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NBS MONOGRAPH 118

'''Its o»

'

Photonuclear Reactions

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UNITED STATES DEPARTMENT OF COMMERCE • Maurice H. Stans, Secretary

NATIONAL BUREAU OF STANDARDS • Lewis M. Branscomb, Director

Photonuclear Reactions

Evans Hayward

Center for Radiation Research

National Bureau of Standards

Washington, D.C. 20234

National Bureau of Standards Monograph 118

Nat. Bur. Stand. U.S. Monogr. 118, 46 pages (August 1970) CODEN: NBSMA

Issued August 1970

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402 (Order by SD Catalog No. 013.44:118). Price 50 cents.
NATIONAL BUREAU OF STANDARDS

SEP 1 2 1971

Library of Congress Catalog Card Number: 70-606490
Contents

Page

I. Fundamentals 1

1. The Trend of a Photonuclear Cross Section 1

2. The Interaction Hamiltonian 2

3. Effective Charges 2

4. The Electric Dipole Sum Rule 3

5. The Bremsstrahlung-Weighted Cross Section 5

II. The Scattering and Absorption of Photons by Nuclei 6

1. The Optical Theorem and the Dispersion Relation 6

2. The Kramers-Heisenberg Formula 7

3. The Absorption Cross Section 7

4. The Elastic Scattering Cross Section for Spherical Nuclei 8

5. The Scattering Cross Sections for Deformed Nuclei 8

III. Photonuclear Experiments 10

1. X-ray Sources 10

2. Photonuclear Experiments 13

2.1. Total Absorption Experiments 13

2.2. Elastic Scattering Measurements 13

2.3. Partial Cross Sections Obtained by Activation Curve Analysis 14

2.4. Radiative Capture Cross Sections 14

2.5. Spectroscopy 16

2.6. Angular Distributions 16 IV. Heavy Nuclei and the Hydrodynamic Model 17

1. Survey of Experimental Data 17

2. The Hydrodynamic Model 17

3. Deformed Nuclei 18

4. The Dynamic Collective Model : 20

5. Photon Scattering Experiments 22

6. The E2 Giant Resonance 24

V. Light Nuclei and the Independent Particle Model 25

1. Survey of Experimental Data 25

2. The Independent Particle Model 25

3. The Particle-Hole Model and Its Extensions 27

4. Caldwell's Experiment 29

VI. The Decay of the Dipole State 31

1. Introduction 31

2. Energy Spectra 31

3. Angular Distribution Formulae 32

4. Comparison With Experiments 35

5. The Quasideuteron Effect 37

VII. Isobaric Spin in Photonuclear Reactions 38

1. Introduction 38

2. Evidence for Isobaric Spin Impurities in the Giant Resonance 38

3. Consequences of Isobaric Spin Purity 39

References 42

III

Photonuclear Reactions

Evans Hayward

This paper reviews photonuclear reactions in the approximate energy range 10 to 30 MeV.

Various sum rules are discussed and applied to experimental data. Several different theories are described and their predictions compared with experiment, and as often as possible open questions

and discrepancies are pointed out. Key words: Nuclear hydrodynamics; particle-hole calculations; photon scattering; photonuclear; sum rules.

I. Fundamentals

1. The Trend of a Photonuclear

Cross Section

In order to get a general picture of the photon

absorption cross section for an arbitrary nucleus,

I let us consider the interactions that can occur as a

ifunction of increasing photon energy. At the lowest

lenergies only Thomson scattering, the scattering

'from the whole nuclear charge, can take place. This

iscattering is an electric dipole phenomenon. For

lunpolarized incident radiation its differential cross

lection is:

d.- =£)2 -

9. L + cos^ B (1.1)

where the scattering amplitude D = — Z'^e'^/AMc^.

At slightly higher energies the photon can disrupt

the internal nuclear coordinates and excite nuclear

^pnergy levels. The first excited state can decay

;tDnly to the ground state and its ground-state

Radiation width, y, is its total width, V. Each

successive state has the possibility of decaying to

all of the states below it. Since the total width, Tk,

of a state at excitation energy, Ei, is the sum of the

oartial widths associated with its decay to all lower

states, the states at higher excitation become

orogressively wider and wider. The cross section

or the absorption of photons by a single isolated

evel is

rt(£)=2irXt2

2h+l yk {ETk

2/o+l n iEk'-E^y+{ETk)

(1.2)

lere Ik and lo refer to the spins of the excited and

{round states Through its width, Tk, this cross

lection depends on the properties of all the states letween Ek and the ground state. On the other

hand, the integrated absorption cross section for a

level depends only on the photon energy and the

ground-state radiation width. It is:

/

(TkiE) dE= (irXi) 2/,+l

2/o +l (1.3)

Above the particle emission thresholds, near eight

or ten MeV, for most nuclei, the levels broaden

much more rapidly until finally they coalesoe

completely.

The giant resonance is theolominant feature of the

photon absorption cross section for nuclei. It is a

large maximum in the cross section, 3 to 10 MeV

in width, located between 13 and 18 MeV for medium

and heavy elements and near 20 MeV for the light ones. The giant resonance occurs in all nuclei and

may be viewed as a general property of nuclear

matter. It results primarily from electric dipole

absorption and its integrated absorption cross section is of the order of the dipole sum. The energy dependence of the giant resonance

absorption cross section for medium and heavy

nuclei has often been approximated by a Lorentz- shaped resonance line:

(Eo^-£2)2_|_£;2r2

(1.4)

We now know that this picture is much too simple,

since the quadrupole oscillations of the nuclear

surface can modify this cross section markedly. For

light nuclei, the giant resonance is the envelope of

a very detailed structure which depends much

more on the individual level properties.

For energies above the giant resonance the most

important absorption mechanism is through the

quasideuteron effect. This is also primarily an

1
electric dipole phenomenon that results from the

strong two body correlations in the nuclear ground

^tate. The high energy photon, which has low

momentum, interacts with a neutron and proton

colliding at high velocity and ejects them with the

dynamics appropriate to the deuteron photodisintegration.

2. The Interaction Hamiltonian

The nuclear Hamiltonian is usually written as:

2M,

(1.5)

where the sum is over all the particles. When radiation is present its interaction is usually treated as a perturbation to the nuclear Hamiltonian. The

nonrelativistic Hamiltonian associated with the

electric interactions is:

H' = H,+H,= £ (1.6)

where here the sum is over all the charges. The

first term. Hi, can describe transitions in which only one photon is absorbed or emitted. It is the term

used to calculate the photon absorption cross section and describes the interaction that results in the disruption of internal nuclear coordinates. The

second term, Ho, is smaller. First order matrix ele- ments of H2 are of the same order as the second

order matrix elements of Hi. Since Ho contains two

powers of Ct, it can describe scattering events in

first order. The second order term may be rewritten as:

NZ Z' li+A

Ze" 2Mc'

N+Z Ze'~ A 2Mc2

2Mc2

:i.7)

where the second term is nuclear Thomson scattering,

the scattering from the whole nuclear charge. The

first term is scattering that results from the disruption of internal nuclear coordinates and must be

combined with second order terms in Hi to obtain

the complete nuclear scattering amplitude, i.e. the

Kramers-Heisenberg dispersion relation. The absorption cross section depends on the

matrix elements of Hi. In evaluating these matrix

elements we make use of Siegert's theorem which

allows us to replace (prG) by (r,'y), where y is the

photon polarization vector and r; is the coordinate

of the ith proton with respect to the nuclear center of mass. This approximation is valid for electric multipoles when the photon wavelength is large

compared to the nuclear radius. At high energies the

detailed structure of nucleon and meson currents is important and Siegert's theorem no longer applies.

3. Effective Charges

\\ hen a photon interact with a neutron or pro

ton in the nucleus, the rest of the nucleus recoils The magnitude of the cross section thus depends on

the size of the nuiltipole moment set up between

the nucleon and the recoiling nucleus. This dynamical

effect can be taken into account bv making use of

the concept of effective charges.

To obtain the effective charges [1] for neutron and

proton transitions in electric dipole approximation we need include in Hi only the first term in the

multipole expansion of (5.

Hi =

Replacing p , by My , we have:

Hi 2^-— (v,-y)

,=1 c

(1.8)

(1.9)

(1.10)

The total nuclear current is ZeV, where V is the

velocity of the center of mass:

tiA A 1=1 y=i (1.11

The energy associated with the center of mas

motion is: — Ze(V-y) =

c cA E (v,-y)+ Z (v,-y)l

j=i >=i J

(1.12

If we add and subtract this from Hi, we have

6nc Hi=-

c

E (v,.y)(i-ZA4;

1 E (v,-y)+Z(V.y)

N ^ 7E(v,-y)

i=l

7 E (vry)+Z(V.y) (1.13

The energy associated with the center of mas

motion of the nucleus, — do Ze(V' y)/c, is responsiitl for Thomson scattering and is not connected wit

internal nuclear coordinates. The first two terra describe the electric dipole transitions of protons an

neutrons having the effective charges N/A an — Z/A. The effective charges for transitions ii

1 Figures in brackets indicate the literature references at the end of tl monograph.

2
volving higher electric multipoles are

g,= [U-l)/^]^+(-)^[(Z-l)/^^]. (1.14)

q„={-)^Z/A^ (1.15)

The magnitudes of these effective charges have some consequences for interactions in which the

excited nucleon emerges directly without colliding

with the other nucleons. Then for the most important

electric dipole case, the ratio of the (7, n) to the

(7, p) cross section for a given photon energy

would be NZ-/ZN- = Z/N. In addition, since the

effective charges of the deuteron and alpha -particle are respectively {N~Z)/A and 2{N-Z)/A, their

cross sections for direct emission are very small and

identically zero for self-conjugate nuclei. Violations

I of both of these rules have been observed experimentally but they can always be explained away by

j including a very small Coulomb interaction.

! The effective charges for electric quadrupole

Uransitions are: q,= 1-2/A +Z/A^-

qn^Z/A^-

(1.16)

(1.17)

i The direct proton emission is then much more

'likely. It is also worthwhile to note that for E2

transitions the neutron and proton effective charges

jhave the same sign whereas for £1 transitions they

_ are opposite. As a consequence, when the absorption

process involves a mixture of £1 and E2 and is followed by direct emission, the proton distributions

lilshould be peaked forward and the neutron dis- Lfributions backward of 90°.

i|H 4. The Electric Dipole Sum Rule

ifi, The sum rule is a model-independent conservation

haw which gives for atomic systems the total inte- i^ated absorption cross section. This rule is a conj sequence of the commutation relations and is usually

! Hscussed in terms of the oscillator strength. For a

transition from the ground state, | 0), to an excited

iitate, (k I , the oscillator strength is defined as

?M.|(i|,|o>|. = 2^(„ z I k){k I z I 0)

(1.18)

j vhere (A1 z| 0) is the matrix element of the dipole

operator connecting the two states and M is the

oass of the particle imdergoing the transition,

iince

it

(0 I 2 I k) = io}ok{Q \ z \ k)

{k\ z \ O) = io)ko{k I 2 I 0)

J f

is straightforward to show that

and (1.19)

(1.20)

M

i=- — (0 I i I k){k I z I 0)-{0 I z I k){k I i I 0)

in

= -j{0\lH,z-]\k)(k\z\0)

-(Ol2|fe)(A:|[if,2]|0) (1.21)

The sum over all the states, k, yields:

Zf>^=-~ {o\LLH,zlz:\\o}

= - ^

(0 I {Hz'-2zHz+z'H) I 0) (1.22)

/r For an electron in an atom

H=py2M+V{r) (1.23)

where V{r) is the Coulomb potential. Using the

relationship p = — ifi d/dz, the double commutator

becomes — fi-/M so that the summed oscillator strength is 1. For an atom with Z electrons.

(1.24)

The integrated absorption cross section into the

Ath level is proportional to the oscillator strength

and is

/

27rV/i akiE)dE=-^f,. (1.25)

Summed over all the transitions this becomes

/

2Tr-e-fi (1.26)

When applied to nuclei this sum rule must undergo

two important modifications. The first is a dynamical

one that results from the fact that when a photon

interacts with a nucleon, the rest of the nucleus

recoils; whereas when an electron is excited the

recoil of the atom is negligible. The second difference

stems from the fact that the nuclear potential is more complicated than the Coulomb potential of the

atomic system. The nuclear potential has terms for which the double commutator with z does not vanish.

The first modification can be made by means of the

effective charges already discussed. Then the nuclear

absorption cross section integrated over all electric dipole transitions becomes

/

aiE) dE==

27rV^

Mc

+ i:(-z/A)%

27rV^. McNZ

A

(1.27)

fi and fj being just the summed oscillator strength

associated respectively with proton and neutron

transitions. This result will be referred to as the

classical dipole sum. It gives the magnitude of the

integrated cross section based simply on the kinetic

energy term in the nuclear Hamiltonian.
It is worthwhile to consider how various kinds of

nuclear potential affect the dipole sum. It is often

stated that velocity dependent or attractive exchange

forces act to increase the dipole sum. By considering

the expansion of the double commutator it is evident

that a potential that contained the nucleon velocities to at least the second power would enhance the

dipole sum; it would contribute in the same way

as the kinetic energy term. Next consider the central potential which may

be written in terms of the four two-body exchange

operators:

Vc= Vim) lwP'^+xP''+yP"+zPB^ (1.28)

where w-\-x-\-y-\-z = 1. The Wigner and Bartlett

operators involve no exchange and spin exchange

respectively. Therefore their commutators with the

coordinate, z, yield nothing. The operators, and

P^, represent space exchange and spin and space

exchange; their commutators contribute and contribute in the same way to the dipole sum. Since

either operator changes the proton coordinate,

Zi, into the neutron coordinate, zy, the expansion of

the double commutator becomes:

{xP^+yP")z^-2zi{xP''+yP")z,+z^{xP''+yP") = zfixP''-\-yP") -2ziZjixP''+yP")

+ZiHxP'^-{-yP") = izj-ZiyixP^-\-yP") (1.29)

The summed oscillator strength resulting from the

two-body central potential would be

M

llh=-Y.^ (z-z.)2F(r.,) (0 I xP^+yP» \ 0) a M = - rr-^ E r./^(r.v) (0 1 xP^+yP" \ 0) (1.30)

For an attractive potential ( V negative) the dipole sum is increased.

The nuclear potential may contain ^-s or £•£

terms. How do they contribute to the dipole sum?

If F=C^-s = c(rXp) 'S = C[_{xpy—ypx)<Tz

-\-{ypz —zpy)(Tx-{-{zpx—xp,)<7y] (1.31)

Since only terms containing pz — —ih 5/52 contrib- ute to the double commutator eq. 1.22 may be writ- ten as:

2^Jk= — {ypz<Tx—xpzay)z^ —2z{ypz(Tx—xpz<ry)z

iMC

[ (yci —xcTy) 2z —2z{yax —xay) ] = 0

(1.32)

Similarly

V=D£'£^D{rXp)'{rXp)=Dl{xpy-ypxy

+ (ypz-zpyy+{zpx-xpzr:\ (1.33)

Again keeping only terms that involve pz we have

MD

(/+;c2)/j,V = 2MD(;c2+y2) (I.34)

where y/ + is the distance of the nucleon , from the z-axis. The integrated electric dipole absorption cross section has then a minimum value resulting from

the kinetic energy term in the nuclear Hamiltonian.

The sum is augmented further by potential energy

terms that do not commute with the dipole operator. In the following, the factor by which the dipole sum

exceeds its classical value will be called (3 so that:

/

adE== — 7^ = 0.06 —18 MeV barns.

Mc A A

(1.35)

The integral in the above expression is over all electric dipole transitions, but since we have neglected all meson currents, a logical upper limit is the meson production threshold. It has been assumed

throughout that the photon wavelength is large

compared to nuclear dimensions. Violations of! this condition will act to decrease the dipole sum.

'

Because data do not generally extend to thej meson production threshold, the integrated cross

sections are usually quoted only over the giant I resonance, i.e. to 25-30 MeV. Figure 1 summarizes

j

experimental data on the photonuclear cross sections! integrated to 30 MeV as a function of Z. It shows

that for the lightest elements only about half of a "dipole sum" is in the giant resonance. The magnitude of the integrated cross section increases gradually until for the nuclei heavier than tin the

integrated cross section is approximately one

dipole sum. The indications are that the amount of

y stci

Figure 1. Photonuclear absorption cross sections integrated

to 30 MeV.

The data for the light element come from the total absorption experimeni

of reference [25], The data for the heavy elements come from the Livermor* neutron yield data [43-52]. ,

"i
Figure 2. The bremsstrahlung iveighted cross section divided by A'" as a function of A.

The data come from the total absorption experiment [25] and the neutron

production eiperiments [49-52].

integrated cross section above 30 MeV is comparable

to that below. hen making these comparisons one

should remember that higher multipoles undoubtedly

contribute especially at the higher energies.

5. The Bremsstrahlung-Weighted

Cross Section

Levinger and Bethe [1] have also discussed a second integral, 5<t/ E dE, and related it to the mean square radius of the nucleus. Their result:

J J

a/EdE=a^,= ' ' ^^'^^^

assumes that there are no nucleon-nucleon correlations in the ground states. It is also independent of

the presence of exchange or velocity-dependent

forces in the nuclear Hamiltonian.

Subsequently Foldy [2] pointed out that the

symmetry properties of the ground-state wave function were more important than correlations between

nucleons and specifically that

o-_i = 3 TcA

NZ

(0 I 0). (1.37)

This rule applies even in the presence of correlations

for nuclei for which the ground-state wave function

is symmetric in the nucleon space coordinates; i.e. the Is shell nuclei. Levinger has also combined the classical dipole sum rule, JcrrfE = 60 NZ/A MeV mb, with the harmonic oscillator energy, £ = 42 A~'^'^, to obtain

(r_i = 60

NZ

A 42.-1"'/' = 0.36-4^/3 mb (1.38)

setting N = Z = A/2. This result yields the ex- perimentallv observed dependence on A while eq

1.36 docs not. See figure 2. The empirical value for the coefficient is better approximated by 0.16 than

0.36. This enormous discrepancy stems largely

from the fact that the giant resonance energy is, in nature, nearly twice the harmonic oscillator spacing.

5
at

ti

i

an

II. The Scattering and Absorption of Photons by Nuclei

1. The Optical Theorem and the

Dispersion Relation

Associated with every absorption process there is a higher order coherent scattering process; the total absorption and coherent scattering cross section can both be expressed in terms of the same complex

forward scattering ampHtude, R{Efi). FamiHar

examples of this connection are the coherent scatter- ing processes, Rayleigh and Delbruck scattering,

which are associated respectively with the absorption

processes, the photoelectric effect and pair production. The relationship between the total absorption

cross section, ai{E), and the forward scattering

amplitude R{E,0) is called the optical theorem:

(r,(£)=47rMmi?(E, 0)

The coherent scattering cross section is

d(x da {E,e)= \R{E,d) p

(2.1)

(2.2)

The dispersion relation gives the connection between

the real and imaginary parts of the forward scattering

amplitude:

Rei?(£, 0)

2ir^hc I

dE'atjE')

E'^-E^

+D (2.3)

where D = — Z^e"^/AMc"^, the energy independent

Thomson scattering amplitude. This equation per- mits the evaluation of Re R{E,0) if the cross section,

cFt{E), is known at all energies where it is important.

It has already been pointed out that the photonuclear giant resonance may often be adequately

represented by a Lorentz -shaped resonance line. Then the forward scattering amplitude is:

R{E, 0) = — E'

Eo''-E^+iTE

AMc" {Eo''-E^)^-\-T^E^

(2.4)

where Eq is the giant resonance energy and T its width. Using the optical theorem we find that: E^T

Mc A {Eo^-E^y+T^E^ (2.5)

The amplitude, R{E,0), has been normalized so that the integral of the absorption cross section will yield the dipole sum

:

/

at{E) dE =

litmi NZ^

Mc A.

(2.6)

The scattering cross section must include the

Thomson scattering amplitude since it interferes with the nuclear scattering amplitude:

da R{E,0)+D (2.7)

If the absorption cross section is known but does not

fit the Lorentz line, then the real part of the coherent

scattering amplitude may be obtained from the

dispersion relation. The shapes of the absorption and coherent

scattering cross sections for a hypothetical nucleus

with A'^=Z = 50 are compared in figure 3. The

giant resonance exhausts the dipole sum with 13 = 1.3 and is represented by a Lorentz line with £'o = 15

MeV and F = 5 MeV. The scattering cross section is asymmetrical because of the interference with

Figure 3. A comparison of the absorption cross section with the\. forward scattering cross section for a nucleus with N = Z = 50r for ivhich the dipole sum is enhanced by a factor 0 = 1.3.

|| The peak in the scattering cross section is shifted to a higher energy as a

result of interference with nuclear Thomson scattering. The symbol C t stands il for the magnitude of the nuclear Thomson scattering cross section which is effective at very low energies. The symbol (Toq represents the high energy limit of the scattering at an energy well in excess of all absorption which has been assumed to be entirely of an electric dipole character.
Thomson scattering which interferes destructively on the low and constructively on the high energy

side of the giant resonance. The low and high

energy limits of the coherent scattering are indicated

in the margin. The lower limit is just the Thomson

scattering cross section, {Z^e^/AMc^y. The high

energy hmit is

(2.8)

This limit is obtained by assuming that the absorption cross section is contained entirely by the Lorentz line and that the latter approaches zero as £—>t». Since these conditions are not realized in nature, it is not possible to use the high energy

limit of the scattering cross section to evaluate /3. Note that for /3—>1 this expression reduces to the

Thomson scattering cross section for Z free protons.

2. The Kramers-Heisenberg Formula

The preceding discussion relates to the connection

between the total absorption cross section and the

coherent scattering cross section. By coherent

scattering is meant that scattering in which the

nuclear system returns exactly to its initial state of energy and angular momentum. No angular

momentum is transferred to the nucleus itself. There

is, of course, additional elastic scattering in which

the nucleus may take up angular momentum but the

scattered 7 rays lie so close in energy to the incident ones as to be indistinguishable experimentally. These we designate as elastic but incoherent. In order to absorb angular momentum in the

two-step scattering process the nucleus must have some intrinsic asymmetry. This asymmetry is most

often provided by its spin but it may also result from

the intrinsic deformation for nuclei having large

quadrupole moments. The nuclear scattering cross section is obtained by calculating the cross section

for one orientation of the nucleus relative to the

photon's polarization vector and then averaging over

the orientations the nucleus can assume in the

laboratory. A final average is then made over the

polarizations in the incident photon beam. If the

nucleus has no intrinsic asymmetry, the first average

need never be performed. The scattered radiation is coherent with the incident radiation and in the dipole

case has the familiar l+cos^0 angular distribution.

The usual treatment of angular correlations concerns itself with transition between states of

well specified spin and parity. The sequence /-^

is regarded as a two-step process and the spin

of the intermediate state, Ik, is an essential parameter. Fano [4] has treated the scattering process by an alternative description in which the angular

momentum transfer is the essential parameter.

Considering only electric dipole transitions, Fano

has shown that the nuclear scattering cross section can be written as the sum of three independent

cross sections each characterized by the angular

momentum transferred to the nucleus. The angular

momentum transferred, v, can have the values 0, I, or 2; the corresponding scattering cross sections are labeled scalar, vector, and tensor. Each of the three

cross sections has its own scattering amplitude, A^,

and angular distribution factor g,{9) so that the

whole scattering cross section for an unoriented

nucleus is:

dff _ ^\ AA^

^2y+l (2.9)

The angular distrbution factors, g,id), depend

only on the angle between the polarization vectors

of the incoming and outgoing photons, X and fi. Then

g2 = |[l +M^-v)^] (2.10)

For unpolarized incident radiation these become:

go = i(l+ cos'd)

gi=K2+ sin^e)

g2 = T^(13+ cos^e) (2.11)

The scattering amplitude, A^, is the KramersHeisenberg dispersion formula generalized to include

the inchoherent scattering associated with v = l and

2.

1 Ek — fia} — ^irk

+

(-1:

and

(/.||r||/o) [[' + V3Z)5.o5o/

1 hi

(2.12)

a= (-l)^''+^/+^(e/c)2(coVa))'/2

IS

X coco'[(2. +l)/(2/o +l)p2

The summation is over all the intermediate states k

that can be reached in electric dipole transitions and

{Ik \\r\\ lo) is the reduced matrix element of

the dipole operator connecting the states lo and Ik. The symbols w and w' represent the frequencies of the

incoming and outgoing photons and

| ^ j

the 6-j coefficient that weights the different components that comprise the total scattering cross

section. The delta functions insure that the Thomson

scattering amplitude interferes only with the part of

the nuclear scattering that is coherent with the

incident beam.

3. The Absorption Cross Section

Since the forward scattering amplitude is A^/S,

it is straightforward to show by the use of the optical

7
theorem that the absorption cross section into a resonance described by this relation has the Lorentz

shape. This manipulation requires setting Io = Ij,

fiu> — h(x>' = E, and v = 0. Then

since

2EkVk

he 3(2/o+l) , {Eu'-E-^y^V.m-'

lo) (2.13)

(/o II r II II r II /o)= (-1)^0+1-^^

I (/, II r || h)

and

J/n /o 0 ] (-1)^0+^^+1

1 1 [3(2/o+l)]i'

In this expression the reduced matrix element may

be replaced by either the oscillator strength, fk, or the ground-state radiation width jkThe oscillator strength of the Ath energy level has

already been defined as: 2ME,

f'c=—

^ \ {k\r\ 0)1'^

2MEk

{hruk I rl I /omo) 1^ (2.14)

Making use of the Wigner-Eckhart theorem this becomes:

2ME, \{h

Jk= —~ (lAiomo I hnik)^

2h-\-l

(2.15)

Summing over the components, nik, and averaging

over mo, we obtain:

2ME, I (J.||r||/o)|^

Jk= ^, /or I - ^ /t^ r i i > 2^ (1X^0^0 | iim*)''

(2/,+l)(2/„+l)

2ME, I </, II r II /o) |2

Th€

<T,,(E) =

3(2/„+l)

47re'^

(2.16)

(2.17) Mc ' iEk'-E'y+Tk'E'

Similarly, the ground state radiation width is:

4e2 Jk =

Ek

he 2h+\

So that:

(Tt{E)=2Tr\k^ 2h+lyk E'Tk'

2/o+l r, {Ek^-E^y+Tk'E^

(2.18)

(2.19)

4. The Elastic Scattering Cross Section

for Spherical Nuclei

The scattering cross section includes not only the

scalar term (^ = 0), but the vector (i'=l) and tensor

(v = 2) terms as well. The energy dependence

associated with the tensor and scalar scattering are the same while that associated with 1^ = 1 is slightly a different. In terms of the oscillator strength, fk, the elastic scattering amplitude, A„ is:

^.= a Z (-l)'°+^-'*3(2/o+l)

2 ME,

X

|/o In V X

(-

Ek — hu — \iVk Eic +hu +¥Tk]

+ V3D5o. (2.20)

The electric dipole scattering from discrete energy

levels includes contributions from all three of these terms. The radiation pattern is determined by the

two spins, /o and Ik, and the reduced matrix ele- ments. In the continuum that makes up the giant resonances, at least for the heavy elements, the

situation is quite different. Here there are many

overlapping energy levels. The sum of their contributions consists of terms of the form:

Ik=Io-i

lo+i-Ik j/o

ll

/o

1 Ik

If we make the usual assumption that here fk is proportional to 2/, + l (or the ground-state radiation

widths, yk, of these overlapping states are all the

same) then since

Z (-l)'°+'-^K2/,+l)

'

//t=/o-i 1 h V

\ 1 /J = 0 (2.21)

for i/>0, the only contribution to the elastic scatter- ing cross section is that resulting from coherent

scattering. As a result of this simplification the

elastic scattering cross sections for heavy spherical

nuclei are related to the giant resonance absorption

cross sections through the optical theorem and

dispersion relation and have the angular distribution

typical of a classical dipole: l +cos^0.

5. The Scattering Cross Sections for

Deformed Nuclei

The nuclei having large intrinsic deformations

have an important additional component in their scattering cross sections. It is incoherent with the

incident radiation and results from the tensor

scattering amplitude associated with the transfer of two units of angular momentum to the nucleus.

It is an experimental observation that the giant resonances for the highly deformed nuclei consist

of a superposition of two resonances having a 2:1

ratio of areas. These are associated classically with

charge oscillations along the one long (AK = ()) and

two short (AK = + 1) axes of the nuclear ellipsoid; in other words the index of refraction of the nucleus

depends on its orientation. It is also well established

that these nuclei are characterized by rotational

l?te
spectra, and that the ground-state is a member of a

rotational band. The radial parts of the matrix

elements associated with transitions between the

giant resonance and all the members of the groundstate, rotational band are the same. The relative

intensities of these lines are given by simple angular

momentum factors. This scattered radiation is known

as Raman scattering. In the intrinsic nuclear coordinate system the

transition matrix elements depend not only on I and

m, but also on K, the projection of / on the symmetry

axis, and electric dipole transitions are specified

by the additional requirement that AK= e = 0, + 1. The laboratory matrix elements may be expanded

in terms of the intrinsic matrix elements, the expansion coefficients being integrals over D functions

which may, in turn, be expressed as products of vector coupling coefficients. The scattering amplitude. Ay, may be written in a compact form [5] if it is assumed that all of the oscillator strength, NZ^/A,

associated with the transition AK = e, is distributed

in a resonance located at E^. The energies of in- coming and outgoing photon are set equal, then: ^ NZBE?

+1 ( — 1 )

' X •! E (lel-eUO)

E,-E-

+

(-1)"

E,+E+^iT, (2.22)

In the Danos-Okamoto model the transitions that

make up the giant resonance of deformed nuclei are associated either with AK = 0 or AK= ±1. For

a nucleus having positive deformation the former

are at the lower energy and the higher energy

resonance contains two-thirds of the area. If A and

B are the intrinsic scattering amplitudes associated

with the major and minor axes of the nuclear

ellipsoid and are analogous to eq. (2.4), then the

scattering cross sections can be written in a simple

form. That associated with = 0 is:

da A+2B +D 1+ cos^e

(2.23)

I The scattering cross section for f = 1 is identically

^ zero. The Raman scattering cross section is:

d(T

I

(/oXo20| //Ko)M 1(^-5)

13+ cos2 0 40

ll (2.24)

The weighting factor, (/oXo20 | I/KoY, gives the

Figure 4. The scattering cross section for a deformed nucleus.

The lower curve is the coherent scattering associated with the absorption

cross section and calculated from eq (2.23). The center curve is that obtained when the elastic scattering for a spin 7/2 (Ta or Ho) nucleus is added to it. The upper curve is the total coherent plus Raman scattering cross section. This is the cross section that is actually measured in a poor resolution experi- ment. It is independent of the spin of the nuclear ground state and as large as can be obtained for a classical system.

relative intensities of the Raman lines, i.e., the

relative contributions to the scattering to those

members of the ground-state rotational band that can be reached in dipole-dipole transitions. For a

spin zero nucleus all of the transitions involving

v = 2 populate the 2+ state. For odd nuclei there are contributions to the ground-state as well as those

having Jo+l and Jo +2. The total intensity in the

Raman spectrum is, however, a constant by virtue

of the sum rule:E (/oKo20| //Xo)2 = l

l/=lo (2.25)

Since the final states all lie within 200 keV of the

ground-state, the Raman lines have not been

separated experimentallly. The measured result is independent of the ground-state spin and as large

as would be expected for a nucleus with /o— for which the weighting factor approaches 1. Figure 4 shows the scattering cross section for a deformed nucleus. The lower curve is the coherent

scattering associated with the absorption cross section and calculated from eq (2.23). The center curve is that obtained when the elastic scattering for a spin 7/2 (Ta or Ho) nucleus is added to it. The

upper curve is the total coherent plus Raman

scattering cross section. This is the cross section

that is actually measured in a poor resolution

experiment. It is independent of the spin of the

nuclear ground state and as large as can be obtained

for a classical system.

9

I
III. Photonuclear Experiments

Photonuclear experiments differ from other ex- periments in nuclear physics only in that the re- actions are initiated by x rays and that the cross sections are generally somewhat smaller. The follow

ing paragraphs describe (1) the different kinds of x-ray sources and (2) the different kinds of photonuclear experiments.

1. X-ray Sources

The kind of x-ray source available largely deter- mines the experiments which will be performed.

These sources fall into two main classes, x rays produced in nuclear excitation and x rays generated

in electromagnetic processes. The former are usually

the 7 rays that follow neutron or proton capture

and as a result they occur at a specific energy with an

energy spread that is often determined by Doppler

broadening or the target thickness. The latter are produced by bremsstrahlung or positron annihilation

in flight and their energy can therefore be controlled

at will. The practical energy resolution is, however, no better than a few hundred kilovolts. In general,

the nuclear 7 rays are continuous in time since they

are produced by a Van de Graaff or nuclear reactor; whereas the electromagnetic radiation is pulsed with

a duty cycle often as low as 10~^. The latter is a consequence of the pulsed nature of high energy

electron accelerators and is a disadvantage except

in time-of-flight experiments.

There are three important sources of proton

capture y-rays that have been used to study photonuclear reactions: The F^^(p,a7), Li^(p,7) and the

H^(/),7) reactions. Their properties are comjjared in table 1. The F(p,ay) reaction has been used by

Reibel and Mann [6] in a resonance fluorescence

experiment. This reaction produces three 7 rays that come from the excited states in O^^ at 6.14, 6.92,

and 7.12 MeV. Their relative intensities can be

changed by varying the incident proton energy and

as much as 80 percent of the intensity is at 7.12 MeV for proton energies of 2.05 MeV. The energy

Table 1. Proton capture y rays

Reaction £;^(MeV) AE(keV)

Fis(p, ay) 7.12 130 LF(p, 7) 17.6 12.2 mp, 7) >20 40

spread of about 130 keV results from the fact that a particles are also emitted in this reaction.

The Li'(/>,7) reaction is an important source of

17.6 MeV 7 rays. Its great disadvantage is that it is contaminated with a broad band of x rays centered ^ near 14.2 MeV. It is however, very useful in ex- I periments where the incident photon energy can be

determined from that of an outgoing nucleon. r The H^(p,7) reaction has been used primarily by

Stephens [7-10] and his collaborators. It represents

a source of 7 rays which may be continuously varied

from 20 MeV upward by increasing the bombarding

proton energy. The practical upper limit depends on the experiment and results from the serious

neutron background generated in the H^(/>,n)

reaction. The energy resolution is determined by the

thickness of the gas target and can be made as low as 40 keV.

Neutron capture 7 rays have been used by

Donahue [11—13] and more recently by Ben-David

[14-15] to study the (7,n) and (7,7) reactions for excitation energies near 8 MeV. These 7 rays occur^

in a rather restricted energy range but the sprea

^ of each line is only a few volts, being determined

largely by the Doppler width, A = Eo(2kT/AMe-y

associated with the thermal motions of the targetj atoms. Table 2 shows a list of useful neutron capture

7 rays and their relative intensities.

Table 2. Neutron capture 7 rays

Element

Hg

CIFeCo

Fe

Al

Ni

Cu

Cr

CICr

Ni

FeCr

FeN

E{MeY)

5.44

6.12

7.28 7.94 7.64

7.73

7.82

7.91 8.449 8.56

8.881 8.997 9.30

9.72

10.16

10.83

7 rays/neutron capture*

0.038

.177

.042 .029

.30

.22 .06

.20

.06 .021

.14

.27 .024

.06 .0006

.13

* The errors on these intensities are of the order of : 30 percent.

10
Table 3. Resonancefluorescence y rays

Nucleus £(MeV) Ground-state radiation width (eV)

3.56 7 Mg" 9.92 6 Mg" 10.66 18 11.4 23

C12 15.1 36

Knowles [16] has described a method for producing

monochromatic, partially polarized photons by

Compton scattering of neutron capture y ravs. The

1 source is a Ni or Ti target placed in a reactor in a position of high thermal neutron flux. The capture

7-ray beam emerges from the reactor and is scattered

from an extensive aluminium plate. The scattering

' angle determines the photon energy. These mono-

- chromatic photons were then scattered from other

' targets and the resonant radiation studied. In this

• way an x-ray beam continuous in time and coni tinuously variable in energy from 1 to 10 MeV is ^produced.

= Monochromatic y rays can also be produced by

resonance fluorescene. Here an intense bremspstrahlung beam is incident on a suitable target which

A picks out the resonant radiation and reradiates it in

'" all directions. Table 3 contains a list of energy levels

- having large ground-state radiation widths [17]. ^With the exception of Li% for which the 3.56 MeV

state is O""", the photons scattered from these levels are plane polarized at 90°. The production of such

•'.a source suitable for doing experiments is still in the

•''future. The classical photonuclear experiments have been

j performed with bremsstrahlung x rays produced by

allowing the internal electron beam of a synchrotron

or betatron to strike an internal target. The resultant

"X-ray spectrum is continuous in energy extending

. 'Up to the kinetic energy of the radiating electron. Figure 5 shows the shape of the bremsstrahlung

spectrum according to the formula of Schiff [18] (where the spectrum has been integrated over the

E. Me V

""iGURE 5. The cross section for the production of X-ray energy

{left-hand scale) and photons (right-hand scale) between E and

E+dE when 19 MeV electrons strike a tungsten target.

\

I

angles of the outgoing photons. This averaging

approximates the multiple scattering of the electrons

in practical targets. In most bremsstrahlung experiments the yield of some reaction is measured as the peak bremsstrahlung energy, Eo, is advanced in small steps. An

involved analysis process is then used to obtain the

cross section. In such experiments only those photons

in the top energy interval, say 0.5 MeV, are used to produce the desired effect while the bulk of the

spectrum produces a "background" that must be

subtracted off. The bremsstrahlung monochrometer represents a method for selecting a very small energy interval out of the continuous bremsstrahlung spectrum.

This AE can be taken several MeV below the peak

energy and in this way a reasonable x-ray intensitv

obtained. This scheme has recently been used in photon scattering [19,20] and photoneutron experiments at the University of Illinois and their arrangeDETECTOR 1—

V

COINCIDENCEAFigure 6. The bremsstrahlung monochromater.

Monoenergetic electrons are deflected from a circular accelerator and

allowed to radiate in a thin foil. The electrons are then analyzed magnetically. The bremsstrahlung photons strike a second target and some product radiation

is detected in coincidence with the electron. In this way the photon energy can

be specified.

ment is illustrated in figure 6. The electron beam

is deflected from the circular orbit of the betatron

and radiates in a thin, external target. The electrons are then analyzed magnetically. The bremsstrahlung

photons are allowed to impinge on a target and some

product radiation detected in coincidence with the

analyzed electron. The energy of the absorbed photon

can then be specified by requiring that hci: = Eq — E. This technique is basically limited by the chance

coincidence rate. When higher intensities are available, as with the

linear accelerator, effectively monochromatic photon

beams can be produced by allowing monoenergetic

positrons to annihilate with electrons at rest. In the

center of mass system two photons each having an

energy of mc^ are emitted isotropically in the anniI
tittj, M e V

t'iGURE 7. The cross sections for the production of annihilatioril^ and bremsstrahlung photons in the energy interval d(hw)

when 19 MeV positions strike a beryllium target.

hilation of a positron. If the positron has a very high

velocity and annihilates in flight then the photons

emitted in the forward hemisphere come off within

a very small angle in the laboratory system and

with a spectrum that increased to a maximum

energy of approximately £++Hmc^ The backward

going radiation is concentrated in a similar cone with

a lower limit on the laboratory energy of approximately }/2mc^. The relationship between the laboratory photon energy and its angle, 6, relative to the

positron direction is:

^co = mcV(l-/3cos0) (3.1)

Here 0 is the velocity of the center of mass system,

[(E+-mc2)/(£++mc2)]i/2. The exact energies of the

maximum and minimum photon energies can be

obtained by setting d equal to 0 and ir. As the positron

energy increases the peaks at either end of the

spectrum become sharper and the photon beam is concentrated in a smaller cone. The cross section for this process differential in the

energy of one of the outgoing photons is:

da _ /_e^Y _J_ Twi

_^ ^2 ^

2(mc^)^

X(l +7) ^ {ho}i)iho)2) ).

where ho}i+ho}2 = mc^{l+y) and 7 is the positron

energy in units of its rest energy E'^/nic^. The

spectrum of photons emitted when a positron of

total energy E+ = m(?-\-\9 MeV annihilates in a Be

target is shown in figure 7. In a target of practical

thickness this spectrum is modified primarily by

ionization losses and secondarily by multiple

scattering before annihilation in the target.

A second troublesome feature is that the positrons

also radiate in the annihilation target. For relativistic

particles the ratio of the annihilation to the bremsstrahlung cross sections (per atom) is essentially

\?tlir/^yZ. This shows the importance of using

a low Z target to suppress the bremsstrahlung. The

energy distribution of the bremsstrahlung photons

produced in a Be target by electrons having a kinetic

energy of 19 MeV is shown for comparison in figure

7. When annihilation radiation is used as a source of X rays in an experiment, it is obviously necessary to take this bremsstrahlung into account. This is done

by measuring the backgrounds with negative elec- trons striking the annihilation target and assuming

that electrons of positive and negative charge

produce the same bremsstrahlung spectrum.

The contemporary electron linear accelerators

have sufficient intensity to make experiments with

positron annihilation radiation practical. Three

such systems have been developed. In all, the posi- trons are created in a shower when the electron

beam strikes a high-Z target, but two different loca- tions have been used for this target. In one case [22

the shower is produced at the end of the accelerator

and monoenergetic positrons selected and focused by

a system of slits and magnets. The lithium annihilation target is located in the field of the final bending

magnet and shaped so that the forward radiation is focused on the experimental target while the posi- trons are bent into a shielded area. In the other

case [23] the positrons are created after the first section of the machine and then accelerated up to the desired energy; in this way the positron beam

may be carried by the same magnet system that

is used to deflect and focus the electron beam. An

additional magnet is required for removing the

positrons after the annihilation target just as would;

be used after a bremsstrahlung target. This scheme

has the additional advantage that the background^

generated in the positron-producing shower is at a

relatively low energy and far away from the experiment.

Table 4 shows a very rough comparison of the

yields and energy spreads for the three x-ray sources

just discussed.

The electromagnetic sources always rely on some

kind of an energy control device which must be

calibrated. The most common calibration points

are the thresholds for the production of neutrons (oi; the resulting radioactivity) when a target is irradiated

with x rays. In order to produce an activation curve

that begins sharply at the neutron binding energy,

it is highly desirable to select a target which can emit s-wave neutrons and leave the residual nucleus

Table 4. Rough comparison of electromagnetic x-ray source:

Source photons/electron A£(MeV)

Bremsstrahlung 10-5 0.5 Monochromator 10-6 .1 Positron annihilation 10-10 .5

12
in its ground-state, since near threshold

(3.3)

where E is the energy above threshold. The slope of

this curve is then

da/dE=S{f+i)E'-'''- (3.4)

which is infinite for outgoing s-wave neutrons and

varies much more slow ly for higher angular momenta.

Table 5 lists some targets that have proved useful

for this purpose. ^ ith the exception of deuterium

they are all relatively heavy nuclei from which the s-wave neutrons are simply evaporated. The neutron

binding energies in the light elements come at higher

energies (~ 15 MeV) so that it would be highly

desirable to use one as a calibration target. For these

elements the neutrons are emitted primarily in direct interactions and the matrix elements for the

transitions that result in s-wave nucleons are small

and consequently the thresholds are less sharp. An

extreme example is Al-^ which cannot absorb

electric dipole radiation, emit s-wave neutrons, and

leave Al-^ in its ground state and at the same time

conserve parity.

A very excellent alternative is to use the resonance

fluorescence of the 15.1 MeV state in C^-. This level has a large ground-state radiation width and the

scattered photons are easily detected. The disadvantage is, of course, that the dectection apparatus

is different from that used to measure the (7,0)

thresholds so that it is often very inconvenient.

2. Photonuclear Experiments

The objective of photonuclear experiments has

been first of all to determine the gross features of the

photon absorption cross section, such as its magnitude and energy, and then to explore the detailed

shape of the cross section with better energy resolu- tion or by examining the spectra of outgoing nucleons. Angular distributions have sometimes been

used to infer the multipolarity of the absorbed

radiation. In the following we will simply list some

different kinds of photonuclear experiments and

point out a few of their sailient features.

Table 5. Calibration Energies*

Threshold (MeV)

D{y, n) 2.22

Pt'«H7, n) 6.13

Bi"'(7, n) 7.39

Au'9'(7, n) 8.05 Cu"(7, n) 10.8 OHy, y) 15.1

80 100 120 CHANNEL NUMBER Figure 8. The spectrum of photons transmitted bv 4.5 m of

ifoter when 90 MeJ ' bremsstrahlung is incident upon it.

The insert shows the total photonuclear cross section iliat has been derived from it (reference [15)).

2.1. Total Absorption Experiments

The continuous bremsstrahlung spectrum is usually used as a source of x rays in these experiments.

A well defined beam is filtered through a long

absorber of the material under study and the

spectrum of transmitted photons recorded. This

spectrum is then compared with that which would

be expeceted if only the electronic processes, Compton scattering and pair production, were removing

photons from the beam and the cross section for nuclear absorption obtained. This analysis requires

a detailed knowledge of the electronic absorption

coefficients since the nuclear absorption is never more than 10 percent of the total. The results obtained by this method are quantitatively consistent with other measurements.

Figure 8 shows the pulse height distribution

produced in a sodium iodide pair spectrometer [24] when a 90 MeV bremsstrahlung spectrum filtered through 4.5. m of water was incident on it. The

oxygen cross section [25] derived from it is also shown.

2.2. Elastic Scattering Measurements

Here a sodium iodide spectrometer has been

used usually with bremsstrahlung sources to measure

the elastic scattering cross section. First the spectrometer is placed in the x-ray beam and compared with

the transmission monitor so that A^i is the number

of counts measured in the energy interval, AE, near

the peak of the bremsstrahlung spectrum when

monitor reading is Mi. A second measurement

N2/M2 is then made with the spectrometer viewing

the target from some large angle. The elastic scatter- ing cross section is then:

* The neutron binding energies are taken from V. J. Ashby

and H. C. Catron, UCRL 5419.

(3.5)

13
Figure 9. The elastic scattering cross section for lead.

The broad peak in the scattering cross section in the energy range 10-20 MeV

is associated with the giant resonance absorption and is related to it by the

optical theorem and the dispersion relation. The peak in the scattering cross section below 10 MeV results from the resonance fluorescence of discrete energy levels.

where K is a constant that depends on the target

properties and geometrical factors. The elastic scattering cross section for lead [26] shown in figure 9 displays two peaks. The higher

energy one is the scattering that is related to the

giant resonance absorption by the dispersion

relation already discussed. The lower energy maximum occurs below the threshold for particle emission

where the nuclear energy levels are discrete and

well defined. The observed elastic scattering results simply from their resonance fluorescence.

2.3. Partial Cross Sections Obtained by

Activation Curve Analysis

In the past, by far the most common photonuclear experiment was one in which the number of

outgoing particles or the radioactivity generated in a

target was measured as a function of incident

bremsstrahlung energy, Eq. Each point on such an

activation curve is the ratio of the number of events,

N{Eo), counted in some nuclear detector to the

reading on an integrating monitor, M{Eo) :

7V(Eo)/M(Eo) =nldEe{EME)I{E, E,)/E (3.6)

In this expression I{E,Eo)dE/E is the number of

photons in dE in a bremsstrahlung spectrum extending to Eo, n is the effective number of atoms/cm^

in the target and fi{E) is the detector efficiency. The

latter must be made independent of E so that it need not be included in the integral. In addition, the

monitor must be calibrated so that its reading can be related to the amount of energy incident upon it. Then, the activation curve in the units of events

per unit energy incident on the monitor can be

converted to a cross section by means of the inverse

matrix of the bremsstrahlung spectrum which has

been tabulated by Penfold and Leiss [27]. These

tables are now almost universally used in the

reduction of photonuclear data.

Since this reduction process is essentially one of

differentiation, great care must be taken not to introduce artificial wiggles in the activation curve nor to destroy those that result from structure in

the true cross section. The monitor response function must be smoothed but the details in the acti- vation curve should not be removed since they contain the interesting nuclear information.

The most important limitation on this kind of

experiment is the statistical inaccuracy resulting

from the finite number of counts acquired. The

statistical errors in the cross section may be eval- uated by propagating those on the activation curve through the inverse matrix used to reduce the data.

Figure 10 shows an activation curve obtained by

counting the total number of neutrons emitted

by a gold sample [28]. Approximately 5 X 10^ counts per point have been obtained. The cross section resulting from this analysis is also shown

along with the statistical errors. The latter are as

large as 25 percent on the high energy side of the

resonance. In a heavy element such as gold, photon absorption is followed almost exclusively by neutron emission, since the Coulomb barrier essentially elimi- nates proton emission. The total absorption cross section may then be obtained by correcting for the

multiplicity of neutrons emitted in the {y,2n) process. The lower set of points in figure 10 shows the importance of this effect. This correction is usually

quite uncertain. In a few cases the {'y,2n) cross section has actually been measured and these data can be used. More often the magnitude of the effect

is estimated from the statistical theory with an

appropriate contribution for direct emission.

2.4. Radiative Capture Cross Sections

The giant resonance of nucleus. A, can be excited by nucleon capture into the nucleus, A-\.

If the ground-state gamma radiation is observed.

E, M e V

Figure 10. The activation curve or number of neutrons

generated as a function of bremsstrahlung energy incident on a

^ gold target.

The neutron production cross section derived from it is also shown. On the high energy side of the giant resonance there are two sets of points. The lower ones represent the photon absorption cross section and were obtained from the upper ones by correcting for the emission of two neutrons in the (y. 2n) process where it is energetically possible. The errors indicated are statistical.

14
Table 6. Ground-state, proton capture experiments

Target Excited nucleus T(MeV) Reference

19.80 29

Li' 17.25 30 B" Q12 15.95 31, 32

Q12 1.94 33 W' 016 12.11 34

pi 9 Ne^i 12.87 35, 30 Mg" 11.69 36 Al" Si2S 11.59 31, 37

psi S32 8.86 38, 39 K" Ca" 8.33 30, 40

then the principle of detailed balance can be apiphed to infer the inverse photo-nuclear cross section. The advent of the tandem accelerator has made

possible the exploration of the giant resonance

Tvdth an energy resolution of a few tens of kilovolts

by studying the ground-state, proton capture

cross sections. Sufficient energy is available here

to excite the giant resonances of the light nuclei

where the Coulomb barriers are still low enough

1o make the measurements practical. This is ac- complished by measuring the {p,yo) cross section

las a fimction of incident proton energy, the y ray

being selected with a sodium iodide spectrometer.

Often this is made difficult by the proximity of the

first excited state to the ground state of the final nucleus. In order to make comparison with photonuclear

reactions it is necessary for the two adjacent nuclei to exist in nature. A list of those that have been ex- plored is given in table 6 along with the binding

energies, T, and references to contemporary ex- periments.

The cross sections for two reactions the exact

inverse of one another are related through the principle of detailed balance. This rule, which is a con- sequence of general S -matrix theory, states that

(3.7)

where ai and <T2 are the cross sections for the two

processes referred to the same excitation energy

Eg. The quantities, gi and gi, are the statistical weights associated with the two interacting particles,

and p stands for the momentum of their relative

motion. This equality is only valid for unpolarized

'beams and imoriented targets.

Consider a{y, po) for the nucleus A. Then

g = gygA = 2(2lA +l) and p = ho:/c. For the inverse

trip, 7o) reaction g = 2(2/^_i+l) and /j = Mfrei =

\^2MEp{A — l)/A. Substituting in the above

expression we have:

{2/^ +1) (Wc)V(7,Pu) = (2/a-i+1)

X{A-l/Ay{2ME,)a{p,yo) (3.8)

The excitation energy in a (p,yo) reaction is:

where T stands for the proton binding energy. In the (7,/>o) reaction the excitation energy is:

(3.9)

-'Ee=hco{l-hco/2AMc^) (3.10)

Substituting for and Ep in the above expression we obtain:

(2J^_i+l) A -I {E-T)

'^'^P'^^l^fl^-A ^2McV(p,7o)

(3.11)

Thus the proton-capture experiments yield the

cross section for populating the ground state of the

residual nucleus in a {y,p) reaction and sets a lower

limit on the total {y,p) cross section. More than anything else these experiments have

demonstrated the detailed structure that exists

in the giant resonances of the light nuclei and shown

that its character changes drastically as we progress from C^2 to Ca^°. Figure 11 shows a comparison of

the cross sections [33, 37] for C^^ and Si^^ and is

15

b cj

Ee, MeV

20 2 2 24 26 28

~i I I

I

I I

1—I—I—

r

b"(P,x)c'^

oU ^ L

8.0 10 E„(Me V)

J \ L

- 2

~i r

16 Ee , MeV

AI (P,ro)Si

Ep , MeV

FiGUEE 11. A comparison of the ground-state, proton-capture

cross sections [32] for and Al^'.

There is an important qualitative difference between the amount of structure seen for these two nuclei.

15
Figure 12. A comparison of the photoproton [42] (upper) and photoneiUron [41] (lower) spectra obtained when oxygen targets are irradiated with bremsstrahlung X rays.

The bremsstrahlung spectra extended to 25 MeV and in both cases it has been assumed that only

the ground state of the residual nucleus is populated.

meant to illustrate the large qualitative difference

between these two nuclei.

2.5. Spectroscopy

A vast amount of data exists and is still being

accumulated on the energy distributions of photonucleons. The classical spectroscopy experiments were those in which a target was irradiated with

bremsstrahlung from a betatron and the outgoing

nucleons detected in nuclear emulsions. Other more

tractable detectors are now coming into use especially with the higher intensities available from linear accelerators. Nevertheless, even now very few

spectra have been measured at more than one

bremsstrahlung energy. We can look forward in the next few years to a great deal of exciting new data

obtained using either positron annihilation radiation or with varying bremsstrahlung energies.

Experiments that make use of high resolution

detectors have been complementary to the radiative proton capture experiments in revealing a

large amount of structure in the giant resonances

of the light nuclei. When irradiated by the contin- uous bremsstrahlung spectrum the nucleus absorbs

out narrow bands of x rays corresponding to its excited states and emits particle groups to the

available states in the residual nucleus. Figure 12 shows a comparison of the neutron [41] and pro- ton [42] spectra from 0^^ where the level density

s I(

« lies

tlDf

in the residual nuclei is exceptionally low and afifio great deal of structure is observed. In both cases I"

it has been assumed that the outgoing nucleon leaves TM

the final nucleus in its ground state; it is evideni

that the same giant resonance structure is repro duced in both.

aea

j Stic 2.6. Angular Distributions

The angular distribution measurements suffei. from the same criticisms as the spectroscopy ex-^ periments; namely, that the precise energy of the,j

j, absorbed photons is not always known In factr, these two kinds of data are often obtained in thq same experiment. Sometines fast neutron angulai^

distributions are measured with threshold detectors,

the Al"(n,/>)Mg^^ and Si^*(n,/?)AP* reactions being

iijl

Brl

aci;

;

ttioi

lbs

favorites. The latter, for example, samples neutrons

essentially in the energy range 5 to 10 MeV. The

qualitative result of these experiments is that neai

the peak of the giant resonance the photonucleoB

angular distributions peak at 90° as would be ex-, pected for an electric dipole interaction. As the excitation energy increases these distributions begin to peak slightly forward as a result of interference

with radiation of opposite parity, presumably E2. ' The most accurate angular distributions, yielding

the most detailed information, are those obtained

from the ground-state, proton capture experiments*

These are found to be almost energy independeni

throughout the giant resonance.

kill.

16
IV. Heavy Nuclei and the Hydrodynamic Model

1. Survey of Experimental Data

Three gross quantities can be determined from

he photonuclear cross section measurements, the

ross section magnitude, the giant resonance energy, md its width.

Some data on the photonuclear cross sections

ntegrated to 30 MeV have already been presented n figure 1. The data on the light elements came rom the total absorption experiment of Wyckoff

t al. [25]. These results are consistent with the

alues obtained by summing the separately meaured (y,n) and (7,/)) cross sections. For the heavy

lements only the neutron producing cross sections

re important and the data presented here come rom the Livermore neutron yield measurements

43-52]. The neutron yield data for nuclei having

i <100 are excluded, because for them the iy,p) cross

.ections are appreciable. The general conclusion rom these and other data is that the integrated

ross sections for nuclei as light as carbon and oxygen

i approximately half of the dipole sum. This value

jacreases gradually with atomic number and finally

bove Z=50 levels off at a constant value. These

articular data show the constant to be 1.0 but it lay be somewhat larger. The reason for this large

ncertainty has to do with the systematic errors ssociated with bremsstrahlung beam monitoring

^d the determination of the neutron detector

teciency. In any case, the heavy element cross

-ictions integrated over the giant resonance are

It least as large as the dipole sum.

Figure 13 shows a plot of EqA^'^ vs A. The data

»r A<40 are taken from reference [25] and those

CintE 13. The giant resonance energy times A^'^ as a function

of A.

Iht data for the light elements come from reference [25] and those for

iditun and heavy elements from many recent neutron yield measmrements

-521.

for the heavy elements from the Livermore experi- ments. The horizontal line corresponds to the dependence

Eo = 80A-'i' MeV,

the prediction of the hydrodynamic model. The

resonance energies lie fairly close to this line for the heavy elements but begin to fall lower for A < 100. Figure 14 shows a plot of the giant resonance width vs A. The data have been taken from the same experiments as for figure 13. These data show

that the average giant resonance width is about

five MeV and that the dependence on A proceeds

through several maxima. These are associated with

the three major regions of large intrinsic nuclear

deformation, near A = 25 160, and 240. Closedshell nuclei have the narrowest giant resonances.

2. The Hydrodynamic IModel

Many of the features of the giant resonances of

the heavy elements may be described in terms of

the hydrodynamic model. In this model these

many particle systems are assumed to consist of

interpenetrating neutron and proton fluids of constant total density confined within a rigid boundary.

The giant resonance is then associated with the

lowest mode of the electric dipole counter-oscilla- tion of these two fluids. The attraction between the

two fluids produces a restoring force the magnitude

of which is related to the symmetry energy, K(N—Z)'^/A, of the semiempirical mass formula.

Figure 14. The giant resoance width as a function of A.

The data come from the same experiments as those used for figure 12.

17
The model [53] establishes a connection between

the resonance energy and the integrated absorption

cross section associated with this lowest dipole

mode. The resonance energy obtained on the assumption that the fluids are contained within a rigid

boundary is

2.0Sh

R

SKNZ

MA"

1/2 (4.1)

where 2.08 is the solution of the eigenvalue equation

for a sphere of radius R. These assumptions also lead to the conclusion that only 86 percent of the

oscillator strength is contained in the lowest mode,

the rest being at higher energies.

Therefore

/

adE= — 7 (0.86). Mc A(4.2)

This classical description has no mechanism for the inclusion of the exchange forces that are re- sponsible for the enhancement of the dipole sum.

This effect may be artificially inserted by replacing

the nucieon mass by an effective mass that is some- what smaller, i.e., M* = M//3 where /3>1. Then

eqs (4.1) and (4.2) become:

2 .08^

R

8KNZ0

MA-"

1/2

/

adE= —— (0.86) —

Mc A

(4.3)

(4.4)

The softness of the nuclear surface tends to in- crease the nuclear radius, 7?= 1.2^^'^ Fermi, by«;10

percent and perhaps to increase the fraction of the

oscillator strength in the lowest mode. There is thus some arbitrariness in the choice of parameters

to be used. One selection might be K= 23 MeV,

R = 1.2A''\ and /3 = 1.1. Then

£0 = 80^-1/^ MeV. (4.5)

Figure 15. The integral, <r_2, in units of .00225 A^'' as a

function of Z.

The data come from the Livermore neutron yield experiments [43-52],

Using the same model Migdal [3] has shown that

the static polarizability of the nuclear ground state

eo = e'R'A/iOK. (4.6)

The polarizability is proportional to the forward'

scattering amplitude and is simply

(4.7) In order to obtain the static polarizability we need

only consider Re R{E^), since lmi?(£',0) = 0 at zero frequency. Taking the limit of Re R{E,0) as the

energy approaches zero, we have from eq (2.3):

€0 = lim

he f a{E') dE' he f he f

<j{E) dE

^0 27r2 J E'^-E^

he r 2^ i

r odE _2t^ _ 27r2 J E^ ~ h^ ¥2tv^^RM

h^ 40K

= 2.25 10-M5/3

(4.8:

barns

MeV

(4.9; Figure 15 shows a plot of this experimental integra

in units of 0.00225 A^'^ mb/MeV for the heav}

elements that can be described by the hydrodynami(

model.

3. Deformed Nuclei

The hydrodynamic model predicts that the gian resonance energy varies inversely as the nucleaj

radius or as A~^'^. A natural consequence of thil prediction is that for deformed nuclei having twi characteristic dimensions, the giant resonance shouh

be a superposition of two resonances. Most deformei

nuclei are prolate ellipsoids. The higher energ resonance corresponding to charge oscillations alon,

the two short axes would then comprise two-third

of the integrated cross section. The separation of th' two resonance energies is a measure of the quadrupol

moment of the ground state. ' For deformed nuclei, the eigenvalues for exci tation for the lowest £1 mode differ by only a fe^ percent from the value KR= 2.0S obtained for tb spherical case. Danos [54] has shown that the cor nection between the resonance energies, Ea an

Eb, and the lengths of the long and short axei a and 6, is

f;6/£a = 0.911a/6+0.089 (4.10,'

In terms of d = a/b and the mean nuclear radiuij

R, the intrinsic quadrupole moment is:

Qo = iR'ZA^i^

d^-1

d"^

(4.11

Careful measurements of neutron yield croi sections have shown that the giant resonances fi, the rare earth nuclei do indeed have two maxim,

and these cross sections can be fit by the sum of tM'

iOlll

18
1 ' r 1 ^ r

20 22 24 E, Mev

Figure 16. The neutron yield data for hohnium and erbium.

The solid curve is the sum of two Lorentz lines having the parameters <rOa=318 mb, =2.2 MeV, ro=2.33 MeV, 0-06=328 mb. Eft = 16 MeV, and r6=4.5 MeV. The dashed curve is a smooth line drawn through the data. A better S? [56) i« obtained with the data if the coupling [66] with the surface vibrations is taken into account.

resonance lines having approximately a 2:1 ratio of areas.

r(E) = 0-a" {ETa

{EJ-E'y+E'Ta^

(Eny

{Eb'^-E^y+E^Tb''

(4.12)

,siJ

'

where o-(,Ti = 2craTa. As an example figure 16 shows

the results of such an analysis for holmium and

erbium [55]. The solid curve is the sum of two

Lorentz lines having the parameters <ra° = 318 mb,

£:a = 2.2 MeV, Fa = 2.33 MeV, (76« = 328 mb, £,,= 16

MeV, and Tb = 4.5 MeV. The dashed line is a smooth

curve through the data points. In the region of large nuclear deformation beyond

the closed shell at Pb^°* only the giant resonance of

U^^* has been studied in detail [46]. The total photon

absorption cross section was obtained by summing

the {y,n), {y,2n) and {y,F) cross sections. All were

obtained using positron annihilation radiation as a

source. The fission cross section was measured in a multipleplate ionization chamber and the neutron- producing cross section with BF3 cou ters embedded

in paraffin. The (7,^) and y,2n) cross sections have

been obtained from a statistical analysis of the time

distribution of the neutron counts relative to the

accelerator burst. The result is shown in figure 17. This total cross section has also been fit with the sum of two resonance lines. In the region of large deformation between the

closed shells at O'^ and Ca*°, the hydrodynamic

model is certainly not applicable. Nevertheless,

the vestiges of the double-humped absorption cross

section are still apparent.

The determination of the giant resonance energies for a deformed nucleus is one of the most accurate ways to obtain its intrinsic quadrupole moment. This method measures the intrinsic rather

than the spectroscopic quadrupole moment because

the measurement takes place in a time determined

by the giant resonance width, fi/6 MeV. This is very small compared to the time required for the

nucleus to roll over in the laboratory system, a measure of which is the spacing of the levels in the

ground-state rotational band, ^100 keV.

Table 7 summarizes the data on the giant re- sonances of the deformed nuclei. It contains the resonance parameters used to fit the data in the

various experiments as well as the intrinsic quadrupole moments, ^0, derived by means of eq. (4.11).

The experimental results just discussed in no way

establish the spatial correlation assumed to exist between the nuclear axes in the intrinsic system.

There are two ways to demonstrate this correla- tion experimentally. The first is to show that the

photonuclear cross section depends on the nuclear

orientation relative to the direction of the incident

photon beam. The second is to demonstrate that

the nuclear polarizability is a tensor through the

observation of the nuclear Raman effect. Both of

these experiments have now been performed.

The yield of photoneutrons from an aligned

holmium target has been measured [56] as a function of bremsstrahlung energy and as a function of

the angle between the aligimient axis and the photon

beam direction. The scattering cross sections from holmium

(/o = 7/2) and erbium (mostly /o = 0) have been

measured with poor energy resolution so that they

include all the Raman components. These data are shown in figure 18. The coincidence of the two

measured cross sections demonstrates the validity

of the sum rule of eq. 2.25. The lower curve is the

coherent scattering cross section obtained from the

measured absorption cross section (dashed curve of

figure 16) through the optical theorem and the dis- persion relation. The upper one includes the tensor

scattering cross section calculated using the re- sonance parameters that fit the neutron yield cross

section. The results of the alignment experiment as well as the measurement of the Raman scattering definitely show that the nuclear polarizability is a

Figure 17. The photon absorption cross section for U"^

The measured partial cross sections [46] that make up the total are shown. The resonance parameters that fit the data are given in table 7.

19

1

I
Table 7. Giant resonance parameters for deformed nuclei

Nucleus i a 1 h VO (V_>OUl. J1.X.) net.

ivie V \t„V iVlC T MeV MeV "Fa barns barns

IT.,153 1 9 ^ ^ Ti 70 2 75 5 83 QO A- f\ Q 1 0 o [o/]

1^; xo . ot 2 4 4 0 9 1/ O o .y rooi [28] 19 Ifi 4. 3 4 3 4 1X 1 Q f\ 1 0

19 1 Ifi lU . 0V 3 3 4 9 1 77 ft O-i-O o lol] 19 99 1 fi7 2 64 4 97 9 7 9 -i-O [C91 19 9^ 1X O . Qf» 7U 2 77 5 28 9 07 7 7_i_0 oO 7 fi lc:91 no 19 9 1 0 2 3 4 5 9 7 7 1 0 o 7 Q 19iZ 1 . 1

7'^ xo . i J 2 65 4 4 9 07 7 i.-uO Q

l4-4j 19iZ 1 . I

1xo fi . u 2 7 4 8 9 0^ 7 1 1 0 D [Oil

1IZ9 . Zo9H 1XO . 7H1 o 2 57 5 0 9 7 7 1 0 oo [52

1

x^r 19IZ . z9 Ifi 0 2 3 4 5 9 u 7 7 _uO O [t)«| la 1

9

IZ . o X o . o 2 3 4 4 9 1 c ID fi Q_i_0 o 0 . o

f9Ql 19IZ . o X o . o J, o0

gO O 9 IQ U . y it J O

1XZ9 . O X o . o 2 0 4 0 1 oo U 1 . i [OUJ

1 9 7^; XZ . i J xo . o 3 0 5

1

L 0 7 -uO 7 [4-4| 19/1 xo . o 2 4 3 8 1X oo 7 1 1 0 oo 12.4 15.3 2 6 4 5 2 1 6.8±0 6 [61] 12.59 15.13 1 .94 4 98 3 96 6.4±0 3 [521 ^186 12.59 14.88 2 .29 5 18 3 58 6.0±0 2 [52] U236 10.85 14.1 2 .45 4 00 12.8±1 3 ~10 [46]

tensor. These experiments also indicate that the

absorption cross section has a component that is not associated with the tensor polarizability and is not sensitive to ahgnment. In the results presented

here it is most clearly seen as the difference between

the actual data points between 14 and 16 MeV in

figure 16 and the sum of the two Lorentz lines used

to fit the data.

4. The Dynamic Collective Model

In recent years there has been an enormous flurry

of activity that stemmed from the simultaneous

recognition by several theorists that the quadrupole

oscillation of the nuclear surface could couple with

the giant resonance and perhaps produce observable

effects. Calculations and estimates of these effects on both the photonuclear absorption and scattering

cross sections have been made by Semenko [62, 63],

Le Tourneux [64, 65] and Danos and Greiner and

their collaborators [66-73]. Their magnitudes de- pend on the softness of the nuclear surface, the

parameters entering the calculations being the

transition rates and energies of the low-lying, col- lective states. This theory makes use of the adiabatic approximation which assumes that the nucleus changes its shape in a time^/i/l MeV long compared to the

giant resonance oscillation time, ~/i/15 MeV.

For the deformed nuclei the coupling of the giant resonance with the oscillations of the nuclear sur- face has two effects. First the higher energy, AK—l

mode is split in two; i.e., the nucleus acquires a dynamic triaxiality. Secondly, each of the three

modes is divided up so that the main line shares

approximately 10 percent of the strength with a

satellite. The predicted absorption cross sections

based on this redistribution of strength is qualitatively very similar to that obtained in the simpler

model; perhaps it provides a better fit to the existing

data.

The coupling of the giant resonance with the sur- face vibrations, of course, implies that the giant resonance decays through the low-lying vibrational

states that can be reached in dipole-dipole transitions. Consider only even-even nuclei for simplicity.

Then the most important such contribution comes

from the scattering to the 2+-state of the 7-vibra6 X 10" O Er • Ho

E, Mev Figure 18. The scattering cross sections for holmium and

erbium.

The lower curve is the coherent scattering cross section calculated from the measured absorption cross section of figure 16. The upper one includes the contribution from Raman scattering calculated from the two-resonance fit to the neutron yield data. The experimental result shows that the total scattering

cross section is independent of the spin of the ground state according to the sum rule of eq 2.25.

20
2* K = 0

yS-vib o

-3"

K = 0

jFlGURE 19. A comparison of the energy level diagrams for

even-even, spherical and deformed nuclei.

tional band. These ideas are illustrated in figure 19. This can be as much as 30 percent of the total scattering cross section. The latter consists of the

coherent scattering plus that which populates the

ground-state rotational band. Figure 20 shows the

scattering cross sections predicted by Arenhovel

and Greiner [68] for Er^^^. The main component

is the coherent scattering associated with the absorption cross section; it is augmented on the high

energy side owing to constructive interference with

I nuclear Thomson scattering. Also shown is the cross Bection for populating the 1st 2+ state of the groundstates rotational band and the cross section for populating the 2+ state of the 7-vibrational band

(K = 2). The latter two components are not coherent

with the incident radiation and result from twostep scattering processes in which two units of angular momentum are absorbed by the nucleus.

It is worthwhile to point out that the scattering to the 2nd 2+ state is associated only with the upper

\UAK= +1) resonance. This is required by the

I'selection rule for electric dipole transitions AK = +

1

jjsince the 7-vibrational band has K= 2. It might be

interesting to demonstrate this in an experiment by

linjecting monochromatic photons at the energies of

the two peaks in the absorption cross section (12 and

16 MeV) and measuring the spectra of the deexcitation y rays. The softness of the nuclear surface produces much

,more dramatic effects on the giant resonances of the

! spherical vibrational nuclei. These nuclei have a dynamic deformation. The effect of the coupling is

Figure 20. The predicted scattering cross sections [68) for

Er'*^, showing the different energy dependence for the coherent

scattering cross section, the Raman scattering to the '2''' state of the ground-state, rotational band, and the scattering that populates the 2+ state of the y-vibrational (K = 2) band.

to jjroduce bumps on the giant resonance, the number and amplitude of which depend on the nuclear

(lack of) rigidity. The theory of this effect, explored

by Le Tourneux and Greiner and his collaborators,

is based on the hydrodynamic model and assumes

that the surface vibrations are harmonic.

As an illustration the {y,n) cross section of

Pr'^^ is shown [74] in figure 21. It displays a striking

amount of structure, a far cry from the single Lorentz line of the simple model. At least part of this structure can be explained by the dynamic collec- tive model. The heights of the vertical lines are pro- portional to the strengths of the transitions found

using the low energy spectrum of the nearby even- even nucleus Nd'"*®. The theory still does not ex- plain the large amount of structured oscillator strength on the rising side of the experimental cross

section. This may represent the vestiges of the in9 10 II 12 13 W 15 EXCITATION ENERGY, MeV

Figure 21. The (7, n) cross section for Pr'".

The heights of the vertical hnes are proportional to transition strengths obtained using the dynamic collective model. The unexplained structure of the

rising side of the cross section may result from single particle transitions.

21
Figure 22. The predicted [72] scattering cross sections for Pd'"^ showing the coherent scattering cross section and those for

transitions to the 1st and 2nd 2'^ states.

dependent particle model transitions that are so important for light nuclei. Very careful work is required to see the structure demonstrated here. In addition to the difficulties associated with the analysis of bremsstrahlunginduced cross sections there is often a fundamental

lack of information concerning how to correct for the emission of two neutrons above the iy,2n)

threshold or how to include the iy,2n) cross section

if it is not measured. In this instance the data were

obtained by measuring the residual radioacitivty

so that the (y,2n) cross section is not included.

Figure 22 shows the prediction of Arenhovel and

Weber [72] for the different components that make

up the scattering cross section for the spherical

vibrator, Pd^°®. The upper curve is the coherent

scattering cross section and the two lower ones the

cross sections for populating the 1st and 2nd 2+ states

(see also fig. 19).

5. Photon Scattering Experiments

The calculations in the dynamic collective model

are so far available for even-even targets which

occur in nature only in isotopic mixtures. The ex- periments, on the other hand, are usually performed

using odd-^, monoisotopic targets. The data for the odd target are then compared with the calcula- tions performed for its even-even neighbor, the

justification being that the dipole excitations of the

core are by far the most important and that the

softness of the nuclear surface is a slowly varying

function of A. In the following we show that it is also legitimate to make the same comparison for photon scattering cross sections provided the measurement is made with poor energy resolution

(Nal(Tl) spectroscopy) such that the transitions to all the final states are recorded. In so doing we

assume that the addition of an extra nucleon to the even-even core serves only to produce a multiplicity

of states, separated somewhat in energy. These

become the final states populated in photon scatter- ing by the odd-A nucleus.

The electric dipole scattering amplitude. A,, of' eq. (2.12), can be rewritten as:

^^=(_l)/o+//+.[;(2,+i)/(2/o+l)]i/2

II r XE(//I|r II /o)F., I k

where

F,k= (e/c)-wX {("(^jj + V3£>5.o5o/ (4.13)

1 . (-1)-

and D = —Z-e-/AMc^. Here the incoming and out- going photon energies, fios, have been set equal; i.e., they are unresolved experimentally. The summation

is over all the intermediate states, k, that can be

reached in electric dipole transitions. The second

term is the nuclear Thomson scattering amplitude.

For an even-even nucleus /o = 0, /*= 1, If = v, and

.4.= (2.+ l)'/^-^<. ||r||l)(l

0 V v]

1 1 ll + \/3Z)5,o5o/

l)<l||r||0)F..

+ \/3l>5.o5o/ (4.14)

The odd--4 nucleus is formed by the addition of an exterior nucleon, having spin j, to the even-even

core, having collective states /. We assume that the

only consequence of adding this single nucleon is that it couples with the states of the core to produce

a multiplicity of states, separated somewhat in energy and characterized by the angular momenta,

/. The ground-state spin /o =j- The low-lying collec- tive states have the wave functions | Ij; J) and the

states formed by the electric dipole excitations of the core have the wave functions] Ikj; Jk)- Neglecting core-particle interactions the reduced electric dipole matrix elements for the odd-A nucleus can

be expressed in terms of those for the adjacent even-even nucleus through the use of the projection

formula of tensor algebra [75].

</y;/. II r II /„;•;/) = ( - 1 ) V(2/,+l)(2/o+l)

X

Jk Jo I

lo h j

(/.||r||/o) (4.15)

Since /o = 0 and /a:=1, the reduced matrix element [ connecting the ground-state, Jo=j, of the odd-^ '

22
system with the dipole states, Jk, is: A li r II 0;-;y)=(-l)^V(2J.+l)(2; +l)

7 ll Xi [{l||r||0)

0 ly = (_l)y.+3y+V(2j,+i)/3

X<l||r||0). (4.16)

Similarly, the reduced matrix element connecting

Jk with the final state, //, is:

{vj;Jf \\r\\\j;Jk) = ( _ 1 ) V(2//+1)(2A+1)

xK'-^' H (HlrllD (4.17)

[1 f jJ

Combining eqs (4.13), (4.16), and (4.17), we obtain for the scattering amplitude of the odd-^

nucleus:

/ (2.+l)(2//+l ^ 3(27 Ky+i)1)

X Z (" II r II 1)(1 II r II 0)F,a(2/,+1)

k

^ ;I+V3Z)W,., (4.18)

11 J vWJf J Jk]

Summing over Jk and making use of the orthogonality relation of the 6y-symbols, we obtain:

(2//+1)

"^3(2y+l)(2f+l)

XZ <" II r II 1)(1 II r II 0)F.,+ V3D5,o5;y, (4.19)

k A comparison of eq (4.14) and eq (4.19) reveals that

^,(odd-^)

4

(2//+1)

(2y+i)(2^+i)

A (even-even)

(4.20)

The scattering cross section is proportional to

I A, 1^. In a poor resolution experiment one would

detect the scattered photons without distinguishing

between the final states, //; one would sum over J/. And since

(2//+1) ^ ^ (2.+i)(2y+i)

' (4.21)

the scattering cross sections for the even-even and

odd-^ targets are identical in this model. This

result permits the comparison of measurements

made on an odd-A target with a calculation made

for an even nucleus.

It is, of course, important to measure the branching ratios to the various final states throughout the

giant resonance. Since the giant resonance absorption cross section is a continuous function of energy, monochromatic photons are required to attack this problem at all. The sources already mentioned,

positron annihilation radiation, the bremsstrahlung

monochrometer, or the 17.6 MeV radiation from

the Li^(p, 7) reaction, are all candidates, provided

the energy spread of the source can be made small

compared to the differences in the energies of the

7 rays being examined. Shimizu [76] has already

made an attempt in this direction using the Li 7

rays on Ta. Another approach is not to try to separate the de- excitation 7 rays according to their energy, but to separate them according to their angular momentum This can be done by using a plane polarized

photon source such as is obtained by fluorescing the

1+ state in C'^ at 15.1 MeV. Fortunately, this energy

occurs in the giant resonances of many nuclei. The

experiment simply consists in measuring the ratio of the number of ~ 15 MeV photons scattered in the

plane and perpendicular to the plane of the polarization vector, X. If we use sodium iodide detectors

having poor energy resolution then we are integrating over all the final states and can apply an even- even nucleus theory to an odd-^ target.

Rewriting eq (2.9) which gives the scattering cross section in terms of the angular momentum, absorbed by the nucleus in the two-step scattering

process, we have

da ^ P

(Kl 2f+l

(4.22)

The scattering associated with i' = 1 is unimportant

so we need consider only v = 0, 2. If X and are the polarization vectors associated

with the incoming and outgoing photons, then the

factors gv{Q) are:

gi = i[l-(^-y)a

^2 = |[l +M^-l»)a (4.23)

If we have plane polarized incident radiation with

X perpendicular to the plane of scattering.

For X in the plane of scattering.

(4.24)

(4.25)

The average is the familiar distribution for unpolarized radiation:

go=i(i+cos2 e) (4.26)

23
Figure 23. The sum of the cross sections [61] (7, n), (7, 2n) and (t, 3n) for Tb^^\

The emooth curve is the sum of two Lorentz Unes that fit the data. This

curve has been subtracted from the data for energies above 20 MeV. These

data are then compared with the predicted [78] E2 cross section.

Similarly,

^2" = 1 +i cos^ 9, and

g2=iV(13+cos2 9). The total scattering cross section is:

For plane polarized incident radiation,

(4.27)

(4.28)

3 30

Ao P cos^ 6 3 5 At d = T, the latter becomes

dil

(4.29)

(l+icos^e). (4.30)

(4.31)

Thus a measurement of the 90° scattering cross sec- tions using photons polarized perpendicular and

parallel to the plane of scattering yields the relative

intensities of the scalar {v = 0) and tensor (v = 2)

components. Unfortunately, there are very few potential sources of plane polarized radiation. Besides

the carbon line, the level in Si^^ at 11.4 MeV is the

most promising.

6. The E2 Giant Resonance

In a further treatment Ligensa et al. [783 have

extended the quantum hydrodynamics of deformed

nuclei to quadrupole multipolarity. This theory in- cludes rotations, (3 and 7 vibrations, the odd particle,

giant dipole phonvrns, and giant quadrupole phonons

and yields for the giant quadrupole absorption cross

section:

A hcMc"

where

x(i+«)i:^

2ir A 2M

Ek') /ET,J+l (4.32:

l<^ll<?l|0)|^

3 NZ 5(2/o+l) R^{\+a)

are the quadrupole strengths and Q is the quadrupole

operator.

There are five main E2 giant resonances. The

surface oscillations cause some of the modes to split.

The parameters in the problem are fixed by the position of the lower giant dipole peak and the low energy

spectrum of the nucleus with no free parameters.

Figure 23 compares the prediction for Tb'^' with

the results of Begere et al. |]61]. The smooth curve

is the sum of two Lorentz lines that fit the data. This

has been subtracted from the data points of energies

in excess of 20 MeV to obtain the experimental E2

cross section. The latter resembles remarkably well the theoretical curve shown dotted.

24
V. Light Nuclei and the Independent Particle Model

1. Survey of Experimental Data

The data for the nuclei with A<40 can be contrasted with those for the nuclei with A > 100 already

presented. We assume that nature knows how to make a smooth transition between the two. For the

light nuclei the (7. n) and {y, p) cross sections are of comparable magnitude in the energy region of the

giant resonance. The total absorption cross section can be obtained directly or by summing the separately measured (7, n) and (7, p) cross sections. hen this is done, it is found that tlie total absorption cross section integrated to 30 MeV may contain

as little as 50 percent of a dipole sum.

Since the electric dipole sum rule is a conservation

law that sets a lower limit on the integrated absorption cross section, it is necessary to look elsewhere

for the remainder of the dipole strength. An important contribution can be found in the quasideuteron

cross section which is important at high energies and

results from the high momentum components in the

nuclear ground state which are associated with strong

two-body forces. A selection of data on the integrated partial cross section is given in table 8. The second and third

columns contain the separately measured (7, n) and

{y, p) cross sections and the fourth column their sum. The fifth column contains the integral of the

total absorption cross section measured at the Lebedev Institute and the sixth those measured at NBS

[25].

The giant resonances for the light nuclei between

carbon and calcium are all located near 20 MeV;

there is apparently no systematic dependence on A.

The width of these giant resonances increases in the

middle of the s —d shell where the nuclei are known

to be deformed:

The giant resonances of the light nuclei display

different kinds of structure. If it is fair to make any

generalization at all, we might say that the nuclei in the Is shell have perfectly smooth giant resonances owing to their low level densities and general simplicity; the nuclei in the Ip shell have giant reso- nances usually consisting of several peaks about

1 MeV wide suggestive of single particle transitions;

in the 2s — IJ shell the giant resonances retain the

~1 MeV structure, the latter being fractured into

many sharp states.

2. The Independent Particle Model

The independent particle model description of the

nuclear photo-effect has been emphasized by Wilkin- son [92, 93] and is quite fundamental to many calculations currently in vogue. In this formulation the

giant resonance was ascribed to the electric dipole

absorption of photons by the valence nucleons and

those in the uppermost filled shell. These particles

make transitions upward to the next unfilled shell of

opposite parity according to the selection rule A/'=±l. The transition energies are all nearly the same and are thus grouped together to form the giant resonance. Furthermore, their summed oscillator strength is just equal to the dipole sum.

Let us consider the simplest case in which we have

pure shell-model states and for convenience intro- duce a harmonic oscillator potential. Then the electric dipole transition probability for a single particle to

Table 8. Integrated cross sections for light nuclei in units of (27rV/i/Mc) X(NZ/A)

The exponents represent the upper limits of the integration in MeV.

{y, n) (7, p) z

Li 0.47'» 0.143" o.ep"

C .29 .51 0 .19" .37" .56^6 Mg .23" .5032 .73'° Si .19=^ .682* .87^8 P 2728 .75'^ 1.0230 s .17^ .77" 0.953" Ca 122s .7328 .8528

Leb NBS

y, p

y, n

Reference

0 . 783"

.713" 1.02

0 . 633"

.623"

.6530

.783"

1.3 1.95 2.2 3.6

1 . 1930 1.5430

.833"

.733" 5.2

6.1

[79, 80]

[81, 32]

[82, 83, 84]

[81, 85, 86]

[87, 88]

[89, 85]

[89, 85]

[91, 88, 90]

25
Table 9. Radial integrals for the harmonic oscillator

Transition Jdr i^RuRil'

l/aV372

Ip^ld 1/a \/5/2

l/>-»2s 1/a

ld-*lf l/aVl/2

La—>Zp — 1 / i/a

2s—>2p

proceed from state 1 to state 2 is:

(5.i:

where q is the effective charge, Ne/A for protons and —Ze/A for neutrons. The radial and angular parts of

the matrix element may be separated by replacing z by \/4x/3 rYio, then:

= E (/2m, I Yio lyimO^ll drrm.R.)^

(5.2)

where 7 = /±|.

We can dispose of the dependence on m by making

use of the Wigner-Eckart theorem and summing the

Clebsch-Gordan coefficients:

X

272+1

4^ ,(7211 yiiiji)^ 3^3 j drrmiR2\ (5.3)

Making use of the relationship [94]:

(72 II Yi 1171)^= (271+1) (yiilO 1721)^3/4^), (5.4) we have the fairly simple expression:

D' = ^ (271+1) (71IIO I72I) [/drrm.R, ' (5.5)

The radial functions, jRi and JR2, depend of course on the nuclear potential. Those associated with the

harmonic oscillator are very frequently used since

they can be expressed analytically. For the first two

shells they are [95]]:

(2^+3)!!

]l/2 {arY

2^+3

-(ar)' -l/2(ar).

(5.7)

where t is the orbital angular momentum of the

nucleon and a= [iVfwo/^l^'^- The values of the integrals.

/

drr^RuRn, (5.8)

are given in table 9 for the electric dipole transitions

from the Is, Ip, \d, and 2s shells. These are the ele- mentary transitions that must be included for the

elements up to Ca*". The nucleus, O^^, has received more attention than

any other from both the theorists and experimentalists. The structure is simple enough to be tractable. In the independent particle model its electric dipole

absorption cross section consists of the five transitions

from the filled p-shell to the 2s and \d shells (see

fig. 24). These are listed in table 10 along with the

transition probabilities, Z)^, evaluated by means of

eq (5.5). The radial integrals of table 9 have been

used. For the harmonic oscillator, of course, all the

transitions take place at the energy ^coo = 41y4~^'^

MeV, 15 MeV in O^^

The electric dipole absorption cross section inte- grated over all the transitions yields the unenhanced

dipole sum. For a single transition at energy Ek the

integrated absorption cross section is:

/

Ok dE =

he

(5.9)

By summing the transition probabilities, such

as those given in table 10, it is straightforward to show for simple nuclei that:

A Mo)o

(5.10)

In so doing it is necessary to weight the transitions

from partially -filled shells by the ratio of the number

2s,,

—•- Is,

SHELL-MODEL LEVELS FOR 0

Figure 24. Shell-model levels for O'*.

16

26
Table 10. Transitions in O^*

Transition

l/>3/2—»2si/2

1jP3/2—» 1^5/2

l/>l/2—> 1^3/2 lj5l/2^2si/2

l/>3/2—>l<^3/2

4/3

6 10/3

2/3

2/3

115028

5.5

5.5

E(Me\)

18.53

17.65

16.58 12.38 22.73

JE;'(MeV)

19.6

22.2

18.1 13.5 25.2

273

14

20

0.951

.898

.903

.996

.963

of particles, n, to the number of nucleons the shell can

contain, (2yi+l). Since eq (5.5) represents the

transition probability from a filled shell, that from a

Dartially filled shell would be

(5.11)

|rhe integrated absorption cross section is then:

2irV^ NZ

I

dE=

Mc A

(5.12)

More realistic choices for the nuclear potential alter the radial integrals and change the transition

energies, Ek, by splitting the basic shell-model states nto their various components. The effect of the spinprbit force, for example, is to remove the degeneracy

etween the two possible j-values for a given i.e..

In the extreme, when we consider nuclei having

arge intrinsic deformations, each value ofj is further

•plit into components labelled by the quantum

lumber Q, the projection of j on the nuclear symmetry axis. Here the transition strength is divided up

imong the many single-particle transitions that can

•ccur, and for electric dipole transitions we have the

idditional selection rule AK= 0, ±1 [K is the proection of Iq on the nuclear symmetry axis). The

ransition energies then fall quite naturally into two

joups, corresponding toAK = OandAJt=rfcl. These

nay be associated with particle motion along and

uerpendicular to the symmetry axis. In fact, Wilkinson ^96] has used the Nilsson wave functions to prelict the giant resonance shapes for the rare earth

iuclei and obtained results quite analogous to those

Iready presented in the hydrodynamic model.

: The independent particle description of the nuclear

|»hotoeffect as outlined above has a very important

Tialitative shortcoming, namely that it gives the

iant resonance energy as essentially 41A''^'^ MeV,

jhe harmonic oscillator spacing. We have already

jeen that the expression, Eo = SOA~^'^, applies well

I) the heavy elements, and the light elements all vave their giant resonances near 20 MeV. The latter

5 still appreciably higher than the oscillator spacing.

3. The Particle-Hole Model and Its

Extensions

I The particle-hole calculations, introduced by Elott and Flowers [97] and pursued by Brown and

his followers, made substantial improvements in the

situation. These improvements take place in two in- dependent steps. The first is to take for the basic

transition energies not the harmonic oscillator spac- ing but energies based on experimentally observed

spectra. The states that make up the giant resonance

differ from the ground state in that one particle has

been elevated to the next open shell of opposite

parity. The energy of such a state is given by the sum

of the hole energy and the particle energy. When

considering nucleus A, the energies of the hole states manifest themselves in the levels of nucleus A — 1, the energies of the particle states in those of nucleus

A-\-l. These states can usually be identified by their spins and parities. In this way the harmonic oscil- lator energies are replaced by those prescribed by

nature. They are not exactly right because the hole- core interactions in the (/I — 1) -particle system and

the particle-core interactions of the (^ + 1) -particle

system are not precisely the same as those in the Aparticle system. Nevertheless, the single particle

energies obtained in this way are (for light elements)

several MeV greater than the oscillator spacing and

a much closer approximation to the true transition

energies. The second step is to take into account the

particle-hole interaction by diagonalizing the matrix

elements for all of the elementary transitions with an appropriate force. In this process the amplitudes

of the basic transitions add coherently in such a way

that nearly all of the dipole strength is concentrated

in one or two transitions which are those most ele- vated in energy. Returning once more to the example of O^^, the

elementary, unperturbed transition energies are taken from the known energy levels of O'^ and C.

These are illustrated in figure 25 and the transition

energies tabulated in table 10. At this stage it was

assumed that there is complete charge symmetry

and that the proton transiticn energies are the same

as those for neutrons. The transition probabilities

for the unperturbed situation are those given by eq

(5.5) and evaluated for O'^ in table 10. Several different particle-hole calculations [97-

101 J have been performed for O'^ and they show that

the position and strength of the electric dipole states are not critical to the choice of interaction. In all cases most of the dipole strength becomes concentrated in two states that have been pushed up in energy to 22 and 25 MeV. Gillet's transition energies

and relative transition probability, ^)'^ are given in table 10 for his approximation II which includes some

ground-state correlations.

27

II
2s,

Figure 25. The energy levels in O'^ and O^' from which the unperturbed transition energies are obtained.

Each of the dipole states is a mixture of the five hasic shell model transitions. Nevertheless, each one

consists primarily of the one indicated in the first column in tahle 10. The final column gives the fraction, I'\ of the amplitude resulting from this hasic

transition. Notice that the two important transitions are primarily of the p—*d type and that the less in- tense is the one in which the nucleon spin is flipped. In general, the most important shell-model transition

is the one involving the largest angular momentum.

After diagonalization this transition collects an even

greater fraction of the transition strength and locates

it at a higher energy.

It has already been pointed out that for the harmonic oscillator all of the transitions take place at the same energy and that the sum ^ EkBi? has its classical value. In the particle-hole calculations the

degeneracy in the Ek is removed. If correlations in the nuclear ground-state are neglected,

^

Dl^ is the same both before and after the diagonalization process and one is led to the inconsistency that

^ EkDi} exceeds the classical dipole sum even for forces that commute with the dipole operator.

This difficulty stems from an inconsistency in the

calculations as they are performed. The singleparticle transition energies are closely related to the

true nuclear Hamiltonian yet the transition matrix

elements used, D/t, are those appropriate to the

harmonic oscillator. The strong two-body forces which spread the nuclear energy levels also modify

the ground-state wave function in such a way that

the matrix elements are decreased. The inclusion of

ground-state correlations, as has been done by Gillet and others, represents an attempt to alter these wave functions appropriately. Drastic changes will, however, be required to decrease the predicted

summed oscillator strength,

^

EkDk^, for the light

elements to the point where the experimentally observed magnitude of approximately one-half the

dipole sum is obtained.

150

100:

1—

r

"1—

r

J \ I—

f

J L

Figure 26. The upper dashed curve is the result of the continuu

calculation of reference 102.

The solid curve is the result [111] of including the interference with t 2p—2h quasibound states listed in tahle 11 as well as the 4p— 4ft states' 21.05 MeV. The lower curve shows the experimental total absorption cr< section [84] as well as the result of Kluge [105| who assumed the O^^ grou

state to have a slight positive deformation. '

28
total absorption experiment [84] done at the Lebedev Institute. The details of this experimental cross

section are, after all, what the theories are trying to

; emulate; its magnitude is just a little too large, howi ever, to be consistent with other measurements (see table 8). A comparison of the dashed curve with the

' experimental result shows that the theoretical cross

section magnitude is too big and that the experimental data show much more structure. These structures in the experimental cross section

have been explained in part by two apparently dif- ferent ideas, both suggested by experiment. First,

Kluge [105] has taken the ground state of O'*^ to have a slight (==10%) positive deformation. This means that the 6 MeV 0+ state, which supports a

rotation band, is partly mixed into the ground state. His result is reproduced in figure 26b where the posi- tions of the K= 0 and K=l transitions are indicated

with heights proportional to their intensities. The hypothesis of Kluge is bolstered by the experimental fact that the giant resonance state does not decay exclusively through the and f ground and

third excited states of O'* and N'^. Approximately

7 percent of the time it decays through the and

first and second excited states in these nuclei

[106-110]. In all the theories, based on the simple

model described above, O^^ emits only s and rf-wave

particles. In order to emit p-wave particles and popu2p-2h Correlation in ^^O

Table 11. The 2p2h quasi-bound states in O'^

2Pi2

2 If,

2

Idi - 2 2si

-

2

Ids.-

2

ip.-

IPA2 ISi-

+ p3^ particle

Figure 27. This diagram illustrates how a 2p -2h fluctuation

in the ground stale on irueracting with an electric dipole

photon can leave the residual nucleus in its 5/2+ state.

This ground state is momentarily deformed.

T = 0, + parity T = 1, — parity Energy (MeV)

0+ 1- 23.0

2+ 1- 26 0 2+ 2- 24.2

2+ 3- 23.9

late the positive parity states, the O'^ ground state must be contaminated by more complicated configurations.

As an example, we now show how a simple 2

particle-2 hole correlation in the O^^ ground state permits the f

+ state to be populated and at the same time look like a deformation. Since the nuclei

in the middle of the s — d shell have large intrinsic deformations, a ground-state correlation in O'^ that

resembles a deformation is one in which two particles are promoted from the lpi/2 to the lrf,,/2 shells leaving

the ground state with 0+. When the electric dipole

photon interacts with this system it can promote one

of the nucleons momentarily situated in the lrf,5/2 orbit to the 2pzi2 shell. The latter then emerges leav- ing the residual nucleus in its f+ state (see fig. 27).

An analogous scheme exists for populating the

state. It is straightforward to concoct more elab- orate fluctuations involving 3p — 3h, 4p — 4/?, etc. which surely exist in nature.

Gillet, Melkanoff, and Raynal [111] have noticed, on the other hand, that at least two of the minima

in the O'^ absorption cross section occur at just the

energies where there are peaks in the N'"* {d, 70) O^® and C'^ {a, 70) O'^ cross sections [112, 113]. In both cases the angular distribution of the outgoing

7-ray labels the excited state as 1~. Because these

states are excited by deuterons and alpha particles

they must have T=0 unlike the main giant reso- nance which has T= 1; furthermore they must repre- sent 2p —2h and 4p — 4/^ configurations respectively.

The dominant 22 MeV state is then broken into

three parts by destructive interference with the

4p-4/^, r=0 state at 21.05 MeV and the 2p-2h,

T=0 state at 22.7 MeV. Gillet et al. point out that

there are, in fact, exactly four 2p —2h quasi-bound

states in the energy region of the giant resonance. These are made by coupling two Ip — lh states to T=l and 1~. This requires one ph state to be T=0

with positive parity and the other T=l with negative parity. Table 11 lists the four possible 2p —2h

quasi -bound states and their excitation energies.

There is a remarkable correspondence between these

energies and the dips in the cross section of Dolbilkin

et al. [84] (see fig. 26).

4. Caldwell's Experiment

Using the annihilation radiation as a source Caldwell [110] has measured the cross sections for the

29
Table 12. Integrated cross sections [110] /or O^^ photoreactioiis

Final state / <Trf£(MeV-mb)

.2.87 Fractional / crdE Reaction (MeV) Til

(7, p) Ground state 34.42 0.287

5.3 (1 2+. 5 2+) "4.94 .041 6.33 (3 2") 22.30 .185 7.30 (3, 2+) 5.47 .047 9.1 2.03 .017 9 22 1 .50 .013 9.9 2.36 .020 10.8 '•2.34 .020

(7, ") Ground state 26.67 .223 5.2(1 2+. 5, 2+) 3.49 .029 6.18 (3 2-) 9.43 .079 6.79 (3 2''") 4.50 038

(7. «) 15.11 (1+. T=l) 0.23 .002

I

" The 9.22 MeV cascade transition cross section has been subtracted from the 5.3 (1 2"*". 5 2"'') values.

' Observed 10.8 MeV yields multipHed by — = 3.0 ± 0.5 from Ref. 114.

photodisintegration of 0^^ leaving 0^'' in its groinid

state and three excited states and N^^ in seven excited states. This was accomplished by embedding

Nal(Tl) detectors in the 47r neutron detector and

measuring photon spectra in coincidence with

neutrons and in anticoincidence with neutrons (to

obtain the {y, p) partial cross sections). Using the

ground-state cross section obtained by others [42,

22 24 PHOTON ENERGY, MeV

Figure 28. The total photonuclear cross section for O'^ as synthesized by Caldwell [110] compared with the directly measured cross section (dashed curve) of reference [25].

34], the total photonuclear cross section has been

constructed and found to agree with the total absorption cross section of Wyckoff et al. (see fig. 28).

Caldwell's integrated partial cross sections are listed in table 12. Caldwell's residting partial cross sections are compared with the predictions of the calcidation of Buck

and Hill [103]. The latter is one of a munber of

cather sophisticated treatments of the O'" problem

that ^ake into accoimt the continuum and recognize

the difference between neutrons and protons. The

calcidation suct«ds, really for the first time, in obtaining some correct gross section magnitudes and

the gross shape of all the partial cross sections. The

theory, however, overestimates by 40 percent the

magnitude of the decay through tile |~ states. Part

of this strength is siphoned off through the positive

parity states and the rest is located above 30 MeV;

neither of these possibilities are, of course, included

in the theory.

Using his experimental data and the penetrability

factors given by Buck and Hill, Caldwell has esti- mated the amplitude of the T— 0 contamination of

the T=l giant resonance to be ~0.08.

30
VI. The Decay of the Dipole State

1. Introduction

We have just seen that the nuclear absorption of

X rays in the giant resonance region takes place

through the excitation of a single particle. The excited nucleons may emerge directly without further

interaction, and under these circumstances their energy and angular distributions reflect the absorp-

' tion process. Alternatively the excited nucleon may

j undergo one, two, or many interactions before it escapes the nuclear surface. In this way it loses all memory of the absorption process. In light nuclei (.4<40) the nucleon emerges with

I only a few interactions. For heavier nuclei evaporaI tion is the dominant decay mode. At Ca^" the ld-t/2

j shell is filled for both neutrons and protons. New

I nuclei are built by filling the higher angular momen1 tum / and g shells. When these nucleons are excited

by giant resonance photons, they are unable to penetrate the angular momentum barrier and hence

I remain in the nucleus making collisions and sharing

their energy with the other nuclear particles. EvenII tually sufficient energy is concentrated on a neutron

; having low angular momentum and it is evaporated

from the nucleus; protons belonging to this distribu-

' tion have insufficient energy to surmount the Coulomb barrier. As a result, neutron emission becomes

the dominant mode of decay, and for heavy nuclei

j photon absorption results almost exclusively in

j neutron emission. How the transition is made be- tween the dominance of direct emission for AKiO

to that of the evaporation process for A>100 has not

! yet been established in detail.

' The relative magnitudes of the (7, p) and (7, n)

I cross sections for the nuclei in these two extreme

. ranges of atomic number provide an interesting

' contrast. For the special case of the self-conjugate

I nuclei nature has conspired to make the neutron and

1 proton escape probabilities equal by adjusting the

' depth of the nuclear potential felt by the protons to

j compensate almost exactly for the Coulomb barrier. The proton binding energies are thus lower than

! those for neutrons. The (7, n) and (7, p) cross sec- tions for these nuclei should therefore be identical in both magnitude and shape. The experimental fact

is that the ratio Jff(y, p) dE/ / cr(7, n) dE increases

from one for He* to about six for Ca*° (see fig. 29).

This rather striking effect must result from the mixing of states of different isotopic spin both in the_

ground and excited states. For heavy nuclei, on the other hand, the (7, p)

cross sections are almost insignificant and are located near 25 MeV, an energy where both E\ and E2 absorption participate. The protons are emitted with

high energies in direct interactions. As an illustration figure 30 shows a comparison of the (7, n) and (7, p)

cross sections [115] for Pd^-^

2. Energy Spectra

The evaporation of neutrons from heavy nuclei

produces a spectrum that is predicted by the statistical model and may be approximated by an expression of the form

Ene-^^i^

where En is the neutron energy and T the nuclear

temperature. The evaporated neutrons proceed

through many overlapping energy levels and have es- sentially isotropic angular distributions. Except for information concerning the nuclear temperature

these distributions contain no information and are therefore uninteresting.

Direct emission results from most interactions in

light nuclei and from a small fraction in heavy nuclei. These produce particle groups that come from the

highly excited giant resonance states and populate

the ground or excited states of the residual nucleus.

The energies of these groups are: hui—T—Eh

where fiu) is the absorbed photon energy, T is the

binding energy and Eh is the excitation energy of the

hole.

4 8 12 16 20 24 28 32 36 40

A

/30 /-SO <r(7, p)(lE/ / <t(t, n)dE up to Ca^".

These data come from table 8.

31
80

1 \ 1 ^

^

I T

a(y,n)-

<^ ' °°V-o-(x,p)x 10

16 20

E, MeV

IFlGURE 30. A comparison [115] of the neutron producing

cross sections of Pd'"' with its (y, p) cross section.

3. Angular Distribution Formulae

The spins and parities of the final states as well as the composition of the excited states can be studied

by investigating the angular distributions of the out- going nucleons. The dependence of the angular distri- butions on various angular momentum parameters

for transitions through isolated intermediate states of

specified spin and parity has been exhaustively

treated in the reviews [116, 117] on angular correlations. Here we will cite those formulae which may

pertain to the photoeffect with particular emphasis on electric dipole transitions. In angular correlations the transition is regarded

as a two-step process through the intermediate

state, Ik, each link being described by its own independent factor, Ay, so that: ^{e) = Z A^{hh)A,{IJf) prices 6) (6.1)

V When specialized to photonuclear reactions the

quantity, A^iloh), describes the transition from the

initial state, Iq, to the state, h, through the absorption of a photon of multipolarity, L. In terms of the

3—y and 6—/ symbols it is:

where the primed symbols stand for interfering

quantities and where Ik, for example, means

(2/a:+1)^'^. (R is the transition amplitude.

The decay of the state, h, through particle emission to form the final state, //, is more complicated

since it involves the coupling of both the particle

spin, s, and the orbital angular momentum, t. There

are two different representations that are commonly

used to couple s, (, and // to give Ik. These we will

, / L V v\\L L' v\ ^ (6.2)

call the channel-spin representation and the /-representation (see fig. 31) . In the channel spin representation s and // are coupled first to give the channel spin

S, and then S and ^coupled to yield Ik. In the 7 -repre- sentation s and ( are coupled first to givey, the total angular momentum of the outgoing particle, and then

j and // are coupled to yield Ik- In the channel spin

formalism the transition between Ik and // is described

by:

^,(/i//) = (-r^-

and in the /-representation by:

A.{hif) = {-\y>'"'+'"''+^'-"Ukh'jj

The advantage of the channel-spin representation

is that the distribution for each S is independent;

there is no interference between the distributions

corresponding to the two channel spins. The advantage of the /-representation is its relevance to

calculations done in /—/

coupling, such as the

particle-hole calculations of Brown or Gillet. These

theories yield directly the amplitudes from which the

angular distribution can be evaluated. If it is desir- able, one can then obtain the weighting factors for the different channel spins by recoupling. For ex- ample, the amplitude associated with the channel

spin, S, (Rs, is just a linear combination of the two

possible amplitudes in the /-representation.

" ' \\-\ 0

]

Ik

J

Ik

hlfj

s i S

(6.3)

(6.4)

(6.5)

It is worth noting that the complexity of the angular distribution formulae is limited by the condition

that V is less than or equal to the smallest of 2L, It,

2Ik. It follows that reactions that proceed through

intermediate states of spin zero or one-half of those

that yield only s-wave particles have isotropic

angular distributions.

For intermediate states of well-defined parity only even powers of v occur in the sum. For electirc dipole

transitions then there would be only terms involving

32
S=2 ^=1,3; S=3 1=3 S=0,1 1=1

' Figure 31. The energy-level diagram for a simple {y, n) or

(7, p) reaction.

The coupling of the angular momenta in the channel epin and y-representations are shown. Electric dipole absorption by C'^, Si^', and S'^ followed by

proton emission populating the ground states of B", .Al^'.and P^'are illustrated, as specific examples.

j Po and P2. On the other hand, if the excited state

I results from two overlapping levels of opposite

i parity, terms odd in v, such as Pi, appear. This effect

I is very important in photonuclear reactions where

forward peaked angular distributions are common for excitation energies above the giant resonance. These

are presumed to result from the interference between

. the states formed by the absorption of electric dipole

I and electric quadrupole radiation.

• Tables 13 and 14 give some examples of simple

j angular distributions for particles of spin ^ obtained

using the two formulations. These are very unrealistic

since they contain no interference terms. Some time ago Courant [118] published the for- mulae for the angular distributions of nucleons

emitted in so-called direct interactions following

electric dipole absorption. The interaction was as- sumed to take place between the photon and a single

nucleon, the rest of the nucleus being treated as a

spectator. These distributions then depend only on

the orbital angular momentum of the nucleon and

are independent of the spins of the initial or final

nuclear states. They are:

n""(e) = ^(^+i)+i(^+i)(^+2) sin-e

for Ar=+1

W-{6) = f(f+l)+^,{f-l) sm'e

forA/"=-l (6.6)

where f is the nucleon's orbital angular momentum

before the interaction. Wilkinson [92. 93] has emphasized that the A^=+l transitions are nuicli more

likely to occur and the first of these two fornnilae

has often been successfully compared with experiment. It is Morth noting that this formula is identical to one of the two possibilities in the channel spin

formalism. The channel spin S is //±^. The Courant

formula corresponds to the choice S = I/ — ^. These

are labeled by an asterisk in table 13. Tables 13 and 14 give some examples of simple

angular distributions for outgoing particles of spin

h These are very unrealistic since they contain no

interference terms. To include them consider electric dipole absorption by a spin-zero nucleus. Then in the

channel spin representation we can write the amplitude (Rfs as Bise'^' where S( is the nuclear plus

Coulomb phase shift and Sir = 8(' — dt. If we can have /= 0, 2 and S = 1, 2, 3 then

W'id) =3(Boi-+B,i=) +3[V2B2iBoi cos Ao-, +h{B;^-B,,-) -15,3-] P,( cos e) (6.7)

For ^=1, 3 and S = 0, 1, 2, 3

+3 [ -5io- +^Bn'- i'.BrP -#^32=

+iV()BnB32 cos Ai3]P. (cos e) (6.8)

Similarly, in the 7 -represent at ion 6\j = B(je'^', the

spins of the final states are specified

JF{e, i) =3[Bo./.^+B.3//]-3[iiB../r

+ V2Bo./.B-.3/, cos An.>]P2 (cos e) (6.9)

W{d, I) =3[Bo./,.-+523/.2+52vr]

+3[f-B2v.- —fB2V2^~f-B23/2B2V2+ (1/ 's/5Boi/,B23/2

-3/V5B0./2B2V2) cos Ao2]P2(cos0) (6.10)

The above expressions may be useful in understanding the angular distributions in the ground-state

proton capture cross sections where the spins of initial and final states are known.

Transitions between discrete excited states and

well separated final states are rare in photonuclear

reactions. They do occur in the light elements where

the level densities are low and they may also occur

in some special cases for heavy elements. By and

large, however, the giant resonance absorption is into a large number of overlapping excited states in the continuum. The decay takes place mainly by

neutron emission populating a large number of final
Table 13. Angular distributions in the channel spin formalism

Jo L h // S = s +// £

0+ £1 l- 0 1 3/2 sin2 e* 1/2-. 3 2- 1 0 -WsPo Isotropic

H-,3/2- 1 2 3U-HP2I ^(1+3/2 sin^e)*

3/2+ 1 1 P0+Y2P2 3/2(1 -Hsin^ e) 3/2-, 5/2- 2 1 Po-l/lO Po 9/10(1+1/6 sin2 e) 3/2-, 5/2- 2 H /2n — Vi^ sin' fl"! 3/2+' 5/2+ 2 3 Po-2/5 Po 3/5(1+ sin^e)*

5/2-, 7/2- 3 2 Po-1/7 Pi 6/7(1 + ^ sin^e) Ml 1 + 0 1 Po-Pi 3/2 sin^d Vr 1 1 Po + KP: 3/2(1 -Hsin^e)

E2 2+ 1 1 5P0-5/2 P, 15/2(1 -Hsin2 Q) 1 3 5Po+20/7(Po-P4) 5(1+2 sin' 9-5/2 sin< 9)

El Vr 0+ 1 -2Po Isotropic

3/2- 0+ Yi 1 -4Po+2P. -2(1+3/2 sin' e) Ml 0+ 0 -2Po Isotropic

3/2+ 0+ Y2 2 -4Po+2P.> -2(1+3/2 sin'e) E2 3/2+ 0+ Y2 2 -4Po-2P.. -6(1 sin' e) 5/2+ 0+ Yi 2 -6Po-24/7(P2-P4) -6(l+2sin'9-5/2 sin^fl)

States that are also overlapping. The angular distributions of these neutrons is then expected and observed

to be isotropic.

The expression for the angular distribution result- ing from plane polarized incident radiation has been

given by Satchler [119]. For this we simply replace

the P>.(cos 6) in eq (6.1) by CP^(cos 6) where

(P. (cos e)=(P, +OOL- /.(LL')piP;-(cOS d) COS 201

(6.11)

where col' is ±1 according to whether the transition

is electric or magnetic, pi is the degree of polarization, P,"^ is the associated Legendre function, and

f{L, L')

-2L(L +i: (.-l)(.-2)[.(v+l)-2L(L + l)]

(6.12)

The symbol d\ stands for the angle between the reac- tion plane and the electric vector.

If we include only electric dipole transitions and

assume 100 percent plane polarization in the inci- dent beam, then <jJL' = +l, /)i = l, /2(1, 1) = — |, P;'" = Pfi = 0, Pj/2 = 3 sin2 e, and

r(e) =AoPo+A£P.-i sin2 6 cos 20i]. (6.13)

[ We wish to evaluate the intensities perpendicular

to the beam direction, d = ir/2, and at the azimuthal

angles such that the electric vector is in the reaction !' plane /I I, 0i = O; cos 2di = \ and the electric vector perpendicular to the reaction plane, /-'-, 0i = ir/2; cos 2^1=— 1. Recalling that Po(cos9)=l and

P2(cos 6) =i(3 cos^ 6 — 1), we obtain

/(7r/2) =^oPo+^2[P2-| sin^ e cos 2e{]

l'- = Ao+A2 (6.14)

In previously made measurements of photoneutron

angular distributions the ratio A2/A0 varies from

Table 14. Angular distributions in the j -representation for I = 0+, It = 1 and L = 1

// t i W{e) W{e)

Y2 1 Yi 9Po Isotropic Y2 1 ?,/2 3V3(Po-KP2) 3/2 V3(l+3/2 sin'0) 3/2 2 3/2 -3\/3(Po+2/5 P.) -21/5 V3(l-3/7 sin'e) 3/2 2 5/2 3\/3(Po-2/5 P2) 9/5\/3(l+ sin'e) 5/2 3 5/2 3\/3(Po+16/35 P2) 153/35 \/3(l-8/17 sin' B) 5/2 3 7/2 3\/3(Po-5/14 P2) 27/14 V3(l +5/6 sin' 8)

34
1

1

'

' 1 /

I'yiy ~

/ —

lOO'j/KSO")

1 I I 0.4 0.6

) Figure 32. The sensitivity of a measurement made with plane

polarized photons compared with a normal angular distribution measurement as a function of the ratio — Aj/Ao.

The Hymbol /" represents the nucleon intensity in the photon polarization

,'plane and that perpendicular to it.

— 0.1 to —0.7 depending on the nucleus. Figure 32

shows a plot of /"//-'- versus —A2/A0 over this .range. For comparison is shown a plot of the ratio /(7r/2)//(30°) as would be obtained in a simple

distribution experiment. It is apparent that angular

. the measurement using polarized incident photons af- fords a more sensitive method of measuring —A2/A0.

shapes for C^^ and Si-^ are compared. The envelope

of the fine structure consists of three or four peaks

having widths of ~ 1 MeV. These have been associ- ated with the particle-hole transitions that form the entrance channel into the giant resonance. The excited nucleons are held in the nucleus by the angular

momentum barrier for a time which is long enough for the initial Ip-lh state to mix with the many 2p-2h 1~

states existing at the giant resonance excitation energy. The width of the spikes in the cross section

tells us that these complicated configurations last for

a time keV. This phenomenon is also seen in the giant resonances [36, 38, 40] of Mg, S, and Ca. This situation is further dramatized when we pass

into the 2plf shell. Here the dominant shell model

transition is of the f^g type. These nucleons are confined within the nucleus where they mix with an even greater density of 2p-2h, 3p-M, etc. states until

finally the excitation energy is transferred to an s- wave neutron which is then evaporated. Figure 33

is included to illustrate this point. It shows the

neutron yield produced by 30 MeV bremsstrahlung

[120] divided by NZ/A as a function of Z. For nuclei

up to Ca*° this quantity is almost constant. As soon

as we enter the 2plf shell the neutron yield makes a dramatic jump. This is a demonstration of the importance of the angular momentum barrier in keeping the nucleus together while the thermalization

process occurs. The emission of s-wave neutrons then becomes dominant.

These ideas are supported by the angular distribution measurements. It has been found that, in general, the ground-state photoproton angular distribu- tions are nearly constant functions of energy as though they were produced by a single excitation. For carbon and oxygen the distributions for out- going ground-state protons are essentially those ex- pected for outgoing c?-wave particles; in both cases

4. Comparison With Experiments

It has already been pointed out that the electric dipole transition that dominates the giant resonance

is always the one in which a nucleon, having the

highest available angular momentum, makes a transition in which its angular momentum is increased by

one more unit. The interactions that take place fol- lowing photon absorption depend on the level density

at the appropriate excitation energy as well as on the

magnitudes of the Coulomb and angular momentum

barriers. In the Is-shell the level densities are so low

that the nucleons emerge without interacting. In the

' l/)-shell the dominant transition in the absorption

process is lp3/2—^1^5/2. These particles survive the

angular momentum and Coulomb barriers, though

interactions with the excited 2p-2h, 3p-3h, etc. 1~

states produce occasional depressions in the cross

section as we have seen in the case of O'®. In the 2s-ld shell the giant resonance absorption

cross section is qualitatively very different. It is fractured into a large number [37] of sharp peaks

having widths of ^=70 keV. This striking feature is illustrated in figure 11 where the giant resonance

Figure 33. The neutron yield divided by NZ/A.

This quantity is essentially constant up to Ca*" after which it makes a dramatic jump. This is the point at which s-wave neutron emission becomes

the dominant mode of decay for the giant resonance.

35
B/A

1 1 1 1 i 1

1 1

1 1

1 1 1

1

1

1 — Eq =22 Mev

0 —

o o - -

o —

0 _ 0 0 -

o

n ° o °o o o

o°

o

0 1 nl 1 1 1 , 1 , 1 ,

1 ,

Figure 34. The anisotropy, Ti / A, in the angular distributions

[121] of the fast neutrons as a function of neutron number.

some s-wave emission has to be mixed in. For these

nuclei the excited nucleon emerges directly. In the 2s-ld shell the ground-state {y, p) cross section for Ne^" and S^^ must emit p-wave particles

in order to populate the ^+ ground state of F'^ and

P^^ These proceed almost entirely by p-wave emission in channel spin 0. Even though their dominant

excitations would lead to /-wave emission, Mg^^ and

Si^^ also emit p-wave particles in populating the

ground states of Na^^ and AP. Here the configuration mixing that fractures the cross section transfers the excitation energy to a p-wave proton which can

leak out of the nucleus.

It is worthwhile pointing out that two of the

minima [35] in the F'^ (p, 70) Ne^° cross section occur

at energies where the angular distribution coefficient,

A2/Ao, deviates from its constant value. This cer- tainly suggests that at these energies more complicated configurations interact with the particle-hole

state. Gillet, Melkanoff, and Raynal [111]] have al- ready shown that 2p2/i states in the giant resonance

of O'^ can produce minima in the cross section by

interference with the Ipl/i excitation. The excursions

of the angular distribution coefficient, A2/A0, may

also be associated with this phenomenon.

For heavier nuclei about 10 percent of the photonucleons are emitted in direct interactions. Since the

interaction takes place with the valence nucleons and

those in the last filled shell, we might expect to see a correlation between the angular distributions of the more energetic nucleons and nuclear shell structure. On the basis of the Courant formula an outgoing

nucleon with ^=1 has an angular distribution proportional to sin^ 6 with no isotropic component. As i increases the isotropic component becomes more and more important. When one considers that the angular momentum barrier inhibits the emission of nucle- ons having large i, it is possible to understand the

experimental results at least qualitatively.

Figure 34 shows the results of a survey published

by Baker and McNeill [121]]. They have irradiated

many targets with 22 MeV bremsstrahlung and used

silicon threshold detectors to measure the angular

distributions of the outgoing neutrons. These detectors respond primarily to neutrons in the energy

range 5 10 MeV. The ordinate, B/A, is the anisotropy, defined by the expression:

W{d)=A+B sin' e+C cos esin^d (6.15)

The maximum in B/A for neutron number 35 cer- tainly results from the filling of the low angular

^ momentum 2p3/2 shell. The anisotropy decreases as higher angular momentum neutrons are added to the

l/:,/2 and lgg/2 shells. It increases once more with the

filling of the 2d:-,/2, 2^3/2, and 2si/2 shells. It then de- creases once more as particles are added to the

l/jii/2, I/19/2, 2/7/2, and 2/5/2 shells. The final peak at neutron number 126 may be attributed to the filling

of the 3pi/2 and 3p3/2 shells. In order to discuss the asymmetry in the photonuclear angular distributions it is necessary to recall the effective charges that result from the relative motion of the outgoing nucleon with respect to the

recoiling nucleus. For electric dipole absorption

these are:

qp = N/A

qn=-Z/A

qD={N-Z)/A

q^ = 2(N~Z)/A (6.17)

I

From these it follows incidentally that electric dipole

absorption cannot result in the direct emission of deuterons or a particles from a self-conjugate nucleus, i' These heavier particles may, however, be evaporated

from heavy nuclei and the observed numbers are consistent with the predictions of the statistical model.

For electric quadrupole absorption the effective charges are: q^=l-2/A+Z/A'

qN = Z/A'

9D = 1-2/^+22/^2

I g« = 2gz> (6.18)

It is important to note here that the effective charge ' for direct neutron emission is first of all very small

and secondly, is opposite in sign to the effective charge associated with electric dipole absorption.

The E\ and E2 effective charges for direct proton

emission have the same sign. We would therefore expect that in the energy range above the giant • resonance in heavy nuclei ( ~25 MeV), where both

£"1 and E2 participate, the photoproton angular

distributions might be peaked forward of 90° and

the photoneutron distributions peaked backward

with a much smaller asymmetry. The experimental

facts bear out this prediction onl)^ partially. The

photoproton distributions are certainly peaked for- i ward of 90° and the asymmetries observed in the

photoneutron distributions are very much less pro- nounced but there is no doubt that they too are peaked forward of 90° [121, 122, 123]. Now for the

j first time Gorbunov [124] has found a backward j

'1

36
peaking of photoneutrons from He^ following the

absorption of photons having energies in the range

30-36 MeV. At higher energies the distribution is found to peak forward. This reversal presumablv re- sults from nucleon-nucleon correlations: it would be

instructive to know if it occurs for other nuclei.

5. The Quasideuteron Effect

The dominant mode for the absorption of high

energy x rays is through the quasideuteron mechanism. This becomes important for photon energies

above the giant resonance and results from the in- teraction of the photon with a two-particle cluster rather than with a single nucleon. The interaction occurs with a neutron-proton pair since protonproton and neutron-neutron pairs have no dipole

moment, and hence the quasideuteron designation.

The photon, having high energy and low momentum,

picks out the high momentum components in the

nuclear ground state, associated with the colliding

neutron-proton pair, and disrupts this cluster so that

both particles can emerge from the nucleus with the

energy and angular correlation appropriate to the

deuteron photodisintegration.

Levinger [125] pointed out that the quasideuteron

inside the nucleus differed from a real deuteron in that the two particles must interact in a smaller

volume. The ground-state wave function for the

quasideuteron has effectively the same shape as that

for the deuteron, the difference being that the inter- action takes place in a much smaller volume so that

the amplitude of the wave function must be corre- spondingly larger. He estimated the quasideuteron

cross section to be

NZ

(TQD = 6A—aD (6.19) A

where NZ/A is the number of neutron-proton pairs

in the nucleus.

Levinger also pointed out that one could expect

the outgoing nucleons to reproduce the energy and

angular distributions associated with the deuteron

photodisintegration. Let us consider photon energies

in the range 30-140 MeV, i.e., above the giant reso- nance but below the meson production threshold.

Here the deuteron photodisintegration cross section

consists mainly of an electric dipole contribution with an electric quadrupole component that increases

with energy. The interference between the El and

E2 components makes the proton angular distributions peak forward of 90° and the neutron distribu- tions backward of 90° in the deuteron center of mass

system. In the laboratory system both distributions

peak forward of 90°; the backward peaking of the

center-of-mass neutron distribution is more than

compensated by the forward motion of the deuteron

center of mass. The same is at least qualitatively true for the quasideuteron. It is never obvious, for instance, how to specify the quasideuteron binding

energy, even if it is known that the residual nucleus

is left in its ground state, for the neutrons and proton

have a positive energy resulting from their motion in- side the nucleus. The momentum associated with

this motion also produces a smearing of energy spectra of the outgoing nucleons compared with those

produced in the deuteron photodisintegration.

Insofar as the experiments (which are very diffi- cult) have been performed they bear out most of the

predicted features of the quasideuteron effect. In the

first place the spectrinu [126] of protons produced

when a target was bombarded with high energy

bremsstrahlung of maxiuunn energy. En, varied ap- proximately as E~- with a very sharp break at En/2,

decreasing very rapidly at higher energies. The E~-

dependence results from the fact that both bremsstrahlung spectrum and the deuteron photodisintegration cross section vary approximately as l/E' and

the cut-off at En/2 occurs because the highest energy

photon can only give half its energy to the proton.

This result indicated very early that there was no

other important source of high energy protons in the

photoeffect.

The angular distribution [127] of neutrons in coincidence with protons of a specified energy has

also been measured and shown to have the correla- tion dictated by the dynamics of the deuteron photodisintegration. The width of the angular distribution curves for complex nuclei have been found to be

wider than those obtained with deuterium targets

and this difference used to infer the internal energies

of the nuclear particles.

For a long time the experiments implied that

Levinger's estimate of the quasideuteron cross sec- tion magnitude was too large. New experiments,

however, indicate that perhaps the constant in eq (6.19) is closer to 10 than 6.4. The newer experiments [128, 129] take into account all of the outgoing

nucleons that do not populate the ground state of the

residual nucleus.

37
VII. Isobaric Spin in Photonuclear Reactions

1. Introduction

Isobaric spin has some consequences for photonuclear reactions. If we express the electric dipole

operator in terms of the isobaric spin operator t- we

have

= e/2 X! J", —e X) r,fr' (7.i:

where t' has the eigenvalues +^ for neutrons and — I

for protons. The first term of the operator is associated with the center of mass motion and is responsible for Thomson scattering; the second term, —e 2 Titr, induces electric dipole transitions aci cording to the selection rules AT=Ozbl for those

nuclei having = (N— Z)/2 t^O in their ground

states and AT=1 for those having T2 = 0. These

selection rules can be violated in that the Coulomb

interaction can mix excited states of different iso- baric spin. If isobaric spin is a good quantum

number and the selection rules are obeyed we should

expect the electric dipole (7, n) and (7, p) cross sec- tions for the self-conjugate nuclei to be identical in

size and shape. In addition these nuclei would not emit deuterons or alpha particles in direct interactions following £'1 absorption. In the following we

cite various examples of violations of these rules. With that caution we will then consider the conse- quences of taking isobaric spin seriously.

2. Evidence for Isobaric Spin

Impurities in the Giant

Resonance

If we have perfect charge symmetry and electric dipole interactions in which the excited particle comes directly out of the nucleus, then the (7, n)

and (7, />) cross sections should be identical in both

magnitude and shape. The (7, p) thresholds are, in general, lower than the (7, n) thresholds by just

such an amount as to compensate for the Coulomb

barrier. For a given excitation energy, then, the out-_

going protons do have slightly higher energies than

the neutrons but only enough to make the (7, p)

cross sections ~20 percent larger than the (7, n)

cross sections. Experimentally the ratio of the (7, p)

to the (7, n) cross sections integrated over the giant resonance varies from 1 at helium to 6 at calcium.

This is illustrated in figure 29 where p) dE/

<t(7, n) dE is plotted for the self-conjugate nuclei. This is a gross indication of isotopic spin mixing in the giant resonance. Quantitatively the ratio of 6

for Ca^° corresponds to 40 percent contribution of

the T=0 amplitude to the excited state—or a 15

percent contribution to the intensity.

C^- is an example of a nucleus for which the (7, 7j) and (7, p) cross sections [32, 130] are conspicuously

different. These are compared in figure 35. Notice the

huge dip in the (7, n) cross section at exactly the

energy where the (7, p) cross section peaks. The cor- responding cross sections for O"', on the other hand,

show no such obvious difference; the ground-state

{7, n) and (7, p) cross sections are apparently

identical. In connection with the 2p2ft excitation in O'*" it has already been pointed out that the N'"* (rf, 70) O^^ process does occur [^112] with a rather sharp maximum at 22.7 MeV. Its angular distribution establishes it as an electric dipole phenomenon which must

result from isobaric spin mixing.

The isobaric-spin forbidden reaction [[131]

Mg-* (a, 7o) Si'** also takes place in electric dipole

transitions. This result is also based on an unambiguous angular distribution measurement.

24MeV

Figure 35. A comparison of the (7, n) [130\ {open circle) and

the (7, p) cross sections [32] for C'^.

38
3. Consequences of Isobaric Spin Purity

The ground states of the stable nuclei have their Z-component of isobaric spin (N—Z) /2, and

'T=ztTz. The dipole state has three isobaric spin

components Tk=T, T+1, and T— 1. Their relative

' strengths are given in part [132] by a geometrical

factor: (TTJAT. I nn+AT,)2

Here AT';=+1 describes the electric dipole part of

/i-capture on the target; ATz= —1 represents the

electric dipole part of n, p reactions; and AT, = 0

is the electric dipole giant resonance. The last then

has two components Tk=T and Tk=T-\-\. These

have the relative geometrical intensities:

(TTIO I TTY' T/T+l^^^NTTIO I T+IT)' l/T+l (7.3)

This statement is consistent with the well-known fact that for the self-conjugate nuclei with T=0 the giant resonance consists only of AT=1 transitions. If we

consider only the geometrical factor, we find that for the nuclei having T=|, of which He' and C are ex- ampl. =. the T+l giant resonance is the more important. The T=l nucleus, Ni^^ would have T=l

and T=2 giant resonances of equal strength. As we

progress through the periodic table T, or the neutron

excess, increases and the T+1 giant resonance be- comes less and less important. For Zr^° T is 5; for

pjj208 jg 22. For Ca"*" the T+1 resonance is everything; for Pb^"* it must be less than 5 percent. In figure 36 are illustrated the T and T+1 giant resonances for an arbitrary nucleus, {N, Z), having

A'^ neutrons and Z protons. The splitting of these two

giant resonances is a few MeV; Lane's estimate [133^

of 100(T+1)/A MeV would be ^6 MeV for Zr^". Also

shown are the ground states of its neighbors, (A'^— 1,Z)

and (iV, Z— 1) having T— ^ and T+§ respectively.

Both giant resonances can emit protons to populate

the ground state of the residual nucleus but only the

lower can emit neutrons; neutron emission from the

upper states involves AT=f which is forbidden.

Neutron emission from the T+1 resonance can occur when the excitation energy exceeds the threshold for populating the first T+^ state in the residual nucleus,

i.e. Q„+E., = Qp+AEc. The first T+^ state is the

analogue of the ground state of the (A^, Z— 1) sys- tem and separated from it by the Coulomb energy, A£c=1.44 Z/^'/''-1.13 MeV. This scheme is also illustrated for the specific nucleus, C.

One might think that we could test these ideas by

looking for low energy neutrons as the excitation

energy is increased across the threshold for populating the T+§ state. This situation, however, is muddied by the fact that either the (7, 2n) or

(7, pn ) threshold usually occurs at this energy too. Experimental evidence for the T+1 giant reso- nance is still paltry. The best example lies in tfie photodisintegration of C'^ Here the giant resonance

[134], made up of the sum of the (7, n) and (7, p)

cross sections, consists of a broad resonance centerea near 23 MeV plus a smaller peak near 13 MeV.

Presumably it contains both T=| and T=| components. The T=| part of the N^' giant resonance

has been excited [135] in the reaction C- (p, 70) N".

Its shape resembles very closely the lower energy

part of the C''(7, n) cross section (see fig. 37). It has

been inferred [136] that the T= ^ C^' giant resonance

has this shape and that the remainder is the T=f

component.

Some evidence has also been cited in the photodisintegration of the Ni and Zr isotopes. Min [137]

associates the large proton emission cross section of

Ni5» at 19.5 MeV with the T+1 = 2 resonance, the

lower T= 1 resonance being at 17 MeV. The Zr iso- topes have their ftnain giant resonance at about 16

MeV, but on the falling side of the neutron production cross section there is a shoulder near 21 MeV

that has been ascribed [50] to the T+1 giant resonance. O'Connell [138] has derived a sum rule relating

the strengths of the two giant resonances to the mean

square radius of the excess neutrons and their correlaN-l, Z N, Z N, Z-l

20 MeV

Qb= 17.5 MeV

Figure 36. An illustration o) the T and T+1 giant resonances

oj the nucleus (N, Z).

The ground stales of the adjacent nuclei (/V— 1, Z) and (N, Z — I) having T— J4 and T+J/^ are also shown. Both giant resonances can emit protons to populate the ground state of the (AT, Z—l) system whereas only the lower resonances can emit neutrons to populate the ground state of the (iV — 1, Z)

system. An excitation energy of Qp+AEc is required for neutrons from the r+l giant resonance to populate the first T+J^ state of the {N — \, Z)

system.

39

!
tions. This sum rule includes not only the geometrical

factor already mentioned but also the dynamical

factor discussed by MacFarlane [132]] and Goulard

and Fallieros [1393- The derivation of this sum rule assumes that isobaric spin is a good quantum number and uses the dipole operator in its long wavelength approximation. It is:

She

[2T(/?K') + r(2T-l)(i?r')]

= cr_i(T)-T^_i(T+l) (7.4)

The notation cr-i(T) stands for J (t{T)/E dE, often

called the bremsstrahlung weighted cross section but

is, in fact, proportional to the sum of the squares of

the transition matrix elements, Dk^{T), populating the giant resonance having isobaric spin, T:

k flC (7.5)

The quantities, Rv and Rt, are called respectively

the isovector radius and the isotensor radius. They

are defined in terms of the reduced matrix elements

of their respective operators.

The isovector radius depends on the mean square

neutron and proton radii:

= N{Rn')-Z{R,-^} (7.6)

(O'Connell's operator, r,^ has the eigenvalues ztl.) If we assume that the neutron and proton radii are approximately the same, then the mean square iso- vector radius is just proportional to the number of

excess neutrons, 2T, times their mean square radius.

For making numerical estimates we assume that the mean square radius of the excess neutrons is the same as the measured mean square charge radius.

Then

2T.</?y2) = 2T(i?eh^) (7.7)

The isotensor radius, Rt, is defined by the relationship:

[3r.2-T(T+l)]<i?7.2) = (TT. E (rrr.)(3rA-^ ^i)\TT.) (7.8)

The mean square isotensor radius, (Rt), then de- pends on the correlation between pairs of nucleons,

i and j. The factor, (rrry), represents their spatial

correlation function. The weighting factor, (3rj'r,~ —

t^'T,), has the value zero for a neutron-proton pair

with t = 0. For t = l pairs it is 2 for neutron-neutron

pairs 2 for proton-proton pairs, and —4 for neutronproton pairs. In other words, this factor yields zero

for the T= 0 core of the nucleus. The isotensor radius

then measures the correlation between pairs of excess neutrons. To examine the consequences of this sum rule, set

the term involving (Rt'^} equal to zero. This can' happen for a nucleus with T=^ or if there are no

correlations between pairs of excess neutrons, i.e.,' in a literal independent particle model. Then

|

T<7?eh^ ) = c7_i ( T) - r(T_i ( T+ 1 ) ( 7.9)

!

3fic Combining eq (7.9) with

<T_i = a_i(T)+(r_i(r+l) we have

(T-i(T+l)

(7.10)

1 T+1

1 r+1

1-

1-

27rV r(i?eh-)

She <T-i

0.48T(/?eh^)

(7.11)

Here we recognize the factor, 1/(T+1), as the

geometrical factor that determines the importance

of the (T+1) giant resonance. The term in the

square bracket is the dynamical factor that remains,

f or a harmonic oscillator

dE

60 NZ

OMA"^ mb (7.12)

for N=Z= A/2. If we take {R\h) = A^'' {m\ then

<T-l{T+l)

T+1 (7.13)

This is essentially the result Goulard and Fallieros

[139] obtained in the harmonic oscillator model. It

is imperative to point out that the harmonic oscil- lator model certainly does not apply to the real world since no giant resonance peaks at 42^"''^

MeV. In fact, the data imply that the coefficient

in eq (7.12) is 0.16 rather than 0.36 (see fig. 2).

This would make the dynamical factor even more

important in decreasing the size of the T+1 giant resonance. Using cr„i = 0.16^4*'^ mb and the em pirical

dependence [1403 of ^4 on T

T= 0.2^7(^+200),

we find that the strength of the T+1 giant resonance becomes zero at A'—'110. For y4 > 110 it transpires that a-i is much too large to be consistent with the nuclear radius. The correlations between the excess neutrons, represented in eq (7.4) by {Rt'}, must therefore play an important

role. This term in the sum rule has to be appreciable

and negative.

An investigation of the properties of (Rt'^} shows,

indeed, that it is negative [141]]. Furthermore, it stems from the anticorrelated pairs of excess neutrons occupying orbits of opposite parity. The sum

rule then states that even if £r_i(T+l) vanishes for large A, there must also be large anticorrelations of

the excess neutrons moving in shells of opposite

40
parity in order to make tlie experimentally observed

bremsstralilung weighted cross sections, ja/EdE,

consistent with the nuclear radii. Having established

that the correlation term exists, we are no longer

1 confined by the harmonic oscillator result that

o-_i(T+l) is zero for .-1>110: it can be small but

I finite. Pb [208J represents an extreme case. In this nuI cleus there are no A7'=l electric dipolc transitions

' and the entire giant resonance has the isobaric spin

of the ground state [142]]. The sum rule may then

i be written as

TT-p- 3^ l2T{Rv-}+T{2 T- 1 ) = <r_i( T) = a_

I

(7.14)

Using the experimental result ff_i = 202 mb and the

j charge radius of 5.5 fin for Ry, we obtain — T(2T-l)(7?r-)= -116 mb.

J 3/ic

I This quantity is a measure of the correlations be-

' tween the excess neutrons. The sum rule may also be applied to the T=|

! nuclei for which the correlation term vanishes be- cause of the (2T— 1) factor multiplying it;

' ^

[7V(i?„2 ) - Z(i?/ ) ] = 2a_i ( T) - a^i ( T+ 1 )

.

(7.15)

We recognize the Cabibbo-Radicati [143] sum rule

in its nonrelativistic form. Omitted is a term containing magnetic moments that stems from the usually neglected spin-flip parts of the electric dipole

operator. This rule has already been combined [144,

145] with a formula due to Foldy [2] and applied

to the three-nucleon system. Foldy gives the bremsstrahlung-weighted cross section for nuclei having

symmetric ground states, i.e., those in the Is shell, in terms of their mean square matter radii, R-:

47r2e2 NZ

,_,=._,(r)+,_,(T+i) =-- —«'

The resulting relationship is: —--i?2=<T_i(T)=<T_l(T+l)

3 nc Experimentally He^ has two break-up modes:

HeW+p

The first can have only T=\\ whereas the second

can have both T=% and \. However their measured

values are substantially the same:

(r_i(2) =1.34±0.05 mb

(r_i(3)=1.42±0.07 mb

10 IS 20 25 30 35 40 EXCITATION ENERGY, MeV Figure 37. A comparison of the results of the calculation of Easlea [Ub] with the measureil C" (7, 11) and C'^ (p, 7o)N"

cross sections [134, 736].

Both T = y^ and T — 3/2 etatea were impulated in the former; while only T = }^ states were formed in the latter.

If equal, we are led to the conclusion that the three- body break-up mode is to be associated with T=f.

The sum rule can also be applied to C'^ which has

7'=^. The giant resonance should therefore consist

of 7'= I and 7 = 5 parts. Cook [134] has synthesized

the total cross section for C'^ by summing his measured (7, ») and (ff, p) cross sections. The calculation

of Easlea [146] as well as the C'-(p, 70) N'^ experiment of Fisher et al. [135] can be used as a guide

to decide on a dividing line between the T= 5 and

T=f strength. Figure 37 shows the data from the

(p, 7o) experiment which populates only T=| states along with the (7, n) cross section that populates both

T = ^ and T=| states. The energy, 22.5 MeV, was

then taken as the boundary between the two giant resonances and the integrals evaluated from the total cross section which is not shown.

These were

cr_i(^) =3.3 mb and (r_i(|) =4.3 mb

(r_i(|)-i<T_i(|)=1.15 mb

Using the charge radius as the radius of the valence

neutron 7?^ = 2.36 fm, the left hand side of the equation becomes 1.34 mb. These two numbers are cer- tainly in satisfactory agreement.

These notes are based on lectures given under the

auspices of a Sir Thomas Lyle Fellowship at the

School of Physics of the University of Melbourne.

The author thanks Professor Brian Spicer for his hospitality during that visit.

41
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