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General introduction
Physica XXlI 941-951 Arnsterd~tm N u c l e a r Reactions Conference
B e t h e , H . A. 1956
GENERAL
INTRODUCTION
b y H. A. B E T H E *) Laboratory of nuclear Studies, CorneH University, Ithaca, U.S.A.
Introduction. Various people are interested in m a n y different phases of nuclear reactions. Some are most interested in the properties of light nuclei, in finding energy levels to which you can assign specific quantum numbers. Others are more interested in special levels of heavy nuclei, such as the levels of the collective model. Still other interest centers about the more classical part of nuclear physics, reactions of heavy nuclei at higher energies where you want to obtain statistical information about nuclear levels. Quite a lot of progress has been made on that in recent years. Some reactions with heavy nuclei and fast particles do not obey statistics and most of these can be attributed to surface reactions. Another point which has been added to the statistical considerations of reactions in heavy nuclei is the one particle model, which has been very successful and ties the theory of nuclear reactions together with the theory of the shell structure of nuclei. I want to give a brief survey of some of these phenomena and then I want to spend a little more time in discussing some features of the one particle model and some features of the statistical information which one can obtain. Light nuclei. In this case we normally have selection rules a n d we can obtain quantum numbers, spin, parity of nuclear levels, either of a compound state or of the residual nucleus which is formed after disintegration. A great deal of information on the reactions with light nuclei has been obtained and much of it has been assembled in a form so as to be available. Particularly interesting are comparisons between the states of light nuclei which you derive from experiment and the theoretical calculations with the shell model, including all the fine points of the shell model, configurational interaction and so on. In the last years quite a lot of success has been achieved b y the consistent and thorough application of the shell model to reactions with light nuclei. Another hne of determining quantum numbers of light nuclei is that of *) This paper has been prepared by Prof. C. C. J o n k e r and Mr. A. V a n d e r V e g t, Natuurkundig Laboratorium der Vrije Universiteit, Amsterdam, Nederland, from a recording of Prof. B e t h e ' s introductory lecture, delivered at the Amsterdam Nuclear Reactions Conference on July 2, 1956.
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finding isobaric triplets. We know a great number of such triplets from which we g e t m o r e confidence in the charge independence of nuclear forces and more detailed information about the general structure of nuclei. It seems to me that the concept of isobaric spin is much more far reaching in nuclear physics than was anticipated and much more powerful even than W i g n e r in his first papers realized. In heavier nuclei most of the reactions go b y means of the well known theory of the compound nucleus. We first have the formation of the compound nucleus which, as we now believe, has many features of the one particle model. Then follows the desintegration of the compound nucleus, which has mostly statistical features. I want to say more about both matters later on. Many reactions with heavy nuclei cannot be classified in the way I have now outlined and this has given rise to the thought that perhaps the whole idea of the compound nucleus and the statistical treatment is wrong. I think that this is a much too far reaching conclusion. One must not lose sight of the fact that most of the cross-section is still very well described b y the statistical picture. There are certainly other reactions which have to be treated in a somewhat different way, but these are exceptions. The most important of these are those due to surface phenomena.
Sur[ace type reactions. The first surface phenomenon which has been studied in great detail is the stripping reaction. This was started b y the very simple theory of B u t l e r , which seemed to make it possible to understand the angular distribution of the nucleons which come out of the stripping reaction and to deduce from the measured angular distribution the angular momentum of the nucleon which was added to the nucleus in the reaction. This beautiful theory was obviously not complete and has unfortunately undergone a lot of complications in recent years. M e s s i a h and H o r o w i t z in particular, pointed out that it was necessary to take into account the boundary condition at the surface of the nucleus on the particle which leaves the nucleus after the stripping reaction. In a (d, p)-reaction e.g. you need the boundary condition on the proton w a v e function. They were able to show that according to the particular boundary condition assumed on the proton, you g e t very different angular distributions, so much so that you can easily be deceived in deducing the angular momentum of the neutron which has been added to the nucleus. This was further developed b y T o b o c m a n , who also included the effects of the Coulomb scattering of the deuteron and of the proton in the theory in an effort to make the theory applicable to nuclei of relatively high charge and to incoming deuterons of relatively low energy. Unfortunately T o b o c m a n ' s numerical calculations showed an even greater dependence
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on the boundary condition assumed for the outgoing and possibl3~ also for the ingoing particle, than those of H o r o w i t z and M e s s i a h . At the end of this time it seemed that the method would lose its value in all but the simplest circumstances, namely that of very fast deuterons on relatively light nuclei. Fortunately one simplifying feature has recently been pointed out b y B o w c o c k , namely that it is possible to analyse the outgoing particles, the protons in a (d, p) -reaction, in spherical harmonics. One gets first the differential cross-section. Then one tries to deduce from this the angular dependence of the matrix element. This is particularly simple and straightforward when the neutron is added to the nucleus in an s-state. B y taking in the analysis of the matrix element the spherical harmonic of rather high q u a n t u m number, one treats particles which never came near the nucleus and are therefore free of the doubtful boundary condition at the surface of the nucleus. B y doing it in this way, one is able to get a much closer comparison to B u t l e r s theory. B o w c o c k was able to show that the high spherical harmonics had indeed the same ratio as B u t 1e r s theory predicted, regardless of the boundary condition on the outgoing wave function. With the right value of this ratio one can then use the measured angular distribution to deduce, for instance, the neutron width of the nucleus which absorbed the neutron. Among the surface reactions are surely the reactions which yield protons from a heavy nucleus, because we know very well that once the compound nucleus is formed, protons can essentially not be emitted. Experimentally, protons contribute usually a few percent to the total cross-section. Similarly we know reactions in heavy nuclei which give neutrons of relatively high energy. For instance it was observed b y R o s e n that heavy nuclei bombarded b y 14 MeV neutrons will yield neutrons not only of a temperature distribution, but also neutrons of energies of the order of 6, 8 or 10 MeV. These values are comparable to the incident energy and much higher than would follow from the temperature, which is about 1 MeV. Such neutrons undoubtedly arise from surface interactions in which the incident neutron never enters the nucleus but collides as it were with one particle outside the nucleus. Fortunately for the understanding of surface reactions we know now a great deal more about the structure of the nuclear surface.
Knowledge o/nuclear sur[ace. This knowledge has been obtained particularly from the experiments at Stanford b y H o f s t a d t e r et al. on the scattering of electrons b y nuclei. From these experiments one can obtain very thorough information on the distribution of charge inside the nucleus. It is found that the effective radius of the nucleus is about 1.2 × 10"-13 × A ~ cm. The factor 1.2 is determined with great accuracy to about 2 or 3% and the proportionality of the nuclear volume with A is established to the same
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accuracy. It is even found t h a t a nucleus as light as hehum still follows this formula, which is something of a surprise and has of course no deeper significance. Futher we know the thickness of the surface. The experiments show t h a t the density falls from 0.9 to 0.1 of the central density in a distance of about 2.4 × 10-13 cm. This is quite a large thickness. For a heavy nucleus such as lead the mean r a d i u s is about 7.2 × 10-13 cm, but the surface thickness is 2.4 × 10-13 cm, so t h a t even for such a heavy nucleus most ~)f the nucleus is surface. There has been some question as to the relation between the nuclear radius observed in electron scattering and in nucleon scattering experiments. Perhaps the best analysed experiments on the scattering of nuclear particles are those on t h e scattering of protons of energies up to 20-30 MeV. These experiments have been analysed quite carefully by S a x o n at U.C.L.A. a n d the best radius is about 1.33 × 10-13 × A i cm. One should not pay too much attention to this radius because it depends on the assumption you make about the depth of the potential for the proton inside the nucleus. Moreover, one has what seems rather direct evidence on the size of nuclei from the inelastic scattering of neutrons. From these processes one generally •gets the impression t h a t the radius is 1.4 to 1.5 × 10 - 1 3 × A ~ cm rather than 1.2 × 10-13 X A ~ cm. More about this later. Turning now to special levels in heavy nuclei, I want just to mention briefly that, as is well known, the lowest levels can best be understood from the collective model of B o h r and M o t t e l s o n . This understanding, is in terms of a non-spherical shape of the nucleus, viz. the rotation of an ellipsoidal excrescence, which you have on top of the sphere. Sometimes, the nucleus actually has an ellipsoidal shape in its lowest state. This, however, is only true for a veDr limited range of nuclei e.g. round A = 150. In general you merely get a spherical shape which can easily be distorted into an ellipsoidal shape by Slight external forces. In this case the levels are more of a vibrational than of a rotational nature (cf. A. B o h r ' s paper). The spacing of these low lying levels is clearly not significant as an extrapolation of the levels at higher e n e r g y . These levels are of a different nature and are therefore cUfferently arrange d and spaced from the levels at higher energies, where you talk more about the excitation Of single nucleons and about levels you can derive from the single nucleon picture. However, on the other hand, the collective model should certainly be derivable from the single nucleon model by considering the interaction of the nucleons carefully and correctly. I think t h a t one of the tasks of the theory of nuclear physics will be to make the connection between these two viewpoints and to establish how one viewpoint follows from the other. I want to discuss now what I would c ~ the normal type o[ nuclear reaction, t h a t is a nuclear reaction with a heavy nucleus at high energy, which follows
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more or less the old picture of the compound nucleus. I discuss first the one particle aspects of these reactions.
The one. particle aspect of these reactions goes back to the beautiful experiments of B a r s c h a l l et al. at Wisconsin and to the model proposed by F e s h b a c h , P o r t e r and W e i s s k o p f a few years ago. We say in this model that the incident nucleon retains its identity inside the nucleus for quite a while. We describe the motion of the nucleon in the nucleus by a complex square well potential Vo + iV1; the depth of the real part is about 40 MeV and that of the imaginary part ! to 10 MeV. The peculiar thing is: why is the imaginary part so small. To determine the constants in this one particle model one uses the elastic scattering of protons, which I mentioned before, and the elastic scattering of neutrons, which can equally well be investigated. The data are mostly compiled in the paper of F r i e d m a n and W e i s s k o p f (in "Niels Bohr and the development of physics"). According to this, the imaginary part of the potential goes from about 1 MeV for slow neutrons to something like 15 MeV for 30 MeV protons and there is no great difference between neutrons and protons. Generally the trend is an increase of [VII with increasing kinetic energy. At very high energies it will of course go the other way again and one gets smaller V1, that is smaller absorption of the incident nucleon. The depth o~ the real part o/the potential. You can derive the depth of the real part of the potential from a consideration of bound states or from a consideration of /ree nucleons. For the derivation one has to agree on a certain value for the nuclear radius and I will take the value of S a x o n and W o o d (1.33 X 10-13 X A~cm), simply because among the scattering experiments with nucleons the proton scattering is perhaps the best analysed of all. We know that at A : 120, with 70 neutrons and 50 protons in the nucleus, the 3s-neutrons are beginning to be bound. The phase of the wave function of these neutrons just inside the edge of the potential can be found simply by fitting boundary conditions. You get kR : 2.64~. Since R is known you can compute k and from this depth of the potential. The depth is about 43 MeV. On the other hand, looking at neutron scattering you find that there is a resonance which you have to ascribe to 4s-neutrons at about A = 150, which is very soon after A = 120. Using the normal wave mechanical analysis you will say now that the inside wave function will have a flat tangent at the boundary, so the phase at the boundary should be given by kR : 3.5~. From the values of R and k you find now a depth of 50 MeV. This is very disturbing and very surprising, because according to any Physica X X I I
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theory we can devise the depth of the potential should be less when the energy of the nucleon is higher or the potential should rise with increasing kinetic energy. This is also one of the results of B r u e c k n e r ' s theory: you have generally the greatest binding for nucleons which move with nearly zero velocity. For proton scattering I cannot give you a quantitative argument but I will simply take the values of S a x o n ' s analysis. You find a depth of 56 MeV, which differs even more from the value for bound neutrons. This value is extrapolated from results at higher energy: Proton energy in MeV 31 Depth of V0 in MeV 36
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5 ~ 0 52.5 ~ 56
The potential depth exhibits the trend I mentioned and postulated b y B r u e c k n e r ' s theory, namely, the potential becomes deeper w i t h lower energy of the scattered proton. B y extrapolating this to zero energy you get 56, which is more than 50 and a lot more than 43. The accuracy of the experiments is sufficient to make these numbers really significant and the discrepancy between the values derived from bound and free nucleons real. And further, there are more paradoxes. Consider the difference in the neutron and proton kinetic energy. Look for instance at the 7sPtlg4 nucleus, on which proton scattering experiments were done. According to ordinary Fermi-statistics the kinetic energy, that is the top of the Fermi-sphere, is 23.6 MeV for protons and 30.8 MeV for neutrons. (Here r0 = 1.33 × 10-13 cm is always assumed because only b y keeping r0 fixed you get significant comparisons). The difference of these is 7.2 MeV. But if you calculate the Coulomb energy of an average proton in this heavy nucleus you find as much as 17.4 MeV. This shows that the difference in kinetic energy is far less than the Coulomb energy difference. Since the total energy of neutron and proton is nearly equal there must be a difference in the nuclear potential energy for neutrons and protrons, with the proton having a nuclear potential energy that is 10.2 MeV greater. This is significant. For instance, you want to calculate according to T e l l e r and J o h n s o n the difference in radius of the neutron and proton sphere. Then if you want to get a large difference in radius you have to assume that the difference in potential energy is the full Coulomb energy and our calculations show that this cannot be so. This paradox is most easily explained. In fact, this is exactly what B r u e c k n e r is talking about when he says that particles low down in the Fermi-distribution are more strongly bound. In fact this has been used as one of the chief arguments for the small effective mass of nucleons in the nucleus. With effective mass 0.5 the difference in kinetic energy becomes twice 7.2 MeV and the difference in potential energy will be about equally
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large. Actually you have to make the effective mass less than 1/2" in order to get a difference in potential energy of 17.4 MeV. I think, however, that there is one additional reason for this difference in potential energy of protons and neutrons, which is more powerful and typical and which we just have forgotten over the last twenty years, viz. a proton interacts mostly with neutrons and a neutron interacts mostly with protons. In fact when you write down the interaction in the B r u e c k n e r theory (or in perturbation theory for that matter) you can assume that the forces act mostly in even states, i.e. are essentially of the S.erber-type (ordinary plus exchange). There is evidence that at least a large part of the forces must be of this type. If you say this it follows that there cannot be any average interaction between particles with the same charge and the same spin, because in that case the wave function must be antisymmetric in the spatial coordinates and we have just assumed that there are no forces in the odd states. Therefore one should say that a given proton interacts on the average with one proton and two neutrons and vice versa. Taking then the usual numbers for the ratio of singlet and triplet forces one can show that about one quarter of the binding of a proton is due to interaction with protons and three quarters are due to the interaction with neutrons. Then, fixing the average interaction energy from the known total binding energy and the values of the kinetic energy (as given above) one finds that this proton-neutron difference gives a much deeper potential for protons because there are many more neutrons in the platinum nucleus than there are protons. In fact this effect gives about 9 MeV difference in potential depth for neutrons and protons. This is almost equal to the 10.2 MeV difference in nuclear potential energy which, as we showed above, should exist in the P t nucleus. Thus only a negligible amount (1.2 MeV) is left over for Brueckner's effective mass effect. In other words, the difference between the binding energies of protons and neutrons in heavy nuclei does not give evidence in favor of a low effective mass of nucleons inside these nuclei, and the main experimental evidence for a low mass comes at present from the scattering of protons. Our argument about the neutron and proton binding energies leads also to a resolution of another paradox I mentioned, namely the difference of potential energy between protons and neutrons found from scattering. This difference was quite differently explained b y S a x o n et al. in their paper. I think it is merely the well known difference between the potential energies of proton and neutron in the nucleus. Now what about the first paradox, the difference in potential found from bound ariel, free neutrons. This can be explained if you take seriously the fact that the nucleus does not have a sharp boundary. Taking a potential that is constant up to a certain point and then decreases exponentially (c./. the potential used b y G r e e n, P h y s Rev. 102 (1956), 1325) the Schr6dinger
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equation can be solved. For bound 3s neutrons the phase of the inside wave function, where it is fixed to the exponentially decreasing part, becomes now k R = 2.73zc instead of 2.64~ as for the rectangular well. If you have scattering of low energy 4s neutrons then even a small potential has a large influence on the wave function and therefore the maximum of the wave function occurs a bit further out, giving k R = 3.08z~ instead of 3.5 ~. The phase numbers have now come very much closer together and as a matter of fact a potential depth V0 = 41 MeV from the bound neutron case and 39 MeV from the free neutron case is found. Assuming a binding energy of 6 MeV in this case, the kinetic energy difference between bound and free would be 4 MeV and the difference in potential energy is 2 MeV. This relation between the change of potential and kinetic energy is compatible with that derived from proton scattering. This shows that even in discussing phenomena which seem to be completely in the realm of classical nuclear physics, in which you consider the bulk of the nucleus, it is extremely important to consider the gradual drop of the density of the nucleons at the surface. When you do this you can at the same time explain a paradox which was noted e.g. by W a l t and B e y s t e r in measuring both the elastic and inelastic scattering of neutrons of about 2.5 MeV by nuclei. They found t h a t as far as the elastic scattering was concerned they could assign a certain radius and a depth of potential of the conventional magnitude and an imaginary part of the potential of about 4 MeV. But when they then calculated the inelastic scattering, they got numbers which were hopelessly too small. If, however, the nuclear surface is assumed to be smoothed out, then everything becomes much more reasonable. To see this, consider nucleons of some high angular momentum l which is so chosen t h a t they will barely be able to reach the nucleus. This means t h a t their kinetic energy will be nearly zero at the nuclear surface and therefore even a small nuclear potential will have a great influence on them. To make this more quantitative, let us introduce the radial kinetic energy T(r) = E - - V(r) ~
l(l + I) h2/2mr ~
This will have a minimum at a certain value r = R. Let us now choose l in such a way that T ( R ) = 0; then the particles l can just enter the nucleus while particles l + 1 would not be able to do so. Now assuming V(r) to be an exponential whose range is chosen in accord with the Stanford electron scattering experiments, and taking a nucleus of A = 120, we find R-------- 8 . 5 × 10-13 cm, as compared with the official radius of 6 × 10-x3 cm. This leads to a doubling of the effective geometric cross-section, amply sufficient to explain the observed inelastic scattering. And now one more point about these potentials. One can extrapolate these potentials of about 40 MeV depth to zero energy by assuming an
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effective mass. I take the value found from the proton scatteri@ experiments, about 0.6 times the actual mass. Then the potential depth for the nucleons of lowest kinetic energy is about 60 MeV, a value considerable greater than found from the top of the Fermi-sphere. Talking about the spacing of energy levels, one can take, as you know, the oscillator potential for the shell model. The levels are then about equidistant. For a nucleus with A = 150 there are about five levels and the spacing between two levels should really be I/5 × 60 MeV = 12 MeV. This is a lot more than the spacing usually assumed and the difference is to be accounted for b y polarization effects. If you have a half filled shell you can lower the energy very much by taking into account the polarization of the nucleus according to B o h r and M o t t e l s o n . For a completely filled shell you cannot do so and you get no lowering of the energy.
The imaginary part o/the potential. The first attempt to calculate the imaginary part of the potential, made b y L a n e , T h o m a s and W i g n e r , resultedin a very large value of about 20 MeV. The next calculation, giving about the fight result, was made by L a n e and W a n d e l . They use what was once referred to as "the frivolous model", viz. they simply assume that one should take each nucleon as a free particle, that the Fermi-sphere is filled and that there is no particle outside this sphere. Taking then an incident nucleon one considers how it raises the nucleons from the Fermi-sphere to higher levels. In order to calculate the probability for this process they use the observed nucleon-nucleon cross-sections. This frivolous model has been rather well established b y the work of B r u e c k n e r et al. They have shown that indeed it is quite legitimate to consider a nucleon as occupying a one-nucleon state in infinite nuclear matter and in particular to consider the nucleons as occupying all the states within the Fermi-sphere. This assumption does not give the actual wave function of the nucleus but it does give essentially correctly all the energies and all the matrix elements of the Hamiltonian. It is therefore legitimate to consider a nucleon in a heavy nucleus as moving more or less freely, to consider the excitation process in the manner of L a n e and W a n d e l but with the difference that one should not take the observed cross-section, but instead the interaction which the nucleon feels inside the nucleus. Now it can be shown and I will talk about this tomorrow, that this interaction inside the nucleus is essentially just the ordinary potential used in the Born approximation. So one should get a good approximation to the imaginary part of the potential b y using Born approximation with the ordinary nuclear interactions and proceed otherwise according to the frivolous model. The calculation was done b y B r u e c k n e r , E d e n and F r a n c i s with the result that the imaginary potential is now smaller than the observed one. It is consistently too small b y a factor of at least two: 0.5 MeV at low
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energy and about 4 MeV at a neutron energy of 20 MeV. The energy dependence is correctly given, except for a constant factor of two or more. I have just one suggestion, viz. that at the surface of the nucleus there m a y be an effect which has not been taken into account, an effect similar to the B l o c h - N o r d s i e c k theory of the emission of radiation. You have a free nucleon coming into the nucleus without any accompanying distortion effect. Once it goes into nuclear m a t t e r it carries with it a polarization of nuclear matter around it and therefore the wave function of the nucleon really changes discontinuously as it enters the nucleus. This m a y give rise to additional excitation of particles,
Statistical aspects o[ heavy nuclei. Now a word about the statistical aspects. First of all we say in the statistical theory t h a t the probability of reaching a given level is the same for all levels. This is certainly not exactly fight but I think it is not too far wrong if you have very m a n y levels. So we need a theory of the level density. To my knowledge the best theory developed is that of C l a u d e B l o c h . When you have the level density and represent it as a function of energy you can define a temperature. Then you can t r y to measure the temperature in nuclear reactions by determining the energy distribution of the emitted neutrons. G u g e l o t in particular has been doing a lot of experiments of this kind and he and others have contributed to our knowledge of the temperature of the nucleus. For m a n y years it was almost an axiom that the temperature of the nucleus from such experiments was always 1 MeV. Fortunately this axiom has now been proved false. C r a n b e r g at Los Alamos has done experiments, using a time of flight method, in which he measured the distribution in energy of the neutrons made by neutrons of 2.5 MeV incident on 28sU. The result of these experiments, which give much greater emphasis to the low energy end of the spectrum than any experiments done previously, is that the nuclear temperature in this case is only 0.3 MeV. So we have the result that indeed the temperature does seem to depend on the energy, so that for an energy of let us say 10 MeV one gets a temperature of 1 MeV but a temperature of only 0.3 MeV for energies of 2.5 MeV. If you compare these two numbers you find that the temperature goes roughly as the square root of the energy, maybe even a little faster with energy, which is the energy dependence we used to be fond of. Another fact about the level density is that it is not a completely smooth function of the mass number. There are strong variations between even and odd nuclei and also for magic nuclei. They are best investigated by means of the levels of slow neutron resonances with the help of velocity selectors. H u g h e s et al. have given us a large amount of very valuable information on this subject. It seems that the difference between even and odd nuclei and the difference between magic and non-magic nuclei is perhaps a little
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greater than is explained on the basis of energy alone. Additional and interesting confirmation has been supplied recently at Harwell by means of the fast neutron experimental reactor. In this reactor you can measure average capture cross-sections for neutrons in the neighbourhood of a few 100 keV. You find very great variations between even and odd nuclei and very low minima near the magic nuclei. Another important question about the level density is the spacing of individual levels in the sl~w neutron region. Here again the Brookhaven experiments are the most extensive and we know that the levels are very far from equally spaced. However, at least at one time it seemed as if the distribution was rather more even than a random distribution. There are, on the other hand, several examples where two very strong levels are very close together, which m a y be because strong levels are more easily detected. A particularly interesting problem is the question of the distribution o/the widths o/the levels, because that gives us information about the properties of the nuclear states. In particular we are interested in the neutron and gamma-ray widths of these levels. These behave apparently in very different fashions. The neutron widths show extremely wide fluctuations from level to level. Taking all the evidence from the Brookhaven and other experiments and with the help of a thorough statistical analysis, P o r t e r and T h o m a s recently came to the conclusion t h a t the best representation is by means of a Gaussian distribution of the square root of the reduced widths. This is also very reasonable theoretically, viz. the matrix elements for the one channel of neutron decay are randomly distributed. On the other hand the experimental evidence seems to show that the gamma widths are very nearly constant. The gamma widths are surely due to a superposition of very m a n y channels. The gamma rays m a y lead to any level of the residual nucleus and therefore one has a superposition of some 10,000 random variables which gives a very narrow statistical distribution for the total gamma width of the levels of a given nucleus. In a case which was investigated particularly well by S a l l o t the gamma widths seem to be the same even for levels of different nuclear spin. A final class of widths we have to deal with is the fission width of fissionable nuclei. They seem to take an intermediate position. The analysis of the fission widths by T h o m a s and P o r t e r , at any rate, seems to indicate t h a t there are about two and a half channels for fission. I want to leave it to the fission experts to tell us what this means. In this brief survey of the field of nuclear reactions, I am perfectly aware that I have left out many interesting things and even more that I have emphasized matters which I have worked on recently to the detriment of those which others have worked on. Received 7-9-56.
B e t h e , H . A. 1956
GENERAL
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b y H. A. B E T H E *) Laboratory of nuclear Studies, CorneH University, Ithaca, U.S.A.
Introduction. Various people are interested in m a n y different phases of nuclear reactions. Some are most interested in the properties of light nuclei, in finding energy levels to which you can assign specific quantum numbers. Others are more interested in special levels of heavy nuclei, such as the levels of the collective model. Still other interest centers about the more classical part of nuclear physics, reactions of heavy nuclei at higher energies where you want to obtain statistical information about nuclear levels. Quite a lot of progress has been made on that in recent years. Some reactions with heavy nuclei and fast particles do not obey statistics and most of these can be attributed to surface reactions. Another point which has been added to the statistical considerations of reactions in heavy nuclei is the one particle model, which has been very successful and ties the theory of nuclear reactions together with the theory of the shell structure of nuclei. I want to give a brief survey of some of these phenomena and then I want to spend a little more time in discussing some features of the one particle model and some features of the statistical information which one can obtain. Light nuclei. In this case we normally have selection rules a n d we can obtain quantum numbers, spin, parity of nuclear levels, either of a compound state or of the residual nucleus which is formed after disintegration. A great deal of information on the reactions with light nuclei has been obtained and much of it has been assembled in a form so as to be available. Particularly interesting are comparisons between the states of light nuclei which you derive from experiment and the theoretical calculations with the shell model, including all the fine points of the shell model, configurational interaction and so on. In the last years quite a lot of success has been achieved b y the consistent and thorough application of the shell model to reactions with light nuclei. Another hne of determining quantum numbers of light nuclei is that of *) This paper has been prepared by Prof. C. C. J o n k e r and Mr. A. V a n d e r V e g t, Natuurkundig Laboratorium der Vrije Universiteit, Amsterdam, Nederland, from a recording of Prof. B e t h e ' s introductory lecture, delivered at the Amsterdam Nuclear Reactions Conference on July 2, 1956.
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finding isobaric triplets. We know a great number of such triplets from which we g e t m o r e confidence in the charge independence of nuclear forces and more detailed information about the general structure of nuclei. It seems to me that the concept of isobaric spin is much more far reaching in nuclear physics than was anticipated and much more powerful even than W i g n e r in his first papers realized. In heavier nuclei most of the reactions go b y means of the well known theory of the compound nucleus. We first have the formation of the compound nucleus which, as we now believe, has many features of the one particle model. Then follows the desintegration of the compound nucleus, which has mostly statistical features. I want to say more about both matters later on. Many reactions with heavy nuclei cannot be classified in the way I have now outlined and this has given rise to the thought that perhaps the whole idea of the compound nucleus and the statistical treatment is wrong. I think that this is a much too far reaching conclusion. One must not lose sight of the fact that most of the cross-section is still very well described b y the statistical picture. There are certainly other reactions which have to be treated in a somewhat different way, but these are exceptions. The most important of these are those due to surface phenomena.
Sur[ace type reactions. The first surface phenomenon which has been studied in great detail is the stripping reaction. This was started b y the very simple theory of B u t l e r , which seemed to make it possible to understand the angular distribution of the nucleons which come out of the stripping reaction and to deduce from the measured angular distribution the angular momentum of the nucleon which was added to the nucleus in the reaction. This beautiful theory was obviously not complete and has unfortunately undergone a lot of complications in recent years. M e s s i a h and H o r o w i t z in particular, pointed out that it was necessary to take into account the boundary condition at the surface of the nucleus on the particle which leaves the nucleus after the stripping reaction. In a (d, p)-reaction e.g. you need the boundary condition on the proton w a v e function. They were able to show that according to the particular boundary condition assumed on the proton, you g e t very different angular distributions, so much so that you can easily be deceived in deducing the angular momentum of the neutron which has been added to the nucleus. This was further developed b y T o b o c m a n , who also included the effects of the Coulomb scattering of the deuteron and of the proton in the theory in an effort to make the theory applicable to nuclei of relatively high charge and to incoming deuterons of relatively low energy. Unfortunately T o b o c m a n ' s numerical calculations showed an even greater dependence
GENERAL INTRODUCTION
943
on the boundary condition assumed for the outgoing and possibl3~ also for the ingoing particle, than those of H o r o w i t z and M e s s i a h . At the end of this time it seemed that the method would lose its value in all but the simplest circumstances, namely that of very fast deuterons on relatively light nuclei. Fortunately one simplifying feature has recently been pointed out b y B o w c o c k , namely that it is possible to analyse the outgoing particles, the protons in a (d, p) -reaction, in spherical harmonics. One gets first the differential cross-section. Then one tries to deduce from this the angular dependence of the matrix element. This is particularly simple and straightforward when the neutron is added to the nucleus in an s-state. B y taking in the analysis of the matrix element the spherical harmonic of rather high q u a n t u m number, one treats particles which never came near the nucleus and are therefore free of the doubtful boundary condition at the surface of the nucleus. B y doing it in this way, one is able to get a much closer comparison to B u t l e r s theory. B o w c o c k was able to show that the high spherical harmonics had indeed the same ratio as B u t 1e r s theory predicted, regardless of the boundary condition on the outgoing wave function. With the right value of this ratio one can then use the measured angular distribution to deduce, for instance, the neutron width of the nucleus which absorbed the neutron. Among the surface reactions are surely the reactions which yield protons from a heavy nucleus, because we know very well that once the compound nucleus is formed, protons can essentially not be emitted. Experimentally, protons contribute usually a few percent to the total cross-section. Similarly we know reactions in heavy nuclei which give neutrons of relatively high energy. For instance it was observed b y R o s e n that heavy nuclei bombarded b y 14 MeV neutrons will yield neutrons not only of a temperature distribution, but also neutrons of energies of the order of 6, 8 or 10 MeV. These values are comparable to the incident energy and much higher than would follow from the temperature, which is about 1 MeV. Such neutrons undoubtedly arise from surface interactions in which the incident neutron never enters the nucleus but collides as it were with one particle outside the nucleus. Fortunately for the understanding of surface reactions we know now a great deal more about the structure of the nuclear surface.
Knowledge o/nuclear sur[ace. This knowledge has been obtained particularly from the experiments at Stanford b y H o f s t a d t e r et al. on the scattering of electrons b y nuclei. From these experiments one can obtain very thorough information on the distribution of charge inside the nucleus. It is found that the effective radius of the nucleus is about 1.2 × 10"-13 × A ~ cm. The factor 1.2 is determined with great accuracy to about 2 or 3% and the proportionality of the nuclear volume with A is established to the same
944
H.A.
BETHE
accuracy. It is even found t h a t a nucleus as light as hehum still follows this formula, which is something of a surprise and has of course no deeper significance. Futher we know the thickness of the surface. The experiments show t h a t the density falls from 0.9 to 0.1 of the central density in a distance of about 2.4 × 10-13 cm. This is quite a large thickness. For a heavy nucleus such as lead the mean r a d i u s is about 7.2 × 10-13 cm, but the surface thickness is 2.4 × 10-13 cm, so t h a t even for such a heavy nucleus most ~)f the nucleus is surface. There has been some question as to the relation between the nuclear radius observed in electron scattering and in nucleon scattering experiments. Perhaps the best analysed experiments on the scattering of nuclear particles are those on t h e scattering of protons of energies up to 20-30 MeV. These experiments have been analysed quite carefully by S a x o n at U.C.L.A. a n d the best radius is about 1.33 × 10-13 × A i cm. One should not pay too much attention to this radius because it depends on the assumption you make about the depth of the potential for the proton inside the nucleus. Moreover, one has what seems rather direct evidence on the size of nuclei from the inelastic scattering of neutrons. From these processes one generally •gets the impression t h a t the radius is 1.4 to 1.5 × 10 - 1 3 × A ~ cm rather than 1.2 × 10-13 X A ~ cm. More about this later. Turning now to special levels in heavy nuclei, I want just to mention briefly that, as is well known, the lowest levels can best be understood from the collective model of B o h r and M o t t e l s o n . This understanding, is in terms of a non-spherical shape of the nucleus, viz. the rotation of an ellipsoidal excrescence, which you have on top of the sphere. Sometimes, the nucleus actually has an ellipsoidal shape in its lowest state. This, however, is only true for a veDr limited range of nuclei e.g. round A = 150. In general you merely get a spherical shape which can easily be distorted into an ellipsoidal shape by Slight external forces. In this case the levels are more of a vibrational than of a rotational nature (cf. A. B o h r ' s paper). The spacing of these low lying levels is clearly not significant as an extrapolation of the levels at higher e n e r g y . These levels are of a different nature and are therefore cUfferently arrange d and spaced from the levels at higher energies, where you talk more about the excitation Of single nucleons and about levels you can derive from the single nucleon picture. However, on the other hand, the collective model should certainly be derivable from the single nucleon model by considering the interaction of the nucleons carefully and correctly. I think t h a t one of the tasks of the theory of nuclear physics will be to make the connection between these two viewpoints and to establish how one viewpoint follows from the other. I want to discuss now what I would c ~ the normal type o[ nuclear reaction, t h a t is a nuclear reaction with a heavy nucleus at high energy, which follows
GENERAL INTRODUCTION
945
more or less the old picture of the compound nucleus. I discuss first the one particle aspects of these reactions.
The one. particle aspect of these reactions goes back to the beautiful experiments of B a r s c h a l l et al. at Wisconsin and to the model proposed by F e s h b a c h , P o r t e r and W e i s s k o p f a few years ago. We say in this model that the incident nucleon retains its identity inside the nucleus for quite a while. We describe the motion of the nucleon in the nucleus by a complex square well potential Vo + iV1; the depth of the real part is about 40 MeV and that of the imaginary part ! to 10 MeV. The peculiar thing is: why is the imaginary part so small. To determine the constants in this one particle model one uses the elastic scattering of protons, which I mentioned before, and the elastic scattering of neutrons, which can equally well be investigated. The data are mostly compiled in the paper of F r i e d m a n and W e i s s k o p f (in "Niels Bohr and the development of physics"). According to this, the imaginary part of the potential goes from about 1 MeV for slow neutrons to something like 15 MeV for 30 MeV protons and there is no great difference between neutrons and protons. Generally the trend is an increase of [VII with increasing kinetic energy. At very high energies it will of course go the other way again and one gets smaller V1, that is smaller absorption of the incident nucleon. The depth o~ the real part o/the potential. You can derive the depth of the real part of the potential from a consideration of bound states or from a consideration of /ree nucleons. For the derivation one has to agree on a certain value for the nuclear radius and I will take the value of S a x o n and W o o d (1.33 X 10-13 X A~cm), simply because among the scattering experiments with nucleons the proton scattering is perhaps the best analysed of all. We know that at A : 120, with 70 neutrons and 50 protons in the nucleus, the 3s-neutrons are beginning to be bound. The phase of the wave function of these neutrons just inside the edge of the potential can be found simply by fitting boundary conditions. You get kR : 2.64~. Since R is known you can compute k and from this depth of the potential. The depth is about 43 MeV. On the other hand, looking at neutron scattering you find that there is a resonance which you have to ascribe to 4s-neutrons at about A = 150, which is very soon after A = 120. Using the normal wave mechanical analysis you will say now that the inside wave function will have a flat tangent at the boundary, so the phase at the boundary should be given by kR : 3.5~. From the values of R and k you find now a depth of 50 MeV. This is very disturbing and very surprising, because according to any Physica X X I I
60
946
H . A . BETHE
theory we can devise the depth of the potential should be less when the energy of the nucleon is higher or the potential should rise with increasing kinetic energy. This is also one of the results of B r u e c k n e r ' s theory: you have generally the greatest binding for nucleons which move with nearly zero velocity. For proton scattering I cannot give you a quantitative argument but I will simply take the values of S a x o n ' s analysis. You find a depth of 56 MeV, which differs even more from the value for bound neutrons. This value is extrapolated from results at higher energy: Proton energy in MeV 31 Depth of V0 in MeV 36
17 47
5 ~ 0 52.5 ~ 56
The potential depth exhibits the trend I mentioned and postulated b y B r u e c k n e r ' s theory, namely, the potential becomes deeper w i t h lower energy of the scattered proton. B y extrapolating this to zero energy you get 56, which is more than 50 and a lot more than 43. The accuracy of the experiments is sufficient to make these numbers really significant and the discrepancy between the values derived from bound and free nucleons real. And further, there are more paradoxes. Consider the difference in the neutron and proton kinetic energy. Look for instance at the 7sPtlg4 nucleus, on which proton scattering experiments were done. According to ordinary Fermi-statistics the kinetic energy, that is the top of the Fermi-sphere, is 23.6 MeV for protons and 30.8 MeV for neutrons. (Here r0 = 1.33 × 10-13 cm is always assumed because only b y keeping r0 fixed you get significant comparisons). The difference of these is 7.2 MeV. But if you calculate the Coulomb energy of an average proton in this heavy nucleus you find as much as 17.4 MeV. This shows that the difference in kinetic energy is far less than the Coulomb energy difference. Since the total energy of neutron and proton is nearly equal there must be a difference in the nuclear potential energy for neutrons and protrons, with the proton having a nuclear potential energy that is 10.2 MeV greater. This is significant. For instance, you want to calculate according to T e l l e r and J o h n s o n the difference in radius of the neutron and proton sphere. Then if you want to get a large difference in radius you have to assume that the difference in potential energy is the full Coulomb energy and our calculations show that this cannot be so. This paradox is most easily explained. In fact, this is exactly what B r u e c k n e r is talking about when he says that particles low down in the Fermi-distribution are more strongly bound. In fact this has been used as one of the chief arguments for the small effective mass of nucleons in the nucleus. With effective mass 0.5 the difference in kinetic energy becomes twice 7.2 MeV and the difference in potential energy will be about equally
GENERAL INTRODUCTION
947
large. Actually you have to make the effective mass less than 1/2" in order to get a difference in potential energy of 17.4 MeV. I think, however, that there is one additional reason for this difference in potential energy of protons and neutrons, which is more powerful and typical and which we just have forgotten over the last twenty years, viz. a proton interacts mostly with neutrons and a neutron interacts mostly with protons. In fact when you write down the interaction in the B r u e c k n e r theory (or in perturbation theory for that matter) you can assume that the forces act mostly in even states, i.e. are essentially of the S.erber-type (ordinary plus exchange). There is evidence that at least a large part of the forces must be of this type. If you say this it follows that there cannot be any average interaction between particles with the same charge and the same spin, because in that case the wave function must be antisymmetric in the spatial coordinates and we have just assumed that there are no forces in the odd states. Therefore one should say that a given proton interacts on the average with one proton and two neutrons and vice versa. Taking then the usual numbers for the ratio of singlet and triplet forces one can show that about one quarter of the binding of a proton is due to interaction with protons and three quarters are due to the interaction with neutrons. Then, fixing the average interaction energy from the known total binding energy and the values of the kinetic energy (as given above) one finds that this proton-neutron difference gives a much deeper potential for protons because there are many more neutrons in the platinum nucleus than there are protons. In fact this effect gives about 9 MeV difference in potential depth for neutrons and protons. This is almost equal to the 10.2 MeV difference in nuclear potential energy which, as we showed above, should exist in the P t nucleus. Thus only a negligible amount (1.2 MeV) is left over for Brueckner's effective mass effect. In other words, the difference between the binding energies of protons and neutrons in heavy nuclei does not give evidence in favor of a low effective mass of nucleons inside these nuclei, and the main experimental evidence for a low mass comes at present from the scattering of protons. Our argument about the neutron and proton binding energies leads also to a resolution of another paradox I mentioned, namely the difference of potential energy between protons and neutrons found from scattering. This difference was quite differently explained b y S a x o n et al. in their paper. I think it is merely the well known difference between the potential energies of proton and neutron in the nucleus. Now what about the first paradox, the difference in potential found from bound ariel, free neutrons. This can be explained if you take seriously the fact that the nucleus does not have a sharp boundary. Taking a potential that is constant up to a certain point and then decreases exponentially (c./. the potential used b y G r e e n, P h y s Rev. 102 (1956), 1325) the Schr6dinger
948
H.A. B E T H E
equation can be solved. For bound 3s neutrons the phase of the inside wave function, where it is fixed to the exponentially decreasing part, becomes now k R = 2.73zc instead of 2.64~ as for the rectangular well. If you have scattering of low energy 4s neutrons then even a small potential has a large influence on the wave function and therefore the maximum of the wave function occurs a bit further out, giving k R = 3.08z~ instead of 3.5 ~. The phase numbers have now come very much closer together and as a matter of fact a potential depth V0 = 41 MeV from the bound neutron case and 39 MeV from the free neutron case is found. Assuming a binding energy of 6 MeV in this case, the kinetic energy difference between bound and free would be 4 MeV and the difference in potential energy is 2 MeV. This relation between the change of potential and kinetic energy is compatible with that derived from proton scattering. This shows that even in discussing phenomena which seem to be completely in the realm of classical nuclear physics, in which you consider the bulk of the nucleus, it is extremely important to consider the gradual drop of the density of the nucleons at the surface. When you do this you can at the same time explain a paradox which was noted e.g. by W a l t and B e y s t e r in measuring both the elastic and inelastic scattering of neutrons of about 2.5 MeV by nuclei. They found t h a t as far as the elastic scattering was concerned they could assign a certain radius and a depth of potential of the conventional magnitude and an imaginary part of the potential of about 4 MeV. But when they then calculated the inelastic scattering, they got numbers which were hopelessly too small. If, however, the nuclear surface is assumed to be smoothed out, then everything becomes much more reasonable. To see this, consider nucleons of some high angular momentum l which is so chosen t h a t they will barely be able to reach the nucleus. This means t h a t their kinetic energy will be nearly zero at the nuclear surface and therefore even a small nuclear potential will have a great influence on them. To make this more quantitative, let us introduce the radial kinetic energy T(r) = E - - V(r) ~
l(l + I) h2/2mr ~
This will have a minimum at a certain value r = R. Let us now choose l in such a way that T ( R ) = 0; then the particles l can just enter the nucleus while particles l + 1 would not be able to do so. Now assuming V(r) to be an exponential whose range is chosen in accord with the Stanford electron scattering experiments, and taking a nucleus of A = 120, we find R-------- 8 . 5 × 10-13 cm, as compared with the official radius of 6 × 10-x3 cm. This leads to a doubling of the effective geometric cross-section, amply sufficient to explain the observed inelastic scattering. And now one more point about these potentials. One can extrapolate these potentials of about 40 MeV depth to zero energy by assuming an
GENERAL
INTRODUCTION
949
effective mass. I take the value found from the proton scatteri@ experiments, about 0.6 times the actual mass. Then the potential depth for the nucleons of lowest kinetic energy is about 60 MeV, a value considerable greater than found from the top of the Fermi-sphere. Talking about the spacing of energy levels, one can take, as you know, the oscillator potential for the shell model. The levels are then about equidistant. For a nucleus with A = 150 there are about five levels and the spacing between two levels should really be I/5 × 60 MeV = 12 MeV. This is a lot more than the spacing usually assumed and the difference is to be accounted for b y polarization effects. If you have a half filled shell you can lower the energy very much by taking into account the polarization of the nucleus according to B o h r and M o t t e l s o n . For a completely filled shell you cannot do so and you get no lowering of the energy.
The imaginary part o/the potential. The first attempt to calculate the imaginary part of the potential, made b y L a n e , T h o m a s and W i g n e r , resultedin a very large value of about 20 MeV. The next calculation, giving about the fight result, was made by L a n e and W a n d e l . They use what was once referred to as "the frivolous model", viz. they simply assume that one should take each nucleon as a free particle, that the Fermi-sphere is filled and that there is no particle outside this sphere. Taking then an incident nucleon one considers how it raises the nucleons from the Fermi-sphere to higher levels. In order to calculate the probability for this process they use the observed nucleon-nucleon cross-sections. This frivolous model has been rather well established b y the work of B r u e c k n e r et al. They have shown that indeed it is quite legitimate to consider a nucleon as occupying a one-nucleon state in infinite nuclear matter and in particular to consider the nucleons as occupying all the states within the Fermi-sphere. This assumption does not give the actual wave function of the nucleus but it does give essentially correctly all the energies and all the matrix elements of the Hamiltonian. It is therefore legitimate to consider a nucleon in a heavy nucleus as moving more or less freely, to consider the excitation process in the manner of L a n e and W a n d e l but with the difference that one should not take the observed cross-section, but instead the interaction which the nucleon feels inside the nucleus. Now it can be shown and I will talk about this tomorrow, that this interaction inside the nucleus is essentially just the ordinary potential used in the Born approximation. So one should get a good approximation to the imaginary part of the potential b y using Born approximation with the ordinary nuclear interactions and proceed otherwise according to the frivolous model. The calculation was done b y B r u e c k n e r , E d e n and F r a n c i s with the result that the imaginary potential is now smaller than the observed one. It is consistently too small b y a factor of at least two: 0.5 MeV at low
950
H . A . BETHE
energy and about 4 MeV at a neutron energy of 20 MeV. The energy dependence is correctly given, except for a constant factor of two or more. I have just one suggestion, viz. that at the surface of the nucleus there m a y be an effect which has not been taken into account, an effect similar to the B l o c h - N o r d s i e c k theory of the emission of radiation. You have a free nucleon coming into the nucleus without any accompanying distortion effect. Once it goes into nuclear m a t t e r it carries with it a polarization of nuclear matter around it and therefore the wave function of the nucleon really changes discontinuously as it enters the nucleus. This m a y give rise to additional excitation of particles,
Statistical aspects o[ heavy nuclei. Now a word about the statistical aspects. First of all we say in the statistical theory t h a t the probability of reaching a given level is the same for all levels. This is certainly not exactly fight but I think it is not too far wrong if you have very m a n y levels. So we need a theory of the level density. To my knowledge the best theory developed is that of C l a u d e B l o c h . When you have the level density and represent it as a function of energy you can define a temperature. Then you can t r y to measure the temperature in nuclear reactions by determining the energy distribution of the emitted neutrons. G u g e l o t in particular has been doing a lot of experiments of this kind and he and others have contributed to our knowledge of the temperature of the nucleus. For m a n y years it was almost an axiom that the temperature of the nucleus from such experiments was always 1 MeV. Fortunately this axiom has now been proved false. C r a n b e r g at Los Alamos has done experiments, using a time of flight method, in which he measured the distribution in energy of the neutrons made by neutrons of 2.5 MeV incident on 28sU. The result of these experiments, which give much greater emphasis to the low energy end of the spectrum than any experiments done previously, is that the nuclear temperature in this case is only 0.3 MeV. So we have the result that indeed the temperature does seem to depend on the energy, so that for an energy of let us say 10 MeV one gets a temperature of 1 MeV but a temperature of only 0.3 MeV for energies of 2.5 MeV. If you compare these two numbers you find that the temperature goes roughly as the square root of the energy, maybe even a little faster with energy, which is the energy dependence we used to be fond of. Another fact about the level density is that it is not a completely smooth function of the mass number. There are strong variations between even and odd nuclei and also for magic nuclei. They are best investigated by means of the levels of slow neutron resonances with the help of velocity selectors. H u g h e s et al. have given us a large amount of very valuable information on this subject. It seems that the difference between even and odd nuclei and the difference between magic and non-magic nuclei is perhaps a little
GENERAL INTRODUCTION
951
greater than is explained on the basis of energy alone. Additional and interesting confirmation has been supplied recently at Harwell by means of the fast neutron experimental reactor. In this reactor you can measure average capture cross-sections for neutrons in the neighbourhood of a few 100 keV. You find very great variations between even and odd nuclei and very low minima near the magic nuclei. Another important question about the level density is the spacing of individual levels in the sl~w neutron region. Here again the Brookhaven experiments are the most extensive and we know that the levels are very far from equally spaced. However, at least at one time it seemed as if the distribution was rather more even than a random distribution. There are, on the other hand, several examples where two very strong levels are very close together, which m a y be because strong levels are more easily detected. A particularly interesting problem is the question of the distribution o/the widths o/the levels, because that gives us information about the properties of the nuclear states. In particular we are interested in the neutron and gamma-ray widths of these levels. These behave apparently in very different fashions. The neutron widths show extremely wide fluctuations from level to level. Taking all the evidence from the Brookhaven and other experiments and with the help of a thorough statistical analysis, P o r t e r and T h o m a s recently came to the conclusion t h a t the best representation is by means of a Gaussian distribution of the square root of the reduced widths. This is also very reasonable theoretically, viz. the matrix elements for the one channel of neutron decay are randomly distributed. On the other hand the experimental evidence seems to show that the gamma widths are very nearly constant. The gamma widths are surely due to a superposition of very m a n y channels. The gamma rays m a y lead to any level of the residual nucleus and therefore one has a superposition of some 10,000 random variables which gives a very narrow statistical distribution for the total gamma width of the levels of a given nucleus. In a case which was investigated particularly well by S a l l o t the gamma widths seem to be the same even for levels of different nuclear spin. A final class of widths we have to deal with is the fission width of fissionable nuclei. They seem to take an intermediate position. The analysis of the fission widths by T h o m a s and P o r t e r , at any rate, seems to indicate t h a t there are about two and a half channels for fission. I want to leave it to the fission experts to tell us what this means. In this brief survey of the field of nuclear reactions, I am perfectly aware that I have left out many interesting things and even more that I have emphasized matters which I have worked on recently to the detriment of those which others have worked on. Received 7-9-56.