On the structure of 225Ra

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On the structure of 225Ra

R. Piepenbring

To cite this version:

On

the

structure

of

225Ra

R.

_Piepenbring

Institut des Sciences Nucléaires, 53, avenue des _Martyrs, 38026 Grenoble Cedex, France

(Re.Vu le

_{16 juillet}

1984, accepte le 7

_septembre

₁₉₈₄₎

Résumé. 2014 Les

arguments habituellement _{avancés pour affirmer que le} 225Ra a une déformation

octupolaire

stable sont discutés en détail.

_{Beaucoup apparaissent}

comme nécessaires mais non

suffisants et sont donc à utiliser avec

_prudence.

Abstract. 2014 Various

arguments are _usually invoked to favour stable

_octupole

deformation for 225Ra.

A detailed discussion of these _arguments is

_given.

Most of them are _{necessary but} not sufficient

criteria for stable

_octupole

deformation, and must therefore be used with caution.

Classification

Physics Abstracts

21.60 - 21.60E

It is well known that the

_octupole

correlations

_play

an

_important

role in the

_region

of Ra and Th. A deformation

_{of Y 30}

_type is needed to understand the low

_lying

level structure of these nuclei. It is however not

_yet

_completely

clear if the

_{octupole degree}

of freedom leads to a stable

deforma-tion or to vibrations. A detailed

_{experimental study}

of

22sRa

is now available

_[1],

whose results

have been

analysed

in two recent letters

_[2,

_3]

and in a more elaborated article

by

Leander and

Sheline

[4].

The last three _papers show some evidence for stable

_octupole

deformation in 22sRa. Various _arguments are

_given

in references

_[2-4]

in favour of this

_{interpretation.}

The aim of the

present Letter is to discuss these arguments in detail and to show

_that,

at

_least,

some of those

appear to be not sufficient criteria for

_octupole

deformation.

Let us first list the different _arguments of the authors of references

_[2-4].

a)

It turned out to be difficult to

_get

a consistent

_description

of the different rotational bands within the Nilsson model

_assuming

no stable

_octupole

deformation

_{[2, 4].}

b)

The fact that the two K =

1/2~

bands have

decoupling

parameters a

of

_nearly

same absolute

values but with

_opposite

_signs

is another criterion for stable

_octupole

deformation

_[3].

c)

The _presence of two rotational bands K =

1/2

and K =

3/2

with

nearly degenerate parity

doublets is one of the most

_simple

criteria for stable

_octupole

deformation in odd-A nuclei

_[3].

d) Despite

extensive

_experimental

studies of references

_{[5, 6],}

one has not found _any vibrational

two-phonon octupole

vibrational 0+ state in 222,224 Ra at _{around double the energy of the}

one-phonon

state

_{[2, 3].}

We shall now

_analyse

each of these

_points

in detail. Five rotational bands are observed

_[1]

with band head

_energies

lower than 250 keV. One of these has

_clearly

been

_interpreted

as built

on the

_5/2

+ 633 Nilsson state. The usual Nilsson

_assignments

for the four other bands with

K =

1/2:J:

and K =

3/2:J:

_are less evident. Two well known

couplings

_are

expected

_to

play

here _an

important

role.

_First,

some of the

single

neutron

levels,

expected

in this mass

_region

for small

quadrupole

deformation,

arise from the

_spherical

_{orbits ~5~}

and

_i11/2

_{having large j}

values and

L-1024 JOURNAL DE PHYSIQUE - LETTRES

hence,

leading

to strong Coriolis

_coupling

between the deformed states. This

_coupling

has not

been

_explicitly

introduced in

_[2-4].

_Second,

the presence of a low

_{lying octupole}

state in the even

isotopes

induces a _strong

_coupling

of the odd-neutron to the

_octupole

mode. This

_coupling

is

just

the

_challenging

_problem

we deal with.

To describe the

_experimental

_data,

one _may use a

_priori

a model either with a stable

_octupole

deformation

_{(Fig. la)}

or with reflection _symmetry

_(Fig.

_lb).

In the first case the

_potential

_energy curve has two minima located at a deformation ±

_{63 5~}

0. The difference between the stable

_(la)

and unstable

_(Ib)

deformation

_depends

on the

_height

of the barrier at _{83 = 0} in

_figure

la. The level order

_{0 +, 2 +,}

_{1-, 4 +, 3 -}

observed in 2 24Ra and the level order

_{3/2 +, 5/2 +, 7/2 +,}

_3/2’

observed

in

_{225Ra suggest}

_(see

Ref.

_[7])

that this barrier is not _very

_high.

In that case the whole difference

between the two versions

_(a)

and

_(b)

becomes not so essential and _may even be reduced to

_using

different

_{representations.}

One _may therefore use either a reflection

_asymmetric

or

_symmetric

basis. The basis which is most

_{nearly diagonal}

_{will appear} to be the most suited to

_give

the best

interpretation.

Let us here start with the

_{symmetric assumption}

and look if a coherent

_description

of the low energy spectrum

given

in reference

_[1]

_is

still

_possible

in the frame of the unified model of Bohr and Mottelson. In this

_approach,

the odd-nucleus basic wavefunction still writes

where

_l/Jv

_represents now the intrinsic wavefunction

_taking

into account the

_coupling

of the

_single

quasiparticle

to the

_octupole

_vibration,

and

_replaces

the usual Nilsson

wavefunction

_/~. As

_usual,

the Coriolis forces

_couple

_{~’1 M K}

states

_with

_{I AK I =}

1.

The

_{phase factor p}

takes a value

_depending

on the definition of the

_{conjugate state 0-,.}

We test the coherence of the

_experimental

level scheme

_by

use of the effective matrix

_method,

developed

earlier

_[8]

and based on the inversion of the

eigenvalue problem.

For the

_{negative parity}

levels,

we restrict the basis

₍₁₎

to two intrinsic states

_having

K =

1/2

and K =

3/2.

The

problem

has then 5

parameters : the intrinsic

_energies

_Bl/2 and _83/2, the effective

~2

inertia

_parameter

_Aeff

=

h2

the effective Coriolis matrix element

_{C~ff = ~ ~ ~3/2}

_{!./+ ! ~i/2 ~ 12}

Fig.

1. -

Potential energy curves for stable

_octupole

deformation

_(part

a) and for reflection

_symmetric

and the effective

_decoupling

_{parameter aeff}

_{= -}

_~1~2

_{1 i I}

_{I ~-1~2}

_~~

It is _{very easy} to fit these five

_parameters

to the six observed

_energies

_{(table I).}

The

_adjusted

values of the

_physical

para-meters of interest here are :

For the

_positive

_parity

levels we can use either a basis

₍₁₎

with two intrinsic states, K =

1/2

and

K =

3/2,

_{or one} with three _states

adding

the K =

5/2

configuration.

With the first choice _we have

eight

levels for the same number of

_parameters

(five)

as in the

negative

parity

case. With the second choice we have two additional

_parameters

_{C’ = 1}

_05/2

_{1 il}

_{~3/2 ~}

_F

and _~5/2’ We then end with seven

_parameters

for thirteen levels. A

_satisfactory

fit can be obtained in the two cases. The

two-band

_coupling

leads however to a better

general

_agreement. We shall therefore restrict our

discussion to this case

(table

II)

where the

adjusted

values of the

interesting

_parameters

are

We _may add that these values are somewhat sensitive to the number

_and/or

_spin

values of the

levels

introduced in the fit.

_Giving

or

_using

too

_precise

values of the effective inertia

_parameter

or effective

matrix

elements

of j+

has

therefore

no

_{physical meaning.}

The

importance

of the

Coriolis

_coupling

can be seen

_{by comparing}

the values of

Aeff

and _aeff obtained here in

equa-tions

₍₂₎

and

₍₃₎

with the values

given

in

_[3]

_(and

_[4]

_{respectively)}

for the

_{negative parity}

band

_1/2

and

for the K =

1/2+

band.

The results shown in tables I and II

_clearly

demonstrate that one can

_obtain,

within the

col-lective model of Bohr and

Mottelson,

a coherent

_description

for the low _energy and low

spin

Table I. -

Observed and calculated

energies

E(I)

in keV

_of

_negative

_parity

in

225 Ra

_for

the two

L-1026 JOURNAL DE PHYSIQUE - LETTRES

Table II. - Same as in table

I,

but

_for

_positive

_parity.

spectrum of

225 Ra. The

_problem

is now to find an

_{interpretation}

of the

_empirical

values

_given

in

₍₂₎

and

_(3).

We first note that the values of the effective inertia

_parameters

do not differ too

much for the two sets of bands. Further

_they

are

_typical

of nuclei with small deformations of the considered transition

_region.

The

_{interpretation}

of the

_empirical

values of the effective

_decoupling

parameters

is more delicate. In terms _{of pure Nilsson}

_{configurations}

this is of course a

_puzzling

problem,

as has been discussed in

_[2-4].

For the

_{negative parity}

K =

1/2

band,

_one

_expects

the

_{1/2 -}

770 Nilsson orbit. For nuclei with small

_quadrupole

deformation

_(say

_{82 =}

_0.15),

the

single particle decoupling

factor _{asp, calculated} with the

_parameters

of Lamm

_[9],

is _{~p ~ 2013} 7.8

which has a much

_larger

absolute value than the effective one _aeff = - 2.22. For the K =

1 /2 +

band the situation is even more difficult. The two

_possible

candidates for the Nilsson

configu-ration are

1/2

+ 631 or

1/2

+ 640. With the usual

_parameters

of

_[9]

the first one lies too

_high

in energy

and the second too low. The same situation occurs if one .uses

_[10]

a Woods-Saxon

potential

with axial

_symmetry.

It has been shown

_{by Ragnarsson [2]}

that the introduction of

an

_octupole

deformation

_helps

to

_bring

the

_1/2

+ 631 down in energy. The use of the

sophisti-cated folded Yukawa

_{single-particle potential [7]}

indicates the same

_necessity

of an

_{~3 5~ 0,}

i.e. a reflection

_asymmetry,

to _get the

_1/2

+ 631 down in _{energy. This appears of} course as a

strong

evidence for stable

octupole

deformation. On the other

_hand,

it is well known that in order to _get the

_{single particle}

relative

_{energies exactly right}

in the calculation of odd-A nuclear

spectra,

it is necessary to make some

_{empirical adjustments,}

which _may be even somewhat

larger

than the

_{expected inaccuracy}

of the used

_{single particle}

scheme

_{[4, 11].}

Earlier work

_by

Rekstad and Lovhoiden

_[12]

has also demonstrated that a re-examination of the

_parameters

K

and ~

of the Nilsson

_potential

for the transitional

_regions

is _necessary. We therefore consider the

present argument

as less

_strong

and continue our discussion

_{by assuming}

that a

_symmetric

shape

may still be used. The

single particle decoupling

factors of

1/2

+ 631 and

1/2

+ 640

with the

_parameters

of

_[9]

are 0.86 and -

5.1,

respectively.

These values have

_nothing

in common

’with the

_empirical

value 1.3. The situation _gets even worse if one uses the

_parameters

of Chi

like in

_[3]

or a Woods-Saxon

_{potential [10]}

for which small

_negative

values are obtained for

asp( 1 /2

+

_631).

The situation becomes much better if one switches from the pure Nilsson

one has :

where

_{Q 3 o}

is the

_{octupole phonon}

_operator. The _operators

_{Cj, Cy , C~}

create

_{quasi particles}

with

_{same Qv}

_{= Qv,}

_{= SZ,,..}

= K values and

_{adequate parities}

_(e.g.

in

_Eq.

₍₄₎

v’ has

opposite

parity

of v or

_v").

The coefficients

_Sv,

_{Sv" Sv"}

can be obtained in a

_microscopic

_way

_{by using}

the

_multiphonon

method

_applied

to odd-mass nuclei

_[13].

The

_empirical

values of the effective

decoupling

factors are then to be

_compared

to the intrinsic

_decoupling

_parameters

_aint defined

_by

To

_push

the discussion

further,

we shall make two other

_assumptions,

which _{may appear} too

drastic,

but which are made

_only

to show that the

_argument

_(b)

on the

decoupling

_parameters

is not sufficient. We first

neglect Sv-.

and

_{higher multiphonon}

terms in

_(4).

Then

_{by taking}

pro-perly

into account the

_{conjugation properties}

of

_4>"

we have shown

_[14]

that

where P =

u;~ M~

+ v~ v~

1 is a correction factor due to the

_pairing

correlations. We

_simplify

further the _argument and assume that we can restrict ourselves to the two Nilsson orbitals

1/2 -

770 and

_1/2

+ 631. The latter choice is

_preferred

to

_1/2

+ 640

_according

to the transfer reaction results of

_[1].

In that

_{over-simplified}

case,

equation (4)

writes :

where

for

_negative

and

_{positive parity, respectively.}

The intrinsic

_decoupling

_parameters

of

_{equation (6)}

are then

simply :

From

_equation

₍₁₁₎

we

_immediately

see that the

_negative

value

_{of~sp(l/2 2013}

₇₇₀₎

_helps

to enhance the too small

_positive

or even

_negative

_{agp(1 /2}

+

631 )

and to _get a value

_comparable

to

_{a;n~(1 /2 +).}

On the other

hand,

in

_{equation (10)}

the fact that a2 I and the small value of

a sp(1 /2

+

631)

explain

a

lowering

of the absolute value of

_a;nt(1/2-)

_compared

to that

_{of asp(1/2 -}

_770).

One could continue the game.

By

choosing single particle

values for

_ap,

_{by using}

the

_empirical

values for

_~(1/2~)

and the normalization condition

_(9),

one could deduce the

_mixing

coefficients a,

~,

y, 5

of the wavefunctions

(7)

and

_(8).

_But,

_according

to the too drastic

_assumptions

made,

this would not have a real

physical meaning.

We

_simply

note

_that,

_by

_using

the

_asp

values indi-cated in

_[3],

one remains with

_plausible

values

for the

_mixing

coefficients. We checked that these values also allow a

_description

of the

_magnetic

moment with reasonable values of the

_{gyromagnetic factors gs}

and gR, as well as an

_explanation

Cri-L-1028 JOURNAL DE _{PHYSIQUE -} LETTRES

terion

_(b)

that the

_decoupling

_parameters for K =

1 /2 t

have

opposite

values,

used in

[3]

as a

criterion for stable

_octupole

deformation,

is

_certainly

_necessary but not at all sufficient as we

have

_just

showed.

_Besides,

this has been

_recognized

later on

_by

the same authors in references

_{[1, 4].}

It has also been claimed

_[4]

that further evidence for

_parity

mixed

_{single particle}

states

_(i.e.

reflec-tion

_asymmetry)

_might

be found in a deviation of the

_decoupling

factor from

_experimental

syste-matics. We feel that even this statement should be used with caution for the

_following

reasons :

it has been found that the calculated

_decoupling

factor

_{a(1 /2 -)}

for

_{83 #-}

0 is a

_{rapidly decreasing}

function _{of 83. A}

_slight

deformation

_change

from one nucleus to _{another may therefore}

_explain

strong

changes

in the

_systematics.

The same fact can as well be

_explained

within the reflection

symmetry

assumption.

A

_{slight change}

of the energy of the

octupole

vibrational state from one

nucleus to another

_changes

the

_mixing

coefficients a

_{and /3,}

_and,

_according

to the

_large

value of

the

~p(l/2 -

770) also

the calculated

_~(1/2’).

Let us come to criterion

(c),

which claims that the presence of two rotational bands K =

1/2

and K =

3/2

with

nearly degenerate

parity

doublets is _one of the

simplest

_ones for stable

octu-pole

deformation in odd-A nuclei. Here too, one can find some _arguments

_against

this assertion.

If this

_reasoning

works for K =

1/2

and K =

3/2,

it should also work for K =

5/2,

and ask for

a

_{5/2 -}

band in the

proximity

of the observed

5/2 +

at 236 keV. Such a

finding

has not been

_reported

in

_[1]

where the first

_possible

candidate appears at 1 225 keV. One could as well _{say that} oppo-site

_parity

states in odd-A nuclei can be attributed to unrelated Nilsson orbitals. This is

_especially

the case when two

_spherical

subshells of

_opposite

_parity

are

_closely

located in energy. And this is

what

_happens

for

_{jl si2,}

_ill/2

and even _g9/2 in this mass

_{region. Finally}

criterion

_(c)

which was

used in

_[3]

as an evidence for near stable

octupole

deformation,

has been rediscussed also

by

the same authors in

_[4]

where

_they

now admit that it is a _necessary but not sufficient criterion.

We now discuss _argument

(d) concerning

the lack of the so-called

two-phonon octupole

states at around double the _energy of the

_one-phonon

state.

_First,

we would remind the reader

that it _may be somewhat hasardous to

_extrapolate

from even nuclei to odd-A nuclei in this transition

_region

where the _presence of one extra odd

_particle

may

polarize

the even core in a

considerable _way.

_Further,

we would also refer to our recent work

_[15]

where we

_clearly

show that the first non-rotational excited levels of

222Ra

and

224Ra

can be

_explained

in terms of

octu-pole

vibrations

exhibiting large

anharmonicities. A _proper treatment of these within the

multi-phonon

method _suggests that the first excited 0+ states observed near 915 keV in these nuclei

may be

mainly

of an

_octupole

nature. It shows also that there are no lower 0 + states _{of this nature,}

explaining by

the _way the lack of the so-called

_{two-phonon octupole}

states at _{low energy. This}

explanation

obtained in a

_{microscopic approach using}

a reflection

_symmetric

basis must however

not be

_used,

_per se, as an _argument

against

stable

octupole

deformation. This could

only

be

done

_if,

in a similar

_calculation,

a reflection

_asymmetric

basis would appear to be less

_diagonal.

To our

_knowledge,

no such result is _{yet available.} We would also like to add that the

_language

used here may be somewhat

misleading.

Instead of

_speaking

of vibrations

_exhibiting

strong

anharmonicities,

one _may

_perhaps

rather refer to as finite

_amplitude

collective motion described in a

_microscopic

_way.

In summary, we have

_analysed

in all the details the various

_arguments

used in favour of stable

octupole

deformation of the transitional nucleus 225 Ra. Most _{of those appear} to be necessary

but not sufficient.

_Equal

success and a

good

overall

description

of the

low-lying

bands in 225 Ra

could be obtained either with or without reflection

_asymmetry.

The actual results do not

_really

tell

_anything

about stable or

dynamic octupole

deformation. The

only slight point

in favour

of stable

_asymmetry

is the

_possibility

to obtain a

1 /2 +

as a

ground

state, but even this

_argument

On the one

hand,

further

experimental

evidence is needed to assert stable

_octupole

defor-mation in this nucleus. On the other

hand,

microscopic

calculations in odd-A nuclei

_taking

into account

_higher

K = 0

octupole

excitations,

which have been shown _to be

important

in

even mass

_nuclei,

have to be undertaken.

Acknowledgments.

I would like to thank R. Chasman and Z.

_Szymanski

for valuable comments and

_interesting

discussions.

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