TRANSITIONAL NUCLEI AND TRIAXIAL SHAPES

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TRANSITIONAL NUCLEI AND TRIAXIAL SHAPES

J. Meyer-Ter-Vehn

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C3, supplément au no 6 , Tome 39, Juin 1978, page C3-15.4

TRANSITIONAL NUCLEI AND TRIAXIAL SHAPES

J. MEYER-TER-VEHN

Swiss institute for nuclear research, CH 5234 Villigen, Switzerland

Résumé.

-

Nous passons en revue dans les noyaux impairs de transition, A = 130 et A = 190, l'évidence de forme triaxiale à partir de familles d'états de même parité. Les données expérimentales sont analysées dans le modèle du cœur triaxial. Des configurations de bandes régulières à 2 dimensions sont trouvées dans l'expérience et nous montrons qu'elles sont une conséquence de la brisure de symétrie axiale. Nous présentons et discutons les développements théoriques récents incluant la question de la stabilité des formes triaxiales.

Abstract. - Evidence for triaxial nuclear shapes from farnilies of unique-parity states in transitional odd-A nuclei around mass A = 190 and A = 130 is reviewed. The experimental data are analysed within the odd-A triaxial core model. Regular two-dimensional band patterns are found in experiment and are shown to be a consequence of broken axial syrnrnetry. Recent theoretical developments are discussed including the question of how stable the triaxial shapes are.

1

.

Introduction : level families of unique parity in odd-A îransitional nuclei.

-

In this talk, we want to present evidence for triaxial shapes in certain regions of transitional nuclei. The new information has been obtained from families of unique parity states in odd-A nuclei which are based on high spin single particle configurations such as h11/2, h9/2, i1312, etc. These groups of states represent a simple case in othenvise'more complicated spectra. They are easily identified experimentally, since they are strongly populated in (a, xn) and (heavy ion, xn) reactions and their members decay predominantly within the family. Because of the almost pure configuration of the odd nucleon, the energy spectrum gives rather direct information on the shape and the collective motion of the core.

We are going to interpret these spectra within a simple theoretical model consisting of an odd nucleon in a single j-shell coupled to a triaxially deformed core. The model represents a particular approximation to the general problem of coupling an

odd nucleon to the collective motion of an even-A core [1 ,2]. For example, anharmonic vibrator models [3, 41 provide a complementary description independent of the concept of intrinsic deformation. However, they al1 imply fluctuating nuclear shapes which are triaxial on the average. A specific advan- tage of the rigid triaxial core model is its relative sirnplicity. It has been introduced b y Davydov [5] and has been applied to odd-A nuclei by several authors [6]. A detailed description of the model and a systematic comparison with recent experimental data has been given by the present author [7]. In this talk we shall demonstrate the qualitative features by discussing some outstanding experimental examples.

Three limiting situations of the particle-core coupling are shown schematically in figure 1. To be definite, we consider a particle in a completely unfilled h1i/2 shell coupled to a deformed core. The shape is determined by the deformation parameter p and the asyrnmetry parameter y. The following cases are shown :

(A) In the limit of spherical syrnrnetry

( P

= O), the

excitation spectrurn consists of multiplets around the core excited states. This is the weak coupling situation which applies to odd-A neighbours of magic nuclei. It has been discussed e.g. by de Shalit [8].

(B) In the limit of axially deformed, oblate shapes ( p

<

O, y = O", or equivalently

P

>

O, y = 609, one obtains normal-ordered rotational bands (e.g. I = j, j

+

1, j

+

2 ,

.

.), based on Nilsson states [9]. The angular momentum j of the odd nucleon has a sharp projection 0 on the symmetry axis (except for slight Coriolis perturbation). For the ground band, it is 0 = j= 11/2. Core rotation (R) occurs about an axis perpendicular to the symmetry axis and j. This gives rise to AI= 1 bands.

(C) For axially deformed, prolate shapes ( p

>

0,

y = O"), decoupled rotational bands appear. The yrast branches of these spectra have a spin sequence

T

= j, j

+

2, j

+

4,

. . .

with energy spacings almost equal to the ground band of the core. The missing

unfavored states (1 = j

+

1, j

+

3,

.

.) are shifted to higher energies. Stephens has interpreted these bands in terms of the rotation alignment mechanism [IO]. Due to Coriolis interaction, the angular momentum of the odd nucleon aligns with the rotation axis. In this case, R is parallel to j. This gives rise to the typical A I = 2 bands. At large

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TRIAXIAL NUCLEI

0 lh

part

( B I

( A ) strongly

'y2 15051

coupled band

( Nilsson

1

prolate

-

core excitation multiplets (de Shalit) decoupled band

-

I

FIG. 1.

-

Schernatic picture of three limiting cases (A) - (C) of particle-core coupling as described in the text.

deformation ( p .> 0.2), these bands approach

= 1/2 Nilsson bands with a decoupling factor

a = j + 1/2 = 6 .

In comparison with experimental spectra, one often encounters situations which fit to none of the limiting cases. The deviations mainly concern non-yrast states not al1 of which are shown in figure 1. The important point of this talk will be that these deviations can be understood, to a large extent, as due to triaxial deformations. Moving through the triaxial plane, a variety of spectra is obtained which are intermediate between the cases (A), (B) and (C). Before showing this, let us add a remark on the distinct difference between case (B) and case (C). In fact, it is of simple geometrical origin.

The odd nucleon in the high-j shell has an oblate mass distribution when seen from the direction of its angular momentum. For low-excited states, the odd nucleon tries to obtain maximum mass overlap with the core. This is optimally accomplished for an oblate core in which case j gets strongly coupled to the symmetry axis (case B). This coupling is much weaker for a prolate core, and j may therefore decouple from the symmetry axis and align with the rotation axis R, driven by the coriolis force (case C). The type of level order is determined by the relative direction of j and R : perpendicular-

A I = 1 bands, parallel-AI = 2 bands. Since axial1 y symmetric nuclei cannot rotate about their symmetry axis, one is bound to have j perpendicular to R in case (B) and, consequently, A I = 1 bands. The new point about triaxial odd-A rotors will be that they can rotate about al1 intrinsic axes. The spectrum will therefore be composed by a superposition of A I = 1

and A I = 2 bands. This will be discussed below in connection with experimental examples.

2. Theory : the odd-A triaxial rotor model.

-

The Hamiltonian describing the coupling of a quasiparticle to a triaxial core can be written as

The rotor Hamiltonian reads

with three moments-of-inertia

and

9,

empirically determined to be

A' - 204 MeV - -

2T0 P 2 A 7 ' 3 '

The interaction between the odd nucleon and the ( p , y) deformed core is

sin y

Vi, =

\1$

k p (cos yY,

+

-

(Yz2

+

~ ~ - 3 )

fi

with the empirical interaction strength -

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C3-156 J. MEYER-TER-VEHN

The Pauli matrices in equation (1) refer to particle and hole subspaces. The energy of the odd nucleon is (ej- A,) for a particle, and (A, - E,) for a hole where A, is the Fermi energy. For a single j-shell, = O. Particle and hole states are mixed by the pairing term where A is the fixed gap parameter which is taken from odd-even mass differences or chosen to be A = 135 MeV/A. There remain three free parameters in the model :

B ,

y , and AF which will al1 be determined from information independent of the experimental odd-A spectra. The total wave- function can be written in the form

+IM,,a =

Ç

CU,L~> (DE&

xg

+

<

- )(I-i> D<r>

< j ) M-K X-0)

K. R

where the summation is restricted to I KI 5 1,

1

0

1

j, (K

+

j) even, and

(LI

+

j) even. DM,, denote the rotational D-functions

,

and ~ ( 8 the wave-function of the odd nucleon. The diagonalisation of Hamiltonian (1) and further details are described in reference [7]. In particular, the Hamiltonian (1) obeys an important particle-hole symmetry which states that a particle coupled to a core with parame-

O" IO" 20" 30' 40" 50" 60"

ters

( P ,

y, A,) has the same energy spectrum

as a hole coupled to a core with parameters Y

( B ,60° - y , - A,). Solutions of the model and appli- _{- -} FIG. 2. - The odd-A energy spectrum as a function of y for

cations to experimental spectra are discussed in the = 5 , hF = cl, and j = 11/2. Al1 states with

E, - E,,/, < 45 h2/2 9, at y = 30" have been plotted. next section.

3. Comparison with expriment : transitional nuclei around mass A = 190 and A = 130.

-

Rich information on unique parity states in transitional odd-A nuclei has been obtained during the last few years. Here, we focus on nuclei around mass A = 190 and A = 130. Families of negative-parity states based on an h91z particle and on an hl,/, hole are systematically observed in odd proton nuclei say between lS60s and zooHg. [Il-211. In addition, some examples of il,/, neutron families in this region have been measured. [22-241. Also systematic work on hl,/, families in neutron deficient Nd, Ce and Ba isotopes [25, 271 and corresponding odd proton nuclei [26,28-301 around A = 130 has been done and is still coming up. The quoted references are far from complete, however may be useful to trace the experimental results. Recent work and many addi- tional references are also found in reference [31]. In the following, some representative spectra are compared with the model calculation.

The calculated spectrum for an hl,/, particle and pA2I3 = 5 is shown as a function of y asymmetry in figure 2. The Fermi energy A, = E, has been

placed on the lowest substate of the splitted j-shell. Due to the particle-hole symmetry, mentioned above, the same spectrum is obtained for an h,,/, hole when replacing y by 60'- y . The results hold also approximately for h,/, and il,/, shells. From figure 2 it is seen that the spectrum changes gradually from a decoupled level order at y = O0

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TRIAXIAL NUCLEI C3-157

med for hl,/, hole data. Note that the theoretical curves are the same as in figure 3a, only y has been changed into 60° - y.

In figure 4, the model calculation is compared with I9Tl. The hg/, families in Tl isotopes are very interesting, since they represent the unique case of strongly coupled bands on an oblate core. Al1 isotopes 191-201Tl show almost the same structure of negative-parity levels [19]. Evidence for a triaxial shape is provided by the second 13/2 state. As one may check with figure 2, the energy of this state changes rapidly with y relative to the ground band. Its experimental position is well reproduced with y = 37O derived from 198Hg. The same y = 37" .f 2" is

obtained from a number of hl,/, spectra in Au

0° 20° 40' 60" isotopes (see Fig. 5) and il,/, spectra in odd Hg

Y

isotopes [24]. In addition to the experimental states shown in figure 4, two 9/2 states at 1 868 keV and

40 5' N

.

,

\

5

W 20 O O0 IO0 20' 30' 40° 50° 60° '::TI p = O 15

FIG. 3a. - Systematic trends of excitation energies of states in y = 37" hg,, families as compared with the calculated results of figure 2. O - _F--O 5 _-

In the lower part, states of the related even-A cores are compared A = 0 7 MeV with the triaxial rotor spectrum to determine the y value of each

_y

- h y2 system

case. The experimental revel energies are taken from:

L

187,1891r [ I l , 121, 189,191Au [14-161, 1g5T1 [19, 211. 912 C c 0 - 4- m c X w 1 0 - Q = -2 4 eb p = 3 3 n m Experiment Theory

FIG. 4. - Negative-parity states in I9Ti 1191. Solid lines indicate the strongest observed and calculated decay transitions of each level, broken lines ali other transitions with intensities of at least

50 % of the strongest transition. Energies are given in keV relative to the basic state. The Fermi energy A, = (A, - E,)/(E] - Q) with first and second s.p. energy E, and s. The calculated quadrupole moment and magnetic moment of the basic state are also given.

Very rich spectra of hg/, and hi,/, families have recently been measured for Ir isotopes [Il-131. In

60" 40' 20 O

O0

figures 6 and 7, experimental and calculated results

r

are shown for lS7Ir. We take this example to give some insight into the band structure of odd-A triaxial

FIG. 3b.

-

Same as figure 3 a , but for h 1 ~ 2 hole families with y _{rotors. ~h~ confusing abundance}_of _low-excited

replaced by 60° - y. Also the comparison for the core states is not

explicitIy shown. The experimental data are taken from : tat te^ which is Seen in experiment may be brought 1851r [13], lS7,l89Ir [ I I , 121, l9l1r [13], l9)Au [14, 151. into a two-dimensional band pattern (see e.g. 1955 keV (relative to the 1/2+ ground state) have

-

recently been reported for '97T1 [20]. They both decay to the first 11/2- state and are good candidates for the second and third 9/2 state of the model

-

calculation. Also, it seems that evidence for the first

5/2- state has been found at almost the predicted

~ r r r y.

- -

a

- -

A - 0 +

-.

shapes in the Hg cores are. This point will be discussed in the next section.

4

- 2;

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C3-158 J. MEYER-TER-VEHN

Experi ment

Theory

1731

2n

-

system

B.0.14 y=37"~-0.5A=0.7MeV

FIG. 5.

-

Negative-parity states in 195Au [16]. Energies, moments and

A,

are defined as in figure 4. Solid lines indicate transitions with 80 to 100 %, broken lines transitions with 20 to 80 % of the strongest decay intensity of each level. Calculated intensities are based on

calculated B(E2) and B(M1) values, but experimental transition energies when known.

T

heory

6.0.23 Y = 1 6 . 2 " & = O

A . 0 . 7

MeV

FIG. 6.

-

The h9,, farnily of negative-parity states in [Il, 121. For details see figure 3.

Fig. 6). It consists of a system of vertical A I = 1

bands with basic states I = j, j + 2, j + 4,

...

,

similar to y-vibrational bands, but at much lower energy. It should be noted that the series of bandheads with

I = j, j

+

2, j

+

4,

. . .

,

represents in some cases the decoupled AI = 2 yrast band. Such a two-dirnensio- nal level pattern is typical for triaxial rotors, reflecting core rotation about two intrinsic axes. A sche-

matic picture is given in figure 8. As we have mentioned above, the A I = 1 bands originate from core rotation perpendicular to j, and A I = 2 bands from core rotation parallel to j. For low-excited states, j aligns preferably with the shortest intrinsic axis which becomes the oblate syrnrnetry axis at y = 60°. The projection

fi

ont0 this axis (not sharp

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TRIAXIAL NUCLEI

Experiment

('Ir11800

h

%

-

system

_B

=O21 Y=240X~0.42

A-0.7

MeV

FIG. 7.

-

The hl,,, famiiy of negative-parity states in l"lr [ I l , 121. For detaiis see figure 3.

system. For example, the hg/, system in figure 6 has

= j = 9/2.

There are also states in the triaxial odd-A spec- trum with f2 = ( j

-

1) corresponding to less complete

Schematic band system

of triaxial rotor (odd A

4

.i

5

Axis of largest moment -of -inertia

FIG. 8. - Schematic picture of the two-dimensional band pattern arising for an odd-A triaxiai rotor. The verticai bands with A I = 1

spin sequence arise from core rotation perpendicular to j, the horizontal bands with A I = 2 spin sequence arise from core rotation paraiiel to j. For low-excited particle states, j aiigns with the shortest intrinsic axis which becomes the oblate syrnrnetry

axis for y = 60". Only a few transition lines are shown.

alignment of j with the f avored intrinsic axis. In fact, there is another band system equivalent to that shown in figure 8, but building up on the basic ( j

-

1) state. Some members of this subsystem are seen in the hl,,, family of 18'Ir, shown in figure 7. The bands based on the 912 (405 keV, calculated energy), 1312 (845 keV), and 5/2 (777 keV) states belong to the

fi

= (j- 1) = 9/2 subsystem, whereas the bands building up on the 11/2 (O keV), 15/2 (438 keV), and 7/2 (385 keV) are classified as

fi

= j = 11/2.

So far, we have considered cases where the j-shell contains more or less one particle or one hole. However, a large number of experimental spectra corresponds to cases with several particles or holes in the j-shell. Within the present model, we can account for this situation by placing the Fermi energy inside the splitted j-shell. The model spectrum as a function of y with A, = E , placed on the second state of the j-shell is shown in figure 9. We see that the states belonging to the f2 = ( j - 1) subsystem are most sensitive to the penetration of the Fermi energy. The basic (j- 1) state is strongly lowered and, eventually, drops below the basic j state. As an example for this situation, we show the spectrurn of lZ9Ba in figure 10. It corresponds to = 5

holes in the hl,/, neutron shell. The characteristic two-dimensional band structure clearly shows up as well as the separation into two fairly distinct subsys- tems, classif ied as

d

= j = 11 /2 (upper part) and

fi

= (j- 1) = 9/2 (lower part) in our scheme. A

detailed comparison with the model calculation is given in reference [25]. The spectrum of lZ9Ba corresponds to y - 21°, a typical yasymmetry for al1 nuclei around A = 130.

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FIG. 9.

-

The odd-A energy spectrum as a function of y for

pA2J3 = 7, Ap = EL, and j = 11/2. Al1 states with

E,- < 40 A2/2 9

,,

at y = 30" have been plotted.

It turns out that B(E2) values and E2/M1 mixing ratios strongly depend on the y-asymmetry and therefore provide another test of the model. In figure 11, calculated B(E2) values are compared with experimental values measured for favored yrast transitions in lZ9J3lLa. These results confirrn y = 22O 2 2" for the A = 130 mass region. The rigid triaxial rotor model yields also good agreement with B(E2) branching ratios and mixing ratios in a number of other cases [7].

4. Recent developments and conclusions.

-

4.1 EXTENSIONS O F THE TRIAXIAL CORE MODEL.

-

Some generalisations of the triaxial core model as presented in this talk have been worked out recently. Toki and Faessler [32] have applied the VMI pres- cription to obtain a more quantitative description of the core energies. This procedure lowers the states at higher energy in agreement with experiment. The sarne authors [33] also have mixed several j-shells to describe the odd nucleon configuration. This results in a lowering of low-spin states which lie systematically too high in the single j-shell calculation. The mixing of different j-shells will be important when extending the present calculations to normal parity states. This has still to be done. The influence of hexadecupole deformation on spectra of decoupled type has been investigated in reference [34]. A

comparison of different rotor models for spectra of the A = 190 region is found in reference [35].

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TRIAXIAL NUCLEI C3-161

FIG. 11. - Caiculated transition probabilities as a function of y. The broken Iines give classical limits for B(E2) : core rotation about the shortest axis (lower part), core rotation about the iiiiddle-long axis (upper part). Also shown are the experimental

data for 129La ( y = 22O) and "'La ( y = 2 3 O ) [30].

Another important extension of the triaxial core model has been performed by Toki et al. to describe groups of two-quasiparticle states such as the negative parity bands in even Pt and Hg isotopes [36] and states of odd-odd nuclei [37]. These calculations confirm the importance of y deformation in nuclei around mass A = 190.

4.2 SOFT-CORE-CALCULATIONS : HOW STABLE ARE THE TRIAXIAL SHAPES ?

-

Aithough the triaxial core model with rigid shapes is rather successful in describing spectra of certain transitional nuclei, this does not necessarily mean that these nuclei have stable triaxially deformed shapes. In fact, no stable triaxial deformations have been obtained so far from microscopie calculations [38]. In general, calculated potential energy surfaces (PES) of transitional nuclei are very flat in y direction ('). Also for spectra of decoupled level order (e.g. the hl,,, system in Au, Fig. 5), anharmonic vibrator models implying strongly fluctuating shapes produce spectra [24, 39, 401 which are very similar to those of the rigid triaxial rotor. However, it seems that strongly coupled

(') One shouid keep in mind, however, that al1 the PES caiculations for heavy nuclei have been performed with rather schematic two-particle interactions of quadrupole-plus-pauing type. We expect this force to smear out possibly existing PES structure. HF caiculations based e.g. on a Skyrrne force and allowing for triaxial deformation might be of great interest.

bands like the hg/, bands in Tl (Fig. 4) are sensitive to core softness and may provide a test on how stable triaxial shapes in certain nuclei are.

In this connection, very interesting work has recently been published by Leander [41]. The thesis work of U. Schneider [42] is dong a similar line. Leander solves Bohr's Hamiltonian [1] for the odd-A case with an arbitrary PES V ( P , y). The PES is parameterised in a sufficiently flexible way and is then adjusted to combined data (energies and B(E2)'s) of a core nucleus and a neighbouring odd-A spectnim. Resulting PES for the combinations (1)

lS60s and 1871r(h9,2) and (2) '%Hg and lg7Tl(h9,,) are shown in figure 12. The corresponding spectra [41] look very sirnilar to those obtained with the rigid triaxial rotor. They are not given here explicitly.

Leander's results give some idea of how the deformation energy of these nuclei may behave in reality. The nucleus Ig60s is found to be still at the border of axial symmetry, but very soft in y direction. Therefore triaxial shapes appear on the average due to dynamics. For '%Hg on the other hand, there appears a pronounced minimum of 3 MeV depth in the PES near y = 35O, compared with the rigid rotor value y = 37O &

ZO.

Leander was not able to obtain a

good fit without assuming such a minimum. We take this result as an indication for a more or less stable triaxial shape of i96Hg at low excitation. We briefly mention that the quasi- y-bands of the core nuclei are also sensitive to core softness. The equal energy spacing between the 2+, 3+, 4+ states in lg2Jg4Pt nuclei [43] suggest minima in the triaxial plane. Unfortunately, these states are not known for lg6Hg and other even-A Hg isotopes. Concerning Lean- der's approach, there remain questions with respect to anharmonicities in the inertial functions. Also a correct treatment of the Pauli principle in cases with more than one particle (hole) in the j-shell has still to

be found.

FIG. .12.

-

Potentiai energy surfaces for (a) Is60s and (b) '%Hg obtained from a fit to data of (a) '%Os and lS7Ir0i,,, system) and of (b) '96Hg and 197Tl(h9,2 system). These resuits are taken from

ref erence [41].

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and broken gauge symrnetry (pairing deformation A). In some sense, y deformation generates the hidden regularity in the complicated anhamzonicities connecting vibrational and rotational nuclei (see Marumori [44]). Such regularity is less apparent in nuclear spectra than that originating from

p

deformation and pairing deformation, but strong evidence for it has now emerged from band patterns of unique parity states in certain regions of transitional nuclei. The present analysis has been based on the triaxial rotor model with rigid shapes. As a function of PA2", the model describes different strengths of core-particle coupling intermediate between weak coupling and strong coupling, whereas the variation of y provides the clue for understanding the variety of different level orders observed in experiment. A most beautiful example for these changes is given by the unique parity spectra in Ir, Au, and Tl isotopes reflecting a gradua1 shape transition from strongly deformed prolate to weakly deformed oblate shapes through a series of triaxial shapes. Clear evidence

Ref el

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(1963) 853.

131 ALAGA, G., Cargese Lectures in Physics, ed. M. Jean, Gordon and Breach, (New York), 1969 3 579. PAAR, V., Proc. Conf. on problems of vibrational nuclei,

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and A 122 (1968) 273.

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for triaxial band patterns is also found around mass

A = 130.

Phase transitions towards triaxial shapes are likely to be very smooth. Recent soft-core calculations indicate that the triaxial shape of Is60s and 1871r(h9/2) is of purely dynamic origin, but that '%Hg and Tl(h9/,) have a relatively stable triaxial shape corresponding to a 3 MeV minimum at y = 35O. For further studies, more experimental results are needed, e.g. on low-spin members of h9/, families in Tl isotopes and quasi-y-bands in even-A Hg isotopes. On the theoretical side, triaxial HF calculations with realis- tic forces should be performed and also more systematic soft-core calculations for odd-A systems. No matter whqt the final answer to the stability of triaxial shapes will be, the y deformations are well defined experimentally for transitional nuclei around mass A = 190 and A = 130 and also in other mass regions. They deserve a central place in the classifi- cation of these nuclear spectra on equal footing with the well established

P

and A deformations.

MLEKODAJ, R. L., CARTER, H. K., SCHMIDT-OTT, W. D.,

Phys. Lett. 58B (1975) 159.

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