AlllOb 21.1371

NBSIR 82-2547

Electromagnetic Nuclear

Interactions:

|

I. Introduction, Operators, and

Sum Rules

Reference

NBS

PubSi - cations

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Radiation Research

Washington, DC 20234

September 1982

-QC

100

. U5G

82-2547

1982

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

NBSIR 82-2547

ELECTROMAGNETIC NUCLEAR

INTERACTIONS:

I. INTRODUCTION, OPERATORS, AND

SUM RULES

ATlONAl, BUBBAV

BTAWDABDI

L1BBAS*

StP a u baZ

QCJOO

RSOJames S. O'Connell

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Measurement Laboratory

Center for Radiation Research

Washington, DC 20234

September 1 982

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director
tfA *< * . • A < 

ry c ie t 

W 2G
ELECTROMAGNETIC NUCLEAR REACTIONS:

INTRODUCTION, OPERATORS, AND SUM RULES

JAMES S. O'CONNELL

CENTER FOR RADIATION RESEARCH

NATIONAL BUREAU OF STANDARDS

FOREWORD

This report is the first of three in a series that will constitute a monograph on the subject of nuclear reactions initiated by photons and electrons.

The present report (I) covers introductory examples, the electromagnetic operators,

and sum rules. Subsequent reports will cover: (II) few nucleon systems and giant

resonances in complex nuclei, and (III) intermediate and high energy reactions.

ELECTROMAGNETIC NUCLEAR REACTIONS

PREFACE

This monograph is aimed at the student or research worker in theoretical

or experimental nuclear physics who wishes to acquire a working knowledge of

the basic concepts and methods of electromagnetic reactions. The point of

view is phenomenological with emphasis on understanding the data in terms of

simple quantum models of the many body system. Prior knowledge assumed of the

reader is a familiarity with the basic concepts of quantum mechanics and a

general familiarity with nuclear physics. An advanced undergraduate course in

these subjects would be an adequate preparation for reading this book.

There has been no book dedicated to the subject of electromagnetic nuclear

reactions since Levinger's Nuclear Photo-Disintegration published in 1960.

monograph gave an overview of the field at a time of transition from betatron

based measurements to electron linacs. The intervening twenty years has seen the

growth of electron scattering, electrodisintegration, the neutron time-of-f 1 ight

technique, monoenergeti c photon generation by tagged bremsstrahl ung and positron

anni hi 1 ati on- i n-f 1 ight , radiative particle capture experiments with tandem

van de Graaffs, semiconductor detectors, sophisticated magnetic analysis

systems, computer assisted data acquisition and numerous other technical

advances. Theoretical advances over the last two decades have also been

enormous: the schematic model with particle-hole interaction, the refinement

of the shell model to include correl ati ons , unification of collective models

of rotation, vibration, and giant resonances, the recognition and treatment of

meson exchange currents and isobar configurations in nuclear wave functions,

Faddeev solution of the three and four nucleon equation of motion, and the

accessibility of high speed, large memory computers for nuclear structure and

reaction computations. Nuclear structure information gleaned from real and

virtual photon reactions has become the foundation for single particle and

That
2

collective motion analysis of hadronic and energetic weak interaction nuclear

studi es

Photonuclear reactions are at the threshold of a new phase of development.

High duty factor electron accelerators coming into use permit for the first

time in this field the widespread use of coincidence experiments to measure

correlations of reaction products with an incident electromagnetic probe of

precisely controlled energy and momentum. This technique has long been profitably used in hadronicaly induced reactions, but the technique holds even higher

promise for el ectromagneti cal ly induced reactions because of the well understood

interaction and single particle nature of the probe. Correlation information is

of great interest in two other fields of nuclear physics: intermediate energy

hadronic reactions (especially the pion-nucleus reactions), and heavy ion

reactions. It is, therefore, an appropriate time to synthesize the state of

common understanding of photonuclear reactions that has emerged from the

"single arm" measurements made with pulsed electron linacs and radiative

capture experiments.

The scope of this book differs from the excellent treatises of LIberall

on Electron Scattering from Complex Nuclei and of Eisenberg and Greiner

Excitation Mechanisms of the Nucleus-Electromagnetic and Weak Interactions. My aim is more modest. I have tried to present the different

facets of electromagnetic nuclear reactions in a unified way. The formalism

of quantum el ectrodynami cs is not developed, but nonrel ati vi sti c operators

are introduced and used to compute reaction cross sections.

The first chapter differs in theme from those that follow. It is meant

to lure the reader not yet convinced he or she can or wants to read an entire

book on photonuclear reactions. In it examples are given of the successful

application of simple quantum mechanics to the understanding of some basic

electromagnetic reactions in nuclear physics.
3

The second chapter on the el ectromagneti c operator plunges into the

nature of the three basic transition operators at a level of development

suitable for the rest of the book. The connection between electromagnetic

and energetic weak interactions is shown. Reaction rates of weak processes

can often be predicted from known rates of the electromagnetic.

Sum rules are introduced in the third chapter as tools with which to analyze

photon and electron reactions. A collection of expressions frequently used is

derived from the closure relation, dispersion relation, and momentum integrals.

Chapter IV looks at how two, three, and four nucleon systems and their

virtual meson constituents are used as a natural laboratory in which to test

understanding of how real and virtual photons operate on interacting nucleons.

Traditionally, electromagnetic measurements on few nucleon targets have been

used to verify the structure and reaction predictions of two-nucleon force

models based on hadronic reaction theories and data.

Nuclear collective motion induced by oscillating electric and magnetic

fields gives rise to giant multipole resonances; these form the subject of the

fifth chapter. The electric dipole giant resonance was the first universal

collective property observed and understood in nuclei across the periodic

table. The systematics of its energy, strength, width and decay properties

occupied the attention of photonuclear scientists for many years. Attention

has now turned to other possible giant resonances.

In Chapter VI we focus our attention on non-resonant reactions. Photon

interactions between 40 and 140 MeV are often character!

- zed as direct reactions

with quasi-free absorption on one nucleon for virtual photons or on two nucleons

for real photons. The study of these reactions gives information about how

average nuclear properties affect the absorption and dissipation of electromagnetic energy. A connection with macroscopic physics is made by discussing

the optical properties of nuclear matter.
4

Finally reactions above the meson production threshold are reviewed. The

dominance of the spin 3/2, isospin 3/2 nucleon resonance on nuclear phenomena

is discussed. At energies above this new giant resonance the hadronic structure

of the photon itself comes into play. Finally, the dispersion relation is

used again to demonstrate that photon reactions at all energies are mathematically related.

A note on units and dimensions used in the book: natural units ( = c = 1)

are used in the equations. SI (system international) dimensions are used for

data except where some other dimension is so common it would be confusing to

change. Energies and masses are usually quoted in MeV while lengths and

momenta are in fm (femtometer or Fermi) and fm
5

ELECTROMAGNETIC NUCLEAR REACTIONS

I. INTRODUCTORY EXAMPLES

A. Single Particle Wave Functions

B. The Current Operator

C. Photoabsorption Cross Section

D. Application to the El Cross Section of Deuterium

E. The Magnetic Moment Operator

F. Application to the Ml Cross Section of Deuterium

G. Photon Absorption in a Resonance

H. The Charge Operator in Electron Scattering

3

I. Application to the Electrodisintegration of He

J. Threshold Pion Photoproduction
6

I. INTRODUCTORY EXAMPLES

Electrons and photons have been favorite probes for studying the structure

of physical systems. Only the wavelengths X (correspond!' ng to energies

uj = Zn/\) change as one proceeds in scale from condensed matter studies to

molecules, atoms, nuclei, and finally elementary particles. Nuclei vary in

-12 -13

size from 10 to 10 cm. To study details of nuclear structure and

transitions one needs photon and electron energies up to several hundred MeV.

In this first chapter, five simple electromagnetic reactions are examined.

Characteristic cross section data are shown together with calculations based

on single-particle Hamiltonians. These examples serve as an introduction to

topics to be treated at greater length in subsequent chapters and demonstrate

how electromagnetic phenomena test nuclear wave functions through the three

basic interactions: current, magnetic moment, and charge.

A. SINGLE PARTICLE WAVE FUNCTIONS

A useful approximation to the wave functions of bound or ejected nucleons

in a photonuclear reaction is given by the solution of the Schrodinger equation

i(i(r) = E up ( r ) (1A-1)

2

2M V(r)

for a proton (mass M, charge e) acted on by a potential V whose range is very

short. If we make the extreme assumption that V(r) = V

Q

r 5(r) the s-wave (£ = 0)

radial equation reduces to
7

1 dfu + v §lh u 2M , 2 or

d r

u = r 4>(r)

= Eu

(1A~2)

which has the normalized solution for the bound state E = -B

u q = (2a) exp(-ar) Y

Q0

(r) (1A-3)

wi th a = (2MB) 2 and

u (0) = - e ^

sin(pr + 6), 6 = tan ^ (-£

p V a (1A-4)

for the s-wave scattering state with momentum p and kinetic energy E =

For £ > 0 a centripetal term

, th

£(£+1 ) is added to the potential and u U)

becomes r times the £ component part of the plane wave exp(ip • r), i.e.,

u (£) = i £

(47r(£+l))"2 r j £ (pr) Y

£o

(r) . (1A-5)

B. THE CURRENT OPERATOR

A real photon is characterized by a unit electric polarization vector

and an energy-momentum four-vector k^ = (k,u) with | k[ = uu. The first order

interaction energy of the photon vector potential with the current j of a

proton at poi nt r is

* -]_

The strength of the zero-range potential is fixed at V

q = -(2M) , but the

binding energy B is a free parameter. An alternative (Be35) to solving the

zero range potential problem is to work with the equation for free motion,

but with the boundary condition = a.

~

r=0

<D>
8

H.j nt = A • j = (s exp (it • r)) • (e

jjj j

. (IB-1)

The photon flux is usually given a boson normalization factor (2tt/uj)

2

. For

our present purposes we will replace the exponential by unity and confine

ourselves to the long wavelength approximation kr << 1. This electric

dipole operator s • p carries an intrinsic unit of angular momentum. Thus,

if the initial proton wave function has zero orbital angular momentum, the

final state must have & = 1 if the proton absorbs the photon.

The relevant radial functions are plotted in fig. (IB-1). The

transition amplitude

final

becomes

e ( 2a

El

1 M \4n

^i nt { ^i ni ti al

> (IB-2)

-ar - -ip-r e ,3

( £ * V

) r d r (IB - 3)

by

= -i v .

The angular integration insures that only the p-wave part of ip f-j na ] * contributes to T^. The integral can be evaluated by applying the gradient

to the plane wave giving

e /2a \

M y47i J

47T£ p

(a

2 + P2

)

'/ l p

• r £ • V f(r) dr = ie*p /

l p* r f(r)

,3 d r

( IB- 4)
9

For future reference we note the zero-range electric dipole transition

amplitude is also easily evaluated with momentum space wave functions as

T

El /

«(P - P

0

) „

£

4tt e • p

(a + p )

(4/i2a)^ 3

, 2 M 2s

(a + P Q

)

( IB- 5

)

The dipole operator a-p/M can be expressed in another form

£• §

= i(E

f

- E.)e-? (IB-6)

where E^. ^ are the initial and final nuclear excitation energies. This replacement is useful when wave functions are expressed in r-space. Fig. 1B-1C shows the

dipole radial integrand in the zero range model using this form of the operator.

C. PHOTOABSORPTION CROSS SECTION

The differential cross section for absorbing a photon and emitting the bound

proton is

da

271

| T

i 2 dn

(f 1 ux) I

I pol . duj ’ avg.

(1C-1)

the flux for a photon is (tu/2n) and ^

is the density of final nucleon

states. An average over the two polarization states of the photon is to be

performed on |T| . Figure (1C-1) shows how the two photon polarizations relate

to the angle 0 the ejected proton makes with the incident photon momentum,

i he average of £ • p is

£

-*

• P

2

pol

avg.

= is [(£ * P)^ + (e

2 x

v y

12 * » 2 = j

P [1 " (k • p/]

]

1

2

2 P sin

2 0 (1C-2)
10

The density of final nucleon states

dn

dui £_

dui

is evaluated using energy conservation

(1C-3)

U) (1C-4)

to give

dn _ Mp dQ

dm

(2n)

3 (1C-5)

Combining all factors the angular distribution of the photoprotons as a

function of photon energy is

d£

= 2e

2 “ 2

dO M u)

2 • 2 n p sine

, 2 ^ 2,2 (a + p )

( 1C - 6

)

* 2 2 The total cross section (using p + a 2Muj) is then

* The corresponding retarded expressions which include the momentum dependence

(i.e. all multipoles) of the photon operator eq. (13-1) are:

= 2e

2 - ^

dQ

2 . 2 p s i n 6

M u) [a

2 + 2,2 (H) 1

(!C-6‘

)

ana

a(w) = 2e4

2 “ £

M uj

r 2. , 2 (a + p + k

2

) & n Pk'

a

2 + (p + k)

2 \ _

27i

.a 2 + (p -k)

2

/

k 2 (1C-7 1 )
11

a(ui) =

47i e

4 a(2M(uu - B))

3

(Muu)

3

3/2

_ 8tt 2e

2 (x - 1)3/2

3 BM 3 x (1C-7)

with x = uu/B and is shown plotted in fig. (102). Note the peak of a is at

-3/2

twice the threshold energy B and approaches zero asymptotically as uu .

Although the cross section rises and falls with uj the reaction is not a

resonance because the transition amplitude eq. (IB-4) does not have a singularity on or near the photon positive energy axis.

The shape of ct(uj) does not change appreciably if finite range potentials

are chosen for V(r), e.g., Yukawa or Woods-Saxon shapes. Furthermore, it will

be shown in Chapter III that the area under ct(lu) is independent of the shape of

V(r). Integrating eq. ( 1C- 7 ) gives

ct(uj) duu = 2tt

This is an example of a sum rule.

(1C-8)

D. APPLICATION TO THE ELECTRIC DIPOLE CROSS SECTION OF DEUTERIUM

These results may be immediately applied to the photodi si ntegrati on of

the deuteron by making two adjustments to eqs. ( 1C~ 6 ) and ( 1C- 7 ) . First, np

* relative coordinates are used to calculate T^. Second, the asymptotic normalization of the ground state wave function must be adjusted to account for the

finite range r^ of the np potential. In the theory of finite range potentials

( Be 50 ) one replaces the zero-range normalization constant (2a) 2 by (2a/(l - ar^)) 2

.

r = r -r , p = H(P -P ) p n’ K v p n
12

This adjustment provides the proper asymptotic normalization for the ground

state. The ijj of a finite range potential is finite at r = o. However, the

exponential shape of the zero range ij; is correct for any finite range potential

when r > r^. With these two adjustments the effective range calculation for the

photodisintegration of deuterium reads

2

e a f

X _ 871 an (.uj; - N D

v J 3 (1 - orr.)

[M(m - B)] 3/2

( ID- 1)

t' (Mw)

The result is compared with D (y,p) n data in fig. ( 1C- 3 ) with a = 0.232 fm

^

and a triplet range evaluated from low energy neutron-proton elastic scattering

data r^ = 1. 76 fm.

E. THE MAGNETIC MOMENT OPERATOR

The nucleon spin operator a = (a

+ ^, a^) is important for many

electromagnetic reactions. The magnetic field B of the photon interacts

# . with the nucleon magnetic moment m to give

us i ng

i nt = m (IE- 1

)

and

§=VXA=VXa exp(ik-r) = i(kxe) exp(ik-r) (IE - 2)

->a (IE-3)
13

gives

H int = (|m * ( i (kxe)exp(i k*r)) ( IE-4)

In analogy to our previous example of the El photo effect on a bound

s-wave nucleon let us calculate the long wavelength approximation [exp(i k*r)-»l]

for the magnetic-dipole (Ml) transition from an s-wave bound state to the

s-wave continuum. This transition depends on the spin structure of the wave

functions

l m ti al = (2a) h -ar

e r

Y oo (IE-5)

u/ f l nal

e~

l6

(4n) h sin (pr+5)y (r) v

2 pr oo

J A

m' (IE-6)

where \\ are the up ( + ) or down (-) nonrel ati vi stic spin functions. The

- 2 spin-flip amplitude is

TM, = ( kxs )

• q I x

2 > Ml 2M m I

x (4n) 2 e '*/ 1 'oo dQ dr sin( P^6). e ‘“ r (2a) h. (IE-7)

The spin matrix element may be evaluated by standard techniques to give

the polarization average

m E1 ,\,m

|<X^

2

, ((kxs) (IE-8)

_ ,2 _ 2 - k - u)
14

We note the radial integral is zero because of orthogonality if

tan 6 = - p/a. Therefore, if the transition is to go at all, the phase

shift in the spin-flip final state must be governed by a parameter different

from a = (2M B)

2

; call it (3 so that

tan 6 = -p/p . (IE-9)

For zero-range potentials the parameters a and p are the inverse of the scattering lengths for the two spin states. The spatial part of the transition amplitude

i s

i j [cos 5+-sinS] . ~ T = 2H (40)-* ( 2 c()'5

<u - 2 e

‘ l6

'Ml 2M 1 iu ( 1

, 2 , 2, (a + p )

= §g

(411)* (2a Y'

2

u, (P~°0 e

" i5

(a

2 +p

2 )(p

2 +p2 f

2 (IE- 10

)

The Ml differential cross section is

d(JMl =

4FT^

dQ uj Ml

2

Mp

po!.

^ 2TT

)

avg.

The angular distribution of the ejected particle is isotopic,
15

dcrM

, 2 2 , _.2

Ml _ e p coup (a -ft)

( IE- 11

)

giving a total magnetic dipole cross section

Ml

_ 7ie 2

M

2 (2MB)^(2M(uj-B)

1

'1 (a-p)

2

M

3

uj[2M(ut B)+ft

2

]

(IE- 12

)

Figure (IE- 1 ) illustrates the zero-range model wave functions and their

product. The lack of a spatial part to the Ml operator in the long wavelength

limit causes the integrand in eq. ( IE- 7 ) to peak at r=0. This behavior is in

contrast to the El integrand which peaks at a relatively large value of r (see

fig. (lB-lc) )

The magnetic dipole cross section for the zero range model is shown in

fig. (IE-2) for three values of the final state potential strength parameter (3.

singlet s-wave continuum near the np breakup threshold is a good example of

the action of the magnetic moment part of the photon operator. Both the

neutron and proton contribute through their magnetic moments (presumed to

equal their free values

p^

= 2.79 and p

n = - 1.91 in units of e/2M). The

spin matrix element yields

* 2S+1

The notation is Lj where S is the magnitude of the vector sum of the neutron

and proton spins S = (ct + (?n ), L is the orbital quantum name (S,P,D,F, etc. for

L = 0, 1, 2, 3, etc.) and J is the total np angular momentum.

F. APPLICATION TO THE MAGNETIC DIPOLE CROSS SECTION OF DEUTERIUM

3 1

* The S^ Sq transition from the deuteron bound state to the spin
16

|<

1

S

0 |(kxe)-[|Jpap+p

nan ]|

3

S

1

>|

2

pol . avg.

= 3 VM

n )2 • (1F_1)

The spatial part of the transition amplitude is obtained from eq. (IE-11) with

* the replacement of M and p by their reduced values to give

da

D

(Ml)

dfi

e

2 (p-p

n

) 2

ofujp(a-p)

2

6M(a 2

+p

2

) 2

(p

2

+p

2

)

( IF- 2

)

and

2rce 2 (p -p ) 2

(MB) 2 (M(urp)) z (a-f3)

2 a

n (Ml) = (IF- 3

)

u

3 M'

3

uj [M(uj-B)+r]

2 with a = MB for the Ml contribution to D+y^p+n. The parameters appropriate

for the bound deuteron (B = 2.22 MeV) are a = 0.232 fm , r^ = 1.76 fm while

p of the singlet s-wave continuum is the inverse of the singlet np scattering

length p = a

^ = -0.042 fm The Ml photoabsorption cross section is much

smaller than the El except within a few keV of threshold.

The reaction which is most sensitive to the Ml transition is the inverse

of the photoabsorption viz. radiative capture n+p -» D + y of very slow neutrons.

For neutrons thermal ized to room temperature (T = 300 K) the lab momentum p

n =

2p is

P

2M

kT = 0.026 eV (IF- 4)

* -]_

Note the effective range correction to the normalization factor (1-a r^)

is not used for the S->S transition because the overlap integral peaks at r=0.

However other effective range corrections modify eqs. IF- 2 and 3 slightly

(No 65).
17

1 3

and the S Q

-> S^, transition is dominant. The relation between the photodisintegration and radiative capture cross sections for this transition is

given by

CT cap 2 2CT dis

P

( IF- 5

)

ment yields 334 mb (No 65). The 10% disagreement between these two numbers

was a puzzle for more than 20 years. The missing 10% of theoretical cross

section has now been found to be due to explicit meson and nucleon resonance

effects in the interaction of photons with nucleons in the presence of other

nucleons as discussed in Chapter IV. These two (or more) nucleon effects,

collectively called meson exchange corrections are explored in more detail in

the next chapter.

where the J's are the spins of the respective particles and uj and p are centerIn general the detailed balance for the photoreaction A+y t B+C is

of-momentum variables.
18

G. PHOTON ABSORPTION IN A RESONANCE

A nucleon bound in a harmonic oscillator potential,

V(r) = j

M tu 0

r 2

(1G-1)

where uj^ is the energy spacing between major shells, can absorb a photon by

making a transition to the next higher unfilled shell. The cross section for

El absorption on a proton is given by

2 2

a(uj) = ^nr-K'h P-p/M l Vi

2

• (ig-2)

The density of final states for a resonance transition is unity. As an example

an S-*P transition has an amplitude

El

eN N P s / 'io

(f > -

e • r Y (r) * f

r e -T/2b2

J_

. oo b 3r J o

-r

2 /2b 2

2,

e r dr

V2

b

(1G-3)

-2

where p = - i V has been used for the current operator and b = relates

the oscillator parameter to the energy level spacing. Evaluation of the cross

section gives a spike at the inter shell spacing

ct(uj) =

2

-|vj

e 5 (uj-uj

0

) . (1G-4)

The harmonic potential is a good approximation for bound single-particle

states in a nucleus, but it lacks the damping due to decay channels for continuum

states. If we add a damping term to the oscillator Hamiltonian u)q -> uJq -i T/2

the photon absorption cross section has an energy dependence given by the

Lorentz form
19

a(io) =

4ne‘ 2 r uj I M . 2 2.2 2 r

2

(.UJq “10 J +U) 1

preserving the area

I

a(uj) duj =

, 2 2

2n e M

( 1G- 5

)

( 1 G - 6

)

Note the area is independent of uJq and V.

Complex nuclei have particles in many oscillator levels, but nucleons can

only make dipole transitions across one major shell to unfilled orbitals.

Thus only the last neutron and proton shells contribute to electric dipole

absorption. Since all photon transitions occur with uj = uj^ the total cross

section is a superposition of single-particle Lorentz lines. Figure 1G-1

shows the total absorption cross section for gold and a fit to the data by a

Lorentz resonance line.

The phenomenon of the concentration of electric dipole absorption cross

section around one photon energy is called the El giant resonance. The interpretation of the resonance as a collection of independent harmonic oscillator

transitions suffers from the fact that the observed resonance frequency is at

twice the energy as that inferred from other measurements of ujq. This problem

is discussed in Chapter V.
20

H. ELECTRON SCATTERING - THE CHARGE OPERATOR

When high energy electrons scatter from the nucleus the electromagnetic

quanta exchanged in the interaction are very similar to real photons. As will

be discussed in the next chapter, one may assume only one virtual quantum with

an energy above a few keV is exchanged between the electron and the nucleus.

This quantum differs in two respects from a real photon. First, the momentum

transfer to the nucleus

k = ki - k

2

(1H-1)

(where k^

^ are the initial and final electron momenta) is always greater than

the energy transfer to the nucleus

uj = E x

- E 2 ^ k x- k

2

. (1H-2)

This is in contrast to a real photon for which energy and magnitude of momentum

transfer are equal, uj = | k[ . In this respect, electron scattering gains a new

degree of freedom over photoabsorption since the momentum transfer in a reaction

can be varied independently of the energy transfer. This feature of the

electron scattering quantum gives it the name "virtual photon."

The second difference between real and virtual photons is that the virtual

quanta have a component of the vector potential A^ parallel to the direction

of propagation k of the virtual quantum and a non-zero fourth component cf> in

the four-vector potential A^ = (A,p). The scalar field p is the Coulomb

potential created by the electron at the position of the nucleus. Thus, in

addition to the transverse current and magnetic moment interaction terms j - A

j

and m*(VxA), the virtual photon has a longitudinal current interaction j*An

and a Coulomb interaction ep with each proton.
21

To round out our illustrative examples of electromagnetic interactions of

a nucleon bound in a potential, we calculate the Coulomb contribution to electron

scattering from a point proton bound by an energy B in a zero-range potential.

The operator for Coulomb interaction with a proton is simply

Hnt = e<j> = e exp(ik-r) . (1H-3)

Since one of the final nuclear states possible with the Coulomb transition operator is the ground state, elastic electron scattering is possible. The transition

amplitude is the Fourier transform of the ground state density

T

ei = O0

( r ) |exp (ik-r)| 4< o (r)> . (1H-4)

For the zero-range wave function

T , = ^ tan"

1

-f- . (1H-5)

el k 2a

The continuum states in the spin independent zero range model are

ip • r

e K +

s i n ( p r+6

)

pr

* giving an inelastic amplitude

sin p r pr

(1H-6)

T

i nel

<i]i(p,r) |exp(ik*r)| ^Q

(r)>

* Note that although the s-wave continuum state (without spin flip) is orthogonal

to the bound state, the spatial dependence of the s-wave part of the operator

makes the contribution to the overall amplitude non-zero.
22

(2a)"5 (4n)^

2 ^ ,r -,2

a + (k-p)

(e

1(^cos 6-1) . a

2 + (p+k)

2 4pk 2 , . ,2 a + (p-k)

i6 . . . e sin 6 /, -1

+ -—--——— I tan

2a k

2pk 2 2 .2

,a +p -k

+ zn (1H-7)

where

tan 6 = - p/a, sin 6 = -p/(p

2 +a2 )^, cos 5 = a/(p 2 +a

2 )^ (1H-8)

and

s =

2 2 2

0 for a + p

- k > 0

2 2 2 . 1 for a + p

- k^ < 0 ) .

The electron scattering Coulomb cross section (whose derivation will be

discussed in the next chapter) has an elastic part

da

el

dQ

a

o el

(1H-9)

where a is the Mott cross section

o

a

e cos 6/2

L

2k

x

2 sin 2 0/2

and an inelastic part

(1H-10)

,3 d a dQdk„ CT o ^i nel

2 d P

3

6 ^" B "2 m)

(27i) J ^

(1H-11)

f

P

2 6(uj-B-^) dp = Mp
23

If the recoil proton is not detected the scattered electron spectrum is

d 2 2a

dQdk, = a F. , ( k ,ui) inel v ’ J T. ,

1 nel

9 Mp dfi 2 P

( 2tt ) (1H-12)

ion.

? At higher

= The

scattering is then called quasi- free .

2 i 1

As the momentum transfer increases, the elastic form factor F ,(k) = T ,

decreases while the area under the inelastic spectrum increases. A remarkable

sum rule governs the interplay of elastic and inelastic scattering from a particle

bound i n a wel

2 Figure (1H-1) shows] F^ i (k,uj)| the inelastic response funct

momentum transfer (k>>a) the peak shifts to p=k making uu

el

(k) F. ,

l nel

( k,w) duj = 1. (1H-13)

This results from the completeness relation of the eigenstates of a Hamiltonian.

A nuclear system with more than one proton has a slightly more complicated sum

rule (see Chapter III).

I. APPLICATION TO THE ELECTRODISINTEGRATION OF

3

He

3

The breakup of He into p + d and p + p + n by electrons is dominated by

the Coulomb interaction for scattering angles away from the extreme forward

3

and backward directions. A calculation which treats the He ground state as

a proton bound by a zero-range potential to a deuteron can be used to compute

tha He (e,e‘) p d cross section near the two-body breakup threshold at

3 uu = B , = 5.5 MeV. Above tu- = B = 7.7 MeV, however, the He ( e ,

1 ) p p n

2 pd 3 ppn

’ *

reaction also contributes to the inelastic electron scattering cross section.
24

The three nucleon breakup channel can be calculated by treating the deuteron as

a proton bound to a neutron by another zero-range potential with B = 2.2 MeV.

3

The He ground state wave function is then a product of zero range wave functions

4*0 = ex P (-Yr-crp) . (11-1)

The pd and ppn breakup channels are represented in this model by products of continuum wave functions (eqs. lA-4,5) in the r and p coordinates. The electron

scattering calculation using the charge operator acting on the two protons can

then be carried out as described in the previous section. The result is compared

with the data in fig. ( II - 1) . The initial sharp rise in the inelastic scattering

cross section is explained by the theory as due to s-wave p + d breakup and is in

agreement with the known p + d s-wave scattering length.

This reaction is treated in Chapter IV using more realistic three-nucleon

wave functions, but it is interesting that the qualitative features of the

data can be understood in this simple zero-range model.

J. THRESHOLD PION PHOTOPRODUCTION

The liberation of pions from nucleons is the dominant reaction of photons

above uj = m - 140 MeV. Figure ( 1 J- 1 ) indicates the relevant kinematic variables,

71 fluxes, and coupling constants. The nonrel ati vi sti c amplitude near threshold

for charged pion photoproduction

V

; = <N 'I vri * (*

1 2r)l N> (1>1)

involves the nucleon spin operator and the isopin raising and lowering operators.

The ( + ) on the right hand side of (1J-1) goes with yn-pn: and the (-) with

yp-n7i +

. The center-of-momentum differential cross section can be calculated

from
25

da =

2n

V

" V Kl| Y N| m) t

2

,3 d q po1 ' ( 2;1

)

3

dW

avg.

(1J-2)

UJ where W = uj (1 +

-^j] is the total CM energy available. Reduced mass factors

appear from the initial [1 + and final fl +

-pj (relative momenta. The

m

resulting expression near threshold is

dn )

\ m 2 ) / \

/ m * y (i*8)(W

using f = 0.08

3

, k,

(10-3)

k da

q dQ (10-4)

15 pb/sr (yp->n7T )

20 pb/sr (yn^p7t )

Figure (10-2) shows a comparison of eq. (10-3) with the data.

In Chapter VII, we shall explore the influence of nuclear structure and

pion-nucleus interactions on the (y,n) process.

CONCLUSION

We have tried to show in this introductory chapter some examples of how

the simplicity of the electromagnetic charge, current and magnetic moment

operators permit us to calculate the cross sections of real and virtual photon

reactions. Many fine points have been glossed over, but the agreement with

data on hydrogen and helium photoreactions is impressive. In the following

chapters we shall elaborate on: the electromagnetic operators for real and

virtual photons, sum rules, more realistic wave functions for few nucleon

system, many-nucleon systems and collective motion, direct reactions and the

influence of the nuclear medium on elementary processes, the optical

properties of nuclear matter, the explicit role of mesons in photoabsorption,

and the high uj dependence of photonuclear reactions.
26

References for Chapter I

(Be50) H. A. Bethe and C. Longmire, Phys. Rev. 77, 647 (1950).

(Sk74) D. M. Skopik, Y. M. Shin, M. C. Phenneger, and J. J. Murphy, Phys. Rev

C9 , 531 (1974).

(Be76) B. L. Berman, "Atlas of Photoneutron Cross Sections Obtained with Mono

energetic Photons" UCRL-78482

(Ka75) P. T. Kan, G. A. Peterson, D. V. Webb, Z. M. Szalata, J. S. O'Connell,

S. P. Fivozinsky, J. W. Lightbody, and S. Penner, Phys. Rev. C12,

1118 (1975).

(Ad68) M. I. Adamovich, V. G. Larionova, S. P. Kharlomov, and F. R. Yagudina,

Sov. J. Nuc. Phys. 7, 360 (1968).

(No65) H. P. Noyes, Nuc. Phys. 74, 508 (1965).
27

FIGURE CAPTIONS

Fig. (IB-1) a) Bound state wave function with a = 0.694 fm ^ corresponding to

10 MeV binding. R marks the nuclear radius, b) The £ = 0 and £ = 1 continuum

wave functions with p = 0.694 fm ^ corresponding to 10 MeV kinetic energy, 5 =

-0.785 radians, c) Integrand of the electric dipole amplitude at 20 MeV photon

energy. Note the main contribution to the radial integral comes from outside

the bound state radius R.

Fig. (1C-1) Coordinate system for calculating the average of the square of the

transition amplitude over the two photon polarization vectors using the relation

(s x

‘ P)

2 + (e

y

* P)

2 + (k-p)

2 = p2

Fig. (1C-2) Electric dipole cross section for a photon bound by energy B in a

2

zero-range potential well, eq. (1C-7). The cross section is in units of (2e /BM)

and the photon energy is in units of B.

Fig. ( 1C- 3 ) Comparison of expression ( ID- 1 ) with data for the reaction D(y,p)n

(Sk74)

Fig. ( IE- 1 ) Radial wave functions for an s-wave transition. The parameters are:

a = 0.694 fm \ (3 = - a, p = 0.694 fm \ o = + 0.785 rad. The units are: p

Q

in

fm in fm, and the overlap integrand p^ p

Q

in fm

+2

.

Fig. ( IE- 2 ) Magnetic dipole cross section for a nucleon with magnetic moment p

bound by energy B in a zero-range potential well, eq. ( IE- 3 ) . The cross section

2 /

is in units of (2e BM) and the photon energy in units of B.
28

Fig. (1G-1) Photon cross section of gold as measured by neutron emission. A

Lorentz resonance with parameters uu Q = 13.82 MeV, r = 3.84 MeV, ct(uj o

) = 560 mb

gives a good representation of the data (Be76).

Fig. (1H-1) Inelastic Coulomb form factor in the zero-range model as a function

of energy transfer for two values of the momentum transfer with B = ^

= 10 MeV.

3

Fig. ( II- 1) Cross section for the reaction He(e,e') compared with a calculation

in which the Coulomb interaction is used with zero-range wave functions for the

^He ground state (Ka 75).

Fig. 1J-1) Feynman-type diagram for threshold photoproduction of charged pions.

The nucleon is treated nonrelativistical ly but relativistic kinematics are used

for the pion.

Fig. ( 1 J- 2 ) The differential cross section in the center-of-momentum system

(Ad68) compared with the theoretical expression of eg. (1J-3).
fm 2

FIG. I B- I

OJ

I

CD

X

FIG.

1C
006

roI o o O o O o

00 ID iO rO

Oro

00

C\J

<o

C\J

^3“ 00

00

00o

oo

CO

CO ( qr/) JD >

QJ 3

FIG.

1C
I UJ

o

Lt.
=

10

MeV

o

fOO

OJ

o

I o

o

-

CJ/B
U) (MeV)

Fig.* (1G _ 1). Photon cross section of gold as measured by neutron emission. A

Lorentz resonance with parameters u> Q = 13.82 MeV, r = 3.84 MeV, a(ui ) = 560 mb

gives a good representation of the data (Be76).
(MeV

o

I

o

Ljl

CJ(MeV)
0.8

. 3 Fig. (11-1) Cross section for the reaction He(e,e') compared with a calculation

in which the Coulomb interaction is used with zero-range wave functions for the

3

He ground state (Ka75).
<UJ

fc\J
ft

b/sr

OJ

(MeV)
40

CHAPTER II. THE ELECTROMAGNETIC OPERATOR

A. Real photon transition operators

B. Virtual photon transition operators

C. Meson exchange currents

D. Coulomb distortion and radiative processes

E. Multipole analysis

F. Long wavelength expansions

G. Coincidence measurements

H. Relation of weak and electromagnetic interactions

I. Real and virtual bremsstrahl ung
41

CHAPTER II. THE ELECTROMAGNETIC OPERATOR

The formal development of the electromagnetic interaction operator with

nuclear constituents follows from quantum field theory and the Feynman analysis

of quantum electrodynamics ( B j 6 4 ) . In practice, photonuclear calculations are

performed with the nonrel ati vi sti c reduction of the elementary currents to

give operators for use with model si ngl e-particl e wave functions (De66,Do75).

The intent of the present chapter is to introduce enough background to give

physical insight for calculations to be discussed in subsequent chapters.

A. REAL PHOTON TRANSITION OPERATORS

The interaction energy of the electromagnetic field (described by the

vector potential A (r\t) = ( A ,

cf» ) with a nucleon current j (r,t) = (j,p) is

H H

H Y(2A-1)

where p is the Coulomb potential and p the charge density. For a real photon,

i.e., a wave packet of electric (E=-3A/3t-Vp) and magnetic (B=VxA) fields for

which the energy tu and momentum k are equal in magnitude and the E, B, and k

vectors form an orthogonal set, the p can be chosen zero (transverse gauge).

The vector potential of a plane wave is written as

A = e exp (ik-r-iwt) (2A-2)

where £ is a unit vector in the direction of the electric field and at right

angles to the direction of propagation (£*k=0). The magnetic field unit

A vector is (k x e).

The single nucleon current has two sources j = j + m^ when treated nonrel ati vi sti cal ly . The orbital current due to the velocity of the proton
42

J, = £R

'p M

and the magnetic moment of the proton or neutron

(2A-3)

— _ euo

m — —xn

a 2M (2A-4)

where the nucleon magnetic moment is expressed in nuclear magnetons (e/2M).

The orbital current has an interaction energy j *A while the magnetic moment

interaction is iff *B = m^-(VxA). The total interaction energy of the nucleon

with a plane wave photon is

H =

Y

ef

l£ + 1 Im ^ kx^’ CT

~W * 1 Im (3xk 4

exp( i k* r- i tut

)

exp(i k* r-imt)

The photon absorption cross section is given by

da = 2n fe)E |<"f|£ H(J)|VI 53

(2A-5)

(2A-6)

where the bar over the sum is a short hand for averaging over spin components

of the initial state summing over the final state components )

and averaging over the two values of photon polarization A = ±1 for helicity

states (or A = x,y for plane polarization)

E . M

f

Z ,A,M.

2 (2J.+1)

£, i k • r\ . r* i k • r V(e ) = ike

/• - i k • r\ • / - i k * r V* (se ) = l (e • k)e

- i k* r N ./h M i k • r Vx(ce ) = i(kxc)e

(kxe) -a = e • (axk)

Useful identities
43

The factor (4ti/2uj) is the square of the photon normal i zati on factor and dn/dcu

is the density of final nuclear states. The leading 2tt is from the Fermi

Golden Rule for transition rates. One may associate the transition amplitude

<4i f | H |t|i.> with the diagram shown in fig. (2A-1).

We saw in Chapter I how model wave functions for 4^. and and approximations

for lead to cross section expressions for a number of reactions that agree

with measurements.

The usefulness of photon reactions derives in large measure from the

simplicity and reliability of eqs. (2A-5,6). Compared to strong interactions

the photon has the virtues of not being distorted as it approaches a nucleon

inside the nucleus and interacting with only one nucleon at a time. This last

2 -1

property follows from the smallness of the coupling constant e = (137)

The expressions (2A-5,6) can be derived by a nonrel ati vi Stic reduction of

the S-matrix expressions of quantum electrodynamics. The nucleon current is

j = eu[F,(k^)y + ^ F J 0 (k^) a k ]u (2A~8)

p

L lv p

/3

p 2M 2

v p' pv

v '

where u is the nucleon two-component spinor, F^

^ the Dirac and Pauli form

factors, k the nucleon anomalous magnetic moment, y the Dirac matrix, and

r a = i/2 [y ,y ] (Pe74).

Elastic photon scattering by a nucleus involves terms of second order in e

from The amplitude f(uu,0) for this process can be calculated from the

three diagrams of fig. 2A-2. Diagram la is the coherent scattering by the total

charge of the nucleus. At moderate energies this is the energy- i ndependent

Thomson amplitude

= - (Ze)

2

a AM

F ( k 2

-k

2

) (2A-9)
44

modulated by the nuclear form factor. In diagrams lb and 1c the intermediate

nucleus is excited to state n> with energy E ,

< f !

- > : v n>< n | H I i>

' Y

1

n

E + uj n v ^ <f H - > 1 Y n>< n I H 1 i>

1 Y* L-J

n

E - uj n (2A-10)

The coherent nuclear scattering of photons interferes with other extranuclear

and atomic processes: Delbruck scattering (associated with e +

e pair production)

and elastic scattering from electrons (atomic Rayleigh scattering). The

Delbruck and atomic Rayleigh amplitudes are peaked very strongly forward and

can often be neglected at backward angles.

In principle the photon scattering cross section could be computed with

model wave functions for f., ¥ , 4^; however in practice one uses the optical

theorem relating the imaginary part of the forward scattering amplitude f to

the photon absorption cross section for each multipole L,

4ti Im f^(uj) = uj ct *"(uj) , (2A-11)

and a dispersion relation (see Chapter III) between Re f'

_ (uj ) 0) and Im f^(iu,0)

to compute each fk Then

do ' ' UL '

L,L

g i_ L

i (cose) f f (2A-12)

where g^, is a geometric factor ( E i 7 0 ) . The absorption cross section is

easier to compute than f(uu) with model wave functions since the sum over

intermediate states is not required.

Photon scattering and absorption and their relation can be seen in the

S-matrix expressions
45

f L (ui,e) = (2L+1) P

L (COS0) [S

L (io)-l] (2A-13)

o^

Qt = (2L+1) 2(l-ReS

L

(u))) . (2A-14)

k

Both processes tend to be dominated by resonances (poles in f(w,0) in the

lower half of the complex energy plane E

n = u) -ir/2). A Breit-Wigner form

gi ves

S

ir

(uj q

~ ir/2)-tu (2A-15)

B. VIRTUAL PHOTON TRANSITION OPERATORS

The analysis of electron scattering in the one-photon exchange approximation

leads to a factorization of the reaction amplitude into a purely leptonic part

(the eye

1 vertex) times a hadronic part (the NyN' vertex) (De66). The electromagnetic quantum that propagates between these vertices has a momentum greater

than its energy (k>uj). It is called a virtual photon to distinguish it from a

real photon for which k=uj. The distance over which a virtual quantum propagates

can be estimated from the uncertainty principle. The mismatch between momentum

k and energy uj is allowed to travel a distance Az = (k-w) ^ before attaching

to a nucleon. Typically this distance is smaller than the nuclear radius

implying the lepton vertex is inside the nucleus, that is, the impact parameter

for electron scattering is a few fm or less. It is assumed that the eye

1

vertex is unaffected by the nuclear medium.

Electron scattering from a nucleon in a nucleus can be considered as a

generalization of the real photon-nucleon interaction. Using the kinematic

variables k and uu defined by fig. (2B-1) the vector potential A = (A,(j>) now

r
46

has a non-zero Coulomb part and A has a longitudinal component k*A plus two

transverse components (A=±l) which add incoherently. The interaction energies

become

H = j A = ep

e p K

+ e

+ e 'A

+

!m

exp(i k* r) (2B-1)

The first term on the right of (2B-1) is the Coulomb interaction, the second

the longitudinal current, the third the transverse current, and the fourth the

magnetic moment interaction. The Coulomb and longitudinal current operators

* are not independent. By current conservation one can replace their sum by one

operator

A —a. p + ^ = (1 - f

)p (2B-2)

A nonrel ati vi sti c reduction of j^A^ in powers of k/M and p/M yields the

following expression for the inelastic electron scattering cross section on a

nuclear target in the one-photon exchange approximation with m

g = 0 (Mc62,De66)

V • i + = 0 , k • j + u>p = 0

J p at P V*m =0.
47

mtz (k > u, ’ e) = CT

k

4

k

4

F c (k,u.)|

2 k

2 e

2 + tan2

t| F

B

< k *“> F

(J (k,u) )|

4

]

e cos 0/2

2 sin 0/2

2k1 sin 0/2 . , 9 9 1 * \ j

f

N

^ '

2 1

cl 2J.+1 M M l f

,M

i E

a

'

l<

J

f

M

fl 2 e

j

exp(ik-r.)|j.M.>

I 2 _ X

F P +F

K

ctI i, M , X1 i M^,M. S A=±l S wie -L^

L— exp ik*r.

2 m

+ 9^ (cr-xk)^ exp ik-r. |J.M,.> Jlii

2 /dn

(2B-3)

where a

Q

contains the Mott cross section, target mass M^ recoil factor, and

the average charge and magnetic nucleon form factor given approximately by

9 p G

P g"

f

/ k

2x = G

P = _Ji_ = _M_ Vp; r E p p *p *n

1 + J

2

a (2B-4)

with a = 855 MeV and G" = 0. The density of nuclear states dn/dE is unity

for a discrete level. The charge projection operator e. = 1/2 ( l+x

z ) j

gives

1 for a proton and 0 for a neutron. The Coulomb, current, and magnetic

moment form factors F are averaqed and summed over initial and final

c,p,ct a angular momentum states, but in contrast to a real photon, the polarization

projections k = ±1 of the transverse components of p and cr are summed (not

averaged). The Coulomb part has the k = 0 longitudinal component.

The dipole form corresponds to an exponential distribution in space. The

2 2 ^ radius parameter associated with this value of a is <r > = 0.81 fm.
48

The single-particle expression given in eq. (25-1) is an approximation

2 2 since small terms of order (k/M) and (p/M) were neglected. A form correct

2 2 through order (k/M) and (p/M) is (De66)

P ik-C^x^

cF

4M

Z /

1 + 2kF2

)

(2B-5)

r" _ . axk fr J P N J

ct

1 2M(F

1

kF

2

4

The non-rel ati vi stic operators through the order (k/M) are given in (Gi80). The

Dirac and Pauli nucleon form factors are related to the electric and magnetic form

factors by

g

e - f 2 + n k f 2

Gm = F 1

+ K f

2

(2B-6)

2 2 where n = k /4M . M For reference, we give the ful ly-rel ati vi sti c elastic and inelastic

electron-nucleon scattering cross sections,

2

da

dQ

Gm 2n

e cosQ/2

2k-^si n 2 6/2

1

G

E /(i+n) +

2(i+n)

+ t£n

2 (2B-7)

d

2 a

dQdai

e cos6/2

2k-^si n 20/2

W

2

(k^ ,uj) 2V/ 1

(k

|J ,uj)

. 2 6

tan

2 (2B-8)

An alternative expression for the inelastic electron scattering cross

section (for nucleons or nuclei) can be given in terms of two fictitious total

absorption cross sections Oj and ct^ for virtual photons (Pe74)
49

dQdtu

1~8

2

(2B-9)

r

L = e r

T

, £

The r's can be interpreted as the number of virtual photons associated with

the inelastic scattering process, s is related to the polarization of the

i.e., the transverse term becomes the real photon absorption cross section at

the same energy transfer ui.

C. MESON EXCHANGE CURRENTS (MEC)

Other currents in the nucleus are due to charged mesons (mostly pions)

and to isobars, excited states of the nucleon, (mainly the delta). The electromagnetic E and B fields couple to the meson and isobar currents to give contributions to beyond the single-nucleon impulse approximation discussed

in the previous two sections ( R i 7 9 ) . The electromagnetic interaction operators

are constructed by making a nonrel ati vi sti c reduction of those two-nucleon,

one-pion diagrams considered to be the most important. The products of vertex

functions and intermediate propagators are taken.

2 photons and K = - k^/2My is the energy of a real photon leading to the same

hadronic final state. The cross sections have the low k limit
50

The vertex operator for the N71N interaction, fig. (2C-la), is

jjp- °*q * (2c-i)

7T

2 where f is the coupling constant (f = 0.081), a and x the nucleon spin and

isospin operators, and q the pion momentum. Fig. (2C-lb) shows the NnA vertex

in which the J

71 = 3/2 , T = 3/2 isobar resonance (M = (1236 - i60)MeV) is

formed. The interaction operator is

— S-q T

m

M (2C-2)

71

where S and T are the spin and isospin operators for a 1/2 to 3/2 transition

f = f in the quark model (Ri 79).

The long range (r > 2 fm) two-nucleon force is thought to arise from

one-pion exchange (OPE), fig. 2C~2a, with an interaction potential given by

the product of the two NnN vertices divided by the pion propagator

5

q)(i

1

-x

2 Hcr 2 q)

OPE

q

2 + (2C-3)

The intermediate range potential (1 < r < 2 fm) is thought to arise from

the intermediate-delta two-pion exchange shown in fig. 2C-2b. Four vertices

and four intermediate propagators are now involved.

The principal electromagnetic couplings to the pion and delta currents are

* shown in fig. 2C-3. Diagram 3a is called a pair current; its operator is

(Ho73)

* The name "pair" comes from the nucl eon-anti nucl eon intermediate state at the

•ytx vertex before the nonrel ati vi sti c reduction. This diagram is also called

the "sea gull" or "gauge" term.
51

ef2

£ - , -Jpair

= — < T

lx

K m Vz

71

(e-CT

1

)(CT2 *q

2

)

2 , 2 q 2 + m

7i

(e*a

2

)(a

1

*q

1

)

2 2

q-, + TM

F x

v (k)

where is the nucleon isovector form factor.

Diagram 3b is the pion current

(204)

* • ef ,-*> — . £

' 2

t, - — (t l

X

m Vz

n

e * ( 51

'- ) ( a“i

• ) ( o

z

• q 2

)

t 2 ^ 2 W 2 a 2v

(q l + V (q 2 + V

F n(k) (2C-5)

where F is the pion form factor.

71 K

The creation of a virtual delta shown in fig. 203c is given by (Mo78) a

magnetic coupling

eff

(exk) • j = —j

m

71

Z/*

P( T

1

*X

2

) T

1

(exk)-S

1 (q 2

*CT 2 )(q

2 *S

1

)

(M

2 - M2

) (q

2 + m2

)

FA(k) (2C-6)

where F A is the delta transition form factor.

These three operators have the following features:

(a) They are transverse and isovector in character. Their forms follow

from the dominant photopion production mechanisms discussed in Chapter VII and

the NN pion exchange potential.

(b) They are two-nucleon operators and are to be evaluated between

two-nucleon wave functions in analogy to the NN potential where meson coordinates

are also suppressed.

(c) The charge distribution of the nucleus is not altered by these

isovector currents. The pions can be thought of as moving instantaneously

between the nucleons.

The connection between the OPE potential and pion exchange currents can

be seen by considering the expression for total current conservation expressed

in energy-momentum space
52

k*j =

up = -i [K+V, p] (2C-7)

where K and V are the one-nucleon kinetic and the two-nucleon potential energy

operators. The charge density is given by

P(r) = e 2 6 ( r- r\ )

2

When the commutator of p is taken with the OPE potential eq. 2C-3, using

(208)

[(i

1

-i

2

),i

1z ] = i] il - i+ i? = 2i (f

1x

;r

2

) z one finds that if F = F^

v in eqs. (2C-4 and 5) then

+ r„) = ->'

t V

0 PE‘ P ] and Hp = "I C*. p] •

(2C-9)

(2C-10)

The commutator of the two-nucleon potential with the charge density is related

to the two-nucleon pion currents while the commutator of the one-nucleon kinetic

energy with the charge density is related to the single-particle current j

.

The isoscalar MEC are less certain. They are usually smaller and of the

2 2 same order as relativistic corrections (p /M ). Currents due to higher mass

mesons and isobars are thought to lead to smaller transition amplitudes because

of the anti correl ation of two-nucleons at small interparticle distances.

Additional one-body currents are generated by resident nucleon isobars,

fig. 2C-3d.

D. COULOMB DISTORTION AND RADIATIVE PROCESSES

Eq. (2B-3) for electron-nucleus scattering is derived using plane waves

for the incident and scattered electrons. Although this approximation is

valid for few nucleon systems, it becomes increasingly inaccurate for larger Z.

The distortion of the electron wave by the Coulomb field of the nucleus can be

found by using the Dirac equation with the monopole part of the nuclear charge
53

distribution. The equation is expanded in partial waves and solved numerically.

The resulting asymptotic phase shifts are summed to obtain the complex electron

scattering amplitude as a function of scattering angle (Ub71).

Additional distortions due to deformation of the charge distribution,

magnetic moments of the nucleus (for J > 0), and virtual excitation and deexcitation of intermediate nuclear states are not included in the usual treatment, but are estimated to be small.

An approximate correction to the plane wave Born approximation (PWBA) is

the replacement of the asymptotic value of the momentum transfer by its value

inside the nucleus

k

eff = k (1 + 1.16 ZeVC^R)) (2D- 1)

2 t

where R is the charge radius. The "experimental form factor" (d ct/ct )

2 plotted

as a function of k

g

^

will correspond more closely to the nuclear response

F

L

2

(k,u>) + (

1-2+tan

2 0/2)F

T

2

(k,uJ ).

For inelastic electron scattering the distorted wave Born approximation

(DWBA) is the most widely used approach. In the DWBA the incoming and outgoing

electron waves are calculated using the phase shift method discussed above

while the nuclear excitation is calculated treating the virtual photon as a

plane wave exp(ik*r.). The distortion mixes the Coulomb and transverse response

J functions so that a model independent separation of the two form factors

cannot be made.

A further complication in the calculation of the electron scattering cross

section for a nuclear transition is the radiation of real and virtual photons

during the scattering process (Ma69, Mo69). These photons affect the shape of

2

d ct for a discrete transition (with internal nuclear excitation E ) adding a

radiative tail that extends from

k

2

E x (i - E x

/2M

t )]/(1 + 2kx sin 2 |/M

T

) (2D-2)
54

to lower energies. This tail complicates the extraction of the transition

strength of the level (which in real photon scattering would be fit by a

Lorentz line) and forms a background under all nuclear excitations of higher

E . The estimation of discrete peak strength (radiative correction) is usually

accomplished by fitting the data near the peak with an exponential form.

The estimation of the background cross section underlying a peak from

bremsstrahl ung accompanying transitions of lower nuclear excitation (radiation

tail subtraction), as given by evaluating fig. 2D-1, can be expressed approximately

by

= E

dQduj x m ovV f(k

i

} + i (k T «k

2+ V

(2D-3)

where dcr/dQ is the calculated radiationless cross section for exciting a

specific level and F is the radiation function giving the spectrum of real

photons. (See Section I.) The summation adds the tails of all nuclear transitions with less than the level in question.

In addition to the radiative corrections to the data, ionization energy

loss, bremmsstrahl ung in the field of another nucleus (thick target effects),

and detector resolution must be taken into account before comparison with

nuclear model calculations (Be71).

E. MULTIPOLE ANALYSIS

The notational simplicity of the nucleon charge, current, and moment

operators

[e, e'p/M, iep(crxk)/2M] exp(ik-r)

must be sacrificed when these operators are to be sandwiched between singleparticle wave functions of definite angular momentum. The (e,e‘) cross section

for a discrete transition J. -» (analogous to eq. (2B-3)) is written in terms

of multipole amplitudes as
55

da

dQ

a

o (2J.+1)

.[J]

J.>!

1

M

CJ] J.>

1

1

(2E-1)

r j 1 jr j i The reduced matrix elements of the Coulomb C

L

, electric E , and magnetic M

L J

multipole operators are formed by decomposing the operators in eq. (2B-3) into

partial waves carrying angular momentum J and definite parity.

The usual formula for the expansion of a plane wave into spherical Bessel

functions and spherical harmonics is

exp(ik-r) = £ (4n)'2 L (i)

L j

L (kr) Y

LQ

(f) (2E-2)

A where L = (2L+1) and the z~axis has been chosen along k. In this section, we

will work with the contrastandard elements which are related to ordinary

spherical harmonics by

y m

L] = ( " 1)L

Y LM

(9 ’ $) (2E' 3)

This notation permits us to treat the angular momentum properties of operators

on the same footing as wave functions (Fa 59, Da71 ,0c72a , Le73)

Defining the Coulomb multipole operator as

c£

J]

= (4n) % E(-1)J ej j J (kr

j

) Y^

J]

(f' J.) , (2E-4)

the Coulomb form factor can be written as

2

F c (k)| = | <4< f | Eej exp(ik-r\.)| • > £ IrCJ]

2 J • + 1 M i. (J) I <J

f

M

f

|

Q

J . M . > i l i

• (2E-5)

* Components of vector operators are written as v^
56

Using the Wi gner-Eckart theorem

F c<k) l = 2J.+1 E <j

f !|c [J:l

ij j^l

2

(2E-6)

The current and moment operators are more complicated because they involve

the coupling of a (from the plane wave) with one unit of angular momentum

(from the vectors p or crxk) to form an operator with total angular momentum J,

i k - r

e pA

= (4n)"2 E 0) L

L J L (kr) Y

LQ

(r) p^

= (4I0

1 * E (-d l+1

* L j L y^

l]

p[

1]

= (4ti)"2

i E L j

LJ

L 1 J

0 A -A

j L [Y [L]

x pTM]TM .

(2E-7)

In the last step the uncoupled product of two elements was replaced by its

expansion in terms of elements coupled to a definite angular momentum.**

Similarly, for the moment operator

e ik ' r (5xk) = (4n) h

kX E L J (

L 1 J

1 j, [Y

[L]

x a

[1]

],

CJ] (2E-8)

A

LJ \0 X

1

J A

where we have replaced the vector cross product by an angular momentum coupling

(Gxk)^ = l

2 [a [1 ^ x k [1]

][

1] = A k . (2E-9)

<j

f Mf

: 9

f J] |J.H.> = mEV Jf ° E <jJ|e^l'j.>

f fI A ' 1 1 \-M

f

X M .)

f

' 1

**

[j-,] [j

A

1

B

J *

3 = E

J , A

< J

" 1

m

1

J

* 2 m

2

[j-|] [jo] r j

"I JA> [A x 3

]j^

JJ
57

If we hold J fixed, L takes on three values: J+l, J, J-l. The parity of

each current term is n = but each moment term is 71 = (-1)*" since a is

an axial vector. Therefore, to form an operator of definite parity, we combine

the L = J ± 1 current terms with the L = J moment term to form an operator of

parity 71 = (-1)^. This combination is called an electric operator because

the classical electric multipole field has this parity. The n - or

magnetic combination is formed by the sum of the L = J ± 1 moment terms

with the L = J current term.

* Defi ne

Z

[J] - i ,- lVJ f_L_)

2 E

\

1 ( 1}

\ 2J+1/ M

/\

e(J+l) J P1

J

M j J+1 [Y^ x pW]fJ1

/\

+ e(J-l) J \J-1 1 J

v 0 A -A,

[J-l] w

j J_ -L [Y L X p

L J

] A

+ ^ JJA (

J 1 3

) jj [Y

[J] xo[1]

][

J]

|, (2E-10)

0 A -A

The single particle label j and its summation has been suppressed in e.,

<jj , j(kr -), and Y(fj) for notational simplicity.

)
58

**

J] = i <- 1 > J Uk

J, J- 2 (4ti)

2 M

tjk

2

/\

\J+1 1 J

0 A -A.

(j+i) j a r

,x A w

i ji*. x ct laj] [J + l]

x _[1] 1 [J]

J+l L ' CT J A

+ ^ (Pi) j X

\J-1 1 J

0 A -A,

j,., [Y^ x

+ e j j I J 1 M j, [Y

^ x p

Ci]]CJ3

0 A -A

/ J A (2E-11)

Evaluating the 3-j coefficients, we obtain

eTM = -i iM!

A

M

[J-l] v „[1]-,[J]

jn_T C

L X pL J

] A

+ M|

j [Y W1 x (2E-12)

tfP] = -i IM.

A

1 M

x A

HJ< M)2 j ry^ x ^13]^]

2 \ 2J+1/ J J+1 LT x Q J A

EiS

i [Y [J_1]

x a

[1]

] [J]

2 l 2 J+l/ J J-1 L a J A

• rvCJ] [1]-,[J] + e jj [Y L x p

L J ]^ (2E-13)
59

r j 1 rji

The F^ and M

^ operators never connect to the same final state because

they differ in parity

|F t OO| 2

=X) |<Kf|4]

The transverse form factor is then

e,P

k J MMx +

iM

i (oi xk)x

M

, 1 J 2M exp(i k • r j ) J »Jj.>

1

2 J .+1

i E^'k- .[J]|

f

J.M.>

i l

2

+ J . M . > I 2

} (2E-14) 1 T T I A 'll*

the Wi gner-Eckart theorem and summation over M^, M. , and k then yield

l F T

(k)

|

2 = 23^1 S||<J

f || £

CJ]

|| J,-)! 2 + |< J

fl|

«

[J]

|i Ji>l

2

|. (2E-15)

The matrix elements may be further reduced in isospin by using the WignerEckart theorem in isospin space,

<

j

f Tf T zii°o

T]

ii j iV2 > = ( - i)Tf Tz

( o °

CT]

; (2E ~ i6)

z z

Here, oj^ is the isovector part of C^, or and oj^ the isoscalar

component.

* The Coulomb, electric and magnetic operators given in (De66) and (Do75) are

related to those presented here by

**

c [j] =

o E

(4n) hM

2 ° u1

, 2r[

J] =

2 jA 2

, X M J

' 2J+1

= (4n)'2 Tmag

J,\
60

The total absorption cross section for real photons integrated over a

resonance line can also be written in terms of the transverse multipole

operators

a (id) =

Y

9 2 2

271 e

ID 2 t

+i E

j

+ !<J

f || «

[J]

|| jpi

|<j f

l| 5

CJ]

I| jp|

k^uj (2E-17)

Table 2-1 reviews the terminology used in the classification of electromagnetic single-particle operators.

We are now required to compute the reduced matrix elements of the

operators

jj(kr) Y

[J]

, j L (kr) [Y [L]

x p

C1]

] CJ]

, j L

(kr) [Y (L]

x a

tl]

] CJ]

. (2E-18)

This type of analysis is most easily performed using the diagrammatic recoupling

technique of ref. (Da71*). Writing a single-particle state as

4* ^ = R (r) [Y^ (r) x X

[2]

] [j] (2E-19)

where x is the spin state and the radial wave function of orbital £, the

reduced matrix elements are:

(i)

r ' J

- £ <^j']||

jj Y

[J]||

4,[J]> = ( .)J +J +4

J ,1

' ,i £' i U'

j

'

1 J

£'

(4/t)' j £ L) \0 0 0, <

Rr

|jj(kr)|R

£

>, (2E-20)

(i) (£'-L-l-£)

|| j L [Y [L]

x p

[ 1 ]

] [J] |U[j] >

= +J*-1 J L j' i

1 i (£' j

1 h)

(471) j £ J

L 1 J £ L £+l\ /£+l 1 £

£ £ ' £+ 1 ) \0 0 0 0 0 0

£+1

2£+l < R

( 3

L \ 3r -m.> r

L 1 J / /£ ‘ L £-l\ /£-l 1 £

' £ £'£-l) \0 0 0 0 0 0

2£ tr <Mi §?

+ 4f1i r£'.>

(2E-21)
61

(i)

(£' - L - i - ^[j']||

jL

[Y CL]

x G

C1'J

] [J]||

A . A = (~)

£l+2 j' 6 J L j1 j £' £

’£' a L L £

( 471 ) j

1 j J) \o 0 0

<R£I

| j L (kr)| R £ >

(2E-22)

The point to be gleaned from these complicated expressions is that the

form factor of a transition to a state of specific £ , j , J is given by the

product of vector coupling coefficients and a radial integral of j^(kr). The

momentum dependence of the transition resides here. This fact is extensively

used in spectroscopic studies to assign quantum numbers to discrete states.

Shell model calculations often involve harmonic oscillator wave functions

because of their simplicity and i ntegrabi 1 i ty . As shown by eqs. (2E-20,

21, 22) one is left with radial integrals of three types:

Charge and Moment:

< R

n '£' I Ji ( kr ) I R „>

Current:

< R n'£' I j L

(kr)

(i

n£

8r

"

~r Pn£>

(2E-23)

(2E-24)

and

<R

n'£' I j L (kr) (§7

+ T

1

)

R n£> (2E-25)

Oscillator radial wave functions are of the form

R n£ ( r ) = P(z) exp(-z) (2E-26)

where P is a polynomial in z = (r/bj and b is the oscillator parameter. The

derivatives in (2E-24, 25) operating on give back the same form with £

changed by one unit. The remaining integral can be evaluated as

< R

n

>

1

| J L

( kr )| R nc>

= p, (y) exp(-y)

n£

(2E-27)
62

Where P‘ is now another polynomial (confluent hypergeometric function) in the

2 variable y = (bk/2) . The function P

1 depends on the multipolarity of the

transition operator, the principle quantum number of the shell, the orbital

angular momentum, and the total angular momentum of both single particle wave

functions. These polynomials can be found in ref. (Do79a).

Shell model wave functions require an additional correction for the

motion of the center-of-potenti al about the center-of-momentum of the nucleons.

For the harmonic oscillator this correction is given by multiplying all transition

amplitudes by the form factor

f

CM

(k) = exp(y/A) (2E-28)

We illustrate this formalism with an example to show how the sum over parT

tides . works and to show the effect of considering a transition between

J states of definite isospin T. ^

C has a ground state with and T=0 and an

excited state at 15.1 MeV with J^l"*" and T=l. The only el ectromagnetic transition operator that can connect these two states is one with J=l, n=+ which is the

magnetic dipole operator

12 4 8

If the l ground state is taken as the configuration (ls^) (lp anc* + 4 7 1

the 1 excited state as (ls a ) (lp , (lp2 ) , then the transition can be PICtured as simply one nucleon in the p-shell recoupling its orbital angular momentum

and spin from j = 1 + h to j ' = 1 ~ h- Both neutrons and protons in the p-shell

can make this transition. Because of the way single particle states are normalized

(total state functions normalized to 1 and single particle wave function also

normalized to 1) we have the simple relation for closed shell nuclei

<df] |l£o(i)l|

i X)<^']|| 0 [0]||^[j ] > (2E-23)
63

i . e . , the sum over single particle operators between the total state wave functions is the same as the sum of possible single particle transitions. In our

* example j = 3/2 and j' = \ so that only one reduced matrix element is involved

Y^x x

[ "2]

ihl

N

[ 1 ] Y^x x

C "2]

[3/2]

Electromagnetic transitions can,

in general, change the isospin quantum number of a nuclear state by 0 or 1, an

isoscalar or isovector transition. The fundamental interaction operators can

be divided into their isoscalar and isovector components by the replacement

of e . = h(l + i • Z

) where x

Z |p> = + p> in the Coulomb and current operators and

u . = %(u + i

Z

u ) where u = u + u = 0.88 and u = u - u = 4.70 in the

*j vt s j

H

s M p *n Mv *p *n

moment operator. In the present example only the isovector parts of the operators

will contribute to the transition. The single particle operators are therefore

multiplied by the factor hz

Z and p

. is replaced by for all particles. The

isospin part of the single particle matrix element is always of the form

zl n ^ M X>(j)<l 0\h I- 0 0> =

' J (2E-30)

where the M(j) are the single-particle space-spin parts of the matrix element and

M = y. M(j). Thus, the additional specification of the isospin quantum numbers of

J the initial and final states reduces the transition amplitude. Explicit evaluation

of the 3-, 6-, and 9-j coefficients and the use of harmonic oscillator radial wave

functions gives (with q = kb, b being the oscillator parameter)

“Cl]..

II J 0

(kr) v [0] [1]Y

L J x cr L j lp 3/2

> = - 1 (3)'

,, 1 2, e c ( 3. c n ) -n

2

/4

(471)

<lpp

|| J 2 (kr) Y^x

[ 1 ] lp 3/2

> = +i

n

2 /4

3C6)*

5 (4n)^

(2E-31)

The symbol x denotes the transpose of x-
64

«ap,

i! ji(kr) yUlx p

Cl]

[1] " (2)^ e

“ n 74

lp 3/2

> = -i

3

" k “—r~

3/2 3

(4rc)

2

,C1]

'3/2'

C\J rH r

.2 ,2 *1

1 3 L

1 - M

v (2 - 4 )J

2

- k

2

b

2

/ 2 e

Taking the isospin factor into account

da , n + ,+x _ fjj - 2 a/ A k

2

1

( 0 -> 1 ) - a I x + tan 0/2) —r 3 dQ 0

1 2k

2

M

29 1

" M

v (2

. 2 . 2

k b .

-k

2

b

2 /2+k2 b2

/24 2,

s f M ( k _) N p

This equatio-n has the momentum dependence of the nucleon form

center-of-mass correction for the oscillator model. Other examples

(2E-30)

. (2E-31)

2

(2E-32)

factor f., and

N can be found

in ref. Do79a.
65

F. LONG WAVELENGTH EXPANSIONS

The multipole expressions for current and magnetic moment operators given

by (2E-15) can often be simplified for real photons by expanding the spherical

Bessel functions and keeping the first nonvanishing term

j L (kr) =

(kr)

L 1-3-5- • -(2L+1)

(kr)

2

2(2L+3)

0(kr) (2F-1)

This approximation will be valid for kr << 1, (e.g., r < 5 fm, k = uj < 40 MeV). In

particular the electric-dipole current operator is often used in its unretarded

form (replacing exp(ik*r.) by unity)

J

H

E1 =. E £ * P .

j

6

j M (2F-2)

and the magnetic-dipole moment term as

i|J H Ml E 2M (sxiO-CTj + j (exk) • Fj (2F-3)

where £ . = (rxp) ..

<1 3 It is often convenient to replace the nucleon momentum operator by the

position operator

<f

by letting the nuclear Hamiltonian operate on the initial and final states,

°> = <f 1^1 o> = i<f

j [H , r]| o> = i w<f|r|o> (2F-4)

uj = E^ - E q (nuclear recoil has been neglected). This replacement is sometimes referred to as Siegert's theorem. Then

H El.1. E e . e • r . J J

2 2

da =471 e uj Y

E km E ~ | N | 2 dn

e . £ • r . o> -rJ 3

I 1 ckJ

(2F-5)

(2F-6)

where e . = 1 for a proton, 0 for a neutron. Using isospin notation (x | n - - jn>,

x

z

!

p

=+|p» one can write e. = ^(l+x

2

) . and specify the r. as center-of-mass

J J J coordinates. (2F-6) can be written in the familiar form
66

2 2

da =471 e uj Y

|<f *5 ct. x

J

J J 0>

2 dn

duo (2F-7)

Neutrons, although uncharged, can make single-particle electric dipole

transitions. As can be seen in the expression for the dipole operator in

center-of-mass coordinates D = h Y £*r. x

Z

. The dipole moment is

J

J J created by the shift of the center-of-charge C = h Y r • 1 • away from

J 3 the center-of-mass R = Y r. = 0. Neutrons are as effective as protons

j

J in making this shift.

Two relations between form factors and cross sections can be seen in

the low momentum transfer limit. The transverse form factors Fp(k,uu) and

F a

(k,uu) become the same as the real photon transition amplitudes as k -» uu.

This gives the following connection between inelastic electron scattering

and the photon absorption cross section

Fj (k,w)

2

w cr^(uu)

k+uu =

0 2*2

2 7i e (2F-8)

for transverse transitions of any multipolarity.

In this same limit the current and Coulomb operators can be related

through current conservation to give

F T,p^> k^uu

uj\2 L+l uL r, n k — | F c (k * u>) (2F-9)

G. COINCIDENCE MEASUREMENTS

When a nuclear decay product x (x = y, n, p, n, etc.) is detected in coincidence with an inelastically scattered electron in the reaction A(e,e'x) the

cross section becomes a three-fold differential and, in general, depends on four

form factors which are functions of k, uj, p^, k-p^. The Born approximation

result can be written (Am79) as an extension of the single arm expression (2B-9)

,3 / in da

d a(e , e x) _ r

v dQduudQ T dQ

x x (2G-1)
67

The da is the anqular distribution of x about the direction of momentum transv

a

fer k

da

Px 1

dQ 16tt K MW

x

r 2 k

£ -4* M + ~ (M + M ) 2 zz 2 xx yy

7

U)

+ i (M -M ) e cos2({) + \ (M +M )(2 -4} e(l+£))'2 cos<}> 2 xx yy

Y x 2

v zx xz 2 Y

x

UJ (2G-2)

The azimuthal angle $ is defined in fig. (2G-1). W is the total energy available (ai + Mt ). The M . . are products of transition amplitudes in the transverse

(x and y) and longitudinal (z) directions,

terms vanish and £ becomes the degree of 1

single arm (e.e 1 ) cross section (2B-9) by

of the unobserved product x

For real photons the first and fourth

inear polarization. One regains the

integrating (2G-2) over the direction

71 da~~ sin© dG ( 2 G - 3

)

dQ x x

v 7 x

o

then the two interference terms with cos2dY and cosb vanish.

x • x
68

Coincidence measurements of the type (e,e'x) are often limited by the

accidental detection of a scattered electron with a decay product x from an

unrelated event. If the average counting rates of the e' and x detectors are

N

g

and respectively, the accidental rate is given by

N =(-7-H N , N (2G-4)

acc \d.f./ e x

v J

< -8

where x is the resolving time of the coincidence circuit (typically = 10 s)

and d.f. , the duty factor, is the fraction of the time the electron beam is on

< -2

the target (typically = 10 for pulsed linacs). The average true coincidence

rate for detecting a scattered electron and the associated decay product is

AQ AE

X X N . = N , —

5 p—

coin e 471 E x (2G-5)

where AQ is the solid angle acceptance of the product detector and AE /E is

X XX

the fractional energy acceptance. A coincidence measurement can be considered

feasible when

N . coi n

N acc

This usually requires d.f. = 1 unless there exits a strong energy and angular

correlation between the scattered electron and the decay product (as there is

for example in quasi-free electron-nucleon scattering).

(2G-6)
69

H. RELATION OF WEAK AND ELECTROMAGNETIC INTERACTIONS

Inelastic electron scattering in the one-photon exchange approximation is

an example of the larger class of semileptonic reactions that share the common

feature that the leptonic and strong interaction vertices can be evaluated

separately. The charge changing weak interactions are: beta decay (e \> e

),

•f ** TM +

(e v ), lepton capture (£ , v^) and neutrino reactions (v^,^ )> )>

2. - p or e.

The weak transition operators have a richer structure than the electromagnetic. Besides the vector currents

y^

there are axial vector currents from

Vs’

j = u [F-. y J + F- ct k ] u p 1 'p 2 pv v ±

(2H-1)

jcJ = u[F A y c y - i F y. y ] r u p5 A *5 p

3 5 ±

The conserved vector current hypothesis identifies the weak vector coupling

constants F^

^ as the same functions of k^ as the nucleon electromagnetic form

factors. The axial vector weak coupling constants and form factors are

F = - 1.24 (1 + k

2

/b

2 )" 2

, b

2 = (855 MeV)

2

A M (2H-2)

F

p = 2M F A

/(k

M

2 .2

)

.

2

The overall weak coupling constant (the analog of e ) is

G = 1 x 10~ 5 /M

2

. (2H-3)

The weak interaction transition operators are similar in structure to the

electromagnetic single particle operators

H weak

= (!><?>P>tf-p) t ± exp(ik-r) (2H-4)

and their multipole decomposition is carried out as in the previous section

for the H

p operators (0c72b). The experimental confirmation that the nuclear
70

part of a weak interaction amplitude has the same numerical value as the

corresponding

j H 1 4* .> el ectromagneti c form factor has been verified by

many examples.

The correspondence between weak and electromagnetic amplitudes is useful

for predicting the cross sections of energetic weak interactions. For example,

the reaction D(v

e ,e )pp (see fig. 2H-1), has been measured (Wi 80) and found to

be in agreement with the expression

d

2a

dOdE, ^^2

Mp

(2n)

Z (2H-5)

- 2 sin (6/2)

where the last factor can be deduced from model fits to the el ectromagnetic

3 1 . . S - Sq magnetic dipole transition from D(e,e')np data. The neutrino breakup

of the deuteron is closely related to the weak interaction p+p^d+e

++v

e which

governs the rate of proton fusion in the sun and other stars (a reaction that

cannot be measured directly in the laboratory).

The weak neutral current due to Z° exchange between a lepton and nucleon is

responsible for neutrino elastic scattering and for a weak component in electron

and muon scattering from nucleons. Although the weak scattering amplitude is muc

smaller than the electromagnetic amplitude the parity nonconservation of the weak

interaction gives rise to an a asymmetry in the scattering of polarized electrons

(Pr78). Transitions between discrete nuclear states due to neutral currents have

total reaction cross sections for energetic neutrinos in the range

-40 -4i 9 10 - 10 cni (Do79b).
71

I. REAL AND VIRTUAL BREMSSTRAHLUNG

Three closely related cross sections that involve the radiation process are

shown in figs. 2B-1 and 2D-1. Diagrams 2D~la and b are the emission of a high

energy real photon ( k , u>= I k | ) by an electron undergoing elastic scattering in the

Coulomb field of a nucleus (momentum transfer K=(k^~k-k

2

)> energy transfer

2

uj=K /2Mt

= 0). The i ntegrated-over-angl e spectrum of the real photons is

dg

dk (1 + e

2

2 e) £n 183 ^ £

z l/3 9

( 21 - 1 )

where £ = E^/E^. This is the completely-screened Bethe-Hei tl er form valid for

E^=100-1000 MeV on high Z radiators (Ko59,Ma73).

Most real photon measurements are done using bremsstrahl ung spectra. The

number of photons per MeV per microampere of electron beam incident on a

radiator whose thickness is one percent of a radiation length (r.l.) is shown

in fig. 21-1. Radiators much thicker than 1% r.l. generally do not generate

much additional flux in the forward direction due to multiple scattering of

the electrons in the target.

The disadvantage of bremsstrahl ung as a source of photons is that photons

of all energies are incident on the target at once. Other schemes to generate

monoenergeti c photons are: tagged bremsstrahl ung (0c62), positron annihilation

in flight (Be79), and Compton scattered laser light (Fe79).

The energy spectrum and angular distribution of the scattered electron

A A A after bremsstrahl ung is given in the peaking approximation (k = k^ or k^) as

d

2

g

dfidE, = f(E ) — ^ 1

; dQ

(E 2> 0) E 2

) da

dQ

(E re) ( 21 - 2 )
72

where da/dft is the elastic scattering cross section at the appropriate incident

electron energy and

F{ E)

2

e

rck (1 + e

2

) £n —m

e (21-3)

an unscreened expression valid for 0>>m

e /E^ ^

. Wide angle bremsstrahl ung

forms a background to A(e,e') measurements (Mo69,Ma69).

Inelastic electron scattering observing only the scattered electron was

discussed in Section B. Now, however, we are interested in the cross section

1 for this reaction when the scattered electron is not observed i.e., A(e,x)A e'

where x is some nuclear decay product (y

1 etc.)- To obtain this cross

section we need to integrate over the angular distribution of k^ at fixed uj.

Eq. (2B-3) was derived using the simplifying assumption that the mass of the

electron could be neglected. This is accurate except for electrons that are

scattered in a very forward direction 0^m /E, A more accurate formula (but

e 1 ,

still treating the electrons as plane waves) is

d

2o _

dQ dE„

CT c (0) F c (q,w) + ct t

( 0) F T

(q,u) (21-4)

where the transition form factors are multiplied by "Coulomb" and "transverse

Mott cross sections"

0 2 k 0 k _ _ , ^ f < E 1

E 2 + Vk

2 * ^

M 1

-T = f

2

? S < E

1

E 2

' <V £) ( V*>

- m

e

) k 1

(21-5)

( 21 - 6 )
73

and

E

i = (k? + m

2 )/ k2 = 2(E

1

E

2

- - m

2

) = k

2 2

UJ

UJ = E x

- E 2

, t = ^

- t

2

, k

2 _ 2

~ m

e When m

g = 0, we obtain the familiar forms

k

4 /k 2

? \ 2 2 _ /«

0

C = a

M ”^4 aT = aM\ // + tan 6/2 )> °M

= “2

4 (2I ’ 7)

k \2k / 4k^ sin 0/2

k

2 = 4k

T

k

2

sin 2

0/2 . (21-8)

jj 1. £

The Coulomb and transverse Mott cross sections are plotted in fig. 21-2 for

k^ = 100 MeV, k^ = 80 MeV. We note the strong forward peaking of a-j. at

m uj

) max

„

= k^k^ , l (21-9)

This peaking is responsible for the fact that the integrated over angle

inelastic electron scattering cross section comes mainly from the

far forward transverse component (i.e., 1 i ke a real photon).

For electric transitions at angles away from the extreme forward and

backward scattering angles Coulomb transitions dominate current transitions,

both because > Oj and because, in general at low momentum transfer,

F c

> F-p as can be seen in the relation (2F-9).

When the angular integral over the unobserved electron is made to obtain

the (e,x) cross section at fixed excitation energy

da

duj

^

=J

[<Jc (8) F

c

2

(k, u ) + a

T

(0) F T

2 (k,m)] dn ( 21 - 10 )
74

most of the contribution to the integral comes from small values of 0. In this

region k-njj i . e . , the virtual photon is almost real. If the (e,x) reaction is

dominated by one multipole L the integral can be evaluated using low-momentum

transfer approximations for F c and Fy. As noted in Section E the momentum

dependence of F c and Fy (and, therefore, the angular dependence resides in the

* radial integrals. The form factors can be expanded at low k as

i \ k*

-

I L <Rr |j L (kr) f(r)| r £ > ~

( 2 l?I)TT <R

£ ,

| r

L f(r)| R

£

>. (21-11)

Thus eq. (2F-8) can be rewritten as

F T

L (k,ui)

2

k+o>

uxj (lu)

Y

0

2 2 Zn e

( 21 - 12 )

Low k transverse form factors are usually dominated by the current part so

eq. (2F-9) can be written as

F„

L (k,uj)

k->uj L+l

/. \ 2L+2 uxj (oi)

>“/ 2n

2

e

2

(21-13)

Now the integral eq. (21-10) can be evaluated. The result (Gi71,Dr76,Sh79) is

usually expressed as

* . Monopole transitions are an exception since the leading term is zero by

2 orthogonality; the next term (kr) applies then.
75

da

e ,x

duo = a (uj)

Y

N" (E

1

,ui) (21-14)

where N^" is interpreted as the spectrum of virtual photons seen by the nucleus

during inelastic electron scattering. For electric dipole transitions

N

E1

(Erk) -Sjj

~ 2E-,E~

(1 + e )J Hn - 2s

m k

e (21-15)

i.e., a function similar to that found for real bremsstrahl ung. Other multipoles differ in terms linear in s.

When Coulomb distortion of the electron waves is taken into account

(So77) the spectra show a great sensitivity to L. Fig. 21-3 shows the

intensity kN(Epk) vs k for a Z = 28 target. This sensitivity has been

exploited to look for E^ giant resonances in the presence of the much stronger

E^ giant resonance (Ha80).
76

REFERENCES CHAPTER II

Am79 E. Amaldi, S. Fubirvi, and G. Furlan, 11 Pi on-El ectroproducti on" SpringerVerlag (1979).

Be71 J. C. Bergstrom, H. Crannell , F. J. Kline, J. T. O'Brien, J. W. Lightbody, and S. P. Fivozinsky, Phys. Rev. C4, 1514 (1971).

Be77 R. Bergere, Proceedings of International School on "Electro- and Photonuclear Reactions" ed by J. Ehlers et al .

Spri nger-Verl ag 1977, p. 68.

B j 64 J. D. Bjorken and S. D. Drell, "Relativistic Quantum Mechanics and

Relativistic Quantum Fields", McGraw-Hill, New York (1964).

Da71 M. Danos, Ann. Physics 63, 319 (1971).

De66 T. deForest and J. 0. Walecka, Advances in Physics, 15, 1 (1966).

Do75 T. W. Donnelly and J. D. Walecka, Ann. Rev. Nuc. Sci. 25, 329 (1975).

Do79a T. W. Donnelly and W. C. Haxton, Atomic Data and Nuclear Tables 23,

103 (1979) and 25, 1* (1980).

Do79b T. W. Donnelly and R. D. Peccei , Phys. Reports 50, 1 (1979).

Dr76 E. T. Dressier and E. L. Tomusiak, Nuc. Phys. A273 , 383 (1976).

E i 70 J. M. Eisenberg and W. Greiner, "Excitation Mechanisms of the Nucleus",

p. Ill (1970), North-Hoi 1 and Pub. Co.

Fa59 U. Fano and G. Racah, "Irreducible Tensorial Sets", Academic Press, New

York (1959).

Fe79 L. Federici et al_. , Proceedings of Mainz 1979 Conference on "Nuclear

Physics with Electromagnetic Interactions", Spri nger-Verl ag , p. 234.

G i 71 B. F. Gibson and H. T. Williams, Nuc. Phys. A163 , 193 (1971).

G i 80 C. Giusti and F. D. Pacati , Nuc. Phys. A336 , 427 (1980).

Ha80 E. Hayward, Proceedings of 1979 Oak Ridge Conference on "Giant Multipole Resonances" ed by F. E. Bertrand, Harwood Academic Publishers,

p. 275.
77

Ho73 J. Hockert, D. 0. Riska, M. Gan' and A. Huffman, Nuc. Phys. A217 , 14

(1973).

Ko59 H. W. Koch and J. W. Motz, Rev. Mod. Phys. 31, 920 (1959).

Le73 D. R. Lehman and J. S. O'Connell, National Bureau of Standards, NBS

Monograph 136 (1973).

Ma73 J. C. Matthews and R. 0. Owens, N. I.M. Ill, 157 (1973).

Ma69 L. C. Maximon, Rev. Mod. Phys. 41, 193 (1969) •

Mc62 K. W. McVoy and L. Van Hove, Phys. Rev. 125, 1034 (1962).

Mo69 L. W. Mo and Y. S. Tsai

Rev. Mod. Phys. 41, 205 (1969).

Mo78 E. J. Moniz, Proceedings of 1977 Les Houches Summer School

Physics with Heavy Ions and Mesons", North-Hoi 1 and Pub. Co., p. 432

(1978).

0c62 J. S. O'Connell, P. A. Tipler, and P. Axel, Phys. Rev. 126, 228 (1962).

Qc72a J. S. O'Connell, National Bureau of Standards, NBS Technical Note 713

(1972).

0c72b J. S. O'Connell, T. W. Donnelly and J. D. Walecka, Phys. Rev. C6, 719

(1972).

Pe74 M. L. Perl, "High Energy Hadron Physics", John Wiley and Sons, N.Y.,

(1974).

Pr78 C. Y. Prescott et <3l_. , Phys. Lett 77B , 347 (1978).

Ri 79 D. 0. Riska, "Mesons in Nuclei", p. 757, eds M. Rho and D. H. Wilkinson,

North-Hoi 1 and Pub. Co. (1979).

Sh79 A. C. Shotter, J. Phys. G. : Nuc. Phys. 5, 371 (1979).

So77 C. W. Soto Vargas, D. S. Onley and L. E. Wright, Nuc. Phys. A288, 45

(1977).

Ub H. Uberall , "Electron Scattering from Complex Nuclei" (2 Volumes),

Academic Press N.Y. (1977).

Wi80 S. E. Willis et al .

Phys. Rev. Letts. 44, 522 (1980).
78

Table 2-1. Classification Scheme of Electromagnetic Operators.

Four-vector interaction energy

Coulomb

1 ongi tudi nal

convection current magnetic moment

or spin current

transverse

multi pol esCoulomb el ectri

magneti

jj(kr) Y

J rvJ+l 1..J

J J+1

[V xp ]

rvJ-l 1,J

j J

_ 1

[Y xp ]

jj CY

J

xa

1

] J jj CY

J

xp

1 ]'
79

Fig. 2A-1 Diagram for the single particle amplitude for absorption of a photon.
(a) (b) (c)

Fig. 2A-2 Diagrams for computing the photon scattering amplitude.
81

e — — e*

Fig. 2B-1 Diagram for computing the electron scattering amplitude.
82

7

O <L

(a) lb)

Fig. 2C-1 Piorrnucleon vertices (a) NflN, (b) NflA.
83

TT

A ~9~~

i 7T !

I

I 7T

<b 6

(a) (b)

Fig. 2C-2 Two-nucleon interactions (a) one-pion exchange, (b) two-pion exchange.
84

Fig. 203 Principal electromagnetic couplings to the pion

and delta currents, (a) pair or gauge term, (b) pion current

term, (c) delta creation, (d) delta current.
85

Fig. 2D-1 Two diagrams which contribute to bremsstrahl ung

and the radiation tail of electron scattering.
86

Fig. 2G-1 Coordinate system for the kinematic variables of the

reaction (e,e'x). l^, and ]< lie in the horizontal plane.

The nuclear reaction product x is measured relative to the

momentum transfer £ and the horizontal plane.
87

Fig. 2H-1 Variables for the reaction D(v

e ,e)pp.
88

Fig. 21-1 Bremsstrahlung intensity spectra of 4 elements

as radiators. The ordinate is the number of photons per

MeV per radiation length multiplied by the photon energy

in MeV. The endpoint region is illustrated in the inset

(Ma73)
FIG. 21-2
90

Fig. 21-3 Intensity spectra versus photon energy of

El, E2 , E3 virtual photons generated when 50 MeV

electrons are inelastical ly scattered by a nickel

target (Ha80)
91

CHAPTER III. SUM RULES

A. Closure Sum Rules

1. Electric Dipole Sum Rules

2. Si egert 1 s Theorem

3. Other Electric Sum Rules

4. Magnetic Sum Rules

5. Electron Scattering Closure Rules

B. Dispersion Relation Sum Rules

C. Momentum Transfer Sum Rules
92

CHAPTER III. SUM RULES

The electromagnetic interaction in the nucleus can be treated as a

-i- perturbation because of the small coupling constant e = (137)

2

. A real or

virtual photon is absorbed on a single nucleon (or meson) without interacting

with the other nuclear constituents. Let us denote the charge, current, or

moment operators generically by (H) . The transition amplitude

fi = <Vf

l'

?

2 z® (rrV VFi

rA» (3-1)

will have only one term active at a time out of the A possible choices. In

contrast, the multiple interactions of hadronic probes (p,n,7t, etc.) displace

more than one nucleon during their interactions with the nucleus.

Sum rules are statements about integrals over electromagnetic cross

sections. They are used to relate reaction data to ground state properties or

to relate data from different types of reactions. Sum rules are also used to

test the internal consistency of theoretical models of nuclear reactions. The

discussion in this chapter will cover sum rules based on closure over excited

nuclear states, dispersion relation sum rules, and momentum transfer sum rules

for electron scattering. Review articles on aspects of this subject can be

found in refs. (0c73, We74, No78, Ar79).

A. CLOSURE SUM RULES

Completeness requires the eigenstates of a Hamiltonian (H|¥ > = Ej ^

n >) sum

to unity when written as a projection operator

Ziv^J = 1 • (3A_1)

n

This closure property can be used to eliminate intermediate states in a sum

over the absolute square of a transition amplitude if the operator is independent

of the transition energy uj = E^ - E

q

,
93

Y^y<f|®l o> <0| ®

+

| fxf I® I Q> = < 0 |® + ®jo> (3A-2)

where ®

+

is the Hermitian conjugate"*" of ® (interchange of rows and columns

and complex conjugation of the matrix). Frequently, some power of the transition

energy appears in the amplitude. It can be removed by successive application of

the relation

<f|iu

n ®jo> = <f|uj

n

' 1

[H , ® ] | 0> . ( 3 A- 3

)

The time derivative of a matrix element whose operator does not explicitly

depend on time is given by its commutator with H,

d_

dt

<f | ®I °> = i <f [H,®] 0 > . (3A-4)

This allows the replacement of a particle momentum operator "p = M dr/dt by the

position operator

<f|jjj| 0> = i <f|[H,"r]|0> = io)<f

j "rj 0> (3A-5)

by letting H operate on the initial and final states. Equation (3A-4) leads

to the useful theorem

f

“| < f|®|0>| 2 = \ <0|[®

+

, [H,®]]|0> . (3A-6)

Equations ( 3A- 2 and 6) are the progenitors of most closure sum rules.

^Reversing the order of a matrix element brings in the Hermitian conjugate of

the operator <0 | ® | f> = <f|®|0> where (*) is complex conjugation (change

of sign of imaginary components). Although the Hamiltonian is Hermitian H =H,

as are all operators that generate physical observables, the commutator of two

Hermitian operators is anti -Hermi ti an <0 J [ H , ® ] +

| f> = - <0j[H,®]|f> =

- <f | [H, ® ] | 0>* = - <f I [H , ® ] | 0> since (H ® ) + =® +

H +.
94

1. Electric Dipole Sum Rules

The oldest and most venerable sum rule is that for the electric dipole

cross section

CO

(3A-7)

where D = h *-r*‘ e*r. i z + • = D is the long wavelength approximation to the El

J J J operator. Equation (3A-7) is derived as follows: writing the energy integral

of eq. (2F-7) as

CT o 4„

2

e

2 E <f dI o> uj <f D 0> (3A-8)

and removing the energy dependence (replacing wD by [H,D] symmetrically) one

obtains

o 2 2 o = 271 e , o E

f {<f j [H , D] | 0> <

f j D | 0>+<f j D| 0> <f | [H,D] I 0> } . (3A-9)

Turning around the matrix elements <0 j D

j f> = < f | D

( 0> and <0j[H,D]|f> = -

<fj[H,D]!0> and applying closure gives

g = 27i 2

e

2 {<0|D[H,D] - [H , D]D j 0>

}

and eq. ( 3 A- 7 ) results.

(3A-10)
95

The double commutator of the dipole operator with the Hamiltonian can be

eval uated

H . Efi!,.

j 2M 2

Z

j^k

VCr-.r^) = K + V . ( 3 A- 11

)

The kinetic energy term gives the Thomas-Reiche-Kuhn (TRK) formula

TRK _ 27i 2e

2

NZ „ cn NZ , G

o -M “A

" 60

“A

(MeV' mb) • (3A-12)

The potential energy contribution depends on details of the effective interaction of nucleons in the nuclear medium and on the two-nucleon wave function.

The long range one-pion-exchange potential, eq. (203), gives a contribution

since z Z

. in the dipole operator does not commute with "t.-t. in the OPE potential

J J K see eq. (209). Meson exchange currents also contribute to < 0 J [ D , [V,D]]|0>.

The net effect of the potential contribution to ct q

is usually written as a

multiplicative constant (1 + k) times the TRK result. In practice the upper

limit of the energy integral is taken as the threshold for pion production

uj = m = 140 MeV.

7tApproximating the real photon operator by D is not as bad an approximation

as one might expect. The full current single-particle operator exp(ik-r)

leads to

E i, >0 2 2 a = 27i e

o <f|rr euj ' I M

0> (3A-13)

where Px = c-p and z = k*r.

£-1 = [x,P

x

] = i, [x,

[£•? , k-r] = 0.

r x rE_ x ii = 1 r£_ rE x

2 i] = -2

> l 2M’ l 9m> a 2M ’ 2M !
96

The exponential can be expanded as

+ikz 12? e~

* = 1 ± ikz - j

k z (±i kz)

The current operator can then be expanded as

x lp

x

z p

x

z exp(ikz) = w (ix + - h [H, 17— ] + . ..) = uie-J

M (3A-14)

Closure gives

a

Q = 2 7i 2

e

2

< 0 |[(£-jy, [H , e

- J ]]| o > . (3A-15)

Using only the kinetic energy parts of H in the commutator one finds

0=0TRK

0 0

0 2 N 41

1 + 1

+ 0

J (VNT -J (3A-16)

The average momentum in nuclear matter is p = 208 MeV/c leading to a 3%

correction to (Fr75)

TRK Relativistic and potential energy corrections to a

Q

decrease the sum by a 2/2 term proportional to <p >/M so the net change is probably smaller than the correction quoted above. Magnetic moment contributions to ct q are also at the few

2 2

percent level since the leading term in the cross section is k /M times the

current term.

Dipole sum rules with other energy weightings can be constructed

cr =

n

CO

/.

n , x , _ , 2 2

u) cr ( uj ) duj - An e o

v Ef

u, n+ 1|<f| d|o>| 2 (3A-17)

using closure. The most useful are n = -2, -1. The n = -2 sum rule is given

in terms of the nuclear pol ari zibi 1 i ty a defined classically as the induced

dipole moment divided by the electric field strength. Quantum mechanically
97

_ o_2 / 2 _ Re f(uu) - Re f(0)

a = 2e u>

j <

| D

j U> | = — ——

UJ

where Re f(uj) is the real part of the photon elastic forward scattering

amplitude (as will be discussed in Section 3B). Thus

(3A-18)

ur>o

ct (uj)du>

/

<_< '-*-u o

-XT— = 2 " ° (3A-19)

HO 2 2

In the harmonic oscillator model V (r) = (Mu> )r ; the polarizability is the

inverse of the spring constant

HO NZ

“ =

"A

M uu 2

* ( 3 A- 20

)

In general, the polarizability is related to the nuclear symmetry energy

2

constant K (= 25 MeV) entering the semi empi ri cal mass formula as K(N-Z) /A. A

.simple model (Le60) gives

e

2

A

40K

2 <RS

TRK (3A-21)

The n = -1 dipole sum rule

'-1 I = e

2 <0

U)

2l 2 2 i u 0> = tt e <0 Ei J

i

z

i . z x.x . |0>

1 J1 J

( 3 A- 22

)

This sum is sensitive to position and isospin correlation among the nucleons

in the ground state generated by the residual interaction among nucleon pairs

V

res (f

1

,r"

2

) = V(ir

1

,r

2

) - VCrp - V(r

£

) (3A-23)

where V(r) is the average one-body potential well (e.g., a harmonic oscillator

4/3

or Woods-Saxon radial dependence). In general one expects ~ A and the
98

4/3 data bears this out. The ratio cr_^/A is twice that predicted by the

oscillator model without residual interactions.

The n = +1 sum is readily evaluated by using the s-p form of the dipole

operator

uu M-/•' rs

u) a (uj)duj = —I—

22

<0 I y

Y u2M

i J

wh ich is seen to be the momentum analog of cr_^.

-3/2

cross sections fall off as u> (deuteron-1 i ke)

convergent (neither is that of a Lorentz line).

of °+r

t/tj 2 p/Pj

X

|o> (3A-24)

High energy photon absorption

so the integral is not

This fact limits the usefulness

The dipole cross section can be split up among final state channels that

differ in or T^ (for nuclear ground states having J. or T. r 0). Sum rules

relating the cr_^ moments of different channels have been derived (Ha72). For

nuclei with T^ = ^ the dipole resonance may have T^ = or 3/2. As will be

discussed in the next chapter the location of the T^ = h, El resonance is at a

lower energy than the T^ = 3/2 resonance. A relation exists between the a_^

moments of these two channels and the body radii,

°-l (1)

- 2

CT -1 (l) = I

(NR

n

2 ' Z R p2

> (3A' 25)

where are the mean square radii of the neutrons or protons in the nuclear

n,P

ground state. This relation (and a more complex one for J

z >h) is a useful

boundary condition for identifying isospin split El giant resonances.

2. Si egert 1 s Theorem

The replacement of the current operator by the position operator, p/M -*

iujr" for the electric-dipole operator in the long wavelength approximation can

also be made for other electric multipoles. This leads to the relation
99

<J

.[J]

J.>

l J+l

<J

.[J]

!!V (3A-26)

for k = u>, ujR <<1,J>0. The ability to replace the current density by the

charge density is known as Siegert's theorem (Si37)„ Although the true

rj]

operator for real photons is E , the more mathematically tractable Coulomb

rjl

operator C

L J is useful for model calculations and sum rule derivations.

In the long wavelength approximation the single particle multipole operator

i s

c [J] = (-D

J

J (Jnil Y\ e (kr y Y (f ) o (2J+1)!! ^

e

j

tKr

j

J Y J,O tr

j

J (3A-27)

The derivation of eq. (3A-26) follows from replacing the momentum term in

= V (i)

L

L j, X L (kr) V

Lj0

(-r) PX (3A-28)

with

p^/M = e-p/M -* i uie*r = iujr 'j ^ ^(r) . (3A-29)

The two spherical harmonics are combined to one Y , . by the addition theorem

J , A

and the L = J-l component selected as the lowest order term in k. After the

polarization average eq. (3A-26) results.

The content of Siegert's theorem is that by current conservation each

transition current is accompanied by a transition charge density given by

it*j(r) = - ujp(r) and one can compute either transition amplitude to compare

with real photon data. An advantage to using the charge density operator

rather than the current density is that the matrix elements of p(r) are not

affected by meson currents in the nucleus. Mesons are pictured as moving

instantaneously in the nonrel ati vi sti c approximation so that charge always

resides on the nucleons, but meson currents exist between nucleons.
100

3. Other Electric Sum Rules

The reduced transition probability of a multipole transition is often

defined as

B ( C J ,uj) 20. +1 M M

f ’ 1

| < JfMf | ^ ^ e . MM 11 J r.

J Y 1M (r.)|j.M.>| 2

J JM V j

y

i l l I

<J E

j

e.r,

J Y , | ! J . >

2

. (3A-30) 2J.+1 I -fll j j j i

The photon absorption cross section (integrated over a discrete level) is then

given by

9i-i r

( j +1 ) 1 ] B ( CJ ,uj) (3A-31)

, , f , ,3 2 2J-1

cr(uj) = (27i) e uj 1

J [( 2 J + 1)!!]

2

Sum rules are formed by taking various energy moments of cr(iu). The relation (for

J. = 0)

y UJ B(CJ,U)) = \ <0| y [e .rJ Y* 1M ,[H,e .r

J Y 1M ]] f ’ 2 I j J J JM ,L ’ j j JM JJ 0> (3A-32)

follows from closure on eq. (3A-30). Using the identity

E r J v/ * r 2 J v J(2J+ir 20-2

[r Y

JM’ [ P > r Y JM ]] =

Zn ~ r (3A-33)

the kinetic energy part of the double commutator is evaluated leading to a

TRK-like energy weighted sum rule

or

uj B(CJ,uj) = J(2J+1) 2

471 Z2M

2J-2

r

% < EJ > = /

a^j(uj)du) =

( 0 + 1 ) [(20-1)!!]'

;i 2 e

2

7 — , , 23-2

1 (uJ R r)

(3A-34)

(3A-35)

where uj d is the resonance energy. This result does not include contributions

K from the potenti al
101

The sum in eq. (3A-34) can be divided into a part from the isoscalar part

of the transition operator and the isovector part (for a T? = 0 ground state).

For example the E2 sum rule for photon absorption reads

oo

a„

2

(E2) = /

q(E2)

2

dtu

U)

2 2

7i e 3M (3A-36)

o w

where the terms in the bracket are the isoscalar and isovector contributions

respecti vely.

4. Magnetic Sum Rules

The magnetic dipole operator in the long wavelength limit

M = i e + e,£

2M J J

C 3 A- 37

)

J has a spin and an orbital contribution. The Ml cross section for photon

absorption is

r >. _ , 2 2 \ s •? \ m i * \ ! dn aM1 (uj) = 4n e u; <f Hh> -r- . Ml I I I I duj

As with the electric dipole, closure leads to (Ku 63)

(3A-38)

a (Ml)

o

03 J[

o^(uj)du) = 27i^e^<0

| [M

+

, [H

j 0> (3A-39)

In contrast to a (El) however the kinetic energy part of H commutes with the

Ml operator because there is no spatial dependence to M. This feature makes

the ct q (M 1) sum rule more model dependent than the q

q (E 1) rule.

The potential terms which do not commute with M are the NN spin-dependent

terms and the single particle spin-orbit term

V ct.'O, + V 2 "O . s J k so j j

(3A-40)
102

The spin-orbit term is usually the larger and contributes to ct q through isovector transition as

V so (Mp-M n -|)

2

<0| E J2 . *CT . 0>

J J

1

(Ml) mainly

(3A-41)

since (p +p )/(u -u ) - 0.19. This term has its maximum value for nuclei which

p n p n have within-orbit spin-flip transitions £,j-*£,j-l. Closed shell nuclei with both

, V - _ , spin-orbit partners have <0 Lr‘ jZ.-ct.J0> = 0.

0 3 0

5. Electron Scattering Closure Rules

2 2 Rules for the integrals of F^(k,iu) and F-|.(k,w) over uj are a continuation of

the real photon results into the region of virtual photons. Since two dynamical

variables k and u> are involved, one must choose a path of integration in the

(k,tu) plane. Here we discuss J

dtu with k constant.

When closure is applied to the electron scattering cross section eq.

(2B-3) at constant k one obtains

= (k

2

) + (| + tan

2

|) [F

2

, P (k

2

) + F 2

, o

(k

2

)] (3A-42)

where DF is the Darwin-Foldy term from eq. (2B-5). Note the sum includes

elastic scattering. The sum over final states of the form factors is

facilitated, without much loss in accuracy, by neglecting the terms in
103

2 2 2

ui /k or by using an average <uj >. This step is justified because the

2 2 matrix elements are usually small when iu /k =1. The sum rule is expected

-1 -I

to hold in the range of momentum transfer 0.25 fm <k<2„5 fm . The lower

limit is determined by the minimum excitation energy (uj = k) for which one

has a fairly complete set of final states available (= 50 MeV). The upper

limit follows from the fact that the interaction operator j A is correct only

M M

2 2 through order k /M . With these approximations closure gives

F? (k

2

) = <o| Y] (e. + (e.-2p.)) (e + (e -2p )) e

1 k

' (

j' r

k

5

1 0>

L L-J J 8M J J K 8M

F

2

(k

2

) = <0|

I .P ^ Z e

j

e kPj

Xp

i x ik* (r .-r, )i e j k

y 0>

j>k

A

F T,a

(k2) = <0

2M 2^jMk

G

j

j »k

x x i k* (r .-r, )i

a

k e j k | 0> (3A-43)

The z-axis has been taken in the direction of k, and the x and y matrix

elements in the transverse direction have been taken equal (Dr58, Mc62, Cz67,

0c70)

The charge sum has a particularly simple form (neglecting for simplicity

the k

2 /M2

terms).

F 2(k

2

) = Z + Z (Z-l) f

2

(k

2

) (3A-44)

where

Z(Z-l) f. (k

2

) =. z

j^k

<0 e .e. J k

i k (r .-r )i J k 0> (3A-45)
104

is the proton pair-correlation form factor. This function is constrained by

f

2 (0) = 1 , f

2 (») = 0 . ( 3 A- 46

)

2 2 f^(k ) modulates the total charge scattering between the coherent (z ) and

incoherent (z) limits.

The transverse current and moment sums are

f t(k2) = I

1 T

* 2Z(

o~ 1} UP(k2)

l,p J

f-r l*r L

(3A-47)

Fj (k

2

) = ^

[Z M n

2 + N p

2 + A(A-l) f°(k2

)] (3A-48)

1 > u 2M r 11 ^

where f,^ and f^

0 are similar to the charge function with e^.e^ replaced by

x x x x 2

e.p. e.p, and p .ct . p.a, , respectively. At their moment limits the F tend JJKK J J ^ K toward their coherent and incoherent values as shown in Table 3-1.

An energy weighted sum rule, for inelastic scattering only, can be formed.

The Coulomb part is

L

2

d a

dujdQ

iudu) -Z-n£ i k • r . e

j e J 0>

For the kinetic energy term

( 3 A- 49

)

L

dujdQ

uidu) _ 7 k (A“l) a

~ L M A

, Z-l , , . 2 n 1 " A-l

f 2 ^ ^ ( 3 A- 50

)

This expression is the analog of the TRK sum for real photons and reduces to

2 TRK 2 2 k CT /27I e in the k^O limit. The enhancement factor (1 + k) of real photon

o reactions will apply to electron scattering at low k, but will approach unity at

high k as the coherence disappears (To80).
105

B. DISPERSION RELATION SUM RULES

A second method for deriving integrals over the photoabsorption cross

section uses the mathematical properties of a real photon's forward scattering

amplitude f(uu) in the lab system. The absolute square of this complex function

of photon energy is the zero degree elastic photon scattering cross section

da

el

f(u))|

2 = (Re f)2 + (Im f)2

. (3B-1)

0

‘

The imaginary part of f is related to the total absorption cross section

through the optical theorem

T rr n U) O (U)) ,~ n Im f(u)) =

4^ (3B-2)

which states that the depletion of the incident photon wave exp(ik*r), due

to absorption by the target nucleus, is accounted for by an out-of-phase

(therefore imaginary) forward scattering wave. The total wave function is

exp(ik°r) + f(0) exp(ikr)/r . (3B-3)

The dispersion relation expresses Re f in terms of an integral over Im f.

Thus, given the total absorption cross section ct(id) , we can calculate the

differential elastic scattering cross section in the forward direction at all

energies. To find da/dfi at other angles one needs further information about

the multipolarity of the absorption and its related scattering process.

The derivation of the dispersion relation between Re f and Im f starts by

allowing the photon energy to become complex (although all final results will

be for real positive values of uj). Cauchy's theorem from function theory states

that the real and imaginary parts of a normalizable function with no singularities

* Total absorption includes elastic scattering.
106

in the upper half plane of its complex argument are related through a principal

Our scattering amplitude f

(

uj) satisfies this criterion of having no singularities

(poles or cuts) for Im uj > 0. This follows from the principle of causality

which states that a scattered wave cannot propagate faster than the speed of

light. In reference (To56) this is shown to be equivalent to the statement

that f(Im u) > 0) has no singularities.

Two steps are needed to put eq. (3B-4) in a practical form: the negative

frequencies must be eliminated, and the integral must be convergent when

Imf(u) 1 ) is replaced by tu'cr(u)

1 ) using the optical theorem (3B-2). For zero-spin

nuclei f

(

-uj) = f*(uj), i.e., the imaginary part changes sign for negative

frequencies, but the real part does not (by time reversal arguments). The

amplitude for spin one-half targets, such as the proton and neutron, has two

components, the non-spin flip f^ and the spin flip f ^

photons. The operator o’ is evaluated between the initial and final target

value integral

Re f (uj) = - P

71 (3B-4)

f(w) = <x f | e

2 + f

2

^ u) ^

>

| x

i > (3B-5)

where and are the polarization vectors of the incident and scattered

spin states x- The forward elastic scattering cross section averaged over

photon and target polarizations is given by

da

el

dft

( 3 B - 6

)

0

°

Recall P (-) = —i— + i7i5(x) so that poles on the real axis are passed in

V X X+l£

the upper complex plane.
107

and satisfy the following crossing relations

f

1

(-uj) = f

1

(uj) , f

2

(-uj) = -f

2

(uj) (3B-7)

theorems

Im f = ~ (a + a )/ 2 = ~ a+ 1 4n v a p' 4-71 t

(3B-8)

Im f 0 = ~ (ct - a )/2

2 4n a p' (3B-9)

where a

a and are the total absorption cross sections for photon and target

heli cities anti parallel and parallel and ct^ is the total cross section for

unpolarized photons. We shall mainly be interested in the spin-averaged

forward amplitude f^ (hereafter called f) although f^ is needed to construct

the scattering cross section.

The negative frequencies of (3B-4) are eliminated by using Im f

(

uj) = -Im f(w)

to obtain

Re f(w)

CO

uj

1 Im f (uj1 ) du>‘ ,2 2

uj - UJ (3B-10)

It is known from measurements of a on nucleons and nuclei at uj > 2 GeV

that ct is constant or a slowly rising function of photon energy so that integrals

CD /

of the form / cr(uj) duj/uj n must have n>l to be convergent. This can be achieved

o

by evaluating the disperson relation (3B-10) at a fixed value of uj for which

Re f(uj) is known. For example, the scattering amplitude at zero frequency has

2

the well known real Thomson limit equal to -(charge) /mass. For a nucleus

f(o) = - (Ze)

2 _ AM i'/ Im f

(

uj

1 ) duj1

UJ (3B-11)

If (3B-11) is subtracted from (3B-10)
108

Im f(u)

1 ) duj

1

, f ,2 2 ,

ID (u) "ID )

= -4?f g(

?0)

«*»' . (3B-12)

2n J

Q

id

1 - uj

Equation (3B-12) is now in a practical form for application to photoabsorption

Re f(uj) - Re f(o) =

2uj_

7T

03 /

03 data at finite id since cj(id) duu/u)

2

is convergent.

The nucleon spin-flip amplitude is thought to satisfy an unsubtracted

dispersion relation. The low energy limit of for nucleons is

lim

Re f (u))/id = - (e

2 /M) (k

2

/2M) (3B-13)

ID^O

where k is the anomalous magnetic moment.

In reference (Ma74) eight sum rules are derived for elastic scattering and

photon absorption for the nucleon and J=Q ,h nuclei. We discuss here a few of the

more useful expressions. In the previous section on closure sum rules the static

polarizability a was introduced (induced dipole moment divided by the field

strength). The magnetic polarizability a

m

is given by the low energy limit of

the spin-flip amplitude (3B-13), whereas the electric polarizability is a

coefficient in the expansion of the spin averaged amplitude

Re f (uu) = f ( o ) + of uj 2 + 0(id

4

) . (3B-14)

Taking the u^O limit of (3B-12) the following sum rule for the electric

polarizability is obtained

f“ dw = 2n

2 a J UJ (3B-15)
109

In the past the assumption was made that Re f(») for a nucleus was Z

times the Thomson amplitude of the proton (Ge54). (3B-12) would then lead to

the TRK sum rule (3A-12). However, we now know that this assumption is incorrect

for reasons discussed in Chapter VII. Equation (3B-12) may be used to predict the

spin-averaged forward elastic scattering cross section in the giant resonance

regi on.

A useful model that leads to an analytic expression for Re f(uj) is the

classical absorption and scattering of an electromagnetic wave on a damped

harmonic oscillator. The absorption cross section is the Lorentz expression

ct(id) = a

2 r2

id

(.uj -u>o

j + id r (3B-16)

where a

Q

is the absorption cross section at the resonant energy id q and f << id q

is the width. The corresponding scattering amplitude is

* Note this form of f (uj) appears to have poles in the upper complex energy

plane (id = uj q + if/2) which violates causality. However, this expression comes

from combining the contributions of two scattering diagrams (see fig. 2A-2b

&c), whose poles are in the lower plane by

f(w) ~ — ~ —

ID - (-U) - i-^) w - ( UJ 0 "i 2)

Using only the first term leads to a Breit-Wigner resonance form.
110

Im f(u>) = a 3 r

2

o 0)1

4n , 2 2,2 , 2r 2 {w -uj^ j + uj r

Re f(u>) = ^

2, 2 2, r UJ (uJ q

-O) )l

471 ( 2 2x2 2 r2

(UJ -u)

Q

) + O) 1

+ Thomson .

In the long wavelength approximation (for a J = 0 nucleus)

(3B-17)

(3B-18)

da

el

dQ

f(uj,e)

2a

a

_o

47T,

4r

2

uj f

( 2 2x2 2 r

2

(O) -U)

Q

) + UJ 1

0

uj 2(u) o

2

-uj

2

) r Re f ( o

)

471 , 2 2x2 2 r

2

(u) -lu q

) + uj r

+ (Re f ( o )

)

l + cos e (3B-19)

with Re f(o) = - (Ze) /AM. Note the scattering cross section becomes constant

at uj >> uj . o Im f depends on a at one energy, but Re f depends on an integral over a(uj)

at all energies. The reason is Im f comes from processes in which the intermediate state satisfies energy conservation, but Re f comes from transitions

between nuclear states whose energy difference is not equal to uj. These states

contribute to absorption at frequencies other than uj. The integral sums these

contributions to Re f(uj).

C. MOMENTUM TRANSFER SUM RULES

A second type of electron scattering sum rule involves the integration

of form factors over momentum transfer (Oc75). One such rule relates the

total volume under the Coulomb form factor surface (elastic plus inelastic) to

the nuclear Coulomb energy E c
Ill

co S L

F c (k,w)| 2 dm - (Z f p

2

(k) + N f

n

2 (k))

d

3

k

2 2

ItCYT

2 c (3C-1)

where f (k) are the nucleon charge form factors and

P . n a

e

c = e2< °i E

j>k 5 /

d,3.h3„ rd r

1 0> (3C-2)

r - r

is the Coulomb energy of interacting nucleons in the nuclear ground state. An

alternate statement of this rule is that the momentum integral of the proton

pair correlation function is

Z ( Z - 1) [f, (k) -4-%- = 2= E r . J 4itk^ e

c

Another simple rule relates (in plane wave approximation) the momentum

integral, of the elastic form factor to the central charge density of

the nucleus

j

F c (k) d

3

k = (2n)

3 p(o). (3C-3)

For inelastic Coulomb form factors from a spin-zero ground state,

03

Fj ( k) = (4tt)"

2 J j

Pj(r) j J (kr)r 2

dr (3C-4)

a number of sum rules can be formed using the identity

where

f k

n

j (kr) dk =

o r

r _ h 9 n-l T (H( J+n+1)

)

n,J

“ 71 21 F (4( J-n+2)

)

(3C-5)

(3C-6)
112

for -(J+l) <n<J+2, J = + integer, n continuous. The k

n moment of the

transition form factor can be related to r moments of the elastic

or transition charge density

CD l

k

" Fj( k)dk= (W*3 c n>J ^

pj(r)r dr

n+1

= (4ti)^ J C . r‘

(n+1)

n , J (307)

-1 -2

For example, the r and r moments can be formed for elastic scattering

and monopole transitions.

The n = -1 moments are particularly interesting

CO CD F j(k) j, . r 2 —^ dk = (4n) 2 J C_

1 j J

Pj(r)r dr

O/ ’ *0

(3C-8)

This sum rule applies to J = 1 transitions. (For inelastic monopole

transitions P

Q

(r)r2dr = 0.

)

The first few coefficients are:

C-l 1 ~ 7I// ^’ 2

-i 2 =^ = ft/16- The integral on the right of eq.

(308) represents the net amount of nuclear charge which participated in the

transition. If we define a transition charge Qj as

then

The net charge change Qj

restriction "net charge"

Qj

= e(47i)"2 J

pj(r)r2 dr

C ,J

-1, J /o

Fj(k)

dk

is a measure of the strength

is necessary since Pj(r) can

(309)

of the transition. The

have positive and negative

regions.
113

REFERENCES CHAPTER III

Ar79 H. Arenhovel , Proceedings of International Conference on "Nuclear Physics with Electromagnetic Interactions" Mainz 1979 ed. by H. Arenhovel

and 0. Drechsel , Spri nger-Verlag.

Cz67 W. Czyz, L. Lesniak, and A. Malecki, Ann. of Phys. 42, 97 and 119 (1967).

Dr58 S. D. Drell and C. L. Schwartz, Phys. Rev. 112 , 568 (1958).

Ge54 M. Gell-Mann, M. L. Goldberger, and W. E. Thirring, Phys. Rev. 95, 1612

(1954).

Ha72 E. Hayward, B. F. Gibson, and J. S. O'Connell, Phys. Rev. C5, 846 (1972).

Ku63 D. Kurath, Phys. Rev. 130, 1525 (1963).

Le60 J. S. Levinger, "Nuclear Photodisintegration", Oxford University Press

(1960).

Ma74 L. C. Maximon and J. S. O'Connell, Phys. Letts. 48B , 399 (1974).

Mc62 K.‘ W. McVoy and L. Van* Hove, Phys. Rev. 125, 1034 (1962).

No78 J. V. Noble, Phys. Report 40C, 241 (1978).

0c70 J. S. O'Connell, Phys. Letts. 32B, 323 (1970).

0c73 J. S. O'Connell, Proceedings of International Conference on " Photonucl ear

Physics and Applications", Asilomar, California 1973 ed. by B. L. Berman,

Vol . I, p. 73.

0c75 J. S. O'Connell and J. W . Lightbody, Jr., Nuc. Phys. A237, 309 (1975).

Si 37 A. J. F. Siegert, Phys. Rev. 52, 787 (1937).

To56 J. S. Toll, Phys. Rev. 104, 1760 (1956).

To80 V. Tornow, G. Orlandini, M. Traini, D. Drechsel and H. Arenhovel, Mainz

preprint KPH 7/80.

We74 W. Weise, Phys. Reports 13C , 53 (1974).

Fr75 J. L. Friar and S. Fallieros, Phys. Rev. Cll, 274 (1975).
114

Table 3-1. Momentum limits of the total form factors.

| k Charge Current Moment

1

0 Z

2 a

1

(El)

k

2 J+l 2

2 2

2 7i e 2M

2

V

CT_

o.

3J m A

1

(M1)

'

-

7i 2

e

2

/M

2

_

00 Z

§

z <e!>

2

M^

k

2

(ZM

p

2 + N,

n

2

) 2 M

Z
NBS-114A irev. 2 -ac) U.S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR-82-2547

2. Performing Organ. Report No, 3. Publication Date

September 1982

4. TITLE AND SUBTITLE

ELECTROMAGNETIC NUCLEAR REACTIONS: I. INTRODUCTION, OPERATORS, AND SUM RULES

5. AUTHOR(S)

James S. O'Connell

6. PERFORMING ORGANIZATION (If joint or other than NBS, see in struction s) 7. Contract/Grant No. NATIONAL BUREAU OF STANDARDS

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Same as Item 6.

10.

SUPPLEMENTARY NOTES

PH Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

This monograph covers the data and theoretical models of nuclear reactions

initiated by electrons and photons. The main emphasis is on the nonrelativistic

calculations of the differential cross sections of the major reaction channels.

12.

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolons)

Deuteron; electromagnetic; electron; helium; nucleus; photon

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

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