Direct reactions to many-channel, many-level final states

Nuclear Physics A24Q (1975)

413-424;

@

North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilm without written permission from the publisher

DIRECT REACTIONS

TO MANY-CHANNEL, MANY-LEVEL (II). (d, p)n and (?He, d)p

FINAL STATES

P. B. TREACY Department of Nuclear Phvsies, Research School of Physical Sciences, The Austrdlian National University, Canberra, ACT2600

Received 9 September 1974 Abstract:

In an earlier paper, the cross section for a direct reaction to a generalised positive-energy tinal state, described by an R-matrix wave function, was derived. Here a distinction is emphasised between two classes of such a reaction, depending on whether it can or cannot decay back to the target state. A major contribution in the latter case can be via direct break-up. This is computed for the reaction 7Li(3He, d)*Be(p)‘Li, treated by a stripping mechanism. The dependence of the cross section on the final-state (sBe) channel energy is discussed. The.present work uses shellmodel wave functions for R-matrix basis states, and this is shown to be appropriate and useful for direct reactions.

1. Introduction

In an earlier paper of the same title ‘), a theory was outlined for a direct reaction to a particle-unstable final state described by an R-matrix wave function for a manychannel many-level state. The mechanism was shown to be applicable to the process (d, p) fission, where fission succeeds a direct stripping react&n. In that work use was made of a rather accurate approximation proposed by Barker 2). The present work discusses reactions for which Barker’s formula cannot be assumed (although, as will be shown, it may still be a useful approximation). It is useful to distinguish just why direct reactions to positive-energy final states fall into two distinct classes illustrated by their representatives: I (d, p) fission; II (d, p)n. Basically, in class I the initial state of target plus deuteron, and the final state of two fission fragments plus a proton, have no overlap; in II the final state may be exactly the target plus a split-up deuteron. The latter system can obviously be created by direct break-up outside the usually-assumed nuclear radius. For I it was seen in ref. ‘) that the final-state energy dependence of the cross section has a simple resonance form 3). The theory of ref. ‘) also implies a unique energy dependence for the process II, and this will be discussed in the present work. The basic distorted-wave theory for direct reactions induced by light ions is wellestablished, and goes back to papers such as ref. 3, and well-known computer programs such as JULIE and DWUCK. Where final states have positive energies and direct break-up is possible, there is a numerical complication in computing radial integrals with slowly decaying oscillating amplitudes. There is an adequate contourintegration technique4) to enable evaluation of such quantities to high accuracy. It is however remarkable that the basic question of the final-state channel-energy 413

4I4

P. B. TREACY

dependence of the cross section seems to have been treated in a cavalier fashion by most authors - the usual procedure being to attach, without justification, a Lorentzian amplitude to the reaction amplitude derived from the bound-state theory. In ref. ‘) it was shown just where this would be permissible, namely for a reaction of class I to an isolated single-level resonance well away from channel thresholds. In the case where threshold-energy dependence cannot be neglected, the accuracy of the standard R-matrix amplitude 5, is well established down to parts per million [cf. ref. 6, for an extreme example of thisl, and its approximation by Lorentzian [sometimes called Breit-Wigner or “Gamow” forms ‘I)] or by single-level S-matrix amplitudes 12) is scarcely justified where a detailed experimental line shape is available for analysis +. We note that another approach i3,r4), using perturbation methods, is of interest here. A brief outline of the theory is given in subsect. 2.1 below. In subsect. 2.2 (and the appendix) the use of shell-model wave functions for R-matrix basis states is proposed and discussed. Detailed computation and comparison to ex~~ental data for 7Li(3He, d) are given in sect. 3. We emphasize that this is conventional DWBA, apart from the inclusion of radial integrals for positive-energy final states, and correct energy dependence of wave functions. It uses a zero-range approximation, which immediately removes “off-energy-shell momenta ” ’ 3, 14) from the computation. In sect. 4 are detailed some purely theoretical line shapes and their approximation by various formulas, and sect. 5 s~marises our conclusions and discusses impli~tions for extraction of spectroscopic information from these types of data. 2. Summary of theory 2.1. CROSS SECTIONS; INTERNAL AND EXTERNAL CONTRIBUTIONS

In ref. ‘) an expression was obtained for the cross section for a direct stripping reaction in a general form common to classes I and II of direct reactions. As will be mentioned in sect. 3 below, the reaction 7Li(31-Ie,d) via interfering li levels in ‘Be is a suitable example for illustrating this general process. We are interested here in the theory of the direct stripping reaction [cf. ref. ‘) eq.

ml : A+a + B+b,

B+C+c,

(1)

where B is formed via capture of the transferred particle t by the target A and the observed decay to channel (C + c), of energy E,, may or may not comprise (A + t). The t The work of ref. 6, relates to the well-known Be ghost’), and the analysis in ref. 6, relies on experimental data of Hay et al. s). The numerical magnitude of the ghost, as there analysed, has been questioned by Berkowitz et nt. ‘), who completed a number of experiments on this nucleus. It must be remarked that their published beam profile line shapes were far too impure for accurate estimation of the ghost as established in ref. *). Other objections to the conclusions of ref. ‘) by Lorenz lo) have been discussed in ref. %

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415

latter must always be present among the outgoing channels of B, but it could be inhibited, either by its being a closed (negative-energy) channel, or by competition from other channels. In the notation of ref. ‘), the differential stripping cross section per energy E, may be written as d2a ~ da, dE, =

1 Q~~Z)Z

in terms of the Born-approximation T,,[DW]

papbg

Pb y a

IT,,[DW]l’

(2)

matrix element

= (xb-%J

tibi ~bkd+k4

(3)

$a>-

Eq. (3) may be evaluated in terms of the collision matrix [ref. ‘) eqs. (25a) and (25b)]. An appropriate form for eir is given by

= dicOi-

uiczi

tri > ai),

(4’4

with R-matrix level parameters in the usual notation of Lane and Thomas 5). Of particular interest is XrJeMB, an energy-independent normalised basis state, obeying a constant boundary condition at the radius ri = ai of each of the ‘7” channels of B, open and closed. Eq. (4b) refers to all of these, however in the channel region the only finite contributions to (3) come from the channel (A + t), because of the orthogonality of the channel wave functions. Evidently if no such contribution occurs, the reaction amplitude is proportional to the internal contribution A,,,F&, which is essentially Barker’s formula. If it does occur, the energy dependence is compounded by the channel wave functions 0, and I, as E, passes through the resonances of the system as represented by the level matrix Apl. The possibility of an external contribution, for r, > a,, identifies reactions of class II such as (d, p)n. As will be seen from the computations below, this retains an identifiable resonant part (contributed via the U, of eq. (4b)) but also a significant nonresonant amplitude which may be identified physically with direct break-up. This sharp distinction between contributions is to some extent an artefact of the assumptions made (zero-range DWBA): inclusion of finite range and coupling to other channels, would certainly render less sharp this division. Eq. (2) using (3) and (4a, b) provides a recipe for our computing problem. A more specific form is required for XaJeMBin terms of the radial wave function u,Jr) of the transferred particle [ref. ‘) eq. (25c)]. In subsect. 2.2 below this relation is given specifically in terms of shell-model functions. 2.2. SHELL-MODEL

REPRESENTATION

OF THE BASIS STATES

X#JRMn

In order to compute the matrix elements (3) it is necessary to specify the XPJeMe

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P. B. TREACY

of eq. (4a). It is tempting to identify these R-matrix basis states for r, 5 Ui with shellmodel wave functions ’ “), As is well known ’ 6), the use of such a truncated set of states makes the usual R-matrix expansion incomplete. Certain remedies for this exist, such as the various modifications I’- lq ) of the R-matrix to allow for level-dependent ~~omogeneous”) boundary conditions, and some successful computations on these lines have been made ‘**21). However, looking forward to our application of sect. 3, for states which are well represented by pure shell-model configurations, these difficulties do not arise and conventional R-matrix theory can be used. The chosen boundary conditions must be compatible with those of R-matrix theory. These remarks are amplified in the appendix, together with a justification of the boundary condition chosen. Returning to the results of Lane r5) it can be shown that a shell-model representation for a state such as XPJflB is expansible quite generally in the form

wherej is the transferred angular momentum, the J and M are spin and components and the Tand M, are isospins and components. The /? refer to the expansion of er,, of con~guration v), into pairs, and are defined in ref. “1. We note that the expression C

@~.Jcjolc .

*.

l%z#A,Bc#t

represents all expansions of I,& other than that into (A+ t), the only part contributing to T,,[DW], eq. (3). The wave function of the transferred particle is written ~(~~j~ Mj MT*) = C NM, MlljMj)~(sM~lxr(P I&MI

(6)

in terms of the intrinsic and orbital functions $(sM,) and [cf. ref. ‘), eq. (21)] Xl@,)= i’I;M’(dt9 cbt)%(~t).

(7)

The radial function ur(rJ, which is used in computing radial integrals, must be constrained so that XIJeMB satisfies the R-matrix boundary condition [ref. 5, eq. (2-l), p, 2831

at each channel boundary ai of surface channel function r&. Substituting for X,,JeMB from eqs. (5)(7) in (8) it follows that (9)

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In the appendix, it is pointed out that the ur(r,) may readily be made to satisfy such a condition by choosing them as eigenfunctions of an infinite square well potential. The example of sect. 3 below illustrates a specific case for a final state comprising lp nucleons. 3. Cross section for 7Li(3He, d)‘Be at 17.64 and 18.15 MeV 3.1. REASONS

FOR CHOICE

OF THIS REACTION

The energy dependence of the cross section for a reaction of class I is very simple, and requires only the A-matrix (eq. (4.a)) using known or fitted level parameters (E,, y,,). A single-channel, many-level such case was computed in ref. ‘). For a process of type II to interfering states, it is necessary to consider final states which can decay back via nucleon channels to targets. Several appropriate nuclei exist in the lp shell”): for example the $- levels in ‘Li at 6.56 and 7.48 MeV, or their analogs in ‘Be, and the well-studied 2+ states in *Be at 16.63 and 16.92 MeV. In neither of these cases do both final states decay strongly via any channels to target states. The 1+ levels in ‘Be at 17.64 and 18.15 MeV both decay strongly to (‘Li + p), and have wellestablished shell-model assignments 23), behaving essentially as ‘Li plus a p, proton 24). Both levels thus decay to the same channel. For each level the existence of a large nucleon reduced width enhances the accuracy of a DWBA treatment. The reaction ‘Li(jHe, d)‘Be was chosen, as it is expected to be of stripping type and there are some accurate data available on the energy spectra of outgoing deuterons 25). In certain respects the (3He, d)p reaction is a simpler example than (d, p)n. The latter must always be in competition with (d, n)p, whereas for the former the competing process (3He, p)d may frequently be inhibited. A stripping calculation provides both the energy and angle dependence (eq. (2)), although in practice only the former has been measured over the region of interference. The general line shape, and in particular the relative peak heights, are of specific interest for any realistic theoretical fit. 3.2. METHOD

OF COMPUTATION

The theory of eqs. (2) and (3) was programmed for a Univac 1108 computer, with the zero-range, non-spin-orbit approximation 3, for stripping to the 17.64 and 18.15 MeV levels in Be. Optical potentials used for the entrance and exit channels at c.m. energies 25) 10.5 MeV, 3.84.5 MeV, are listed in table 1 [refs. ““*“‘)]. The program incorporated the Vincent-Fortune contour integration technique 4), using Simpson’s rule and fifth difference corrections 28). High accuracy of integration was essential, and the routine was checked out to accuracies of order lo-* on tables of the sine integral, which is similar to, but less involved than, a typical radial integral. Detailed checks were also made of the stability of solutions against variations in integration

P. B. TREACY

418

TABLE1 Optical potentials for 3He and d channels Channel

cm. energy (MeV)

3He + 7Li d+aBe

10.5 3.8-4.5

Y

rv

ay

W

rw

aw

140.0 105.5

4.026 3.260

0.72 0.79

7.5 16.2

4.429 4.776

1.12 0.484

The definitions U = - V[l +exp(u)]-‘-i4W exp(w)[l +exp(w)]-‘, are employed, with energies m MeV and radii in fm.

with v =

(r--J/a.,

and w =

(r-rw)/aw

step and maximum L,, values of 0.2 fm and 7h being found satisfactory. An important extra check consisted in confirming that the computed cross section was insensitive to a chosen real starting radius for the contour integral. It was found that, with the potentials of table 1 a radius of 30 fm was necessary for neglect of the nuclear force. However, at even this radius the channel Coulomb wave functions had not reached their sinusoidal asymptotic forms, and it was found expedient to compute Coulomb functions at complex radii up to (30225 i) fm, by analytic continuation and direct integration of the Schroedinger equation away from the real axis, for the Coulomb potential. In the asymptotic region the final-state wave function Jig of eq. (4b) was used directly in radial integrals. As discussed in sect. 3 and justified in the appendix, for radii less than the channel radius, lp shell-model eigenfunctions in a square-well potential were chosen. In effect, for each level only the p-wave radial dependence j,(Kr)

=

SF - cos (Kr)

(10)

was needed, with K constrained to fit the chosen boundary condition. The levels’ reduced widths [proportional to the 8 of eq. (S)] were obtained by fitting individual line-shape functions ‘) to the 17.64 and 18.15 MeV levels, assumed to have observed widths of22) 10.7 and 147 keV respectively. For the present computations, a channel radius of 4.2 fm was chosen. The choice of boundary conditions 2p) was made as discussed in the appendix; in the present case it was chosen to have zero shift function at the peak of the 17.64 MeV level (‘Li + p channel energy 0.4 MeV). With these, the redu~d-width amplitudes of the 17.64 and 18.15 MeV levels were found to be 0.540 and 0.826 MeV*, and the constant K = 0.758 fm-‘. 3.3. RESULTS AND FITS TO DATA

A deuteron spectrum 25) from 7Li(3He, d) at a beam energy of 15 MeV, at a laboratory angle of JI = lo”, is shown in fig. 1. A lit is shown by the curve of fig. 1, com-

419

DIRECT REACTIONS (II)

2000 -

15-00_ 5 = : g1000_ z 5

500 _ ‘4 0.3

a5 Channel

0.7 energy

0.5 (MeV)

1.1

0

20

C.m. angle

60

60

60

(degrees)

Fig. 1. Differential cross section for the 7Li(3He, d)*Be reaction to excitations in *Be from 17.5 to 18.3 MeV (‘Li+p channel energy 0.3 to 1.1 MeV). Points are data from ref. is) with a beam energy of 15 MeV observed at a deuteron laboratory angle of 10”. The full curve is a calculated fit as described in subsect. 3.3, with an assumed background given by the dashed curve. Fig. 2. Angular distributions corresponding to the calculated tit of fig. 1. The curves are as follows: full curve: channel energy 0.4 MeV (*Be excitation 17.64 MeV); dashed curve: 0.6 MeV (17.84 MeV); dotted curve: 0.9 MeV (18.15 MeV).

puted for a c.m. angle of W, which encompasses $ = (10.0 +O.l)’ over the energy range 17.64-18.15 MeV. A smooth background, evident in the data of fig. 1, was added to the calculated cross section, and the curve was normalised (to 2500 counts per channel) at the 17.64 MeV peak. The fit is satisfactory; there is little jnterference between the levels except within about 20 keV of the 17.64 MeV peak, which anyway is broadened in the data. The predicted heights, and characteristic asymmetry of the 18.15 MeV peak, are in good agreement with the data. Experimental data on angular distributions are lacking. In fig. 2 are shown calculated angular distributions at several energies in the region of interest. These vary little, in fact the curve of fig. 1 differs insignificantly from the shape of the integrated cross section. This confirms a similar effect remarked in the 160(d, p) reaction4). 4. Comparison of line shapes for different mechanisms The computations of fig. 1 contain both internal and external contributions, as the final *Be state decays to the target nucleus ‘Li. It is of interest to compare the relative

420

P. B. TREACY

contributions of the inner and outer wave functions (4a, b), as well as to examine in more detail the energy dependence of the different line shapes. 4.1. SINGLE-LEVEL

LINE SHAPES

Some purely theoretical ~lc~ations were made of line shapes predicted by the theory of sect. 2, and these were compared to the well-known R-matrix forms. First, it was assumed that the 17.64 MeV level alone exists, and its contributions in the inner and total (inner plus outer) regions compared. The results are shown in table 2. TABLE 2 Theoretical single-level line

Channel energy WeV)

(a) Ratio of inner : total contributions

shapes for ‘Lit3He, d)sBe

(b) Integrated cross section (arbitrary units)

(c) Ratio eq. (11) to col. (b)

0.375 0.385 0.395 0.400”)

0.0461 0.0444 0.0424 0.0413

3.453 10.38 53.07 100.0

1.127

0.405 0.415 0.450 0.900

0.0401 0.0371 0.0295 0.0029

55.96 13.85 2.225 0.904

0.968 0.906 0.714 0.082

1.043 1.028 1.000

(4 Ratio eq. (11’) to col. (b)

1.012 0.968 0.999 l.ooo

0.996 0.991 1.001 1.055

“) Indicates peak.

Column (a) shows that the cross section is dominated by contributions from the asymptotic region (by a factor of about 29, i.e. external contributions constitute over 95 % of the reaction. Under these conditions one might expected the energy dependence of the cross section to depart significantly from that of a single-channel, singlelevel R-matrix line shape

In column (b) of table 2 the computed (“exact”) cross section is listed (peak = 100) and in column (c) the ratio of the form (11) to the exact cross section is shown. Deviations of some 10% are seen. Such deviations as those mentioned above are to be expected, and they do invalidate the use of a simple line shape for reactions of class II. However, a very useful and accurate approximation 3o) to a true single~ha~el, single-level line shape may be ob-

DIRECT

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421

tained by using not a(E,)[I], eq. (ll), but

(11’) where B, is a complex constant and P, the c-channel penetration coefficient. A tit using (11’) is shown in table 2 column (d), and this reduces residual deviations, even in the wings some ten half-widths away, to less than 1 ‘A. The form (11’) is very satisfactory as an approximation for class II direct reactions. It has been used successfully in recent data in this laboratory 31) for the process 6Li(d, d’) leading to 6Li excited states which decay to (4He+d). 4.2. TWO-LEVEL

LINE

SHAPES

Reverting to the more realistic case where the 17.64 and 18.15 MeV levels are fed in proportion to their reduced widths, some computed cross sections of similar quantities are listed in table 3. Again, column (b) shows that the ratio of inner to outer contributions is about 1 : 25, while column (c) represents the relative contribution of an R-matrix many-level line. shape in the form given by Barker

4%)[Il = I1 G$$tJT%12, al.

(12)

TABLE 3 Theoretical Channel energy (MeV)

two-level

(a) Ratio of inner : total contributions

line shapes

(b) cross section (arbitrary units)

for 7Li(3He, d)8Be

(4

(4

Ratio

Ratio eq. (12’) to col. (b)

eq. (12) to col. (b)

0.375 0.385 0.395 0.400 “)

0.0437 0.0430 0.0419 0.0412

4.04 10.96 53.98 100.0

1.150 1.095 1.033 1.ooo

1.099 1.019 1.019

0.405 0.415 0.450 0.600 0.750 0.825 0.9OOa) 0.975 1.050 1.200

0.0406 0.0390 0.0325 0.0103 0.0145 0.0154 0.0146 0.0131 0.0114 0.0084

55.07 12.66 I .60 0.34 1.30 5.38 21.72 12.40 6.07 3.10

0.960 0.879 0.586 0.085 1.051 1.168 1.127 1.012 0.893 0.684

0.976 0.932 0.814 0.963 1.OOo 0.934 0.935 0.966 1.022 1.149

Column (b) corresponds “) Indicates peak.

to the tit of fig. 1.

1.000

422

P. B. TREACY

with real feeding amplitudes G$ (see appendix). Column (d) lists the form

which is the many-channel, many-level generalisation of (11’). (We consider here only a one-channel situation.) The fit of (12’) is seen in column (d) to reduce deviations to better than 10 % except in the valley between peaks, where the actual cross section decreases to less than 1% of that at the 17.64 MeV peak. In summary one may say that eq. (12’) accounts very well for a two-level line shape such as is encountered in 7Li(3He d). It has been used with success in other reactions involving 6Li levels populated in pairs by (p, p’) and (p, a) reactions 31).

5. Summary: discussion of methods of extracting spectroscopic data The theory of ref. ‘) implies that for our class II type reaction, where the final state decays to the target plus transferred particle, direct break-up outside the channel radius is an important part of the mechanism. Such a reaction is 7Li(3He, d)*Be(p), in which the use of conventional optical potentials and shell-model wave functions for the final *Be states, is well accounted for as is seen in fig. 1. A good approximation to the energy dependence of such a reaction may be obtained by using an R-matrix amplitude and a coherent, penetration-dependent amplitude in the channel of the transferred particle (target + t). This recipe, eq. (12’) has been shown to account well for single-level and two-level final states. For the extraction of spectroscopic information on unbound final states, it would therefore seem sufficient to use the generalised formula of Barker, eq. (KY),and fit the constant level parameters (Ea,--ylJ. However, caution is necessary; first, it is necessary to establish whether the approximation (12’) is appropriate - for a direct stripping reaction the present work suggests that it is, though some generalisation of that formula must sometimes be expected. One such example, mentioned by Barker, requires an incoherent sum over feeding amplitudes where the reaction proceeds to two interfering levels having different relative feeds for different channel spins. Another, leading to similar incoherence, might occur where the reaction proceeded via compound states. A second limitation on the method arises because of the essential arbitrariness of the channel radius. This can best be defined as that radius which optimises the chosen type of lit (number of levels and channels) for a given problem. Such a definition demands that one lit not only reaction data, such as 7Li(3He, d) deuteron spectra, but also all other available data on the levels involved - such as in our example ‘Li(p, p) scattering. An example that illustrates the improvement of definition of channel radius - and hence level parameters - is given in ref. 6, for the well-studied low levels in *Be. In the present work this aspect was not important because, as

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423

mentioned above, interference effects are small, and scattering data from ‘Li(p, p) [ref. ““)I can be fitted without critical dependence on the channel radius. In the present work, use is made of shell-model wave functions for unbound states, as is described and justified in the appendix. Such a procedure is particularly appropriate to a direct reaction of class II, where the process itself picks out the overlap between incident and final states, the coefficients of this overlap being the reducedwidth amplitudes. In the present case the use of infinite square-well shell-model states was used, purely for convenience in being able to pre-determine boundary conditions. (There would be no difficulty in similarly using superpositions of oscillator functions.) The employment of such simple shell-model functions seems natural for cases where pure configurations exist. For more complicated model states one of the new extended R-matrix formalisms would need to be used. It seems likely that progress with these methods should follow with a better understanding of the effects of approximations made in these theories, as applied to particular problems. This work has relied heavily on the continued personal advice and guidance of Dr. F. C. Barker, as well as on much of his published work.

Appendix In subsect. 2.2 it was asserted that, provided compatible boundary conditions are used, shell-model wave functions for a pure configuration can be used in conventional R-matrix theory. This is because “) in such a case a shell-model calculation can be done without specifying a boundary condition. The shell-model states can therefore be made to obey homogeneous boundary conditions as do R-matrix states, and are then orthogonal inside the channel radius at which the boundary condition is imposed. A set of R-matrix states may be made to satisfy any constant real boundary condition, as it has been shown 2g) that the actual form of a cross section is independent of that quantity. For model states, it is most desirable to choose a “natural” boundary condition “) at an energy in the region of interest, as this ensures that the wave function is continuous and smooth at a channel boundary at that energy. Such a choice was made for the work of sect. 3, and the R-matrix states made to conform to it. It was thus necessary to choose shell-model states uI satisfying eq. (9). For this work states of an infinite square-well potential, j,(Kr), were chosen, with the wave number K fixed to obey the “natural” boundary condition at the chosen energy. In the example of sect. 3 it is interesting to note that both interfering states have one and the same transferred angular momentum i, and the form of eq. (5) implies that their coefficients O,, jl are both real. This is why in the empirical forms (12), (12’) the relative feeding amplitudes may be chosen as real.

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References 1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11)

12)

13) 14) 15) 16) 17) 18) 19) 20) 21) 22) 23) 24) 25) 26) 27) 28) 29) 30) 31)

P. B. Treaty, Nucl. Phys. Al93 (1972) 97 F. C. Barker, Austral. J. Phys. 20 (1967) 341 G. R. Satchler, Nucl. Phys. 55 (1964) 1 C. M. Vincent and H. T. Fortune, Phys. Rev. C2 (1970) 782 A. M. Lane and R. G. Thomas, Rev. Mod. Phys. 30 (1958) 257 F. C. Barker, H. J. Hay and P. B. Treaty, Austral. J. Phys. 21 (1968) 239 F. C. Barker and P. B. Treaty, Nucl. Phys. 38 (1962) 33 H. J. Hay, E. F. Starr, D. J. Sullivan and P. B. Treaty. Austral. J. Phys. 20 (1967) 59 E. H. Berkowitz, G. L. Marolt, A. A. Rollefson and C. P. Browne, Phys. Rev. C4 (1971) 1564 T. Lorenz, Z. Naturf. 21a (1966) 1196 C. M. Vincent and H. T. Fortune, ref. “); R. Huby, Phys. Lett. 33B (1970) 323 ; G. Baur and D. Trautmann, Nucl. Phys. A211 (1973) 333; H. W. Barz, V. E. Bunakov and A. M. El-Narem, Nucl. Phys. A217 (1973) 141 J. Bang and J. Zimanyi, Nucl. Phys. Al39 (1969) 534; J. Zimanyi and J. P. Bondorf, Nucl. Phys. A146 (1970) 81; P. 0. Dzhamalov, E. I Dolinskii and A. K. Mukhamedzhanov, Sov. J. Nucl. Phys. 15 (1972) 147 R. Lipperheide, Phys. Lett. 32B (1970) 555 K. Mijhring and R. Lipperheide, Nucl. Phys. A211 (1973) 136 A. M. Lane, Rev. Mod. Phys. 32 (1960) 519 C. Mahaux and H. A. Weidenmtiller, Shell-model approach to nuclear reactions (North-Holland, Amsterdam, 1969) W. Tobocman and M. A. Nagarajan, Phys. Rev. 138 (1965) Bl351 L. Garside and W. Tobocman, Phys. Rev. 173 (1968) 1047 A. M. Lane and D. Robson, Phys. Rev. 185 (1969) 1403 J. E. Purcell, Phys. Rev. 185 (1969) 1279 A. A. Ayad and D. J. Rowe, Nucl. Phys. A218 (1974) 307 T. Lauritsen and F. Ajzenberg-Selove, Nucl. Phys. 78 (1966) 1 F. C. Barker, Nucl. Phys. 83 (1966) 418 S. Cohen and D. Kurath, Nucl. Phys. Al01 (1967) 1 C. J. Piluso, R. H. Spear, K. W. Carter, D. C. Kean and F. C. Barker, Austral. J. Phys. 24 (1971) 459 P. E. Hodgson, Ann. of Phys. 17 (1968) 563 P. E. Hodgson, Ann. of Phys. 15 (1966) 329 L. Fox and D. F. Mayers, Computing methods for scientists and engineers (Clarendon Press, Oxford, 1968) F. C. Barker, Austral. J. Phys. 25 (1972) 341 F. C. Barker, personal communication K. H. Bray, A. D. Frawley, T. R. Ophel and P. B. Treaty, to be published