CONCLUDING REMARKS

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CONCLUDING REMARKS

C. Bloch

To cite this version:

C. Bloch. CONCLUDING REMARKS. Journal de Physique Colloques, 1971, 32 (C6), pp.C6-299-C6- 303. �10.1051/jphyscol:1971671�. �jpa-00214889�

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JOURNAL DE PHYSIQUE Colloque C6, supplkment au no 11-12, Tome 32, Novembre-Dicembre 1971, page C6-299

CONCLUDING REMARKS

@. BLOCH

Direction de la Physique, CEN Saclay, France

Heavy ion reactions are an old subject. It is also an extremely rich one : the selection of two nuclei from the periodic table as target and bombarding particle may be done in a very large number of ways (there are, of course, some experimental limitations). With a sufficiently high bombarding energy, the reaction will proceed through very many channels. Altogether this produces an overwhelming amount of experimental data ... which has precisely been for a long time the difficulty of the field.

It is only with some theoretical model in mind (even a crude one) that it becomes possible to << see ⁾⁾some- thing in the mass of experimental curves. A few years ago the idea of the quasi-molecular states gave rise to very interesting interpretations of some experiments.

More recently the suggestion made by Dr. V. Gillet to look for quartet states, followed by the discovery by Mrs. H. Faraggi, Dr. M. C. Mermaz and their collabo- rators of peaks a t the predicted place and with the expected qualitative behavior has been extremely stimulating. This gave a new meaning to a-transfer reactions, which had been observed for years in many laboratories. The comparison of the most recent results obtained with quartets in mind has been the main theme of this meeting, judging from the intensity of the discussions on this topic. I shall therefore, for lack of time, concentrate my remarks on this subject, and apologize for omitting the many other interesting results which have been reported.

I would like first to review some of the most striking experimental data with their << naive ^)> interpretation in terms of quartets, forgetting for the time being all criticism (I shall return to this later).

The most spoken about reaction w a ~ ' ~ C ( ~ ~ 0 , a)24Mg (see in particular the communications by Drs. D. A.

Bromley, R. Middleton, M. J. Levine, J. Gastebois and R. H. Siemssen).

The first striking feature occurring in this reaction is the very selective population of a few states in 24Mg (at 14.14, 16.30, 16.56, 16.84 MeV) when the bombarding energy is sufficiently high, of the order of 30 52 MeV (Yale, Penn. University, Saclay). These states are singled out among many others which form a large fluctuating background but are individually much more weakly excited. The analysis of these experiments suggests that the strong observed states

have an a-cluster structure. For instance, it has been shown that if one tries to produce the same levels in 24Mg by the reaction 1 4 N ( 1 4 ~ , ~ 1 ) ~ ~ M g the cross section is remarkably smooth with no selective excitation at all. Since the initial nuclei 14N certainly have much less of an a-cluster structure than 12C and I6C, one understands immediately that this reaction should produce much less selective excitation of the levels of 24Mg having an a-cluster structure.

The second feature of the reaction is the rapid varia- tion of the cross section for the excitation of each one of the highly populated levels of 24Mg, as a function of bombarding energy. However, there seems to be a good indication of intermediate structure (see the communications of Drs. D. A. Bromley and J. Gastebois) when the excitation function is smoothed over a width of the order of 1 MeV. This may be interpreted as showing that the reaction proceeds through some states in the compound nucleus 2sSi, which are again preferentially selected because of their a-cluster structure.

It must be mentioned that this is somewhat in disagreement with the results of Dr. R. H. Siemssen obtained at about the same energy, which he interprets as evidence of a purely compound mechanism rather than a direct interaction. The disagreement may lie in the fluctuation analysis, which is always a rather delicate matter.

In the region of heavier nuclei, involving f-p shell target nuclei, the (160, 12c) reaction is also very successful in populating selectively a small number of states in the residual nucleus. Systematic accounts of the present experimental situation were given by Drs. H. Faraggi, G. Morrison and F. Pougheon.

The general behaviour of the spectrum of emitted particles is the same as in light nuclei. Some very interesting effects have been reported. For instance, Mrs. H. Faraggi compared the spectra obtained in the reactions (160, 12C) on 54Fe, 56Fe and 58Ni, produc- ing 58Ni, 60Ni and 62Zn. The shell structure of the target nuclei is described on figure 1.

In 56Fe with 2 proton holes in the f712 shell, a quartet introduced into the 2 pSl2 shell produces a state which has necessarily some excitation energy due to the 2 proton holes. The very low lying states should therefore not be populated. The same holds for 54Fe, but the

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971671

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BLOCH

presence of 2 neutrons in the 2 p,,, shell interfers with the incoming quartet : this should decrease the probabi- lity of excitation of the corresponding states. In 5 8 ~ i , on the contrary the injection of quartet may produce the ground state. The excitation spectrum should therefore be shifted toward the ground state by several MeV.

All these predictions correspond exactly to the experimental data.

Another very interesting observation reported by Drs. H. Faraggi and G. Morrison is the (( neutron blocking effect )) occurring in the Ca isotopes. Very clear selective excitations appear in the reaction 40Ca(1 60, 2C)44Ti. The corresponding peaks are gradually washed out when 40Ca is replaced by the isotopes 42, 44, 46, 48, as the additional neutrons interfere more and more with the incoming quartet.

Now comes the question : do we believe these interpretations as evidence for quartet structure ? This gave rise to very animated discussions. I think that the situation may be summarized as follows : there is a majority who believes in quartets, and a minority who pretends they do not.

The minority leader Dr. G. Morrison warned us, and he is absolutely right, that any interpretation of experimental data in terms of nuclear structure is doubtful as long as the reaction mechanism is not quantitatively understood. In particular Dr. G. Mor- rison himself as well as Drs. V. I. Manko and W. von Oertzen showed (theoretically and experimentally) that purely external ⁾⁾factors (i. e. connected with the reaction mechanism but not with the intrinsic wave functions of the target or the residual nuclei) such as the Q-dependence or momentum mismatch, work in the same direction as cluster structure by favoring very much some levels.

This brings me therefore to the second point of the discussion : the reaction mechanism, the precise description of which becomes an essential problem.

Unfortunately the situation here is rather bad.

The best method known so far is the DWBA and we have learnt in the past few years that it is quite reliable an approximation for reactions involving only a nucleon as bombarding or emitted particle. The method has been extended to transfer reactions involving heavy ions by many authors (see communications by Drs.

W. von Oertzen, R. de Vries and Z. Fraenkel). In view of the large number of partial waves, DWBA calculations may become very tedious and a further simplifi- cation has been introduced : the semi-classical approximation (used by Dr. M. C , Lemaire and qualitatively

by Dr. G. Morrison). Dr. A. Winther gave us yesterday a beautiful account of the systematic investigation of this method which he has under way.

Unfortunately, even in its full glory (with finite range interaction, etc ...) the DWBA is a particularly crude approximation when applied to heavy ions. Here are some of the specific complications which it omits and which should be included before any quantitative theory can be set up :

a) Several types of transfer may yield the same final reaction products as shown for instance on the diagrams of figure 2 :

M'S ROTTER 0

The corresponding amplitudes, of course, add coherently, and therefore even small admixtures may produce large effects. The order of magnitudes of the amplitudes are sometimes unexpected. Thus Dr.

H. H. Duhm finds that his second diagram gives an amplitude not very different in magnitude from his first one, although it implies the breaking up of an a-particle.

b) The reaction may be a multistep process in which the transferred particle is broken into several pieces transferred separately, as shown for instance by the following diagrams presented by Dr. A. A. Ogloblin :

c) The transferred particle may be in an excited state and the target or the bombarding nuclei may go through excited states. Such processes are described by higher order DWBA which are clearly very hard to evaluate.

d) Compound nucleus formation may give a non negligible contribution.

All this shows the necessity of a deeper theoretical investigation of the reaction mechanism. It also shows that it is very important, experimentally, to perform the transfer of a given particle by means of dEfSerent bombarding particles. For instance, it is very interesting to compare a-transfer on a given target produced by the reactions ('jLi, d), (160, I2C), etc ... . The first reaction mentioned seems to produce much less selective excitation of quartet states than the second one, for a reason which is not yet understood (perhaps the

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CONCLUDING REMARKS C6-301

deuteron has a very hard time to stay bound once the a-particle is ripped off the 6Li nucleus...).

Let me now come to nuclear models. First of all, I would like to make a very trivial remark, which as shown several times in the discussion seems to have been forgotten by several people : the same nuclear wave function may be represented in many different ways. This is due to the antisymmetrization, or if one prefers to the exclusion principle.

Consider for instance an A independent particle wave function. It is constructed from A independent single particle wave functions q i ( r ) (i = 1, 2, ..., A) by forming the Slater determinant

Y(1, 2, ..., A ) ^Edet ( cp,(k)), (i, k = I , ..., A ) .

If we replace now in Y the q i by A independent linear combinations of them

--

Pi = C aij ^{q j}, (det a i j + ⁰⁾,

J

we obtain clearly the same A-particle wave function (within the irrelevant numerical factor det a i j ) .

This obvious remark should be kept in mind when comparing nuclear models. Take for instance 160 in the shell model with L-S coupling. The 16 basic single particle wave functions are made from the 1 s and l p wave functions (r2 = x2 + y2 + ^{z 2 )}:

combined with the 4 spin-isospin wave functions.

Consider, on the other hand, the a-particle model, in which the basic orbitals are four 1s wave functions centered at 4 points d , (i = 1 , 2, 3, 4). If the distances of these points from the origin are small compared with o ( 1 d, 1 < o), we may approximate the basic orbitals by

These 4 functions are linear combinations of the shell model wave functions (I), and the normalized Slater determinant constructed with them for 160 will be identically the same as with the wave functions (1). It does not make sense therefore to ask whether an experi- ment could decide between shell model and an a- particle model when the a-particles overlap strongly ( 1 d, 1 < ^o).Of course, when 1 d , I 2 o, the expansion (2) should be continued, and this gives rise to admixture of higher shell model configurations, which then diffe- rentiate the cluster model from the shell model.

This remark shows that hand-waving arguments about wave functions may be misleading. Thus, one might think that localizing the 16 nucleons around 4 centers gives rise to a nucleus with a tetrahedral shape,

whereas the orbitals (1) give a spherically symmetrical nucleus.

The a-particle model is an extremely old one, it has never been fully successful in any sense, but nevertheless the interest in it never faded away. This is because it incorporates most directly the four-nucleon cluster- ing tendency which must certainly exist in nuclear matter as a consequence of the nature of nuclear forces : short range attraction, with saturation charac- ter. On the other hand, detailed calculations with this model are never fully convincing. The situation was very well summarized (in 1936 !) by Bethe and Bacher (Rev. Mod. Phys. 8, p. 171) : ^(<... we must say that it can at present not be decided how much truth is in the assumption of a-particles as nuclear subunits. Certainly this assumption must not be taken literally, and the a-particles undergo considerable deformations (polari- zations) in the nucleus. On the other hand, the approximation assuming the elementary particles to move independently (Hartree approximation) is certainly not correct either and must be supplemented by introducing correlations between the particles. Such correlations would lead at least in the direction towards the a-particle approximation. The truth will therefore lie between the two extremes, as Heisenberg has pointed out. However, it seems to us that at present the Hartree approximation offers more prospects for being perfected ^)).

I did some calculations myself on the cluster model in 1952 and was not very happy with the results. I evaluated the ground state energy of, for instance,

160 by the variational principle, using an antisymme- trized 16 nucleon wave function built from 4 gaussian s-wave orbitals centered at the vertices of a regular tetrahedron. The forces were a mixture of Wigner and Majorana interactions with gaussian radial dependence.

The calculation is then elementary and numerical results could be obtained with a desk computer. The two parameters are the sizes of s-wave orbitals (or a-particles) and of the tetrahedron. They turn out to be very sensitive to the force. For a narrow range of Wigner-Majorana mixture (corresponding to saturation) a shallow minimum of the energy was found for a finite value of the tetrahedron size. With a little more Wigner force, collapse would occur leading, according to the above trivial remark, to the shell model. With a little less Wigner force the a-particles would not stick together. There was actually an even more disturbing feature. The very old model of Dennisson for ¹⁶⁰based on a rotating and vibrating tetrahedron of four point a-particles gives a very good description of the first few levels in terms of only two parameters : the size of the tetrahedron and the elastic constant of the bounds between the a-particles. However if one evaluates the zero-point vibration amplitude, it turns out to be of the same order of magnitude as the size of the tetrahedron.

Such an extremely cr soft ⁾⁾tetrahedron is particularly disturbing since Dennisson's calculations are based on the linear approximation assuming small deformations.

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C6-302 C. BLOCH These objections show that the model should not be

taken too literally, but they are not decisive in the sense that the a-particle model may nevertheless be a way of constructing approximate wave functions to be used in a variational principle. There has actually been a considerable revival of the a-particle model in the past few years after the work of Wildermuth, Brink and many others. We heard reports on this subject from Drs. G. Ripka and Y. Abgrall. The fact that Hartree- Fock calculations allowing deformed orbitals tend to produce for 20Ne and 28Si nucleon density distribu- tions which simulate those resulting from the a-particle model is very interesting since the a-particle structure is in no way << forced ⁾⁾into the model. Another interesting point brought up by Dr. Y. Abgrall is the existence of stable asymmetric shapes. Altogether, the a-particle model may turn out to be very useful for suggesting the existence of special types of simple excitations, which should perhaps then be further investigated by other methods based, for instance, on the shell model with configuration mixing.

The shell model calculations correspond to a very different philosophy. No attempt is made to guess beforehand the nuclear wave functions. The assump- tions are introduced mainly in the selection of the shell model configurations retained in the calculation ... and one tries to take into account as many as possible with available computers. Miss M. Soyeur showed us that very satisfactory results could be obtained in the case of 28Si by diagonalizing the Hamiltonian in a space of 839 configurations. This may seem to be an enormous number, but actually it should be considered as relatively small, in view of how rapidly the total number of shell model configurations up to a given energy increases with energy. Nevertheless the wave functions coming out of such calculations require an interpretation. For instance, Miss M. Soyeur finds that usually only a small fraction of the configurations introduced in the calculation occur with large amplitudes. It is then very interesting t o try to understand why this happens, so that one might learn how to construct good wave functions without actually carrying out the diagonalization program. Quite generally, the <( large ^)>shell model diagonalizations are very useful as providing a solid basis for testing the wave functions derived from simple models.

An interesting idea for simplifying the shell model diagonalization problem was presented by Dr. S. K.

M. Wong. It consists in splitting the nucleons outside the inert core into several groups. Then one carries out the diagonalization of the Hamiltonian for each group separately. Wave functions for the whole nucleus are formed by coupling eigenfunctions for the separate nuclear groups. Finally one has to diagonalize the Hamiltonian in the resulting space which has a very small dimension (of the order of 10 or less). Compari- son of this simplified procedure with the complete diagonalization is extremely encouraging.

For instance, in the case of 24Mg consisting of

8 nucleons outside an unperturbed ^{1 6 0}core one diagonalizes first the Hamiltonian of 4 nucleons plus the ^{1 6 0}core (corresponding to the nucleus "Ne). The eigenfunctions are then coupled two by two to form a basis for 24Mg, within which the Hamiltonian is finally diagonalized.

The ideal would be to find that the final diagonalization turns out to be trivial (with almost vanishing non- diagonal elements) corresponding to non-interacting substructures, as in the quartet model, to which I must come now.

The quartet model as explained by Dr. V. Gillet is based on two lines of argument :

a) the analysis of empirical binding energies shows clearly a periodicity which has a very natural interpretation in terms of strongly bound substructures of four or two nucleons, which interact weakly with each other.

b) quartets appear when one tries to construct wave functions which couple the nucleons within a shell in such a way that they interact as strongly as possible.

Of course, the existence of a periodicity of four in the binding energies has been known since the early days of nuclear physics. The new idea here is that the corresponding substructures subsist in the excited states, actually up to quite high excitation energies.

We have heard a lot of discussion on the following basic question : are the quartet model and the cluster model equivalent or not ? I do not think that the issue is settled by now. I would like the answer to the question to be yes, so that the quartet model might be described as an improvement of the shell model allowing groups of four nucleons to form clusters, as much as is possible within a restricted shell model configuration space.

For light nuclei (s-d shell), where j-j coupling does not play any role, Dr. A. Arima showed us in his beautiful talk that there is indeed equivalence between the quartet and the cluster models. First of all, up to 'ONe the supermultiplet approximation holds very well, and for instance the ground state wave function is dominated by the components with highest spatial symmetry (for the Rosenfeld, and Kuo-Brown forces).

Furthermore, as regards to the space part of the wave function, the shell model, the quartet model and the cluster model of Wildermuth turn out to give practically identical wave functions.

For heavier nuclei, when the spin-orbit interaction becomes important, there cannot be such a strict equivalence between the quartet model and the a- particle cluster model (which implies L-S coupling).

Nevertheless I think that it should be possible to show that quartet wave functions describe groups of nucleons which have a tendency to cluster, or in other words have strong spatial correlations.

An important question concerning quartets was raised by Drs. W. Greiner and G. Morrison. Quartets are simple states which, of course, are not exact eigenstates of the Hamiltonian. In very much the same way

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CONCLUDING REMARKS C6-303 1 p, 2 p-1 h or analog states are not exact eigenstates.

In all cases the nuclear interactions produce a ⁽⁽disso- lution ⁾⁾of the simple states among the exact eigenstates.

The evaluation of the spreading width is essential for an understanding of the persistance of simple configurations among a background of other states. Experi- mentally, the spreading width seems to be of the order of 20 keV in light nuclei, whereas in heavier nuclei we only have an upper limit of 300 keV corresponding to the energy resolution. There has not yet been any theoretical evaluation of this width. Presumably the small observed widths are due the small overlap of the quartet states with the other states, resulting from very different couplings.

In conclusion my impression is that we are actually witnessing a real turning point in nuclear physics, with the opening of a wide new field. It may be characterized

as follows. In practically all experiments done so far in atomic, solid state, nuclear or elementary particle physics, the observed object is tested with a one-body operator (the exceptions are very few). For the first time we have with transfer reactions, a systematic procedure for probing nuclei with 2,3,4 ... body opera- tors. This opens the possibility of obtaining much more refined information on the nuclear wave functions. The most remarkable feature of the newly acquired experimental data is the possibility of understanding them (at least partially) in terms of very simple models. This Conference has perhaps not provided us with a fully satisfactory understanding of the nature of quartets. It rather leaves us with a lot of problems of a very interesting kind, and which are certainly soluble in the near future, as well as with many ideas of experiments. What more can one ask from a Conference ?

CONCLUSION

C. BLOCH

Division de la Physique, CEN Saclay, France

Les rkactions d'ions lourds sont Ctudites depuis longtemps. C'est aussi un sujet extrGmement riche : la selection, B partir de la table pCriodique, de deux noyaux, noyau cible et particule incidente peut Ctre faite de beaucoup de f a ~ o n s (il y a bien entendu des limitations exptrimentales). Dans le cas d'une tnergie incidente suffisamment grande, la reaction peut debou- cher sur un grand nombre de voies. Cela conduit B une quantitC tcrasante de rtsultats expdrimentaux ... ce qui a tt6 prdcistment pendant trbs longtemps la difficult6 de ce domaine.

C'est seulement en s'inspirant d'un modele thtorique (in&me grossier) qu'il devient possible de cc voi- quelque chose dans la masse des courbes exptrimi tales. I1 y a quelques anntes, I'idte des Btats quab, moltculaires a donnt lieu B des interpretations tr6s intkressantes de nombreuses exptriences. Plus rCcem- ment, I'hypoth6se des structures en quartets Cmise par V. Gillet, suivie de la dtcouverte par Mme H. Faraggi, M. C. Mermaz et al. de pics aux endroits prtdits et avec le comportement qualitatif attendu ont t t t extrb mement stimulantes. Cela a donnt une nouvelle signifi- cation aux rtactions de transfert a, qui ttaient observtes depuis des anntes dans diffkrents laboratoires. La comparaison des r6sultats les plus rtcents obtenus a partir de la structure en quartets a ttC le sujet principal de cette reunion B en juger du moins par la chaleur des

discussions sur ce thkme. Par suite de manque de temps, je concentrerai mes remarques sur ce sujet et je m'ex-

cuse B l'avance pour l'omission des nombreux autres rCsultats intkressants qui ont kt6 prtsentks ici.

J'aimerais tout d'abord passer en revue quelques-uns des rtsultats exptrimentaux les plus marquants ainsi que leur interprttation << nayve ⁾⁾en termes de quartets, laissant de cat6 pour l'instant toute critique Cj'y revien- drai ensuite).

La rCaction dont on a le plus par16 a kt6 12C(160, a)24Mg (voir en particulier les communications de D. A. Bromley, R. Middleton, M. J. Levine, T. Gastebois et R. H. Siemssen). Ce qui frappe tout l'abord dans cette rkaction, c'est le peuplement t r b stlectif de quelques Ctats dans le 24Mg ( B 14,14 ;

16,30 ; 16,56 et 16,84 MeV) lorsque l'knergie incidente est suffisamment Clevte, de I'ordre de 30 B 52 MeV (Yale, Pennsylvania University, Saclay). Ces Ctats se dttachent de nombreux autres qui forment un fond fluctuant important, mais qui sont, individuellement, beaucoup plus faiblement excitks. L'analyse de ces experiences suggbre que les &tats apparaissant forte- ment ont une structure de particule a. Par exemple, il a t t i montrt que si l'on essaie de produire les mgmes niveaux dans 24Mg par la reaction 14N(14N, a)24Mg, la section efficace est tout B fait rCgulibe sans aucune excitation sklective. Puisque le noyau initial 14N a une