Some themes in the study of very deformed rotating nuclei

lc

Dear Akito, it is ~;o~~~rful to be here to celebrate your bi~hday, to be abIe to say thank you for mom than 20 years of mnst stimuI~tj~~ ~sc~ssio~s and fruitful disa~e~~~~nts, and to ren3~rn~~~~ your friendship and the personal warm& of our many contacts. It is a ~~~~~ plessure for me to be able on this occasion tu ~~~~~ntit discussion rrft ~CXII~of the physical ideas involved in the axrent study of very strongly deformed and rotating nuclei and in the course of that discussion to describe how an idea of yours, Akito, is playing a crucial role in the iu~rpretat~(~~ of one of the fascinating

discoveries that has been made on this active fmntier of nuclear physics. As trxxr of you wiII know ir has become possible for about the past decade to produce nuclei in states wirh very high angular m~men~llrn.

These states are ~r~uc~d

by aflowing

fwo he:lvy

nuclei to cottide with ~~~~~i~ somewhat above the ~~)~Iornb barrier. In such n collision the angu-

tar momentum in the relative motion may be more &an 100 E and if the two nuclei fuse to form a c~~~~~o~~~d system this piece of nuclear matter has ;t cor~~spond~n~ly high angular ~~~(~II~~~~~LII~~, 7’1~ &sed system is ~n~t~aII~in a state of very high

~xc~t~t~(?~ energy

and will &pin to co& by the eva-

goration of neutrons and subse~ue~tIy of y -rays. ~ventualIy the system comes to the states of lowest energy compatible with the existence af the very high angular rn~~~~n~llnl. It is about s~mt af the fascination ~roperi~~s of these rapidly rotating but cold nuclear systems that I shall be talking

today.

3. MotteLFon/ Very deformed rotating nuclei

2c The Super-deformed

Story:

Classical analysis

But now let us get on with the discussion ascertain the most deformed centrifugal

of the Suer-defo~ed

Story.

We are trying to

shapes that rotating nuclei will acquire before being tom apart by the

force generated by the angular momentum.

The first chapter in this story involves a classical analysis of the shapes of rotating liquid drop. ( Cohen, Plasil, and Swiatecki shapes of rotating self attracting

(1975) ). This analysis mirrors the discussion bodies as carried out by the astronomers

of the equilibrium

over the past 300 years

(See Fig. 1). As is well known, the result of this analysis is that at low angular monlentum shape is an oblate spheroid with eccentricity

increasing

as 1’ while at a sufficiently

the angular momentum,

is broken,

the system develops

configuration

the axial symmetry

since for rapidly rotating

systems

the

high value of

toward a bar-shaped

this shape carries the angular momemtum

most

efficiently. Es-----

Fig.1:

Deformations

induced by rotation.

I2

23(a)

+

-2Ca+

As indicated in the side view

(a), an axially sym-

metric flattening leads to a shift of mass of order IX through a distance of order R, with respect

to the rotation axis. Thus for low rotational

proportional

to I* is developed.

frequencies

an oblate deformation

The top view (b) shows the effect of a deformation

that

breaks the axial s~rnet~; here, the deformatjon moves an amount of mass of order y through a distance of order y and thus the effect on the moment of inertia is of order y’. Thus the system angular

momentum

is stabile with respect

exceeds a critical value I,.

to such deformations

until the

B, ~ut~e~o~ / Very defo~ed rotatingnuclei

80 -

60 z I.-.+

A

Fig.2:

Stability limits for rotating liquid drop. (Cohen, Plasil, and Swiatecki,

X7 , 557 (1974) For systems _=

Anl: Phys.

angular momentum

less than 1, the equilibriunl

(ablate).

Between

the Emits II and /,1 the axial symmetry

develops

a bar-shape

rotating with

shape is flattened and axially symmetric is broken as the system

leading to fission beyond the limit I,,. The Z/N ratio for each A

has been chosen to coincide with the valley of [3- stability.

Shell structure For the nucleus, we know already from the study of low angular n~~mcntum states that a classical analysis of the equilibrium

shape can be modified

in a major way by the effect of the shell

structure;

we must extend the analysis in order to recognize

individual

particles

are in general very non-spherical

the fact that the quanta1 orbits of the

and therefore

the lowest configuration

given number of p‘articles may strongly prefer certain shapes over others. the basic concepts the case of spherical

of a

In order to remind you of

involved in the analysis of shell structure I would Iike to begin by considering systems and note that the shell structure corresponds

to a “bunchiness”

in the

B. MotteLron / Very deformed rotating nuclei

4c

spectrum

of single particle

energy gain associated

this bunchiness

orbits;

reflects

the possibility

with an increase in the orbital angular momentum

of compensating

the

by an integer number of

units, by a decrease in energy of radial motion associated with an appropriate reduction in the radial quantum occurrence

number.

The possibility

of this compensation

classical periodic orbits give an oscillatory shells are the especially degenerate

related

tern1 to the level density of the quanta1 spectrum;

stabile configurations

bunches associated

to the

,

f

periodic orbit. For spherical nuclei the most

(2: 1) and the triangular (3: 1) orbits.

I:2

2:3

I:1

3~2

2:1

$

$

4 I,

+

4

I

- 1.0

II,,,, -

closed

that fill all the orbits in one of the approximately

with a (short period)

important periodic orbits are the pendulating

0.5

0 6

osc

Fig.3:

can in turn be simply

of families of periodic orbits in the classical motion (Balian & Bloch (197 1)). Thus the

I,

I

0.5

3:1 4-i

1.0

w,-(JJ, G

Single particle spectrum for axially symmetry harmonic oscillation

potential.

The eigen values are measured in units of W=&cu,_+o,) /3. The arrows mark the deformations corresponding

to the indicated rational ratios of frequencies

w~:o), .

In order to analyse

the shell structure

periodic orbits that occur in appropriately problem has been obtained potential 397

from the examination

(see, however, the interesting and

1115

; oversimplified

(1987)

for motion

? ) idealization

provide surprisingly

of strongly

deformed

deformed

potentiaIs.

nuclei we must examine

of motion in the anisotropic

in a square

well potential

should be obvious,

robust guidance for the interpretation

of spheroidal

all the classical orbits are periodic if the frequencies orbits

the

it seems to

of the main results on superdeformation, In the potential,

U, and UJ_ are in the ratio of integers;

(except for the negligibly

with motion in the equatorial plane;

shape);

but none-the-less

and I shall therefore briefly summarize the main results of this analysis.

space associated

harmonic oscillator

results ohtained by Arvieu & Ayant, Jour. of Physics G, zl,

of this potential

ratio is irrationa1 there are no periodic

the

Most of what we know about this

ie z =i

if the

small fraction of phase

= 0 ). Co~es~ondingly

the quanta1

spectrum

shows major shell structure if and only if ce, and w,, case of equal frequencies built on classical

are in the ratio of (small)

integers.

The

( o, : a,= 1 : 1 ) yields the well known spherical shell structure which is

orbits of simple elliptical

form.

The next simplest

families

are obtained

for

O, : co,= 1 : 2 for which classical orbits vary between figure eight, through banana shapes to a simple cresent

form (see Fig. 4). The quanta1 spectra are shown in Fig. 3 where also the particle

numbers associated super-deformed reflecting

with the major shell closings are indicated.

The observed particle numbers for

nuclei are displaced with respect these (harmonic oscillator)

values by a few units

the effect of the more sharply defined surface in the nuclear potential

orbits with large angular momentum)

(which favour

and the spin orbit force (which further &vours, among the

high angular m(~mentum orbits, those with parallel spin and orbital angular n~oment~lm).

B. Mottelson / Very deformed rotatingnuclei

18

>

Fig.4:

Shape of classical orbits for I:2 potential.

I would like to make a few comments

about these very simple arguments

based on the har-

monic oscillator potential: (i)

The argument sketched above provides at the present time the only qualitative the observed favouring of nuclear shapes that correspond Since the assumed nuclear binding

harmonic

oscillator

field, the understanding

potential

is a very imperfect

of the robustness

potential seems to me to be a matter of considerable

explanation

of

very closely with a ratio of axis 2: 1. representation

of the predictions

importance

of the

based on this

in the program of understand-

ing the nature of shell structure in nuclei (ii)

As can be seen from Fig.3, expected to be even stronger*

the shell structure effect for oblate

shapes

o, : wI=

ing nuclei and in the fission isomers. have not so far been experimentally

rotat-

Despite the strong shell structure effect, the 2 : 1 shapes observed in nuclear spectra;

this is presumably

the cubic term in the surface and Coulomb energy which favours the (prolate) compared

2 : 1 is

than for the 1:2 shapes observed in the super-deformed

a result of

1 : 2 shapes as

with the oblate 2 : 1 by as much as 5 MeV (for A - 150). That this term favours

prolate shapes is of course well known from the study of the fission saddle-point,

and this is

also apparently

in the low

the underlying

reason for the dominance

of prolate deformations

energy nuclear spectra.

*The occurrcncc of larger shells in the 2:l as compwcd with the 1:2 potential can be seen as l~,g~~tuml COIISCof the fact lhat the period of the orbits in the forma potential are shorter by a factor (2) than tbosc in tbe latter.

quence

(iii) The above discussion of shell structure is based on single particle orbits in a non-rotating potential. The effect of rotation is to add a term of order r2/2J0 where .I, is the rigid moment of inertia for the non-rotating mass distribution. Thus the strongly defornled (1 : 2) con~~~lrafion rcccive a very i~lport~lnt ~i(I(iit~(~~~:~I energy

a(Iv~lllt~~~c (compared

with

con~g~rations with smaller defo~ations) when the energetics of l&J systems is considered. Indeed despite strong favouring due to shell structure the 19 shapes are in general not expected to be the Poland states (except for very light nuciei) since the shell structure energy is only of order Aa3 . It is only with the additional advantage provided by its large moment of inertia (and an angular momentum I-A 7’6 ) that the I : 2 shape can becpme the lowest (for a given anguIar momentums.

Discovery of rotational The experimental

bands built on 1: 2 shapes ~stabiisbme~t

of rotational

band structure

based on the sllper-~ief~~r~le~l

(close shell in 2 : I potentials is a saga in its own right (which is more a~prop~ately

a topic for

Peter Twin later this morning) but it may be appropriate bcre to just indicate that the dccisivc clcment that has been exploited in extracting the signal from the noise is the approximate the energy difference of the successive

constancy

of

transitions in such a band;

E(+g(i+l) qO=E

(&-E (I-2)

Thus by searching for coincident pairs of transitions with energy differences that are integer multiplies of a basic unit, the rem,arkable spectrum of Fig. 5 was found by Twin and his collaborators about 3 years ago. The study of this sequence and the search for more have been the focus of vigorous experimental efforts in many laboratories since that time. At present about 20 similar sequences

have been found in neighboring

nuclei around A - 152 and another 20 in the predicted

region around ( 2 = 80 N = 114), I shall not attempt here to survey the variety of effects that have been discovered

in these studies, but rather coj~~er~tr~~tcon a single striking pf~~nomena that has

been observed and that appears to be connected pseudo-spin

that was inaoduced

Adler) more than 20 years ago.

in an interesting

by Arima and his collaborators

manner

with the concept

(and independently

of

by Hechr and

B. MotteLron/ Very deformed rotatingnuclei

8c

8-

SUPERDEFORMED 26

BAND IN

‘52Dy

33 30 32

34 3G 38 40

8

42 44 46

Ey ( MeV) Fig.5:

Gamma rays in ‘52Dy. (Twin et al Phys. Rev. Lett. 57 , 811 (1986)).

Coincident

The indicated gamma rays are all in coincidence band extending

with each other and define a rotational

from 1=24?i to I=6Ofi.

Equality of transition energies in different nuclei and pseudo-spin. As I mentioned have been observed comparison

above, a considerable in neighboring

nuclei

number of rotational bands built on 2 : 1 deformation in the appropriate

regions

of the periodic

table.

Careful

of these bands has revealed that in a large number of cases the observed transition ener-

gies for bands in adjacent nuclei are almost identical (for an example see Fig. 6) the accuracy of the coincidence

is of order

AE, -_ EY

10-3

B. Mottelson / Very deformed rofatingnuclei

9c

and this is indeed remarkable when we remember the relation (3). Even aside from expected variations in the moment of inertia, J, the fact that one member of the pair is an even A nucleus with integer

I while the neighboring

odd A nucleus has odd half integer

I implies

AIT, __,_!__,*-2 E, 21

6)

It is here that pseudo spin comes in.

2 0 s

-2

x

600

800

1000

1200

1400

Ey (rev)

Fig.6:

Con~parison of &r~~nsit~on energies for stiper-deforIi3ed mtation:~l bands in 152~~,151Tb ,and’50Gd.

3. Mottehn / Very deformed rotatingnuclei

1oc

Very briefly the concept of psudo-spin

(Hecht and Adler, NucLPhys. m,

is built on the following observations

129 (69); Arima, Harvey, and Shimizu, Phys.Lett. z,

517

(6%) : a)

The observed

single particle spectra in spherical nuclei exhibit a number of approximate

degeneracies of orbits with the snmc p:wityand Aj=l :

b7/2,

d5/2)

@3/Z,

%/2)

@9/2t

f7,2>

us/z,

P3/2)

for N = 50 - 82

for N = 82 - 126

(~11/z,g9/2)

for N > 126

It should be emphasized that at the present time there is no deeper understanding of the origin of these (approximate) degeneracies. They are just “facts” which, as we shall see, can be exploited. (b) The reduced matrix elements

1 if 4r + e, + A = even 0 if et + & + X = odd

B. MotteLFonJ Very deformed rotatingnuclei

a set of relations that can be very roughly expressed

llc

by saying that the “shape” of a quantum

state /lj> only depends on the total angular momentum

j and not on the orbital quantum

number I . Thus the matrix eIements that describe the coupling of spherical singfe particle states in the presence of a non-spherical

potential will be the same for the pairs in (a) as they

wonld be for spin orbit pnrtners (aside from r:tOi:~lmatrix elements) ”

(f)7/2

5/Z

@)3/2 l/2

G&/2

712

b&/2 3/2

(911/z

for N = 50 - 82 I

for N = 82 - 126

9J2

W/2

S/2

(83/2

1J2

for N > 126

where the pseudo orbital angular momentum is denoted by the usual spectroscopic notation with tilda, I-. The degeneracies of the pseudo spin-orbit partners in the spherical potential thus imply Z-foId pseudo spin degenencies for each intrinsic state in the deformed potential. One particIe motion in a rotating nucleus is described by the cranking model hamiltonian

where h, describes transformation

the non-rotating

to a coordinate

system while the Coriolis term - CD&results from the

system uniformly rotating about the x-axis with frcqucncy (I).

Since

The pseudo spin doublets of the deformed potentiaI IQ, describe in bf above are split by the Coriolis term in (7) which even for a very small rotational frequency o will align the pseudo spin with respect to the rotation axis (x-axis in (7)),

B. Mottekon / Very deformed rotating nuclei

12c

The complete

alignment of the pseudo spin described by (9) implies that in going from nn 1

even to an odd nucleus the extra y Ti in the angular momentum

is provided by the last odd par-

ticle.

Thus equality of transition differences

energies

in even

in the angular momentum

A and neighboring

I can be understood

odd A nuclei despite

the

in terms of the very weak pseudo-

spin-orbit coupling.

In our understanding ing why the moments

of the energy identities we are still left with the necessity of understandof inertia in two different nuclei are so nearly equal.

moment of inertia depend on the mass, radius and deformation,

Indeed since the

6,

we expect

AE,

AJ

1

-F-7A-1a-2 Y

Thus at the present

time we do not understand

although it is clear that the concept of pseudo-spin-

why the y - rays come out so nearly equal, is playing an important role.