Y ukinori SAKURAGI, t Hirofumi KAMEY AMA and Mitsuji KAWAI. Department of Physics ... **Chiba-Keizai College, Chiba 260 t Institute for Nuclear Study, ...
Progress of Theoretical Physics Supplement No. 89, 1986
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Chapter I Projectile Breakup Processes in Nuclear Reactions Masayasu KAMIMURA, Masanobu YAHIRO, * Y asunori ISERI* * Y ukinori SAKURAGI, t Hirofumi KAMEY AMA and Mitsuji KAWAI
Department of Physics, Kyushu University, Fukuoka 812 * Shimonoseki University of Fisheries, Shimonoseki 759-65 **Chiba-Keizai College, Chiba 260 t Institute for Nuclear Study, University of Tokyo, Tanashi, Tokyo 188
Studies of projectile breakup processes with the use of the method of coupled discretized continuum channels are surveyed and introduction to the subsequent chapters is presented.
In nuclear reactions between composite particles, those involving a loosely bound particle, such as deuteron (d), 6 •7 Li, 12 C, etc., breakup of the particle takes place as a real or virtual process. The processes have attracted the attention of many researchers and a number of investigations have been made. Such studies started with those on the deuteron breakup. The deuteron breakup processes are at least of a three-body system composed of proton (p), neutron (n) and the target nucleus (A) in its ground state, A 0 • In principle such a three-body model of the deuteron-nucleus system can be derived as a projection of the A +2 body system onto a function space composed of p, n and A 0 • Since the resultant theoretical three-body Hamiltonian 1> is too complicated for practical purposes, most of the studies have employed a phenomenological three-body Hamiltonian in which the nucleon-nucleus interaction is represented by the optical potentials at half the deuteron incident energy, and an effective nucleon-nucleon potential is used for the p-n interaction. Several different approximate methods for solving the phenomenological three-body Hamiltonian were proposed for theoretical treatment of the process2 > before the adiabatic approximation was introduced by Johnson and Soper3> for virtual breakup of deuteron in elastic scattering and stripping reactions. The approximation was based on the observation that the internal motion of the p-n system, with a weakly bound ground state, is much slower than its center-of-mass motion. It was shown that the breakup process greatly affects the transition matrix elements of those reactions. The approximation has since been used widely for theoretical analyses of the breakup processes4 > and for the analysis of experimental data, not only on reactions involving a deuteron, but also on reactions induced by 6 Li ions, 5 > and has established the importance of the breakup process in those reactions.
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(Received October 30, 1986)
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M. Kamimura, M. Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama and M. Kawai
Method of CDCC
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The method of coupled channels was used by Johnson and Soper3 > in deriving their adiabatic approximation formulae. A major problem in the application of this method to breakup processes is that the breakup states are continuous, and one has to deal with a continuously infinite number of coupled channels. In Ref. 3) the continuous channels of the s-wave breakup states are approximated by a single "channel" with a certain average internal energy, an approximation consistent with the adiabatic assumption. Natural extension of this work is to allow for the spread in the energy of the continuum channels. A general method of dealing with the continuum channels by forming wave packets of the channel wave functions was given by Schmid and Ziegelmann. 6> Johnson and Tandy/> and Anders and Lindner 8 > utilized a method of replacing the continuum by a finite number of "channels" with Strumian wave functions. Rawitscher 9 > used what is now called the method of momentum bins in which the momentum space of the internal motion of the p-n system is divided into a finite number of bins, in which the internal wave functions are averaged. The coupled-channels calculation is carried out with a finite number of such "channels" with the averaged wave functions. The Pittsburg group 10 >proposed a slightly improved version of the method of momentum bins, in which the momentum dependence of the internal wave function within the bins is taken into account by a local WKB approximation. In the works of Refs. 7)"'"' 10), the continuum channels are replaced by a finite number of discretized "channels" which are then used in the coupled-channels calculation. This is what is now called the method of Coupled Discretized Continuum Channels, abbreviated as CDCC. 11 > In subsequent years, CDCC was used by many authors for investigations of the projectile breakup process. Earlier works 9 >,lo),lz> were mainly concerned with theoretical understanding of the mechanism of the process, including examination of the validity, and possible improvement of the adiabatic model. The present authors have developed CDCC so as to apply it not just to theoretical problems, but to the analysis of experimental data. We started this with establishing a reliable model space in which realistic CDCC calculations can be made. This was done in the deuteron induced reactions the model space of which is classified into the deuteron elastic (do), inelastic (d'), and breakup (d*) channels, and the rearrangement proton channel. There were three items to be clarified for that purpose: (i) The role of the rearrangement channels in CDCC calculations, (ii) the role of the d' -channel and (iii) to clarify if the model space is adequate for dealing with the d* channel with good accuracy. We investigated the first item in a straightforward manner ; we compared the results of CDCC calculations with and without the coupling of the proton channel with the d 0 and d* channels. However, a CDCC calculation with a coupled rearrangement channel was exceedingly difficult. We solved this problem by making use of a variational method for CDCC that we had developed. 13> The result of the analysis showed that the effect of the proton channel coupling is important at low energies but decreases rapidly as the incident energy increases, and much less important than the effect of d * channel at energies higher than 40 MeV. At such energies, therefore, the proton channel coupling can be neglected. 14> We investigated the role of the d' channel (the second item) in a manner similar to the previous one; we compared the results of CDCC calculations with and without the coupling of the d' channel with the d 0 and d * channels. The investigation was done for the
Projectile Breakup Processes in Nuclear Reactions
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d+ 58 Ni(Ot, 2t) system. The effect of the 10 Rifm) d' -channel coupling is less important than that of the d *-channel coupling. The situation becomes clearer as Ed increases ; at Ed =80 MeV the effect of the d' channel is quite small whereas that of the d* channel is still sizable. 15 >· 16 > The third item was to establish, after neglect of the rearrangement and inelastic channels, a reliable extension of the model space for realistic CDCC calculations. This was done again first for deuteron induced reactions. The method of momentum bins {d,pl/\ was used for the discretization of the \ \ continuum channels. The model space was \ \ then specified by the maximum linear and 20 0 10 30 J angular momenta of relative motion between the p-n pair, km and lm. respectively, and the Fig. 1. Comparison of calculated partial cross sections for various processes of deuteron on 58 Ni at size of the bin, Llk, or equivalently, number of 80 MeV. The cross sections are shown with discretized channels, N, for a given ken. The respect to the total angular momentum f and the adequacy of the model space was tested 11 > by impact parameter R=] I k, k being the wave the convergence of the calculated transition number of the incident deuteron. Radius of the target nucleus is 4.5 fm. matrix elements with respect to an increase of ken, lm or N. It turned out that the convergence can be achieved by km :-: ;: 1.0 fm- 1 , lm :-: ;: 2 and N :-::;: 8, depending on the particular transition matrix element being calculated, but remarkably independent of the incident energy. From the three ground works on the model space we can conclude that a tractable CDCC framework for the incident energies Ed~ 40 MeV is to treat the d0 and d * channels simultaneously with the coupled-channel method, and treat the transition from the do channel to ad' channel and rearrangement channels with the first-order Born approximation with the use of the CDCC wave function of the d 0 +d* channels. Figure 1 provides a support for this prescription; we compare calculated partial-wave cross sections of (d, d'), (d, p) and (d, pn) reactions for the system of d + 58 Ni(Ot, 2t) and p+ 59 Ni(g. s.) at Ed =80 MeV. Cross sections of the (d, d') and (d, p) reactions are much smaller than the net cross section of the s-wave (l=O) and d-wave (!=2) breakup reactions, as expected, especially at the surface and outer region of the target nucleus. With the model space thus established, it has become possible to carry out realistic CDCC calculations of the breakup process to confront with experimental data. The diagonal and coupling form factors are calculated by folding the p-target and n-target optical potentials into the wave functions of the deuteron ground and discretized continuum states. It is to be emphasized that there is no freely adjustable parameter in the calculations. The use of the nucleon optical potentials at Er, =En= Ed/2 may be justified by the fact that observed proton energy spectra in elastic-breakup (d,pn) reactions show a prominent bump at Er,c::::.Ed/2 and is reproduced by the CDCC calculation (Fig. 2). Using the framework outlined above, we have carried out a number of calculations of
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M. Kamimura, M. Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama and M. Kawai
reactions of deuterons and 3He ions. ll),ls),I 7>- 26 >· Extension of CDCC to light heavy-ion reactions
~ 10 2::
Extension of CDCC to projectiles heavier than deuteron and 3He looks ¢ straightforward, but it is to be noted that 12c (d, pnl12c structure of excited states of projectile nuE~ =56 MeV cleus becomes much more complicated as the 8~ = 15° projectile-mass increases. In this review, we investigate scattering and breakup of 6·7 Li and 12 C ions. There appear excited bound states and resonant states besides non40 20 resonant continuum states of the projectiles. E~ (MeV) This can bring about a variety of reaction Fig. 2. Proton spectrum in C(d, pn) reaction at 8/ mechanisms which are not seen in deuteron =-15° and -45°, 8nLbeingfixed at 15°. The data (3He) induced reactions. It is then essential and the result of the CDCC calculation (solid to prepare good wave functions of those curves) are taken from Refs. 41) and 18), restates of the projectile nucleus at the beginspectively. ning of the CDCC calculation. In order to obtain the wave functions of the internal states of those nuclei including resonant and non-resonant continuum states, we have employed a microscopic cluster modeF 7 > in which the total antisymmetrization between nucleons is taken into account ; it is known that the model is suited for describing both the bound and continuum states of those projectile nuclei. The wave functions well account for the electron-scattering form factors, phase shifts of the scattering between the constituent clusters, etc. The non-resonant continuum states were truncated and discretized in a similar way as in the deuteron case. Real part of the interactions between the projectile and the target nucleus is constructed by the double folding of an effective nucleon-nucleon interaction of M3Y type into the projectile states and the target ground state. Imaginary part of the interaction is simply assumed to have the same geometry of the real part with a constant factor multiplied. This factor is the only adjustable parameter in our microscopic CDCC calculation. Use of the double-folding interactions has been successful in the analyses of the scattering and breakup of 6 •7 Li and 12 C ions in a wide energy range.zs)-33),19),21),23),24) It is to be noted that, as is known from the above, CDCC analysis of experimental data can provide a crucial test of the microscopic cluster model for the state of the projectile nuclei including continuum states. Especially, the analysis is suited for checking the transition densities between excited (continuum) states, since the coupling between excited channels can play an important role in the scattering of light nuclei through a strong nuclear field of the target nucleus; those transition densities cannot be examined by means of electron scattering due to the weakness of electro-magnetic interactions. In fact, it has been found 29 > that the multi-step transitions between the resonant continuum states and the non-resonant ones affect very significantly both the elastic and breakup cross sections. There is another method to construct the interactions between the projectile and 0
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Projectile Breakup Processes in Nuclear Reactions
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Table I. List of reactions studied in the subsequent chapters with CDCC. In the third column, D denotes analysis of the experimental data, while T means theoretical analysis. Analysis
Chapter
4°Ca(d, d) 4°Ca 58 Ni(d, d) 58 Ni 208 Pb( d , d)2°8Pb 58 Ni(d, d') 58 Ni(01+, Z1+)
25.5 56, 80, 200, 400, 700 56 21.6, 80
D D D D
HI HI HI HI
58 Ni(d, d*) 58 Ni 1zc, 51V, nssn(d, pn) 12C, 28Si, 58 Ni(3He, dp) 58Ni(3He, dp) 58 Ni
21.6, 80 56 52 90
T D D D
IV IV IV IV
11, 20, 40, 60
T
v
30"-'170 63"-'148 30"-'170 124"-'178 70, 132 140"-'204 260 168"-'311
D D T D D D D D
VI VI VI VI VI VI VI VI
"O(d, p )"O(ls) 16Q(d, d)160 170(1s )(d, d) 170(1s) 1zc, 1zc, 1zc, 6Li,
l
zssvo,4sca, ssNi, zosPb(sLi, sLi) 4sca, ssNi, zospb(1Li, 7Li) zssi, 4o,4sca, ssNi, zosPb(sLi, ad) 12 C, 208 Pb( 6Li, ad)
1zc, 1zosn, zospb(1Li, at) 12C(1 2C, 12 C') 12C(o1+, 21+, 31-, oz+) 1zc(13C, 13C)1zc 1zceso, 16o)1zcco1+, z1+)
target nuclei. It is called cluster-folding model in contrast with the double-folding model mentioned above. In this case, for example in scattering of 6 Li, the projectile-target interaction is calculated by folding the a-target and d-target optical potentials into 6 Li wave functions given by a simpler cluster model in which the intrinsic structure of a and d is neglected and the antisymmetrization is approximately treated. Studies of light heavy-ion reactions based on the use of the cluster-folding interactions have been also successful in the analyses of scattering of polarized 6•7 Li ions and have elucidated important effects of the projectile excitation and breakup upon spin-dependent interactions between heavy ions. Results of those studies are not reviewed in this article, but are reported elsewhere. 34>-38>' 23 >' 24>
Characteristics of projectile breakup processes Table I gives a list of reactions which are investigated in this review with the applications of CDCC. The breakup processes of deuteron, 3 He, 6 •7 Li and 12 C were studied. The types of reactions and observables and the incident energies were considered widely. The agreement of the CDCC calculation with the experimental data is on the whole satisfactory, in many cases surprisingly good. Characteristic features of the projectile breakup processes found by the analyses are summarized below. First of all, it has been found that the projectile breakup channels affect greatly the elastic scattering cross sections in most cases studied. Furthermore, it has been discovered that the virtual breakup effect is absolutely important in reproducing spin observables in the scattering of polarized deuteron at 56 MeV from heavy targets 16 > and in the
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Energy (MeV)
System
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M. Kamimura, M. Yahiro, Y. Iseri, Y. Sakuragi, H. Kameyama and M. Kawai
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scattering of polarized 6 •7 Li ions at 20'"'--'45 MeV. 34>- 3 s> In the reactions induced by loosely bound projectiles, it is the projectile breakup channels that affect the elastic channel the most strongly. It is found that the projectile breakup takes place in a narrow radial region which is considerably outer the surface of the target nucleus. Figure 1 shows the peripheral nature of the deuteron breakup process which is much more enhanced than the inelastic and rearrangement processes in the region. This fact stems from long range extensions of the coupling potentials of the breakup channels and from the strong absorption of the flux by the imaginary part of the distorting potentials in the nuclear interior. The Pittsburg group 10 > reported a large contribution of the breakup in the nuclear interior region. We note, however, that this inner prominence is due to the use of unrealistically weak imaginary part of the nucleon-target potential and the neglect of the d-wave breakup component whose contribution is dominant as seen in Fig. 1. The peripheral nature of the projectile breakup becomes more striking for 3 He and light heavy-ion projectiles, 39 >· 40 > which is due to the increase of the absorption of the flux in the nuclear interior. This nature is advantageous in studying the surface clusterization of light nuclei which are used as the projectiles. The use of the cluster-model wave functions, which must be most accurate at the surface region, for the projectile nuclei is one of the origins of the success of CDCC in light heavy-ion reactions. 40 > In deuteron breakup processes, transitions among the d* channels themselves (continuum-continuum coupling) are much stronger than those between the do channel and the individual d * channels (bound-continuum coupling). The latter transition can be treated perturbatively, but the former cannot be done so at all,3 9 > Breakup of deuteron projectile is found to proceed dominantly as a one-step breakup followed by multi-step transitions between d* channels, which is symbolized by d 0 ~ [d*~d*]; here the symbol A ~ B means a one-step transition from A to B, while [A~ B] represents multi-step transitions between them. Due to the strong coupling between the d * channels, the process do ~ [d*~d*] deviates greatly from the one-step breakup d 0 ~d*. As for the elastic-breakup (d, pn) reactions, CDCC reproduces well 18>· 39 > the coincidence cross section and the proton spectrum41 > (cf. Fig. 2). The prior-form DWBA, 41 > however, much overpredicts the breakup for which the angle between the emitted p and n is rather narrow; this is due to the neglect of the multi-step process [d* ~d*] which is especially important to reduce the transition to the p-n continuum states with low excitation energies. 18>· 39 > On the other hand, a calculation 4 c> based on the adiabatic approximation (AD) 3>· 4> explained satisfactorily well the (d, pn) reactions both for the narrow-angle breakup and the wide one; this is because AD includes the process [d*~d*] though in an approximate way. It is further found that the [d* ~d*] process is also important to bring about two characteristic processes of the breakup reactions, namely the quasi-free-like process and the stripping-like process, within the framework of CDCC. 20 >· 39> The strong continuum-continuum transitions, [d* ~d*], plays very important role in the elastic scattering cross section, too. It is found 20 >· 39 > that the two-step d0 ~ d *~ d 0 process is not at all a good approximation of the process d 0 ~ [d*~d*] ~do though the former makes calculation very easy, while the latter well simulates the full coupledchannel process of the elastic scattering. Also for light heavy-ion projectiles, the multi-step transitions among the (resonant and non-resonant) continuum states play an important role both in elastic and breakup
Projectile Breakup Processes in Nuclear Reactions
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Dynamically induced potential due to projectile virtual breakup Virtual excitation of projectile (target) nucleus induces generally a dynamical polarization (DP) potential in the elastic channel. It is often stated that if the coupling of the excited channel is weak (and real), effect of the channel on the elastic one can be represented by an absorptive potential to be added to the latter channel, but in the case of strong coupling the excited channel can significantly contribute to the real part of the DP potential, too. A striking example of this contribution to the real part was found in the scattering of 6 Li ions. Coupling of the projectile breakup channels (dominantly of the a-d resonant channels) with the elastic channel is very strong and induces a strongly repulsive real potential and negligibly weak imaginary part. The magnitude of the repulsive potential amounts to about a half of the diagonal potential of the elastic channel with the opposite sign; this explains beautifully the anomalously small renormalization factor (NR ""'0.5) of the real part which is employed in the calculations with the single-channel double-folding model. 42 > Throughout the analyses of the elastic scattering listed in Table I, we understand that characteristic feature of the DP potential induced by a specific excited channel is governed by the ratio of the imaginary part of the coupling potential to its real part. The characteristic features can be summarized as follows : Let y denote that ratio and l1 V +ill W the DP potential. For the 6 Li scattering, we have y""0.6 which brings about L1V>O and llW :::::::o as mentioned above. In the case of 12 C scattering at ""200 MeV, y""0.4 which results llV::::::: -llW >0. For deuteron scattering at 56 MeV, y:::::o:O.Z-----0.3 which induces lJV:::::o:-lJW>O, while at 700 MeV, y)>l which generates llW>llV>O. It is to be emphasized that those DP potentials induced by the projectile breakup channels are very different from the usual type of DP potentials induced by excitation of collective modes, namely llW to estimate a DP potential. Let Vc + iWc denote the coupling potential between the elastic channel and the excited channel concerned. A very qualitative approximation of the DP potential 43 >· 40 > may be given by
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cross sections. 29 >· 30 >' 40 > We have tested the validity of AD in the elastic scattering, breakup reactions and the (d, p) reactions. 15 >- 17 >.zo> It is found that AD becomes better as the incident energy increases (satisfactorily good at 80 MeV), as expected, and the accuracy of AD depends on the type of reactions; among the three-types of reactions AD works best in the elastic scattering and worst in the (d, p) reactions. In AD the excitation energy of deuteron is neglected, and therefore error of the AD wave function is larger in the breakup component than in the elastic component. Since momentum mismatching between the incident and exit channels is the largest in the (d, p) reaction, the error of the breakup component becomes most evident in the reaction. According to a CDCC analysis of 58 Ni (d, p) 59 Ni (g. s.) at 80 MeV, the neutron transfer via the d * channel becomes important at {)P em 0. The DP potential obtained in deuteron scattering at 700 MeV are seen at y"'10 where LlW>LlV>O. The well-known type of DP potential in collectivemode excitation may correspond to the case at y"'0.1 where LlW
