# Intermediate Structure in Nuclear Reactions

Richard H. Lemmer
Leonard S. Rodberg
James E. Young
J. J. Griffin
Alexander Lande
Hugh P. Kennedy
Rudolph Schrils
Pages: 232
https://www.jstor.org/stable/j.ctt130jsxn

1. Front Matter
(pp. i-iv)
2. Foreword
(pp. v-vi)
M. T. McEllistrem

The lecture series on Intermediate Structure in Nuclear Reactions was held in June 1966 at the University of Kentucky. Four lecturers had been invited to present series of three lectures each. Unfortunately, Dr. James E. Young was unable to attend the series. He did, however, submit a manuscript and we are fortunate to be able to have his approach represented in this volume. The three lecturers who did attend, Drs. R. H. Lemmer, L. Rodberg, and A. Lande, gave one lecture on each of the three days. The sessions were distributed through the day to allow ample time for discussion....

3. Preface
(pp. vii-viii)
H. P. K. and R. S.
4. ### Richard H. Lemmer

(pp. ix-x)
• An Outline of Nuclear Reaction Formalism
(pp. 3-18)

As the first topic these lectures, I would like to discuss the questions that come up whenever one thinks of measuring or analyzing nuclear reaction cross sections. One asks what the cross sections are, why they have the shapes they have, and whether they relate to more fundamental dynamics. Those simple questions encompass almost the whole of the physics of the nucleus. If one asks any question, such as what is the scattering cross section for some particular incident projectile, one is really asking a theorist to solve explicitly a many-body problem dealing with the order of a hundred particles....

• Intermediate Resonances and Doorway States
(pp. 19-45)

The first lecture was basically a review of the ideas underlying a particular formulation of nuclear reaction theory, and the relevant formulae. were derived in some detail.* In this lecture we want to discuss the topic of current interest, the intermediate resonance structure⁵,⁶,⁷ For this we shall only need a few of the relevant formulae from the previous lecture. The first is the expression for the transition matrix:

$T\left( E \right) = {T_{{p_{{o_t}}}}} + \langle {\psi _0}( - ),{H_{PQ}}\frac{1}{{E - {H_{QQ}} - {W_{QQ}}}}{H_{QP\psi 0}}( + )\rangle$

The two important aspects of (20) are, first, Tpotis supposed to be slowly varying with energy and, second, the resonance structure comes essentially from the eigenvalues of HQQ.HQQdefines...

• Applications
(pp. 46-62)

After all the formalism that has been discussed, I want to spend the last session discussing what can really be done with this large amount of theorizing. It would be useful, I think, to discuss informally what has been done, what seems to work, and what does not seem to work. The first thing I would like to stress is that contrary to any formal discussion, when we actually try to calculate a doorway state escape width or damping width, we must make the last link between formalism and formulae; namely, we must appeal to a nuclear model.

It is...

5. ### Leonard S. Rodberg

• Development of the Formalism
(pp. 65-84)

The first discussion of what we now call “intermediate structure” was in a paper by Brueckner, Eden, and Francis in 1955,”¹ in which the authors used multiple-scattering techniques to study the effects of what they called “two-particle excited states.” In writing on the related resonances in 1961,² I also called them two-particle excited states. Since then these resonances have been generalized to be called intermediate resonance³, One approach to intermediate structure has been developed by Feshbach and his coworkers.⁴ In discussing here a way of regarding nuclear reactions and resonances, I shall use a slightly different approach drawn from multiple-scattering...

• Introduction of a Model Hamiltonian
(pp. 85-102)

In this section I will apply the techniques developed earlier in order to arrive at a model that describes intermediate structure. The development in principle is exact, but, of course, in any application one must make suitable approximations. We will discuss such approximations, but first let us remove some restrictions that we placed on the previous development.

Rearrangement Processes

Thus far we have treated only elastic and inelastic scattering. The target could be left in any state of excitation after the collision, but no rearrangement of the particles was permitted. For rearrangement processes, as for example,(p, n)or(d,...

• Conclusion and General Remarks
(pp. 103-120)

In the preceding section we worked out a reaction theory based on the shell model. It allows us to calculate the positions and widths of intermediate resonances and, perhaps more importantly, the use of the shell model makes these manageable calculations. The wave functions YDare ordinary shell-model wave functions, with only the slight complication that the residual interaction is complex. In this next section we want to carry this one step further and then make some general remarks about the observation of intermediate structure.

Fine Structure

In deriving the resonance formula for the doorway resonances, we introduced two projection...

6. ### James E. Young

• I. Introduction
(pp. 123-129)

The efforts over the last three years to define and describe nuclear intermediate structure represent an attempt to investigate the explicit energy dependence of nucleon-nucleus scattering amplitudes over and above the momentum-transfer dependence of such amplitudes.* Furthermore, as far as the aspects of reaction mechanisms are concerned, theories of intermediate structure seek to interpolate between the regimes of no energy transfer to the target system, the optical potential, and complete transfer of energy or the Bohr compound nucleus. The theories say in a number of different ways that scattering cross sections measured with good resolution will exhibit Ericson¹ fluctuations as...

• II. Many-Body Theory
(pp. 130-142)

Our aim here is to classify the compound states of the proton channel${}_N{A_Z}(p,p')$so as to identify which of the many configurations occurring is responsible for intermediate structure. Perhaps the most economical method of labelling the configuration is provided by the Green’s functions introduced by Martin and Schwinger.²⁶

(a) Summary of Green’s function method for (p, p')

Nucleon scattering from the ground state targetNAZis described by the exact amplitude

${T_{ab}} = \langle \lambda \hat k,\left( {{x_0}} \right){q_\mu }\left( {{\xi _A}} \right)|\sum\limits_{i = 1}^A {{V_{0j}}} |{\Psi _k}( + )\left( {{x_0},{\xi _A}} \right)\rangle$, (11)

the stationary state satisfying$\left( {E - H} \right){\Psi ^{( + )}} = 0$, (12)

where

$H = {H_0} + \sum\limits_i {{V_{0j}}}$,

$(w - { \in _\mu } - {H_0})\lambda \hat k\left( {{x_0}} \right){q_\mu }\left( {{\xi _A}} \right) = 0$,

$({h^2}{k^2}/2M) = w$,

and$w + { \in _\mu } = E$.

The notation used is self-evident. It is a familiar result that the transition operator...

• III. Intermediate Structure in (d, p) and (p, p’); Isobaric Analogs
(pp. 143-163)

It is our purpose in this section to show that isospin conservation makes it possible to relate intermediate structure in(p, p’)onNAZto(d, p)reactions on the same target. We thus suggest a generalization of isobaric analogs as commonly studied. It is exhibited that the (2p, 1h) compound states occurring inNAZ(p, pf), the states |s’) of Section II, are determined as isobaric analogs of (2p,1h) residual states |s’) occurring inNAZ(d,p). This result follows from the existence of strong (p,h) forces, i.e., the targetNAZmust exhibit low-lying, (p,h) vibrations...

• IV. A Shell Model Reaction Theory of Intermediate Structure in (p, p')
(pp. 164-183)

Since the material in this section is of more than routine technical complexity, we begin with a summary of the gross experimental spectrum as we shall be concerned with it (Fig. 12), showing that spectrum of states relating excitations in ¹¹⁹Sn and ¹¹⁹Sb connected through isospin conservation. The spectrum is discussed in terms of excitations in the bound neutron channel and open proton channel. For sufficiently large energies Tkin the latter we do begin to get charge-exchange neutrons. In reality the proton channel is very complicated, for once we exceed the capture region there is strong possibility for Coulomb...

• V. Conclusions
(pp. 184-188)

In a study of intermediate structure it is exceedingly difficult to provide even primitive numerical and experimental results. Our presentation here has at least delineated the boundaries of a possible approach to a complete theory. We do not pretend that it is the only admissible viewpoint, but it is at least self-contained and as such amenable to experimental comparisons. This is a strong recommendation for even possibly wrong theory. Intermediate structure theories have not heretofore produced numbers or calculable formulas for experimental quantities.

The reason for the great theoretical interest in a description with seemingly little observational support is that...

7. ### J. J. Griffin

• A Statistical Model of Intermediate Structure
(pp. 191-202)

The experiment here is a simple (p, n) reaction—¹¹⁷Sn(p, n)¹¹⁷Sb.¹ The kinetic energy of the incoming proton is 14 MeV. The excitation energy in the compound system ¹¹₈Sb is, therefore, about 18 MeV. We are interested in the emission of neutrons of kinetic energyE₀leaving residual states of ¹¹⁷Sb at an excitation energy, U, so that E* = U + E₀ + BN= Ep+ Bp.

What does such an experiment have to do with intermediate structure? If traditional statistical theory is used to analyze the distribution of neutrons in energy E₀ or alternatively the distribution of...

8. ### Alexander Lande

• Vibrations, Doorways, and Intermediate Structure
(pp. 205-220)

In recent years there has been mounting evidence for oscillatory structure in nuclear reaction cross sections. The width of this structure, ~100 keV, and its average separation, ~300 keV, places it between the compound and single particle resonances of the optical model.* What caused the original excitement was the preliminary data1 for total neutron cross sections over a wide range of energies (2.5 – 15 MeV) in which such “intermediate structure” seemed to appear in many of the natural targets studied. Typically the bumps persisted until ~8 MeV before dying out. Although some have regarded these oscillations as representing nothing more...

9. Back Matter
(pp. 221-221)