fulltext

Exotic cluster model of octupole deformed radium isotopes

) Theoretical Physics, University of Oxford - 1, Keble Road, Oxford OX1 3NP, U.K. (

2

) Department of Physics, University of Cape Town

Private Bag, Rondebosch 7700, South Africa.

(ricevuto il 25 Luglio 1997; approvato il 15 Ottobre 1997)

Summary. — Many properties of the isotopes222;224;226

Ra are well reproduced by a binary cluster model in which a14

C ion orbits the appropriate isotope of Pb. An inter-nuclear potential has been found which accounts for the exotic decay half-lives and also gives a good fit to the energy spacings and large E2 strengths in the ground state 0+

band and the low-lying 0,

band. Cross-band E3 and in-band E4 strengths are appre-ciable as well and a picture of the asymmetric shapes of the isotopes emerges naturally. Coupling a neutron to222

Ra then enables an almost parameter-free calculation of the odd-A nucleus223

Ra.

PACS 21.60.Gx – Cluster models.

PACS 23.20.Lv – Gamma transitions and level energies. PACS 23.70 – Heavy-particle decay.

PACS 27.90 – 220A.

PACS 01.30.Cc – Conference proceedings.

The even radium isotopes222;224;226

Ra all possess a 0+

ground state band interleaved by the levels of a low-lying 0,

band, though the level spacings in both bands are com-pressed relative to a pureL(L+1)rotational pattern. This interpolation of 0

+

and 0,

band states has previously been interpreted as indicating the presence of stable octupole deformations in these nuclei [1], so it is of interest to show that an exotic cluster model can account for their observed properties in a natural way, without such ad hoc hypotheses. Our original reason for using the cluster model was to study exotic decay modes of heavy nuclei in a simple way; but, surprisingly, we obtained good results for half-lives by assum-ing the emitted neutron-rich ions to have preformation factors of the order of unity [2]. This suggested that the model could also be used to explain other properties of nuclei in the actinide region [3].

In earlier work we introduced a cluster-core interaction that fits observations in nuclei ranging from8 Be=4 He+4 He to236 Pu=208 Pb+28

Mg [4]. We have modelled the radium isotopes in terms of the decompositions Pb+14

C [5] since they each decay by emitting

( 

)Paper presented at the 174. WE-Heraeus-Seminar “New Ideas on Clustering in Nuclear and

Atomic Physics”, Rauischholzhausen (Germany), 9-13 June 1997.

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5

Excitation Energy (MeV)

(T) (E) (T) (E) 14 12 10 8 6 4 0+ 2+ + + + + + + 13 11 9 7 5 1_3 _ _ _ _ _ _ (13_ ROTOR) (14+ ROTOR) 222

Ra

K

=0

K

=0

Fig. 1. – Comparison of the measured energies (E) of theK 

=0 

parity doublet bands of222

Ra with those calculated in our model (T). The rigid rotor energies of the 14+

and 13,

states expected from the 0+

–2+

energy separation are indicated by broken lines.

14

C and have similarB(E2) strengths, which are controlled by the cluster charges. Our

potential then takes the form

V(r)=,V 0  x f1+exp[(r,R 0 )=a]g + 1,x f1+exp[(r,R 0 )=3a]g 3  ; (1) with parametersV 0

= 756:4 MeV,x = 0:36anda = 0:73fm, as indicated by previous

experience. The values ofR

0are found by fitting the Pb+ 14

C ground state breakup en-ergies. They are all close to 6.75 fm. This interaction yields bands of states labelled by a global quantum numberG = (2n+L), wherenis the node number andLis the

an-gular momentum. The Pauli principle is satisfied approximately by ensuring that all the nucleons in14

C are above the Fermi surface of Pb, which in the present examples implies

G-values 70 and 71 for the 0 +

and 0,

bands.

With the potential thus completely specified we obtain good agreement, using a semi-classical method, with the measured half-lives, as shown in table I. Predicted energy levels in the lowest positive and negative parity bands of222

Ra are exhibited in fig. 1 and it is seen that we reproduce to good approximation the compression of the spectra [6] relative to rotational schemes based on the 0+

-2+

spacing. A remarkable feature of the radial wavefunctions of the states is that from 1 fm outwards they are all nearly the same; this is illustrated in fig. 2 for levels 0+

(2)10+

. The implicit result is that, like the well-known rotor model, each band is built on a fixed intrinsic state-function; but, unlike the rotor picture, our model also provides a dynamical theory of the effective moments of inertia together with a very natural account of gamma transition strengths [7].

In table II we list the calculatedB(E"; 0 +

observa-TABLEI. – Exotic decay half-lives for222;224;226 Ra. Exotic decay T calc 1=2 (s) T expt 1=2 (s) 222 Ra! 208 Pb +14 C 1:2110 11 (1:010:14)10 11 224 Ra! 210 Pb +14 C 8:1610 15 (8:252:22)10 15 226 Ra! 212 Pb +14 C 9:7010 20 (2:210:96)10 21 tions for226

Ra [8]. They are computed from our wave functions by means of the formula

B(E;0 + !)= 2+1 4  2  hr  i 2 ; where   = Z 1 A  2 +(,1)  Z 2 A  1 (A 1 +A 2 )  : (2)

We find reasonable agreement with experiment on including a modest effective charge

=0:25ein order to compensate for the likely underestimation of the amount of surface

peaking in the radial functions by our local potential model [9]. If we represent the nucleus in terms of a uniform charge enclosed in a surface defined by the radius parameterR = R 0 (1+ P    Y 0 ), withR 0 =1:2A 1=3

fm, then the impliedB(E")-values are calculable

as functions of the’s and hence the shape can be determined from our model (or directly

from experiment up to

4). The model result is shown in fig. 3 along with the shape used

by Sheline et al. [10] in their study of the structure of223

Ra. We emphasise that their results were obtained with parameters fitted to the properties of223

Ra while ours come from the binary model. There is a fair correspondence up to

4, as illustrated, but our

0.0 5.0 10.0 R (fm) -1.0 -0.5 0.0 0.5 1.0 Radial Wavefunction χL (R) 222

Ra

Fig. 2. – Calculated radial wave functions of208

Pb–14

C relative motion, representing states of222

Ra withJ  =0 + ;2 + ;4 + ;6 + ;8 + and10 + .

Fig. 3. – A comparison of the shape obtained by Sheline et al. [10] for223

Ra (dot-dash line) with that calculated in our model using2–4alone (full line) and2–8(dashed line).

model also gives non-negligible values for higher multipoles, leading to a more undulating shape.

TABLEII. –B(E";0 +

!)values for 226

Ra.

Transition Calculated Experimental

(W.u.) (W.u.) 0 + !2 + 4.12 5:150:14 0 + !3 , 1.72 1:100:11 0 + !4 + 1.35 1:080:15

The next logical step is to investigate the structure of an odd-A nucleus on the basis of our cluster ideas. The obvious choice is223

Ra since it has been studied intensively using the deformed rotor model [10] and there is a large database of experimental information [11]. We consider this nucleus to have the structure208

Pb+14

C+n and model it with a set of basis states made by coupling, to good total angular momentum, the states of the

14

C 0+

and 0,

bands of222

Ra and the single neutron states observed in209

Pb. The wave functions of the cluster motion are calculated as described above and energies and state functions for the neutron (1g

9=2, 0i 11=2, 0j 15=2, 2d 5=2, 3s 1=2, 1g 7=2 and 2d 3=2) are easily

fitted by a simple central plus spin-orbit potential [12]. These basis states are then used to construct matrix elements of the effective HamiltonianH =H(

208 Pb -14 C)+H( 208 Pb -n)+V( 14 C-n), assuming that 14 C and208

Pb remain in their ground states. The14

C–n interaction is determined by fitting the1s

1=2and 0d

5=2states of 15

C and in the model of

223

Ra it will appear as a function ofR 0

=jR,rjwhereRandrare the coordinates of the

cluster and neutron relative to208

Pb. Thus on expanding the14

C–n potential as V14 C,n (R 0 )= X V  (r;R )Y  (^r ):Y  ( ^ R); (3)

TABLEIII. – Energies of K =3=2  band members in223 Ra. J 

Ecalc(keV) Eexpt(keV) J 

Ecalc(keV) Eexpt(keV) 3=2 + 0.00 0.000 3=2 , 46.0 50.133 5=2 + 42.0 29.858 5=2 , 88.0 79.722 7=2 + 98.0 61.431 7=2 , 145.0 123.790 9=2 + 176.0 130.172 9=2 , 217.0 174.58 11=2 + 262.0 174.62 11=2 , 303.0 247.39 13=2 , 401.0 315.55

we see that the neutron moves in a deformed field. The basis states are mixed by the above interaction and the eigenvalues and eigenvectors of the Hamiltonian matrix provide possible descriptions of the bands observed in223

Ra.

The calculation is done in two stages. First the levels of222

Ra are taken degenerate, with the relevant wave functions all replaced by theL=0radial function as the effective

intrinsic state (see fig. 2). The eigenstates are grouped into a number of degenerate bands of variousK-values and parities. By plotting the calculated bandheads as functions of the

strength of the14

C-n potential it becomes easy to see the dominant parentage of each

K-band, as shown in fig. 4.

The odd neutron is not allowed to occupy the levels already filled by the eight neutrons in 14

C, which leads to blocking of all states in the lowest four parity doublet bands in the diagram. This predicts that the ground state band in223

Ra should haveK = 3=2,

0.0 0.5 1.0

Fraction of 14C-n potential strength -15.0 -10.0 -5.0 0.0 5.0 E (MeV) 1/2 3/2 1/2 1/2 3/2 5/2 3/2

}

Fig. 4. – Bandhead energies in223

Ra as functions of the14

TABLEIV. – Static moments for223 Ra.

Moment [J

(Energy; keV)] Calculated Experimental

[3=2 + (0:0)] 0.43 (n.m.) 0:27050:0019(n.m.) [3=2 , (50:133)] 0.44 (n.m.) 0:430:06(n.m.) Q[3=2 + (0:0)] 1.30 (e barn) 1:2540:003(e barn)

in agreement with experiment. On lifting the degeneracy of the14

C cluster levels we find that the low-lying spectrum of223

Ra contains slightly split parity doublet bands with

K  =3=2  ;5=2  and1=2 

in line with observation. Theoretical and experimental energy levels in the first two bands are presented in table III.

The calculated eigenstates are then used to evaluate numerous decay rates and static moments, involvingE1,M1andE2operators. We find a reasonable level of agreement

with the data and present our results for the static moments in table IV. We also predict that exotic decay of223

Ra by emission of14

C to the first excited state of209

Pb (with spin

11=2 +

) is structurally favoured over decay to the spin9=2 +

ground state. Although the enhancement we find is not enough to give quantitative agreement with experiment [13], the results of this first attempt are satisfactory.

In conclusion, we can say that our binary cluster treatment of heavy even-even nu-clei, and its extension to even-odd isotopes, continues the long and somewhat mysterious tradition of success for this extremely simple type of model [14, 15].

  

ACM and SMP thank EPSRC, and in addition, SMP thanks the S.A. Foundation for Research, and the University of Cape Town, for financial support.

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