The continuum in cluster decay
()
R. J. LIOTTA
Royal Institute of Technology - Frescativ¨agen 24, S-10405 Stockholm
(ricevuto l’11 Agosto 1997; approvato il 15 Ottobre 1997)
Summary. — Microscopic calculations of cluster decay widths, which have been one of
the recurrent challenges of nuclear theory for many years, require a good description of the preformed cluster in the mother nucleus. Since the cluster being emitted is formed either at the nuclear surface or beyond it, a reasonable microscopic description of the clustering requires the use of a realistic finite single-particle potential, including its continuum.
PACS 21.60.Cs – Shell model. PACS 21.60.Gx – Cluster models. PACS 23.60 – decay.
PACS 23.70 – Heavy-particle decay. PACS 01.30.Cc – Conference proceedings.
In this paper, recent developments in the dynamical theories of cluster decay are re-viewed with special emphasis on the influence of the continuum in the decay process.
The ordinary shell models, including the BCS approach, are discussed. Also cluster-like shell model theories, which incorporate the most important effects of the continuum, are presented.
By discussing diverse calculations, it is concluded that the most essential prerequisite for a realistic model of the mother nucleus is that it should correctly describe the cluster correlation in the surface region as well as the clustering at large distances. This implies, on the one side, that the proton-neutron interaction is indispensable and, on the other, that high-lying configurations, reflecting the influence of the continuum, have to be taken into account properly.
I will mainly present the work that I have done in collaboration with a number of researchers, as shown in the list of references. For a recent and detailed review of this work see ref. [1], from which much of the material to be presented here is taken.
The formation and decay of clusters in nuclei is a time-dependent problem. It is there-fore a very difficult problem, particularly because the initial state is not well defined. One
(
)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and
Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.
should follow its evolution, say, for10 10
years to establish how much time is needed for one half of the packet describing the cluster to filter through a barrier sustained dynamically by the system itself.
A way to avoid solving the time-dependent problem explicitly is based on ideas bor-rowed from reaction theory, as developed by Wigner [2] and Thomas [3]. In this theory the decay width is the residue of theS-matrix. To evaluate this it is assumed that the
de-cay process is stationary and, as a result, both the energies, i. e. the poles of theS-matrix,
and the decay width are in general complex quantities. In the case of alpha-decay the fact that a quantity proportional to a probability (the decay width) is complex is not a practi-cal problem because usually the width is very small and the residues of theS-matrix are
practically real. In fact, in the one-level case (when one can neglect any overlap among resonances) one can show that the residue of the S-matrix is strictly real. But in the
neutron decay of giant resonances this is a serious drawback which usually is ignored. In the Wigner-Thomas theory the alpha decay width is given by
, L (R )=2P L (R ) h 2 R 2M F 2 L (R );
whereM is the reduced mass, P L
(R ) is the Coulomb penetration factor with angular
momentumLandF L
(R )is the formation amplitude of the -particle at the pointR.
In the first calculations done in the framework of this theory one used forRthe sum of
the radii of the -particle and the daughter nucleus while to describe the formation
ampli-tude only one shell-model configuration was used, as it was standard at the time, although mostly to evaluate the relative decay width [4]. Within this limitations, it was found that the calculated absolute decay width was wrong by several orders of magnitude [5].
The weakest point of this calculation was not its total disagreement with experiment but that the penetrationP was so strongly dependent onRthat one could find a rather
acceptable value of the distance for which that agreement improved drastically. It was then obvious that something fundamental was missing since one should expect that,
L
is independent ofRfor distances outside the daughter nucleus, where the alpha-particle
is already formed. The calculated formation amplitude was vanishing small just outside the nuclear surface, indicating that the only configuration included in the calculation was not enough to describe the alpha decay process in the important region where the alpha cluster penetrates the Coulomb barrier. That is, high-lying configurations, which are rel-evant at large distances, should be included in the calculations as had been suggested many years before [6]. This was rather easy to do within the shell model, as later cal-culations showed [5, 7]. It was found that indeed the effect of high-lying configurations was both to make the alpha decay width independent ofRin a region close to the nuclear
surface and, at the same time, to increase the value of,
L by many orders of magnitude.
It was also found that the physical reason of this increase is that, through the high-lying configurations, the nucleons that eventually become the alpha particle are clustered. Yet, the calculated alpha decay width was more than one order of magnitude too small. One then thought that the neutron-proton interaction, which had been excluded in these cal-culations, was important to cluster nucleons of different isospin. This was introduced through a ”giant pairing resonance”, that is a high-lying collective pairing state scalar in isospin [5, 8]. But within reasonable limits for the mixing of this giant resonance in the mother nucleus wave function the alpha decay width could not be increased more than a few per cent. A proper treatment of the neutron-proton interaction would have re-quired a full shell model calculation, as was later performed within a cluster shell-model
configuration mixing model [9]. These calculation not only showed the importance of the neutron-proton interaction, but also the necessity of including the continuum properly.
To treat the continuum exactly would have required computation capabilities beyond what is possible even within the powerful computers of today. An approximate approach to this problem was to replace the proper continuum by the set of outgoing solutions of the Schr¨odinger equation, which are either bound states or resonances in the continuum [10]. Actually, this induced the development of a formalism to study the continuum in general, which is at present been applied in studies of unstable processes [11, 12].
One important conclusion of the calculations mentioned above is that the pairing in-teraction strongly enhances the probability of finding a pair or a quartet of nucleons on the surface, and, for the spin singlet (S =0) component, the probability of finding a pair
shows a sharp peak around relative distance zero. Furthermore, while pure Nilsson con-figurations produce pair and quartet densities peaked at particular points of the deformed nuclear surface, configuration mixing smears these peaks into a more or less uniform den-sity ridge around the nucleus. At the same time, the formation amplitude is very much enhanced in comparison with pure Nilsson orbits [13].
The significance of this is that for more complicated cases the description of pair corre-lations is much simpler, owing to the surprising success of the BCS theory (see e.g. [14]), than the description of all correlations induced by a realistic nucleon-nucleon force. This suggests that the BCS theory with a pairing force may provide a nearly satisfactory model for the description of some cluster-decaying, especially -decaying nuclei. This model is
appealing because it is realistic just in between two closed shells, where the ordinary shell model gets very involved, and because it lends itself to a consistent description of the par-ent and the daughter nuclei. In the BCS model the nuclei are hardly “individualized”, and thus the results change rather smoothly from nucleusAto nucleusA+2, which is
consis-tent with the systematics of the empirical reduced -decay widths of heavy nuclei far from
magic numbers (see e.g. [15]). Recently phenomenological evidence has been found [16] for an even more striking uniformity in even-even actinide nuclei. Experimental branch-ing ratios of -transitions to different final states were used in a coupled-channel
barrier-penetration model of the -decay of these deformed nuclei. The results seem to imply
that the amplitudes belonging to different -core orbital momentaLare essentially the
same for all these nuclei. Such an observation reaffirms that it is promising to apply the BCS theory to the cluster decay of these nuclei. It is to be emphasized, however, that the pairing force only acts between like nucleons, and the feasibility of large-scale BCS calculations hinges on the neglect of the proton-neutron force.
The BCS appoach was introduced long ago in studying -decay [17]. These
pioneer-ing calculations necessarily suffered from the same limitations that were present in the spherical nuclei mentioned previously, namely that they used small shell-model spaces. As a result, the absolute decay rates were undershot by orders of magnitude, and the degree of disagreement depends on parameters that are uncertain enough to diminish seriously the predictive power of the model for the absolute decay rates.
More recently the BCS treatment was applied to studying -clustering in, and the
-decay of, spherical systems with many particles outside the core [18]. In that work the power of modern computers was exploited, and a large number of single-particle states were used. It was found clustering features similar to those present in normal nuclei. The corresponding calculations af -decay widths for light lead isotopes agree well with
exper-imental data. However, the treatment of the neutron-proton interaction was schematical. Therefore, the success of the theory has to be attributed to the fact that that interaction is included effectively through the fitting of the pairing gap to the corresponding
experi-mental value [1].
The theory was also applied to deformed systems [19]. In this approach the alpha-formation amplitude was described within a realistic mean field and a large configuration space. The penetration of the alpha cluster through the deformed barrier was treated in terms of the WKB approximation [14] and the absolute values of the alpha-decay widths for Ra, Rn and Th even-even isotopes were reproduced within a factor of about 3. This approach was later applied to odd-mass nuclei and the anisotropy in alpha-decay from deformed nuclei could be explained [20].
If the BCS model involves uncertainties for -decay, even more does it for
heavy-cluster decay. Therefore, the results can be considered at best as order-of-magnitude estimates. But in view of the complexity of the problem, even such rough estimates are very valuable. The level to be surpassed was set up by the first dynamical microscopic calculation, which assumed pure harmonic-oscillator single-particle configurations for the nuclei involved and underestimated the decay width by 18 orders of magnitude [21].
The decays222 Ra! 14 C+208 Pb,224 Ra! 14 C+210 Pb,226 Ra! 14 C+212 Pb and114 Ba! 12 C+102
Sn were considered. Both the mother and the daughter states were assumed to be spherical. In the structure calculations as well as in the calculation of the formation amplitude the harmonic-oscillator expansion was stopped at18h!. The nucleus
208
Pb was assumed to have the pure closed-shell configuration. The parameters of the BCS approach were adjusted as in the calculations for -decay.
The formation amplitudes obtained resemble very much the familiar -formation
am-plitudes. For 222 Ra! 14 C+208 Pb and 114 Ba! 12 C+102
Sn, the node numbers obey the Wildermuth prescription, which is probably a consequence of the closed-shell structure of208
Pb and of the closed proton shell of102
Sn, respectively. The amplitudes peak on the surface, and their magnitudes are much smaller than those for -decay. For instance, for
the208
Pb+14
C system, the main peak ofF(R )is at 8.5 fm, and its height is3:910 ,5
fm,1=2
. This peak is close to that of the amplitude for208
Pb+ , and is farther out than
any of the BCS formation amplitudes for -decay. There are two competing effects, whose
balance determines the position of the peak: a push by the Pauli exclusion outwards and a squeeze by the Coulomb barrier inwards, and both must be stronger for a heavy cluster than for an -cluster. It may well be that a heavy-cluster–core system is more compact
than an -core system, and the BCS model may work better for a more compact system.
The important point in these calculations is that the width,is practically independent
of the radiusR for a rather large region outside the touching point. This reflects the
moderate reliability of the procedure. The agreement of the widths with experiment is again good. In view of the uncertainties of the model, it is not surprising that one of the theoretical widths exceeds the experimental value. The results are consistent with the observation that some approximations introduced tend to enhance the calculated widths.
Finally, we mention that the BCS-description of14
C-decay was also improved by tak-ing into account barrier deformation. In this calculation the BCS ground state (g.s.) of the mother nucleusB was constructed out of the daughter state A = (Z ,6;N ,8)
stepwise, in the fashion of the multistep shell model. More precisely, the g.s.’s of the nuclei(Z ,6;N ,8), (Z,4;N ,6),(Z ,2;N ,4),(Z ;N ,2), B = (Z ;N), were
constructed consecutively with the BCS, with the preformation of the previous nucleus in this sequence assumed [22]. The method to calculate the14
C-widths of some Ra isotopes was similar to that used in the calculation of -widths. In the calculation of the formation
amplitude the basis size was reduced by using a two-size harmonic-oscillator basis as in ref. [22]. The numerical results for the14
C-decay of Ra isotopes agree surprisingly well (i.e. within an order of magnitude) with experiment. TheR-matrix radius was chosen
from the region beyond the touching distance (10–13 fm), where the widths were found to depend on it weakly. One found that the influence of the deformation is sometimes ap-preciable. With the exception of222
Ra, the theoretical and experimental absolute decay widths agree within a factor of five. The decay of222
Ra is exceptional because its daugh-ter nucleus is208
Pb, which is not described by the BCS but is treated, compellingly yet somewhat inconsistently, as a doubly magic nucleus.
This microscopic description of heavy-cluster decay, with a single-particle basis suited to describe the continuum part of the spectrum, was recently used to analyse the role of pairing correlations and quadrupole deformations [23]. The stability of the results as a function of the number of major shells included in the calculation was attested, giving credibility to the model.
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