A nuclear field treatment of the 3-particle system outside closed-shell nuclei

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A nuclear field treatment of the 3-particle system

outside closed-shell nuclei

R.J. Liotta, B. Silvestre-Brac

To cite this version:

A

NUCLEAR

FIELD TREATMENT OF

THE

3-PARTICLE SYSTEM

OUTSIDE

CLOSED-SHELL NUCLEI

R. J. LIOTTA

_(*)

and B. SILVESTRE-BRAC

Institut des Sciences

Nucléaires,

BP

257,

38044

Grenoble,

France

(Re~u

le 15 novembre

1977,

accepte

le 7 decembre

1977)

Résumé. 2014

Le _{traitement par la} théorie du

champ

nucléaire du

_système

à 3

_particules

est mené à tous les ordres de la théorie des

_{perturbations}

sans toutefois inclure les excitations de trous. Il

est montré _{que la} théorie

_corrige

_proprement les violations du

_principe

de Pauli ainsi _que la redon-dance de la base.

Abstract. 2014 The nuclear field treatment of the

3-particle

system is carried out to all orders of

perturbation theory

but without

_including

hole-excitations. It is shown that the

_{theory properly}

corrects both the Pauli

_principle

violations and the

_{overcompleteness}

of the basis.

LE.

JOURNAL DE

_{PHYSIQUE -}

LETTRES

Classification Physics Abstracts

21.60E

One of the main features of the nuclear field

_theory

(NFT)

is that it treats

_{simultaneously}

the

single-particle

and the

_pairing

and

particle-hole

phonon

degrees

of freedom as

_elementary

modes of

excita-tion. The

_resulting

violation of the Pauli

_principle

as well as the

_{non-orthogonality}

of the

_{elementary’}

modes is corrected within the

formalism,

at least when

_applied

to

_exactly

soluble models where the field

_diagrams

can be summed _up to all orders

_{[1, 2].}

However,

so

far,

it has not been

_possible

to _carry out such a summation in the

_general

case due to the

increasing complexity

of the

diagrammatic expansion

with the order of

_perturbation

in

_{1/~2 (0 = j}

+

_1/2).

In the

_present

letter we consider a

_3-particle

_system outside a closed shell core

_{interacting through}

a

two-body

force. To

_analyze

this

_problem

in the framework of the NFT we will

_only

consider

pairing

and

_{single-particle degrees}

of freedom while the

original

fermion hamiltonian will be

_{replaced by}

the nuclear field hamiltonian

_[3].

where

_r~(~)

is the

_{pairing phonon}

creation

_operator,

c~n(~,)

is the

_{corresponding phonon}

_{energy, A}

is the

particle-vibration coupling

constant

_{given by}

and

where

We do not consider hole-excitations and therefore

only

TDA vertices are

_included,

as shown in

_figure

1.

The value of the

_phonon

vertex A is

while the

_four-point

vertex B is

_given

_by

L-12 JOURNAL DE _{PHYSIQUE -} LETTRES

The

_pairing

field is

_determined,

_according

to rule IV

of the NFT

_{[1], by}

the

_equation

Each state of the basic set

_{

_{OC j ;}

_{I }}

=-

_{{ ~/L,}

_{iji ;}

_{I }}

- where

{ ~, ~ ;

I}

identifies the

nth

_phonon

of

multipolarity ~, coupled

to the

ith

_particle

of

_angular

momentum ji

to

_give

the total

_angular

momentum I

of

the

_3-particle

_system - contains

only

one

_pairing

phonon

and one

_{single-particle}

state

_{(rule I).}

The

Brillouin-Wigner

diagrammatic

expansion

of the

FIG. 1. - Nuclear field

theory expansion of the effective inter-action matrix elements _Wa2a~(E) ’

effective interaction matrix element

_[4]

_Wazal(E)

shown in

_figure

1

_{(rule II)}

includes neither bubbles

(rule III)

nor _any intermediate state which is also a

basic state.

The zeroth order contribution to the matrix

ele-ment W is

To sum the whole

_diagrammatic

series of

_figure

1

we notice that the contribution of a

_diagram

of

higher

order n can be written as

where,

in first

_order,

the matrix element F is

_{given by}

The total matrix element

actually

represents the _propagator of the three

particles

which are

_originally

in the intermediate state

Assuming

that the _propagator F has been evaluated

to

_{order n,}

then one can evaluate the

_following

order, n

+

1,

making

use of the

diagram

of

figure

2 to _get

FIG. 2. - Graphical representation of the

propagator F.

where

Therefore the total

_propagator

F is

_given

_by

which is an

inhomogeneous

set of

_energy

_dependent

linear

_equations

in the set of

intermediate

states

for the _propagator F. For each value of the final basic state _a2 and as function of the _{energy E} one can _{solve eq.}

₍₉₎

to _get the total TDA interaction matrix element as

The

_{diagonalization}

of W

_provides

the wave func-tion

with the normalization condition

_[2]

The matrix elements of a

given

transition _operator

can be evaluated in the same _{way. For}

_example,

the

evaluation of the one

_particle

transfer matrix

basic states

_{{ ~3J,}

_{IB }}

that describe the

_physical

state

I

PB’ IB >

of the residual nucleus B

_according

to

eq.

(11),

can be

_{performed introducing}

the propa-gator T of

_figure

3.

_Proceeding

as for the

_propagator

F

FIG. 3. -

Graphical representation of the _propagator T.

one obtains

_(notice

that the _energy

_EB

is now

already

known

_through

the

_{diagonalization}

of

_W)

which is also a set of

_{inhomogeneous}

linear

_equations

that

_provides

- for

each value of the intermediate

state k(aa’)

AA ;

I > -

the _propagator T. After

solving

the

_relatively

_simple

energy

independent

eq.

(13)

the total TDA matrix element t can be obtained

as

and the

_one-body

transfer transition

_amplitude

to

the

_state

_{~is, IB >}

is

_{given by}

As an

illustration,

we will consider here a

single

particle

level model of

_{spin j}

and _energy s.

The total

_spin

I to be

_analyzed

is such that

_only

two values of ~ are allowed. In this

simple

case

eq.

(9),

which is a 2 x 2 set of linear

_equations

for

each of the two basic states, can

_easily

be

analytically

solved. The

_{phonon energies}

are

_{given by}

~ = 2 s + ;~/; ~ ! ~ ~’; ~ ) = 2 8 + ~

and the

coupling

constants

_by

_{An(jj; ~i)}

=

Vi,

After some

algebra

the matrix elements W are found to have the values

where,

with

and

The

_{diagonalization}

of W

_provides

the

_equation

where

The two values of E

_{given by}

_eq.

₍₁₉₎

may not

always

be

_physical.

To

_analyze

the situation in which both Pauli

_violating

and

_overcomplete

basic states

are involved we first consider the case

_{I = 3 j - 4}

(À1 = 2j - 1,~=2~2013

3)

where no

physical

solu-tion is allowed

_by

the Pauli

_principle

as can

_easily

be checked

_{by counting}

the allowed number of states

in the m-scheme.

_Utilizing

the Racah formula

_[5]

one finds that in this case

which

_{implies A}

= B = 0 and there is _no solution

to _eq.

_(19).

Another

_interesting

case is I =

3 j -

3

(also, here,

~,1

=

2 j -

1,

A2

=

2 j - 3)

where the

only

solution

allowed

_by

the Pauli

_principle

has the value

_(cf.

eqs.

(26.45)

and

(26.11)

of ref.

[6])

In this case one finds

which

_yields

A = 0 and the

_only

solution

_given

by (19)

coincides with the exact result.

_{Nevertheless,}

this

_only

_physical

state is described

by

the NFT in

terms of a two-dimensional

_overcomplete

basis.

Finally,

it is worthwhile to

_point

out that the NFT

gives

the correct answers in these cases

_through

the non-trivial and

_general

_{6 - j}

relations

₍₂₁₎

and

_(22).

In the above

model,

it is shown that the

NFT,

provided

that it is carried out to all

orders,

gives

exactly

the results of the shell-model. We have some reasons

to

feel

that this is a

_general

feature of the

theory

L-14 JOURNAL DE _{PHYSIQUE -} LETTRES

References

[1] BES, D. R. et al. Phys. Lett. 52B ₍₁₉₇₄₎ 253. [2] BES, D. R. et al. Nucl. Phys. A 260 (1976) 1.

For a _general review of the NFT as well as references see

BROGLIA, R. A., BORTIGNON, P. F., BES, D. R. and LIOTTA,

R. _J., Phys. Rep. 30C ₍₁₉₇₇₎ 305.

[3] BES, D. R. et al. Nucl. Phys. A 260 (1976) 77.

[4] DussEL, G. G. and LIOTTA, R. J., Phys. Lett. 37B ₍₁₉₇₁₎ 477.

[5] RACAH, G., Phys. Rev. 62 ₍₁₉₄₂₎ 438.

[6] DE SHALIT, A. and TALMI, I., Nuclear Shell _{Theory (Academic}