quantities in
that theory. It would appear that there is a non-vanishing
contribution to the moment of the triton, but the calculations
were not carried far enough to indicate whether or not the
contribution is significant.

In connection with the problem of the triton, it should be
noted that Villars18 has carried through computations for the
exchange moment predicted by the pseudoscalar theory. His
results appear to be of the right order of magnitude and 0f the

right sign for the dipole moment.

 

*fl, = 2.7896 (Nuclear Magnet0n8)16

16
up: 0.85647 1 .0003

A": 4.9103 1: .0012 16

**(eQ) : 2.73 x 10‘27 cm? 17

-66..

It is significant that a field theory of nuclear forces,
postulating a mechanism for the origin of those forces,;yields
interaction terms in the Hamiltonian that agree qualitatively with
experimental evidence. In fact, considerable attention has been
given to a phenomenological approach to the problem of the
deuteron in which the form of the interaction potential is taken
to be that predicted by the neutral theorylg: 20: 21

V': I, (r) + 11 (r)a;-c; +.Jz (r) 312
where

312 = 3 (6', . r03or)/r3.

Another approach to the whole problem of nuclear forces
representing an extension of field theory to its logical extremes
seems indicated in the various attempts to develop a‘Unitary
Field Theory22 in which fundamental quantities like mass and
charge folloW'as consequences of mathematical consistency of

proper field eouations.

l.

2.

4.

6.

'7.

-57 ..

BIBLIOGRAPHY

Schiff, Leonard 1., Quantum Mechanics, McGraw Hill Book
Company, Inc., New York,"1949.

Pauli, W., "Relativistic Field Theories of Elementary
Particles,” Reviews ofModern Physics, vol. 13 (July, 1941),
pp. 203-231.

Yukawa, Hideki, "0n the Interaction of Elementary Particles,
1," Proceedings gthithsico-Mathematical Society 2.1%
vol.-l'l (1935), p. 48.

Press, AL, "Sur La Theorie O‘ndulatoire Des Electrons Positifs
Et Ne‘gatifs," Journal _1_D_e_ Physique, May, 1936, pp. 545-355.
Proca, A1., 'Theorie Non Relativiste Des Particules A Spin
Ectier," Journal 22 Physique, 19;”, pp. 59-67.

r16h1ich, H., Heitler, 17., and Kemer, 11., "On'tne Nuclear
Forces and the Magnetic Moments of the Neutron and the Proton,"
Proceedings 21 the. 32E]; Society 2_f_ London, vol. 166A (1938),
pp. 154-177.

Kemmer,N., "Quantum Theory of Einstein-Bose Particles and
Nuclear Interaction," Proceedingg g}; the m Society 91
London, vol. 166A (1933). pp. 127-153.

Bhabha, H.J., "0n the Theory of Heavy Electrons and Nuclear
Forces," Proceedingg 9.333.125.9111 Society a: London,

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ll.

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l4.

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Kellog. JOBOM.’ Rabi, IoIo’ Ramsey, NoFo, Jr., and
Zacharias, J .11., "An Electrical Quadrupole Moment of the

Deuteron," Physical Review, v01. 55 (February,.1939),

 

pp. 318-319.

Morgenau, Henry, and Murphy, George Moseley, The Mathematics
93 Pysics and Chemistgy, D. Van Nostrand Co., Inc., New
York, 1943.

Miller, 0., and Roseni'eld, L., .0_x_1. _t_t_1_e_Field TheorypiNuclear
Forces, Det Kgl. Danske Videnskabernes Selskab., Mathematisk-
fysiske Meddelelser, XVII, 8, Kaibenhavn, 1949.

Ma, S.T., and Yu, F.C., ”Electromagnetic Properties of Nuclei
in the Meson Theory," Physical RevieW, vol. 62 (August, 1942),
pp. 118-126.

Weatherburn, 0.15)., Advanced Vector Analysis, G. Bell and Sons,
Ltd., London, 1928.

Stratton, Julius Adams, Electromagnetic Theoyy, McGraw Hill
Book Company, Inc., New York, 1941.

Cassen, B., and Condon, E.U., "On Nuclear Forces," Physical
Review, vol. 50 (November, 1936), pp. 846-849.

Bethe, H.A., "The Meson Theory of Nuclear Forces, " P sical
Review, vol. 57 (February, 1940), pp. 260-272.

Arnold, Wayne R., and Roberts, Arthur, "The Magnetic Moments

of the Neutron and the Deuteron," Physical Review, vol. 71

'(June, 1947), pp. 878-886.

Rosenfeld, L., Nuclear Forces, North-Holland Publishing 00.,

Amsterdam Interscience Publishers, Inc., New York, 1948.

18.

19.

20.

21.

22.

-69..

Villars, Felix, "0n the Magnetic Exchange Moment for H3 and
11.33," Physical Review, vol. 72 (August, 1947), p. 256.
Volkoff, G.M., "Tensor Forces and Heavy Nuclei," Physical
Review, vol. 62. (August, 1942), pp. 126-133.

Rarita, William and Schwinger, Julian, "On the Neutron-

Proton Interaction,” Physical Review, vol 59 (March, 1941),

 

pp. 436-452.

Bethe, 11.11., "The Meson Theory of Nuclear Forces," Physical
Review, vol. 57- (February, 1940), pp. 390-413.

Finkelstein, R.J., "On the Quantization of a Unitary Field

Theory,” Physical Review, vol. 75 (April, 1949), pp. 1079-1087.

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