IL NUOVO CIMENTO VOL. 110 A, N. 9-10 Settembre-Ottobre 1997
Challenges in nuclear molecular physics
()
R. R. BETTS(1)(2) (
1
) Physics Division, Argonne National Laboratory - Argonne IL. 60439 USA (
2
) Physics Department, University of Illinois at Chicago - Chicago IL, 60607 USA (ricevuto il 15 Agosto 1997; approvato il 15 Ottobre 1997)
Summary. — Despite over thirty years of experimental and theoretical effort in the
study of molecular resonance and cluster phenomena in nuclei, fundamental questions still remain to be answered. The most important of these relate precisely to those experimental quantities which would directly prove the underlying nature of the phe-nomenon – decay widths and transition strengths. The status of the understanding of these quantities is reviewed and new experiments suggested.
PACS 25.70.Ef – Resonances.
PACS 25.70.Pq – Multifragment emission and correlation. PACS 01.30.Cc – Conference proceedings.
1. – Introduction
The study of nuclear molecular phenomena goes back over thirty years and a wide body of experimental information and theoretical analysis exists. Despite this consider-able effort, however, fundamental questions remain to be answered. Specifically, we still do not have any firm information on those quantities which would directly prove the un-derlying nature of the phenomenon although the body of circumstantial evidence is quite compelling. In this paper a brief summary of the experimental results and theoretical ideas which define the field is presented together with what, to my mind, are some crucial questions. Three possible experiments which might shed some light on these questions are then proposed.
2. – Experimental results
The original measurements of the energy dependence of elastic scattering and reac-tions of12C+12C carried out at Chalk River [1-3] displayed a number of surprising nar-row structures at energies in the vicinity of the Coulomb barrier. Over the intervening
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
976 R. R. BETTS
Fig. 1. – Energy dependence of the12 C+12
C nuclear structure factor.
years, these data have been refined and the compiled results, shown as a nuclear struc-ture factor, are displayed in fig. 1 [4]. The above-mentioned narrow resonances are clearly visible. These resonances correspond to highly excited states of the compound nucleus
24Mg which have widths for decay into12C which are both much larger than expected
for the statistical decay of the equilibrated compound nucleus and represent substantial fractions of the value expected for a12C single-particle or molecular state. Unfortunately, little other systematic information is available on the properties of the individual reso-nances such as their decay widths into nucleon or alpha-particle channels. In fact, as these data largely result from measurements of gamma rays from20Ne,23Mg and23Na, contributions to the resonance yield from particle decaying states of these residual nuclei are completely missed. I shall return to this important point later.
Measurements of elastic and inelastic scattering of12C+12C and16O+16O at energies above the Coulomb barrier are shown in figs. 2 and 3 [5-10]. These data show series of broad structures whose energy as a function of angular momentum follows a grazing or molecular trajectory characteristic of the rigid rotation of a touching configuration of the two colliding nuclei. The broad structures are therefore considered to represent a family of single particle molecular resonances which are fragmented by mixing with some more complex configuations intermediate between the simple molecule and the average com-pound nuclear states. Note, however, that the assumed structural relationship between these resonances is based on nothing more than their energies and angular momenta.
At the time, it was generally assumed that these phenomena were confined to a few light, symmetric and near symmetric systems. The observation [11, 12] of a strong back-ward rise in the elastic scattering of 16O+28Si as shown in fig. 4 and an associated resonance-like behaviour of this large angle scattering cross-section therefore indicated a much wider occurrence of nuclear molecular phenomena than was previously thought possible. On the other hand, there were striking differences between these data and the results for the lighter systems. No clear regularities of the angular momenta associated
CHALLENGES IN NUCLEAR MOLECULAR PHYSICS 977
Fig. 2. – Elastic scattering excitation functions for12 C+12
C and16 O+16
O.
with the structures in the excitation functions were evident and, when analyzed as single isolated resonances, the partial widths for molecular decay channels were considerably smaller than those for the lighter systems. In part, this may be associated with the non-identical particle nature of this system allowing all partial waves to contribute rather than just the even angular momenta required for the scattering of identical bosons such as
12C+12C.
In fact, some relative simplicity appears to be restored for even heavier but symmet-ric systems such as24Mg+24Mg and28Si+28Si which represent the heaviest systems for which resonance behaviour has been convincingly demonstrated [13-18]. A series of detailed measurements for these systems have shown that the molecule-like sequences of resonances similar to those of the lighter systems also appear in 24Mg+24Mg and
28Si+28Si where the measured angular momenta (J = 30,40) represented, at the
time, some of the highest spin nuclear excitations observed. A sample of these data for
24Mg+24Mg scattering are shown in fig. 5.
It is still true that the partial decay widths in the elastic scattering channel for these heavier systems are considerably smaller than for the lighter systems. However, when the resonance yields to the inelastic scattering channels are included, the situation is re-stored to something approaching that for12C+12C at the Coulomb barrier. In fact, the resonances now appear in the total inelastic scattering yield as shown for 28Si+28Si in fig. 6. This total large-angle inelastic scattering yield has many of the characteristics of complete damping of energy and angular momentum similar to that expected for fission of the excited compound nucleus.
Another interesting feature is apparent in a comparison [16] of data for28Si+30Si and
30Si+30Si which is also shown in fig. 6. The prominent structures observed in28Si+28Si
are completely absent in the28Si+30Si and30Si+30Si data and this disappearance is ac-companied by the drop in the overall level of the yield. This feature is suggestive of a
978 R. R. BETTS
Fig. 3. – Energy dependence of the cross-section for inelastic and mutual inelastic excitation of the 4.43 MeV 2+
state in12
C. Also shown is the12 C+12
C fusion cross section.
reduction of shell effects with increasing neutron number which result in the disappear-ance of a secondary minumum in the fission barrier and also raise the fission barrier with a consequent fall in the fission cross-section.
3. – Theoretical ideas
The very first papers [2, 3] already contained the suggestion that the resonances ob-served in12C+12C scattering might correspond to a molecule-like configuration of the two
12C nuclei stabilized by the existence of a pocket at large distance in the potential
describ-ing their relative motion. At the time, however, no basis existed for nuclear structure or other effects which might produce the postulated quasi-stable molecular configurations.
Subsequently, it has become evident that a number of related mechanisms are capable of producing the highly-deformed, long-lived states which are manifest as the observed resonances. The extremes of these mechanisms are represented by models based on
po-CHALLENGES IN NUCLEAR MOLECULAR PHYSICS 979
Fig. 4. – Angular distribution of elastic scattering of16 O+28
Si showing a strong backward angle rise over the expectations of an optical-model calculation.
tential scattering in which the resonances originate from quasi-bound states in the ion-ion potential [19-21], to models whereby the resonances are viewed as something similar to fissioning shape isomers stabilized by shell effects at large deformations [22, 16].
These seemingly dissimilar viewpoints can be conceptually unified within the frame-work of the two- or multi-center shell model [23-25]. In this way, the scattering models can be thought of as a limit of the two-center model where the two nuclei are sufficiently separated that the appropriate wavefunctions consist of the internal wavefunctions of the participating nuclei coupled to their relative motion. The shape isomer extreme corre-sponds to the limit of large overlap of the nuclei such that the two-center potential appears very much like a deformed one-center potential. Both of these limits have their good and bad points. The scattering models are able to give quite a good account of the gross and intermediate width structures observed but require input of the scattering and inelastic coupling potentials and, in general, have not been able to produce the narrowest of the observed structures. The shape isomer viewpoint, although intuitively appealing and ca-pable of providing insight into the variety of observed structures and their symmetries, is so far unable to provide quantitative information on cross sections and widths.
980 R. R. BETTS
Fig. 5. – Elastic and inelastic scattering excitation functions for24 Mg+24
Mg. The assigned reso-nance spins are indicated.
From an empirical point of view, attempts have been made to characterize the consid-erable amount of12C+12C data in terms of energy eigenvalue expressions suggested by molecular rotation-vibration models [26, 27]. One example of this approach is shown in fig. 7. The expressions used to fit these data have their origins in the spectra of atomic molecules in which case the energies of the intrinsic excitations, vibrations and rotations are quite different and these degrees of freedom are therefore largely uncoupled. In the case of nuclear molecules, the energies of the intrinsic excitations and rotations are quite comparable and, although there is no a priori information on the scale of the energy as-sociated with vibration along the internuclear coordinate, this too should be similar. We therefore expect that the degrees of freedom which are so cleanly separated in the case of atomic molecules will be strongly mixed in the nuclear case. Nevertheless, establishing the quantum numbers implied by the empirical fits would clearly be a major step forward. 4. – Crucial questions and possible experiments
From the above brief summaries it should be clear that there exists a compelling body of evidence in favor of an interpretation of the data in terms of nuclear molecular configu-rations. The most important pieces being the existence of apparently rotational sequences of resonances with large moments of inertia and with large partial widths for binary frag-mentation. What is equally clear, however, is that the molecular hypothesis is not yet proven. The crucial questions which remain to be answered are:
1) What is the shape of the system ? This could be addressed by a determination of a transition quadrupole moment of a radiative transition between two resonances.
alter-CHALLENGES IN NUCLEAR MOLECULAR PHYSICS 981
Fig. 6. – Total inelastic scattering excitation functions for28 Si+28 Si,28 Si+30 Si and30 Si+30 Si.
natively, can patterns be observed in the decay widths of resonances which allow us to definitively associate them into families with the same intrinsic structure?
In the following subsections, three experiments are outlined which may resolve some of the above issues.
4.1. Nuclear structure of resonances. – In calculations of the potential energy sur-face of24Mg as a function of deformation [22], the results of which are shown in fig. 8, highly deformed minima appear which have been suggested [22, 28] to correspond to the resonances observed in12C+12C. These minima can be categorized by their shell model wavefunctions written in terms of excited multi-particle multi-hole configurations relative to a closed16O core. For example, the strongly deformed oblate and triaxial minima have [1],4
[2]12and [1],4
[2]8[3]4configurations where [N]M
refers to the harmonic oscillator shell and the number of nucleons in that shell. The non-axially symmetric minimum has the configuration [2]4[3]4. If these assigments are taken at face value, it is expected that the alpha-particle decays of resonances associated with the minima should populate states in20Ne with similar structure but with one alpha-particle removed. States with such con-figurations are well known [29, 30] in20Ne and the 0+band heads lie at 6.72, 7.20 and 8.30 MeV excitation, respectively. In fact, these states are so well studied that the degree of mixing between them is quite well determined. It might therefore be thought that data on these obviously important decay chanels would already exist. However, as pointed out earlier, the most detailed information in the Coulomb barrier energy region comes from gamma-ray measurements which cannot include states which particle decay, as happens to be the case for these specific states in20Ne.
There are some interesting data at rather higher bombarding energies [31] for the 6.72 MeV 0+state and the 7.83 MeV 2+state, some of which are shown in fig. 9. The excitation functions for these states show a considerable enhancement of the cross sections over the
982 R. R. BETTS
Fig. 7. – Classification of the low-energy12 C+12
C resonances according to a rotation vibration scheme.
expectations of the statistical model and also display strong resonances which appear to be somewhat different to those observed in data for the ground state and 1.63 MeV 2+ state, the latter of which is of course included in the gamma ray data.
The first of the proposed new measurements is therefore to revisit these questions and obtain data for the resonance decays to these interesting final states in the energy region close to and below the Coulomb barrier. It may well be that such data will provide some surprises and allow a further elucidation of the structure underlying the12C+12C barrier resonances and perhaps also allow the existing empirical band assignments to be placed on firmer ground.
4.2. Radiative decays of resonances. – Obviously, the most direct demonstration of the molecular and rotational nature of the observed resonances would be a determination of the radiative transition rate between two of the postulated molecular structures, as this depends on the quadrupole moment of the charge distribution. Of course a measurement of this type is made extremely difficult by the competition of the radiative decay with the much faster particle decay widths of the resonances. For example, assuming a quadrupole moment of 200 fm2 typical of a molecular or superdeformed 24Mg nucleus, a radiative decay branch of 1 part in 105 is expected for the 6.5 MeV transition between the 1 MeV wide,J =14Ecm=25.5 MeV12C+12C resonance and the lowerJ =12resonance. Two
attempts to measure this transition [32, 33] have resulted in upper limits which call into question the assumed molecular nature of the resonance. However, it is possible that, for such highly unbound virtual states, that the density overlaps of the rotational states
CHALLENGES IN NUCLEAR MOLECULAR PHYSICS 983
Fig. 8. – Potential energy surface for24
Mg calculated by using the deformed shell model. The quasi-stable cluster configurations associated with the minima in this surface are also shown.
are considerably smaller than the value of unity assumed in the estimated width even though the structure of the states is still molecular. It is also true that, in the period since these measurements were made, detection and data acquisition technology has improved so much that it is now time for a new measurement to be attempted.
My own prejudice is that such a measurement would best be made for one of the lower energy resonances. There are several reasons for this, the low-energy resonances are much narrower than those for which the first measurements were attempted and thus the sought for gamma ray would also be much narrower and therefore be more visible over background. The background itself should be reduced as the total reaction cross section is much smaller than at the higher energies. On the negative side, the transition energies
984 R. R. BETTS
Fig. 9. – Excitation functions for12 C+12
C leading to several final states in20 Ne.
are lower which results in a much smaller radiative decay width for the same quadrupole moment due to theE5dependence of the transition rate — however, this is partially offset by the smaller particle decay widths.
There are three possible schemes for such an experiment — all require high beam intensity, optimized charged particle detection and highly segmented, high rate, modest resolution 4gamma-ray detection. The earlier attempts utilized population of the
res-onance via the12C+12C entrance channel and searched for the12C decay of the lower resonance following the gamma emission. Detection of all the emitted particles overde-termines the kinematics of the final state and allows efficient rejection of backgrounds by suitable kinematic cuts. The disadvantages of this scheme are the extremely large cross sections for processes which also result in a final state consisting of two12C nuclei which, through random coincidences, can mimic the sought-for signal. At the lower energies, the
CHALLENGES IN NUCLEAR MOLECULAR PHYSICS 985
Fig. 10. – Energy level diagram for8
Be showing the expected rotational sequence for a4 He+4
He molecule.
known large alpha-particle decay widths of the resonances raise the possibility of search for the radiative decays through12C+12C population of a resonance and detection of a fi-nal state consisting of a photon, an alpha-particle and a20Ne. The asymmetric mass split in the particle channel makes their detection more difficult to optimize but the much lower cross section of this channel ( compared to12C+12C elastic scattering ) will improve the random backgrounds significantly. Probably the optimum scheme, from a purely physics point of view, is to reverse the latter and populate the initial resonance via the alpha +
20Ne channel and detect two12C nuclei following the photon emission. In this way the
particle detection can be easily optimized, the problem being of course a20Ne beam and
4He target.
I hope that a serious discussion of some of these suggestions can be a consequence of this workshop, perhaps leading to new efforts to solve this problem. Particularly inter-esting in this respect are reports [34] of positive results for one of the lower resonances contained elsewhere in these proceedings.
4.3. Where is the rotational excitation of the12C 0+2 state ? – The best example of a
nuclear molecular configuration is found in the nucleus8Be, the low-lying states of which are shown in fig. 10. The lowest 0+, 2+ and 4+ states are observed as resonances in
4He+4He scattering at energies which are in excellent agreement with the expectations
for the rotation of a molecule-like configuration of the two alpha-particles. The reduced widths of these resonances indicate an alpha-particle spectroscopic factor of unity sup-porting the conjectured structure. It has been suggested that the 7.65 MeV 0+ state in
986 R. R. BETTS
The reduced width of this state gives an alpha-particle spectroscopic factor of 2 — there being two equivalent alpha-particles which can be removed, leaving the8Be ground-state configuration.
Despite the long history of this state, no conclusive evidence exists regarding the rota-tional band built on this configuration. It has been suggested that the linear three-alpha chain configuration assignment for the 7.65 MeV state is incorrect although the most re-cent calculations [35], which reproduce the observed width of this state, do indicate a substantial component of this configuration in the wavefunction. Based on scaling the en-ergy of the 2+state in8Be and assuming the same spectroscopic factor as the 7.65 MeV state, the 2+chain state is expected to lie at about 8.6 MeV in12C and to have a width of
>50 keV — no such state is known. The closest candidate state lies at 10.3 MeV and the
question then is, could a lower 2+state have been missed in earlier studies ?
Most of the relevant [36, 37] investigations of12C have been carried out using decay
to populate variousJ =0;1and 2 states in12C which subsequently alpha decay. As the 12C produced in this way is essentially at rest, the primary alpha-particle carries away 1/3
of the total decay energy and can therefore be used to determine the excitation energy of the decaying state. The remaining energy is carried by the8Be, which itself subsequently decays into two alpha-particles. To cut a long story short, none of the experiments carried out to date have been exclusive in the sense that all three alpha-particles were uniquely detected and identified, and the excitation spectra therefore have large backgrounds due to incompletely measured events. Consequently, it is possible that the postulated 2+state could have escaped identification, particularly if it is weakly populated in the initial decay. With advances in detector technology, such as silicon pixel devices, it is possible to conceive of a new experiment in which all three alpha-particles are uniquely measured and much cleaner excitation spectra reconstructed. One possible arrangement would be to make the -decaying12B or12N in inverse kinematics and separate and identify these nuclei
using one of the new generation recoil mass spectrometers which now exist at a number of laboratories. The decays of these nuclei could then be studied with a silicon array in the focal plane of the spectrometer with the primary beam switched off. In fact, the measurement of the in coincidence with the decay alpha particles could give information
on the spin of the alpha decaying state in addition to the clean excitation information given by the exclusive measurement of the three alpha-particles. In this way, the question of the location of the 2+state built on the 7.65 MeV configuration could be answered and perhaps another excellent example of a nuclear molecular configuration identified.
5. – Summary and conclusions
The long history of efforts in the study of nuclear molecules has been briefly summa-rized and the body of evidence which supports the conjecture of the underlying molecular nature of the observed resonances outlined. We have seen that although this body of ev-idence provides a strong basis for the molecular hypothesis, the “ smoking gun “ has, so far, eluded us. At this point in time, rather specific and well focussed experiments are required to get to the essential point. I have suggested three possible avenues for fur-ther investigation which, I hope, will provoke some renewed discussion and collaboration toward the goal of proving the true nature of the nuclear molecular resonances.
This work was supported by the U.S. Department of Energy, Nuclear Physics Division. Many useful discussions with K. LISTERand A. WUOSMAAare gratefully acknowledged.
CHALLENGES IN NUCLEAR MOLECULAR PHYSICS 987
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