Shape isomerism in mercury isotopes

Nuclear Physics A245 (1975) 205--220; ~ ) North-Holland Publishing Co., Amsterdam

Not to be reproduced by photoprint or microfilmwithout written permissionfrom the publisher

SHAPE ISOMERISM

IN MERCURY

ISOTOPES t

D. KOLB tt and C. Y. WONG Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830

Received 20 March 1975 Abstract: We study the energies of the mercury isotopes in their oblate, prolate, and bubble shapes within the framework of the single-particle K-matrix theory. A change from an oblate ground state to a prolate ground state is observed when the mass is decreased from 184 to 182. Self-consistent stable bubble configurations were also obtained for these isotopes. They lie a few MeV from the ground deformed state, coming down lower for the mass region around A "~ 200 and 182. For the isotopes of ~SSHg, laeHg, and la2Hg, the oblate and prolate minima are nearly degenerate; a strong mixing of the prolate and oblate configuration is expected. This mixing is occasionally enhanced for states with non-zero angular momentum when their energies become accidentally near-degenerate due to their differences in moments of inertia. We study such hybridization of shapes in terms of a phenomenological two-center collective model. Good qualitative agreement with experimental energy spectra and branching is observed, indicating a decrease of the prolate-oblate energy difference as the mass number decreases from 188 to 184.

1. Introduction T h e s t u d y o f the n u c l e a r p r o p e r t i e s o f the m e r c u r y i s o t o p e s has recently g a i n e d w i d e s p r e a d theoretical a n d e x p e r i m e n t a l interest 1-15). T h e a n o m a l o u s increase in the rms r a d i u s going f r o m 1s 7Hg to ~s SHg o b t a i n e d in optical p u m p i n g e x p e r i m e n t s ~) suggests s t r o n g l y a t r a n s i t i o n o f g r o u n d state shapes. E x p e r i m e n t s in the even m a s s isotopes reveal t h a t even t h o u g h t h e s u d d e n decrease in t h e energy o f the first excited 2 + state does n o t o c c u r a t m a s s 184, the s p a c i n g between a d j a c e n t excited levels a r e r a t h e r peculiar. R e c e n t experiments o n the energy levels o f ~8SHg show the a p p e a r a n c e o f t w o s e p a r a t e b a n d s with s t r o n g m i x i n g 6). A l l the e x p e r i m e n t a l facts p o i n t t o a n u n u s u a l t r a n s i t i o n in the light m e r c u r y isotopes. T h e o r e t i c a l w o r k o n m e r c u r y i s o t o p e s has been p e r f o r m e d with m a n y different m e t h o d s a n d emphasis. S t r u t i n s k y - t y p e calculations ~) using d e f o r m e d single-particle states, as well as H a r t r e e - F o c k calculations with S k y r m e interactions s), indicate a c h a n g e o f g r o u n d state d e f o r m a t i o n f r o m a n o b l a t e s h a p e to a p r o l a t e shape as the mass n u m b e r decreases. T h e exact l o c a t i o n where this t r a n s i t i o n occurs is, however, different f r o m the t w o calculations. F a e s s l e r et al. t o o k the g r o u n d state shape o f 1S4Hg to b e p r o l a t e even t h o u g h t h e y f o u n d b o t h o b l a t e a n d p r o l a t e m i n i m a t o lie at the t Research sponsored by the US Atomic Energy Commission under contract with Union Carbide Corporation. tt Permanent address: GSI, Darmstadt, Germany. 205

206

K. KOLB AND C. Y. WONG

same energy; the transition of ground state shape was taken to occur between ~s6Hg and la4Hg. On the other hand, Cailliau et aL a) found lS6Hg to be prolate while 18aHg oblate. Obviously, the exact location of the transition depends delicately on the single-particle spectrum. Because of the strong polarizing tendency of the neutron state 1521 K = ½) at the top of the Fermi energy, as pointed out by Nilsson et aL 14), it is not surprising that one can end up with different conclusions on the exact onset of the transition. For the same reason, it is also not unexpected that the onset of oblateprolate transition occurs at slightly different masses for the even- and odd-mass isotopes. In the heavier mercury isotopes, recent theoretical interest has centered on density non-uniformity in nuclei 9-14). If one considers the particular configuration of 2°°Hg by removing the topmost s½ protons and the p~ and p~ neutrons from the closed shell 2°spb, the self-consistent field which sets in has a tendency to cause an even deeper central density depression (the bubble configuration), as is indicated by theoretical calculations with different interactions. Recently, a new formulation of the single-particle K-matrix theory was put forth as a very promising method to study properties of nuclei in different configurations 16,17). Applied to a large number of nuclei throughout the periodic table, the theory predicted the correct binding energies, rms radii, single-particle spectra, fission barriers and secondary minima, etc. It is of interest to extend the K-matrix theory to study the Hg isotopes in order to assess further the success of the theory when compared to experimental results. Furthermore, as previous studies of deformed Hg isotopes do not include the spherical bubble configuration while the studies of bubble shapes do not compare them with deformed configurations, we shall study all three shapes in these nuclei, thereby providing us a general idea on the energy range where these bubble states may be looked for. Finally, we wish to understand the peculiar branching of the high angular momentum states in the light Hg isotopes in terms of a phenomenological two-centered collective model. The differences in the heights of the two minima can then be deduced and compared with results from the single-particle K-matrix theory.

2. Single-particle K-matrix calculations The detailed formulation of the single-particle K-matrix calculation for the evaluation of the potential energy surface and equilibrium minima has been presented previously 16,17). We shall summarize briefly the results. By simplifying Brueckner's two-body reaction matrix in the Fermi gas representation, one obtains a singleparticle Hamiltonian of the form

e(k,ko) = h2k2/2m-½Vo[v(k)~(ko)+~(ko)v(k)]+ A(ko),

(1)

where ko is the Fermi momentum. The functions if, v and A are

1 v(k) - 1 + ( a k y '

(2)

207

SHAPE ISOMERISM IN Hg 1 - - ( k o / k l ) 3v

fi(k°)= (~) 31_(kflkl)3, ' A(ko)

(3)

1 t~fi(ko)1

= ½Vo rv(ko)fi(ko)-to(ko) I_

gJ'

(4)

with Fo(ko) =

fOkZdkv(k),

(5)

and k, is the Fermi momentum for nuclear matter at equilibrium. The total energy of the finite nucleus is then

W = E (h2/2m)k2n,+½ ~, ni nj(-½Vo)[v(k)fi(ko)+ fi(ko)v(k)], i

(6)

i.i

where n~ is the occupation number for the ith single-particle state and calculated by a modified BCS formalism re. 17). TABLE 1 Parameters for the effective interaction and related nuclear matter properties Vo

a

Px

2~

C

Eb

A(k®)

~

k,

~:o

96.49

0.5124

0.2476

0.913

26.0

15.95

13.0

303

1.33

0.4

Vo in MeV, a in fm, Px = (3rr2) - tktS in fm -3, ~ is a folding length in fro, C the spin-orbit strength in M e V ' fm, Eb = binding energy per nucleon in nuclear matter in MeV, A in MeV, k in MeV = compressibility in nuclear matter at the Fermi momentum k,, ke in f i n - t , ro = isospin coupling dimensionless parameter 16. iT).

The actual parameters for the effective interaction and the related nuclear matter properties are given in table 1. The corresponding Pb single-particle spectrum is shown in fig. 1, together with that for the spherical configuration of 2°°Hg. One notices that the low-spin states in 2°°Hg lie relatively higher, indicating that this configuration has a bubble structure as shown in the density in fig. 2. As we see from fig. 3, this configuration is, however, not the ground state of Z°°Hg. The density of the ground state, which is slightly oblate, is shown in fig. 4. The required excitation to generate the bubble shape isomer, however, is quite low and strong admixtures of this configuration even to the low-lying excited states in 2°°Hg may be expected. From the systematics in fig. 3, we may also identify the spherical bubble configuration as the first excited shape isomer of 2°°Hg. In fig. 5 we show the deformation of the oblate and prolate configurations varying with the mass number. In fig. 6 the mean square radii of spherical (bubble) oblate and prolate shapes are shown. From fig. 3 we may infer which configuration is the actual ground state. Whereas the deformation as well as radii show a very smooth

208

K. KOLB AND C. Y. WONG Protons

Neutrons E s p. I MeV]

~Hg z~pb

ZOHg2o8[:xo

f 5/2 - - - - _ _

i 'o~2 f 712 h912

d3/2 0

s ~t2

d 3~2-=~-..5 h 11/2 d 5/2

-10

g7/2 ~

-15

g 9J2 P ~12 ~ P 3/2

-20

f 5/2 f 712 ~

p 112

J 1512 g 912 __JJll2

p3/2__~ f 512 I 1312 --"-'-"--f 112 h 9/2 $1/2 d 3,'2 h11/2

-25

g 7/2 -:30.

S 112

-35. d 312 d sl2 -,

Sl/2 g ~ 2 ~ d s~2

g 912 p I/2

p3/2

f5/2

Fig. 1. Single-particle spectra of 2°SPb and 2°°Hg* (spherical configuration).

p [fn~3] Q15

2OOHg* 0.1

(105

r [fro]

Fig. 2. Density of the spherical bubble configuration in Z°°Hg*.

behavior with mass numbers, our results predict a change from an oblate to a prolate ground state as the mass decreases from 184 to 182. Experimental evidence from the isotope shifts strongly suggests that prolatv deformation sets in at ~SSHg, whereas experimental spectra of ~SSHg, 186Hg and 184Hg reveals that the ground state of ~S4Hg remains mostly oblate. From the systematics, the nucleus ~S2Hg, whose spectrum is not yet observed may be the first even-mass Hg isotope to become prolate as the mass decreases. This is in accord with our theoretical results for the even-mass isotopes. The difference between the even and odd isotopes can be understood in terms of the single-particle spectrum presented in figs. 7a, b.

SHAPE ISOMERISM IN Hg

12 .

.

.

8

.

.

.

.

.

.

.

209

.

. . . .

z~ ',' 4

~

L

180 182 184 186 188 190 192 194 196 198 200 202 A(Hg)

Fig. 3. Differences in binding energy of the prolate and spherical bubble configurations with respect to the oblate ones.

2°°Hg

Z [fm] 7

ground state

Protons

6. 5.

\

4.

3

2

1 Rlfm] 0

1

2

3

4

i

i

i

L

5

6

7

8

IlL

Fig. 4. Two-dimensional density contour lines o f the ZO°Hg oblate ground state. Axial symmetry is assumed with respect to the z-axis, reflection symmetry with respect to the R-axis.

For the light Hg isotopes with neutron numbers less than 102, the ground states are obtained by populating neutrons in states with low K-values arising from the spherical i~ orbital and the 1651 K = ½> state which have strong tendencies to polarize the nucleus towards a prolate shape. For even-mass isotopes with neutron numbers greater than 102, the ground states are obtained by populating neutrons in those states of i¥ which have large K-values, and a tendency to polarize the nucleus to-

210

K. K O L B A N D C. Y. W O N G

- deformation 0.4 0.3. 0.2' 0.1 A [Hg]

0.0

;8/,

-010.

188

196

192

200

:___.___._._.-----

¢

-020, x prolate Minimum ® oblate Minimum

Fig. 5. Deformations of the prolate and oblate configurations varying with the mass nmnber.

5.70

.~-----K

560

/

550

/

rp :fr.l 5./.5

~

x

~

\,

535

180

J~"/ 9"

~,I"

.....

• oblate

~ . ~

x prolate

~

• bubble

I

18/*

~ 1~ ~"

192

196

200

(

20/~

A [Hg]

Fig. 6. Mean square charge radii for spherical, oblate and prolate configurations as a function o f mass number.

wards the oblate shape. Because of pairing interactions, many states at the top of the Fermi energy contribute in this polarization for the even Hg isotopes. For the oddmass isotopes the odd and unpaired neutron may specialize to specific single-particle states, thereby providing extra weight to the polarization due to these states. For tS3Hg and is SHg ' the configuration of a prolate core with an odd neutron occupying the 1521 K = ½> state, which appears at the top of the Fermi energy and has a strong

SHAPE ISOMERISM IN Hg

211

~a~Hg Protons Esp[MeV] ~ h a 2

-15

|

- 0.2

- 0.1

0

(11

0.2

0.3

Fig. 7a.

~84Hg Neutrons Es IMeV] 0 -

~2

-5

-10

- (12

- 0.1

0

0.1

0.2

0.3

-defornmtion Fig. 7b. Fig. 7. (a) Proton s.p. spectra for ~a*Hg; (b) neutron s.p. spectra as a function of the fl-deformation [ref. 16)] for la4Hg.

212

K. KOLB AND C. Y. WONG

polarizing tendency towards the prolate shape, apparently has a lower energy than the configuration o f an oblate core coupled to an oblate orbital. These two configurations should, however, be very close in energy. For laTHg, this [521 K = ½) is probably so deeply embedded under the Fermi surface that similar coupling does not lead to the ground state. Thus, the difference in the onset o f prolate deformations in the even and odd-mass Hg isotopes can be attributed to the specialization of the singleparticle states in the case of the odd-mass isotopes.

3. Phenomenological two-center collective models The results obtained in the previous section indicate a strong coupling in the collective motion for the light Hg isotopes. The coupling is particularly important when the two minima are nearly degenerate with respect to each other. Such accidental degeneracies may occur for the 0 + states built on the two shapes. In cases where the minima for the 0 + state are quite far apart, near-degeneracy may also occur for states with non-zero angular momentum as a result of the difference in their moments of inertia at the different minima. It is of interest whether the experimental data for the light isotopes are consistent with such a picture of a strong coupling between the two configurations. For this purpose, a complete analysis using the full collective model does not seem warranted in view of the scanty experimental data available. There are also theoretical difficulties associated with the quantization of a Hamiltonian with a variable effective mass. We should instead attempt to analyze the experimental data of the excitation spectra and transitions in terms of a simple one-dimensional twocenter model with a constant effective mass. It is assumed that the collective deformation energy surface contains a minimum in the oblate degree of freedom and a minimum in the prolate degree of freedom. One wishes to focus attention along a cut in the fl-~ plane joining the two minima and to study collective levels arising from collective motion in this degree o f freedom. Along such a one-dimensional cut, the collective potential can be described by a parabolic potential well situated at ~, = - 6 0 ° and a height of hi with a vibrational frequency hco1, and another potential well situated at ~, = 0 ° and a height of h2 with a vibrational frequency of hco2 (fig. 8). Explicitly, the collective Hamiltonian we adopt is of the simple form

n = _1 p2_ +

(7)

2B where p~ is the collective momentum conjugate to ~ and B is the effective mass. The potential is taken to be of the form

~½Bo~(~,-~h)2+h~ =/½8o, where ~,~ =

for~, =< ~

(8)

for

(9)

____

3 , ~2 = 0 and ~,. is the location where the two parabolas meet. To

SHAPE ISOMERISM IN Hg

213

>

1

(OBLATE)

(PROLATE)

Fig. 8. Shape of potential assumed in the phenomenological collective model.

make the problem simple, we do not even have to smooth out the kink between the two parabolas as it will not affect the main features of the results. We further assume that the effective mass parameter B is a constant. As a function of the angular momentum, this potential surface will change according to the increase in rotational energy at each point along the cut. We know, however, that for the mass region around A ~ 182, there should be a big difference in the moments of inertia at the two different minima. This can be readily observed by considering the single-particle states around the Fermi energy at the two minima. For the prolate minima as the states from the higher shells come close to the Fermi energy forboth neutrons and protons, the number of single-particle states contributing to the moment of inertia is large. By comparison, for the oblate minimum, the deformation is only of the order/~ ~ -0.15, so the states in the next shell are appreciably far from the Fermi level, being separated by the 82 (or 126) proton (or neutron) gaps. As a result, in the oblate minimum the number of single-particle states contributing to the moment of inertia is relatively small. This reduction is even all the more drastic for the proton contributions as only one proton state lies near the Fermi energy below the magic 82 single-particle gap. One expects that the moment of inertia at the oblate minimum is many times smaller than the moment of inertia at the prolate minimum. Our limited knowledge of the moment of inertia does not allow an extensive extrapolation on the energy surface for different deformations. We shall instead assume that the effect of the angular momentum is to shift the height of the oblate potential well from hao to hll = hxo+ 1(I+1)~i2, 2.f, and the prolate potential well from h2o to

h2t = h2o+ I(I+1)h2,'" 2J2

(10)

214

K. KOLB A N D C. Y. W O N G

while all other potential characteristics such as hoot and htO 2 remain the same. In this simple model, knowledge of h2 0 - h t o, J r , o¢2, hcox, hco2 and B allows a prediction of the energy spectra and the wave functions of the low-lying states. We shall apply such a simple model to Hg isotopes near the transition region. The details of how a two-center potential of different heights can be solved has been presented previously zs). We shall summarize briefly the important results: The eigenenergy E is calculated by solving the eigenvalue equation ½U(~t , -- q l ) U ( ~ 2 , q2)(C~2 q2 -- C1~ q l ) -

C2~: U ( c q , - q l ) U ( c t 2 - 1, q2) U(~ 2, q 2 ) U ( o q - 1 ,

-C~

-ql)

= O,

(11)

where

oti = (-- E + h,)/hco i

)

q, C, = B o,

)

i = 1, 2,

and the U are the cylinder parabolic functions. The unnormalized wave functions obtained from solving the eigenvalue equations are [U(~I, - ¢ 1 ) ¢(7) = [OU(tx2, +~2)

for 7 < 7m for 7 > 7m,

(12) (13)

where ¢, = (

- r,),

D -- u( x,

q2).

In our analysis, we take hoot = hco2 - I MeV, which is approximately the energy at which a//-band is expected for this mass region, and we set B ---300 m . fm2/rad 2 where m is the nucleonic mass. W e adjust h2o -hxo; Jt and :2 to obtain a fitto the observed spectra. Good agreement with experimental spectra of ISSHg, IS6Hg and ~84Hg was obtained with parameters shown in table 2. W e shall discuss the different isotopes separately, beginning with 18SHg. W e show in fig.9 the calculated spectrum of tSSHg together with theirwave function along the cut in the [/-7plane under consideration. For the 0 + states, the prolate minimum is assumed to lie at an energy of 0.87 M e V above the oblate minimum, while the moment of inertia constants at the two minima are taken to be h2/2,/= 0.062 M e V for the oblate and 0.0133 M e V for the prolate minimum, respectively.As TABLE 2 Parameter values (MeV) used in the two-center model to obtain good fit of the low-lying spectra

Oblate Prolate

h2o--hlo ?~2/2.ft ?~z/2.f2

a S4Hg

I e6Hg

I s SHg

0.35 0.064 0.014

0.53 0.066 0.013

0.87 0.062 0.0133

SHAPE ISOMERISM IN Hg

215

one can see, the energy spectrum is quite well reproduced by such a simple model. The wave functions shown at the left side of the spectrum indicate that while the first 0 +, 2 +, 6 + and 8 + are well localized in their respective minima, the second 0 + and 2 + states as well as the two 4 + states are not. In fact, these states (especially the 4 + states) are so well mixed that it is difficult to classify them as belonging to one of the bands built on either minimum. Theoretically, such strong mixing also shows up in perturbing the energies of states of the same angular momentum so that it is hard to recover the I(I+ 1) rule for states in the same band with different angular momenta. The fact that for some of the states the deformations are not well localized also explains the peculiar branching in these nuclei, namely, there appears to be a strong "back-bending" effect if one follows the sequence of strongest transitions. The matrix element for electromagnetic transition depends on the overlap of the collective wave functions. We can get a qualitative idea of the size of this overlap by considering the integral in our model 1,1, =

(14)

where the wave functions ~,(~) are normalized according to 2

1.

For the states under consideration, the overlap integrals were calculated and their values are indicated in figs. 9-11. From the overlap integrals, one can judge that if one starts with a 14+ state localized in the prolate minimum, then the predominant transitions will lead to the first 6+ state. Now, because of the hybrid nature of the two 4+ states, the overlap integrals differ only by a factor of two. Furthermore, because of the strong dependence of the E2 transition on the energy difference, the predominant transition leads to a decay to the first 4+ state. There again, even though the overlap integral of this 4+ state to the two lower 2 + states are of the same magnitude, the one with the larger energy difference is favored, leading to the first 2 + state. Thus, the peculiar "back-bending" type branching can be explained in this simple model. A very interesting prediction of the model is the presence of a 0+ state at about 760 keV. It would be interesting to search for such a 0 + state about this energy. We show in fig. 10 the calculated 1s 6Hg spectrum together with their wave functions. For the 0 + state, the prolate minimum is assumed to be 0.53 MeV above the oblate minimum while the moment o f inertia constants at the two minima are taken to be h2/2j = 0.066 MeV for the oblate and 0.013 for the prolate minimum, respectively. As one can see, the energy spectrum is quite well reproduced. The wave functions for these states indicate that except for the first 4 +, 6 + and 8 + states, the wave functions are not as localized to varying degrees. The first 0 + state is nearly localized in the oblate minimum; there is a small tail at the prolate minimum.

216

K. KOLB A N D C. Y, W O N G

144--

3.5

1.0

WAVE FUNCTION

12"/--

3,0

1.0 8+

2.5

6+ O.89 --

2.0

(6'r) ll.O

0.45

0.70 0.97 ~

6+

1.5

2~-------~ ¢

'LO

o.o51 o.7,2

--

0,5

--

0

O.06 04-

o.S OBLATE MINIMUM

'I88Hg

EXP.

PROLATE MINIMUM

Fig. 9. Shape o f wave functions, energy levels and overlap integrals calculated with the phenomenological collective model for lSSHg. Experimental energy levels axe shown at right.

The amplitude of the first 2 + state is predominantly in the oblate minimum;however, its value at the prolate minimum is not small. From the wave functions, we calculate the overlap integrals between states with angular momenta I and I:1:2. One finds that ff one starts with the first 14 + state, it will feed into the first 4 + state. For this 4 + state, the overlap integral with the first 2 + state is 0.59 and is 0.81 with the second 2 + state. Because of the strong dependence on energy difference, transition into the first 2 + state dominates. This may explain why the second 0 + and 2 + states at about 600 keV have not been observed because of the small branching from the first 4 + state.

SHAPE ISOMERISM IN Hg

217 3.5 t4 +

WAVE FUNCTION 3.0

t.0 8+

12 +

2.5 1.0

10+ /

-

~

2.0 >

t~ L5

1.0

.4 t ~0.13

0.99

O.BO

6" l.O

05

4"

O.B1~ o.97 t 0.59 ~/0,~0

o+

i O.9B

0+

'18SHg '

0.5

0.20

,

OBLATE MINIMUM

( P-"

EXP.

T

PROLATE MINIMUM

Fig. 10. Shape of wave functions, energy levels and overlap integrals calculated with the phenomenological collective model for i UeHg" Experimental energy levels are shown at right.

Finally, we show our calculated spectrum and wave functions for 184Hg in fig. 11. For the 0 ÷ state, the prolate minimum is assumed to lie at an energy of 0.35 MeV above the oblate minimum. The moment of inertia constants are h2/2J = 0.064 MeV for the oblate and 0.014 MeV for the prolate minimum, respectively. The energy spectrum is quite well reproduced. The branching of the first 4 ÷ to the first 2 + can also be explained by the strong energy dependence of the E2 transitions and the fact that the overlap integrals are of the same order of magnitude. By comparing the three isotopes, one notices that as the mass number decreases, the smaller is the difference in the height of the prolate minimum relative to the oblate minimum. One sees first the hybridization of 4 + states in 'SSHg and then the 2 + state in ~StHg which becomes even stronger in ~S4Hg. The values of the moments of inertia obtained in the phenomenological fit for the prolate minimum is about four times that for the oblate minimum. It will be of interest to see whether future calculations on the moment of inertia Dead to the same result.

218

K. KOLB AND C. Y. WONG WAVE FUNCTION

~

+

3.0

--

2,5

IO+-

--

2.0

8+

--

t.5

12+

~4~~~ "

ILO

.o

I °'09

!: ,:i:o

l

> v

O.

OBLATE MINIMUM

--

184Hg

o+

0+

--

1.0

--

0.5

EXP.

PROLATE MINIMUM

Fig. 1]. Shape o f wave functions, energy levels and overlap integrals calculated with the phcnomenologica] collective mode] for lS'*Hg. Experimental energy levels arc shown at right.

4. Discussion The results of the phenomenological analysis can be compared with the theoretical predictions of the oblate-prolate differences in the light Hg isotopes. One finds from fig. 3 that the difference in height is very well predicted for lS4Hg; they are a little too large for lS~Hg and lSSHg. However, the trend of a decrease in separation is reproduced and the onset of a permanent prolate deformation is predicted to locate at mass 182, both from our theoretical results and the systematics of the phenomenological collective model This location of onset of prolate deformation differs from that of 184 obtained by Faessler et al. and 186 obtained by Cailliau e t al. which appears to be, however, excluded from experimental spectra as the first even-mass Hg isotopes to be prolate in their ground states. Looking at the various models to explain theoreticagy the systematics in Hg isotopes, we find that they mainly differ in their single-particle part. As is well known, one cannot yet achieve the same degree of precision as one does with the liquid drop model. However, the exact details of the singie-particle spectrum have usually only little influence on gross ground state properties of a nucleus such as the total energy,

SHAPE ISOMERISM IN Hg

219

the deformation and the density. Not so in a transitional nucleus whose ground state may assume one or the other shapes depending on small differences in their respective binding energies in the different minima. Thus, the ground state properties in this region depend sensitivity on the single-particle spectra. It is not surprising that different models lead to different conclusions on the exact onset of prolate deformation because of the inevitable differences in their single-particle spectra. Turning it around, these nuclei provide a good probe concerning, the single-particle spectrum in the deformed region. It seems to us then if we are aiming for relative changes in gross properties of nuclei, one can no longer get along with a fair s.p. spectrum but better make sure to have a good s.p. spectrum at least concerning the relative position of levels as long as one is not really interested in very strong deformations. If the relative position of the single-particle states is correct, but the average spacing around the Fermi sea is too big (typical problem of this model and other interactions a)), then relative structures in the potential energy surface are exaggerated but at least qualitatively correct. If, however, the relative position of single-particle levels is incorrect, wrong structures of the potential energy surface are predicted even if the average level spacing is quite good, as in the oscillator or Woods-Saxon model, for example. For calculations in the fission region, we have to demand that average spacing and relative position of single-particle states be good. That means, e.g., that we should be able to calculate the particle and hole spectrum in Pb quite correctly, a demand no existing self-consistent single-particle model meets so far, even approximately. Looking into transitional nuclei with such a nice experimental signature for the radius as in the Hg isotopes we may get some good ideas, how far one can trust the validity of our necessarily rather imperfect model. One certainly can also not exclude that one might have to worry about a better continuum coupling, and possible left out correlation effects than is presently taken into account in a density-dependent effective interaction for Hartree-Fock calculations s), if one wants to describe transitional nuclei correctly. Unfortunately, one runs into similar problems again for superheavy nuclei, where barrier heights get completely dependent on shell effects, after the liquid drop part gives no barrier against deformations any more. Work to explore the sensitivity of fission barriers for superheavies on changes of model parameters within allowed limits is in progress t9). As there is experimental evidence in lead already of a density depression at the center, we may expect an increased tendency for bubble configurations in a superheavy nucleus which has indeed been calculated t 7, t 9, 20) making it essential to use a self-consistent model. The computed low excitation energy of the bubble configuration in 2°°Hg is a consequence of a self-consistent rearrangement of the nuclear potential. Finally, the predicted low excitation energies of the bubble configuration around 2°°Hg and tS*Hg makes it interesting in future work to consider the density or transition density leading to the low-lying 0 + states in electron scattering experiments for these isotopes. The reportedly strange 0 + state at 1.029 MeV of 2°°Hg as observed

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by Breitig et al. 5) is worthy of a detailed investigation to see how the density o f that state may be different from a uniform distribution. References 1) J. Bonn, G. Huber, H. J. Kluge, L. Kugler and E. W. Otten, Phys. Lett. 38B (1972) 308 2) P. Hornshoj, P. G. Hansen, B. Jonson, A. LindaM and O. B. Nielsen, Phys. Lett. 43B (1973) 377 3) D. Proetel, R. M. Diamond, P. Kienle, J. R. Leigh, K. H. Maier and F. S. Stephens, Phys. Rev. Lett. 31 (1973) 896 4) N. Rud, D. Ward, H. R. Andrews, R. L. Graham and J. S. Geiger, Phys. Rev. Lett. 31 (1973) 1421 5) D. Breitig, R. F. Casten and G. W. Cole, Phys. Rev. C9 (1974) 366 6) J. H. Hamilton et aL, Int. Conf. on reactions between complex nuclei, Nashville, eel. 1 (1974) p. 178 7) A. Faessler, U. G6tz, B. Slavov and T. Lederberber, Phys. Lett. 39B (1972) 579 8) M. Cailliau, J. Letessier, H. Flocard and P. Quentin, Phys. Lett. 46B (1973) 11 9) C. Y. Wong, Phys. Lett. 41B (1972) 451; Ann. o f Phys. 77 (1973) 279 I0) K. T. R. Davies, C. Y. Wong and S. J. Krieger, Phys. Letto 41B (1972) 455 11) K. T. R. Davies, S. J. Krieger and C. Y. Wong, Nucl. Phys. A216 (1973) 250 12) X. Campi and D. W. L. Sprung, Phys. Lett. 46B (1973) 291 13) M. Beiner and R. J. Lombard, Phys. Lett. 47B (1973) 399 14) S. G. Nilsson, J. R. Nix and P. MOller, Nucl. Phys. A222 (1974) 221 15) F. Dickman and K. Dietrich, preprint 16) D. Kolb, R. Y. Cusson and M. Harvey, Nucl. Phys. A215 (1973) 1 17) D. Kolb, R. Y. Cusson and H. W. Schmitt, Phys. Rev. CI0 (1974) 1529 18) C. Y. Wong, Phys. Lott. 30B (1969) 61 19) D. Kolb, in preparation 20) R. Y. Cusson and C. Y. Wong, in preparation