Microscopic description of rotational spectra including band-mixing. - I. Formulation in a microscopic basis

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Microscopic description of rotational spectra including

band-mixing. - I. Formulation in a microscopic basis

F. Brut, S. Jang

To cite this version:

Microscopic description

of rotational

_spectra

_including

_band-mixing.

I.

Formulation

in

a

microscopic

basis

F. Brut and S.

_Jang

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

(Reçu le 24 mai 1982, accepté le 5 _{juillet 1982)}

Résumé. - La théorie de

projection

du mouvement collectif est _{utilisée pour décrire}

_l’énergie

de rotation avec

mélanges

de bandes. _L’énergie est écrite comme un

_{développement}

_{perturbatif en}

fonction des

_puissances

inverses

d’une

_quantité

_{reliée à la valeur moyenne de}

_{l’opérateur}

_Jy2.

Le

_{présent approche}

est discuté en relation avec les

différences sur les fonctions d’onde individuelles obtenues par resolution des équations variationnelles _qui sont écrites avant ou _après

_projection.

En

_plus

des différentes

_quantités

familières

_qui

_apparaissent dans la formule

phénoménologique

de _{l’énergie,} tels que le paramètre d’inertie, le facteur de

_découplage

et l’élément de matrice de _mélange

_pour

_{|0394K| = 1,} d’autres

_quantités

moins familières _ayant les facteurs de

_{phase particuliers,}

(-

1)J+1

J(J + _{1), (-}

_{1)J+3/2(J -}

_{1/2) (J}

+

_{1/2) (J + 3/2),}

_(-

_1)J+1/2(J

+

_{1/2) J(J}

+ _{1), (-}

_{1)J J(J}

+ _{1) (J - 1) (J} + ₂₎

et

_[J(J

+

_1)]2

sont _également obtenues. Le terme de _mélange de bandes _{avec |0394K| =} 2 est aussi nouveau. Toutes

ces

_quantités

sont

_exprimées

en fonction de l’interaction à deux corps et _{des valeurs moyennes de}

_{l’opérateur}

_Jym,

où m est _entier, en utilisant le formalisme

_{particule-trou.}

La différence entre les moments d’inertie d’un noyau

pair-pair et du noyau

pair-impair

voisin, aussi bien que les effets du

mélange

de bandes sur le moment d’inertie sont étudiés. Tous ces résultats sont

_présentés

dans le but de faciliter les

_comparaisons

avec les termes

phénoménolo-giques correspondants

et de _permettre leur utilisation dans des

_applications

futures. Abstract. - Within the framework of the

projection theory

of collective _motion, a

_microscopic

_description

of the

rotational energy with

band-mixing

is formulated _using a method based on an inverse _power

_perturbation

expan-sion in a _quantity related to the

_expectation

value of the _operator

_Jy2.

The

_reliability

of the _present formulation is

discussed in relation to the difference _between the individual wave functions obtained from the variational

equa-tions which are established before and after

_projection.

In addition to the various familiar

_quantities

_{which appear}

in the

_{phenomenological}

_{energy formula,} such as the moment of inertia _parameter, the

_decoupling

factor and the

band-mixing

matrix element

_for

_{|0394K| = 1,} other unfamiliar

_quantities

_having the factors with

_{peculiar phases,}

(- 1)J+1 J(J

+ 1),

(- 1)J+3/2(J -

1/2) (J

+

_{1/2) (J}

+

_3/2),

_{(- 1)J+1/2(J}

+

_{1/2) J(J}

+ _{1), (-}

_1)J

J(J + _{1) (J - 1) (J} + ₂₎ and

_{[J(J + 1)]2}

are obtained. The

_band-mixing

term _{for |}

_0394K|

₌ 2 is also new. All these

_quantities

are

_expressed

in terms of

_two-body

interactions and

_expectation

values of the _operator

_Jym,

where m is an _integer, within the

framework of

_{particle-hole}

formalism. The difference between the moment of inertia of an even-even and a

neigh-bouring

even-odd _nucleus, as well as the effect of

_band-mixing

on the moment of inertia are studied. All results are

put into the forms so as to facilitate

_comparisons

with the

_{corresponding phenomenological}

terms and also for further

_application.

Classification

Physics Abstracts 21.10-21.60

1. Introduction. -

The purpose of the

present

paper is to _present a

_{general microscopic description}

of

rotational bands and their mutual

_coupling

for

strongly

deformed nuclei within the framework of the

projection

method. While the results are not all new,

the

_present

_approach,

which is neither a technical

method of numerical calculation nor a

sophisticated

formal

_{theory, provides}

in a

_systematic

_way a

tenta-tive

_{microscopic interpretation}

of the

phenomeno-logical

model

_description

of the rotational _energy and

_{band-coupling.}

The Peierls-Yoccoz

_{theory [1-3]}

is a method of

incorporating

nuclear collective motions within a

shell-model

_description

_regarding

the shell-model

wave function as a trial function in a variational

procedure.

One

_purely

quantum mechanical

_approach

for

_describing

the nuclear rotation is that of

project-ing

the deformed Hartree-Fock trial function onto

the

_subspace

of

_angular

momentum

_eigenstates.

This model has been

widely

applied [4-10]

to the

study

of rotational

_properties

_{of many nuclei.}

However,

the

original

Peierls-Yoccoz

_projection

method has the defect of

_leading

to an incorrect

nuclear mass. Peierls and Thouless

_{[ 11 ]}

_subsequently

proposed

a double

projection

method

yielding

wave

functions

_satisfying

both translational and Galilean

invariances and thus

_giving

the correct total mass.

Nevertheless,

the

_application

of the double

_projection

method to the rotational

_problem

is _very cumbersome and further

developments

of the Peierls-Thouless

theory

for numerical

_application

have not been

pursued

to date.

_Rouhaninejad

and Yoccoz

_[12]

and Zeh

_[13]

have then shown

_that,

when the variational

principle

is

_{properly applied}

so as to minimize the energy of the

projected

wave

function,

the total mass for the centre of mass motion comes out

_correctly.

The

formulation from the variational

_procedure

after

projection

leads, however,

to _very

_complicated

equa-tions and the relevant

_approximate

_way of

_handling

this

_approach

has been

_fully

discussed

_by

Kam-lah

_[14],

_Beck,

_Ring,

Islam and

_{Mang [15],}

Onishi

_[16]

and Villars and

_{Schmeing-Rogerson [17].}

If one

con-fines oneself

_only

to numerical calculations of the rotational

_problem,

the variation of intrinsic functions after

_projection

can be

_performed

without

_difficulty

provided

that the _{model space is} not so

_large.

Thus,

it has been shown that the numerical calculations in this line

_yielded

much

_improved

wave functions for

mildly

deformed nuclei

[18].

It is noted that the

_high

spin

states have also been studied

_[19]

in the frame-work of the

_microscopic

variational

approach.

For

highly

deformed

_{nuclei, however,}

it does not

appear

to matter much whether one

_performs

the variation

before or after

projection [20],

_except

probably

for

high spin

states. In this _sense, a

_{comparison [21]}

_J

between these two methods of variation used for the first three states of

2°Ne

is _very instructive. Kelson

[6]

has

_{already pointed}

out that _very little difference

occurs in the intrinsic wave functions for different J values of the states

_{in 160,}

obtained from the variation before or after

_projection.

_Indeed,

it can be shown

that when all

_quantities

in the

_equation

derived from the variational

_procedure

after

_projection

are

expand-ed in _powers of the

_quantity,

r =

2/(

j2

>,

then the lowest order

_equation

is

_nothing

but the

_ordinary

Hartree-Fock

_equation

and that the solutions of

_higher

order

_equations

can be considered as corrections to

the lowest order solutions. It has been further shown

_[12]

that as

_long

as one

_keeps

the constraint of

unique generating

function

_having

K =

0,

the

Peierls-Yoccoz moment of inertia is valid. However if one

takes into consideration

_band-mixing

with K =

1,

this will be modified.

_Therefore,

when the value of T is small

_enough,

which is the case for well deformed

nuclei such as the sd-shell

_nuclei,

it would be

_enough

to consider

_only

the lowest order or at most _up to

first order

_expansions.

The

_{overlap integrals}

in the

_{projection theory}

are

generally

small for nuclei of sizable

_deformation,

except

in the

_region

of Euler

_{angle B}

= 0 and also

in the

_{vicinity of B =}

n. It thus turns out that the

expansion

of the

_integrands

in both

overlap

and energy

integrals

in powers

of fl yields [3]

in first order

in

_p2,

the _{energy spectrum} formula

_analogous

to the energy formula of the Bohr-Mottelson model

[22].

A

_simple

_way of

_{investigating}

the two

_band-mixing

in the

_projection

method has

already

been

_discussed,

for

_{example, by}

Verhaar

_[23]

and

_{by Gunye}

and Warke

_[8].

_They

have both

_employed

the method of

writing

the K-mixed wave function as a sum of two

K-pure

Peierls-Yoccoz

_projected

wave functions with

mixing amplitudes.

The latter authors used Hartree-Fock

_projected

wave

functions,

whereas the former

author contented himself with a _pure formal

_approach

based on both the

possibility

of

partially expanding

the

_overlap

and energy

integrals

in _powers

_{of B}

and

using

the

_{single-particle approximation. Petry [24]}

has also

_investigated

the two-band

_coupling

_problem

within the framework of the Peierls-Yoccoz

_theory,

but this

_study

considered

_only

_{the very limited} case

of two K

_{= 2}

and K

_{= 2}

bands and the

_expansion

of the matrix elements in the

_vicinity

_{of fl}

= 7r was

completely ignored.

It is thus believed that a

systema-tic and

_{general microscopic investigation}

of such

band-coupling

deserves further

_exploration.

In the

_present

work we show a method of

develop-ing

the matrix

_equation

in the

_{projection procedure}

so as to expose

explicitly

the rotational

_ingredients

involved and

subsequently

formulate a

_general

_theory

of rotational

_band-mixing

within a

_microscopic

basis.

The discussion on the difference between the individual wave functions obtained from the variational

equa-tions established before and after

_projection

takes also

_part

in the

_present

_study.

In order to

_gain

an

insight

into the

_{physical meaning}

of the

_results,

comparisons

with the Bohr-Mottelson model are

made where the occasion arises. In section

2,

the

expansion

of the

_overlap

and _energy

_integrals

in powers of Euler

angle B

is discussed. In the

following

section,

the

_expansion

of these

_integrals

in inverse powers of a

_quantity

_{(designated by X),}

related to the width of the curve

_representing

the matrix element

K

I exp( -

ipJ y)

I K >,

is

_given.

In section

4,

the energy

spectra

are

_{finally expanded}

in inverse _powers

of

X,

using

a

perturbation

expansion.

The

cor-respondence

between the

_present

_description

and the Bohr-Mottelson model is discussed. In the last section the

_reliability

of the

_present

formulation is

first examined in relation to

_generalized

Hartree-Fock

type

equations

obtained from the more relevant

pro-jection

method. The rest of the section is then devoted

to detailed calculations of various

_quantities,

obtained

in the

_preceding

sections,

in terms of

_two-body

inter-actions and

_expectation

values of the

_operator

_{Jym ,}

where m is an

_integer,

within the framework of the

particle-hole

formalism.

2.

_Expansion

in Euler

_angle

_{powers. -}

_Let

_{I nK >}

be the intrinsic state of

_quantum

number K which is the

_eigenvalue

of

_J.,,

and of additional

_quantum

number n needed to

_specify

the intrinsic state. The

with

_{single-particle}

orbits. We confine ourselves here

to

_{axially symmetric}

Slater determinants and shall

not take into consideration more released

_symmetric

conditions.

Since the matrix element of the

_operator

_{exp( -}

_{ipJ y)}

between the intrinsic

_states

_{I nK >}

is

_generally

small

except

for fl

=

0,

_{one uses} often the

approximation

Actually,

this is too

_rough

and an

improved

estimate can be found

_using

a correction factor which

multi-plies

the Gaussian function on the

_right

hand side of

equation (1).

It is noted that the

_operator

_Jy

is the sum

of the individual

_{operator jy}

over all nucleons.

Ver-haar

_[23]

and Villars

_[4]

have shown that the matrix element in

_{equation (1)}

is also sizable in the

neigh-bourhood

_{of p =}

n and its

magnitude

varies

accord-ing

to the values of K. This

_implies

that the matrix elements of the _operators

_{exp( -}

_{ipJ y)}

and

_{H. exp( -}

_i#Jy)

_between

_InK>

_{and In’}

_{K’ >}

may be

expanded

as

and

where x

= t P

when

expanding

in the

neighbourhood

of fl

= 0

_{and x = t p’}

_{with p’ = 1t - P}

for the

expansion

around

_{B =}

n. The

_expansion

coefficients

aKKt

and

_{s(m) ±}

in which the

_plus

and minus

_signs

stand for the

_{expansion near}

= 0 and

_j8

= 1t,

respectively,

are then to be determined. The

_{signification}

of the

developments (2)

and

₍₃₎

as well as the

_meaning

of the

quantity

X can be understood

_by

considering

the

expression (2),

for

_example,

for the case n = n’ and K = K’.

Thus,

when _we

_put

X =

2 j y 2 >

in equa-tion

_(1),

all terms in the summation on the

_right

hand

side of

_{equation (2)}

amount to correction factors

having

various

_degrees

of

approximation

to

exp( - i p2

Jy

>).

It can be shown that the

_quantity

X is

_related,

in first order in

X2

of the

_{expansion (2),}

to the

_expectation

value of the

_operator

_Jy

_by

with

Equations

(4)

and

_{(5) imply}

that the

_quantity

X _may be considered as a mean value of the

expectation

value

Xnx averaged

over the rotational bands under

consi-deration. It is

_{quite important}

to remark that the

quantity XnK

has an order of 20 or more for

_strongly

deformed nuclei and its

largeness

insures

_quick

convergences of various

expansions

we

_performe

in

the

_present

formulation.

The

_expansion

coefficients

_aKKt

and

_EKKt

in

equation (2)

and

₍₃₎

_may now be determined

_by

expanding

the

_exponential

function

exp( -

_{ipJ y)}

on

the left hand sides and then

_picking

the same order terms of x on the both sides of the

equations.

We thus

obtain the relations :

and

The

_higher

order

_EKKt

can be obtained

_{by replacing}

Jy

by

HJy

in

_Equations

_(6).

In

_deriving

the results

(6)

and

_(7),

use was made of the definition of the

time-reversed state

and

where the

_plus

or minus

_{signs depend}

on whether K is

integer

or

_{half-integer.}

In the case where K =

0,

we have

with r = ₊ 1 for the

even-parity

band and r = - 1

for the

_odd-parity

band. In

_Appendix

_A,

some

_higher

order coefficients

_a(m)±

and

_e(m)±

are shown.

By referring

to the

_properties

of the

_operators

_Jy

with m =

0, 1, ...,

_some

important

selection rules _can

be established from

_{equations (6)}

and

_(7).

We see thus

The

_Wigner

rotation matrices

_dj

and

dkK,(j3’ = 1t -

fl)

can also be

_expanded

in _powers of x

= t P

and x =

fl’,

respectively, using

the

relations,

dkK,(j3) =

KK’(fl) j

(-

I)K - K’

dk, K(j3)

dK’K( j

and

_{dkK’( 1t -}

dKK’(7E

_B’)

=

-

I)J+K j

as

(-

1)J+K

_{dk,-K,(j3’)}

as

where

withK>K’>0.

Some

_higher

order factors

_yKK

which were used in the _present

_description

are

_given

in

_Appendix

B. The lower order results of the

_expansion

of

_diK,(f3)

for the

case with K = K’ are

already

known

[4,

22,

23].

In

view of the

_development

_(11),

we can

_put

and

The

_expansion

coefficients

_aKK±

can be obtained

using

the

_properties

of the

_Wigner

rotation matrix in rela-tion to

_{equation (11).}

These coefficients are

widely

used

_throughout

the

_present

_study,

of which two

coeffi-J+ 1

cients

_{aKK- ðx.l = -}

KK

_(-

1) 2(J

+

2) and aKK+

-2 KK

- {J(J

+

1) -

K2 }

are well known but the

_higher

order coefficients are not familiar

_(see

_Appendix

_B).

An

_important

fact is that the coefficients

_aKK±

follow the similar selection rules to those for

_aKK’

and

_EKKt.

3. Perturbation

_expansion

in inverse _powers of X. -The fact that the

_quantity

X is

_large

can be ensure the

convergence of

perturbative

series,

provided

that the

expansion

could be

_performed

in inverse _powers of X. When we substitute the various results of

_expansions

we have made so

far,

into the

_overlap

and energy

integrals,

we see

immediately

that the

only

factor

which contains X is the

_exponential

function

exp( -

Xx2)

and that the power of x in the

_integrals

is odd.

Indeed,

the

_integration

of the

_{overlap integral}

over

_fl

can be transformed into the

_integration

over

the variable x, and we

_get

where

with s

_{= 2 ’(q}

+

_r),

which are

_integers

_{because q}

and r

are both even or both odd. Since the

_quantity

X is

sufficiently large,

the main contribution comes from the first few terms of the summation on the

_right

hand side of

_{equation (14), whether p}

+ s is small or

large.

In view of this

fact,

the

_{overlap integral}

_NKx,

can now be written in inverse _powers of X as

-

with

These matrices

_nKK,

can in turn be described in terms

of known matrix elements

_by

means of the

rela-tions

_(6),

₍₁₂₎

and

_(13).

The _argument used for

_deriving

the result

₍₁₆₎

can

equally

be

_employed

for the _energy

_integral

in order to

expand

it in inverse _powers of X. Thus

The matrices

_hKK,

have the same forms as those of

nKK,,

except

for

_ai.K’

which are to be

_{replaced by}

_EKx-,

and these matrices can be

_{subsequently expressed}

in

terms of known matrix elements

_by

means of the

for-mulas

_{(7), (12)}

and

_(13).

_Thus,

for

_example,

_hKK’

=

( nK I H I n’ K ’ ) nKK.

The rotational _{energy of the} state

_{having spin}

J and

belonging

to band K is

_{given by}

If we want to

_incorporate

the

_perturbative

effects

arising

from the _presence of other rotational

_bands,

it is necessary to consider both the

_diagonal

matrix

elements and the

_off-diagonal

ones of the matrix

constructed from the ratio of the matrix

_[H]

to the matrix

_[N].

Here we

_distinguish

the difference between the ratio matrix and the ratio of the two matrices. The immediate aim is now to

_expand

such a ratio

needed in

_constructing

the ratio matrix

_owing

to the

antisymmetric

character of the

_product

matrix.

_Indeed,

the

_product

matrices

_[N]-l[H]

and

_[H]

_{[N ] -1}

are not

symmetric

with each other. Since the matrix

_{[N ]}

is not

singular,

we have an inverse matrix

_{[N ] -1}

which is

symmetric

if

_[N

_is

_symmetric.

Because the matrix

_{[N ]}

_]

is defined to be

_positive,

there exists a matrix

[N ] - 112

which is also

_symmetric.

As a _consequence, we cons-truct the ratio matrix as

which is now

_symmetric

and _possesses the same

eigen-value as of

_equation

_(19).

When substitutions of

equations (16)

and

₍₁₈₎

are made into

_equations

_(20),

we _may

_{finally expand}

the _energy matrix

_[JC]

in inverse _powers of X as

with

where

n’(0)

is the inverse of

n(o).

The matrices

n(m)

and

h(m)

are defined in

equations (16),

(17)

and

(18)

and

here we have for

_simplicity

omitted the matrix

sym-bol

_[

_]

in

_{equations (21.1 )}

and

_{(21. 2).}

The

_higher

order matrices

H(m)

_up to m = 4 _are

given

in

Appen-dix C. It is remarked that when m =

0,

equation

(17)

reduces to the

_identity

matrix for even-odd

nuclei,

and becomes

_diagonal

for even-even nuclei with

nKK-

=

bxx-

for non zero

_integer

values of K and with

n(O) = [I

+

_{r( -}

_{1 )’}

_6KO]

_6KK’

for K = K’ = 0. The

matrix

n’(O)

is

_diagonal

and non

singular,

and the

matrix

H(o)

is also

_diagonal

in the _present represen-tation.

4.

_Energy

level - In the

presence of several

rota-tional

_bands,

the

_perturbed

_energy level of

_spin

J

belonging

to the rotational band

_Ki

is now

_expressed

in inverse powers of X as

,

with

Here,

the notation

_Hijm)

stands for the matrix element between the states

_Ki

and

_Ki

of order m in _the expan-sion of the matrix

[JC]

of

_{equation (21).}

_{We, see}

that

Ekm)

with m >, 1

_represent

correction terms to

unper-turbed _energy

_E’O).

It is worthwhile at this _step to

mention

that,

because of the

_expansions

₍₂₎

and

₍₃₎

we made at the

beginning,

the usual

_expanding

factors

like

f sin p

dfl ft’

N o(fJ)/

f sin #

dfl

NO(fl)

which stroll

through

the formulation in other studies of the

pro-jection

method do not _appear in the _present work. The first order correction

_E(K,)

has

_only

_diagonal

matrix element but the

_higher

order corrections have

_off-diagonal

matrix elements as well as the

diagonal

ones. It is noticed that the

necessity

of

having

the factor

_[J(J

+

₁₎

-

K 2]2

as well as the

band-mixing

term

_for

_{I AK}

_I

= 2 urges _us to the evaluation

of the

_{perturbation expansion}

_up to fourth order in

1/X.

Generally, perturbation theory

can be

applied

to the

_{band-mixing problem}

with confidence when the

_off-diagonal

matrix elements are small

compared

to the difference between the

_diagonal

matrix elements for different K. In other

_words,

the

_perturbation

expansion

is valid

_mostly

for the case in which the

heads of the bands in consideration are located

sufficiently

far from each other

_and/or

the

_off-diagonal

matrix elements are small.

Naturally,

the

applicability

of the

_present

_{approach depends}

to a _great extent

on the value of X.

4.1 FIRST ORDER ENERGY CORRECTION. - In the first _{order energy}

_correction,

we have

_only

the

diagonal

matrix element which is written in terms of

n(m)

and

h(m)

_{having m}

= 0 and 1.

It is remarked

that,

except

for the case

_{of Ki}

=

t,

the first order correction _to

EK°

is

independent

of

spin

J and this

term

_{multiplied by}

_{1 /X}

is known

_[12,

13, 15,

22]

_{as _ j y 2}

> / e,

where e is the moment of inertia of Peierls-Yoccoz. The terms in the second bracket is related to the

_decoupling

_parameter of the Bohr-Mottelson model.

4.2 SECOND ORDER ENERGY CORRECTION. -

With the aid of the matrices

h(2)

and

n(2)

in addition to the matrices

h(1)

and

_n(l),

the second order _energy correction terms _may be written as

with

_Ei(lo)

=

2-(E(Ko,)

₊

E(O»

II

2B

x, x,

This formula can be

_{subsequently expressed}

in terms of known and familiar matrix elements as in

equa-tion

_(25).

To

_{simplify lengthy expressions,}

we introduce the abbreviated

notations,

With these notations we have

with

Here we

find,

in addition to the famous factor

_J(J

+

₁₎

-

Ki2,

the

_band-mixing

element

_for

_{I AK}

_I

= 1. It is

interesting

to note that the

_present

_{study gives}

a

_particular

term which

_couples

two

_Ki

_{= 2}

bands

_having

diffe-rent structures.

_{Correspondences}

between the terms from the Bohr-Mottelson and the

_present

result are

dis-cussed in detail at the end of this section.

4.3 THIRD ORDER ENERGY CORRECTION. - Instead of

writing

down

_complete

and

_{lengthy expression}

of the third order energy

correction,

we will confine ourselves here to some

particular

results contained in

E(’)

in

order to understand the various contributions

_coming

from the third order correction. The correction

_{EK; (3)}

can be classified into terms which are

_independent

of

_spin

_J,

those terms which contain the factor

_J(J

+

₁₎

-

K 2,

the terms which

_couple

the different bands and

_finally

those terms which are

_{specific only}

to

_K,

_{= t, 1}

and

_t.

One

surprising

result of the

_present

_description

of the rotational energy is that even the third order energy correction in

_{1 /X}

does not

_yield

the terms

_containing

the factor

_[J(J

+

_{1) -}

_Ki2]2

and it is necessary to calculate the fourth order correction in order to find the square

of J(J

+

₁₎

as well as the

band-mixing

term

_for

_{I AK}

_I

= 2.

Actually,

E )

contains the elements like

_{aK;Ki and(a (2)+)2}

which are

respectively proportional

to

[J(J

+ 1) -

Ki2]2

but the sum 8

oe KiKi - 2(aKiKi

which appears in

E(i K

vanishes and there is thus no term

_having

_{the square of}

J(J

+

₁₎

and this is the reason

_why

several studies in the

_past

have

predicted

this term but have never formulated

1

There are other terms which are

_proportional

to

_{( -}

I)J+

2 (J

+

t)

J(J

+

_{1 )}

and which are

specific

to

_Ki

=

1

only, namely,

1

The

_quantity

is

_specific

to

_Ki

_{= 2}

and these terms _{may be}

_compared

with the terms

_having

the same J factor in the

phenome-nological

energy formula.

The

_off-diagonal

matrix elements such that

_{hil(2) , h)§ (1)}

etc... are related to

_band-mixing

terms between

_Ki

and

_Kl,

satisfying

the relation

_Ki

=

Ki

_±

1 or Ki + Kl

= 1. It is also _seen that the

band-mixing

terms

having

the matrix

elements

_{h)§ (1), hlm}

and

_h(1)

are allowed

_only

for

_Km

=

K,

and

Ki

=

K,

± 1 or

_Ki

+

_K,

= 1. These

band-mixing

terms are rather

_lengthy

and less

_important

than the one obtained in the second order correction so that we do not

_give

any

explicit

forms of them.

4.4 FOURTH ORDER ENERGY CORRECTION. - The fourth order

energy correction is

composed

of terms

similar to those found in the third order correction : terms which are

_independent

of

_spin

_J,

terms with the factor

J(J

+

1) -

K 2,

band-mixing

terms for

_Ki

=

Kl

± 1 and

_Ki

+

_K,

=

_1,

and those _terms

specific

_to the

Ki

=

1

1, 2

and 2 bands. In

_addition,

we have the term

_having

the square

of J(J + 1)

of the

form,

There is no terms

_proportional

to the cube

of J(J

+ 1 )

and we would have to _go to

_higher

order corrections in order to find them.

We have three more new _terms, of which one is

specific

to

_Ki

=

2,

and the second is

_specific

to

_Ki

=

1,

The third is the

_band-mixing

element between

_Ki

and

_K,

satisfied

_by

the

_conditions

_K1-

_Kll

_I

= 2 _or

Ki + K, = 2, viz:

with

Rowe

_[25]

has studied the

_possibility

of two-band

_{mixing for}

_{I AK}

_I

= 2 from _a

phenomenological

point

of view

and concluded that the effects of this

_coupling

are

_negligible

for 183W

nucleus. Since this

_mixing

term is in the fourth order correction we have

_actually

to divide the

_{expression (35) by}

X 4

and the effect of the

_band-mixing

with

I AK

I

= 2 is thus _seen to be _very small.

Gathering

all the results we have established so

_far,

we are now able to write the rotational _energy level for

where the

_band-mixing

matrix elements

_M(2)

and

_MiB4)

are

_given

in

_{equations (28)}

and

_(36).

_Explicit

forms of some of the functions

_{designated by W(m),}

where m stands for the

expansion

order in

X,

are shown in

_Appendix

D. It is noticed that for the purpose of

understanding

the

_physical

_meaning

of the

_quantities

contained in the _present

description only

the

_leading

order contributions to each

_geometrical

factor of J are

_explicitly

shown in equa-tion

_(37).

At this _stage, it is _very tentative to _compare the formula

₍₃₇₎

with the

_{phenomenological expression}

given by

Bohr and Mottelson

[22], namely

or with similar forms which can be obtained

by

replac-ing

J(J

+

1) by J(J

+

1) -

K 2. However,

much

care is needed whenever a direct

_comparison

is made

between two

_{corresponding}

terms in these

_formulas,

since it is not obvious that these terms _{carry the} same

weight

in their

_respective

contributions. For

_example,

the term with the factor

_[J(J

+

1) -

K 2]2

in the

present

study

appears

only

in the fourth order in

1 /X

and therefore the contribution of this term to the total energy should be,considered with relation to all other

contributions from the fourth order correction.

More-over,

_{equation (38)}

contains no

_band-mixing

terms

neither

_for

_OK

_I

= 1 _nor

for

I AK

I

=

2,

whereas the

present

description

takes into account the effects of

band-mixing

without

_introducing

the Coriolis force.

Therefore,

the

_{interpretation}

of the factor

_{A l,}

for

example,

as the

_diagonal

effect of the Coriolis forces

acting

in the

_rotating

coordinate

system

may not be

applied

without caution to the

_{corresponding}

matrix

element - 2

HiJ y

>

in

_{equation (37).}

Various

terms

_arising

in the

_{particle-rotor}

model _may

_provide

however a _way of

_{understanding}

the

_{corresponding}

terms in

_{equation (37).}

The

_{explicit expressions}

similar to the factors

_B1

and

_B2

in

_{equation (37)}

can be obtained from the

functions

_W(2)

and

_W(3),

and these have been

_already

shown in

_{equations (30)}

and

_(34).

The factors

_B3

and

_B4

cannot be obtained in the

_present

work as it is limited to fourth order in

_{1 /X}

and it would be

neces-sary to _go to

_higher

order

_expansion

in order to

include them.

For even-even

_nuclei,

the _energy

_spectrum

_up to

lIX2

order becomes

In

_deriving

these

_formulas,

use is made of the moment

of inertia of

Peierls-Yoccoz,

0 =

(XI2) 2 /

HJy

>ii.

The function A’ is the correction terni to the inertia

parameter

A and this comes from the

_mixing

of the

Ki

= 0 band with

K,

= 1 bands. The

expression

(39. 3)

is a concise form of the correction which has

_already

been

_{predicted by}

Yoccoz

_[12]

but has not

_yet

been formulated.

_Apart

from the

_band-mixing

_term, the formula

₍₃₉₎

is well known and this is to be

_compared

with the more

_general

formula of Beck et al.

_[15]

and Villars et al.

[17].

Because of the severe _symmetry condition we have

_{preserved throughout}

the

formula-tion,

we cannot have the terms

_having

the

_expectation

values of the

_operators

other than

_J;a.

5. Self-consistent field - Before

pursueing

further it is now _necessary at this state to discuss the

diffe-rence between the intrinsic wave functions obtained

variationally

before and after

_projection.

For

sim-plicity

we confine ourselves here to the case of even-even nuclei with axial

_symmetry

and we follow

_closely

the method of reference

_{[1 2].}

The variational

_{principle applied}

to the state

pro-jected

out of the Slater determinant constructed with

individual wave functions u

yields

where

_{P J}

is the

_{Legendre polynomials}

and

_N(fl)

is the matrix element of

_{equation (2)}

for even-even nuclei with the intrinsic states which _may now

_depend

on J. The function

_F(M,

_fl),

which is also a function of various

_energies

_involved,

is shown

_explicitly

in

Appendix

E. When all terms in this

_equation

are

expanded

in powers of

X,

and further the wave

func-tion u is

developed

in inverse _powers of X as

we obtain then a rather

_{complicated equation}

which

contains different orders of

_{F(u, /3),}

where b2

is the coefficient of the

_#2

term _{in the}

expan-sion of

_N(fl)

and the

_averaged

_7P

is

_X/4

times the factor shown in the discussion which follows equa-tion

_(23).

In view of the

_explicit

form of

_{F(u, B) given}

in

_Appendix

E,

it is seen that the lowest order

equa-tion,

F(o’)(u’) =

0,

is

_nothing

but the

_ordinary

Har-tree-Fock

_equation.

If there exist a solution of the lowest

_order,

then the first order

_equation,

which

becomes now

F 0 (’)(uo,

ul)

+ F

F 2 (0)(uo)

= 0 and which

is

_independent

of

J,

can be solved

_by

the matrix inversion method. Since the

_quantity

X is

_large

for well deformed

nuclei,

it is believed that the

considera-tion of the soluconsidera-tions of u _up to first order would be sufficient for the correction to the lowest order solu-tion. This shows

_clearly

the limit and the

_validity

of the method which use the

_ordinary

Hartree-Fock

solutions in the

_projected

formulas. For

_example,

it was shown

[12]

that the old formula for the moment

of

inertia,

e,

is still valid when the

_ordinary

Hartree-Fock solutions are

used,

since the

J-dependent

correction to

u(O)

appears

only

in the second order in

_{1 /X}

and thus

_modify

the rotational energy in the

terms in

_I/X4

order. It should

_{be, however,}

noticed

that the _argument used for the discussion of

reliabi-lity

of the lowest order solution in the rotational

problem

makes not much sense for the centre of mass

problem,

since the first order solution is as

_important

as the lowest order solution in the calculation of

the translational constant.

In what

_follows,

the intrinsic

_state

_{I nK >}

will be considered to be a self-consistent solutions of the Hamiltonian of

_strongly

deformed nuclei in an

_axially

symmetry

field. It is desired that

_the

_{I nK >}

is at least the first order solutions of

_equation

_(42).

When the self-consistent basis stands

_only

for the solutions from the

_ordinary

Hartree-Fock

_equation,

then the for-mulation below has to be

_regarded

in the restricted

sense.

The zero order _energy in

_equation

₍₂₂₎

now becomes

which

_implies

even values of J for K =

0,

_{as was}

expected.

Suppose

now we have a set of states in which

i >

= ai

0 >

represents

a

_one-particle

_state,

I rx >

= b:

0 >

represents

a one-hole state and

I irx >

= at b/

I 0 >

represents

a

one-particle

one-hole state and so _on,

_where

_{I 0 >}

stands for the

_ground

state of an even-even nucleus which _may

subsequently

become the core for the nearest even-odd nuclei under consideration.

Here,

italic indices

_{i, j,}

k... are

associated with

_particle

states and

_greek

indices

a,

fl,

y... are associated with hole states.

The nuclear Hamiltonian is now written in the form

where si and e,,,

are

_{respectively energies}

of the

_particle

namely, Hmn

stands for the

two-body

interaction term

having

m creation and n annihilation _operators

(par-ticles

_{or/and holes).}

The _operator

_Jy,

written in the same

_{particle-hole}

formalism,

is

_{given by}

where (

where three

_operators

on the

_right

hand side

_represent

a _constant,

_one-body

and

_two-body

_operators,

res-pectively (see Appendix F).

The

_operators

_j3

and

_j4

which have been used for

_explicit

calculations of some

matrix

_elements,

can be

_expressed

in second

quantiza-tion forms.

Indeed,

the

_operator

_j3

is

_composed

of

one-body, two-body

and

_three-body

_operators,

whilst the _operator

_j4

is

_composed

of a _constant,

_one-body,

two-body, three-body

and

_four-body

_operators. For actual

_{calculations,}

we could

_equally

use the

_product

forms of these

_operators,

_namely,

_j3

_{= Jy}

x

j2

and

j4

=

jl

x

Jy

_{by introducing}

a

complete

set

of

intermediate states between each _component

_operator.

Since H and

_Jy

commute each

_other,

the matrix element

_{(27. 2)}

becomes

where the

_prime

on the

sigma

indicates

_(n’

K’) #

(ni Ki), (n, Ki).

Generally, E I n’

K’ > n’ K’

I

repre-sents the sum of the

_complete

set and satisfies the closure condition. Because of the selection rules

implied by

the _operator

_{J y m}

, not all intermediate

states,

expressed

as a sum of

_n-particle

and v-hole

states, are allowed and thus the

_{specification}

of the

initial and final states determines the nature of the intermediate states.

5.1 BASIC _QUANTITIES. - 5.1.1

XnK

and X.

-We have shown that the

_{quantity X}

is

_closely

related

to

_{X nK}

which is the

_expectation

value of the

_operator

j2

with

_respect

to the intrinsic

_state

_nK

_>.

In the

_case

_{I nK >}

_represents

the

_ground

state

vacuum

10), XnK

is

_{given by}

whereas,

if I nK >

represents

the

_one-particle

one-hole

state,

_ap

ba +

I 0),

we have

The

_prime

on the

_sigma

denotes the omission of the

terms k = _p and

y = _a from the summation.

A

_simple

but

_important

case is the

_one-particle

states of an even-odd nucleus

_for

_{nK ),}

which is constructed

_{by applying}

a

_particle

creation

_operator

to the vacuum

_{represented by}

the

_neighbouring

even-even nucleus.

_For

_InK)

= a’

_{10 >,}

we have

The first term is

_nothing

but the contribution from the _core, which is the

_neighbouring

even-even

_nucleus,

and two other terms

_represent

corrections

_arising

from the removal of a

_one-particle

state from the core.

When we

_put

_{XnK=X +åXnK}

with

åXnK= -ai.k.+,

we _may assume that the X _represents a core

contri-bution and

_AX,,K

is a small correction. In most cases,

LBX nK

is seen to be small and we have in

_particular

X =

X nx

for the

ground-state

band of an even-even nucleus.

5.1.2

₍

_{Hj y 2 > ii}

and

₍

_HiJy

_{> i - j.}

-

The first order energy correction which contains two basic matrix

elements (

_{H y >}

and

₍

_{HiJy >}

is _very instructive for

_{understanding}

the

_meaning

of various matrix elements in the different orders of the

_expansion.

The first

_part

of

_{equation (25),}

for which we use the

nota-tion

_Ej/)+

in accordance with the

_meaning

of the

sign ( + ),

is now

where the

_prime

on the summation denotes that

(n’K’) :0 (nK).

The above form of the

_expression

appears

throughout

the

present

theory

and we there-fore treat it with several

_{specific assumptions}

about the intrinsic

_state

_{I nK >.}

Because of the

_{operators H}

and

_J’,

the choice of the intermediate states follows

from the nature

_of

_nK

_>.

For

_example,

the constant

operator

of

_j2

in

_{equation (46)}

as well as the kinetic

energy

part

of

_{equation (44) give}

no contributions to

_E(1)+

a)

Even-even

_nuclei,

- An

interesting

case is when

nK >

represents

the vacuum with K = 0. The

allow-ed intermallow-ediate

_{states n’ K’ >}

are then either

one-particle

one-hole states or

_two-particle

two-hole states.

_But,

the

_one-particle

one-hole states

_give

no

contribution,

since the matrix elements of the

Hamil-tonian between the vacuum and

_one-particle

one-hole states vanish in the

_present

_{approximation.}

As a

consequence

Fig.

1. - The

diagram representation

of

_£11)+

for the

ground

state vacuum,

I nK > =

0 >.

intrinsic

_state

_{I nK >}

_may

_be,

for

_example,

an excited

band such as

_one-particle

one-hole state or

two-particle

two-hole state. In the former case, we can also

express

Ej/> +

in terms of V and

_Jy

and this is

perform-ed in

_Appendix

G where we have also shown in

figure

6 the

_{diagramatic representation}

for the

one-particle

one-hole case.

b)

Even-odd nuclei. - A

simple

form for the intrin-sic

_state

_{[ nK )}

for an even-odd nucleus is the

one-particle

state

_a’

₀

_),

_generated

out of the

neighbour-ing

even-even nucleus core. The allowed intermediate states then consist of

_one-particle

states,

two-particle

one hole states and

_{three-particle}

two-hole states.

Of these the

_one-particle

states which should be different from the

_one-particle

state

_of

_{I nK )}

cannot

contribute to

_E1/)+

in the

_present

_{approximation.}

The contributions from the

two-particle

one-hole

states and from the

three-particle

two-hole states

yield

These four terms are

_represented

_{by figure}

2. The first

term, illustrated in

_figure

2a is

_nothing

but the contri-bution from the core. As will be seen

_later,

the inertia

parameter

of an even-even nucleus is

_directly

related

by

the first term of

_equation

₍₅₃₎

and it thus

_implies

that the next three terms can be

_interpreted

as

correc-tion to this

_parameter,

arising

from the

_change

from the even-even nucleus to the even-odd nucleus. This

point

will be discussed

_again

in the next section.

I

Fig.

2. - The

diagram

representation

of

_{Ejl) +}

for the

one-particle

state,

I nK >

=

a’

0 ).

The second

_part

of

_equation

_(25),

which we shall denote

_E(K)-,

can be evaluated in the same manner as in derivation of

_{equation (53),}

_except

that the

inter-mediate

_{three-particle}

two-hole states are not allow-ed because of the

_one-body

_operator

character of

_Jy.

We have thus

The interaction matrix element

_Vpa,pk

is shown in

figure

3. In

_deriving

this

_result,

we have used the time-reversed state

Fig. 3. - The

diagram representation

of

_{Ell) -}

for the

one-particle

state, I nK >

_{= ap+}

0 ).

The energy

E(K) -

is related to the

_decoupling

para-meter.

5.1.3

_Decoupling

_parameter. -

By

analogy

with the

_{phenomenological}

_energy formula in which the

with

_{a = - t I J + I - t),}

we can _express the

corresponding

decoupling

parameter in the present

formalism. This can be

accomplished by combining

the first order _energy correction

_£11)-

of

_equation

₍₅₄₎

and the term

_containing

the factor

_J(J

+

_{1) -}

K 2

in the second order energy correction. Thus

This

_{microscopic decoupling}

_parameter

is to be

compared

also with the calculation

_performed

within the formalism of the

_approximate

_projection

method

_[26].

when we assume that the

two-body

interaction

_part

in the denominator has

_{approximately}

equal magnitude

to that of the numerator, equa-tion

₍₅₆₎

is

_{roughly proportional}

to -

_{2 (}

_iJy),

which is the

_{phenomenological}

result for the

decoupl-ing

parameter.

5.2 MOMENT OF INERTIA. - The second order

energy correction has a term which is

_proportional

to

_J(J

+

_{1) -}

K 2 and

which _may be

_simply

written as

In accordance with the

_{phenomenological}

model the factor

_multiplying

_J(J

+

1)

is

_{generally interpreted}

as the moment of inertia factor. It has been shown

_{[ 15]}

that the self-consistent moment of inertia of Thou-less and Valatin

[31]

reduces under certain conditions

to the moment of inertia of

_{equation (57).}

There exist various _ways of

_{obtaining expressions}

similar to

equation (57).

For

_example,

it has been shown that the Yoccoz formula of the moment of inertia follows also from the classical statistical mechanics

_[27].

For even-even

_nuclei,

the moment of inertia

para-meter for the

_ground

state band can be

expressed

as

where use has been made of the result

_(52).

The modi-fication to

_{equation (58)}

has been

_proposed

in a

num-ber of different ways

[28, 31]

in which additional

assumptions

have been made for the intrinsic states.

The discussion on the

_relationship

between _equa, tion

₍₅₈₎

and the

_Inglis

formula

_[29]

is _very useful.

Hu

_[30]

and also

_Rouhaninejad

and Yoccoz

_[12]

have related the

_Inglis

formula to that of Peierls-Yoccoz.

Let us first note that the Hamiltonian H commutes

with

_Jy

_{in every} basis. In the Hartree-Fock

approxima-tion,

that

is,

when we use

_equations

₍₄₄₎

and

₍₄₅₎

in the commutator

_[H,

_Jy]

=

0,

_we obtain then three

equations

amongst which one is

_particularly

interest-ing

at this _stage. It takes the form .

Thouless

_[31]

_]

has

_already

considered this kind of

equation

in different forms. The first interaction term

with V

_corresponds

to the

_one-particle

one-hole interaction and the second interaction term

corres-ponds

to the

_two-particle

two-hole interaction. When we relate

_{equation (59)}

to

_{equation (58),}

we

_get

When we

neglect

the

one-particle

one-hole

inter-action,

equation (60)

reduces,

in unit

_of1i2,

to

This result is to be

_compared

with that of

_Inglis.

There is

_{however, a priori,}

no reason that the second term

in

_equation

₍₆₀₎

is

_neglected,

_except

that the kinetic energy part is much

_larger

than the

_one-particle

one-hole interaction. An alternative _way of

approximat-ing

the interaction

_V,,,P,ki

in

_{equation (58)}

was shown

in reference

_[12].

In a similar way, we can also express the inertia

parameter

of even-odd nuclei in terms of the

two-body

interaction and this can be

_accomplished

_using

the results

₍₅₃₎

and

_(57).

_However,

what is

_interesting

is the amount of

_{change brought}

about

_by

the transi-tion from the even-even nucleus to a

_neighbouring

even-odd nucleus which is now

supposed

to be a

one-particle

state

_generated

from the core. Since

E,(,’)’

for the even-even nucleus is

composed

of the

contribution from the core and the correction due to

the addition of an extra

_particle,

the difference

bet-ween the inertia _parameter of the even-odd nucleus