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The formation of the compound nucleus
Weisskopf, V.F. 1956
THE
Physica XXI I 952-958 Amsterdam Nuclear Reactiohs Conference
FORMATION
OF THE
COMPOUND
NUCLEUS
b y V. F. W E I S S K O P F *) Physics Department, Massachusetts Institute of Technology,Cambridge, Mass., U.S.A.
Synopsis The problem of the compound nucleus can be divided into two parts. One is the formation of a compound system, and the second is the decay of the compound system into its products. In the last years progress has been made in respect to our knowledge concerning the formation of the compound nucleus. Especially the optical model with a complex potential has given us a tool to distinguish compound system formation from potential scattering. Hence, we have collected some knowledge as t6 the probability of compound formation as a function of the energy of the incident particle. In particular, the transition was observed from the small interaction known from the shell model to large interaction at higher energies. The detailed theoretical explanation of these effects is still unknown. However, the qualitative trend is easily understandable. The second problem of the decay of the compound system is a much more difficult one. Here the question is raised as to whether the decay is independent of the way of formation. A critical discussion of this problem shows that the conditions for this independence will only rarely be fulfilled. Hence, conclusions based upon this inde.pendence are idealizations which cannot be expected to describe the details of the observed nuclear reactions. Deviations from the B o h r hypothesis of independence are caused not only by direct interactions between the incoming particle and the outgoing nucleon. They are also caused by the lack of "thermal" equilibrium in the compound state. Some general reasons for this conclusion will be given. U p to a few years ago our t h i n k i n g a b o u t nuclear reactions was d o m i n a t e d by N. B o h r s suggestion, which considers a reaction t a k i n g place in two stages: The f o r m a t i o n of a c o m p o u n d nucleus (C. N.) a n d its s u b s e q u e n t decay. I t was a s s u m e d t h a t the s t r o n g i n t e r a c t i o n b e t w e e n the nuclear c o n s t i t u e n t s causes a complete coalescence b e t w e e n the incident particle a n d the t a r g e t nucleus when the two come within the range of interaction. A c o m p o u n d nucleus is f o r m e d i m m e d i a t e l y after collision, a n d the inc o m i n g particle forms a p a r t of it, which is indistinguishable from a n y o t h e r nucleon. The e n e r g y a n d m o m e n t u m of the incident particle are q u i c k l y d i s t r i b u t e d over the whole s y s t e m and, hence,, the properties of the C. N. do not d e p e n d u p o n the detailed w a y of formation. This i n d e p e n d e n c e is an integral p a r t of the B o h r a s s u m p t i o n : The second stage, the d e c a y of the *) Paper read on July 3d, 1956 at the Amsterdam Nuclear Reactions Conference. --
9 5 2
--
T H E FORMATION OF T H E COMPOUND N U C L E U S
953
C. N., depends only upon its over-all properties, as its energy or total angular momentum. Recent developments have shown that the B o h r assumption is an idealization, which is not valid in all cases. The detailed observations of nuclear reactions have revealed m a n y important deviations from the predictions based upon the Bohr assumption. Therefore, it is necessary to use a more general scheme for the description of nuclear reactions, a scheme which includes also the phenomena which do not fall within the frame of the B o h r assumption. First of all, it became probable, that the incident particle does not necessarily coalesce immediately with the target nucleus forming a compound nucleus. The experiences with the shell model have shown that nucleons can move like free particles in thin nucleon matter. Furthermore, certain properties of nuclear reactions strongly indicate that sometimes the interaction takes place directly between the incident particle and one or a few other constituents of the target, without forming a C. N. as an intermediate stage. In this case the energy or momentum transfer from the incident to the particle takes place more or less directly without having the incident energy distributed among all constituents of the target. We therefore use a three-stage description for a nuclear reaction, which is based upon the following ideas. As a first approximation the target nucleus acts upon the incident particle as a whole. This action can be described in form of a potential. In this approximation the motion of the incident particle is described by a one particle problem, in which the particle moves in a potential V(r) depending only upon its position r relative to the center of the target nucleus. We know from the shell model that a motion of this type plays an important role, and we consider this state of motion of the incident particle as the first stage of the nuclear reaction: the "independent particle stage" (I. P. stage). In this stage, the incident particle is influenced by the nucleus, its path is deviated, but it still can be considered as distinct from the target nucleus. The motion of the incident particle under the influence of V(r) is a description of what happens in the entrance channel of the reaction. If the potential V(r) were an ordinary real potential, it would cause only a scattering of the particle. The fact that reactions do occur can be expressed by ascribing to V(r) an imaginary part which gives rise to an absorption. Any reaction appears as an absorption from the point of view of the entrance channel. This absorption leads to the second stage of the reaction: The Compound System (C. S.). The C. S. is the state of the systems after the particle has been removed from the entrance channel. This can happen in m a n y ways. It can exchange its energy and m o m e n t u m by collision with some other nucleon, it can set up some surface vibration or some other collective
954
v.F. WEISSKOPF
motion. The C. S. is characterized b y the fact that the state of affairs cannot be described b y a one particle picture, in which the incident particle moves in a potential. It should be emphasized that the concept of C. S. is somewhat more general then the concept of C. N. as used before in the B o h r description. The C. N. is characterized b y the fact that the incoming particle has "coalesced" with the target and has formed an entity in which it does not play a role distinguished from any other nucleon. We consider the C. S. to be the state in which some energy exchange between target and projectile has taken place, regardless of the role of the incident particle. Hence, the C. S. includes the C. N., b u t it also refers to states created b y direct collisions, or b y surface excitations etc., in other words, to any states in which the incident particle is removed from the entrance channel. The exact formal distinction between the first and the second has not yet been given. There are problems involved which have been the objects of m a n y papers on nuclear interaction, in particular b y B r u e c k n e r and his collaborators. The difficulties come from the fact, that in the first stage also, there is an interaction between the incident particle and the constituents of the target ; however, the target acts only as a whole and remains in many respects undisturbed since the interaction can be described in terms of a potential. The third stage of the reaction is the break-up of the C. S. into the residual nucleus and the emitted particle. This stage is in many ways similar to the first stage running in the opposite time direction. The emerging particle finds itself under the influence of the residual nucleus before it leaves; an ~nfluence which can be described b y a complex potential. The B o h r picture of a nuclear reaction is included in his description. It is a special case which would occur under the following two conditions. Firstly, the potential in the I. P. stage must have a large imaginary part all over the volume of the nucleus, in order to insure immediate formation of the C. N. Secondly, the C. S. must be such that the incident particle shares its energy with all constituents. In general, neither of these conditions will be fulfilled. There are cases, however, in which they are approximately valid. Fig. 1 shows a schematic representation of the different stages in a nuclear reaction. Tl~e left side represents the I. P. stage, in which we distinguish the incident beam, the shape elastic scattering and the absorption. The latter contains all the possible reaction processes. The I. P. stage can be represented b y what is usually referred to as the optical model: The incident beam is scattered and absorbed b y a complex potential. The C. S. stage contains all the processes which produce the absorption. Several different schemes are indicated. For example, the incident particle might collide with one of the constituents of the target and force it to leave the boundaries of the nucleus. Such processes are called direct interactions since they lead directly into the third stage. The properties of these events
T H E F O R M A T I O N OF T H E C O M P O U N D N U C L E U S
955
depend upon the place of the collision, at the surface or in the interior of the nucleus. Apart from direct interactions, we expect multiple collisions, excitations of collective modes, surface vibrations or rotations etc., which all can lead into the third stage. Finally, the energy exchange of the incoming particle can be such t h a t the energy is thoroughly interchanged among all constituents before the break-up. In this case a C. N. is formed which has lost " m e m o r y " regarding its formation. The break-up of the C. N. takes place according to the well known statistical probability rules. It includes also the possibility of a breakup into the incident particle and the original target nucleus in its ground state. In this case we get compound elastic scattering, which adds coherently to the shape elastic scattering. I.R Stoge
C.S Stoge ~.
"'~,
Surfoce Direct Ini'n.
L/ ~
~t
~"
~
Vi ¢ t V 2 / ~
r~
Fino¢Stoge
/ // //
VolumeOirectInf'n. • Multiple ~. /-. . . . . . . .Coll'ns. . / CollectiveExci,. ~
I
"'
Opficol Model Fig. 1. Nuclear Reaction Scheme.
The I. P. stage is the one about which most is known. The description of this stage in terms of a complex potential has been quite successful. In fact, the optical model is a satisfactory w a y to describe a nuclear reaction as long as one restricts oneself to an over-all description, in which everything occurring in the C. S. and final stage is lumped together into the absorption process of the first stage. The complex potential which serves well for reactions initiated b y neutrons or protons has a very simple form: The real part and the imaginary part are assumed to be proportional to each other, and are given as follows: V=VI+iV2
V2=~V1
Vl(r) = -- Vo(l + exp[(r -- R)/a]) -1 R = roAt + rl
where r is the distance from the center of the target nucleus. This potential is a wall with rounded edges, and a is a measure of the width of the rounding. Certainly it is an idealization which is neither uniquely determined nor necessarily the best choice. Some authors 1) have added spin-orbit coupling
956
V. F. WEISSKOPF
terms to the potential which probably improves the usefulness of the method. There are quite a number of studies available now comparing the crosssections following from this potential with the experimental ones and thus determining the constants. The first suggestion of such a potential was given b y S e r b e r , F e r n b a c h and T a y l o r 2) who restricted their considerations to high energies. More recent studies are made b y F e s h b a c h , P o r t e r and W e i s s k o p f 3), N e m i r o v s k y 4), W a l t and B e y s t e r 5) and unpublished work b y F e s h b a c h , P o r t e r and C a m p b e l l 6) on neutron reactions, b y S a x o n and W o o d s 7), F u j i m o t o and H o s s a i n s) on proton reactions. The optical model can only cope with the I.P. stage of the reaction and, therefore, predicts only the total cross-section, at, the shape elastic crosssection, a,e and its dependence upon the angle, and the cross-section ac of the formation of a Compound System (absorption). In the case of protons the total cross-section does not exist. The comparison with the experiments is complicated b y the fact that only at is directly observable. As long as there is appreciable compound elastic scattering (ace) the observed scattering cross-section is ane + ace and the observed reaction cross-section is a c - ace, rather than abe and a,, respectively. However, the effect of the compound elastic scattering is negligible at higher energies. The following constants seem to give the best agreement with the experiments with neutrons between 0 and 14 MeV: R = (1.27 A ~ + 0.6) 10-is cm a = (0.5-4-0.1) X 10-13cm Vo = 43 + 3 MeV (indications of slight decrease with increasing energy) = ~0.13 4- 0.5 for energies > 4 MeV t0.7 + 0.3 for energies < 1 MeV (continuously changing between 1 MeV and 4 MeV) For protons the values for lower energies are not well determined. At higher energies one finds: R and a roughly equal to the proton values, and
V0 =
47 at 17 MeV 36 at 31 MeV [0.18 at 17 MeV "t t 0.4 at 31 MeV
The values obtained b y the Soviet-researchers are slightly different since they used a different potential shape: V=
V0forr
V = V0 exp [(R -- r)/b] for r > R
T H E F O R M A T I O N OF T H E C O M P O U N D N U C L E U S
957
T h e y find as the best fit for n e u t r o n reactions: Vo = 42 MeV b = 1.4 × 10 - 1 3 c m $ =
0.05
It is quite satisfactory that so m a n y results can be reproduced with relatively few constants. The experiments indicate definitely an increase of $ with energy although this increase is somewhat less pronounced then previously noted. The first calculations were made with a rectafiguL~r well (a = 0) and this idealization depressed the values of $ at low energy. The proton results show a definite decrease of V0 with increasing energy. This is indicated also in the neutrons at higher energy by T a y l o r 9). Unfortunately the second stage of the nuclear reaction is very m u c h less understood at present then the first one. It is still a matter of conjecture what happens when a C o m p o u n d System is formed. The accumulated experience of the last years has shown that processes other than the formation of a C. N. play a more important role than generally assumed. It is not yet possible to identify the nature of these processes with any certainty. The predominantly forward peaking of the outgoing particles in m a n y reactions indicates the importance of direct interactions. F r o m simple considerations regarding the mean free path of a nuclear particle within the nucleus (taken from the imaginary part of the potential) and the reflecting effect of the nuclear surface, it seems probable that the volume-direct interactions will be lessimportant than the surface ones. This will be true to a greater extent for proton ejections because of the barrier effects.In fact, a recent calculation by E It o n 14) has shown that even for incident particleswith energies as high as 31 MeV, volume-direct interactions will be negligible for proton emission in m e d i u m heavy elements. Surface-direct interactions are probably the most important competitors to the actual C. N. formation. Since they take place at the rim of the nucleus as seen from the incident direction, a proportionality to the first power of the radius is expected. The C. N. formation is unquestionably present in all nuclear reactions with not too light nuclei and at not too high energies. (E < 50 MeV). There is good experimental evidence, mainly from the Los Alamos Laboratory lO), that the energy distribution of neutrons emerging from a nuclear reaction is a "Maxwellian" distribution at low energy and is isotropic. This is just what one would expect from the statistical theory of the compound nucleus. F r o m these observations, it seems that in the case of incident energies of 15 to 20 M e V (neutrons or protons) a C. N. is formed with 80-90% probability 11). The "temperature" of the C. N. stillrepresents a problem since it is low (0.7 MeV) and apparently not very dependent on A. Also irregularities have been found in the dependence of the temperature on the e x c i t a t i o n e n e r g y 12).
958
T H E FORMATION OF THE COMPOUND N U C L E U S
Recently a number of authors (B. Cohen, N. R o s e n 13)) also reported irregularities in the distribution of protons emerging from nuclear reactions. Sometimes their number is larger and their energy lower, than the Coulomb barrier would allow. These and similar effects are poorly understood and indicate that nuclear reactions, particularily with charged particles, occur frequently in the surface region. The increasing amount of experimental knowledge about nuclear reactions has shown that the mechanism of energy exchange within the nucleus is quite complicated and is less amenable to a simple statistical description than we were previously led to believe.
Short communications directly/ollowing this paper were read by E isberg, p. II61, A roster, p. 1162 and Emmerich, p. 1163 o/ this volume. Received 23-8-56.
REFERENCES 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) ll) 12) 13) 14)
C u l l e r , F e r n b a c h and S h e r m a n , Phys. Rev. 101 (1956) 1047. F e r n b a c h , S e r b e r and T a y l o r , Phys. Rev. 75 (1949) 1352. F e s h b a c h , P o r t e r and W e i s s k o p I , Phys. Rev. 96 (1954) 448. N e m i r o v s k y , P., J. exp. theor. Phys. U.S.S.R. 30 (i956) 551 ; Dokl. Ae. Nauk. U.S.S.R. !01 0955) 257. W a i t and B e y s t e r , Phys. Rev., to be published soon. F e s h b a c h , P o r t e r and C a m p b e l l , low energy calcniations of neutron reactions with rounded potential wall; to be published soon. S a x o n , R. D. and W o o d s , D. S., Phys. Rev. 95 (1954) 577; Phys. Rev. 101 (1956) 506. F u j i m o t o , Y. and H o s s a i n , A., Phil. Mag. ~6 (1955) 542; B u r g e , F u j i m o t o and H o s s a i n , Phil. Mag. ! (1956) 19. T a y l o r , T. R., Phys. Rev. 92 (1953) 831. G r a v e s , E. R., and R o s e n , L., Phys. Rev. 89 (1953) 343; R o s e n , L. and S t e w a r t , L., Phys. Rev. 99 (1955) I052. G r a v e s , E. R. and D a v i s , R. W., Phys. Rev. 97 (1955) 1205. C o h e n , B., Phys. Rev. 92 (1953) 1245. C o h e n , B. and N e w m a n , E., Phys. Rev. 99 (1955) 718; R o s e n , N. (private communication). E l t o n , L., to be published soon.
THE
Physica XXI I 952-958 Amsterdam Nuclear Reactiohs Conference
FORMATION
OF THE
COMPOUND
NUCLEUS
b y V. F. W E I S S K O P F *) Physics Department, Massachusetts Institute of Technology,Cambridge, Mass., U.S.A.
Synopsis The problem of the compound nucleus can be divided into two parts. One is the formation of a compound system, and the second is the decay of the compound system into its products. In the last years progress has been made in respect to our knowledge concerning the formation of the compound nucleus. Especially the optical model with a complex potential has given us a tool to distinguish compound system formation from potential scattering. Hence, we have collected some knowledge as t6 the probability of compound formation as a function of the energy of the incident particle. In particular, the transition was observed from the small interaction known from the shell model to large interaction at higher energies. The detailed theoretical explanation of these effects is still unknown. However, the qualitative trend is easily understandable. The second problem of the decay of the compound system is a much more difficult one. Here the question is raised as to whether the decay is independent of the way of formation. A critical discussion of this problem shows that the conditions for this independence will only rarely be fulfilled. Hence, conclusions based upon this inde.pendence are idealizations which cannot be expected to describe the details of the observed nuclear reactions. Deviations from the B o h r hypothesis of independence are caused not only by direct interactions between the incoming particle and the outgoing nucleon. They are also caused by the lack of "thermal" equilibrium in the compound state. Some general reasons for this conclusion will be given. U p to a few years ago our t h i n k i n g a b o u t nuclear reactions was d o m i n a t e d by N. B o h r s suggestion, which considers a reaction t a k i n g place in two stages: The f o r m a t i o n of a c o m p o u n d nucleus (C. N.) a n d its s u b s e q u e n t decay. I t was a s s u m e d t h a t the s t r o n g i n t e r a c t i o n b e t w e e n the nuclear c o n s t i t u e n t s causes a complete coalescence b e t w e e n the incident particle a n d the t a r g e t nucleus when the two come within the range of interaction. A c o m p o u n d nucleus is f o r m e d i m m e d i a t e l y after collision, a n d the inc o m i n g particle forms a p a r t of it, which is indistinguishable from a n y o t h e r nucleon. The e n e r g y a n d m o m e n t u m of the incident particle are q u i c k l y d i s t r i b u t e d over the whole s y s t e m and, hence,, the properties of the C. N. do not d e p e n d u p o n the detailed w a y of formation. This i n d e p e n d e n c e is an integral p a r t of the B o h r a s s u m p t i o n : The second stage, the d e c a y of the *) Paper read on July 3d, 1956 at the Amsterdam Nuclear Reactions Conference. --
9 5 2
--
T H E FORMATION OF T H E COMPOUND N U C L E U S
953
C. N., depends only upon its over-all properties, as its energy or total angular momentum. Recent developments have shown that the B o h r assumption is an idealization, which is not valid in all cases. The detailed observations of nuclear reactions have revealed m a n y important deviations from the predictions based upon the Bohr assumption. Therefore, it is necessary to use a more general scheme for the description of nuclear reactions, a scheme which includes also the phenomena which do not fall within the frame of the B o h r assumption. First of all, it became probable, that the incident particle does not necessarily coalesce immediately with the target nucleus forming a compound nucleus. The experiences with the shell model have shown that nucleons can move like free particles in thin nucleon matter. Furthermore, certain properties of nuclear reactions strongly indicate that sometimes the interaction takes place directly between the incident particle and one or a few other constituents of the target, without forming a C. N. as an intermediate stage. In this case the energy or momentum transfer from the incident to the particle takes place more or less directly without having the incident energy distributed among all constituents of the target. We therefore use a three-stage description for a nuclear reaction, which is based upon the following ideas. As a first approximation the target nucleus acts upon the incident particle as a whole. This action can be described in form of a potential. In this approximation the motion of the incident particle is described by a one particle problem, in which the particle moves in a potential V(r) depending only upon its position r relative to the center of the target nucleus. We know from the shell model that a motion of this type plays an important role, and we consider this state of motion of the incident particle as the first stage of the nuclear reaction: the "independent particle stage" (I. P. stage). In this stage, the incident particle is influenced by the nucleus, its path is deviated, but it still can be considered as distinct from the target nucleus. The motion of the incident particle under the influence of V(r) is a description of what happens in the entrance channel of the reaction. If the potential V(r) were an ordinary real potential, it would cause only a scattering of the particle. The fact that reactions do occur can be expressed by ascribing to V(r) an imaginary part which gives rise to an absorption. Any reaction appears as an absorption from the point of view of the entrance channel. This absorption leads to the second stage of the reaction: The Compound System (C. S.). The C. S. is the state of the systems after the particle has been removed from the entrance channel. This can happen in m a n y ways. It can exchange its energy and m o m e n t u m by collision with some other nucleon, it can set up some surface vibration or some other collective
954
v.F. WEISSKOPF
motion. The C. S. is characterized b y the fact that the state of affairs cannot be described b y a one particle picture, in which the incident particle moves in a potential. It should be emphasized that the concept of C. S. is somewhat more general then the concept of C. N. as used before in the B o h r description. The C. N. is characterized b y the fact that the incoming particle has "coalesced" with the target and has formed an entity in which it does not play a role distinguished from any other nucleon. We consider the C. S. to be the state in which some energy exchange between target and projectile has taken place, regardless of the role of the incident particle. Hence, the C. S. includes the C. N., b u t it also refers to states created b y direct collisions, or b y surface excitations etc., in other words, to any states in which the incident particle is removed from the entrance channel. The exact formal distinction between the first and the second has not yet been given. There are problems involved which have been the objects of m a n y papers on nuclear interaction, in particular b y B r u e c k n e r and his collaborators. The difficulties come from the fact, that in the first stage also, there is an interaction between the incident particle and the constituents of the target ; however, the target acts only as a whole and remains in many respects undisturbed since the interaction can be described in terms of a potential. The third stage of the reaction is the break-up of the C. S. into the residual nucleus and the emitted particle. This stage is in many ways similar to the first stage running in the opposite time direction. The emerging particle finds itself under the influence of the residual nucleus before it leaves; an ~nfluence which can be described b y a complex potential. The B o h r picture of a nuclear reaction is included in his description. It is a special case which would occur under the following two conditions. Firstly, the potential in the I. P. stage must have a large imaginary part all over the volume of the nucleus, in order to insure immediate formation of the C. N. Secondly, the C. S. must be such that the incident particle shares its energy with all constituents. In general, neither of these conditions will be fulfilled. There are cases, however, in which they are approximately valid. Fig. 1 shows a schematic representation of the different stages in a nuclear reaction. Tl~e left side represents the I. P. stage, in which we distinguish the incident beam, the shape elastic scattering and the absorption. The latter contains all the possible reaction processes. The I. P. stage can be represented b y what is usually referred to as the optical model: The incident beam is scattered and absorbed b y a complex potential. The C. S. stage contains all the processes which produce the absorption. Several different schemes are indicated. For example, the incident particle might collide with one of the constituents of the target and force it to leave the boundaries of the nucleus. Such processes are called direct interactions since they lead directly into the third stage. The properties of these events
T H E F O R M A T I O N OF T H E C O M P O U N D N U C L E U S
955
depend upon the place of the collision, at the surface or in the interior of the nucleus. Apart from direct interactions, we expect multiple collisions, excitations of collective modes, surface vibrations or rotations etc., which all can lead into the third stage. Finally, the energy exchange of the incoming particle can be such t h a t the energy is thoroughly interchanged among all constituents before the break-up. In this case a C. N. is formed which has lost " m e m o r y " regarding its formation. The break-up of the C. N. takes place according to the well known statistical probability rules. It includes also the possibility of a breakup into the incident particle and the original target nucleus in its ground state. In this case we get compound elastic scattering, which adds coherently to the shape elastic scattering. I.R Stoge
C.S Stoge ~.
"'~,
Surfoce Direct Ini'n.
L/ ~
~t
~"
~
Vi ¢ t V 2 / ~
r~
Fino¢Stoge
/ // //
VolumeOirectInf'n. • Multiple ~. /-. . . . . . . .Coll'ns. . / CollectiveExci,. ~
I
"'
Opficol Model Fig. 1. Nuclear Reaction Scheme.
The I. P. stage is the one about which most is known. The description of this stage in terms of a complex potential has been quite successful. In fact, the optical model is a satisfactory w a y to describe a nuclear reaction as long as one restricts oneself to an over-all description, in which everything occurring in the C. S. and final stage is lumped together into the absorption process of the first stage. The complex potential which serves well for reactions initiated b y neutrons or protons has a very simple form: The real part and the imaginary part are assumed to be proportional to each other, and are given as follows: V=VI+iV2
V2=~V1
Vl(r) = -- Vo(l + exp[(r -- R)/a]) -1 R = roAt + rl
where r is the distance from the center of the target nucleus. This potential is a wall with rounded edges, and a is a measure of the width of the rounding. Certainly it is an idealization which is neither uniquely determined nor necessarily the best choice. Some authors 1) have added spin-orbit coupling
956
V. F. WEISSKOPF
terms to the potential which probably improves the usefulness of the method. There are quite a number of studies available now comparing the crosssections following from this potential with the experimental ones and thus determining the constants. The first suggestion of such a potential was given b y S e r b e r , F e r n b a c h and T a y l o r 2) who restricted their considerations to high energies. More recent studies are made b y F e s h b a c h , P o r t e r and W e i s s k o p f 3), N e m i r o v s k y 4), W a l t and B e y s t e r 5) and unpublished work b y F e s h b a c h , P o r t e r and C a m p b e l l 6) on neutron reactions, b y S a x o n and W o o d s 7), F u j i m o t o and H o s s a i n s) on proton reactions. The optical model can only cope with the I.P. stage of the reaction and, therefore, predicts only the total cross-section, at, the shape elastic crosssection, a,e and its dependence upon the angle, and the cross-section ac of the formation of a Compound System (absorption). In the case of protons the total cross-section does not exist. The comparison with the experiments is complicated b y the fact that only at is directly observable. As long as there is appreciable compound elastic scattering (ace) the observed scattering cross-section is ane + ace and the observed reaction cross-section is a c - ace, rather than abe and a,, respectively. However, the effect of the compound elastic scattering is negligible at higher energies. The following constants seem to give the best agreement with the experiments with neutrons between 0 and 14 MeV: R = (1.27 A ~ + 0.6) 10-is cm a = (0.5-4-0.1) X 10-13cm Vo = 43 + 3 MeV (indications of slight decrease with increasing energy) = ~0.13 4- 0.5 for energies > 4 MeV t0.7 + 0.3 for energies < 1 MeV (continuously changing between 1 MeV and 4 MeV) For protons the values for lower energies are not well determined. At higher energies one finds: R and a roughly equal to the proton values, and
V0 =
47 at 17 MeV 36 at 31 MeV [0.18 at 17 MeV "t t 0.4 at 31 MeV
The values obtained b y the Soviet-researchers are slightly different since they used a different potential shape: V=
V0forr
V = V0 exp [(R -- r)/b] for r > R
T H E F O R M A T I O N OF T H E C O M P O U N D N U C L E U S
957
T h e y find as the best fit for n e u t r o n reactions: Vo = 42 MeV b = 1.4 × 10 - 1 3 c m $ =
0.05
It is quite satisfactory that so m a n y results can be reproduced with relatively few constants. The experiments indicate definitely an increase of $ with energy although this increase is somewhat less pronounced then previously noted. The first calculations were made with a rectafiguL~r well (a = 0) and this idealization depressed the values of $ at low energy. The proton results show a definite decrease of V0 with increasing energy. This is indicated also in the neutrons at higher energy by T a y l o r 9). Unfortunately the second stage of the nuclear reaction is very m u c h less understood at present then the first one. It is still a matter of conjecture what happens when a C o m p o u n d System is formed. The accumulated experience of the last years has shown that processes other than the formation of a C. N. play a more important role than generally assumed. It is not yet possible to identify the nature of these processes with any certainty. The predominantly forward peaking of the outgoing particles in m a n y reactions indicates the importance of direct interactions. F r o m simple considerations regarding the mean free path of a nuclear particle within the nucleus (taken from the imaginary part of the potential) and the reflecting effect of the nuclear surface, it seems probable that the volume-direct interactions will be lessimportant than the surface ones. This will be true to a greater extent for proton ejections because of the barrier effects.In fact, a recent calculation by E It o n 14) has shown that even for incident particleswith energies as high as 31 MeV, volume-direct interactions will be negligible for proton emission in m e d i u m heavy elements. Surface-direct interactions are probably the most important competitors to the actual C. N. formation. Since they take place at the rim of the nucleus as seen from the incident direction, a proportionality to the first power of the radius is expected. The C. N. formation is unquestionably present in all nuclear reactions with not too light nuclei and at not too high energies. (E < 50 MeV). There is good experimental evidence, mainly from the Los Alamos Laboratory lO), that the energy distribution of neutrons emerging from a nuclear reaction is a "Maxwellian" distribution at low energy and is isotropic. This is just what one would expect from the statistical theory of the compound nucleus. F r o m these observations, it seems that in the case of incident energies of 15 to 20 M e V (neutrons or protons) a C. N. is formed with 80-90% probability 11). The "temperature" of the C. N. stillrepresents a problem since it is low (0.7 MeV) and apparently not very dependent on A. Also irregularities have been found in the dependence of the temperature on the e x c i t a t i o n e n e r g y 12).
958
T H E FORMATION OF THE COMPOUND N U C L E U S
Recently a number of authors (B. Cohen, N. R o s e n 13)) also reported irregularities in the distribution of protons emerging from nuclear reactions. Sometimes their number is larger and their energy lower, than the Coulomb barrier would allow. These and similar effects are poorly understood and indicate that nuclear reactions, particularily with charged particles, occur frequently in the surface region. The increasing amount of experimental knowledge about nuclear reactions has shown that the mechanism of energy exchange within the nucleus is quite complicated and is less amenable to a simple statistical description than we were previously led to believe.
Short communications directly/ollowing this paper were read by E isberg, p. II61, A roster, p. 1162 and Emmerich, p. 1163 o/ this volume. Received 23-8-56.
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