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 amplitude, R(E,0). Familiar examples of this connection are the coherent scattering 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, (E), and the forward scattering amplitude R(E,0) is called the optical theorem: σ,(E)=4TX Im R(E, 0)

The coherent scattering cross section is

(2.1)

3.0

σ, (E)

=

E2r Mc A (E2-E2)2+г2E2

(2.5)

16

E, Mev

24

FIGURE 3. A comparison of the absorption cross section with the forward scattering cross section for a nucleus with N =Z=50 for which the dipole sum is enhanced by a factor ẞ= 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 σ ↑ stands for the magnitude of the nuclear Thomson scattering cross section which is effective at very low energies. The symbol σ 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.

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 y 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 1+cos20 angular distribution.

The usual treatment of angular correlations concerns itself with transition between states of well specified spin and parity. The sequence I→ II 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

(2.16)

(2.17)

Σ (-1)+1-Ik (21x+1)

Ik-10-1

Io Io 川 1 1 Ik

=0 (2.21)

for >0, the only contribution to the elastic scattering 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: 1+cos20.

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=0) 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

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 I 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, A,, may be written in a compact form [5] if it is assumed that all of the oscillator strength, NZB/A, associated with the transition AK=e, is distributed in a resonance located at E.. The energies of incoming and outgoing photon are set equal, then:

In the Danos-Okamoto model the transitions that make the giant resonance of deformed nuclei are up 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 v=0 is:

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 experiment. 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 Io+1 and Io+2. The total intensity in the Raman spectrum is, however, a constant by virtue of the sum rule:

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 Io 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.

Photonuclear experiments differ from other experiments in nuclear physics only in that the reactions 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 determines 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 they 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 y 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 10-3. 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 F19(p,ay), Li'(p,y) and the H3(p,y) reactions. Their properties are compared 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 y rays that come from the excited states in O16 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

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

The Li(p,y) reaction is an important source of 17.6 MeV y 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 experiments where the incident photon energy can be determined from that of an outgoing nucleon.

The H3(p,y) reaction has been used primarily by Stephens [7-10] and his collaborators. It represents a source of y 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 H3(p,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 y rays have been used by Donahue [11-13] and more recently by Ben-David [14-15] to study the (y,n) and (7,7) reactions for excitation energies near 8 MeV. These y rays occur in a rather restricted energy range but the spread of each line is only a few volts, being determined largely by the Doppler width, A= E。(2kT/AMc2)1 /2 associated with the thermal motions of the target atoms. Table 2 shows a list of useful neutron capture y rays and their relative intensities.