A model for the decay out of superdeformed bands

Nuclear Physics A520 (1990) 179c-186c North-Holland

A MODEL FOR THE DECAY OUT OF SUPERDEFORMED BANDS E. VIGEZZI, R.A. BROGLIA and T. DOSSING The Niels Bohr Institute, University of Copenhagen, DK-~IO0 Copenhagen, Denmark and Dipartimento di Fisica, Universiti~ di Milano and INFN Sez. Milano, Italy

The decay out of superdeformed, rotational bands is studied in terms of the spreading of the superdeformed strength on states corresponding to normal configurations. The mixing of the superdeformed state depends on the tunneling through the potential energy barrier separating the two shapes. Simple, yet accurate estimates of the decay probability are presented in terms of the transmission coefficient of the barrier, and the electromagnetic width and level density of the normal states.

Abstract:

1. I n t r o d u c t i o n Several superdeformed rotational bands have been observed during the last few years. Among other properties, they are characterized by a fast drop in intensity, that takes place over a few transitions. With the exception of 15°Gd, the transition energies are very regular along the decay, as is testified by the constancy of the moments of inertia. Moreover, the fact that the decay does not take place in a single transition, makes it difficult to interpret the decay as due to a resonance phenomenon. We propose to consider the decay at each spin as due to the mixing of the superdeformed state with states at normal deformation. This coupling is due to the transparency of the potential energy barrier separating the superdeformed and normal minima. The exponential increase in the transmission coefficient with lowering angular m o m e n t u m should account for the fast decay. The spectrum of normal deformed states, to which the superdeformed state can couple is expected to be very complex, due to the rather high excitation energy above yrast at which the depopulation takes place. A statistical t r e a t m e n t of the mixing process and decay is then justified. This approach should also be able to describe fluctuations in the intensities, superposed to the generic behaviour common to all superdeformed bands. The decay of superdeformed bands has several points of contact with the phenomenon of v-decay of shape isomers, found in actinide nuclei. In that case, however, the electromagnetic decay to normal deformed configurations competes against fission; in the present case it competes instead against the collective E2 decay along the superdeformed band. As the decay extends over several transitions, important information on the mixing and on the transmission coefficients as a function of angular m o m e n t u m can be extracted from the data. The results can be used to test 0375-9474/90/$03.50 ~) 1990 - Elsevier Science Publishers B.V. (North-Holland)

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E. Vigezzi ef al. / Decay our of superdeformed bands

barriers and inertial parameters of the large amplitude motion connecting superdeformed and normal shapes, predicted by nuclear models. We first describe the decay in simple terms, neglecting the spreading of the superdeformed strength, and show the dependence of the decay on the transmission coefficient. We then consider the decay as an admixture problem, and deduce some analytic expressions for the average intensity, which put in evidence the relevant parameters that determine the depopulation process. Finally, we briefly report on the results of a comparison of the model with the experimental data [cf. also ref. 1)].

2. A simple formula At a given spin, the superdeformed state is viewed as a zero-point vibration at an energy Es = 2x-hWsin a well, separated from the region of normal deformation by a potential energy barrier along the quadrupole deformation f12. The simplest approach is to consider the tunneling through the potential barrier as a decay mode in itself [cf. ref. 2)]. The electromagnetic width for collective quadrupole transitions along the superdeformed band is denoted by Fs. The tunneling width I'tunn, according to WKB theory, can be written as Ftunn = hwsT/27r,where T is the transmission coefficient. On the average, the fraction of nuclei leaving the band at spin I, N-out(-/'), is then given by No~,(I) = rt~n---2"-

Ftotal

rtunn

(1)

Ftunn + Fs

The relative intensity in the band can be written as Int (I) = Int (I + 2)(1 - No.t(I)).

(2)

We expect that Ftunn will display an exponential dependence on angular momentum, since T ~ exp (--(Eb -- Es)/hWb), Eb and Wb being respectively the height and the frequency of the barrier; the latter quantities are expected to have a smooth dependence on angular momentum. One then writes

rtunn = Ce ~I,

(3)

and determines, using eqs. (1) and (2), the values of C and cr that give the best fit to the experimental intensities. The resulting values of a associated to several superdeformed bands are listed in column 2 of table 1. In the case of 152Dy it was pointed out in ref. 2) that the dependence of the transmission coefficient on angular momentum requested to explain the corresponding value of er is much stronger than the one displayed by the energy barriers calculated by standard theories, keeping the inertial parameter constant. In this case agreement with experiment could only be achieved taking into account the changes in inertial mass due to the increase of the static pairing gap with lowering angular momentum.

E. Vigezzi et al. / Decay out of superdeformed ban&

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3. A description of decay through a d m i x t u r e The model discussed above does not take into account explicitly the coupling of the superdeformed and normal states. Moreover, it neglects the fact that the decay from the superdeformed state also implies the electromagnetic decay from normal states, and that therefore also the related electromagnetic width should enter into eq. (1). A more detailed treatment of the admixture requires a treatment of the complex spectrum of states in the first well. A reasonable description is provided by the Gauasian Orthogonal Ensemble (GOE). According to the GOE, nuclear levels can be obtained as the eigenvectors [A) of a symmetric matrix A whose elements follow a Gaussian distribution, such that (A,?i) = 2a 2 and (A~i) = a 2 for i # j. The variance a 2 can be adjusted, so as to give the desired average distance D , between normal states. The average level distance at the center of the ensemble is given by Dn = 7ra/v/-N, where N is the dimension of the matrix A [cf. e.g. ref. 4)]. The states [A) have a width due to statistical and rotational electromagnetic decay, whose average value we indicate with F , . Typically, F,, is expected to be somewhat smaller, but of the order of, F,. We then place the superdeformed state [0s) at the center of the ensemble, and turn on the coupling v between [0s) and the states [A). The resulting eigenvectors, [m) will have an overlap c,n -= (m[0s) with the superdeformed state. Each state Ira) will possess two widths. One for decaying into the superdeformed and one for decaying into normal configurations, given respectively by rs(m) = [cm[2Fs, and Fn(m) = (1 -[c,,[2)F,,. Let us now assume that the superdeformed cascade reaches spin I. We assume that the flux populates the state [m) with a probability Pfeed(rn) given by its superdeformed content, Pteea = [cm[2- The width of the state [m), due to electromagnetic decay to normal configurations, is equal to ( 1 - [cm [2)F,,. Once it has been populated, the state Ira) will have a probability P,,(rn) to decay to a normal state, leaving the band, given by

(1 -l~l~)r. P,,(m) = (1 -Ic.~12)r. + IcmlW,

(4)

The total probability is obtained summing over all the states: Nout(I) = E

Pf~¢d(m)P,,(m) = E Erl

Ic~12

(1 - I c , 1 2 ) r , (I -Ic~12)r. + Ic

rr,

(5)

Both the sum over m and the factor Pteed were absent in the simpler eq. (1), since in that formula it was supposed that the full superdeformed strength was concentrated in a single state. Since Nout depends on the particular realization of the GOE, it is a random variable, and one can describe fluctuations in the intensity around the average value 1). The quantities [cm [2 depend only on the ratio [v[/Dn between the absolute value of the coupling matrix element v and the average distance between normal states D , . Consequently the probability distribution of Nout depends only on this ratio apart from the ratio Fs,, -- F~/Fn. To complete the discussion, an expression for the coupling v is needed. This requires a more detailed picture of the mixing which is not available at present.

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E. Vigezzi et al. / Decay out of superdeformed bands

It is however possible to individuate the relevant physical parameters on which v depends, using the Fermi "golden rule" to obtain the width F due to the coupling between the superdeformed and the normal states. On one hand, the mixing takes place through the potential energy barrier, coupling the superdeformed state to neighbouring states in the first well of vibrational character. The relevant width is Ftunn, that was already used in eq. (1): F t . n n = ~.vs__~T 2r

(6)

On the other hand, the spreading width associated with the coupling of the state ]0s) with the states IS} is given by the Fermi golden rule: 2Try2 r

=

-

-

D,

(7)

Due to the rather large excitation energy, it is reasonable to assume that the vibrational states of the first well are in turn mixed thoroughly into the dense background of normal states IS}. One can then equate the two widths (6) and (7), obtaining

v2 - hwsTD,

(8)

47r2 A more elaborate discussion of eqs. (6) and (8) can be found in ref. 3) (cf. especially pp. 765, 769, and eq. (5.8)). We may expect that eq. (8) is at most valid on the average. In the calculations using the GOE, we consider v as a Gaussian random variable, whose variance is given by eq. (8). The different steps in the process of mixing and decay are represented in fig. 1. While the preceding treatment of the decay also allows for the evaluation of fluctuations in the intensities, the quantity of most immediate interest for comparison with the experiment is the average value Nout. In fig. 2 we show N o , t as a function of F/D, for a fixed value of the ratio of the electromagnetic widths Fs, = 2. It can be approximated rather accurately by simple expressions in the limits of weak (F/D, < 1) and strong (F/D, > 1) coupling. In what follows we briefly describe these two limits. A simpler description of the normal states can be given employing an equidistant level spectrum, with a distance between levels equal to D , , rather than the GOE. One must take into account the fact that when the coupling is weak, the superdeformed strength is spread appreciably only on an energy interval much smaller than the average level distance. Fluctuations in the distance between the superdeformed state and its nearest neighbour are in this case of decisive importance. Accordingly, one can shift rigidly the equidistant spectrum, and let the distance z between the superdeformed state and its nearest neighbour be randomly distributed, uniformly between 0 and D,/2. For each value of x, one can diagonalize the coupling, as in the GOE case, calculate the overlaps c,, and obtain No,t. We see from fig. 2 that the resulting average over x approximates well the GOE result for Nout in all the range of F/D,.

E. Vigezzi et aL /Decay out of superdeforyned bands

183c

a)

>

>

/ 0.2

c)

012

0.6

1+2

016

d)

At _~-.

p' out

-'I I-2

Fig. 1. (a) The potential energy barrier separating the superdeformed minimum (/~2 ~ 0.6) from the normal minimum (/~2 ~ 0.2) is shown, together with the vibrational states in the two wells; the distance between vibrational states is of the order of 1 MeV, while the energy of the superdeformed minimum is about 4 MeV above the normal minimum, at the spins relevant for the superdeformed band decay. (b) The wavefunction ~bcorresponding to the zero-point vibration in the superdeformed well is shown; it has a tail in the normal well, allowing the coupling of the superdeformed state with the complex spectrum of compound states in the first well (dashed), whose average distance is of the order of 10 eV. (c) The eigenstates resulting from the coupling of the superdeformed state with the compound states at spin I possess a superdeformed component (full) and a normal component (dashed). The probability to populate one of the eigenstates in the superdeformed, collective rotational decay from spin I + 2 is proportional to its superdeformed component. (d) Part of the flux coming from spin I + 2 will decay to spin I - 2 by a collective transition (full arrows), and part will leave the superdeformed band, decaying to normal states (dashed arrows), according to eq. (5).

As a f u r t h e r a p p r o x i m a t i o n , valid only i n the weak coupling limit, one can neglect the c o u p l i n g of the s u p e r d e f o r m e d s t a t e with all the other states, b u t the n e a r e s t n e i g h b o u r . O n e has t h e n a simple two-level p r o b l e m t h a t c a n be solved exactly. Using eq. (5), for a given distance z b e t w e e n the two states, the q u a n t i t y Nout(z)

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E. Vigezzi et al. / Decay out of superdeformed ban&

0.0

J ¢xO 0

-1.0

-3.0

- 1.0

-2.0

0.0

1.0

2.0

log (r/D) Fig. 2. The logarithm of the average value Tout is shown as a function of the logarithm of the ratio F/D, as calculated in the GOE model (continuous Line), in the equidistant level model (dashed line), in the two-level model, cf. eq. (12) (dash-dotted line), and in the Breit-Wigner approximation (dotted line). turns out to be 1

Nout(x) = (1 + Fsn) + ( r . . / ( 1 + r,.))=~/v~'

(9)

where Fsn - F s / r . . Considering t h a t x is distributed uniformly b e t w e e n 0 and Dn/2, one can deduce the probability distribution of Nout, P ( N o u t ) . P ( N o u t ) is different f r o m zero only for a < Nout < b, with 1 a = ( r . . + 1) + F s . D U 4 ( F . .

+ 1)v 2;

1 b = F s . +-----~'

(10)

.

(11)

and is given b y

P(gout) = F ~ . + V

FZ

1 I~'1

1

D . No~t ~/Nout - ' ( F , . + 1)No2ut

E. Vigezzi et al.

/ Decay out of superdeformed bands

185c

From this distribution the average value Nout can be calculated. One obtains

--_1

i-~_~_~_(asin ( - l + Fs,D2,/4(l + F,,)2v2

This estimate is shown in fig. 2, for F/D, < 1. It gives a fair approximation to the GOE result. For v2(1 -t- Fs,)2/D~Y.. < < 1 one obtains from eq. (12)

No.t ~ ~

--

1

I F ~-~'.

(13)

It turns out from fig. 2 that eq. (13) describes the GOE result better than eq. (12) when F/D, is smaller than, but rather close to, unity. For values of the ratio F/D, larger than unity, one expects 5) that on the average the superdeformed strength s(E) should follow a Breit-Wigner distribution: 1

F

s(E) = 2~r(E - E.) 2 + (r/2)2"

(14)

Using eq. (14), we can calculate Nout from eq. (5), making the substitution

n

One then gets, for F , # Fs, the relation N--o.t =

F . . - ~/1 + ( 2 / . ) ( D . / r ) ( r . . - 1) ( r . . - 1 ) , / 1 + ( 2 1 , O ( D . I r ) ( r . . - 1)'

while for r , = Fs, Nout ~- 1 reduces to

D./~F. In the

(16)

limit of very strong coupling, eq. (16)

= 1 - ~-~rs..

(17)

From fig. 2 it is seen that for F/D, > 1 the values of No, t obtained from the Breit-Wigner distribution agree very well with the equidistant level model and give a good description also of the GOE result. One should consider, however, that in the case of the decay of the superdeformed bands, the strong coupling limit is of a rather academic interest, since in this case No,t ~ 1 and the band cannot be observed. As we have done in sect. 2, we can parametrize the ratio F/D, according to r/D. = ~e ~

(18)

and find the values of & and C that give the best fit to the data, using this time eqs. (2) and (5). This is done for 5 in column 3 of table 1 [cf. also ref. 1)1. Comparing column 2 and column 3 in table 1, it is seen that the values of the exponents determined by eqs. (1)-(3) are about one-half of those obtained in the more detailed

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E. Vigezzi e$ al. / Decay out of superdeformed bands TABLE 1

The values of the exponents a (cf. eq. (3)) and & (cf. eq. (18)), giving the best fit to the experimental intensities observed for the decay of superdeformed bands in several nuclei in the mass region close to 152Dy Nucleus 146Gd 149Gd lS°Tb 151Tb 151Dy lS~Dy

a -0.3 -0.4 -0.5 -0.5

& -0.7 -0.9 -1.0 -1.0

-0.3

-0.7

-0.8

-1.5

model. This can easily be understood, considering t h a t for not too strong coupling, eq. (1) gives N--out ~ r , , n n / r , , while from eq. (13) one gets N-.ut ~ X/Fton./Fs.D,,. T h e dependence of the electromagnetic widths on angular m o m e n t u m is weak, as c o m p a r e d to t h a t of the transmission coefficients, and can, in the first approximation, be ignored. One then obtains from eq. (1) d lnN-out ~ a, while eq. (13) leads

to A ln ou, Once the best value of & has been determined, it is possible to deduce, using eq. (6), the angular m o m e n t u m dependence of the transmission coefficient. These values can then be directly c o m p a r e d with nuclear structure calculations of inertial p a r a m e t e r s a n d potential energy barriers.

References

1) 2) 3) 4)

E. Vigezzi, R.A. Broglia and T. Dcssing, Phys. Lett. B, submitted B. Herskind eL al., Phys. Rev. Lett. 59 (1987) 2416 S. Bjcrnholm and J.E. Lynn, Rev. Mod. Phys. 52 (1980) 725 C.E. Porter, Statistical Theory of Spectra: Fluctuations, ed. C.E. Porter (Academic Press, New York, 1965) p. 3 5) A. Bohr and B.R. Mottelson, Nuclear Structure, vol. 1 (Benjamin, New York, 1969) p. 302