Walter D. Loveland
Oregon
State University
Corvallis, 97331
Compound Nuclear Reactions
Chemists have long recognized the many similarities between atomic and nuclear structure. have pointed Gordon and Coryell (I), in THIS JOURNAL, out how ideas drawn from the study of atomic structure can be used in trying to understand nuclear structure. In a similar vein, many authors have pointcd out similarities between chemical reactions and nuclear reactions. These similarities have become more apparent with recent interest in studies of chemical reactions with beam techniques and "chemical accelerators." The theoretical techniques and phenomenological models used to interpret nuclear reaction data are now being applied routinely to the study of chemical reactions. For this reason, as well as for general scientific interest, it is important that chemists have some familiarity with the techniques and models used to discuss nuclear reactions. This article deals with a review of current developments in the "compound nuclear model" of nuclear reactions. In the study of nuclear reactions, unlike many studies of chemical reactions, a great emphasis is placed on detailed knowledge or measurement of the initial and final states of the reactants and products. For example, let us consider a typical nuclear reaction as shown in Figure 1. "Incoming projectile" nucleus a strikes stationary "t,arget nucleus" A giving rise to a nuclear reaction with products "emitted particle" b and "residual nucleus" B. Typically one might measure the nuclear charge and mass of all species as well as their kinetic energy, the direction they are moving in space (usually measured relative to the incoming projectile direction) and in some cases, the nuclear polarizations. The information gathered in these measurements is usually summarized by expressing the probability that a given reaction will occur with the emitted particle b having kinetic energy E and moving into solid angle dn at plane angle 0 with respect to the incident beam (see Fig. 1). This probability is called the double differential cross section (dZu/dndE) and has units of (cm2/MeV steradian). The number of reactions occurring per unit time with emitted particle b having energy E and moving into solid angle do is given as number of reactions (unit time-energy-solid angle) = N '
dzc
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dFTER
BEFDRC
Figure 1. A schematic view of o nuclear rsoction showing incoming proiectile a striking target nucleus A giving rise to residual nucleus 0 and emitted particle b moving to solid angle do.
emitted particle energies and angles) is the total cross section, a, which is given by
[For a more detailed discussion of the cross section concept and its application to chemical reactions, the reader is referred to the article by Greene and Kupperman in THIS JOURNAL (IS).] AS we shall show, the quantity (dZu)/dQdE) can tell us much about many detailed features of the nuclear reaction mechanism, such as the angular momentum transfer during the reaction, the nature of the reaction intermediate, etc. In addition, nuclear reactions can be used as nuclear spectroscopic tools giving much information about the quantum states of reactants and products. Nuclear reactions, like chemical reactions, can occur via different rcaction mechanisms. Weisskopf (8) has presented a simple conceptual model (shown in Fig. 2) INITIAL STAGE
I I
INTERMEDIATE SWE
FINAL I I
SmGE
(')
where N is the total number of A nuclei present in the "target" and $ is the number of a nuclei striking the target per second per unit area. The total probability that the nuclear reaction will occur (integrating over all Work supported in part by the U.S. Atomic Energy Commission.
Figure 2. A conceptual view of the stages of a nvcleor reaction (after Weisskopf 1211
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BEFORE
BEFORE
AFTER
DURING
A
AFTER
Figure 3. Schematic view of the various types of nuclear reaction mechonitms. IAI Shope elmtic scattering [the energies ore given in the center of mars svteml, (81 Direct reostionr, (Cl Compound nuclear reaclion.
for illustrating the relationships between the various nuclear reaction mechanisms. Consider a general nuclear reaction such as that shown in Figure 1 (bearing in mind that for some cases, the nuclei b and B may be identical t o a and A). As projectile a moves near target nucleus A, a will have a certain probability of interacting with the nuclear force field of A causing a to suffer a change in direction but no loss in energy (see Fig. 3a). This reaction mechanism is called shape elastic scattering. If shape elastic scattering doesn't occur, then a may interact with A via a two body collision between a and some nucleon of A, raising the A nucleon to an unfilled level (see Fig. 3b). If the struck nucleon leaves the nucleus, a direct reaction is said t o have occurred. If the struck nucleon doesn't leave the nucleus further two-body collisions may occur and eventually the entire energy of the a nucleus may be distributed among the nucleons of the a A combination leading t o the formation of a compound nucleus C (see Fig. 3 4 . Because of the complicated set of interactions leading to the formation of the compound nucleus, loosely speaking, it "forgets" its mode of formation and its subsequent breakup only depends on the excitation energy, angular momentum, etc., of C and not the nature of a or A. Sometimes the compound nucleus C may emit a particle of the same kind as a (or maybe even a itself) with the same energy that a had when the original reaction process started. If this happens, we say that compound elastic scattering has occurred. Also C may decay into reaction products B b, D d which are unlike either a or A. Detailed theoretical formalisms and phenomenological models have been erected to describe these different reaction mechanisms. This article deals with the compound nucleus stage or model because of the similarity between the "activated complex" of chemical reaction theory and the compound nucleus. (Similar
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analogies exist between other nuclear reaction mechanisms and features of chemical reactions in molecular beams.) I n the next section of this article, a qualitative introduction t o the features of compound nuclear reactions is given while in the following section attention is given t o the occurrence or resonances in compound nuclear reactions and statistical features of these reactions. The final section deals with a special topic in the study of compound nuclear reactions:.the occurrence of intermediate structure. The literature dealing with nuclear reactions and in particular, compound nuclear reactions (or CN reactions as they are called) is extensive. General introductory level treatments are found in standard nuclear chemistry textbooks such as Harvey (3) and Friedlander, Kennedy, and hKiler (4). A highly recommended more advanced treatment is found in the textbook by hlarmier and Sheldon (5). Many excellent review articles have appeared including those of Bodansky on statistical aspects of compound nuclear reactions (6),and the article of Thomas (8) on heavy ion-induced compound nuclear reactions which also contains an excellent general review of compound nuclear reactions. Intermediate structure and resonances have been treated in a recent book by Lynn (9). General Description of Compound Nucleus Model
The compound nucleus model of nuclear reactions is used to discuss those nuclear reactions that seem to involve a complicated set of two-body interactions that lead to a relatively long-lived reaction intermediate, the compound nucleus. How long does the compound nucleus live? From our definition above, we can say that the compound nucleus must live for/at least several times the time it would take a nucleon'to traverse the nucleus sec). Thus the time scale of CX reactions is of the order of 10-'6-10,~'8 sec. Lifetimes as long as lo-'& sec have been obderved. These relatively long lifetimes should bc compared to the typical sec. time scale of a direct reaction of Another feature that is sometimes attributed t o the compound nucleus is that its mode of decay is independent of its mode of formation (the independence hypothesis or amnesia assumption). While this statement is not t,rue in general, it remains a useful notion for understanding reactions involving the excitation of a single level of the compound nucleus or the cross section for reactions involving a vide range of compound nuchar and product energies. For example, let us consider the classical work of Ghoshal (10). Ghoshal formed the c~mpouudnucleus 64Znin two ways, i.e., by bombarding WUwith protons and by bombarding W i with n particles (see Fig. 4). He then examincd the relative amounts of "CU, W n and "Zn found in the two bombardments and within his experimental uncertainty of lo'%, he found that the amounts of products were the same in both bombardmeuts, thus "verifying" the independence hypothesis. (Later experiments have shown smaller scale deviations from thc independence hypothesis.) Mathematically we can express the ideas described above using the Hauser-Feshbach formalism (11). Neglecting split-orbit coupling, we say that the total cross section for a compound nuclear reaction, uCN involving projectile a striking target A leading t o a
angular momentum conservation is not trivial. I n the following section we shall discuss how some of these operations are carried out for special situations. Special Cares of Compound Nuclear Reactions
compound nucleus C which in turn .decays t o residual nucleus B and emitted particle h is given by
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where the letters a, P, and r refer to A a, B b and B b' combinations which are called the "channels" of the reaction. a A is t,he entrance channel, while B b is an exit channel as is the combination B' b'. I n the "channel picture" of a nuclear reaction, we think of the compound nucleus like a small lake where the entrance channel lets particles in and the exit channels carry away particles. Each channel is described by the nature of the particles involved and their quantum states. The quantities 1 and E refer t o the orbital angular momentum and energy involved in a given reaction channel. X is the rationalized de Broglie wavelength of a,' The "T's" are transmission coefficients and reflect the probability of either forming the compound nucleus via channel a(T,, (E,)) or the probability of compound nucleus decay via channel P or r(Tca (EB),TI, (E,)). Said in words, eqn. (1) canhe written as
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s : ;
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(probability of forming the compound nucleus) X (relative probability of decay into channel 8 )
Note that the probability of forming the compound nucleus is taken as the product of the geometrical area swept out by the incoming projectile (rXa2) times the statist,ically weighted sum of the probabilities of a penetrating A is a given channel ar(Z(21, l ) T h (E,)). Note further that the relative probability of decay of the compound nucleus is simply the probability of decay into a given channel (3 divided by the total decay probability. The problem of discussing CN reactions and predicting their properties is then one of calculating the transmission coefficients. This is done by solving the Schrodinger equation for various projectile-target or emitted particle-residual nucleus combinations. The actual performance of the summations indicated in eqn. (I), and the proper accounting for energy and
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This statement is true only if the mass of s, M., is much less than the mass of A, AL. Strictly speaking, X, = h/(ZM,E,)'/l where ,Wmis the reduced mass ( M . M A / M , M A ) and Em is the ehmnel energy of a.
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Figure 5 shows a schematic view of the quantum levels of the compound nucleus reaction intermediate. Note the increasing number of levels as the CN excitation energy increases. Quantitatively, the number of levels per MeV of excitation energy, E, increases approximately exponentially as ELI2. The interesting categories of CN reactions can be defined by the ratio of the width of a compound nucleus level, r, t o the average spacing between compound nuclear levels, D. (Recall from the Heisenberg uncertainty principle that F . T 2 h, where T is the lifetime associated with a given compound nucleus level.) The categories of reaction are (a)r/ D > 1, the case of many overlapping levels in the compound nucleus (see Fig. 5 ) . Intuitively category (a) reactions are those in which the excitation energy of the compound nucleus is low while category (b) reactions are those in vhich the excitation energy is high. Let us consider first the case of r / D
