Atomic, molecular and nuclear collisions

Nuclear Physics A353(1981)185c-198~. 0 North-Holland Publishing Co., Amsterdam Not to be reproduced by photoprint or microfilm without written permission from the publisher.

ATOMIC, MOLECULAR AND NUCLEAR COLLISIONS

R.C. JOHNSON

Department of Physics, University of Surrey Guildford, Surrey GU25XH, England

1. Introduction

An excellent review of some of the major interconnections between the fields

of atomic, molecular and nuclear collisions was given at the Graz Conference by E.F. Redishl). This reference is recommended as background reading for the following report.

An important region of overlap between theories of collisions involving a few degrees of freedom is in the use of approximate methods based on semi-classical ideas and on Glauber theory. These topics were not represented in papers submit- ted to this session. This probably reflects the fact that these methods and their expected range of validity are rather well understood. On the other hand calculations based on the distorted-waves method and closely related optical model ideas have a less firm foundation, although highly sophisticated numerical methods are available for the extremely complicated calculations which are often associated with this approach. I believe that one of the reasons that this conference is of interest to "traditional" nuclear, atomic, and molecular reaction theorists is the hope that n-body scattering theory methods could throw some light on the validity of their methods. So far this hope has not heen realized. The situation is, of course, quite different for systems which can be well approximated by a 3-body model. For these systems calculations based on the Faddeev approach have been very successful. Several examples illustrating these remarks will be found in this report.

One of the aims of the conference organizers in setting up this particular session was to bring together researchers in the three fields. In this they were

very successful and the discussion sessions proved to be stimulating and rewarding for those involved.

Not all authors of contributed papers were directly involved in the discussion session, but I have tried to interweave their contributions into the present report if relevant. The section titles reflect the broad categories into which the discussion was split by the moderator. The latter (E.F. Redish) is to be congratulated for the excellent job he did not only in providing a good framework for the discussion, but also for the stimulation he provided in the form of ideas and questions.

2. Methods for rearrangement collisions

2.1 Truncated channel coupling array theories and coupled channels calculations

A significant fraction of the first session was concerned with the question of the best way to calculate Z-cluster to Z-cluster rearrangement amplitudes. Colli- sions of this type in atomic, molecular and nuclear physics are often treated using the Coupled Reaction Channel (CRC) method. This method introduces into the Schr8dinger equation an approximate wavefunction composed of pieces having the form appropriate to selected open channels. In the Channel Coupling Array (Bound State Approximation) (CCA(BSA)) approach2) a similar truncation is introduced into the Faddeev-type components of the wavefunction. The resulting equations differ from

the CRC equations in that non-orthogonality terms present in the CRC equations are absent.

185~

186c R.C. JOHNSON

Fig. 1. Coordinates for 3-body system.

The differences between the two approaches can be understood within the framework of the simple 3-body system discussed by Austem3) and shown schematically in Fig. 1. In this model only particles 0 and 1 and 1 and 2 interact directly,through the potentials U, and V, respectively. The SchrGdinger equation to be solved is

(E - K - U - V)'i' = 0, (1)

where K is the total kinetic operator for the system. I shall take particle 0 to be infinitely massive in order to simplify the specification of coordinates,

In both the CRC and CCA(BSA) approaches the approximate wavefunction associated with B has the form

where @l and $2 are bound states in the potentials U and V and satisfy

(s 1 - Kr - U)Q, = 0,

(E 2 - Ko - V)$l, = 0.

The two methods eive different urescrintions for the functions f and e.

(2)

(34

(3b)

whose in the two channels. Fig. 1, the wavefunction Y is split

asymptotic forms give the scattering amplitudes In the CCA approach to the system shown in

into two components

Y = Yl + Y2,

where Yl and Y2 satisfy the coupled equations

Pl> = IOl'k> + GlUI'Y2>,

P2> = G2VIYl>.

In equations (5) the incident channel has been assumed 2.is incident on a bound state @l of particle 1 and 0. G2 are defined by

(4)

(5a)

(5b)

to be one in which particle The Green functions Gl and

ATOMIC, MOLECULAR AND NUCLEAR COLLISIONS 187~

Gl = (E + io - K - U)-1, (6a)

G2 = (E + io - K - V)-1. (6a)

Equations (5) incorporate correct boundary conditions and provide an exact

solution of the SchrGdinger equation (1). In my opinion the CRC and CCA(BSA) methods correspond to different ways of

approximating the complicated three-body inverse operators Gl and G2 of equations

(6). A natural approximation in some circumstances is to truncate the expansion

(E+iotK-U) = 90 E+io-c;-Ks

in terms of eigenstates, 19 “> 1'

of Kr + U, to a single term 495)

1 E+io-K-U E+io-El-K,

where 1$1> is one of the set cl@?}. An analogous approximation for G2 is

G2 " 1@2> <q

. E+io--E2-KR

(8)

(9)

These truncated expressions for Gl and G2 lead to the CCA(BSA) equations when they are inserted into eqs (5). In differential equation form they read

(E - Ed - Ks)Iyl> = U/y2>, (loa)

(E - ~~ - KR)IY2> = vI'yl>, (lob)

It also follows from (8), (9), and (5) that

PIIY1> = Iy%

P21Y2> = I’y2>’

where Pl and P2 are projectors corresponding to the two states $land $2. Eqs. (10) and (11) constitute the CCA (BSA) equations.

In the CRC approach a different approximation is made for the Green functions

Gl and G2?. In this case Gl and G2 are replaced by suitable inverses of the operators (E+io - K - U) and (E+io - K - V) calculated in the space spanned by the set

{I$1 s' rrar;-,;>!+2Q~ such that, for example, Gl]$> is a vector in this space for arbi- These inverses are very complicated to calculate directly, but the

approximate Gl must be defined so that it satisfies the two conditions

Q$,/(E - K - U)G = +$,1, (12a)

<@2i (E - K - U)C, = <@,I. (12b)

With a similar approximation for G2 it then follows that (5a) and (5b) are equiva- lent to the differential equations

'F-The following analysis expresses the view of the rapporteur and-did not arise in the course of the discussion.

188c R.C. JOHNSON

+lI(E - K - U - V)[Y> = 0, (13a)

<+21(E - K - U - V)IY> = 0, (13b)

with Y of the form of eq. (2). These are the CRC equations3). Thus the CRC and CCA(BSA) equations as discussed by Austem in reference 3 and

in the discussion session correspond to different ways of evaluating the Green functions Gl and G2. No generally useful way emerged from the discussion of decid- ing which approximation is superior. Levin2) has discussed individual cases where the two approximations have been compared with more exact calculations, but no useful general insight seems to have been produced. In many of the cases considered the coupling between the two channels involved turned out to be weak so that the calculations gave very similar results,

A general characteristic of the CCA(BSA) approach is that non-orthogonality terms do not appear. This has definite practical advantages. On the other hand, the CCA(BSA) equations, unlike the CRC equations, do not satisfy reciprocity in general, indicating that'the (CCA)BSA truncation introduces elements that violate basic physical reo,uirements. smal12).

In individual cases this violation appears to be

Perhaps the predilection of many for the CRC approach is best summed up by Austern's remark to the effect that experience with approximations in quantum mechanics suggests that increasing the space in which the operators of a system are allowed to act tends to lead to more accurate calculation.

2.2 Atom-diatom collisions

Some further details were presented by Kupperman of the techniques described in this review paper?) in the treatment of atom-diatom rearrangement and scattering. In the Born-Oppenheimer approximation the relative motion of the three nuclei involved in, for example,

H + H2 *H+H 2' (14a)

+H+H+H, (14b)

is described by a Schr'ddinger equation similar to equation (1) for the relative motion of the nuclei, but with U + Y replaced by a much more complicated for-ml/). The latter usually includes three-body terms associated with the dependence of the electron motion on nuclear separation. Examples of the form of the potential energy surfacer are given in ref. 6. In the Discussion Session the specific example of a collinear collision was mentioned. A choice of coordinates and typical forms for the potential energy surfacev in this case are shown in Figs. 2(a), (b) and (c).

The solid lines markedv = m in Figs. 2(b) and 2(c) correspond to coincidence of a pair of nuclei. In Fig. 2(a) the plateau in the upper right hand corner can be considered infinitely high for collisions well below the threshold for reaction (14b).

The method outlined by Kupperman is a CRC-type approach in which the basis set used is allowed to change as the integrations region in R, p space is altered. Thus, referring to Fig.2(b),a discrete set of cuts Rl, R2, . . . are defined between points where v = m. The potentials v(p', Ri) are then used to define a sequence of square integrable complete sets of eigenfunction $,(p', Ri) which depend para- metrically on the Ri* These sets are well adapted to the expansion of the three- body wavefunction in the region of Ri. The associated CRC-type equations are solved numerically in the region between Ri and Ri+l and matched across the bound- aries between this region and the next.

By using polar coordinates (E,$) in the (p', R') plane (see Fig. 2(c)) (or hyperspherical coordinates in the three dimensional casee)), and defining basis functions which are eigenfunctions of the potentialv(si,$) associated with a circular cut, the method is applicable to energies above dissociation thresholdb). In this case the basis functions.have the oscillatory form in the plateau region required to describe diatom break-up.

ATOMIC, MOLECULAR AND NUCLEAR COLLISIONS 189c

Y P’ p’. / , L I -

/

--__

(b)

R' Fig. 2. (a) Coordinates for collinear atom-diatom collision. (b) Potential energy surfacev for atom-diatom collision well below break-up threshold. The coordinates P' and R' are proportional to p and R and are defined in ref. 6. The dashed lines are equipotentials. (c) Potential energy surfacev showing cuts appropriate to

energies above break-up threshold.

Kupperman's discussion made an interesting contrast with the earlier discussion of nuclear CRC and CCA calculations in this session. The calculations described by Kupperman are clearly much more detailed than those typically carried out in nuclear physics. In the latter case a fixed basis set $1 , $2 , . . . is nor- mally employed. It would be extremely interesting to see how far the approach of ref. 6 could be adapted to cope with a nuclear three-body problem.

Alternative approaches to rearrangement in atom-diatom collisions were also discussed b review T

Micha. Many of the points he mentioned were covered in his invited talk ) and will therefore not be discussed here.

3. Applications of few-body methods in chemistry

Haftel and Lim discussed their work on the reaction

He + He2 + He + He + He, (15)

at milli-eV energies (< 10°K). They assumed the He2 system is bound and used a rank-one separable approximation to the He-He interaction (Mitra model). An exact numerical solution of the Faddeev equations was obtained and predictions were presented of reaction rates as a function of temperature. They found that in some cases Born approximation gave correct orders of magnitude; but in general they found that the multiple scattering series diverged and multiple scattering corrections were very important. They reported a predicted behavior of the break-up cross-section

19OC

near threshold of the form

RX. JOHNSON

For work at higher temperatures a rank-one representation of the (He-He) interaction is not satisfactory. A more general approach to the representation of this interaction below lOOoK and for momenta up to 5Ao-1 tributed paper by the same authors8).

Haftel and Lim reported that at 1°K they obtained a that obtained by Grehen, Thomas and Berlinsky9) in their bination process

is described in a Con-

similar reaction rate to calculation of the recom-

H + H + He * 112 f He (171

at leK, In this calculation the inverse break-up matrix element was evaluated in single scattering impulse approximation and time reversal invariance was used to predict the recombination rate for process (17). Greben reported a calculated result close to the measured rate, This calculation is of particular interest in connection with the stability of atomic hydrogen gas when He is used as a coolant. Recent experimental results have demonstrated the feasibility of stabilizing very high densities of atomic H gas at low temperatures and in high magnetic fields").

4. Scattering and reaction experiments in atomic and nuclear physics

4.1 Electron-hydrogen scattering

Weigold discussed recent experimental work on electron-atomic hydrogen scattering at 54.4-eV electron e*ergyll*12), exciting the 2s and 2p levels:

e f H(ls) -t Hfn = 2,L,M) + e'. (18)

A feature of the experiment was the separation of the cross section for 2p and 2s excitation by the detection of scattered electrons in coincidence with Lyman a photons from the decay of the 2p levels.

Analysis of the electron-photon angular correlation yields measured values of the quantities

A = 0(10)/0(2p~, (19)

and *

R = Re 'fllf20', o(2p)

1201

where ELM is the amplitude for exciting the L,M state, o(10) is the cross section for exciting the L=l, M=O state, and the angular brackets denote an average and sum over unobserved spin substates.

The cross-section ratio U(2s)/5(2p) was also measured 12 ). Coupled channels

calculations for low partial waves plus a unitarized Born approximation for the higher partial waves give a good account of this observable but at angles larger than 20° the measured values of Rand A cannot be explained by currently available theories. A full accountof these experiments and a detailed comparison with theoretical calculations can be found in refs.lland12 and will not be reproduced here.

It was noted in discussion that at the energy of this experiment the ionization cross section is quite large. It is not clear how far current theories include effects of these channels in an adequate fashion.

4.2 Proton-nucleus scattering

What happens when a 100 MeV proLUll nits a nucleus? Does it interact with the


nucleus as a whole and excite some sort of collective excitation, form a compound nucleus, or bounce off a single target nucleon? A contribution from the Maryland group13) addressed itself to this question and was presented by Holmgren. The Maryland experiment and analysis strongly suggests that a number of features of the inclusive proton spectra, the inclusive neutron spectra and the coincident 2p inclusive spectra &an be understood in terms of the expected results of a collision of the incident proton with a single nucleon in the target. It has been known for some timel4) that the quasi-free peak seen at small angles can be understood in these terms, but Holmgren suggested that even in regions of energy and angle (beyond the evaporation peak) where no quasi-free peak is evident, this mechanism could be playing a dominant role.

10

Nuc;Lon energy (MeV) II 100

Fig. 3. Proton and neutron inclusive spectra produced by 90 MeV protons on 27A'L13). The solid line is a plane wave impulse ap roximation

by Wu P3 , quasi-free scattering estimate

).

Figure 3 is taken from ref. 13 and shows the yield of protons and neutrons produced by 90 MeV protons on 27A!L at 30" and 90". The ratio of proton yield to neutron yield above the evaporation peak has the value 2.5, independent of angle. This is close to the ratio

ZU f NU pn

NU pn

(21)

expected if collisions with a single proton or neutron with cross section u is the dominant mechanism.

"VnEW.

More detailed calculations b PP and

Similar results are obtained for all targets between J,W$~J~~;CX: this

Additional evidence is provided by (p, p'p) coincidence spectra. The spectra deduced by fixing the detector for one proton at 15" and considering coincident events corresponding to an energy loss from the incident proton to the nucleus of 60 MeV are shown as the experimental points in Fig. 4, for the case of a "Ni target.

The figure shows a remarkable correspondence between the distribution in energy and angle of coincident protons and the associated cross section for the 62 MeV (p, p') inclusive reaction on the neighbouring nucleus 54Fe15).

These results reinforce the conclusion expressed by Redish in his Gras reportl): the dominant process in proton-nucleus scattering at these energies involves a three-body system (two nucleons plus a core). The implications of this for theoretical work are discussed in ref. 16

An important ingredient of theories of nuclear reactions involving protons is an adequate proton optical potential. Most analyses of proton elastic scattering

192c R.C. JOHNSON

data assume that the optical potential is local or nearly so, In a Contribution to this Conferencel'), Okumusoglu et al. produce evidence from 32.5 MeV proton scattering measurements on "Ar that this assumption may not be valid. Following earlier work by Kobos and Mackintoshl'), Okumusoglu et al, suggest that anomalies in the large angle scattering might be best explained in terms of an orbital-angular- momentum dependence in the optical potential in addition to that given by the usual spin-orbit potential.

4.3 Inclusive spectra from o-o scattering and other reactions involving light nuclei

A set of inclusive spectra of protons and deuterons produced in a-cx scattering with incident CL ener ies in the range 110 to 172 MeV were discussed by Paic and form a Contributed paperl$. The results were compared with two mechanisms:

1) A Serber-type break-up model of the kind found successful for describing a-particle break-up on heavy targetszO), 2) A statistical partition modell'f).

Perhaps because of the lack of coincidence measuremehts,it was difficult to come to any definite conclusion about the relevant importance of the two mechanisms. According to Paic, mechanism 2) alone does not give an adequate account of the deuteron and proton angular distributions.

Evidence for the observation of the proximity reaction n + 3He + 'H + p fed by the reaction gBe(3He, Hilber, et al.*l).

3He n)'Be at 7.1 MeV is presented in a contributed paper by

The excitation function for the reaction 6Li(~, 2cr)d for a energies between 6,6 and 13.2 MeV has been measugrzd in a kinematically complete experiment and is reported in a Contributed paper ). The two a-particle detectors were placed so that the condition for zero spectator momentum was satisfied. A plane-wave impulse approximation formula modified to include absorption and Coulomb effects was able to give a good account of the measured energy dependence of the cross section, except at the lowest end of the energy range.

4.4 Heuteron Scattering and break-up reactions

h number of kinematically complete c1 + c1 + c1+ n + p break-up reactions were contributed to Session II and reported on by Nefkens 23). was discussed in Session VI by Wickz4).

One paper in this class In this experiment with 28.3 MeV incident

cl-particles, p articular emphasis was placed on the formation of the ground state resonance 'Li(3/2 -). Wick presented Faddeev calculations by Koike and resonating group calculations (RGM) by Schutte, et al.25).

Fig. 5 compares both types of calculation with data in the neighbourhood of the resonance maximum corresponding to well-defined intervals in the relative energy, E in the o-p system. The 5Li ground state corresponds to E, tl%'resonance maximum both theories reproduce the data quite we I:

= 1.97 MeV. Near 1, but significant

discrepancies are found for E,,, values far from the resonance. It should be noted that the RGM calculations contain an overall normalization factor (which has been fixed at the resonance eak) because of the incomplete way the decay of the 5Li ground state is treated $4). The Faddeev calculations contain no such normalization and give the cross section at the resonance maximum very well. The RGM calculations include Coulomb and antisymmetrization,1s well as channels corresponding to break-up of the o-particle, in a rather complete fashion, whereas the Faddeev calculations do not.

Because of the kinematically complete nature of their experiment this group was able to present their results in terms pf the polarization state in which the 'Ll(3/2 -) ground state was produced. Their results for the second rank polarization tensors t20, t21, and t22 and the differential cross section, too, as a function

of the 'pi production angle 6$ are shown in Figa 6. The theoretical curves are

RGM calculations from ref. 25. It would be extremely interesting to See how far these observables could be accounted for in a Faddeev approach.


(AWI * JWW) NOl.kXK?- SSOXI

194c R.C. JOHNSON

l-kxkenbroich et al.

w LOO 800 1200 160°

t 0 * 3 8 s I I I I LI 00 LOO 80' 1200 160'

Fig. 6. Polarization tensors for 5Li(3/2 -) produced by 28.3 MeV incident a-particle cle24) as a function of 5Li production angle eLic

An experimental study of 0x3~ en induced deuteron break-up with 42 HeV I60 is described In a Contributed Paper ). The dominant features of the spectrum of out- going coincident I60 and protons are peaks associated with excited states in 170. Presumably these states are the same as those excited strongly by 5 MeV deuterons in '60(d,p)'70* leading to unbound states in 170. Methods of analyzing stripping to unbound states are discussed in the contributed paper by S.N. Mukherjee and Rama Dasz7).

AII interesting set of measurements linking deuteron elastic scatterpgg from o-particles and from a range of heavier targets was discussed by Clement 1.

Just as (d,p) tensor analyzing power measurements are known to be sensitive to the deformation of the deuteron, a similar sensitivity is expected in deuteron elastic scattering tensor analyzing powers. This effect can be interpreted as a term in the deuteron optical potential of the form

(22)

where R is the deuteron center-of-mass coordinate and 5 is the deuteron spin operator. %he Watanabe folding model gives the approximate resultzg)

a 13 U&R) = Qd R z ii x u,(R), (23)

ATOMIC, MOtECUtAR AND NUCLEAR COl.L.iSlONS 195c

where Qd is the deuteron quadrupole moment and U,(R) is the central part of the deuteron optical potential in the folding model.

Eq. (23) predicts both a real and an imaginary part for UT(R). In agreement with an earlier observation 30) by Goddard and Haeberli at lower energies the elastic tensar analyzing power measurements of Clement, et al. for 20 and 22 MeV deuterons are best fitted by setting the real part of the predicted UT to be very small, but retaining the imaginary part. This is illustrated for 22 MeV deuterons on '08Pb in Fig. 7.

Fig. 7. Tensor analyzing power component A,, measured in 22 MeV elastic deuteron scattering from 20*Pb2g). The dotted curve is obtained using both the real and imaginary part of tne folded tensor potential UT of eq. (23). The

dashed curve uses the imaginary part of UT~ only -$.

J

Fig. 8. Tensor potential +R for 22 EIev elastic deuieron scattering from cy- particles. The dotted curve is the prediction of the folding model while the solid and dot-dashed curves are obtained from fits to tensor analyzing power&). The dashed curve is obtaZned by fitting wLth an alternative radial dependence31),

and gives an Inferior fit.

196c R.C. JOHNSON

In contrast, earlier measurements31) of the tensor analyzing powers in d + 4He elastic scattering at 20 MeV indicated that a real TR potential much stronger and less diffuse than the predictions of the Watanable folding model were required by the data (see Fig. 8).

The data reveal a strong target mass dependence for the real part of UTR which is hard to understand on the basis of the folding model. It was suggested by Ioannides and Johnson32) that the reduction of the deuteron D-state produced by Pauli principle effects on a deuteron travelling through nuclear matter would reduce DTR from its folding model predictions, but quantitative estimates yield too small an effect at the energies of interest33). Rawitscher and Mukherjee34) suggest that effects of break-up of the deuteron by the nuclear surface will in- crease the tensor potential. However, it should be noted that the analysis of Clement, et sl28,31) does not include (~&2 or (a*s)2 terms in the deuteron -- optical potential, where 2 and & are the deuteron center-of-mass momentum and angular momentum operators. A strong term of the (~2)' type is predicted to be the result of Pauli principle 32935) effects on deuteron propagation in the nucleus.

It would be very interesting to see how far the Faddeev equation approach (which would have to include non-central forces) could account for the 4He data. This might help to throw some light on the difficulties experienced with the folding model for the heaviex targets.

5. Acknowledgements

I should like to thank all those authors who provided me with figures and other material associated with their contributions and who patiently answered my questions about their work.

References

1.

2.

3. 4. 5. 6.

7.

8. 9. 10. 11. 12. 13,

14.

15. 16. 17.

:;:

E.F. Redish in Proceedings of the VIII International Conference on Few-Body Systems and Nuclear Forces, Ed. by H. Zingl, M. Haftel and H. Zankel (Springer-Verlag, Berlin, 1978), p. 427. F.S. Levin, "Some recent developments in n-particle scattering theory," Review paper, this volume. N. Austern, Physics Letters 90B (1980) 33. J-M. Greben and F.S. Levin, whys. Lett. 71B (1977) 252. J.M. Greben and F.S. Levin, Phys. Rev. C?%-(1979) 437. A. Kupperman, "Energy transfer and rearrangement processes in few body molecular collision: theoretical," Review paper, this volume. D. Micha, "Few-body processes in atom-diatom collisions," Review paper, this volume. M.I. Haftel and T.K. Lim, Contributed paper VI-12. J.M. Greben and A.W. Thomas, Contributed paper VI-lo. 1-F. Silvera and J.T.M. Walraven, Phys. Rev. Lett. 44 (1980) 164. E. Weigold, L. Frost and K.J. Nygaard, Phys. Rev. A21 (1980) 1950. L. Frost and E. Weigold, Phys. Rev. Lett. 45 (198On47. H.D. Holmgren, C.C. Chang, A.A. Cowley, J.?ilk, and J.R. Wu, Contributed paper VI-14. P.G. Roos and N.S. Wall, Phys. Rev. 140B (1965) 1237; N.S. Wall and P.G. Roos, ibid 150 (1966) 811. F.E. Bertrand and R.W. Peele, Phys. Rev. C8 (1973) 1045. P.C. Tandy, E-F. Redish and D-B. Boll&, Phys. Rev. c16 (1977) 1924. N.T. Okumusoglu, J. Birchall, N. Videla and J.S.C. McKee.Contributed paper VI-20. A.M. Kobos and R.S. Mackintosh, Ann. Phys. 123 (1979) 296. G. Paic, B. Antolkovic, A. Djaloeis, .I. Bojzld, and C. Mayer-B&icke, Contributed paper VI-21.


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J.R. Wu, C.C. Chang, and H.D. Holmgren, Phys. Rev. Lett. 40 (1978) 1013. Ch.Hilber, D. Kamke, and J. Krug, Contributed paper VI-13: D. Gola, H. Eichner, C. Heinrich, H.J. Helten and H. Paetz gen Schieck, Con- tributed paper VI-8. B. Nefkens, "Non-mesonic few-nucleon phenomena," Rapporteur report, this volume. U. Berghaus, 0. Bohn, H. Br&kmann, P. Lara, K. Wick, A. Schitte and Y. Koibe, Contributed paper VI-3. W. Schiitte, H.H. Hackenbroich, H. StGwe, P. Heiss and H. Aulenkamp, Phys. Lett. 65B (1976) 214. G. Flodkvist, A. Johansson, L. Glantz, I. Koersner and B. Sundqvist, Contri- buted paper VI-7. S.N. Mukherjee and Rama Das, Contributed paper VI-17. H. Clement, R, Frick, G. Graw, P. Schiemenz, N. Seichert and T.-H. Sun, Con- tributed paper VI-4. P.W. Keaton and D.D. Armstrong, Phys. Rev. C8 (1973) 1692. R.P. Goddard and W. Haeberli, Nucl. Phys. A317 (1979) 116. R. Frick, H. Clement, G. Graw, P. Schiemenz,xd N. Seichert, Phys. Rev. Lett. 44 (1980) 14. KA. Ioannides and R.C. Johnson, Phys. Rev. Cl7 (1978) 1331. R.C. Johnson in Proceedings of the Int. SympT on Nuclear Direct Reaction Mechanisms (Ed. M. Tanifuji and K. Yazaki, INS Tokyo 1979) p. 200. G.H. Rawitscher and S.N. Mukherjee, Ann. of Phys. 123 (1979) 330. J.A. Tostevin, R.C. Johnson and M.H. Lopes (In prezation).

Atomic, molecular and nuclear collisions

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