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Evaporative data on intermediate resonances in nuclear reactions
Nuclear Physics A96 (1967) 273--278; (~) North-Holland Publishing Co., Amsterdam
Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
EVAPORATIVE
DATA ON INTERMEDIATE
RESONANCES
IN NUCLEAR REACTIONS DONALD W. LANG t Arqonne National Laboratory, Aryonne, Illinois tt Received 10 October 1966
Abstract: The data on high-energy tails of evaporative spectra are interpreted in terms of "intermediate resonances", and the widths and spacings of these resonances are estimated. The estimated widths are close to the limits of present techniques of observation. It appears that the levels should overlap strongly. Cross correlations and other properties that help in identification are discussed.
1. Introduction T h e r e has recently been interest in features o f a nuclear cross section associated with times i n t e r m e d i a t e between those for c o m p o u n d - n u c l e u s events (up to 10-~4 sec) a n d those for direct reactions ( a r o u n d I0 -21"5 see). Associated with events t r a n s p i r i n g in times o f i n t e r m e d i a t e d u r a t i o n , features with an energy width that is likewise i n t e r m e d i a t e should p r e s u m a b l y a p p e a r in curves o f cross section versus energy. Experiments involving e v a p o r a t e d particles have been interpreted as offering evidence for the existence o f such " i n t e r m e d i a t e r e s o n a n c e " features; this p a p e r discusses the validity o f this evidence. Sect. 2, which is concerned with the interp r e t a t i o n o f experiments, leads to an u p p e r a n d lower limit on the widths involved. Sect. 3 is a discussion o f other p r o p e r t i e s that may, a c c o r d i n g to evidence from particle spectra, be associated with intermediate resonances.
2. Particle spectra and intermediate widths F o r a nuclear r e a c t i o n initiated by an energetic nuclear projectile, two types o f reactions have a l r e a d y been widely studied. In a " d i r e c t r e a c t i o n " , the incident projectile influences a n u c l e o n or a few c o r r e l a t e d nucleons. The rest o f the nucleus m a y be c o n s i d e r e d inert except as required by c o n s e r v a t i o n laws. The r e a c t i o n p r o d u c t s n o r m a l l y are energetic, a n d their a n g u l a r d i s t r i b u t i o n s are f o r w a r d p e a k e d ; the single r e a c t i o n is associated with a short i n t e r a c t i o n time. If the incident projectile is c a p t u r e d in the initial event, m o r e collisions m a y follow until the extra energy associated with the event is spread over a s u b s t a n t i a l n u m b e r t Present address: Theoretical Physics Division, A.E.R.E., Harwell, Didcot, Berks., England. tt Work performed under the auspices of the U.S. Atomic Energy Commission. 273
274
D.w.
LANO
o f excited nucleons in the "compound nucleus". Decay by particle emission occurs when enough energy to permit escape becomes concentrated on a single nucleon or cluster of nucleons. The interaction time is long, and the sense of the vector associated with the incident projectile is "forgotten." There may be a non-isotropic component of the cross section, but fore-and-aft symmetry is usual. Data that have been accumulating for some years t - 5 ) provide evidence for a third region of interest. There are more high-energy reaction products than would be predicted from a compound-nucleus picture, but the extra particles are fairly close to isotropy or at least fore-and-aft symmetry; and this conflicts with a directreaction description. Describing these observations as "evaporation from local hot spots" implicitly proposes a model for the reaction. The simplest idea conveyed by such a model presents difficulties; since the extra motion of the excited nucleons is not large in comparison with the ordinary motion of nucleons near the nuclear Fermi level, any "hot spot" in a nucleus in which the nucleons move incoherently should quickly dissipate its energy over the nuclear volume. The time for the nucleus to rotate sufficiently to "forget" the forward direction is considerably longer. Consideration of the imaginary part of the optical potential for alpha particle6, say, makes it difficult to credit the coherent motion of a "spot" through the nuclear volume. Such difficulties can be obviated 6, 7) by considering particular modes of excitation rather than particular localities. An individual nucleon in an excited but bound state has a considerable mean free path in nuclear matter, corresponding to a small imaginary optical-model potential in an ordinary nucleus. On a compound-nucleus model, this would correspond to a considerable time interval between the "first capture collision" and the subsequent collisions that lead to a statistical distribution of energies. While in the "doorway state" between the first and subsequent collisions, there would be a higher probability that events leading to particle emission would result in the emission of a high-energy particle. The lifetime to be associated with such a state has been estimated on the basis of particular models 7) of the state. The purpose here is to establish upper and lower empirical limits on the lifetime. The Fourier relation between lifetimes and widths of resonances then delimits the width of the associated intermediate resonances. In this paper the intermediate width is given the symbol g to distinguish it from the compound width F. At the lower limit, the motion of a nucleon near the Fermi level carries it about once around the nucleus in the lifetime of the state. The lifetime for A ~ 125 is then t, ~ 8 x 10 - 2 2 see, and the corresponding width of an intermediate-resonance feature would be g < 0.6 MeV. A lower limit on g may be obtained from the slope of the apparently evaporative spectra in the high-energy region. The high-energy spectrum of protons from a typical (~, p) experiment 2) at 30 MeV is much the same for all nuclei heavier than, say, iron. For events leading to the high-energy proton spectrum from gold, it is therefore reasonable to associate a lifetime (and hence an
NUCLEAR REACTIONS
275
intermediate width) equal to that appropriate for an iron nucleus struck by a 30 MeV alpha particle. The same idea is expressed in the assertion that the states emitting the high-energy tail of the apparently evaporative spectrum are assumed to have attained configurations roughly as complex as those in the nickel compound nucleus at 36 MeV and therefore decay selectively to lower energy states which again have relatively simple configurations. If this matching of isothermal spectra is justified, then it is possible to extract an isothermal lifetime, width and strength function. For the example cited (a compound nucleus of nickel at 36 MeV), the level density is of the order 109.5 MeV-~. From the Weisskopf evaporation formula a), the appropriate value of the width F for the nickel nucleus (and of g for heavier nuclei) is
F ~. F. = 8m"a~z2"°3(U-Q") h2o3(U) ,
(2.1)
where Fn is the partial width for the emission of neutrons, mn the reduced mass of the neutron, ~c, the cross section for the reverse process, i.e. capture of a neutron by what is here the residual nucleus, o~(U) the level density in the compound nucleus at energy U, m ( U - Q , ) the level density in the residual nucleus at the maximum energy available after emission of a neutron whose binding energy is Q~, z, the slope of the logarithm of this level density, and h Planck's constant. On simplification, eq. (2.1) is replaced by F ~ 2 • 10-2(A~+ 1)2"c2 exp (Q,/r.),
(2.2)
where A is the nucleon number of the nucleus, and all energy quantities are expressed in MeV. The expression in the right-hand side of eq. (2.2) is very sensitive to the value of Q/z used. A lower limit in the case treated would be set at F ~ 15 keV; but by a few small variations of parameter, this can be altered to 30 keV. Since it is a lower limit, however, this does not concern us. 3. Further deductions about compound resonances
An immediate deduction from the figures given is that the resonances so far described have greatly overlapping widths; F/D ~ 108. For heavier nuclei g/Dg is then assumed to be 10a also. In this region, then, discussion of the detection of such levels must depend on the sort of analysis used to find the mean widths of ordinary compound-nucleus levels. One such analysis is by means of auto-correlations. An immediate question then concerns the differentiation between compoundnucleus fluctuations and "intermediate fluctuations." If one examines the bombardment of gold with 30 MeV alpha particles, for example, the value of F for the compound nucleus is calculated to be of the order 1 keV. Thus in any reaction that exhibits a hot spot evaporation tail, one expects to detect the intermediate resonances as fluctuations coherent over widths much greater
276
D.W. tANG
than the compound-nucleus fluctuations. In particular, for some favourable reactions measured with good Iesolution, the yield to a fixed region of low excitation energies (or even to a single state in the residual nucleus) should show an auto-correlation width that corresponds to the intermediate resonances; it would be much larger than that calculated for evaporation from the compound nucleus. In the example used, such a correlation would be predicted for gold bombarded by 30 MeV alpha particles if the energy spread in the beam were less than 15 keV. Given the existence of bounds on the width predicted for the intermediate structure, it is still questionable if detection is possible. We note first that there are many possible angular momenta in the intermediate state, so that the sum will exhibit smaller relative fluctuations than appear if only one angular m o m e n t u m is available. On the other hand, with the immense number of contributing partial widths, the total width is comparatively well defined. Some features of ordinary resonances are missing in a region of such marked overlap. There is, for example, no reason to expect isolated features for which a good J-value can be inferred. Similarly, the individual terms making up an "intermediatewidth Ericson fluctuation" may be subject to laws with more than simply statistical relevance but it appears hopeless to predict the exact shape - now or ever. There is no available reason to suggest that any particular observed shape of a cross section should be repeated at higher energy or in a nucleus with different A. Evidence already exists, however, that there should be cross correlations between various modes of decay. In calculations on the compound nucleus, Bodansky 9) has compared ratios of cross sections and has found that . . . P') . > .o(n, ~(P' . . . p) a(p, n) a(n, n')"
(3.1)
This inequality holds after all the appropriate energy factors have been removed. It suggests that there is a definite probability (higher than the purely statistical value) that the outgoing particle is the one that went in. This suggests that the yields of protons to various residual energies or separate states of the final nucleus should be cross correlated with an energy width corresponding to the intermediate resonances but not with the width corresponding to the purely compound states. The sort of calculations required to establish this correlation are as follows. Define
C^B(A)
<~-A-(E) I\~(~-A-) 1]/,
(3.2)
where the angular brackets denote an average over a considerable range, and 6A(E) and tYB(E+ A) denote the averages of the cross sections for processes A and B over a range that is much smaller but still large compared with the correlation variable A. Then considering first the auto-correlation so that A and B are the same process, C A A ( A ) is expected to have a term proportional to F2/(F2--~-A2), where F is the
NUCLEAR REACTIONS
277
width of the simple compound-nucleus states. There is also a somewhat smaller term ~2/(g2.4-A 2) having the same functional form, where g is the width of the intermediate states. For processes in which A is not the same as B, the term in F is assumed 1o) to vanish. The evidence from (p, p') experiments indicates that there is a non-vanishing term of some sort. It is reasonable to expect this term to involve g. Bodansky 9) also cites other evidence indicating that more protons are emitted in ~t-particle reactions than would be predicted on purely statistical grounds. Miller 11) has found a different proton spectrum in (p, p') and (ct, p) reactions. It is therefore postulated that as the beam energy of any particular incident particle is varied there should be cross correlations of width g between different parts of the high-energy tails of the spectrum of any one type of outgoing particle. In such a case, the emitted proton may be suspected to have been part of the incident alpha. It appears to be an open question whether the same phenomenon should occur at all, or to a lesser extent, in the high-energy tails associated with the emission of different particles. It would be of considerable interest to test all the possible cross correlations in an experiment in which or-particles, protons and neutrons were detected. The present discussion and the evidence cited are concerned entirely with intermediate resonances in a region in which these resonances overlap very considerably. No evidence is given for the existence of isolated intermediate resonances, although the arguments to explain the existence of high-energy tails by means of doorway states do not rule out the existence of such relatively simple configurations in isolation. To this extent, the discussion also allows us to place a limit on the range of widths expected for doorway states. It should also be noted that the structure of the states is assumed to be simpler than is usual in the energy range, but it is not specified. It is to be expected, therefore, that estimates of the number of states should be larger than those of Le Couteur 12). In particular, the sort of state produced by a collision of an ~-particle with a nucleus is likely to be more complicated than the two-particle one-hole states often considered to be the archetype of the doorway states. It is best to conclude by pointing out a pitfall in interpreting a given bump as an intermediate resonance. Even in a cross section synthetically generated by the random addition of contributions from a very large number of compound-nucleus terms, it has been shown 13) that there are always energy regions containing aggregations visually indistinguishable from intermediate structure. It is a pleasure to acknowledge the hospitality of the University of Kansas for the period during which the results of this paper were transformed from numerical to written form. References 1) P. C. Gugelot, Phys. Rev. 93 (1954) 425; R. M. Eisberg, G. lgo and H. E. Wegner, Phys. Rev. 100 (1955) 1309; J. B. Mead and B. L. Cohen, Phys. Rev. 125 (1962) 964
27S
D.W. LANO
2) V. A. Sidorov, Nuclear Physics 35 (1962) 253; A. Alevra et aL, Nuclear Physics 58 (1964) 108 3) C. H. Holbrow and H. H. Barschall, Nuclear Physics 42 (1963) 264; R. M. Wood, R. R. Borchers and H. H. Barschall, Nuclear Physics 71 (1965) 529; R. R. Borchers, R. M. Wood and C. H. Holbrow, Nuclear Physics 88 (1966) 689 4) L. W. Swenson and N. Cindro, Bull. Am. Phys. Soc. $ (1960) 46; L. W. Swenson and C. Gruhn, Bull. Am. Phys. Soc. 8 (1963) 356 5) R. W. West, Phys. Rev. 141 (1966) 1033 6) K. Izumo, Progr. Theoret. Phys. 26 (1961) 807; K. Izumo, Nuclear Physics 62 (1965) 673 7) A. K. Kerman, L. S. Rodberg and J. E. Young, Phys. Rev. Lett. 11 (1963) 9 8) V. F. Weisskopf, Phys. Rev. 52 (1937) 295 9) D. Bodansky, Ann. Rev. Nucl. Sci. 12 (1962) 79 10) T. Ericson, Ann. of Phys. 23 (1963) 390 11) J. M. Miller, private communication 12) K. J. Le Couteur, Phys. Lett. 11 (1964) 53 13) P. P. Singh, P. Hoffman-Pinther and D. W. Lang, Bull. Am. Phys. Soc. 11 (1966) 350 and to be published
Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher
EVAPORATIVE
DATA ON INTERMEDIATE
RESONANCES
IN NUCLEAR REACTIONS DONALD W. LANG t Arqonne National Laboratory, Aryonne, Illinois tt Received 10 October 1966
Abstract: The data on high-energy tails of evaporative spectra are interpreted in terms of "intermediate resonances", and the widths and spacings of these resonances are estimated. The estimated widths are close to the limits of present techniques of observation. It appears that the levels should overlap strongly. Cross correlations and other properties that help in identification are discussed.
1. Introduction T h e r e has recently been interest in features o f a nuclear cross section associated with times i n t e r m e d i a t e between those for c o m p o u n d - n u c l e u s events (up to 10-~4 sec) a n d those for direct reactions ( a r o u n d I0 -21"5 see). Associated with events t r a n s p i r i n g in times o f i n t e r m e d i a t e d u r a t i o n , features with an energy width that is likewise i n t e r m e d i a t e should p r e s u m a b l y a p p e a r in curves o f cross section versus energy. Experiments involving e v a p o r a t e d particles have been interpreted as offering evidence for the existence o f such " i n t e r m e d i a t e r e s o n a n c e " features; this p a p e r discusses the validity o f this evidence. Sect. 2, which is concerned with the interp r e t a t i o n o f experiments, leads to an u p p e r a n d lower limit on the widths involved. Sect. 3 is a discussion o f other p r o p e r t i e s that may, a c c o r d i n g to evidence from particle spectra, be associated with intermediate resonances.
2. Particle spectra and intermediate widths F o r a nuclear r e a c t i o n initiated by an energetic nuclear projectile, two types o f reactions have a l r e a d y been widely studied. In a " d i r e c t r e a c t i o n " , the incident projectile influences a n u c l e o n or a few c o r r e l a t e d nucleons. The rest o f the nucleus m a y be c o n s i d e r e d inert except as required by c o n s e r v a t i o n laws. The r e a c t i o n p r o d u c t s n o r m a l l y are energetic, a n d their a n g u l a r d i s t r i b u t i o n s are f o r w a r d p e a k e d ; the single r e a c t i o n is associated with a short i n t e r a c t i o n time. If the incident projectile is c a p t u r e d in the initial event, m o r e collisions m a y follow until the extra energy associated with the event is spread over a s u b s t a n t i a l n u m b e r t Present address: Theoretical Physics Division, A.E.R.E., Harwell, Didcot, Berks., England. tt Work performed under the auspices of the U.S. Atomic Energy Commission. 273
274
D.w.
LANO
o f excited nucleons in the "compound nucleus". Decay by particle emission occurs when enough energy to permit escape becomes concentrated on a single nucleon or cluster of nucleons. The interaction time is long, and the sense of the vector associated with the incident projectile is "forgotten." There may be a non-isotropic component of the cross section, but fore-and-aft symmetry is usual. Data that have been accumulating for some years t - 5 ) provide evidence for a third region of interest. There are more high-energy reaction products than would be predicted from a compound-nucleus picture, but the extra particles are fairly close to isotropy or at least fore-and-aft symmetry; and this conflicts with a directreaction description. Describing these observations as "evaporation from local hot spots" implicitly proposes a model for the reaction. The simplest idea conveyed by such a model presents difficulties; since the extra motion of the excited nucleons is not large in comparison with the ordinary motion of nucleons near the nuclear Fermi level, any "hot spot" in a nucleus in which the nucleons move incoherently should quickly dissipate its energy over the nuclear volume. The time for the nucleus to rotate sufficiently to "forget" the forward direction is considerably longer. Consideration of the imaginary part of the optical potential for alpha particle6, say, makes it difficult to credit the coherent motion of a "spot" through the nuclear volume. Such difficulties can be obviated 6, 7) by considering particular modes of excitation rather than particular localities. An individual nucleon in an excited but bound state has a considerable mean free path in nuclear matter, corresponding to a small imaginary optical-model potential in an ordinary nucleus. On a compound-nucleus model, this would correspond to a considerable time interval between the "first capture collision" and the subsequent collisions that lead to a statistical distribution of energies. While in the "doorway state" between the first and subsequent collisions, there would be a higher probability that events leading to particle emission would result in the emission of a high-energy particle. The lifetime to be associated with such a state has been estimated on the basis of particular models 7) of the state. The purpose here is to establish upper and lower empirical limits on the lifetime. The Fourier relation between lifetimes and widths of resonances then delimits the width of the associated intermediate resonances. In this paper the intermediate width is given the symbol g to distinguish it from the compound width F. At the lower limit, the motion of a nucleon near the Fermi level carries it about once around the nucleus in the lifetime of the state. The lifetime for A ~ 125 is then t, ~ 8 x 10 - 2 2 see, and the corresponding width of an intermediate-resonance feature would be g < 0.6 MeV. A lower limit on g may be obtained from the slope of the apparently evaporative spectra in the high-energy region. The high-energy spectrum of protons from a typical (~, p) experiment 2) at 30 MeV is much the same for all nuclei heavier than, say, iron. For events leading to the high-energy proton spectrum from gold, it is therefore reasonable to associate a lifetime (and hence an
NUCLEAR REACTIONS
275
intermediate width) equal to that appropriate for an iron nucleus struck by a 30 MeV alpha particle. The same idea is expressed in the assertion that the states emitting the high-energy tail of the apparently evaporative spectrum are assumed to have attained configurations roughly as complex as those in the nickel compound nucleus at 36 MeV and therefore decay selectively to lower energy states which again have relatively simple configurations. If this matching of isothermal spectra is justified, then it is possible to extract an isothermal lifetime, width and strength function. For the example cited (a compound nucleus of nickel at 36 MeV), the level density is of the order 109.5 MeV-~. From the Weisskopf evaporation formula a), the appropriate value of the width F for the nickel nucleus (and of g for heavier nuclei) is
F ~. F. = 8m"a~z2"°3(U-Q") h2o3(U) ,
(2.1)
where Fn is the partial width for the emission of neutrons, mn the reduced mass of the neutron, ~c, the cross section for the reverse process, i.e. capture of a neutron by what is here the residual nucleus, o~(U) the level density in the compound nucleus at energy U, m ( U - Q , ) the level density in the residual nucleus at the maximum energy available after emission of a neutron whose binding energy is Q~, z, the slope of the logarithm of this level density, and h Planck's constant. On simplification, eq. (2.1) is replaced by F ~ 2 • 10-2(A~+ 1)2"c2 exp (Q,/r.),
(2.2)
where A is the nucleon number of the nucleus, and all energy quantities are expressed in MeV. The expression in the right-hand side of eq. (2.2) is very sensitive to the value of Q/z used. A lower limit in the case treated would be set at F ~ 15 keV; but by a few small variations of parameter, this can be altered to 30 keV. Since it is a lower limit, however, this does not concern us. 3. Further deductions about compound resonances
An immediate deduction from the figures given is that the resonances so far described have greatly overlapping widths; F/D ~ 108. For heavier nuclei g/Dg is then assumed to be 10a also. In this region, then, discussion of the detection of such levels must depend on the sort of analysis used to find the mean widths of ordinary compound-nucleus levels. One such analysis is by means of auto-correlations. An immediate question then concerns the differentiation between compoundnucleus fluctuations and "intermediate fluctuations." If one examines the bombardment of gold with 30 MeV alpha particles, for example, the value of F for the compound nucleus is calculated to be of the order 1 keV. Thus in any reaction that exhibits a hot spot evaporation tail, one expects to detect the intermediate resonances as fluctuations coherent over widths much greater
276
D.W. tANG
than the compound-nucleus fluctuations. In particular, for some favourable reactions measured with good Iesolution, the yield to a fixed region of low excitation energies (or even to a single state in the residual nucleus) should show an auto-correlation width that corresponds to the intermediate resonances; it would be much larger than that calculated for evaporation from the compound nucleus. In the example used, such a correlation would be predicted for gold bombarded by 30 MeV alpha particles if the energy spread in the beam were less than 15 keV. Given the existence of bounds on the width predicted for the intermediate structure, it is still questionable if detection is possible. We note first that there are many possible angular momenta in the intermediate state, so that the sum will exhibit smaller relative fluctuations than appear if only one angular m o m e n t u m is available. On the other hand, with the immense number of contributing partial widths, the total width is comparatively well defined. Some features of ordinary resonances are missing in a region of such marked overlap. There is, for example, no reason to expect isolated features for which a good J-value can be inferred. Similarly, the individual terms making up an "intermediatewidth Ericson fluctuation" may be subject to laws with more than simply statistical relevance but it appears hopeless to predict the exact shape - now or ever. There is no available reason to suggest that any particular observed shape of a cross section should be repeated at higher energy or in a nucleus with different A. Evidence already exists, however, that there should be cross correlations between various modes of decay. In calculations on the compound nucleus, Bodansky 9) has compared ratios of cross sections and has found that . . . P') . > .o(n, ~(P' . . . p) a(p, n) a(n, n')"
(3.1)
This inequality holds after all the appropriate energy factors have been removed. It suggests that there is a definite probability (higher than the purely statistical value) that the outgoing particle is the one that went in. This suggests that the yields of protons to various residual energies or separate states of the final nucleus should be cross correlated with an energy width corresponding to the intermediate resonances but not with the width corresponding to the purely compound states. The sort of calculations required to establish this correlation are as follows. Define
C^B(A)
<~-A-(E) I\~(~-A-) 1]/,
(3.2)
where the angular brackets denote an average over a considerable range, and 6A(E) and tYB(E+ A) denote the averages of the cross sections for processes A and B over a range that is much smaller but still large compared with the correlation variable A. Then considering first the auto-correlation so that A and B are the same process, C A A ( A ) is expected to have a term proportional to F2/(F2--~-A2), where F is the
NUCLEAR REACTIONS
277
width of the simple compound-nucleus states. There is also a somewhat smaller term ~2/(g2.4-A 2) having the same functional form, where g is the width of the intermediate states. For processes in which A is not the same as B, the term in F is assumed 1o) to vanish. The evidence from (p, p') experiments indicates that there is a non-vanishing term of some sort. It is reasonable to expect this term to involve g. Bodansky 9) also cites other evidence indicating that more protons are emitted in ~t-particle reactions than would be predicted on purely statistical grounds. Miller 11) has found a different proton spectrum in (p, p') and (ct, p) reactions. It is therefore postulated that as the beam energy of any particular incident particle is varied there should be cross correlations of width g between different parts of the high-energy tails of the spectrum of any one type of outgoing particle. In such a case, the emitted proton may be suspected to have been part of the incident alpha. It appears to be an open question whether the same phenomenon should occur at all, or to a lesser extent, in the high-energy tails associated with the emission of different particles. It would be of considerable interest to test all the possible cross correlations in an experiment in which or-particles, protons and neutrons were detected. The present discussion and the evidence cited are concerned entirely with intermediate resonances in a region in which these resonances overlap very considerably. No evidence is given for the existence of isolated intermediate resonances, although the arguments to explain the existence of high-energy tails by means of doorway states do not rule out the existence of such relatively simple configurations in isolation. To this extent, the discussion also allows us to place a limit on the range of widths expected for doorway states. It should also be noted that the structure of the states is assumed to be simpler than is usual in the energy range, but it is not specified. It is to be expected, therefore, that estimates of the number of states should be larger than those of Le Couteur 12). In particular, the sort of state produced by a collision of an ~-particle with a nucleus is likely to be more complicated than the two-particle one-hole states often considered to be the archetype of the doorway states. It is best to conclude by pointing out a pitfall in interpreting a given bump as an intermediate resonance. Even in a cross section synthetically generated by the random addition of contributions from a very large number of compound-nucleus terms, it has been shown 13) that there are always energy regions containing aggregations visually indistinguishable from intermediate structure. It is a pleasure to acknowledge the hospitality of the University of Kansas for the period during which the results of this paper were transformed from numerical to written form. References 1) P. C. Gugelot, Phys. Rev. 93 (1954) 425; R. M. Eisberg, G. lgo and H. E. Wegner, Phys. Rev. 100 (1955) 1309; J. B. Mead and B. L. Cohen, Phys. Rev. 125 (1962) 964
27S
D.W. LANO
2) V. A. Sidorov, Nuclear Physics 35 (1962) 253; A. Alevra et aL, Nuclear Physics 58 (1964) 108 3) C. H. Holbrow and H. H. Barschall, Nuclear Physics 42 (1963) 264; R. M. Wood, R. R. Borchers and H. H. Barschall, Nuclear Physics 71 (1965) 529; R. R. Borchers, R. M. Wood and C. H. Holbrow, Nuclear Physics 88 (1966) 689 4) L. W. Swenson and N. Cindro, Bull. Am. Phys. Soc. $ (1960) 46; L. W. Swenson and C. Gruhn, Bull. Am. Phys. Soc. 8 (1963) 356 5) R. W. West, Phys. Rev. 141 (1966) 1033 6) K. Izumo, Progr. Theoret. Phys. 26 (1961) 807; K. Izumo, Nuclear Physics 62 (1965) 673 7) A. K. Kerman, L. S. Rodberg and J. E. Young, Phys. Rev. Lett. 11 (1963) 9 8) V. F. Weisskopf, Phys. Rev. 52 (1937) 295 9) D. Bodansky, Ann. Rev. Nucl. Sci. 12 (1962) 79 10) T. Ericson, Ann. of Phys. 23 (1963) 390 11) J. M. Miller, private communication 12) K. J. Le Couteur, Phys. Lett. 11 (1964) 53 13) P. P. Singh, P. Hoffman-Pinther and D. W. Lang, Bull. Am. Phys. Soc. 11 (1966) 350 and to be published