Intermediate Structure in Nuclear Reactions 9780813163314, 9780813152929

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Intermediate Structure in Nuclear Reactions

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Intermediate Structure in Nuclear Reactions Lectures by Richard H. Lemmer Leonard S. Rodberg

James E. Young J. J. Griffin Alexander Lande Edited by

Hugh P. Kennedy and Rudolph Schrils

UNIVERSITY OF KENTUCKY PRESS LEXINGTON, KENTUCKY

Copyright @ 1968 by the University of Kentucky Press Library of Congress Catalog Card No. 67-29341

Foreword THELECTURE 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. The fact that the conference was small helped to stimulate exchanges between both lecturers and auditors. Some of the discussion sessions were nearly as long as the lectures which they followed. The main purpose of the conference was to provide an informal setting for a pedagogical review of the formalisms and physical content of this relatively new subject in nuclear reactions. It offered some of the originators of different points of view toward Intermediate Structure an opportunity to present their different approaches. Professors Lemmer, Rodberg, and Young presented three different formal developments of the subject. They illustrated their developments by presenting sample calculations and by relating the subject of intermediate structure to other topics in nuclear reactions. Professor Lande offered evidence in support of a particular identification of those nuclear configurations which should be considered if one attempts to construct states intermediate between single particle states and the complex states associated with the compound nucleus. We were fortunate to be able to persuade Dr. James Griffin to present evidence for the interpretation of nuclear reactions in terms of progressive stages of complexity; such an interpretation is very closely related to the concepts emphasized in intermediate structure. Since the primary purpose of this conference was a pedagogical

one, each lecturer was invited to make his lectures reasonably complete and self-contained. The resulting repetition of some of the introductory ideas was judged to be quite helpful to those who were introducing themselves to the details of this subject for the first time. The lecturers were also invited to retain the informality, directness, and detail which they used in their lectures, rather than carefully rewriting them into a series of succinct manuscripts. In this connection, applications of the theory to specific nuclei was based on experimental information available in the spring of 1966 for the three principal lecture sets, and in the winter of 1966 for Dr. Lande's contributions. What develops then is not a concise review of the formal developments in Intermediate Structure theory, but an informal and detailed discussion of the attitudes which led to these developments, a discussion of the physical content of different formal developments, and the presentation of many of the details usually omitted in the original literature or in review papers. We hope that the result will be as useful to the uninitiated reader as it was to those who had the privilege of attending the lectures. This small conference was attended by 85 people, with visitors from many institutions and laboratories. Several of the visitors contributed in a spirited way to the discussions and thus to the content of the conference. Special mention in this connection should be made of the contributions of Professor Kirk McVoy of the University of Wisconsin and Professor Kamal K. Seth of Northwestern University. As chairman of the conference it is my duty and privilege to thank the many people who helped to make it the informative and pleasant experience it was. The conference was supported entirely by the Graduate School of the University of Kentucky, and it was a special privilege to have Dean A. D. Kirwan welcome the attendees to the University. Special thanks go also to Professor F. Gabbard, Acting Chairman of the Physics Department, for his interest and support of arrangements for the conference. W. E. Coomes and Abdul Waheed contributed by handling the recording of all lectures and discussions.

M. T. MCELLISTREM

Preface THE ANALYSIS of resonances in nuclear reaction cross sections has provided much evidence for two aspects of highly excited nuclear states. In a few nuclei broad resonances are successfully interpreted as indicating single-particle states. These are found especially in light nuclei. Most resonances, however, have small single-particle amplitudes. This is consistent with the usual attitude toward compound nuclear states, which characterizes them as complicated configurations of many particles. In 1961, the proposal was explicitly put forward that nuclear configurations "intermediate" between these extremes ought to be identifiable in the energy dependence of nuclear reaction cross sections. Since that time, a large amount of work has been completed by several theoretical groups to establish appropriate formal frameworks for these intermediate resonances and to elucidate those properties which would permit them to be identified experimentally. The report of isobaric analogs in energy-averaged cross sections gave special impetus to the field since they were interpreted by many as a special type of intermediate structure. Their discovery encouraged a search for other types of intermediate structure. The main purpose of this lecture series was to provide a pedagogical review of the theoretical developments and the underlying physical ideas of intermediate structure. Many expositions which are more detailed than would normally appear in review articles are included here. The original manuscripts were revised to include some clarifying points from the floor and some material not presented in the lectures was added for purposes of amplification. It was not intended to review the present state of experimental evidence for intermediate structure. Those specific calculations and data comparisons which do appear are offered only as illustrative examples. Lemmer, Rodberg and Young all proceed toward the same objec-

viii

INTERMEDIATE STRUCTURE IN NUCLEAR REACTIONS

tive. That is, the subdivision of the scattering amplitude into two parts. One part displays the intermediate resonances. The other, when energy averaged, contains little or no energy-dependent structure. Lemmer subdivides the scattering amplitude in a purely formal manner without the explicit introduction of a model. He achieves this by expanding the wave function of the system in terms of the actual states of the target nucleus. In Rodberg's development nuclear models play a more fundamental role. He defines the central concept of a doorway state in the context of a model Hamiltonian. Young presents his concept of intermediate structure in the context and language of many-body theory. Because of the different frameworks used by the authors, there is little connection that can be made between their notations. Specific calculations bearing on intermediate structure by Griffin and Lande are also included. We wish to thank the lecturers for their cooperation in the preparation of the manuscripts. Our thanks also go to the chairman of the conference, Professor Marcus McEllistrem, for affording us the opportunity to consult extensively with him, and to Mrs. Betty Hoskins for her typing of the manuscripts. H. P. K. R. S.

Contents Foreword, by M. T. MCELLISTREM Preface

v vii

RICHARD H. LEMMER An Outline of Nuclear Reaction Formalism Intermediate Resonances and Doorway States Applications

3 19 46

LEONARD S. RODBERG Development of the Formalism Introduction of a Model Hamiltonian Conclusion and General Remarks

JAMES E. YOUNG I. Introduction 11. Many-Body Theory 111. Intermediate Structure in (d, p) and (p, p'); Isobaric Analogs IV. A Shell Model Reaction Theory of Intermediate Structure in (P, P? V. Conclusions

123 130 143 164 184

J. J. GRIFFIN A Statistical Model of Intermediate Structure

191

ALEXANDER LANDE Vibrations, Doorways, and Intermediate Structure

205

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RICHARD H. LEMMER

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An Outline of Nuclear Reaction Formalism

As THE FIRST topic of 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. Since the theorist cannot do the two-body problem very well, this is clearly an impossible task. What has developed over the years has been a series of approximations. One tries to extract as much information as one can while putting in as little as one can; clearly, if one puts in as much as he gets out, he is not getting very far. This has led then to the analysis of nuclear cross sections in terms of certain semi-empirical formulae. For example, resonance reactions are described in terms of the positions, widths, and angular momenta of the resonances. What I have decided to do in this first lecture is to develop some basic aspects of scattering theory. Consider the problem of a nucleon scattering from a nucleus of mass A. We can start by writing the A 1 Schrodinger equation. At this point let us recognize that some drastic approximations have to be made just to get beyond this first

+

stage. The purpose will be to get as far as possible with a few approximations that do not depend on intrinsic details of the system. In this way we can arrive at a formalism which describes nuclear reactions. (Notice the word "describe" and not explain; they are two quite different concepts!) Such a formalism would contain in general the form of the energy dependence which a scattering cross section actually has. At this stage, the description of the detailed energy dependence of low energy reactions had stopped until about six years ago. Cross sections were described by the Breit-Wigner formula and extensions of it. From this, resonance widths and energies were extracted but there was no way to relate these properties or the measured cross sections themselves with more fundamental properties of the system. The next step is to try to relate the reaction formalism to the dynamic details of nuclei. For this purpose the link from a general reaction formalism to formulae has to be made by some nuclear model. It is this link which has become so important over the past six years, especially in connection with so-called intermediate structure. With the aid of the model we can also make a comparison with cross sections, but now it is a much more meaningful comparison because comparing the measured cross sections with predictions tells something about the validity of the model. Before actually developing the formalism let us investigate a little further the meaning of the preceding remarks. Although the formalism will be quite general, let us continue to think in terms of nucleon scattering. It is well known that the cross sections show very sharp, very beautiful peaks at certain critical energies, as is shown in Figure 1, and then eventually smooth out at higher energies. These peaks are called resonances, and their energies are denoted by E,. The widths at half-maximum we shall call r,. The interpretation attributed to these peaks is that they reflect long-lived states of the A 1 nucleon system. In other words, they are almost bound states. Therefore, instead of thinking of a particle incident on a target nucleus as a separate problem from the A 1 particle system, it is far more illuminating to think of the scattering problem as just one small aspect of the entire A 1 particle problem. In other words, we will think of it from "inside outwards" instead of from "outside inwards" because this immediately lets us think in terms of a nuclear model.

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Richard H . Lemmer

Figure 1

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If I have the ground state of the A 1 system I might think that I have a relatively simple problem at hand. For example, if I know that the shell model is applicable, I may be tempted to calculate the ground state energy. Even more tempting, I may use models to calculate positions of and transition rates from the first few excited states. We know now from experience that various models provide us with reliable descriptions of states near the ground state in many nuclei. It depends a little on the nucleus what "near the ground state" means. For heavy nuclei it means 1 or 2 MeV. When we get to 8 MeV excitation we should really be very careful about assuming that a nuclear model is going to have any significance because we know that at high

excitation energies the level density increases enormously. The first thing to be decided in fact is whether any sort of model calculation makes sense at 8 MeV. The answer is yes, in a certain specified manner. That manner is the underlying idea of intermediate structure. We will find that microscopic calculations based on nuclear models in this vicinity of excitation energy only have meaning in an average sense. The model calculation gives us some average property of the compound levels at a particular excitation energy. What these average properties are I will discuss later on. Let me just make one point because it is a very important one: we are not saying that we can calculate actual compound nuclear states in general. We are saying that we can go beyond pure formalism. We can put in a nuclear model and get some information that is related to highly excited states of the nucleus. The general formalism to be developed will predict that the shape of the cross section as a function of energy in the vicinity of a single compound resonance is given by the Breit-Wigner formula.

To illustrate the validity of such predictions, let us investigate the data shown in Figure 1. This represents proton scattering on 40Ar, taken at Duke University.l If curve (c) is magnified, we get what is shown in Figure 2. The curve drawn through the points in Figure 2 is just a succession of Breit-Wigner fits. It constitutes an extremely accurate representation of the experimental data. However, if a theorist were asked to calculate in the framework of some model the resonant energies and widths of all the bumps appearing in Figure 1, he would surely give up before he even started. Another point I want to make, however, is that not all experimental data looks as puzzling as this, particularly if the resolution is not as good. Figure 3 shows data of lower resolution. The experiment2 is p ssSr and in this case we notice three very distinct resonances, which, of course, tend to encourage us to try to understand them a bit more because they look very simple. The curve going through them is a Breit-Wigner fit. Again we find that the representation of the data by a Breit-Wigner fit is quite accurate.

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Richard H . Lemmer

Figure 2 PROTON nmo

wmo

RESONANCE

-

FREQUENCY (kc)

amo

- . . *' srae(p.p)

8.90.

-

. m

--

E a

b

m-

t'i

f

INCIDENT PROTON ENERGY IN MeV

Figure 3

arm0

rn

What I want to do next is to give the Breit-Wigner formula, or a slight generalization of it, a firm basis. That, of course, can be accomplished by taking the step from the many-body problem to some sort of formalism. We know that if we have an A 1 particle system, as we have here, then it has a Hamiltonian H which is an A 1 particle Hamiltonian, and we can write this out in the following manner:

+

+

Tois the kinetic energy of the particle that is either incident, or better, the particle that is going to go out. Then there is the Hamiltonian of the target that would be left behind after one particle has gone out. Let us call this H(A). Then these two systems will have an interaction. We write that interaction as Voi, where the 0 refers to a particular particle which we will call the incident one and i refers to all the other particles of the nucleus. This is a many-body Hamiltonian broken up in a convenient way and our entire problem consists of studying the eigenvalues and eigenfunctions of this Hamiltonian: That is just the Schrodinger equation. It looks deceptively simplejust a linear homogeneous equation! What could be easier? The trouble is, of course, that there are A 1 particles. In any analysis of what this basic equation of motion leads to, the most we can hope to do at this stage is to get out the form of the scattering amplitude. Of course, to get from this equation to any cross section, we have to use some form of reaction theory. The one that I am going to present here was developed by Feshbach3 about six years ago. Similar results may be obtained using other standard reaction formalisms. We are only going to study elastic scattering; therefore, if we examine the wave function at infinity, we will never find the flux at infinity associated with the target in any but its ground state. How can we use this fact? We have to examine \k in its various 1 particle wave function energy domains. If we asked for the A for states anywhere between the ground state and the threshold for particle emission, we would find that this wave function vanishes at infinity. These bound wave functions are particularly simple be-

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Richard H . Lemmer .

9

cause they are always associated with discrete energies. In the continuum, as we will refer to the region above threshold, things are a little more complicated because here we do not get exact eigenvalues. As we can see, the bumps in Figure 1 are not perfectly sharp. We are going to rely, however, on the idea that we would like to describe the resonances above threshold as states that are quasibound. As a first approximation, we will say that they are completely bound. In this approximation the states have no width. We will then obtain the widths by including the physical mechanism that causes decay. One useful way to proceed at this point is to expand \k in a complete set of states. Let us assume that H(A) has eigenfunctions 4,, which satisfy the following A-body equation:

In other words we suppose that we know the target energies, both at ground state eo and at all excited states, and that we also know the associated wave functions. Now that in itself is a many-body problem, but let us assume that we have solved it. Then the $, if we take n over all possible states of the target, form a complete set. Consequently we can always write for the following expansion

where ro refers to the coordinates of the incident particle, and u,(ro) is simply an amplitude in the expansion and C$~(A)refers to the ground state of the target. To make use of the assumption of elastic scattering only, we can further write

where the above has been restricted to s-wave scattering only; k is the wave number associated with energy E, and S(E) is the S-matrix. In other words, the form of uo at infinity will give us complete infor-

mation on the S-matrix and therefore complete information for calculations of cross sections. That sounds very nice because all we 1 have to do now is to solve for uo. Of course, because the A particle wave function is made up of such an infinite sum, the uo is coupled to all u,. So we have merely shifted our ignorance from not knowing to not knowing uo. However, we will see that this shift of emphasis has a distinct advantage, formally at any rate, because what we will do is to eliminate all the u,'s in favor of just uo. At least we get an equation that focuses on the quantity that we want and eliminates the quantities that we are not interested in. However, there is one word of warning. As we all know, nuclei are Fermi systems. We have pretended here that we can identify a particular nucleon. That immediately ignores any properties of the symmetry in the wave function that has to be present in the Fermi system. You might suspect that to modify this, all we have to do is to put an antisymmetrization operator in front of the right hand side of Equation (5). This is right except that it essentially negates the whole procedure. Putting an anti-symmetrization operator in front of just the uopiece of Equation (6) and saying the rest does not contribute when we have anti-symmetry present turns out to be wrong. So I would like to emphasize that this naive approach is not true if we antisymmetrize the wave function, but it can be made to be true by a very simple device. In other words, we can end up with the anti-symmetric case by just putting an anti-symmetrizer in front of the uo term, but we will have to do it at the expense of some complication^.^^^ To get ahead of the program: what will happen is, whether we anti-symmetrize or not, the formulas we now derive will be correct in form. There will just be a slight difference in interpretation when anti-symmetry is included. To proceed: instead of carrying sums along like Equation (5), we follow Feshbach3 and introduce projection operators which do no more than tell us that we are working with the uopiece or working with the rest of 9. We introduce a projection operator P which projects onto ~$0and a projection operator Q, where Q = I - P, which projects onto all other states but the target ground state. The uopiece of the wave function can be equivalently written as PQ. As you see, if we operate with P on 9 , by definition it will set all terms equal to zero except for the uo term. P and Q are, however, merely a nota-

+

Richard H. Lemmer

11

tional convenience for isolating different parts of the many-body wave function. All we do now is make use of the fact that if P and Q are projection operators, then P and Q are orthogonal to each other. Since we have a projection operator, P2= P. We simply take the manybody equation again, (E - H)\k = 0, and introduce the operator, P $ Q = 1. We can rewrite Equation (3) like this,

Now what we do is merely multiply in order by the operator P and the operator Q on the left side of Equation (7). Then

and (E - H P P ) =~ HpQe\k ,

where Hpp = PHP, HQp= QHP, HpQ= PHQ and HQQ = QHQ. What do these equations mean? Hppmerely means that no matter what happens, the target must always be in its ground state. This part of the Hamiltonian which is not the full Hamiltonian any more simply describes physically those events where the particle comes means a in and does not excite the target. On the other hand, HQQ system in which you are studying the spectrum or the Hamiltonian 1 particles in which the ground state of the target is always of A excluded. Clearly HpQ and HQP are just coupling terms between these two. We have, for example,

+

In other words, this is the interaction between the open channel and the closed channels. Once we have got that far, the rest is quite trivial to do. What we are after now is P\k, because our complete information lies in knowing UO.We would like io get that piece of the wave function at the expense of eliminating Q\k which is what we will now do. Equation (7) describes the target initially in its ground state so that

solving such an equation would mean a sum of two pieces because there is always a homogeneous equation which corresponds just to the particle coming in without any scattering by the H p g part. We will write P\k out as this homogeneous part plus the inhomogeneous part which would come from the operator 1 E - H P P + iq

'

Now what about the first term I)~(+) in Equation (12)? I),(+) satisfies Equation (11). Now what does Equation (11) mean? It says that the target can never be excited, and so it describes what we would call potential scattering. The particle comes in and simply goes out again without exciting the target. Please recall that this is a stipulation that we have placed upon H p p ;namely, that the actual wave function will always be given by two pieces, one part that corresponds to no excitation of the target and a second part which corresponds to excitation of the target. The asymptotic form of Go(+) will be

where SPot(E)is the potential scattering S-matrix. Notice the difference between S and Spot. S is the full S-matrix of the problem and Spot is the S that relates only to potential scattering when the target is never excited. We know from experience that, although there is no a priori need for this to be so, Spot is a slowly-varying function of energy. This will turn out to be the useful fact in what follows. Clearly it means that any rapid energy variations must be associated with the second term. Upon eliminating P* from Equation (8) we get

The q0(+)is supposed to be known. The fact that + i q appears in the second term on the right is because we want both pieces of P\k to be outgoing waves. To save writing, define

Richard H. Lemmer

13

Then we have a simple equation for Q\E (E - HQQ- WQQ) Q* = HQP$o(+) ,

(16)

which can be solved formally as

Now we have an equation for Q* which contains HQQ,WQQ,HQp, and the known potential scattering wave function A(+).We can insert this back into (12) resulting in

So this completes the manipulation, because now what we have is the open channel part of the wave function. We suspect that the last term in Equation (18) is going to vary very rapidly with energy because whenever E happens to hit an eigenvalue of HQQ (remember now that this is a many-body problem which will have many energy eigenvalues), then (E - HQQ)goes to zero and we are left with just WQQ.It should be noted that WQQcontains the interaction twice so that if for any reason the interaction is weak, then it means that this is going to be a very small number so that the second term is going to give rise to strong resonances. Now, knowing the wave function, all I have to do is examine it at infinity and read off the S-matrix. The S-matrix is often related to the so-called T-matrix, in the following manner:

T is the transition amplitude or the transition matrix. This is useful because all cross sections are proportional to /TI2and not to ISI2 since ISI2also contains the incident waves. So we want the difference between S and 1. Now the transition matrix can be written down very easily for Equation (18) and we will not give the details; the result is the Gell-Mann-Goldberger theorem: *

* Ed. note: Leonard S. Rodberg derives the Gell-Mann-Goldberger theorem.

A(-) also satisfies Equation (I 1 ) but with incoming wave boundary conditions, and Tpotis the counterpart of Spot. This is the main formal result that we will use constantly. As can be seen it has a very simple structure. We now have a slowly-varying part (with energy) in the transition amplitude and a part that is going to resonate because of the poles in the second term. The sum of these two squared will give the total cross section. Let me just illustrate one final point, namely, the question of resonances themselves. We have said that we suspect that because of the energy denominator, the second term is going to have resonances associated with it. Let us ask the following question. Supposing we know the eigenstates and eigenvalues of the operator HQQ:

(E, - HQQ)@, =0.

(21)

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This is also an A 1 problem, but it is a problem in which the target state always appears excited. We can then use the eigenfunctions of

HQQto invert the operator

E

. That is to say, we could write - HQQ

formally,

That is perfectly fair. We notice immediately that whenever E approaches E,, we are going to have a singularity in the answer. We also notice that if the E,'s are widely spaced, then at energy E near a particular E, all of the others give contributions that are really quite slowly varying with energy because they are far from this particular energy. This E, will be the dominant one in determining the behavior of the Green's function (E - HQQ)-' in the vicinity of a particular E,. So we could approximate as follows for E near E,,

where R(E) is a remainder which varies quite slowly with energy. Note that we still have the singularity at E = E,. With this in mind, 1 let us look at the operator we really want to invert. It is not E - HQQ

Richard H. Lemrner

15

but rather the operator

1

E - HQQ - WQQ are, therefore, eigenvalues of

The energies we really want

- HQQ- W Q Q = ] ~0 .

(24) If there is only one E , in the vicinity of the E that we are interested in, then we can use perturbation theory if WQQis small. We say in perturbation theory that [ES

so that correspondingly we would say that in the vicinity of a particular E,,

We now further notice from the equation for WQQthat it is complex. In fact, we can write it out as

where 6 stands for the principal value, and the 6 function term is the energy conserving part. The imaginary part of WQQis very important because it shows first of all that the WQQcontains a negative imaginary part in addition to a real part so that we can write for T(E),

showing that the presence of the WQQis essential in giving to the resonance energy E, both an energy shift A, and a width =

.

2x(as, HQP6(E - HPP)HPQ@~)

(29)

DISCUSSION

RODBERG: LEMMER:

I would like to ask why there is not a homogeneous term in Equation (17). You very carefully included it in Equation (12) but not in (17). In (17) we have a bound channel, and the only way one can excite the bound channel is to come in from the en-

KOSHEL: LEMMER:

trance channel, go to it, and then come back out, so there can be no homogeneous part in (17). That would correspond to having the target initially in an excited state which we do not have here. You see, the Q always refers to the target in an excited state, and the situation that you describe would mean that I have a target that is excited or a linear combination of a ground state and excited states of the target. In your wave equation for the a,, are these essentially states in the continuum which decay? Yes, let us identify them a little more closely. Clearly the A 1 system Hamiltonian HQQhas some set of eigenstates, and they will span some energy space. Some of them will lie below the neutron threshold and some of them can lie above. Now the ones that lie below are of no particular interest to us because they cannot decay. Those that lie above are of interest to us because they will decay and have a width. One can see that perhaps more graphically in Equation (29). You will notice that for the 6 function to be non-zero, the eigenvalues of Hppor equivalently the energy at which you calculate the width must always be above zero. For any eigenvalues below zero energy you will simply find that there is no imaginary part associated with the problem, and you will have a pure bound state. If, however, E, lies above the threshold, it would mean that the 6 function is present, and you have a width. So to answer your question, the E,'s by themselves do not decay even though they lie above the threshold. And the reason why that is so is because you have specifically removed the interaction that causes their decay. So, in a sense, what this does is to keep the nucleus excited because you are not allowing that one remaining interaction to let the particle go out. To save words, what has happened? The particle comes in and the moment the nucleus grabs it, you empirically turn off all the interactions which would let a particle out again. So the captured particle makes all the collisions it wants, in other words, generates a full set of compound states without being able to escape. T o escape it needs an HQP;and, you will notice that the only interaction that supplies an HQPare the HQp7sin the second term of Equation (20), and, of course, in the width itself. So the numerators associated with the connection are associated to the outside and the eigenvalues of HQQ 1 entirely with the definition of the eigenstates of the A system.

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Richard H . Lemmer

17

Referring to the last question, it is not clear to me that the as's cannot decay-certainly not in the particular channel you have indicated-but since the subspace Q is made up of all states of the A particle system, there will be continuum states in which one of the target particles is going to infinity. That is true; however, that is still a bound state, you see, because of the energy restriction. Remember at the very beginning we said that we are only talking about elastic scattering so that while it is true that HQQwill have continuum eigenvalues, at these energies they will still be bound states. That is to say, you can write a bound state as a linear combination of bound and continuum states. It does not mean that if you have a particle in the continuum that the whole state will decay. It still depends on the energy of the system. If I understand correctly, the denominator of the T-matrix is correct only in first-order perturbation theory. Is that true? In the situation I have just written down, I have neglected any off-diagonal matrix elements between different E,'s. I am not sure if you call that first order or not, but that is what has been done. I have made the single pole approximation. But it is more than that. Well, no; if I say the resonance is isolated then I mean that any connection with other resonances is not important. That is true, but let me put it another way. Even after you have made that approximation, you made the further approximation that +, = a,. No, that is actually an identity. If there are no off-diagonal matrix elements to connect to, there is no way to change a,. The energy is different, but the wave function is the same because the only way that you can change the wave function is to go to another a,, and there is only one state in our example by agreement. The point is if you isolate a state, then the wave function that went with it, perturbed or not, is equal because the wave function changes only in second order and the energy already in first order. All right, so what you are neglecting is the second order contributions which do indeed involve other states. Yes, so I would say it this way:

4,

=

a, + "second order"

eS = Es

+ (a,, WQQa8)+ "second order."

MCVOY:

LEMMER:

Right, what I meant was that one could do bound state calculations just as well this way if one wanted to, but if you were to do so, you would normally not quit after just this degree of approximation. You would actually diagonalize. That is correct.

Intermediate Resonances and Doorway States

THEFIRSTlecture 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 s t r u c t ~ r e . ~For , ~ J this we shall only need a few of the relevant formulae from the previous lecture. The first is the expression for the transition matrix:

The two important aspects of (20) are, first, T,,, is supposed to be slowly varying with energy and, second, the resonance structure comes essentially from the eigenvalues of HQQ.HQQdefines a real 1 particle system with many-body eigenvalue problem of the A eigenfunctions +(A 1) and energy eigenvalues E,:

+

+

The main role played by HQQis that, when the energy of the incident particle is equal to one of the eigenvalues E,, we almost get a singularity in the second term of (20). The real singularity is only prevented by the presence of WQQ.WQQ is an operator which couples the compound nuclear states, as,to the entrance channel and also to is given by each other. WQQ * The material in this section is based on the ideas and procedures developed in H. Feshbach, A. K. Kerman, and R. H. Lemmer, Ann. Phys. (NY) 41,230 (1967).

which can be interpreted reading right to left as follows: HpQ is an interaction which causes a transition from the compound state to the entrance channel. The Green's function is the propagator in the entrance channel, as indicated by the P's appearing on the Hamiltonian. The last HQpcauses transitions from the entrance channel back to the compound states. In the previous lecture we saw that WQQhas an imaginary part due to the itl. This imaginary part allowed us to find a width, r,, for the compound state with resonance energy E, given by In the previous lecture we had written (31) as The only point here is that to evaluate this expression one examines what eigenfunctions of H p p are relevant. The 6 function tells us that only eigenstates of Hpp which have positive energy are relevant. Since H p pprojects onto the ground state of the target with a particle incident, only the $,,(+), which together with bound states of H p p form a complete set of states, are relevant here. Inserting into (29) gives (31). We recall that $o(+) are solutions of the potential scattering problem with outgoing spherical waves. Equation (31), it may be noted, looks very much like Fermi's "golden rule" for the transition probability. It contains an initial state, which is a compound nuclear state, a,, and an interaction, which causes a transition into a final state $o(+), which in this case is a particle moving off to infinity and leaving the target in its ground state. A density of states factor may be missed in (31). Let me just comment that the go(+)are normalized so that ($o(+)(E),J/o(+)(Ef)) = 6(E - E')

(33) and the density of states factor is unity. If we use any other normalization, a density of states factor will appear in (31). One error that might occur is to think that (31) is an approximation. This is not true. This is an exact expression for the width of an isolated level.

21

Richard H. Lemmer

No approximation has been made. It may be impossible to calculate a,, but in principle (31) is exact. One other point I want to bring up, because it also tends to be misleading, is that it is wrong to think that the division into potential scattering and resonance scattering is only true when Tpotis slowly varying. This division is always possible but not always useful. If Tpo,and T,,, vary on the same scale, there is no point in breaking them up. The compound state energies at which resonances will occur and the widths of these resonances are given in terms of the solutions of the many-body problem with Hamiltonian HQQ.Once this problem is solved, we have the many-particle as's which can then be inserted in (31) to find the widths. Of course, the many-body problem does not go away. In fact, it never goes away, unless something very drastic is done to suppress its complexity. The next question I would like to discuss is just that. Namely, how does one possibly extract any useful information out of such a manybody system beyond just writing down these expressions? The qualitative answer to this question is that if we consider not actual cross sections but average cross sections, it turns out that we need to know a great deal less about the system than our equations require us to know. Of course, we learn a great deal less, as might be expected. The one additional point I would like to make to close off the formalism discussion is this. Does the Breit-Wigner formula emerge from all this formalism? The answer is immediate if we take a single isolated resonance. This means that I can invert the Green's function near a particular E, as 1

-la8) -(a81 E- HQQ- WQQ E- E8+

Ian) (an1

C m n+s

where the second term varies slowly in the region around E, and may be approximated by setting E = E,. The second term can then be put into Tpot(making it Tfp0,),and we look only at the rapidly-varying part. Writing out the T-matrix near a particular E,

The energy shift A, comes from the real part of WQQ.As far as the potential scattering waves, $o(*), are concerned, they are related to each other for a given partial wave merely by a phase factor That is, e 2 i 6 p is just the potential scattering S-matrix. This shows that the spatial dependence of #o(+)and $o(-) are the same and merely differ by a phase which can be factored out. Let me at the same t h e write out the S-matrix from (35)

Since Spot is given by Spot= 1 - 2.rriTtpot= e 2 $ p ,

(38)

we see that we can write (30) as

Equation (39) is the prototype Breit-Wigner form for any process. In particular, if we consider inelastic scattering, the width appearing in the denominator of (39) would be larger than the width in the numerator. If you have different entrance and exit channels, you would get the appropriate widths in the numerator. Let me just comment that if we have several resonances, then we would use a sum of BreitWigner forms, one for each E,. If, however, the resonances begin to overlap, i.e., r / D >> I, then we would have to do more than that. We would have to diagonalize HQQ WQQrather than HQQalone. Although that just leads to minor complications, it also looks like a sum of Breit-Wigner forms. However, in this case there is no longer a simple connection between the widths in the numerators and denominators. We now come to the question of intermediate resonance structure and how it relates to the topics under discussion. I would like to approach this question by doing a simple problem, rather than by speaking in generalities. We will see in this fictitious, but very instructive example what is meant by the interpretation of intermediate

+

Richard H . Lernrner

23

resonance structure. I would like to point out that from an experimental point of view the observation of fluctuations in energyaveraged cross sections can be directly tested. There are numerous experiments which do show fluctuations that are much wider and are also spaced at much larger intervals than compound states. These fluctuations have been dubbed intermediate resonance structure. But it is an entirely different matter to say whether the interpretation of this structure is correct in terms of the ideas about to be presented; however, I think that I can perhaps demonstrate that the approach used is one way of thinking of these fluctuations. The story starts with examination of the compound states. I would rather like to conduct this tour by pointing out the difficulties before showing the simple facets. The first difficulty is that the a, is an A 1 particle wave function. There is nothing we can do about that. As it stands a, is just as impossible to find as the original wave function for the scattering problem. But do we really need to know the detailed motion of 200-odd particles in order to say anything about average compound state widths? Apparently not. We see this as follows. We assume that the shell model plus residual interactions is a good representation of the nucleus. Now we let a particle come in on our nucleus. Since I have represented the nucleus by a model which consists of a potential well, the H p p part of our problem, this part just contributes to the potential scattering. This occurs because the potential well is a one-body potential, which does not change the state of a single particle other than just scatter it. The effect of the well alone can never be to absorb the particle because that would not conserve energy. The residual interactions, which are not accounted for by the one-body potential, are the interactions between the incident particle and all of the other nucleons. These interactions are responsible for exciting the internal degrees of freedom of the nucleus. Again in the shell model this would mean that the two-body force creates a hole in the Fermi sea by the excitation of a particle to an unoccupied single-particle state, to a first approximation. Simultaneously the incident particle is degraded in energy. If the incident particle falls into the well (see Fig. 4), one has a quasi-bound state. In fact, we have a very peculiar state. This is best seen in an energy diagram (Fig. 5). If the excitation energy for the particle-hole pair is larger than the positive energy of the bombarding particle, then one gets a

+

effect of two body interaction

0 X

- hole - particle

Figure 4

quasi-bound state that is above the threshold for particle emission, which in this example is 2p-lh state. We have an unstable level in the sense that it has enough energy for particle emission, but none of its particles can escape because both particles are bound. The answer to

Ed

-------------- -

threshold

ground state Figure 5

25

Richard H. Lemmer

the paradox is, of course, that I have neglected the residual interaction. The two-body force can now "strike again." One possible result is a return to the initial situation, which would correspond to the decay of the 2p-lh state. We can see then that the quasi-bound 2p-lh state, at an energy above particle threshold for this model nucleus, is a state in which the energy has been shared between the particles in such a way that neither has sufficient energy to escape. The decay of the 2p-lh state requires an additional interaction to throw one of the particles out. Of course, the incident particle may not go into a bound orbit but rather fall into a quasi-bound orbit itself; in that case the width of the resonance will be much larger because the quasi-bound orbit could decay without the residual interaction. We see from this simple picture that the dynamics of a resonance which excites internal degrees of freedom, is quite different from a single-particle resonance. The latter is an interference effect because of the size of the system, whereas the resonance we are discussing is definitely a dynamic effect associated with the transfer of energy to the target. We know that the particles in a nucleus are not moving independently but are interacting with each other quite strongly. The presence of this interaction means that the 2p-lh state is not an eigenstate of the nuclear Hamiltonian. Rather, the 2p-lh is just one excitation, which is then spread by the residual interaction over many other excitations; the other excitations are formed from the 2p-lh state by the two-body force which creates other particle-hole pairs. This can be thought of as a temporal sequence in which various classes of states are formed in succession, i.e., 2p-lh -t 3p-2h -+ 4p-3h . . etc. What we are really saying is that, if we start with a particle outside the nucleus, in order for it to get into the nucleus, it has to get a handle on the excited states. The only handle that it has, because of the assumed two-body nature of the interaction, is through the 2p-lh component. The initial state, where there was no excitation, consists simply of one particle and no holes. Now this interaction, which has only two "hands," can only lift one particle out of the Fermi sea, while simultaneously dropping the incident particle into one of the unoccupied orbits. However, once the interaction has done this, it can now set to work creating other particle-hole pairs. This tells us that the compound states, G,, can be described in this model simply as linear combina-

.

tions of the various multi-particle, multi-hole states. For purposes of discussion, let me assume that there is only one 2p-lh state of importance. That means there is only one excitation at energy, Ed, which has a simple nature. All other such simple excitations are assumed to be far away in energy. This is probably not realizable in actual nuclei, but it will illustrate the point. If we call the wave function for the simple excitation \kd, it will be present with some amplitude in the compound state (P,

where a, is the wave function describing all more complicated excitations. The amplitude asdwill in general be very small because the (Pis will take the strength out of \kd. What has been done in (40) is to push the projection one step further. The projection operator Q has been further subdivided into a projection onto the simple excitation d, and a projection onto the more complicated excitations q. The a, are just eigenfunctions of H,, in our notation. One could perhaps imagine solving this eigenvalue problem. However, this is just as complicated as the original problem. It does not appear that we have made much progress, but that is not quite true. The reason it is not quite true will now become obvious. The part of the compound state that was isolated in (40) is, in fact, the only link between the incident channel and the more complicated excitations and so plays a central role. We can see why all this is so important if we look at the expression for the widths of the compound states. We continue to think in terms of the model. Now in our model the entrance channel is a single particle with the target in its ground state. Because of the two-body nature of the interaction, when we calculate the width we find The width of the compound states in the vicinity of the simple 2p-lh excitation is given by a product of two factors, each of which has a simple interpretation (see Ref. 6). The factor laSdl2measures the probability that the simple excitation will be present in the state at energy E,. The second factor is independent of s and can be interpreted as the width of decay of the simple excitation as though it were

Richard H . Lemmer

27

the only excitation ever formed. This we can call the escape width from the simple excitation rd1, The widths of the compound states are given by

Of course, all we have really done is shift our ignorance from a, to laSdj2. We still do not know (aSdl2. However, as we will shortly see, we can do something with laSdl2 that we could never do with a,. This is where the question of the importance of intermediate structure will arise. We see from (43) as well just why compound resonance widths are so small. It is because the amount of admixture of the simple excitation can be quite small. It is not because the interaction is necessarily weak. (The shell model gives numbers for rdT in the range of 100 to 800 keV. To get down to the few eV widths observed experimentally, quite small. This is qualitatively what we expect we must make laSdl2 for the complicated compound nuclear states. Therefore, the Bohr picture of a compound state as being very complicated is still as valid as ever.) We might now ask, is laSdl2 always small? We know, for example, that in light nuclei the density of states is not as high as in heavier nuclei. Also the widths are generally larger. So that in light nuclei, we =: 1; that is to say, that in light nuclei the might suspect that laSdl2 first excitation that can be made is the only excitation that is ever going to be made. The other classes of excitation are so far removed in energy, because of the small size of the system, that they will not count. This is rather a rough subdivision, but let us take it as a working hypothesis. For heavy nuclei we know experimentally that the compound resonances are very narrow and there are very many of them. Therefore, we would expect that la,dI2 > D. However, let us take the critical case Vd = D. Then one gets an estimate rdl = 2rVd = 2rD. The average spacing of observed levels in a nucleus like 'ONe at an excitation of 12 to 13 MeV is something like 100 keV. This gives r d l =: 600 keV. However, just counting levels from nuclear data sheets is entirely wrong because of the fact that the total Hamiltonian conserves angular momentum and parity. So, in fact, the relevant density of states are for those that have the same spin, parity, and isospin. It turns out that in the few cases we have investigated thus far that the parity restriction, especially, has a large effect on the level density. This will be illustrated more clearly in connection with the isobaric analog states, which "live" entirely on the fact that there is a strong selection rule operating. The isospin

Richard H . Lemmer

35

selection rule demands that only states with the same isospin can couple to the analog states. The last topic I want to discuss in this lecture relates to the question of intermediate structure itself. What I have illustrated thus far is that under certain circumstances, when the damping width is not too large, we expect some concentration of the original doorway state strength near the energy of the uncoupled state. That is our interpretation of an intermediate resonance.' The whole question of intermediate structure then hinges on the magnitude of the damping width. I will now show that, if the damping width is not too large, there will be a corresponding resonance in the cross section. If the damping width is large, no resonance will appear in the cross section. However, in terms of a formalism it is obviously awkward to average a cross section because the cross section depends on the square of the T-matrix, a non-linear expression. It is much easier to average linear expressions. Therefore we make a linear average of the T-matrix, recognizing full well that by so doing we get into the old question of fluctuating versus non-fluctuating contributions to the average cross section. We will consider this problem at the appropriate point. To find the average cross section, I will again use the identity (44). This identity is just as useful here as it was for finding laSdl2. The average of the first part of the T-matrix, T,,, will not be very different, provided the average is taken over an energy interval, I, in which Tpot does not vary appreciably. What I want to do is make an average over the interval, I, which contains many compound states, a,, but at the same time is much smaller than the spacing between doorway states. In that way I see the envelope of Figure 8 as a continuous curve, rather than a very discontinuous series of sharp resonances. In order to do that I have to take averages of the resonance parts of the T-matrix, which look like

The operator in (60) looks very much like the one used in the identity (44) except for the appearance of the additional WQQ.Since the best that the entrance channel can do is couple to the doorway state, we can rewrite (60) as

and for the same reason

Because of Equation (62) any operator on jection q gives zero identically

WQQ

containing the pro-

Equation (63) tells us that W Q Q only enters as an additional interaction between doorway states that can be added onto Hdd in the identity ( 4 4 , which now reads d

1

E-

HQQ

-

1

d=d WQQ

E

- Hdd -

Wdd

- Hdq

1

d , (64)

E - Hqq

By use of (64), Equation (63) becomes 1

E - Hdd -

Wdd

-Hdq E-

1 H q qH q d

Equation (66) has a very simple functional dependence on energy. The numerator is independent of energy, provided we neglect the energy dependence of the potential phase shift, while the denominators give rise to poles below the real axis. Such an expression can be averaged quite easily using a Lorentz weighting factor. It is accomplished simply by replacing E by E il, where I is the averaging interval. The average transition amplitude is thus given simply by

+

Let me call the last term in the denominator

Wddl

Richard H . Lemmer

37

If I again assume only one doorway state, I can find (T,,,)I by taking the diagonal matrix elements of the denominator in (67). The diagonal matrix element of Wddl has the form

where A d l and

rd'

are given by

The matrix elements appearing in (70) and (71) are the 1 V d q l 2 defined previously although here they are not assumed to be constant. If I now assume that the E, are very dense, (71) becomes rdl

= 2 ~v1d q 1 2 ~ Q3

(72)

where the bar indicates average. This is analogous to the form given by the simple model, but the constant matrix element has been replaced by an average. The average T-matrix is now given by

The numerator of the second term we have previously associated with the escape width of the doorway state (42). A d f is the energy shift resulting from coupling with the continuum. Adl is the shift resulting from coupling with more complicated excitations. The total width r d is given by where the rdf and rd1 are the imaginary contributions of the coupling to the continuum and more complicated excitations. In this last result is summarized the whole idea of intermediate resonance structure. It tells us that if is not too large, the presence

of a doorway state will be reflected as a resonance in the average cross section. On the other hand, if rdis too large, this will not happen. The doorway state will just give a constant compound elastic contribution to the average cross section. The important feature is that the doorway state, its width for decay to the continuum, and its width for decay to more complicated excitations are quantities that can only be described in an average sense. The doorway state is not an eigenstate of the system. It only has meaning in the average cross section. There are no doorway resonances in actual cross sections. They only appear in the average cross section under the conditions I have stated. I want to make one final comment, which is an obvious, but very important one; that is, the fact that we are dealing with a Hamiltonian that conserves angular momentum and parity means that the doorway state can also be characterized by its angular momentum and parity. Likewise, all states to which the doorway states couple have the same JT.So that, in fact, the resonances which occur in the average cross section are not necessarily present in all channels but rather in particular channels labeled by the J r of the doorway state, which have been coupled to other compound states of the same Jr. DISCUSSION

McVoy:

LEMMER:

MCVOY :

LEMMER:

MCVOY :

LEMMER: MCVOY :

Have you in any practical applications checked to see whether or not the potential phase shift, 6, of the factor e 2 i 6 ~is, real? That is up to you completely. Certainly in the original problem it must be real. There is no way for it to be anything else but real because it does not involve any inelastic scattering at all. It sort of depends on what your P's and Q's are and how much of the background you have put into TPot. That is right. Whether it is real or not is unimportant because all we want is a connection between +,,(-) and +o(+). This will always be given by the S-matrix of the Hpp channel. Yes, of course. I was asking about the practical question of what the background phase shift on which the resonance sits will be. That is a problem which, to begin with, cannot be answered by the formalism at all. Actually, in one model case where this can be checked,

Richard H . Lemrner

39

where you use an optical model with a complex potential, the background phase is very far from real. In the optical model that is true. However, the formula, Equation (39), is not valid for the optical model. This is a true compound resonance. No averaging has been done, and there can be no reference made at this stage to an optical model. That is true, but we can think of this as one channel where there are several open channels. That is, it does not have to be interpreted as an average. In that case the S-matrix for the single channel is not necessarily unitary. Yes, that can happen, but it will not destroy the form of Equation (39). You also have to specify unitarity in that case explicitly. That is correct. However, we will end up with several relations between the widths in the numerator and denominator to keep it unitary. You say there are a large number of cases where people experimentally see intermediate resonance structure. To what extent have these been successfully fitted with BreitWigner shapes? This has been quite successful for the analog states. I do not know how well the structure is fitted by Breit-Wigner shapes in cases where analog states are not dealt with. However, I have a slide which shows not a fit but comparison with a calculation (Fig. 6). The case that you call trivial intermediate structure is not really intermediate at all. It is the actual resonance. That is why I call it trivial. It is identical with the actual resonance which is an eigenstate of the nuclear Hamiltonian. This scheme for classifying doorway states is excitation energy dependent, isn't it? If we go to higher excitation energy in light nuclei, the resonances are no longer trivial. That is not always right either. The reason it is not right is due to selection rules. For example, the T = 2 levels in "Q at 25 MeV have very little to mix into because of this isotopic spin. States with T = 0 at this excitation energy will undergo much more mixing on the other hand. In order to put things into perspective, it is perhaps worth pointing out that this treatment is just Lane, Thomas, and Wigner all over again but on a lower level. Your essential result, that l?dl is in the "golden rule" form, comes about precisely because of your approximation that the states, a,, are closely spaced. You get the "golden rule" automati-

cally by coupling to a continuum. What you have put in by your averaging is, in essence, the creation of a fictitious continuum of states to which your doorway states decay. It is very interesting because it is precisely that which in the giant resonance context is called the width for compound nucleus formation. As you point out, if that width gets large compared to rdT-which in other contexts is described as the entrance channel width, if you think of this as a two-channel problem-and if the reaction channel widths become large compared to the entrance channel widths, then the maximum in the cross section, which is the ratio of the entrance channel width to the total width, will be a very small fraction of the unitaritylimit. Consequently the maximum will not be seen in the cross section. It is precisely that vanishing of resonances which causes the giant resonances not to be seen. If that fraction turns out to be a large one, then intermediate resonances will be seen, and if it is small, they will not be seen. I might point out that it is really quite fascinating to think about why that continuum should be there. There is a very physical reason for it. When we look at the numerator of your average T-matrix, the width in the denominator is different from that of the numerator despite the fact that we are supposedly dealing with a one-channel problem. In other words, the width in the numerator is just rdf, while in the denominator we have rd= r d T r d l . r d l does, in fact, describe a physical channel in the following sense. Its threshold is identical with the other channel rdT . SOthey cannot be distinguished in energy the way normal channels are distinguished. However, the rdlchannel only opens physically when energy averages are done. The reason is that, if we look with fine enough energy resolution to resolve the fine structure, we are guaranteed that the wave packet is very poorly defined in space. Therefore, its arrival time is large compared to the decay time of the compound states. Your time resolution is so bad that you cannot tell the delayed from the direct channels. Therefore, the two channels are not distinguished. It is only when the energy resolution is spoiled by energy averaging that the two channels can be distinguished by good time resolution. Then the second channel does open. For people who are interested in the analyticity of scattering amplitudes, there is a very amusing aspect of

+

Richard H . Lemmer

LEMMER:

McVoy:

41

this subject. If we think in terms of the complex energy plane and ask what we have been discussing looks like in terms of the position of the poles, the answer is simply that the poles are distributed along a Lorentzian curve (a) in Figure 9. We can see this immediately from which says that the imaginary parts of the compound resonance energies are distributed in energy like laSdl2. Except for one-the one that is feeding them. That one sits at position (b). In the limit of zero coupling, the compound states are bound states in the continuum with zero width, and the width is all in the doorway state. As the coupling begins, the compound state poles move down off the real axis and the doorway state pole moves up. Finally, the situation where they have met is reached.

branch cuts

Re E

* 4*+ +

++++*+ (a)

+ +t

++

+.I.

++

+++

+ +

coupling

Figure 9

Then there is no longer any strength left in the doorway state to draw upon, but as the coupling increases, the range of states which are coupled increases. In the limit of very strong coupling the poles are again very near the real . axis and thus have small widths. The thing I wanted to point out is that if we look at the sheet of energy surface that we get to by going through the elastic cut, we will see just the poles which give rise to the compound state resonances. However, because of the quasi-continuum you have introduced in your energy average, there is a second cut whose threshold is at precisely the same place. If we go through that cut, we will not find this distribution of poles. In fact, the way we get to this cut is by energy averaging, which, as you pointed out, shifts us into the upper half plane by a distance I, so we are farther away from the poles and do not see all the details. If we now come down from the energy averaging through the cut corresponding to the quasi-continuum, we find a pole that would correspond in the optical model case to the optical pole, that is, the single particle pole which describes the distribution of strength; in this case it makes the whole distribution of states act like a single state. That pole is an honest to goodness pole, which sits on the sheet that we get to by going through the quasicontinuum cut. If we go through the elastic cut, we see the actual poles in which case we see the fine structure. These are all very real things so they have to be taken seriously. They have a real physical background. The question of the relationship between the results obtained by the different formalisms has been raised a number of times. I think this is the proper time to discuss that relation in an open conversation. There are a couple of interesting points. I do not quite see all the way through it, but perhaps we can grope at it. It is interesting to note that you got to essentially the same results without explicitly making the assumption that the mean free path is long. You got to the doorway state concept without assuming that the particle had to be trapped in its first collision. Furthermore, you got a result, which looks like perturbation theory, the "golden rule" form for the widths, without explicitly using perturbation theory. The questions are how do we get to the doorway state concept without the assumption of single scattering and then how do we get the result of perturbation theory without explicitly making a perturbation expansion? I think the ---

RODBERG:

Richard H . Lemrner

culprit is the projection operator decomposition Q = d q. Somewhere the transition is made from the exact state a,, which is an exact eigenstate of the total Hamiltonian, to the shell model eigenfunctions. Are the states projected out by d single particle shell model eigenfunctions without configuration mixing? Q was supposed to be a projection onto the true eigenfunctions of the projected Hamiltonian; d looks like the projection onto simple shell model states where the residual interaction is neglected and that is why we get perturbation theory results. No, that is wrong. First of all, \kdis a doorway state. This statement has no other connotation than that \kd is some simple excitation that is directly connected with the entrance channel. What the specific representation of \kd might be depends on the nucleus in question. It could be single particle in the sense of a 2p-lh type state. However, we do not mean a single 2p-lh configuration. That is to say, 2p-lh means a class of configurations. We would allow all 2p-lh states to couple to each other but not with deeper states. In the shell model qdwould be a linear combination of all 2p-lh states which strongly couple to each other. For example, if we choose the 1- state in 016 as a doorway state, we would have a linear superposition of five p-h states. However, we could have a situation where a particle plus a vibration or three quasi-particles interacting acted as the doorway state, but always interacting. The question, "Is XPd a simple shell-model state?" has the answer, "Not necessarily." It is merely the doorway state described in whatever representation was appropriate for the entrance channel. Might there not be a piece of that in the P space? You divided the total Hamiltonian into Hpp and HQQa while ago. The ground state of the target might have some of that configuration admixed. Not if P and Q are chosen orthogonal. What I am saying is that you are not really dividing up HQQin a mathematically rigorous way. You are using physical intuition. If you try to do this with mathematical rigor, it might turn out that you cannot get a particular configuration entirely into the Q space. A particular configuration might be admixed into the ground state of the target. So, you may be jumping over into your P space. Yes, but that is entirely a problem of being able to construct the target states. Those states define the projection

+

LEMMER:

GRIFFIN:

LEMMER: GRIFFIN:

LEMMER:

43

operator P. P in the non-anti-symmetrized version is just P = 40) (boywhere &,is the target ground state wave function. The question is then whether we assume 40is a simple shell-model state or some admixture. However, this is irrelevant because we can always define the projection operator. The more complicated bo is, the more complicated P becomes. It is just a matter of how hard we are willing to work to construct the target states. The other question is, why do you get the result that looks like first order perturbation theory? It is simply because we are effectively describing the decay of the doorway state into a continuum of states because we have averaged over E,. It is not a first order perturbation theory result. He is not using unperturbed wave functions. He is using exact wave functions. That is the whole trick. If someone were to look at that result, he would calculate it in the same way as Lane, Thomas, and Wigner did it, which is first order perturbation theory and wrong. Look, the @, is a many-body wave function. Now if we want to make further approximations to the @, which, of course, we would want to do, then we have an approximate result. The connection between our results, I think, is

(a,, Hpd\kd)= (Y complicated, TeYd) (Lemmer)

(Rodberg)

The thing that was confusing me was that T, is a sum of an infinite series of Y's. What you are saying is that your sum of this infinite series is the correlations of all the particles, which is built into a,. I believe that this is the connection between the two formalisms. In the example you gave of 160we , could in principle have 2p-2h correlations built into the ground state. The doorways would not be exactly 2p-lh doorways but a combination of 2p-lh and 3p-2h states. Yes, that is right. We could. If we do start with a correlated target ground state, how do we then define the doorway state? If we operate on the correlated state with a two-particle interaction, we generate a class of states which is a subset of all the possible excited states. It is the first step we can take from the ground state. These are the doorway states. Can you get any estimate of Vpd?

Richard H . Lemmer LEMMER:

45

The only reliable way to get that is experimentally. We could make a model, of course. However, all the uncertainty of nuclear models enters very strongly in what we use for $,. The reliability of this procedure is questionable. The only place it works well enough to be believed is in isobaric analogs.

Applications

AFTERALL 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 impossible by manipulating formalism alone to solve a manybody problem. It will always come back and bite us at some point, and the point where it bites us here is in asking exactly what nuclear dynamics have to enter the model. The formalism that gives all these nice expressions for the cross sections contains the widths and the resonance energies. The widths and energies are simply stand-in parameters for the underlying nuclear dynamics that determines those parameters. To get anywhere beyond just parameterizing cross sections with such formulae, we have to try to do a calculation, and I think that it would be useful to list what has been done so far. The first calculation, which was given in the second lecture, is C12 plus a neutron by Lovas at Nordita (see Fig. 6). It is, I think, rather an impressive result for a relatively small amount of effort. A second calculation of a doorway state type of system has been carried out for A = 15, that is to say, 15Nplus a proton or 15Nplus a neutron (Fig. 10). This was done by the group at the Massachusetts Institute of Technology, taking excited states which are just the particle-hole state^.^ In fact, this was one of the first computations of this type which suggested that the sorts of width we get for the calculated doorway states seem to be in reasonable agreement with the sort of

47

Richard H. Lemmer

Figure 10

resonance width seen experimentally. Both of these I will characterize as pure shell-model calculations. They are examples of what I called trivial intermediate resonance structure. They do, in effect, the shellmodel calculations that would have been done anyway, and at the end they calculate one more quantity, namely, the escape width. This is similar to what we would do for a low-lying state in the nucleus, i.e., calculate its radioactive decay by beta emission or gamma rays, using some shell-model wave functions. So, in fact, no matter how much formalism we invent-what we really do does not depend primarily on the formalism. Then we have also looked at the case of 19Fplus neutrons. This is the Ph.D. thesis1° of Mr. Afnan at M.I.T. and is a little bit more exotic than a pure shell model, in that it is a rotating shell model. What

Afnan has done is to recognize the fact that in the vicinity of A = 20, there are probably deformations developing. There is sufficient experimental evidence to support this. In fact, a picture involving just particle-hole excitations in a spherically symmetric well would not be particularly accurate for the intermediate states. Instead, we add deformation of the core; i.e., instead of considering a spherical core out of which particles and holes have been excited, we squeeze it so that it becomes deformed. Now particles and holes are excited in this deformed potential. Deforming the average nuclear field merely splits each of the single-particle levels in a characteristic way. Instead of getting a single level for each particle-hole state, we get several components because of deformation. What we have in 19F n is a deformed target with an incident neutron. This neutron can do two things: it can just spin the whole nucleus, that is to say, give rotational states for the target or it can excite the particle-hole pairs in that deformed field. The answers that come out are again relatively encouraging, and Figure 11 shows the result of this computation. As Figure 11 indicates, the solid curve is the experimentally measured total neutron cross section of lgF showing four or perhaps five bumps that have some fine structure sitting on them. So this, in a sense, is an average cross section that has not quite averaged out all the substructure in the intermediate structure. The dotted curve shows what comes out of a particle-hole excitation calculation in the deformed scheme without adjusting any parameters except one energy. The first calculated resonance has been placed at the position of the experimental reso-

+

I

I

Neutron Totol Cross-Section on F"

---

Calculated

- Eaperimental

1

0.50

1

1.OO

1

1.50

1

2.00

1

250 E in MeV

Figure 11

1

1

3.00

3 50

1

400

~

450

~

Richard H. Lemmer

49

nance near 600 keV. The other parameters of the model are taken from G . E. Brown's 160forcell and the results of a Hartree-Fock calculation for the particle-hole energies.12To be fair about the comparison with the data, we should really average out more of the fine structure. The point I want to make is that for this kind of business, the result is very impressive. First of all, to get anywhere near the right kind of energy level separation is already encouraging. Even beyond that, to have associated with those energy levels a width that resembles somewhat the experimental widths is quite impressive. I think that this result can be considered, in the context of the type of approximation we are making, as a reasonable vindication of both the model and the interpretation in this case. However, we do not have enough data yet beyond the total cross section and the (p, r) cross section to say much more than that. We know very little about the properties of these states. Furthermore, we really have no explanation why those resonances look so equally spaced. We draw some comfort from the relatively uniform spacing of levels in a deformed oscillator potential. However, the true potential probably does not have such uniformly spaced levels. Incidentally there is no unique way of selecting the levels that we get by the calculation to compare with the data. There are actually a large number of calculated levels around this energy region, and any slight shift in a particular particlehole energy could bring a completely different set of levels into association with these experimental anomalies. We also calculated the lgF(p,r) cross section (Fig. 12). This is a typical sort of calculated cross section, which does not look in any way like the experimental cross section. It shows in this case that the wave functions used to give the particle escape width do not seem to have even any qualitative connection with the experiment if you use them simultaneously to try to get the gamma ray escape width. We might be able to change the situation by adjusting parameters. That is certainly a possibility; however, it is not in the spirit of the computation to do that. What I want to emphasize here is not any exceptionally good or poor agreement with the experiment but rather the sort of qualitative results that you get when you do a shell-model calculation in the continuum. These calculations are in light nuclei and are associated with T = 1 states. Correspondingly, they are essentially analog states of the type that I will discuss a little later.

E (MeV) Figure 12

Another computation13that has been tried was "Ni plus neutrons, where the doorway states were not represented as particle-hole excitations. Here because of the well-known phonon-like spectra of 60Niin particular a O+ ground state, a 2+ excited state and a possible O+, 2+, 4+ triplet at higher energy-the attempt was made to describe the doorway state as a particle plus a vibration, where the vibration is described macroscopically as some surface oscillation. What we imagine to be happening is the neutron comes in, hitting the surface of the 60Niwhich then starts to vibrate; at the same time the neutron is captured. The resonance is again a bound state in the continuum and consists of phonon excitation in the target with the neutron dropping into some single-particle state in the average field. It is

Richard H. Lemmer

51

completely the same spirit as the other calculations, but now it is more parameterized in the sense that the interaction between the single particle and the surface is a parameter that can be obtained in various ways. The important point of this calculation is that vibrations plus particles, plus particle-surface interactions of the sort proposed by Bohr and Mottelson in 1953, give widths of doorway states that range from 1 to 100 keV at an excitation energy of about 100 keV above the threshold for neutron emission. The large spread in values of the widths depends specifically on the total angular momentum and also on the particular coupling of the wave function. What we are doing is coupling the particle to the 1-phonon and the 2-phonon states, and each doorway state is a linear combination of such states. The wave function can become quite complicated and correspondingly the widths quite small. In particular, we learn that at this low excitation energy the vibrations do not produce significantly larger widths than those of two-particle, one-hole states. In fact, in this same nucleus, we also look at what representative twoparticle, one-hole configurations might give us. We find essentially the same range of widths. So it appears to be impossible to distinguish between coupling to a vibration and coupling just to two-particle, one-hole states in this particular case. However, I do not think that this statement can be generalized at this point. Recently many people have been involved in the question of analog states which form a subject by themselves. The reason for this is that although they are excitations at relatively high excitation energies in a compound nucleus, they do not admix very strongly into the states that surround them. The reason for the small mixing is the approximate conservation of the isotopic spin. The observation of actual resonances in (p,p) cross sections by the Florida State group led Robson14to a compound nucleus theory of analog states which, when averaged, is very much in the same spirit of what we have been discussing. I will not discuss Robson's approach because it is quite similar to what we have already seen, and also there are somewhat simpler presentations that contain essentially the same physics. What I would like to do is to discuss a set of coupled equations,15which is a very simple parametrization of a rather complicated problem. I would like to discuss both quantitative and qualitative aspects of the results that we get from these equations.

Figure 13

I would like to describe typical analog states first?6 Figure 13 is just an elastic proton cross section. It looks pretty much like any other cross section; notice that there are several prominent resonances. I do not recall what the resolution was in this case. It is certainly not high enough to display anything beyond these particularly

Richard H . Lemmer

53

prominent resonances. If these are true intermediate structure resonances, then they should, among other things, show angular distributions that are characteristic of a resonance with a good angular momentum and parity. Consequently, if we change the lab angle and go to angles at which the partial wave of the resonant orbital angular momentum vanishes, we should see suppression or even total disappearance of the resonance. The first resonance in Figure 13 is known from other sources to be a 1 = 3 resonance that shows up at 170°. If we now look at 140.8O, which angle should suppress 1 = 3 resonances, we find indeed that this resonance disappears. Consequently, the theoretical prediction that the anomaly has a definite angular momentum is borne out quite strongly by experiment in this particular example. We also know in this example that all of these resonances have odd angular momentum. Therefore, we expect that the cross section at 90' should be quite smooth because all odd angular momentum resonances are suppressed at this angle, and this indeed happens. This example shows that the orbital angular momentum assignments to these resonances are extremely important even if we understand nothing else about them. Let me discuss in a very simple manner what underlies the description of such resonances. The basic idea is that, if we have a proton incident on a neutron-rich target nucleus with 2-component of isotopic spin N-Z To = 2

'

the proton will just bounce off the system mainly because of the Coulomb field it sees. Alternatively, it can make either proton-proton hole or neutron-neutron hole excitations. However, it is known that low-lying excitations of these types do not change the isotopic spin. Consequently, such excitations would have the same isotopic spin as the ground state of the compound nucleus. We expect very many such states at the proton separation energy, which implies a very large damping width. In contrast to this, there is a unique type of transition that can take place in which the isotopic spin changes. Consequently, the resonance that forms lives as an intermediate resonance for quite a long time because of isotopic spin conservation. This sort of excitation comes about essentially via a charge exchange reaction.17 Instead

n

P Target in ground state

Target in analog state

(4

(a) Figure 14

Figure 14

of the proton being captured as a proton, it is turned into a neutron in some unfilled bound single particle orbit and simultaneously excites a neutron hole-proton particle as depicted schematically in Figures 14a and 14b. We end up again with a 2p-lh state. However, it consists of a proton and neutron with a neutron hole. This charge exchange process leads to a state of one unit higher isotopic spin than the ground state. If we designate the Z component of isospin for a proton as -%, the situation can be depicted as in Figure 15. The ground state of the compound nucleus has isotopic spin and Z component To - %. The excited state has the same Z component, but since there is both a neutron and a neutron hole, the isotopic spin is one unit higher than the ground state. This state is the analog of the ground or some excited state of a nucleus which has the same isotopic spin, To %, but with Z component also To f This just

+

x.

Richard H.Lemmer

T = To

+

1/2

-

T, = TO 1/2 many states

f

T = To

- 1/2

T = To

- 1/2

analog of this state

compound nucleus mound state

T, = To

- 1/2

Figure 15

means that in the analog resonance a neutron has been changed into a proton. A simple way of describing this situation becomes apparent if we consider a target in which both neutron and proton shells are closed. The operation of the charge-raising operator is simply to move a neutron over to the corresponding unoccupied proton single-particle level leaving a neutron hole. This state would be the analog of the target ground state. They are connected to one another simply by the

interchange of a proton for a neutron. In describing the scattering of protons by the target we may again write the wave function as a sum of expansion coefficients multiplied by the wave functions of the target and of the analog of the target:

*=

UnI$.(A)= UO@)I$O+Ua(n)&,+ "the rest."

(76)

n

Of course anti-symmetrization has been neglected, and the ground state of the target nucleus is denoted by &. The other compound states in which the isotopic spin does not change are uninteresting for the moment and are thrown in with "the rest." The only other important state is the analog state of the target which is denoted 4,. This is precisely the same sort of description that has been used for all other doorway states. For simplicity, let me suppose the ground state of the target is a Of state. With this ground state I can associate an average field in which the incident proton travels. Normally there will be many target states which can be excited while simultaneously dropping the proton into some unfilled single-particle state. However, these states are not analogs; they are simply excitations of the target associated with a proton in a single-particleorbit. In contrast with this the analog of the target ground state can be thought of as being associated with a system in which the proton has become a neutron. The analog of the target ground state lies in a different nucleus, of course, but it has the same angular momentum and parity as the ground state of the target. This is denoted cba(A);the Uo(p) and Ua(n)are the single-particle proton and neutron amplitudes associated with the target ground state and its analog, respectively. Of course, there is a difference in energy between the target ground state and its analog. In our simple model it is convenient to ignore the non-analog resonances included in "the rest." Equivalently, we could include their effect by using some sort of optical potential. In either case, we end up with a simple system of two coupled equations, which couple the proton amplitude associated with the target in the ground state to the neutron amplitude associated with the analog of the target ground state. The resonance is associated with the neutron in a single-particle orbital around the analog of the target. This is the basic description of the isotopic analog resonance, which leads to a set of two coupled equations. We

Richard H . Lemmer

57

realize that they depend only on the interaction of the incident proton with the neutron excess because that is the only interaction which can cause a neutron-proton switch. Moreover, this interaction must contain a term of the form

in order to allow the possibility of charge exchange. The interaction may have other dependences on spin and space variables as indicated in (77). Inserting this two-body force as the interaction responsible for absorbing the proton and ejecting the neutron, while at the same time allowing the proton and neutron to interact with the average fields, gives rise to the coupled equations. This set of coupled equations is easy to derive, but I do not propose to go through the derivation. I will only point out that in our simple picture the important parameter is the difference in energy between the ground state of the target and its analog, A. This energy difference is related precisely to the interaction energy of the single proton with the Coulomb force of the other protons in the target. The reason for this is obvious. If I took a neutron in the target and changed it into a proton without allowing this proton to interact with all other protons, the energy of the target and its analog would be identical provided the nuclear force was completely charge independent. However, when the neutron is changed into a proton, it feels the Coulomb interaction of the other protons. Consequently, we expect the energy of the analog to lie higher in energy by an amount equal to the Coulomb interaction of this single proton with the other protons in the target. In addition, we know that the neutron and proton do not quite have the same mass and a correction must be made for this mass difference. This gives A. The analog resonance can thus be pictured as being formed in the following way: we excite the analog state of the target by an amount A and simultaneously drop the neutron back into a neutron orbit bound by an amount, c,. I have added energy A but regained an amount en. In the non-interacting approximation, the resonance energy would be simply

Incident Proton Energy (MeV)

Figure 16

The results of one attemptla at solving such a set of coupled equations for 8sSrplus protons are shown in Figure 16. A discussion of the parameters involved in this fit may be found in Ref. 18. In conclusion I would comment that the analog states are our best examples of intermediate resonance structure. This point was first emphasized by Fallieros17 in 1964. He pointed out that the analog states were a prime example of what we had been calling intermediate resonance structure without, previously, having an example to show of it. The probable reason for their existence is that they have a good isotopic spin to a very good approximation. Estimates of the spreading width have been made by de Toledo Piza, Kerman, Fallieros, and Venter.lg If we assume that the spreading is totally due to the breakdown of charge independence, it is an extremely small effect. The lgFand the 15Ndata both are presumed to show resonances of a T = 1 nature. Their damping just has to be small from the point of view that there is not a large number of states of the same symmetry surrounding them. There is not, to date, a single example of an intermediate resonance structure that "should have died but did not." That is, a state which had the same isotopic spin as the ground state buried in a high den-

Richard H.Lemmer

59

sity of levels surrounding it. No example exists of a state which could have coupled to many states but instead only coupled over a local energy region and appeared as a resonance. From an experimental point of view, it would be extremely interesting to find such examples, particularly ones which could be analyzed in terms of the underlying structure using a shell model or the unified model description. It would be valuable to point to cases where the intermediate resonance theory worked out in practice without the help of a good isotopic spin quantum number. I think the kind of experiment that would do this would be (r, n) on heavy nuclei. In light nuclei the gamma ray always populates the T = 1 because of the equivalence of N and 2. In a heavy nucleus, the dominant excitations caused by an incoming gamma ray would be either excitation of a proton into a region where it would be excluded were it a neutron, or likewise, a neutron excitation to some unoccupied single-particle state. Both of these types of excitations, because of the inequality of N and 2, lead to states of exactly the same isotopic spin as the ground state. So we will, of necessity, be populating states with the same isotopic spin as the ground state. We will, therefore, remove the isotopic spin "life belt" from the resonances that are formed. I think such experiments are really going to lay down the ground rules for the interpretation of intermediate structure of the type that we called true intermediate resonance structure. DISCUSSION

SETH: LEMMER:

SETH: LEMMER:

I have always wondered why in optical models it has not been found necessary to introduce the term - ;z.How can one avoid not using it? It is because the 2; term affects not the depth of the potential but its average value. In this case, because VIz (Eq.77) has a derivative form, it modifies the potential mainly at the surface. It leaves the average well depth essentially the same. It is only when we take its average that we get two different energy values, but the detailed modification is rather a modification in reflection coefficient. The reflection coefficient is a very important property of the optical model. That is right. However, the reflection coefficient is also modified by the diffuseness parameter.

So you are saying that people are mocking up this thing. What I am saying is that I am not sure that we could really distinguish uniquely between using a . 72 dependence and not using it because the optical model we start with is not unique in any way. We have found in the work of trying to fit low energy neutron polarization data that such a term was very important. The polarizations are very sensitive to interferences. To what extent do you think that you can go back over shell-model calculations that have already been done in great number and with a minimum of difficulty and extract the results needed? One would like to be able to say that we could do this. Unfortunately most shell-model calculations to date have been done in an energy region that is not relevant to reactions. They only consider states within 1 or 2 MeV of the ground state. For example, all the quasi-particle calculations which have been done for the Ni isotopes are concentrated completely within the first few MeV. They do not extend high enough in energy. Is there any possibility of probing the two-particle, onehole states below zero energy using stripping reactions? A little of this has already started. For example, Seth's work with (3He,p) reactions. This can be quite interesting if things are arranged properly. For example, in 15N (3He, p) we drop in a neutron and a proton simultaneously. We have already selected the experiment so that there is a hole present initially. Either one of the particles falls into this hole leaving a single-particle state or we get linear combinations of 2p-lh states. I suppose that even below threshold one would like to see fluctuations in the stripping strength as we go to different levels. I do not know what data is available as yet, but people are starting to do such experiments. So far we have done 15Nand 3% (3He,p) going to definite final states. The stripping to some levels is enhanced and that to single-particle states suppressed. Once we take a nucleus that is heavy enough so that we cannot resolve stripping to individual states, we see fluctuations in the envelope of the stripping strength which may be states of a simple structure. I guess that if such an experiment were done in a relatively heavy nucleus, where the density of states even at a few MeV is very high, the results might be interesting. By the way, that would also be a case where the isotopic spin

Richard H . Lemmer

61

was not changed so it would be as good as the (7,n) experiment.

REFERENCES 1. G . A. Keyworth, G. C. Kyker, Jr., E. G. Bilpuch, and H. W. Newson, Phys. Letters 20, 281 (1966); Nuclear Physics 89, 590 (1966). 2. E. R. Cosman, H. A. Enge, and A. Sperduto, Phys. Letters 22, 195 (1966). 3. H. Feshbach, Ann. Phys. (N.Y.) 5, 357 (1958); Ann. Phys. (N.Y.) 19, 287 (1962). 4. A. K. Kerman, Lectures in Theoretical Physics, Univ. of Colorado (1965), edited by P. D. Kunz, D. A. Lind, and W. E. Brittin (The University of Colorado Press, Boulder, 1966); W. Friedman, Ph.D. thesis, M.I.T. (1966). 5. A. K. Kerman, L. Rodberg, and J. E. Young, Phys. Rev. Letters 11, 422 (1963). 6 . B. Block and H. Feshbach, Ann. Phys. (N.Y.) 23, 47 (1963). 7. H. Feshbach, A. K. Kerman, and R. H. Lemmer, Ann. Phys. (N. Y.) 41, 230 (1967). 8. I. Lovas, Nucl. Phys. 81, 353 (1966). 9. R. H. Lemmer and C. M. Shakin, Ann. Phys. (N.Y.) 27, 13 (1964). 10. I. Afnan, Ph.D. thesis, M.I.T. (1966). 11. G. E. Brown, L. Castillejo, and J. A. Evans, Nucl. Phys. 22, 1 (1961). 12. W. Bassichis, private communication, 1966. 13. H. Pecker, Ph.D. thesis, M.I.T. (1966). 14. D. Robson, Phys. Rev. 137, B505 (1965). 15. T. Tamura, Proceedings of the Isotopic Spin Conference, Florida State University, Tallahassee, Florida (1966); J. P. Bondorf, H. Liitken, and S. Jagare, Phys. Letters 21, 185 (1966). 16. P. von Brentano, N. Marquardt, J. P. Wurm, and S. A. A. Zaid, Phys. Letters 17, 124 (1965). 17. S. Fallieros, Nuclear Spectroscopy with Direct Reactions, Vol. I, p. 143, Argonne National Laboratory. 18. E. H. Auerbach, C. B. Dever, A. K. Kerman, R. H. Lemmer, and E. H. Schwartz, Phys. Rev. Letters 17, 1184 (1966). 19. A. F. R. de Toledo Piza, A. K. Kerman, S. Fallieros, and R. H. Venter, Nuclear Phys. 89, 369 (1966).

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LEONARD S. RODBERG

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Development of the Formalism

THEmRsT discussion of what we now call "intermediate structure" was in a paper by Brueckner, Eden, and Francis in 1955,' 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; I also called them two-particle excited states. Since then these resonances have been generalized to be called intermediate resonance^.^ One approach to intermediate structure has been developed by Feshbach and his coworkers.* In discussing here a way of regarding nuclear reactions and resonances, I shall use a slightly different approach drawn from multiple-scattering theory and the optical model techniques associated with that theory, especially as this approach has been developed in a series of articles by Dr. William M a c D ~ n a l d .I~have ~ ~ benefitted much from both these articles and extended conversations with Dr. MacDonald. Before I discuss the formalism, let me say something about my views on intermediate structure, which motivate the kind of formalism I will use. It seems to me, first, that there must be some kind of structure in nuclear excitation functions, that is, variations with energy intermediate between the fine structure observed in low energy neutron reactions and the gross structure observed in average cross sections. The interesting thing about this intermediate structure, if it can be observed-and I think it is unquestioned that it can be, if one includes the isobaric analog resonances as intermediate structure-is that there is some hope of calculating its properties. The fine structure resonances involve states that are too complicated for there to be any hope of calculating the resonance parameters from the interactions or from any kind of nuclear level structure. On the other hand, the gross structure is generally too simple to be interesting, for it

depends only on the size and the shape of the target and does not tell much of what is going on inside the target. Most optical model fits tell a lot about the shape of the nucleus, but do not say much about its structure. Intermediate structure, if it can be observed, allows one to do in the continuum region what has been done for some fifteen years now with bound states, i.e., calculate the positions and the structure of levels in terms of shell-model or other configurations. In principle, one should be able to calculate the positions, the angular momenta, the decay widths and branching ratios for intermediate levels. Some of these things can certainly be computed for the isobaric analog states; e.g., predictions can be made of where they should be, and they are there. So, in that one simple case, this idea has been successful. There have only been a limited number of detailed calculations in other cases. In a sense, this is the biggest problem in intermediate resonance theory today, namely, that not enough work is going on directed at doing some of these detailed calculations. Perhaps we can motivate a little of it here. What I try to do in my approach is to relate the resonance parameters to the interactions between the particles. This is one goal, but it is not the only conceivable goal. We try to begin from a target Hamiltonian and nuclear forces and predict the observed reaction cross sections. First, let me make a couple of general points about the approach I will be using: I will first talk about single-channel processes, that is, cases where the only things that can happen are elastic scattering or inelastic scattering. I will generalize this later, and the generalization is trivial. I will not, at this point, discuss anti-symmetrization; in other words, I will assume that the incident particle is distinguishable from the ones that make up the target. Again, in this particular formalism the inclusion of anti-symmetrization is, in principle, trivial, and I will show what one would do if he wanted to include it. I will use a reaction theory in which the optical model plays a basic role, in which a single-particle interaction between the incident particle and the target is introduced to describe some of the things that happen to the incident particle in the region of the target. I will also introduce projection operators, but they will be, at least in appearance, quite different from those which Dr. Lemmer has introduced. They will be quite conventional; that is, they will select a certain part

Leonard S. Rodberg

67

of a complete set of eigenstates, unlike the rather more general operators introduced in the Feshbach theory. I would first like to discuss the fine-structure resonances, i.e., the resonances one would observe with infinitely-good experimental resolution. Then I will go on to discuss other ways of exploring the resonances in order to expose different features of the reaction process. The use of projection operators in this case is a means of separating those features of the reaction process which one wants to look at closely from those one wants to suppress into some quantity he will not attempt to calculate. It is necessary to exercise great care in order to include all the important effects. Since one cannot solve the many-body problem, there is a great danger in many-body theory of suppressing unwanted terms and simply assuming they are negligible. Usually no one can disprove the assumptions, but they may still be wrong.

The T Matrix I begin with a scattering amplitude rather than a wave function. This has a number of virtues which should become clear as I go along. The main virtues come from the fact that with a scattering amplitude one is looking at the asymptotic behavior of the wave function, and that means that some of the complications of having to look formally at the wave function at close distances are removed. The scattering amplitude is given by the following exact formula:

TI^ = (4,I VI#i+)

(1)

where 41 is the final state wave function in which the projectile is free and is subject only to a kinetic energy operator KO,and the target is described by some target Hamiltonian HT: (E - HT - KO)+/ = 0 . (2) Thus 4, describes the two separated particles. The target Hamiltonian is assumed here to be exact, and we start with a target which, by assumption, we can describe exactly. The state qi+ is a solution of the full Schrodinger equation: ( E - H)#i+= 0 , where the full Hamiltonian H is given by H = HT+ KO+ V

and V is the interaction between the projectile and the target. +;+ is the outgoing wave solution, having an incident free particle plus an outgoing wave:

The interaction V will be assumed to be a sum of two-body forces (although this is not necessary for much of the following development):

Because of the form of the Schrodinger equation, both the initial and final wave functions are just plane waves multiplied by boundstate functions:

The Two-Potential Method Since I want to use an optical model approach, I now introduce a single-particle potential Vo(ro).By "single-particle" I mean that this potential depends only on the coordinates of the incident particle. It is an interaction between the incoming projectile and the target, treated as a single particle. Aside from this restriction, Vo is completely arbitrary. It may be chosen to be the "optical potential," by which I mean a potential that is fitted to the elastic scattering data, but it need not be chosen to be that potential. It is simply an arbitrary single-particle potential. Once I choose Vo,I then have a residual interaction which I call Vl:

The residual interaction is responsible for the most interesting parts of the reaction. What does the scattering amplitude look like when I introduce Vo? I can use the famous Gell-Mann and Goldberger two-potential for-

Leonard S. Rodberg

69

mula. Let us derive this, since it is basic to this approach, and this will enable us to review a bit of scattering theory. The object is to separate the scattering due to Vofrom the scattering due to Vl, to the extent that they can be separated. We start with Equation 1 and introduce an ingoing-wave eigenfunction, xf, of a Hamiltonian that contains Vo. If we let and we have for xf-:

xj- is asymptotically a plane wave plus an ingoing wave. It satisfies, of course, the Schrodinger equation

The wave function hierarchy here is that 4 is a free-particle wave function (an eigenfunction of x0), x is (in other terminology) the distorted wave function (an eigenfunction of Ho), and $ is the full wave function (an eigenfunction of H). Introducing Equation 11 into Equation 1 and using the fact that V = Vo Vl, we then find

+

Note that, if Vo is not Hermitian (for instance, if a complex optical potential is used), then Vo in Equation 11 should be replaced by Vat, but this result for the T matrix will be unchanged. Now we have to derive an alternative form for In Equation 5 I have written gi+ as a plane wave plus an outgoing wave. Now we want to write it as a distorted wave plus a residual term. To do that, we want to change the free-particle Green's function appearing in

+;+.

Equation 5 into a Green's function involving Ho. To accomplish this we use the following identity:

Using Equations 14 and 5, we then have

This is the desired result. The full wave function is now written as the sum of a distorted wave for the incident channel, xi+, and a correction term which is proportional to the residual interaction and contains the Green's function of the distorting potential Vo(i.e., Ho). Now if we compare Equations 13 and 15, we see that the final result becomes

+

.

Tji = (+/IVolxi+) (x~IVl(#i+)

(16)

This is the two-potential formula. The first term describes scattering by Vo alone. The second term describes scattering by Vl in the presence of Vo.The fact that we have the distorted wave xf-, as well as the full wave function gi+, appearing in the second term signifies that Vois also playing a role there. Thus we have in fact achieved only a partial separation. It is easiest to proceed using operators describing the interactions. Thus I would like to define a wave operator Q satisfying From Equation 14 Q satisfies the following integral equation:

Then we simply define a transition operator or T matrix TI to be

This has the standard structure of the integral equation for a T matrix, although this happens to be a very special T matrix. It is of in-

Leonard S. Rodberg

71

terest because the transition matrix element can be written, using the two-potential formula, in terms of Tl as

The single-particle potential Vois seen to occur in two ways in the second term: in the distorted wave functions xj- and xif, and in the Green's function for T I ;that is, while the particles are interacting via Vl, they are always in the presence of Vo. Let me repeat here that Vomust, to yield a solvable scattering problem, be a single-particle potential, but it is otherwise arbitrary. One can choose Voany way he pleases, but the scattering amplitude is always going to be the same. In particular, the sum in Equation 20 is going to be the same, but the division into two terms will be very different. Usually Vois chosen to be energy-independent, or smoothly varying with energy. Then, if one sees fine structure in the excitation function, it is ascribable to the residual interaction term. On the other hand, Brueckner, Eden, and Francis1chose to look at the fine structure as resulting from a fluctuation in the depth of Vo. One of the reasons they did this was that they chose Voin such a way that the TI term would be identically zero. For elastic scattering, one can always ask at any energy what Vo will make the residual term identically zero. That potential is the single-particle potential for that energy. The commonly-used optical potential is one which is instead averaged over some large energy interval. The same remarks apply to distorted wave Born approximation calculations of direct reactions. It has become the custom to choose for the optical potentials in, say, stripping calculations ones that are obtained from fitting elastic nucleon and deuteron scattering data. Although this is an obvious way of reducing the number of free parameters, there is no theoretical reason why the potentials must be chosen in this way. There is a physical point here which enters both in resonances and in direct reactions. The optical potential obtained from fitting elastic scattering data is simply a potential which is required to give the right phase shifts, or, to put it another way, the right asymptotic wave function. On the other hand, in a direct reaction one is asking a very different thing about that wave function. He is asking that it describe the situation in the interior of the nucleus (or perhaps in the surface

region). A very different demand is then being placed on the potential, and there is no reason why the same single-particle potential should serve both purposes. There is a whole host of potentials, of course, which will give the same phase shifts, and there is probably a smaller class of potentials that would give the right wave function within the nucleus. There is, for instance, the "Perey effect," relating local and non-local potentials, that shows quite explicitly that two different potentials giving the same phase shifts will give very different wave functions inside the nucleus, and, consequently, very different predictions for stripping cross sections. Positive-EnergyBound States Now I want to use Equation 20 to discuss resonances in, for example, the scattering of a neutron on a nucleus. For this purpose, the first term in Equation 20 is uninteresting, if we use a single-particle potential which is independent of energy (or nearly so). The second term is of most interest, since it will contain the resonances. We see that it involves an interaction operator TI in which the intermediate states are governed by a Hamiltonian Ho that describes the target exactly and the projectile moving in a single-particle potential. These intermediate states would appear, for instance, if TI were to be expanded in a conventional perturbation series in powers of Vl. Since the only interaction in the intermediate states is via the single-particle potential, the projectile is not interacting with the individual nucleons in the target. There is then no interaction with the internal degrees of freedom of the target through that Hamiltonian. As a result, the wave function which is a solution for this Hamiltonian is a product wave function, and the energies are sums of the separate target and projectile eigenvalues. So we have

and where ET is the target energy and Eo the projectile energy. We now want to look at the intermediate states for various values of these two energies. Let me draw a potential well that is supposed to represent Voand the target in the same picture (Fig. 1). The target is

Leonard S. Rodberg

Initial State

Figure 1

in its ground state when the neutron comes upon it. If the energy scale is chosen so that the ground state energy is zero, then the total energy is just the initial neutron energy, labeled Ek in Figure 1. What can the intermediate states look like? While Ho includes no coupling between the internal degrees of freedom of the target and the incident projectile, Vl certainly does. The intermediate states are eigenfunctions of Ho,but, of course, they include all possible excited configurations of the target. Then what possible states are there, remembering that for intermediate states energy does not have to be conserved? If we denote the neutron separation energy by e,, there are three kinds of states we can have: a) Continuum states: ET > e:

or EO> 0,

b) Bound states with negative energy:

c) Bound states with positive energy:

These three types of states are represented pictorially in Figure 2, and the transitions of the target and the neutron from the initial state to each type of intermediate state are shown. The third kind of state is seldom discussed, but it is of most interest here. It provides a set of states which have positive energy but are bound, and they are bound in the traditional sense; that is, the wave function goes exponentially to zero outside the region of the target. Part of the energy which the neutron brought in has been transferred to the target, enough to cause the neutron to fall into a bound state but still leave the target in a bound, but now excited, state. These states bear a close resemblance to the eigenstates of HQQdiscussed by Dr. Lemmer. Transitions to these bound states can be pictured as follows. Consider an incident zero energy neutron. It may, as in (b), lose more energy than the target gains, leading to a negative energy state. (This state could, of course, be reached, since energy does not have to be conserved in intermediate states.) On the other hand, the target may gain more energy or the neutron lose less, with the resulting state having positive energy, as in (c). However, it does not differ in any essential way from the more familiar negative energy states. Note now that for the positive energy bound states we can conserve energy. That is, the neutron can lose exactly as much energy as the target gains. Of course, the only reason these states are bound is that we have turned off the residual interaction Vl. As soon as we turn on this interaction and allow the individual nucleons to exchange energy, the target can de-excite and the projectile be ejected or, perhaps, a different particle can emerge. Thus these states are stable only because of the particular choice of Hamiltonian Ho we have used. I am now going to make a basic assumption, namely, that all nuclear resonances come from positive energy bound states, in the sense that one can start with such states, "turn on" the residual interaction, and end up with a resonance. This assumption is probably wrong, in the sense that one could find a counter example. We cannot prove

Leonard S. Rodberg

75

this assumption, since we would have to introduce the complete internal dynamics of the target and we would then have to solve a many-body problem; but it is an interesting one and it has led to useful and correct experimental results. In addition to these bound states, there are others that can lead to resonances. They are ones in which the target is excited just above zero energy into a "quasi-bound" state, i.e., one corresponding to a resonance in a target system, with the incident neutron bound. Such a state is not stable, since the target can decay even in the absence of the residual interaction. However, there will be resonances in the A 1-particle system which can be traced back to such states in the A-particle system. As the incident energy is varied, one will reach energies at which the projectile can drop into a bound state, exciting the A-particle system to a given quasi-bound state, conserving energy. As one moves up in energy, there will then be a series of resonances each time one passes a possible bound state for the incident particle which conserves energy.

+

Derivation of the Resonance Formula Now I want to introduce the projection operators used in this approach. One operator P R selects the positive energy bound states and quasi-bound states that we associate with resonances; the other P N R selects the remaining states:

We now introduce another T matrix, denoted by TNn for T-non-resonant :

TNR is identical with TI, except that the only intermediate states which are allowed are those of types (a) and (b) in Figure 2. No positive energy bound states are allowed in this effective interaction operator. What this means is that, if TNR were solely responsible for the scattering, one would never get a resonance, by the basic assumption we have made. To put it another way, TNR has only a weak dependence on energy. In fact, it was to obtain this weak energy dependence that we chose the operator P N R to define the intermediate states.

POSSIBLE INTERMEDIATE STATES Figure 2

The trick in this approach is then to place those features of the process that do not interest us into this interaction operator. In actual calculation, what one does is replace TNR by an energy independent two-body interaction. So, clearly, TNR is of interest because it depends upon two-body potentials, but it is not of interest from the point of view of resonances or intermediate structure. The interaction that enters the actual scattering amplitude is TI.We now want to eliminate Vlfrom the expression for Tlin favor of TNR. We do this in two steps. First, we derive an alternative equation for TI.Equation 19 can be solved algebraically:

Leonard S.Rodberg

77

Further, the inverse operator appearing here can be directly expressed in terms of TI:

The integral equation for TI can then be written

Now, using this and the definition of TNR,Equation 23, to substitute for Vl, we can eliminate the potential:

We have thus achieved our goal of eliminating the residual interaction, but we can go further by solving this integral equation for TI. This can be done most easily by introducing an algebraic identity for the Green's function:

Using this in Equation 27, we find

We have now divided TI into two parts: a smoothly varying part and a part from which we hope to extract the relevant resonance behavior.

This is so because PR projects onto the positive energy bound states which, by assumption, lead to resonances. The denominator in the second term will lead to the familiar resonance denominator. In particular, we can look for the eigenstates of this denominator. These eigenstates, which I denote by Xj, will be solutions of the Schrodinger equation (Ej - Ho - PRTNRPR)~, =0.

(30)

Because of the appearance of the projection operator PR in this equation, the Xj's are linear combinations of the positive energy bound states. That is, the operator Ho- PRTNR is diagonalized over this set of states by introducing the functions

with the sum including all states that have ER > 0 but decrease exponentially at large distances (as well as any nearly quasi-bound states). TheXj's are thus not exact eigenfunctions of the A l-particle Schrodinger equation, since exact eigenfunctions would be linear combinations of the complete set of eigenstates of Ho,while the Xis are truncated linear combinations, including only the positive energy bound states of Ho. The non-resonant interaction TNR is not Hermitian, and Ho may also not be Hermitian since V, may be chosen to be complex. Consequently, the Xis are orthogonal to the adjoint functions 2 j , which satisfy

+

(E* - IFo - PRT+N~PR)Z~ =0.

(32)

Also, the eigenvalue Ej is complex:

Therefore, when we use Equations 31 and 33, and normalize the eigenfunctions according to (Zi[xj) = sij,the projection operator becomes

We then obtain for the resonant part of the T matrix

Leonard S. Rodberg

with the resonance energy given by

and the level width by

if Vois real. (With non-Hermitian interaction appearing in Equation 30, (Xj[Xi) is real but is not unity, if Equation 34 holds.) Equation 37 is a standard and very familiar form for the width, or lifetime, for the decay of the state Xjthrough an interaction TNa into is the phase the continuum states xE,+, the distorted wave states. space density of these states, and 0 includes all the relevant quantum numbers. Earlier we had the positive energy states uncoupled so that they could not decay, but now we have introduced the coupling through TNE, and they are allowed to decay. Since this is a central element in this description of the collision, let us derive this last result. We first assume that Vois Hermitian, so that Vl= (V - Vo)is also Hermitian. Using the definition of the non-resonant interaction, we have

The imaginary (i.e., anti-Hermitian) part of this operator is

The diagonal matrix element of this yields

using the expansion of the delta function

The continuum eigenfunctions, which are degenerate with X j and into which Xi can decay once the coupling is introduced, arise through this delta function. The sum (and integral) over P then includes all possible continuum states that may be coupled to Xi. In case Vois chosen to be non-Hermitian (as with the usual complex optical potential), an additional contribution appears in r,. One may use essentially the same procedure to show that the level width is then

where Im VO=

xi (VO- Vto) and the wave operator

QNR

is defined by

In the neighborhood of a resonance Q,vnX,is an approximate solution of the Schrijdinger equation (E - H)+ = 0. To the extent that this solution can be represented by Xj within the target, the correction term will not be significant. Also, if Im Vois constant over the interior of the target and is not strongly energy dependent, it will not be large. In any case, this represents only a correction to the first term because of the use of a complex potential; it has little physical importance in itself. If the parameters entering in Equation 35 are independent of energy and if the resonances are widely spaced, so that the rj is much less than the level separation (implying that 2, = Xj), this would be the Breit-Wigner formula. However, this result is exact, provided the assumption we have made, namely, that the positive energy bound states are responsible for the resonances, is true. The fact is, though, that the parameters will often depend on the energy, especially when the resonances lie close together. The effective interaction T N R is a many-body operator and in principle is very complicated, but one can make approximations to it that

Leonard S. Rodberg I will discuss in the next section, and these often make it amenable t o calculation. Further, because of the way it is defined, with only nonresonant intermediate states, it usually does not depend very strongly on the energy. As a result, in the region of separated resonances the resonance parameters will also not be strong functions of the energy. Let me close by noting again that, when all the terms that make up the transition amplitude are added together, the same answer must be obtained regardless of how the terms are subdivided. The best subdivision, i.e., the best choice of single-particle potential V, and projection operator PRYis one which makes it possible to identify and isolate the term responsible for the resonances. But there is no uniqueness in these choices, and the details of the results depend on them. Without introducing Vo, we would not have found the positive energy bound states, and so it is essential to this approach that we introduce such a potential, even though the choice of Vois arbitrary. Regardless of the choice (so long as it is "reasonable"), the physical assumption of this approach is that all the resonances will be in the second term of Equation 29, from which we have obtained the resonance formula. DISCUSSION

SETH: RODBERG:

Do you have a better suggestion than using the elastic scattering optical potential for the processes you mentioned? I have no specific alternative to suggest. The proper criterion should be the following. In stripping, for example, you calculate an overlap integral of the form

between distorted waves and a bound state function. Now, you can look at this in two different ways. You can regard it as the first term of a perturbation series, or you can say that you want to make this the closest possible approximation to the exact scattering amplitude. The question you can ask then is, what choice of optical potential will make this the best approximation to the exact scattering amplitude? Or, to say it another way, one should really treat the optical potential as a parameter with which you fit the stripping data. I think you should then be able to use stripping reactions to learn something

about the potentials and the wave functions inside the nucleus. But if you do that, it seems to me you are going around in a circle. Yes, you are left with a lot of parameters. However, the only reason I know of to use the elastic scattering potential is to reduce the number of free parameters. But from the theoretical point of view this is not necessarily the best one to use. In fact, you can make a good case that it is not the best potential to use, but, if you want to reduce the number of parameters, I do not have a better suggestion to offer. Are you including the possibility of a collective excitation of the target? That is included. The target is assumed, in principle, to be treated exactly so that the states selected by PE can include any sort of complicated structure which the target may have. In this formulation, what is the advantage of having Vo complex? There are some formal advantages in keeping VOreal, in that your Ho is then Hermitian; but my guess would be that the best choice-in the sense of including resonance effects most completely in the second term (in Eq. 29)would be a complex potential. You notice that in the calculation of the widths l?, an overlap integral involving eigenfunctions of Ho is evaluated. It is, therefore, very important to have the xi's behave properly in the target region, and this, of course, depends very much on the choice of Voand, for instance, on whether it includes attenuation of the distorted wave function. Is the expression for rjsupposed to be exact, or is it a first-order perturbation expression? It is exact. Could you give some reason why you made the critical assumption about only positive energy bound states contributing to resonances? Basically the notion is that you will get a resonance when you can conserve energy by dropping the incident particle into a bound state and exciting the target to a bound or quasi-bound state. If you can do that, the cross section is going to be large, and a large cross section usually means a resonance. Suppose you are very far off the energy shell, would you still get a resonance?

Leonard S. Rodberg

83

You can certainly occupy intermediate states which do not conserve energy, but you will not get a resonance from such states. So you are really including very few states around the incident energy. No, I am including here all positive energies. The states that are going to give me a resonance at an energy E are the positive energy states at that energy E. There is an energy shift and width coming in from the effective interaction, but the assumption is that you can trace the resulting complicated situation back to this simple picture of an excited target and a bound projectile. That is very dangerous. Oh yes, it is very dangerous, but I think it is fruitful, in the sense that there are probably many correct results which follow from that assumption. The thing is, that in this formulation you have to mock up your projection operators, and, I suppose, ultimately, your TNR,in such a way that the mocking up process may turn out to be very difficult to carry out. You have to make approximations to TNR. In fact, as I will discuss, the whole doorway state concept involves making approximations to TNR. This is so in any formulation. Whether in this one, or any alternative, you have to make approximations to TNR in order to emerge with the concept of intermediate structure. If that is wrong, that says something very interesting about nuclear structure and, in particular, about the mean free path in nuclei. It is very true that here, as in R-matrix theory, or Kapur-Peierls theory, or any other theory, you write an exact expression for the scattering amplitude which involves some energy denominators, and you pick out one of them and say: "That looks like a resonance; let us assume that it is." And I am doing the same thing here. I am saying: "That looks like a resonance," and I am going to fasten onto it, and hope that it is the real thing, and then go ahead and try to do reasonable things-make reasonable calculations-to see if, in fact, it does lead to a resonance. In particular, I attempt to make reasonable approximations to TNR.Once I have obtained a TNR,then, of course, everything else follows, because I can calculate everything in principle, and I get a resonance. Now, I may be doing the wrong things to TNR,but, if SO, then a lot of things about nuclear structure are not understood today. What I am driving at is the same question that Seth asked.

+

RODBERG:

You may have collective A 1 states, e.g. vibrational states. The vibrational states are included. They are included because vibrational states are simply components of the XE's.

Introduction of a Model Hamiltonian

INTHISsection 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, p) reactions, one can in fact carry out essentially the same derivation. The differences are trivial, so let me just indicate what they are. In a rearrangement process the interaction between the separated systems in the final state is different from the interaction between the separated systems in the initial state. As an example, for the reaction p 12C+ n 12N, in the incident channel the interaction V is the proton-I2C interaction, while in the final channel the interaction, which I shall call V', is the interaction between the neutron and 12N. (We will here only be discussing two-particle final states, although three-particle states can also be treated.) The T matrix for such a process is

+

+

T/i = (+/IVfJIC.i+)9

(44)

or, introducing a single-particle potential VIo for the rearranged final state, Tfi = ( x t l VrlI#i+)

(45)

with Vrl = V1 - Vro.No term appears arising from scattering by VIo alone, since a single-particle potential cannot by itself lead to a rearrangement process. We now write the non-resonant interaction as and recognize that the wave operator ONE will be only weakly energy dependent. Then we easily see that the only change that need be made in our earlier results, in order to obtain a resonant expression for a rearrangement process, is to replace TNR by V1lQNBwhenever it is adjacent to a final-state wave function. The result for the T matrix is therefore

We then have the same type of resonance formula as before, and the same positions and widths for the resonances. In particular, we see that the quantity rgj =

2 ~ ! ( x ~v gr 1l ~ ~ ~ ! x j ) 1 2 ~ a ( E )

(48)

can be interpreted as the partial width for decay of the state X j into the final channel P. (/3 identifies the channel and includes all quantum numbers needed to specify the final state.) Anti-symmetrization One other limitation on our previous result was the absence of anti-symmetrization. In some approaches the inclusion of antisymmetrization is quite complicated, because we must deal throughout with anti-symmetrized wave functions. Here, because I am using the T matrix, the inclusion of anti-symmetry is considerably simpler. It is convenient in computing the scattering amplitude to treat the projectile as distinguishable from the rest of the target particles. So I want to go from an anti-symmetrized expression to a form

Leonard S. Rodberg

87

where I can use the kind of formulae I have obtained before, in which the projectile is treated as distinguishable from the other particles. For the case when the incident particle is truly distinguishable from the particles in the target, the T matrix can be written as where I use the index "0" to indicate that 1)~+(0)is a state in which particle "0" is incident, V(0) is the interaction between particle "0" and the target, and +,(0) is a plane wave in which particle "0" is outgoing from the target. Thus the index tells which particle it is that is separated from the target. When the projectile is distinguishable, we do not need the index, but as soon as we move to a situation where we cannot distinguish it from particles in the target, then we have to identify the particles. We can write our original T matrix, for either rearrangement or non-rearrangement processes, in a much simpler way, namely, How would we generalize this when the particles are indistinguishable? The Hamiltonian H is already completely symmetric among all indistinguishable particles. How would we, in the simplest possible way, anti-symmetrize the rest? We could start with a target ground state which is already anti-symmetric among all the identical particles in the target and then directly anti-symmetrize the final state and the full scattering state. This yields for the T matrix where a is an anti-symmetrization operator. This is in fact the correct expression for the T matrix in this case, although I will not give the proof here.' The operator a is simply

where N is the number of particles in the target which are indistinguishable from the projectile and Pi0is the operator which exchanges the particle "0" with particle "i". And so a , acting on each of these states, gives a state that is anti-symmetric under the exchange of any of the indistinguishable particles.

We can now insert the expression for the fact that

Pi02 = 1

a into Equation 51 and use (53)

to obtain a simpler form for the T matrix. For elastic or inelastic scattering in which exchange with the incident particle can take place, this gives

This is the original non-exchange term, minus an exchange term in which the index "I" denotes any of the particles in the target that are identical with the incident projectile. In this term particle "0" is incident, but particle "1" emerges. Since the target is taken to be already anti-symmetrized, we can compute the amplitude for ejecting particle "1" and simply multiply it by N to obtain the full exchange amplitude. Using equation 54, we can very simply extend our previous results to the case of indistinguishable particles, since we have already calculated the direct term, and the exchange contribution is equivalent to the T matrix for a rearrangement process such as we have just discussed. In particular, the resonances in the exchange term will have the same positions and widths as in the direct term. Thus the approach we can use is to compute each of the terms separately and then obtain the final anti-symmetrized result by putting them together. In neither of these terms separately are the Fermi statistics included so that some states are included in each of them which, in principle, cannot actually be there. For instance, in either one of these terms an incident neutron is allowed to be captured deeply in the well, even though such a state may already be occupied. However, such terms will be cancelled when we perform the subtraction. Knowing this, we might as well leave them out in the first place, and this is, of course, what is done. Thus, the results we found earlier apply whether or not the particles are distinguishable, and they can be adapted to rearrangement collisions as well. The Single-Scattering Assumption Now let us begin our discussion of intermediate structure. One important simplification which underlies the concept of intermediate

Leonard S. Rodberg

89

structure and the related doorway states involves an approximation for T N R , the non-resonant interaction. Studies of the optical model during the past decade tell us that the mean free path of a nucleon inside a nucleus is comparable with typical nuclear radii. In particular, from the imaginary potential W as a function of the incident energy, we can calculate the absorption length or mean free path, with the result shown in Figure 3.2 We see, for instance, that the mean free path increases with decreasing energy as a result of the

E (MeV) Figure 3

Pauli Principle, because the number of available states into which the particle can scatter is reduced, and so the probability of scattering is reduced. The result of this reduced scattering is that, from about 8 MeV down, the mean free path is comparable with nuclear radii, and so a nucleon entering at low energy would probably not make more than one collision before it escapes. Viewed another way, if a nucleon is not captured in its first collision, it is unlikely to be captured at all, since it will escape before its next collision. From the viewpoint of multiple scattering theory, that assertion is the essence of the doorway state concept, in which particular shell-model states are identified as providing the "doorways" into the compound nucleus. Let us now relate this to the formalism we have developed. The non-resonant interaction TxR is a many-body operator, while the original residual interaction Vl contains only one- and two-body interactions. If we look at an expansion for TNR, there will be multiple collisions, in which the projectile collides in succession with the target particles. We can write such a series as

where tio is the effective two-body interaction obtained by including all the successive collisions of particles "i" and "0." To specify this effective interaction, we introduce the residual interaction between these two particles V,o = v,o - Vo/A

-the

full residual interaction is

-and

then define the effective two-body interaction by

(56)

We can express TNR in terms of this operator by introducing a set of wave operators. We use the operator QNR defined previously in Equation 46 and similarly express tio as

Leonard S. Rodberg

91

with

Lastly, we introduce the operators they satisfy the relation

having the property that

QNR,~

If we can obtain an integral equation for these operators, we will, in fact, have achieved our objective. We can do this by the following straightforward manipulation, using Equations 59 and 60. The wave operator QNR is

but it also satisfies

i the set of coupled equations Then the operators Q N ~ , satisfy

Using this result, we obtain the expansion given in Equation 55:

The conclusion from Figure 3 is that this multiple scattering expansion is a rapidly convergent series and, in fact, that only the single-

scattering term need be included. This leads directly to the concept of doorway states. The non-resonant interaction T N R is approximated A

by

t, ,the sum of the operators ticwhich are just the effective two-

body interactions used, for instance, in shell-modelcalculations. Their matrix elements can be obtained by fitting energy levels from shellmodel calculations to the actual levels. Alternatively, they can be obtained from the two-nucleon potential. In the shell-model context the effective interactions are generally assumed to be real. This is valid there because the imaginary part, arising from energy-conserving transitions out of the incident channel, is drastically reduced at low energies where the Pauli Principle suppresses such transitions. In our case, also, the imaginary part can usually be neglected in computing level shifts, but it must be used in calculating the widths of the resulting states. It can be estimated by using the formulas developed in the first section. The only assumption needed here is that, prior to capture into a bound state, we have just single scattering. This leads to the conclusion that, in one collision, the incident nucleon must go from the initial continuum state to a bound state, if we are to have any hope of getting a resonance. It must be captured in the first step. Furthermore, that one step must be a simple one, since we have only a twobody interaction and the incident particle has a chance to collide with only one target particle before it escapes. If the second term in Equation 64 were significant, then we could also arrive at a resonant state, by scattering once into some continuum state included in PhrR and then into a bound state. In fact, though, such a process makes a small contribution to most reactions, primarily because the lifetime of the continuum state formed in the first scattering is short compared to the mean time before the second scattering. If the system does enter such a continuum state, it will simply decay back into the entrance channel or into a readily available exit channel. To proceed into a resonant state, the system must instead enter a bound state in the first collision. This then leads to the notion of doorway states, of relatively simple states that you must pass through in this first stage in order to get into the compound nucleus. Such simple doorway states are not

Leonard S. Rodberg

93

eigenstates of the target Hamiltonian HT, but are instead closely related to eigenstates in a shell model or independent-particle model. In order to make use of the simplicity of the doorway concept, we want then to leave the true target Hamiltonian and introduce a shell model for the nucleus, in which we can talk about simple independent-particle configurations. Shell-Model Form of the T Matrix One way to do this is simply to approximate HT by a shell-model Hamiltonian. I do not want to do this, because if we trace through the formalism, we find that HT appears everywhere, and it is therefore not clear exactly what we are doing by such an approximation. So I want to start over and obtain an analogous result for the case of a shell-model description of the target nucleus. Let me introduce a shell-model Hamiltonian HM, defined by

where Ki is the kinetic energy and Vi a single-particle potential for the ith particle. One should, of course, do this using second quantization for the particles that are indistinguishable, but, as in the scattering problem, that is a trivial although formally complicated matter. Note also that, while the potentials for the bound particles, 1 5 i 5 A, could be chosen to be harmonic oscillator potentials (as in most shell-model calculations), Vo cannot be of this form. It will be used to compute the elastic scattering (the so-called "potential" scattering) of the projectile and must therefore vanish at large distances from the target. With this model there is a residual interaction given by

Unlike V,, the residual interaction VM contains the sum of all the two-body interactions, including those in which the projectile does not participate as well as those representing its interaction with the target. (Note that Vl in Equation 66 is a single-particle potential for

particle "1" and is not the residual interaction used earlier.) The eigenstates of HM,denoted by {,+,will satisfy (E - H ~ ) r i += 0 .

(67)

The state ri+ is a shell-model state for the target, with particular single-particle states occupied, and a distorted wave state for the projectile, that is, a plane wave plus an outgoing wave. Using this model, we will now develop some exact formulas for the T matrix. The approximations enter only when we do calculations. The exact formulas make use of a remarkable theorem due to BrenigY8 which is very much like the two-potential formula we have already derived. In that formula we wanted to describe the projectile initially by a distorted wave rather than a plane wave. Now we want to go farther; not only do we want to describe the projectile by a wave moving in a single-particle potential, but we also want to describe the target by a shell-model wave function, rather than by its actual wave function. By analogy with our earlier derivation of the two-potential formula, we note that [/- can be related to the true free-particle state $/ via the integral equation

+

since H M - (V - VM) = HT KO.Introducing this into the T matrix, we have the Brenig result:

This permits us to separate resonant from non-resonant contributions and, at the same time, to use only shell-model wave functions in their description. We can express the full scattering wave function in terms of li+, simply by transforming the Green's function:

Leonard S. Rodberg

95

We may then define a transition operator by

and the T matrix becomes We see that the structure of Equations 69 to 72 is identical with the corresponding formulas that we derived earlier. Again the first term of Equation 72 is uninteresting, in that it will not lead to resonances. Only the second term will contain any resonances. The intermediate states in a perturbation expansion of TM in powers of V M are shell-model states; and so this is the proper framework if we want to discuss intermediate states, such as doorway states, that are shell-model eigenstates.

Doorway States and the Associated Resonances Now we can follow the same procedure as before to obtain the resonant structure associated with doorway states. We first must define the characteristics of a doorway state. The target in the initial state is assumed to be in a shell-model ground state. Then, by definition, li+ is a shell-model wave function in which all the lowest single-particle states are filled. A doorway state is a linear combination of eigenstates of H M , in all of which the target is in a state that differs from the ground state by the excitation of one particle and the creation of one hole. In "quasi-particle" terms, this intermediate state of the target differs from the ground state by the addition of two quasi-particles. The excited particle must be in a bound state so that the system will live sufficiently long that the particles can scatter again and the system proceed into the compound nucleus. Likewise, in a doorway state the projectile must be bound in the potential V,. (In li+ it is in the continuum.) A doorway state is then a positive energy bound state formed out of a particular set of shell-model configurations, all of which have

the same basic structure. For instance, if the target is a closed-shell, even-even nucleus, then the doorway states will be formed from twoparticle, one-hole (or three quasi-particle) configurations of the compound system. The significance of these states is that they are the only positive energy bound states which can be coupled to the initial state by a two-body operator. Thus, if the projectile is to be captured in a single collision, as we have assumed, and if energy is to be approximately conserved so that we obtain a resonant behavior, then the projectile must be captured first into a doorway state. We now define a doorway-state projection operator P D which selects out all the eigenstates of H M that have the above prescribed structure. In defining PDit is also usually desirable to include the negative energy as well as the positive energy bound states having this structure. Corresponding to Po, we define P N D such that PND= 1 - PD. (73) The states selected by P,?iD include all continuum states and the rest of the bound states, which I will call "complicated states," that is, shell-model configurations in which more than one particle has been excited out of the target ground state. These states can have positive energy and thus can also be associated with resonances. However, for the moment they are not of interest since, in the single-scattering approximation, they cannot be coupled directly to the incident channel. Also, they lead to finer structure than the intermediate resonant structure produced by doorway states. What we are doing, then, is placing those things which do not interest us into P N D . Similarly, we define an effective interaction which includes only those intermediate states that are not of interest to us now:

In performing calculations we would assume again that Tn.D was smoothly varying with energy. In fact, since it does contain the complicated states, it is not smoothly varying with energy. It has fine ripples in it. However, if we are looking at doorway states and intermediate structure, what we will do, in effect, is average over an energy interval that includes these ripples and look just at the broader intermediate structure.

Leonard S. Rodberg

97

Now, we can go through the same algebra as we did for the general case and fmd that

The only states that are allowed in the second term are the simple configurations described earlier (for instance, the two-particle, one-hole states in the case of a closed-shell, even-even target). This is what we want in order to exhibit the resonance structure that arises from doorway states. These resonances occur at the eigenvalues of ( H M PDTNDPD), so it is necessary to solve the equation

+

The eigenfunctions Y D are, finally, the doorway states. They are linear combinations of the shell-model states we have described. The introduction of these eigenfunctions will then lead to a resonance expression that describes what has come to be called "intermediate structure." It is

where the sum is over the doorway states, and the positions and widths of the resonances are given by

We see that the doorway states YD associated with the resonances are linear combinations of the eigenstates of H M , since the nonresonant interaction TND mixes these elementary states. In essence, we are diagonalizing the Hamiltonian over the set of bound twoparticle, one-hole states. The resulting eigenstates might, for instance, have a vibrational character, and the Yo's would be vibrational states. For an odd-A compound system, they would have vibrational excitations, that is, phonons of particle-hole excitations, together with single-particle states. If we do not have vibrations, but only simple single-particle excitations, then in effect the mixing due to TND is weak.

If there is strong mixing, then it might be more convenient to use unperturbed eigenfunctions different from the single-particle eigenfunctions we are using. Although we introduced at the beginning a model Hamiltonian H M which was a sum of single-particle Hamiltonians, in fact the derivation of our resonance formula using Equations 67 to 78 does not depend on any particular choice for the model Hamiltonian. But, of course, while the algebra does not depend on this choice, the physical result does. If we want to obtain a resonant structure that arises from doorway states, in the general sense of states which can lead to the formation of a compound nucleus but which are coupled directly to the initial state, then there is some restriction on the choice of possible model Hamiltonians. They should describe the dominant modes of elementary excitation in the target, whether they are of the single-particle form or of the "quasi-particle" or "quasi-boson" form. Furthermore, the Hamiltonian should be such that it has excited states which are coupled to the initial state through the residual interaction. In practice, this means that the initial state must be describable as a simple ground-state configuration of the model Hamiltonian, and it must be coupled strongly to the excited states via a two-body force. We can obtain an approximation to r D by the following reasoning: The imaginary part of TND arises from the singularity of the denominator in Equation 74. The intermediate states appearing in this equation for TND are of two kinds, continuum states and complicated bound states. Assuming that the single-particle potentials are chosen to be real, we are going to get two terms in r ~ :

The second term arises from an average over the complicated states in the neighborhood of the doorway state. Such an average yields the doorway-state width that would be seen in a poor-resolution experiment or in an average over the results of a good-resolution experiment. In either case the averaging interval would be greater than the spacing of the complicated states but less than r ~ .

Leonard S. Rodberg

99

The terms on the right hand side of Equation 79 have been denoted and r by Feshbach,' but I prefer to call them rout and Fin,respecis the width for decay out of the doorway state through the tively. rout two-body operator Tn.Dinto the continuum free-particle state {E,+, while rinis the width for decay of the doorway state into the more complicated states. If we look at the expression for TficreB) for the case when only elastic scattering is possible and compare it with the equation for r~ in that case, we see that the numerator of the expression for Tf,(res) is, in essence, just rout, while the denominator contains both rout and rin.Detailed examination of the factors in the numerator shows that, for resonances having a particular angular momentum J, this T matrix is just

rr

(aJ is the phase shift arising from scattering of the projectile by the potential Vo at the energy E, while p = mk/2nPfi2 is the density of states at that energy.) This resembles, but is noticeably different from, the usual BreitWigner formula for elastic scattering, in which the full width of the resonance has to equal the escape width in the numerator, as required by unitarity. This is manifestly not the case here, that is, rout < Fin rout. What this means is that the doorway states are not in fact eigenstates of the true Hamiltonian, since one would then have the unitary Breit-Wigner formula as derived, for instance, in the first section. The fact that unitarity fails tells us that the states we are looking at are not real states, but are instead simpler model states which, as in the picture developed by Lane, Thomas, and Wigner,g are spread over the actual states. The resonant structure they produce corresponds to the actual cross section averaged over the fine structure. The additional width rinrepresents an effective reaction channel induced by the averaging process. (This effect is familiar from studies of the complex optical potential.) If, in fitting a resonance formula to an experimental cross section, one finds that the escape widths are less than the total width, then we know that the resonance does not arise from the actual states of the system. This is the experimental situation

+

in isobaric analog resonances (one example of intermediate structure), where one finds a proton width r, which is less than the full width of the resonance.

DISCUSSION SETH:

RODBERG:

RODBERG:

The multiple scattering assumption is a very important point. It leads to the question that if you are going to say that the mean free path from the absorptive part of the optical model potential is long, then you should also be able to produce that much of an absorptive potential from just a consideration of two-body interactions. And, these attempts, as we know, have not been successful so far. Usually you get very little W. I think the calculations you are referring to are ones by Lane, Thomas, and Wigner. Brueckner theory calculations, in which you take the full two-body scattering amplitude into account, do give reasonable values for W. You can thus calculate the mean free path by using what you know about nuclear forces and looking at the multiple scattering of a nucleon as it moves through a nucleus. In that way you get the same results as with the imaginary potential. The mean free path is long because nuclear forces, in essence, are sufficiently weak that you get mean free paths as long as those shown in Figure 3. Brueckner, Eden, and Francis did this calculation for W, for instance, and they get quite a reasonable value. Gordon Shaw and others have done this more recently and calculated both the depth and the shape, obtaining reasonable results. The reason I ask this question is that Brown's way of thinking-i.e., vibration plus particle type of excitationmakes quite a point of this, namely, that the damping from the optical model is not obtained fully when you calculate it from first principles. That is, you may need coupling to vibrations in order to get it all. Yes. You have to include the right nuclear states and the right coherence properties of the nuclear states. Perhaps, because this point is so crucial to this development, it should be looked at again in 1966. But I do not suspect that anything will happen to change the picture of nucleons that do not scatter very much inside a nucleus, until you get to very high energies when the mean free path drops considerably.

Leonard S. R

SETH:

No, this is not the point that I am doubting at all-that the mean free path is long-but, it is the explanation in terms of the fundamental two-body interaction of nucleons. There may be problems with it. I think, from the Brueckner approach there are no problems, but there are problems with the Brueckner approach. Various people raise various objections to what happens at higher orders. And, so, this is an important continuing theoretical problem. In your eigenvalue equation for YD(Eq. 76), if the energy is complex, as you stated, what is the boundary condition involved in solving the equation? That is, how do you go about finding the Yo's? The question of the boundary condition does not really arise, since the functions Yo are simply linear combinations of a finite number of bound-state solutions of HIM. They are then found simply by diagonalizing a determinant. This is not quite a shell-model calculation because all you have there is a residual interaction between particles which are in a doorway state. Yes, that is right. It is only a truncated shell-model calculation. I only allow certain specified configurations to enter the calculation. But, that is also what you do in any shell-model calculation. You make a judicious choice of configurations. I have made a particular choice, because of the nature of the intermediate resonance problem. I just want to point out that the approach you outline here has the advantage that it focuses on the doorway, and that, in principle, you can calculate a width. You do this, at the expense of truncating-perhaps in an unnatural manner-the space in which you are going to diagonalize. That is an interesting point. You would like to do two things, and they appear to be impossible to do at the same into two parts, one which is time. One is to divide the Tli resonant and one which is not, and the other is to obtain an accurate description of the resonant states. I have chosen the former and have truncated the set of states I will consider. In principle, if the true energy dependence of TNRcould be retained, then some of the effects of other configurations would be included. But by treating TNR as essentially energy independent, we obtain a far simpler calculation. The price we pay is that the resonance energies are determined by using shell-model wave functions

MCVOY: RODBERG: McVou: RODBERG:

BECKER: RODBERG:

which only include the particular configurations we have allowed in the doorway states. The resonance energies that we get, therefore, may in fact be wrong, because it may be important to include the effects of other configurations. On the other hand, it is not important to include other configurations in order to obtain resonances because other configurations are not doorways. But it may be important, for other reasons, to include them, since they may give large energy shifts and also may affect the decay widths. While we are comparing things, is it fair to compare your formalism to the Feshbach formalism, and say that the two differ only in their choice of projection operators? The projection operators are certainly different, and they enter in a different way. But, I think the spirit is basically the same. Yes, I agree. As far as the projection operators are concerned, I think mine are conventional, in the sense that they are defined over a complete set of states, and I know exactly which part of that set they pick out. Those in the Feshbach formalism, I would say, are more imaginative. Is the anti-symmetrization procedure the same here as you described earlier? Yes, it is. You have to anti-symmetrize the shell-model wave functions properly for the target; this is easy to do in the usual shell-model picture. Then you calculate the direct and exchange terms as I have indicated before.

Conclusion and General Remarks

INTHE 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 Yo are 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 operators, P D and P N D , which act on eigenstates of the shell-model Harniltonian. P D selects the doorway configuration, which we picture, for instance, as a two-particle, onehole configuration. Now we want to go one step further in order to exhibit the next "layer" of resonance structure, that is, to exhibit some of the fine structure which underlies the doorway resonances. At this point we could adopt several alternative approaches. As we have seen, doorway states can be thought of as three-quasi-particle states. In the same context, states which are one degree more complicated (they have been termed "hallway states" by Ferrell and MacDonaldlO) can be thought of as five-quasi-particle states. We could then define a new projection operator PD PH which would project out both doorway and hallway states. To be somewhat more general, we will define a projection operator PBwhich projects out those bound states of the shell-model Hamiltonian which will contribute to the resonances that interest us. These can include all of the bound states, or just the positive energy ones,

+

and it can include states of as much complexity as we wish to take into account. Thus it can include 3, 5, 7, 9, etc. quasi-particle states, or it can be restricted more severely. We have a great deal of flexibility in its selection. The operator which projects out all the states not included in PB will be called PC: This operator selects any more complicated bound states and the continuum states. As in the previous cases we have examined, PCis the operator that selects the states that we are going to "bury" in an effective interaction and not examine closely. Again, as before, if we are interested in structure which is one layer down from the doorway structure but are not interested in still finer structure, we can include this still finer structure in PCand average over the resulting effective interaction. If we do this, the result has exactly the same structure as we have seen before:

where

As before, we assume that TC may be approximated by a two-body operator, so that doorway states are the necessary first step in proceeding to a resonance. Regardless of how we choose PC,we always assume that the energy variation of Tc is weak, and that no fluctuations remain in it. Any fluctuations in Tc are averaged over. So, at this stage, the energy denominator we want to diagonalize is (HM PBTCPB).I shall call its eigenfunctions 8,: (E,-HM-PBT~PB)%,,=O. (84)

+

The 3,s' can be expressed as linear combinations of doorway states and of more complicated bound-state configurations:

Leonard S. Rodberg Similarly, the adjoint state has the form

an =

XD D

+

a * ~ ~ pxmmp . n

.

comp states

To get an idea of the structure of the resonances, let us suppose that we are at one of these eigenvalues and, furthermore, that the eigenvalues are well separated. There are certainly experimental situations where this condition exists, so let us look at one of these resonances. In this case Tfi(re" has the form

where C.

+

(an\HM T ca~n )

= Re ---

(an1 a n )

and

There is only one term in the formula for the width, as opposed to the previous case where we found two terms. The reason is that an is a "complicated" wave function which includes all of the simple and complicated shell-model states. The only decay mode for these states is then out into the continuum. (In fact, they differ from true compound nucleus states only in that they contain no continuum components.) If now we include only the two-body part of the operator, then the only component of an which will be coupled to the incident channel will be the doorway state component so that we then have

where laDn\2 is the proportion of doorway state in 3, and routis the decay width of the doorway state. ( I am assuming here that only one doorway is contributing, but, in principle, Equation 89 should include a sum over all doorway states.)

This same argument also applies to Equation 86, which then becomes

We then have modulation factors which modify both the resonance amplitude and the width. However, at the resonance energy the height of the peak will be nearly independent of n so that it is primarily the width which will be affected by the overlap of the state at E, with the doorway state. Physically, the reason that this is so is simply that the only way the system can get into (or out of) the state 8, is through the doorway. This is then one step up in complexity from the Lane, Thomas, and Wigner ideagof giant resonances coupled to the continuum through the single-particle states. One experimental consequence of this follows from the fact that the angular momentum of the state 8, must be the same as the angular momentum of the associated doorway state. If one looks at the angular distributions for scattering through the states a,, they must all be the same and will be identical with the angular distribution for the doorway state. This is one test of the validity of the doorway state concept and has shown up in the 27Al(p,Y) data.1° Fine Structure Widths Now, what can we say about the width of the fine structure? For illustrative purposes, let us take a simple model in which 8, is a linear combination of one doorway state and a set of complicated states which are each coupled to the doorway state with a strength M = (X~IT~IXcump), but are not coupled to each other. That is, we suppose that the effective Hamiltonian H M Tc has already been diagonalized over the complicated states. The Schrodinger equation, Equation 84, then gives the set of linear equations

+

(En

- Eeomp)acqrnp,n - MUD,= 0 .

The resulting eigenvalues are solutions of

Leonard S. Rodberg En-

107

En-

2 C En -MEwmp

=0

(92)

comp

+

+

with ED= (XIHM TC~XD) and Ecomp = (xeomplHM TCI~eorn~). If the complicated states are equally spaced (the "picket-fence" model), this is

where we have used Ecomp = sD, with D the level spacing, and we have recognized that rin,the width for decay of the doorway state into the more complicated states, is 2nM2/D. This equation can be used to obtain an expression for the fine-structure widths. With r, = -2 Im Em and rout = -2 Im ED (the width for decay of the doorway state to an outgoing channel), one finds, for a # 0, 1 2 rin(rout- rn) r tanh n - = D 1 1 (94) (en - E D ) ~4 FinZ 4 (rout- rnI2

+

+

When the fine-structure states are narrow, that is, rn 1, it is an Ericson

Some fluctuations may well be included in (2), but (3) cannot apply to compound or intermediate resonances. Condition (2) is then necessary, but not sufficient, for intermediate resonances. As we can see, the above is very rough, but, nevertheless it does offer some guideline. Let me close by repeating that what we lack most are detailed calculations of intermediate structure. The same kind of effort applied in the past to the shell model should now be directed to calculations of positive-energy shell-model states, that is, doorway states. DISCUSSION

McVoy:

RODBERG: SETH:

It is an interesting speculation to think of Ericson fluctuations of intermediate structure. Do you know if anybody has tried to look at the Ericson argument again for intermediate structure? No, not to my knowledge. No one has worked out the Ericson theory for intermediate structure. Could you suggest experiments which would shed definitive light on the existence of intermediate structure?

Leonard S. Rodberg RODBERG:

119

I believe (n, n') and (a, p) reactions should be done near closed shells with an analysis of the branching ratios. Of course, in these we do have many angular momenta participating, so their analysis is complex. To avoid this problem to some extent, (p, y) reactions in the same regions should be interesting, but this does not have the advantage of providing you with various exit channels. As you go up in energy, say to 4 or 5 MeV, and start getting direct interactions, you should start seeing the interference between the doorway state resonance and the direct interactions. One of the most interesting properties of the intermediate structure is that it should interfere coherently with direct interactions. This will be difficult to analyze, but very interesting. REFERENCES

1. K. A. Brueckner, R. J. Eden, and N. C. Francis, Phys. Rev. 100, 891 (1955). 2. L. S. Rodberg, Phys. Rev. 124, 210 (1961). 3. A. K. Kerman, L. S. Rodberg, and J. E. Young, Phys. Rev. Letters 11, 422 (1963). 4. See, for instance, H. Feshbach in "Nuclear Structure Study with Neutrons" (Proceedings of the Antwerp Conference) edited by de Mevergnies, Van Assche, and Verier, North-Holland Publishing Co., Amsterdam (1966) and the lecture in this volume by Richard Lemmer. 5. W. MacDonald, Nucl. Phys. 54, 393 (1963), 56, 636 (1964), 56, 647 (1964), and Phys. Rev. 137, B1438 (1965). 6. W. MacDonald and L. Garside, Phys. Rev. 138, B582 (1965). 7. See L. S. Rodberg and R. M. Thaler, Introduction to the Quantum Theory of Scattering, Academic Press, New York (1967), Chap. 7. 8. W. Brenig, Nucl. Phys. 13, 333 (1959). 9. A. M. Lane, R. G. Thomas, and E. P. Wigner, Phys. Rev. 98, 693 (1955). 10. R. A. Ferrell and W. M. MacDonald, Phys. Rev. Letters 16,187 (1966). 11. A. Sugie, Phys. Rev. Letters 4, 280 (1960). 12. K. K. Seth, Phys. Letters 14, 306 (1965) and private communication, has observed such effects. 13. B. Block and H. Feshbach, Ann. Phys. (N.Y.) 23, 49 (1963). 14. B. Block and A. Lande, Phys. Rev. Letters 12, 334 (1964). 15. T. Ericson, Ann. Phys. (N.Y.) 23, 390 (1963). 16. K. K. Seth suggests that it may arise simply from the finite experimental resolution.

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JAMES E. YOUNG

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I. Introduction

THEEFFORTS 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 Ericsonl fluctuations as a function of bombarding energy. As the resolution becomes worse these fluctuations appear in some experiments to give way to a resonance structure. This structure is characterized by parameters which are universal functions of E and A.2$3This is a way of stating that certain specific configurations, the doorway states of Block and FeshbachY4determine the structure described. By worsening the energy resolution further we find that the resonance structure, the so-called resonances of intermediate structure, dissolves, giving way to the giant resonance structure as summarized by the optical p~tential.~ The statistical and optical potential regimes have been thoroughly investigated, and although the transition between the two is experi* The material in these lectures was prepared by the author in collaboration with Dr. Andra's Zuker at the Department of Theoretical Physics, Oxford University. Both of the collaborators are indebted to Prof. R. E. Peierls for his hospitality. The author further thanks Dr. R. E. Schenter for discussion on the material of these lectures. Support of the author by Los Alarnos Scientific Laboratory and Oxford University during the period of a Postdoctoral Fellowship is gratefully acknowledged.

mentally displayed by looking at evaporation spectra as in (n, nf),6it is a persistent source of argument among nuclear physicists as to whether any identifiable phenomenon exists in between.' Hopefully the generation of tandem accelerators currently available will help us to resolve this controversy. The difficulty with which we are faced is that of extracting a small effect (2-10 percent of the scattering cross section), intermediate structure, from a scattering cross section dominated by the elastic background furnished by the optical potential. The effect we hope to see is thus dependent upon the manner in which we choose to observe it. And, generally, we have to ask for good resolution (5-15 keV at 4 MeV or less for neutrons on A = 58 and the same for protons at 10 MeV or less on A = 118) which severely restricts observation. It is surely open to discussion as to whether one ought to investigate the systematics of these small effects. Many of us think that the answer is yes and each time observation is inconsistent with optical potential systematics, in the appropriate regime, we find this answer strongly reinforced. And, measurements taken with increased precision often encounter the difficulty noted. If the departure from the optical potential picture is susceptible to description in terms of a systematics, it becomes possible to say something new about the compound state wave function determining the scattering cross section. Our discussions will adopt a point of view, shortly described, which exhibits the new feature in an economical fashion. We take the viewpoint that average nucleon-nuclear cross sections in the so-called resonance region require an optical potential for their description, which potential extends the shell model to continuum states. This extension of what might be termed the self-consistent or Hartree-Fock potential (H-F) is expected to be valid so long as inelastic effects are small. The strong coupling to inelastic channels such as discussed by McVoy8makes it impossible to label the giant resonances by single particle quantum numbers. In this limit the optical potential is strongly diffracti~e.~Consequently any attempts to describe the position of giant resonances in terms of concentrations of single particle strengths, the picture of Lane, Thomas, and Wigner (L-T-W),1° must fail in the presence of strong inelasticity. We define then a continuum shell model in a region of bombarding energy E and mass number A where the optical potential U meets the

James E. Young

125

restrictions enumerated above. The H-F self-consistency as used here asks that U be formed as the appropriate spin-isospin average of some reaction matrix t in the sense of Brueckner,ll the average taken over the ground state of the target NAZ. This optical potential is made slowly-varying by performing energy averages over the compound states occurring according to methods described by Brown.12With the following definitions

where

where and the stationary state \ki(+)(xo,tA)describing the interaction of nucleon and target is given by the approximation at this point in our discussion. It is usually stated that intermediate structure is obtained by considering those configurations of the (A 1) system which can be connected with the wave function in Equation 4 through a single application of the two-body force Voi. Transfer of energy between these new configurations and Equation 4 is manifested as a resonance when viewed from the elastic channel (Eq. 4). If the target NAZis a doubly closed shell, the new configuration must be those of two particles and a single hole (2p, lh). These are three quasi-particle, (3q), excitations

+

because of diagonalization of the residual nuclear force

Voiamong i

unperturbed (2p, lh) states. At this point the wave function of Equation 4 is to be replaced by

The expansion coefficients IC;s12 select those states s in the vicinity 2A of the energy E of incident systems. The energy spread in the incident systems and also the energy over which energy averages are performed is substantially less than 2 4 which is of the order of the width of intermediate resonances. Our new wave function (Eq. 5) is consistent with the introduction of what we may describe as a theory of the continuum configurationmixing shell model. In effect the (39) configurations are to be taken into account as well as the single particle or optical potential configurations, and therefore coupled equations ought to be solved for the representative wave functions involved. We will not solve these equations, however, but will make reasonable approximations to the full solutions. We imply by the decomposition of Equation 5 that the state $439) has no particle at infinity.13This is a bound-state wave function, a fact of specific utility for our future discussion. To be precise $,(3q) is the bound-state function generating the quasi-stationary state $439; t), $s(39; t) = [exp(iHt)l$s(39), (6) appropriate to the collision problem. This function is required in order to produce a net flux of outgoing particles at infinity in the physical problem. The Hamiltonian H is that defined in Equation 1. Let us put these remarks another way. The expansion coefficients Ci, of Equation 5 are computed from the function of Equation 6 and not $439) of Equation 5. This results in an explicit energy dependence for these coefficients with damping of the state 1s) being included. All of this language from time-dependent perturbation theory can be translated into time-independent statements, yielding Ci, from $,(3ql, by means of the projection operator methods of Feshbach.14 All of the foregoing remarks are obvious when stated another way. If the residual nuclear force is diagonalized among the (2p, lh) configurations to produce states Is) = qs(3q), energy E,, those states Is) satisfying E-AiE,[~',I+~(A)) = 0.

(37)

It is then observed that the singly-excited configurations, lav+,)and la@,,) (Fig. 6), are indeed different as a result of the change in the average field brought about by the presence of the added particle.

(a)

('J)

Figure 6 Singly-excited configurations of the (a) anti-symmetrized product ( A (b) product ( A 1) system; U = U,, UO= UO,.

+

+ 1) system and

The point of all this is that the corrections to the optical potential scattering are to be accounted for by the introduction of states 1s) = I@qk"'%(4~))

(3 8)

into the resonant, or compound, matrix elements

It would appear that we imply by Equation 39 a second-order result. This is not so. The second correction (see p. 136) has shown this cannot be the case. The apparent contradiction lies in the absence of dynamical correlations between particle and target in the wave function (Eq. 38). We correct this situation by considering a many-body perturbation theory for singly-excited, (p, h), states. The essential idea is this. A diagrammatic sum for some (p, h) states of the system may be carried out30even in the presence of the scattering, on the extra nucleon. To reduce the effect of this particle to lowest order in perturbation theory is to provide an external field thereby changing the self-consistency of N A z . In this lowest order it is a good approximation then to replace the wave function of Equation 38 by We proceed now to derive a representation for the state which will turn out to be a vibration of the singly-excited configurations.

James E. Young

139

The class of (p, h) states @,(p, h; A) being considered are taken to satisfy the equation (E,

- T, - ~h - up-

u~)@H

=

Ctlph@H

(41)

with the conjugated interaction31being defined in terms of Equation 37. This being so, Equation 41 can only hold in the sense of perturbation theory as the effective interaction t', in the presence of the extra particle is taken to be the same for ground and excited states. We proceed to solve Equation 41 by what is now a familiar procedure. Essentially we argue that (a, can be generated from one of the states 4, satisfying where by means of adiabatic switching as summarized by the Gell-Mann, Low procedure.32The sense of this method for our purpose here is that writing @, as @x

=

c

c H ~ ,

,

(44)

where we obtain an integral equation

for the correlated state. We can maintain that even when the forces are strong enough to produce a state having correlations not present in c$,, thereby invalidating the switching procedure, Equation 45 nonetheless holds. This equation or its equivalent

can then be solved by the Fredholm resolvant method33to obtain states of frequency not appearing in 4,. The description of the state @, is completed by noting that its eigenvalue of the energy is

following from Equation 44. Finally 4,, is any convenient, i.e., "chosen," (p, h) configuration. We write as in the ground state problem

This is followed by the definition

for the operator J'. for hole-particle interactions so that

This is an approximation to the (p,h)-vertex function occurring as a kernel in the integral equation for density correlations in the Green's function theory of (p, h) states.32The function a, satisfying Equation 49 is a solution of that integral equation. The results (Eqs. 48-50) are not then just a simple transcription of the usual multiple-scattering formalism. The states a, bring J', to diagonal form. The operator is not diagonal in the 4,,representation. We will restrict ourselves to states a, of the set @, which are collective. The projection instruction Qo of Equation 46 is then automatically fulfilled. The restriction to what will be low-lying collective, (p, h) states or vibrations, carrying all or most of the transition strength, is an approximation in our treatment. Its virtue is that of simplicity. The intermediate states Is) of Equation 40 obtained in this way do yield a spectrum which is dominated by reasonably low-frequency components. It is possible to find other states Is) of two particles and a hole at lower energy than these just discussed. These will appear in a discussion in one of the later sections.

141

James E. Young

The ultimate algebraic simplicity of our presentation is achieved by literally taking over multiple-scattering equations, i.e., and to obtain

and

So that even if one used perturbation theory vibrations, retaining J', to first order, i.e.,

Q8 =

0,

Qa

=

1 for the

the expression (Eq. 39) for the resonant part of the transition amplitude, using states (Eq. 48), would not be second order in operators ttojand t', as was to be shown. The transfer of excitation from the external nucleon to individual nucleons of the target occurs via Ctfoj. The target nucleons then exchange the excitation via

the second term of Equation 53. It is just this correlation between pairs of particles that gives a non-vanishing resonant amplitude

where and

It is, of course, considered in all of this discussion that tfoiand tfikare the interactions defined in terms of t'[(Co)],a single interaction operator, for the hypothetical configuration CO= I &rlk(f)(xo)@~(t~)). In summary then elastic scattering is given by

+

Too = (~,+(-)l UoIrlk(+)) TOO(') .

(55)

The corresponding resonant plus non-resonant form for inelastic scattering is

These modifications of the conventional multiple-scattering formalism provide us with expressions for the transition operator adequate to the description of N A Z (p, p') with possibility of intermediate structure. As we have made no explicit reference to the Coulomb force, the formulas (Eqs. 55 and 56) are also suitable for neutrons.

111. Intermediate Structure in (d,p) and ( p , p3; Isobaric Analogs

IT IS our purpose in this section to show that isospin conservation makes it possible to relate intermediate structure in (p, p') on NAZ to (d, p) reactions on the same target. We thus suggest a generalization of isobaric analogs as commonly studied. It is exhibited that the (2p, lh) compound states occurring in NAZ(p, pf), the states Is) of Section 11, are determined as isobaric analogs of (2p, lh) residual states Is') occurring in NAZ (d, p). This result follows from the existence of strong (p, h) forces, i.e., the target NAZmust exhibit lowlying, (p, h) vibrations in its excitation spectrum. We discuss the role of shell effects upon intermediate structure, normally, the strong coupling of states Is) to open proton channels at closed shells. This feature has been emphasized by Kerman.34 Here we can exhibit its occurrence as an aspect of a true configurationmixing shell model. Finally, we discuss in what sense the concept of intermediate structure has relevance for, or pertains to, the (d, p) reaction. (a) Analog states or single-particle states in (p, p') Our primary concern is that of describing N A (p, ~ pf), proton scattering, at energies below the Coulomb barrier, i.e., the Coulomb displacement energy. When one goes above the particle emission threshold of the (A 1) system, as in NAZ(p, p'), it is sometimes possible to single out especially simple configurations of the compound nucleus strongly coupled to the entrance channel. Two such configurations, analog states and intermediate structure states, are well

+

known; both are essentially configurations dominated by (2p, lh) excitations. They get their widths, however, in quite different ways and thus need to be carefully distinguished. It is known1' that the analog states Ip; c) of NAz+~,or the NAZ (p, p') compound states of good isospin, are related to the single neutron states of N+IAz or the NAZ(d, p) residual states, In; c), through charge independence, [H,I] = 0. In the situation ( N - 2) 2 10, that of a large neutron excess, the analog states become pure, being dominated in this limit by (2p, lh) configurations in isospin. Ac - M,,) These states are located at excitations Eii = (-SIN Qm(p;c) in the (p c) system. Here Ac is the Coulomb displacement, Ac = 1.33 (Z/A1D) MeV, and SjNis the separation energy associated with a single neutron state of the In; c) system. The neutron-proton mass difference is M,, = -0.78 MeV and Q, is the mass Q of the Ip; c) system. The core jc), i.e., the ground state of the nucleus NAZ, is characterized as

+

+

+

where C n i = N-Z. There exist neutron configurations Vi consisting of ni constituents in the orbital Ji,the total number of such neutrons being the excess (N - 2). The discussion following is generally restricted to i = 1. The core state is taken to have spin-parity 0+ but is degenerate in isospin, i.e., lof 0. The isobaric analogs

yield two components as depicted in Figure 7. The equivalent wells for the proton and neutron systems have been drawn in detail in Figure 8. It is possible to see that the two components consist of: (a) a proton added to Ic) and (b) a neutron added to I A) = 1 I,, I3= lo- 1), the isobaric analog of Ic). Now the latter component does indeed consist of a neutron (h, p) pair plus a proton and is commonly referred to as a (2p, lh) state. Further, there are ( N - 2 ) ways of achieving (b) and only one for (a). Nevertheless the component (b) appears experimentally18 as a single-particle configuration. The cross section to the analog states in NAZ (p, p') are single-particle dominated. This is not

James E. Young

(a)

("1

Figure 7. Isobaric analogs of the single-neutron state a+.oNlc):(a) extra neutron in space-spin state so is changed to a proton; (b) extra neutron added to analog \ A ) of Ic) formed by changing one of the (N - Z ) core, Ic), neutrons, that in state st, to a proton.

surprising for the component (b) is unmistakably that associated with excitation for single particles. And, this fact owes to the absence of (p, h) forces in b. The component (b) must be interpreted as the state In; A ) as we have indicated above. This is sufficient to support the argument of no (p, h) forces. It is easy to show that nuclear intermediate structure relies upon the existence of (p, h) forces for its manifestation as a physical phenomenon. A more telling argument is that the isobaric analogs identified in Reference 18 correspond to states in (d, p) having large single-particle components. The widths of the analog states are typically 20 keV or so at excitations of 15 MeV in the compound. The states get their widths by mixing of isospin states through the Coulomb force. This is the sort of result we would expect in a literal application of the nuclear shell model to this continuum situation. The states occurring are classified according to the representations of SU(2). Then the Coulomb force is treated as a perturbation mixing the multiplet structure and lifting the degeneracy previously existing. This description is to be contrasted with that obtaining in a configuration-mixing shell model. In the latter, one selects appropriate physical configura-

Figure 8. (a) Single particle wells for NAZ system, ( N - Z ) > 0, which is stable, BN =: BP. Alternatively, the mass M(N, Z ) is less than that associated with the two neighboring isobars (Z f 1). The Fermi energies, t,(i) = h2kf2(i)/2Mi,k,(i) = are fixed by the densities pi of the species, i = N orZ, and their masses Mi.The Coulomb interaction per proton ilVc = (%)(Ze2/Ro) makes the proton well relatively more shallow. The neutronproton mass difference Ma, = 0.79 MeV deepens the well. The displacement energy is ilc = (1.15)(Ze2/Ro)and locates the isobaric analogs. (b) The level sequence in Nk = Zk particles fill the first n orbitals, S,, the excess neutrons ( N - 2)filling up to orbitals S',. Pairing ( N o r 2 2 28), neglected here, dictates a change of language and technicaldetail but is irrelevant to our considerations.

James E. Young

147

tions and then diagonalizes the residual force among these. The physical configurations are thus strongly coupled. This is the situation for nuclear intermediate structure. Isobaric analog states provide an example of what are called symmetry-breaking solutions in elementary particle physics. The nonvanishing expectations of products of nucleon spin are computed in the vacuum state 10). That state is degenerate in isospin and violates a symmetry, charge independence, obeyed by the Hamiltonian H. That operator contains the isospin-dependent potential introduced by Lane.35 Diagonalization of the charge-rotation operator among the basis states of degenerate vacua produces the finite energy shift Ac between the Ip; c) and In; c) configuration. Again, that these results emerge from a discussion of the matrix form of one-particle Green's functions emphasizes the single-particle nature of isobaric analog states. (b) Perturbation theory of intermediate structure in (p, p') We return here to a discussion of some fairly elementary notions of intermediate structure in proton elastic scattering by examining a phenomenological parametrization of the corresponding transition amplitude. This kind of description has been previously explored by Swenson and Izumo36at energies above the Coulomb barrier. We assume that there exists intermediate structure in ( p , p') carried out below the Coulomb barrier, E < Ac, and parametrize the scattering amplitude accordingly. In these circumstances, what, if any, qualitative conclusions are possible? We are in effect making a statement about the compound elastic cross section, which can be illustrated schematically in the following way. We consider that the Smatrix in the elastic channel can be represented as The interpretation is that the compound states that are distant in energy provide a slowly varying background 5. The complex phase 4, q = exp 2i4, associated with 5 is considered to be computed from the usual optical potential Uo.5The remaining compound states, the nearby states, contribute to the rapidly varying energy dependence of elastic cross sections in the resonance region. Our remarks are re-

stricted to this region. The rapid energy dependence is summarized by 67. The optical potential may be chosen so as to reproduce the energyaveraged, (qo),S-matrix12 (770) = v

(60)

Y

the average being performed over an interval locontaining many compound states, lo>> D, as is usually stated. The mean spacing of the states is D. We now perform an average over an interval I rp(say I = 50 keV and r, = 20 keV so that we can drop rp/2) and was defined in Equation 64. No attempts to evaluate Equation 86 exist to date. The object of an experimental investigation would be to study Equation 86 as a function of E between the analog resonances Ep at good resolution. The analog resonances sit on the potential scattering, S6, background. And, the same is true for the intermediate structure resonances we hope to study. It is an experimental fact that that background is stable against the open channel coupling to the intermediate resonances. Intermediate resonances if they exist do not break up the continuum shell model. That the background scattering from the distant compound states is at all associated with a shell model depends upon the inelasticity (Im Uo)through Equation 74.8If the inelasticity is small enough, the nomenclature is appropriate. Otherwise one must state that the resonances sit upon a diffractive, Ramsauerg background. This is also stable against the coupling to the resonances. Any other conclusion is incompatible with experimental observation to date. The background, whatever its description in terms of Uo, is smoothly varying on the scale of energy, I. This is the point. The immediate reaction of the experimentalist to the proposal just made may well be one of bemusement. What is suggested is that (p, p') elastic scattering be performed as a function of E, for the condition E < A, at the "best" resolution possible (e.g., 5 keV at E = 10 MeV, on Wn). The resonances appearing will be: (a) those E, associated with analogs of single-particle, or stripping, (d, p) states; and (b) those E, associated with the analogs of (2p, lh) states whose strength function is inferred from the non-stripping (d, p) states. The averaging of the 03, pf) data over intervals I to produce intermediate

James E. Young

153

resonances E, is to be done on the fine resolution data. If the resolution could be improved further (e.g., 2 keV at E = 10 MeV, on llsSn), the analog states would not have to be left out of the average I; otherwise, they must be suppre~sed.~~ This situation is to be contrasted with that in which we average some (d, p) data to obtain the (2p, lh) strength function, as in point (b) above and in Section 111. d. We are essentially implying that the analog states are less strongly singleparticle states than the stripping states of (d, p). The full notion of a perturbation theory of intermediate structure, connected with stability of the background scattering, mentioned previously, can be examined in the context of many-particle Green's function theory. It is only feasible to discuss certain of the details and leave others, which are principally definitions, to the reader. We will consider a rather strange kind of perturbation theory, for all of the diagonalizations are performed in the (2p, lh) and singleparticle subspaces. The stability of the latter in the presence of the former is a statement that the coupling potentials and thus the widths may be computed in perturbation theory as usually conceived. This is also the kind of result that the theory gives when formulated in terms of Feshbach's projection operators insofar as concerns coupling to open channels.13 Our method is different from his, but the physical statements are equivalent. The propagation of a single nucleon added to a non-degenerate ground-state target, NAZ,qO(cA)= 10) is discussed in terms of the one-particle Green's function41 Gk(t - t') = -i(Tak(t)atk(t')). (88) The time-ordering symbol is T and the fermion operators, ak(t) = e-iHtake"t

(89a)

and satisfy anti-commutation rules [ak, atkl+ = bk.kl, etc. The Hamiltonian of the system is H, and k labels all of the discrete variables necessary to specify a state exclusive of the energy. The Fourier transform of Equation 88 with respect to the time difference

- t'

produces the function Gk(w). The explicit form of this Green's function may be shown to be 7

= t

Gk-'(w) = [GkO(w)]-l- Zk(w),

(91)

where [Gk0(w)]-' = W

- wkO.

The proper self-energy xk(w) was described in connection with Equation 16. The unperturbed propagator Gk(w) carries the spectrum wkOof unperturbed states. It is customary to write spectral representations for the propagator, and these are similar to those discussed in another connection by Lehmann42and I.

The simplest model is one in which we imagine that only certain system interactions between selected excitations of the (N ATAZ) are important, the rest being zero. Such a scheme is presented in Figure 15. Bosons 8 and 8, e.g., (h, p) vibrations of the target NAZ are permitted to interact with the capturing neutron through tri-linear couplings N 0 @ V and V I3 @ N. The excitation IV) is a stable (2p, lh) bound configuration of the (N NAZ) system. The boson I3 reflects the existence of exchange forces. If we consider the excitations to be represented as elementary fields, the formulation constitutes a statement of the extended Lee model.47A great deal can be made of a model of this kind as it is exceedingly simple to visualize the consequences of its assumptions, and intermediate structure for neutron scattering has been discussed from the point of view of such a model.48 It is not our intention to discuss the Lee model in detail

+

+

+

+

James E. Young

Figure 14 Particle-vibration scattering as computed from: (a) the Fermi coupling; and (b) the B-E-T vertex (Yukawa coupling) of Figure 13a and Reference 23.

here but only to present some of its qualitative aspects. Also we can illustrate some possible descriptions of (d, p) states which while not intended seriously are nevertheless instructive. Suppose that we are somehow given the wave function for the IV) state

In = C ~ e x ( ~ ~ > l e > l ~ > , (114) the expansion coefficients having been determined through diagonalization of the particle-vibration force. These states do not strip but may have their strength distributed by coupling to more complex

(b) Figure 15 Particle-vibration scattering in an extended Lee model: (a) corresponding to the trilinear couplings and (b) corresponding to (p, h) vibration--+); (3p, 2h) state--& particle -N.

configurations (a), involving 5 quasi-particles. We define a mixing width, measuring the strength of this coupling, as

The interval I contains 1 V) and N states / U)while Hint.is the interaction responsible for fragmentation of / V). This coupling can also be described in terms of the weak vertex, Figure 16(a), introduced in the Goldberger-Trieman49version of the Lee model when la) is approximated as a product state. In general we write These states are not single-particle dominated; f,is a single-particle state, when la) results from a diagonalization. This is the case for la) = 1 V), V e @ V, Figure 16(b), in Bronzan'sso version of the Lee model. We do not admit this diagonalization for la) but take the C's of Equation 116 to be Icronecker delta functions. The states la) in this approxin~ationare single-particle dominated and have the representation, Figure 17, of a particle added to a vacuum fluctuation. The states 10') in Equation 116 are to be taken as 2+ and 4+ excitations, the latter being considered as (2p, 2h) states when there is strong coupling between the 2+ and 4+ states,18in this approximation. It is necessary that the states la), Equation 116, defined by our ap-

+

James E. Young

169

Figure 16 The mixing of (3q) and (!q) states, 6" = (2p, 2/z), can be computed from the weak vertex (a) of Reference 49. If the (59) states were to be produced by diagonalization of the residual force, then the Bronzan vertex (b) would be employed.

proximations, have only a weak single-particle component in order to make the (d,p) separation of stripping states. This can be arranged. In perturbation theory such as is implied here, the first-order expression for the expansion coefficients is

We assume first-order degeneracy and

1

t ' ~

Figure 17 The 5q states lo) of the theory have the structure of a particle added to a vacuum fluctuation.

which is to say, weak coupling to single-particle components as described above. The degeneracy is removed by going to second order, and we account for the first-order mixing of states la) with those IV) for three quasi-particles in the expression for C.slp,

by taking li) to be in the set IV) of (39) states. We further identify a chosen configuration5l 10'~p,,)for each Iu) state, i.e., noting also that (vlHint.lO'p)= (~1Hi~t.l V)

G,

the coupling constant of Figure 16(a), to obtain

We take the view that a single Nv = 1, IV) state feeds the la) states. The latter make up the line shape of the IV) state as indicated in Figure 18. The spectrum of (2p, 2h) states, the 2+ and 4+ states, and that of the stripping states is used to extract Gqrom the experimental data via Equation 120. With this information the line width for the IV) states is predicted from Equation 115 as The number of la) states in I is Net,. Even though this form of the theory is not to be taken seriously, the value of models such as the Lee model in making descriptions with essentially simple predictions seems clear. (b) The (2p, lh) potential in (d, p) It was noted earlier that the (V) states were eigenvectors of the transformation diagonalizing the particle-vibration force. We want to indicate that the approximation to that force in terms of a separable interaction is simple enough to be valuable for any initial evaluations. The potential in question would be conventionally expressed in the momentum representation as

James E. Young

Figure 18 Microscopic structure on line shape of a (39) state at E, obtaining its mixing width from (59) states.

r,+

where [I] = 21

+ 1,

and is separable if V ~ PP'), = 41(p)q1(Pt)9

the ql's being scalar functions of their arguments. These results are familiar from the work of Yamagouchi.52 We take another approach here-one partially exploited by Mitras3--and construct separable potentials reproducing the Born approximation to the particle-vibration scattering amplitude,

N (c)

Figure 19 The amplitude A (s, t , u) for vibration-particle scatterirg whose Born approximation to direct (a) and exchange (b) processes determines the particle-vibration potential. Symbols are defined in the text.

of Figure 19(a). The variables are the usual ones-s = k2 being the c.m. energy; t = 241 - cos 8) being the momentum transfer for the scattering angle 0-the variables satisfying so that t, describing space exchange, is not independent of the other pair of variables. The amplitude of Equation 223 is a sum of direct and exchange terms. We wish to study each of these in turn. The direct contribution is summarized by the graph of Figure

James E. Young

173

19(b). It gives rise to the potential matrix elements, in the relative

124,

X the strength of the interaction, rn. with & = - t f 2 , &I = isordinarily the mass of the exchanged ( V )particle but here taken as a parameter. The Born amplitude is

where z = cos e

(126)

defined in terms of Equation 125. The partial wave projection is then taken to give

involving the Legendre function of the second kind. It follows that the form factor ql(p) = uz(s), for a separable potential is to be taken as

The exchange is treated in the same way. Corresponding to Figure 19(c) we have (ill v,\lz> =

X

+

+

( i kt):! mu and the partial wave amplitude

-

X 4s - t

+ m,

(129)

The potential form factor becomes

it being understood that the coupling constant X is relatively, i.e., to Equation 125, repulsive.

Being given the N - e potential in the relative system we may diagonalize it to form the ( V )state. The total angular momentum j, of the state is formed as [YbmoXMs8(ls; nl)lmqi.,the vector-coupled product of orbital and spin-angle functions. It is our assumption that Equation 125 acts in the state lo= 0 and Equation 126 is that lo = 1. We have then specified a two-parameter (A, m,) potential, separable and non-local, for our problem. The IV) state we obtain by our calculation is an approximation to the (2p, lh) states of many-body theory depicted in Figure 20. In order to work in the relative N - e system, it is convenient to define a point vibration, namely, a vibration with no internal structure which nonetheless transforms under rotations as an irreducible tensor of appropriate rank. We can carry through this specification in an obvious way when the unperturbed (p, h) basis states are defined in an oscillator potential. Starting from the wave function of the schematic model55

where

Figure 20 The (39)states of many-body theory (labelled by p, m, [) have strong particle correlations through t's and (h, p) correlations through J'.

James E. Young

and

on the unperturbed funcwe make the Moshinsky tran~formation~~ tions la)to the relative and total coordinates. The relative coordinate is taken equal to zero and we obtain

The overlined quantum numbers in transformation brackets and state vectors indicate that the explicit phases required for the recouplings involving hole states are accounted for. We also have the following notation:

x O ~ ,isospin amplitude;

(LS - jj) transformation where [a] = (2a

[I1 J

+ 1);

, rector coupling coeficient;

1(1/2 V)SM,) , spin amplitude; (nlllnfli'l;LlnlNC; L) , Moshinsky bracket; Fmnl(9)= Rnz(r)Yl,(O;) , oscillator wave function.

The point vibration we have formed makes it possible to speak of a particle-vibration potential as in Equations 125 and 129. The particle states IN) are, in a notation similar to that above, the spin angle function. with yfioj:zol/2,

These and the 10) states are coupled to good total angular momentum J and isospin I to form a basis, Equation 114, for the I V) states. Their equation of motion is taken to be

in an obvious notation. Diagonalization of Hint.,a non-local operator in coordinate space, involves the matrix elements

(FL'N~~~(?)F~"~~,z~,(~o)~ Hint .IF'~e(f)F~~n~z,(~o)) which appear in the elements of the regular matrix by virtue of the relation ((l'sj'o)JnM; (sT')IplHint.l(lsjo)JnM;(sT)Ip)

( F X ' ~ ~ ~ ~ ( ~ ) F mHint. ' ~ nJF"~~(i.)F%nOzo(to)) ~~z~~(~o)l (1 36)

where =

(fililjl, fillPljll).

For Hi,,. specified to be the direct interaction of Equation 125 we find

James E. Young

with

and

We have given the potential form factor in Equation 128. It is to be evaluated for I = 0. Similarly, the exchange interaction yields

[" '" 1

= LIT'

K'

mfo

L]

(ntLINfLf'L;L'NT'n'ol'o; L') ir'

n'LL'Lfi'L

e lo

L

where

and the form factor, I = 1, Zil(s),has been given in Equation 131. It is obvious that the magnetic sums over (K'L, u) can be performed in Equation 140 to give the vector equality L = L'. Evidently the determinant of the secular matrix vanishes to give the spectrum of IV) states

I I(E - Es,v(i))Gij - (ilHint.lj)l I = 0.

(142)

+

The unperturbed energies, EON = 60 E,(N), are the sums of particle energies (Ref. 23) and the 2+ and 3- vibration energies, as relevant. Note that in pairing theory the 2+ states are (p, h) states, or at least have such components, in the quasi-particle labels.57Our analysis has been specific to those nuclei, (N - 2 ) >> 1, where pairing occurs. Actually the T = 0 vibrations to which we are restricted are brought about by the action of the long-range specificity force58 in the pairing plus polarization potential models. In solving Equation 142 for E(JuI, I3= I), the energies of the I V) states, we still have to specify (A, m,), the two-force parameters. This is done iteratively, or in any other convenient way, such as to bring the I V) state strength function into agreement with that inferred from the energy averages over the non-stripping (d, p) states. We have worked so hard to develop our 2-body approximation to what is a 3-body problem in the hope that we obtain a good 1 V) state spectrum, and corresponding wave function. (c) The (2p, lh) strength function in (d, p) We have previously stated the major physical ideas pertinent to this heading but it is necessary to reiterate some of these for the sake of coherence. The program of the theory is to obtain a good characterization of the non-stripping residual states of (d, p). These are the states Iu). When the cross sections to these states are summed within an interval I, the resulting excitation function exhibits maxima which are the loci of intermediate structure states Is) = IV). A good characterization of the Iu) states is in reality difficult to obtain as these are states involving at least 5 excitations. It was already hard enough to get some approximation to the Is) states. The best characterization which is available seems to be that of the frivolous model (Eqs. 116-121). That picture says that all of the (d, p) configurations, simple and complex, associated with the observed residual states have single-particle components Ip). These are strong for stripping states and weak, i.e., have no first-order matrix elements to specific configurations, for non-stripping states. That la) contains Is) when summed over I just reflects the excitation chain Ip) -+ IS) + la), the concept underlying Feshbach's notion of a doorway state.4 The scale I is taken to be a fraction of that I,,associated with single-particle averages. Note that Equation 118 plus Reference

James E. Young

179

23 is the mathematical statement of these ideas. It would be concluded that the extensive fragmentation of the single-particle strength in (d, p) through particle couplings to complex excitations is contingent upon the strength of states Is). This has been emphasized by Brown and Lande.59 The strength of the states Is) is just a measure of their transition intensity into states Iu). We may define the strength function as

with D being the mean spacing of states la) and (r,2)a,. the mean mixing into complex configurations given by

Not only does it seem that these definitions are unmotivated, but also we shall have complicated things unnecessarily unless some new information is forthcoming. This will turn out to be the case as we show in the next and last subsection, (d). We are to find there that sum rules can be constructed for intermediate structure parameters owing to an invariance satisfied by ~439). For the present we conclude this part of the discussion with some additional notions about the (d, p) reaction. Recall that the amplitude for the process in question is

This is commonly evaluated in plane-wave Born approximation

or in distorted-wave Born approximation

In the event that either of the approximations furnishes a good description of the experimental data, the intermediate structure or (39)

states Is) play the part of hidden configurations. Let us see how this comes about. A representation of the residual (d, p) states Ir) based upon a series expansion in the system density is

The single-particle states f,,are those of Reference 23. The states $I, = Is') = 1 V) are defined by Equation 142 and those /a)by Equation 118. We make the important point that although ls') can be expanded as in Equation 114 over the basis of particle-vibration states, it does not have the correlation structure of either particle or vibration state because of the diagonalization (Eq. 142). Substitution of Equation 148 into Equation 147 yields: (a) the big contribution from stripping states going with f,;(b) the weak contribution from "non-stripping" states associated with la) of Equation 116, all of that discussion being relevant here; (c) the even smaller and nearly zero contribution from Is1)having the wrong correlation structure on the wrong number of excitations present. It is to be observed in connection with (b) above that the (2p, 2h) excitation 18') of Equation 116 is also found in the second term of q0(tn)as defined by Equation 33. This is the feature that keeps the contribution to Equation 147 from vanishing. These remarks serve to indicate that: (a) the presence of 1s') in Ir) of Equation 148 is not inconsistent with the failure of the states Is) to be observed in (d, p); (b) averages (sums over intervals I) over the non-stripping states la) can produce the Is') strength owing to the configuration mixing implied by Equation 118. (d) Isospin conservation and resonance reaction parameters for (p, p') Principally, here we will examine the consequences of isospin conservation for the open channel proton interactions below the Coulomb barrier. We would like to say that under isospin transformations taking us from the (d, p) to the (p, p') system the strength function for the (3q) states is left invariant, namely,

James E. Young

181

with E ~E = s+Ac,

where i refers to the quantum numbers of the residual and compound states exclusive of the energy. Such a sum rule would constitute a very powerful tool for the investigation of (p, p') below the Coulomb barrier since indeed holds with D(E*)being the mean spacing of (59) states la) at the excitation ES.The ratio D(c,)/D(E,) could be estimated by counting. The damping width is easy to compute since to within an undetermined phase; the elastic-scattering wave function qs(h)is reasonably accessible for energies below the Coulomb barrier; Hint.is a residual interaction to a good extent approximated by a delta-function Hint.

AO 8(ri - rj)

=

9

i