Can we obtain nuclear structure information from exotic decays of heavy nuclei?

Nuclear Physics A512 (1990) 483-491 North-Holland

CAN WE OBTAIN NUCLEAR STRUCTURE INFORMATION EXOTIC DECAYS OF HEAVY N U C L E I ?

FROM

B. BUCK Department of Theoretical Physics, 1, Keble Road, Oxford OX1 3NP, UK

A.C. MERCHANT Instituto de Estudos Avan£ados, Centro T(cnico Aeroespacial, 12.225 S~o Jos~ dos Campos, S~o Paulo, Brazil

S.M. PEREZ Physics Department, University of Capetown, Private Bag, Rondebosch 7700, South Africa

Received 22 January 1990

Abstract: A simple cluster model has shown itself to be outstandingly successful in describing the half-lives for emission of heavy fragments ranging from t4C to 32Si from heavy even-even nuclei. However, its predictions for the half-lives of similar decays from odd-even nuclei are systematically below the corresponding measured values or lower limits set by experiment. In view of recent experimental results on the 223Ra ~ 2°9pb+ 14C system, we suggest that many of the exotic decays of odd-even nuclei so far observed, or searched for, may be proceeding preferentially to low-lying excited states of the daughter nucleus. We show that all the discrepancies between our model predictions and the presently available data for t4C and 24Ne emission would be consistently removed if this should prove to be the case. An opportunity to extract nuclear structure information from exotic decay measurements would thus be available.

I. Introduction

Since the discovery b y Rose a n d Jones in 1984 1) that 223Ra emits

14C fragments

at detectable rates, some 16 heavy nuclei have b e e n e x p e r i m e n t a l l y investigated for signs o f s i m i l a r heavy-particle radioactivity. [See ref. 2) for a recent s u m m a r y a n d refs. 3-5) for the latest d e v e l o p m e n t s . ] O f these, 10 nuclei have b e e n seen to emit h e a v y f r a g m e n t s in the form o f various isotopes o f C, Ne, Mg or Si, a n d lower limits have b e e n set for the half-lives o f similar decays from the r e m a i n i n g 6 nuclei. I n m a n y o f these cases, the e x p e r i m e n t a l mass r e s o l u t i o n is n o t sufficient to d i s t i n g u i s h exactly which isotope o f a given e l e m e n t has b e e n emitted, a n d p e n e t r a bility a r g u m e n t s are u s e d to try a n d resolve this question. Often they are a d e q u a t e to i d e n t i f y u n i q u e l y the e m i t t e d f r a g m e n t (e.g. 14C from 223Ra), b u t s o m e t i m e s d o u b t s 0375-9474/90/$03.50 O Elsevier Science Publishers B.V. (North-Holland)

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remain and it must be admitted that two different isotopes may have been emitted together (e.g. 24Ne a n d / o r 26Ne from 234U). Similarly, the kinetic energy of the recoiling fragment is usually measured well enough to say that it is compatible with a ground state to ground state transition, but not so precisely that a decay to some low-lying excited state(s) of the daughter nucleus can be definitely ruled out. However, other things being equal, the extra energy obtained by decaying directly to the ground state of the daughter nucleus would seem to favour it over the excited states. The perspective on this latter question has recently been radically altered by the observation that the 14C decay of 223Ra is about 15% to the ground state and approximately 81% to the first excited state of 2°9pb [ref. 6)]. The experimental investigations outlined above were foreshadowed by one theoretical model 7) and have subsequently stimulated the development of several others 8-13). Somewhat surprisingly the simpler models with fewer free parameters have been as successful in reproducing the observed decay half-lives as the more complicated ones 14). Indeed, the simplest model imaginable, in which a preformed cluster penetrates the barrier formed by the sum of an infinitely deep square well of radius Ro(A~/3"bA~/3) fm and the cluster-daughter Coulomb potential [with Ro fixed at 0.98 fm, see ref. 15), and Q-values taken from experiment] has given the best description of the ratios of exotic to alpha half-lives of all the approaches investigated to date. This would seem to indicate that the exotic decay half-lives are completely determined by the energy released in the fragmentation and some appropriate radius parameter. For the exotic decays of even-even nuclei, our own model calculations support this conclusion, and we obtain agreement to better than a factor of ten with all the available data 16). However, for odd-even nuclei, we predict values systematically below the corresponding measurements. We also note that the excellent agreement of the square-well model for both even-even and odd-even nuclei mentioned above is achieved only by the introduction of a "relative frequency factor" of forty between the decays of the two types of nuclei. There is clearly something systematically different about the two cases. Since it is well known that many odd-even actinide nuclei a l p h a decay preferentially to excited states of the daughter nucleus (see, for example, ref. 17)), and in view of the recent discovery that 223Ra decays exotically with a similar preference, we propose to investigate here whether the other odd-even nuclei have exotic decay modes which behave in the same way. If this suggestion should turn out to be true, it would offer an excellent opportunity to extract nuclear structure information from exotic decay data, something not previously believed possible. In sect. 2 we present our cluster model and its predictions of exotic-decay half-lives, supposing that all transitions are from parent ground state to daughter ground state. In sect. 3 we examine the odd-even nuclei more closely to determine the consequences of their decaying predominantly to excited states of the daughter nucleus. A summary of our results and our conclusions is given in sect. 4.

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485

2. Cluster model calculations of exotic-decay half-lives We have previously employed the Buck-Dover-Vary cluster model 18), together with the semi-classical limit of the Gurvitz-Kalbermann 19) expression for the decay width of a quasi-stationary state, to obtain values for the half-lives of exotic and alpha decays 13,16,20,21). We may briefly summarise the method of calculation as follows. The parent nucleus is viewed as consisting of a daughter nucleus core (mass A1) and a preformed cluster (mass A~) in relative motion about their centre-of-mass. The nuclear interaction between them would ideally be found from a folding integral involving their respective densities and an effective nucleon-nucleon potential. In the few cases where there are sufficiently well-known charge densities available to carry out this programme, the resulting potential may be accurately parameterised in the form V,u¢(r) -

- Vo[ l + c o s h ( g / a) ] [cosh (r/ a ) + c o s h ( R / a ) ] '

(2.1)

where Vo is the depth of the potential, R is the radius parameter and a the diffuseness. We have found that we can use this potential systematically for all cluster-core combinations, by writing a --0.75 fro, R = 1.04(A~/3+ A~/3)1/2 fm and choosing Vo so as to reproduce accurately the Q-value of the decay when solving the Schr6dinger equation for the relative motion. The main requirements of the Pauli principle are taken into account by choosing the relative motion quantum numbers n (the number of internal nodes in the radial wave function) and L (the orbital angular momentum) to satisfy a Wildermuth condition, 2n + L ~> N. The integer N is chosen large enough to correspond to the microscopic situation in which the cluster nucleons all occupy orbitals above those already filled by the core nucleons. Although this procedure is not rigorously exact when the individual nucleons of the two bodies are not described by harmonicoscillator wave functions sharing a common length parameter, we have checked that our results are not very sensitive to the precise value of N employed over a fairly wide range. The potential acting between cluster and core is thus of the form h 2 (L+½) 2 Vv(r) - Vnuc(r) + Vcou,(r) -~ 2/x r2 ,

(2.2)

where /.t is the reduced mass of the cluster-core system, Vcou~(r) is the Coulomb potential of a point charge interacting with a uniformly charged sphere and the Langer modification [in which L ( L + I ) is replaced by (L+½) 2] is made to the centrifugal term so as to guarantee three turning points, even for s-wave quasistationary states. An expression for the decay width F can now be written down in terms of the three turning points associated with VT(r), (called r0, rl and r: in order of increasing

B. Buck et al. / Exotic decays

486

distance from the origin) and the energy Eo of the equivalent bound state in the related potential VM(r) obtained by flattening off the barrier at its maximum value. It is basically the G a m o w formula with a well-determined pre-exponential factor.

F = Fh2 exp (-2 Irr2 k(r) dr) .

(2.3)

4/~ The semi-classical wavenumber

k(r)

is given by

k(r)= (~221Eo- VT(r)l) 1/2

(2.4)

and F is the semi-classical bound-state normalisation factor F

Iro"~ C O S

r

'o k(r') d r ' -

4) d r = 1.

(2.5)

We have previously written eq. (2.3) with a statistical factor (depending on L and the spins of the initial and final states) multiplying the expression on the right-hand side. We now believe the inclusion of this factor to be incorrect. The reason for this is essentially that our final state contains at most only two non-zero angular momenta (the spin of the daughter and the cluster-core relative orbital angular momentum) which can be uniquely coupled so as to guarantee angular momentum conservation. However, it is still important to know the parent and daughter spins in order to deduce the cluster-core relative orbital angular momentum from triangulation. The centrifugal contribution to the barrier has significant effects upon the penetration probability. We can now find the half-life Tt/: in terms of the width F using the well-known formula

T1/2= h In 2/ F

(2.6)

and compare our predictions with experiment for 14C and 24Ne emission. This includes all cases in which the actual value of the half-life for an exotic decay (rather than just a lower bound) has been measured. We present our results, obtained with the scheme outlined above, in table 1. All the exotic decay half-lives have been computed with Q-values (calculated from the mass defect tables of Wapstra and Audi 22)) appropriate for ground state to ground state transitions. This can be seen to yield excellent agreement for the even-even nuclei to within at most a factor of ten and usually much better. However, the half-lives predicted for the 6 odd-even nuclei are consistently below the 3 measured values and 3 experimental lower limits, sometimes by more than a factor of a hundred and always by at least a factor of ten. Since the divergences from experiment in these cases are so much more marked than for the even-even nuclei, and all the obvious odd-even effects are presumably incorporated in the Q-values, we feel that this must be indicative of some missing physics in our treatment of the odd-even nuclei. A significant clue to the missing ingredient is provided by considering the alpha decays of the same nuclei. With the energy resolution commonly attainable in alpha

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487

TABLE 1 C o m p a r i s o n o f c a l c u l a t e d a n d m e a s u r e d half-lives f o r e x o t i c d e c a y s a n d d o m i n a n t a l p h a d e c a y s Exotic decays

system

T~I~c (s)

221Fr~ 2°7T1+ 14C 221Ra~ 2°7pb+ 14C 222Ra ~ 2°spb + 14C 223Ra ~ 2°9pb + 14C 224Ra ~ 21°pb + 14C 225Ac~ 21lBi+ 14C

2.3x 1 0 1 3 />1.4×1012 7.6 × l0 I0 2.3 × 1013 5.6 × 1015 1.4× 1 0 1 7 7.2 × 1020 1.3×1024 2.2 × 1021 8.9)<1019 4.5 x 1022

226Ra-~ 212pb+ laC 23°Th ~ 2°6Hg + 24Ne 231pa--*.2°7Tl+24Ne 232U ~ 2°8pb + 24Ne 233U--~2°gPb + 24Ne

Alpha decays

T~1~2 p (s) >5.9×1015 > 2 . 2 x 1014 1.0 x 1011 1.6 x 1015 7.9 x 1015 > 2 . 2 × 1018 2.1 × 1021 4.4×1024 1.7x 1023 1.1 xl021 6.6 x 1024

system 221Fr~ 217At q- te 221Ra ~ 217Rn* + a 222Ra_, 218Rn + a 223Ra ~ 219Rn* q- o/ 224Ra ~ 22°Rn + ~ 225Ac~ 221Fr+ a 226Ra--~222Rrl + a 23°Th-. 226Ra + a 231Pa---)227Ac* + a 232U ~ 228Th + a 233U---~229Th + t~

calc(s) T,/: 3.7x 102 />4.5×101 6.8 x 101 5.4 x 105 7.6 × 105 8.0×105 1.7 × 1011 1.1×1013 2.2× 1011 9.5×109 1.6 x 1013

~

(s)

3.5×102 8.5x 101 3.9 x 10 I 1.9 × 106 3.4 x 105 1.7x 106 5.3 x 101° 3.1x 1012 3.9x 1012 3.2×109 5.9 >( 1012

spectroscopy it can be affirmed that in very m a n y actinide nuclei the alpha decay does not go 100% to the daughter ground state. Nevertheless, this state is still the most heavily populated in the decays of even-even nuclei. In the odd-mass actinide nuclei, however, it is quite c o m m o n for an alpha decay to proceed predominantly to one of the low-lying excited states. These favoured states are the most likely candidates for the core states in our picture of an alpha particle orbiting a core to form the parent nucleus. There are three cases (indicated by asterisks on the daughter nuclei) in table 1 where this has to be allowed for. For them, we have calculated alpha-decay half-lives using Q-values appropriate for transitions from the parent nucleus to the most heavily populated excited state of the daughter nucleus. We assume that the lower lying daughter states are less favoured because of the incompatibility of their nuclear sructures with that of the parent ground state. In the experimental half-life column of table 1, we have accordingly quoted alpha half-lives corrected by the branching ratio for decay to the single daughter state which is dominantly populated in the decay (be it the ground state or an excited state) for comparison with our calculations. We can see that this procedure has generally given us agreement between the half-lives obtained from our alpha decay calculations and those obtained from experiment to within a factor of ten, irrespective of whether the parent nucleus is even-even or odd-even. This is about the level of predictability that we expect, and strongly suggests that the o d d - e v e n exotic decays need to be re-examined to see if they might also exhibit this behaviour. We address this task in the next section.

3. Careful examination of odd-even exotic decays Various experimental searches have been made for exotic decay modes from o d d - e v e n nuclei, on the basis of model expectations that they should be readily

488

B. Buck et al. / Exotic decays

detectable. It is therefore significant that although no exotic decays have actually been observed from 22XFr, 221Ra or 225Ac, lower limits for the half-lives have been set which are considerably longer than most predictions 14) (including our own in table 1). Three odd-even nuclei have definitely been observed to decay exotically, (namely 223Ra, 231pa and 233U), and all of them have measured half-lives substantially longer than our predictions. To resolve this problem, we suggest that all 6 of these nuclei may be decaying predominantly to a low-lying excited state of the daughter nucleus. Our idea is, that when we remove the exotic cluster from the parent nucleus, the state of the core left over may be quite different from that of the ground state of the daughter nucleus, but rather similar to that of one of its excited states. Hence, decays would go preferentially to this particular excited state, and be strongly suppressed to the ground (and other) states. Our model is not microscopic enough to predict, a priori, which excited state may be most appropriate for this role, but since it assumes that there should be no hindrance effects due to nuclear structure, we hope to be able to make some suggestions on the basis of Q-values alone. We have already mentioned that the measured kinetic energy of the emitted fragment is not known well enough to rule out the possibility that excited daughter states are being preferentially populated. In order to test our hypothesis, we therefore repeat our calculations so as to obtain half-lives appropriate for 100% decays to each of the several low-lying excited daughter states for these 6 odd-even nuclei. Our results are presented in table 2, and will now be discussed in more detail. There are three cases for which the exotic half-life has been accurately measured, and for each of them the calculated value is about fifty times too small if the decay is assumed to populate the daughter ground state preferentially. For the decay 223Ra~ 2°9pb+ 14C we have firm evidence of an 81% branch to the ~+ (0.779 MeV) state of 2°9pb [ref. 6)]. Our calculated half-life for this decay is 1.6 x 1015 s, in excellent agreement with the experimental value of 2.0 x 1015 s (obtained as the 81% branch of the exotic half-life of 1.6x 1015 s). Similarly, for the decays 231pa~2°VTl+24Ne and 233U ~ 2°9pb + 24Ne, we can again obtain excellent agreement with experiment by assuming that the decays populate preferentially the ~ - (1.34 MeV) and ~ (1.423 MeV) states of 2°7T1 and 2°9pb, respectively. For the remaining three cases in table 2 only lower limits on the exotic half-lives have been set. Theoretical half-lives obtained by assuming preferential population of the daughter ground states are more than an order of magnitude smaller than these lower limits (we can safely include here the decay 221Ra~ 2°7pb + 14C, where we have tested the effect of a variety of plausible values for the unknown spin of 221Ra). Once again, consistency between theory and experiment can be restored if we assume that low-lying excited states of the daughters are populated preferentially. To summarise, our model strongly suggests that odd-even exotic decays may be proceeding preferentially to excited states of the daughter nucleus because of nuclear structure similarities between the favoured state and that of the core. We may well

B. Buck et al. / Exotic decays

489

TABLE 2 Calculations of exotic decays to excited daughter states for odd-A nuclei (The parent spin-parity value is indicated in square brackets) System

J ~ (E*, in MeV) of daughter state

c,l~ Tt/2 (s)

221Fr [3 (-)] ~ 2°7T1+ t4C

½+

(0.00)

2.3 × 1013

(T~I~ > 5.9 × 1015 s)

3+ ~-

(0.351) (1.34)

8.6×10 '3 1.3 × 1016

_5+ 2 ?

(1.67)

5.0×1016

(2.69)

/> 1.1 × 1019

221Ra [.9] -) 2°Tpb + 14C

l--

(0.00)

/> 1.4 × 1012

(T~I~p>2.2x1014s)

3~2

(0.5696) (0.8977)

/>1.8×1013 1>7.8 × 1013

~+

(1.633) (2.340)

/>2.4×10 Is ~>7.4x 1016

I+

(2.624)

/>3.1 x 1017

223Ra [1(+)] ~ 209pb + 14C

9+

(0.00)

expt ---1.6X10 15 s) (T1/2

!~+ ~-

(0.779)

1.6 X l0 ts

(1.423)

4.2×1016

5+

t+ t 2

(1.567) (2.032) (2.152)

2.4× 1016 2.0× 1017 3.2x 1017

(3-)

(0.00)

(7) ?

(0.4048) (0.7663)

225Ac [(3-)] -> 2ttBi+ '4C exp (T1/2>2.2×10

18

s)

(29-) 9

2.3 × 1013

1.4× 1017 7.0×1017 ~>3.7 × 1018

(0.8318)

9.3 x 1018

(0.951)

~>9.6× 1018

231pa [3 ] ~2°TTl+24Ne

2!+

(0.00)

2.2x 1021

( T]I~ = 1.7 x 1023 s)

3+

(0.3510)

7.5 × 1021

121-

(1.34)

4.0 × 1023

5+

(1.67)

9.0× 1023

?

(2.69)

~7.0 × 1025

233U [5+] ..~ 2o9pb + 24Ne

29+

(0.00)

4.5 x 1022

(oxp T1/2 = 6.6 × 102`*s)

~+

(0.779) ( 1.423) (1.567) (2.032) (2.152)

9.2 × 1023 9.9 x 1024 1.3 x 1025 8.1 x 1025 1.2 × 1026

t2s 3+ ½+ ½-

490

B. Buck et al. / Exotic decays

imagine that the presence of a large cluster could polarise the core, and lead to changes in the ordering of the single-particle orbitals from that obtained in the free state. However, we can certainly say that the situation offers a very interesting challenge to more microscopic models, and to experimentalists, to try to confirm and explain this change in level ordering. It seems, that against all previous expectations, we may at last have an opportunity to extract nuclear structure information from exotic decay data.

4. Conclusions

We have compared the predictions of our cluster model with the experimental half-lives (or lower limits) for all the known 14C and 24Ne decays from heavy nuclei. For even-even nuclei the situation is very satisfactory, and we can reproduce the experimental data to within a factor often, and usually very much better. By contrast, the o d d - e v e n decays are relatively badly reproduced if it is assumed that they go to daughter ground states, i.e., we systematically underestimate half-lives, or violate experimental lower limits, typically by factors of 50-100. Our calculated results are, of course, extremely sensitive to the energy released in each decay. We have shown that the above-mentioned discrepancies can be remedied if all the o d d - e v e n exotic decays go preferentially to low-lying excited states of the daughter nucleus. Because of the sensitivity of our calculations (and all others) to the Q-value, and in view of the experimental limits on the resolution of the kinetic energies of the recoiling fragments, we feel that this is a real possibility. Furthermore, a concrete example has recently been demonstrated experimentally in the case of 223Ra-->2°qPb+ 14C [ref. 6)]. The agreement between our model and the data for odd-even nuclei would then be brought to the same high level as for even-even nuclei. The implication of such a state of affairs is much greater than that of merely rescuing our model from systematic disagreement with the data. It is possible, contrary to all previous expectations, that we may be able to extract nuclear structure information from exotic decay data. There seems to be a distinct chance that the single-particle orbitals filled by the daughter (core) nucleons are distorted and change their ordering due to polarisation effects caused by the cluster. This, in turn, influences which excited state of the daughter nucleus is predominantly populated. Many questions remain to be answered concerning the universality of this phenomenon, and whether different clusters cause varying amounts of core polarisation, so that different excited states of the same daughter nucleus are preferred when different clusters are emitted. A.C.M. thanks the Brazilian Conselho Nacional de Desenvolvimento Cientffico e Tecnol6gico (CNPq) for partial financial support.

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491

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