The cluster model and the generalized Brody-Moshinsky coefficients

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The cluster model and the generalized Brody-Moshinsky

coefficients

B. Silvestre-Brac

To cite this version:

The cluster model

and

the

_{generalized Brody-Moshinsky}

coefficients

B. Silvestre-Brac

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

(Reçu le 26 novembre 1984,

accepté

le ler mars ₁₉₈₅₎

Résumé. 2014 Les théories

d’agrégats

éliminant _{rigoureusement} le mouvement du centre de masse font _appel aux

coordonnées

_{intrinsèques}

des agrégats. On montre que les coordonnées de Jacobi des différents _agrégats sont

reliées par une transformation _orthogonale et que l’utilisation des coefficients de

Brody-Moshinsky

généralisés

permet le calcul exact des noyaux d’échange. La

_procédure

est illustrée dans le cas de la

_description

de l’interaction nucléon-nucléon en termes de

_quarks

constituants.

Abstract. 2014 Cluster

theories, which

_rigorously

eliminate the centre of mass _motion, need intrinsic cluster

coor-dinates. It is shown that the Jacobi coordinates of the various clusters are related

_by

an _orthogonal transformation

and that the use of _generalized

_{Brody-Moshinsky}

coefficients allows an exact calculation of the exchange kernels. This

_procedure

is illustrated

_by

the

_description

of nucleon-nucleon interaction in terms of constituent

_quarks.

Classification

Physics Abstracts 21.60G

1. Introduction.

The non-relativistic

_problem

of _many

_interacting

particles

-

even in the

_simplest

case of

_two-body

forces

_only

- has attracted

a

_large

number of studies

for the last

_thirty

years and has initiated many

sophisti-cated methods.

Apart

from the two

_{body problem}

whose

_{SchrOdinger equation}

reduces to an

_ordinary

differential

_{equation, only}

the three

_{body problem}

was

treated with sufficient _accuracy to

_pretend

to be

«

_exactly

solved ». The

study

of four

body

_systems

is

rapidly progressing

and no doubt that their « exact

solution » will be a

_reality

_very soon. More

complicated

systems

are tackled

only through

some kinds of

approximation.

The shell models in terms of

inde-pendent particle

wave functions are

_widely

used because the Pauli

_principle

can be taken into account

rather

_{easily (especially}

if the formalism of second

quantization

is

_employed).

_However,

these models violate an essential

_symmetry

of the Hamiltonian : the space translation invariance. One thinks that this violation is not _very

_important

for

the

bound states

of

_heavy

_systems.

On the other

hand,

a

good

treatment

of the centre of mass motion is crucial for the

scattering

problem

or for the

description

of

_light

_systems.

If the

independent particle

scheme is conserved

_(in

this case, the Pauli

_principle

is

_easy),

the restoration of the

broken

_symmetry

is a _very

complicated

task

_[1].

Another

_possibility

is to use harmonic oscillator

functions as a basis of

independent particle

motion.

In that case the centre of mass motion comes also

into a harmonic oscillator motion and to eliminate the

spurious

centre of mass motion it is _necessary to

diagonalize

matrices

bigger

than the

_physical

_problem

due to the

_possibility

of excitation of the centre of

mass in several

_quanta.

In

fact,

the

_only

_way to

eliminate

_rigorously

the centre of mass motion is to

separate

the total wave function of the n

_particles

in

terms of a centre of mass

_{function fc.m.(R)}

_{(a plane}

wave for

instance)

and an intrinsic function

V/int(p,,..., Pn-l)

expressed through

n - 1 intrinsic

coordinates pi. In that case the

_{antisymmetrization}

of

_{t/J int}

is

_particularly

difficult. As a matter of fact a

permutation

P of the

_particles

concerns the

_original

coordinates ri and not the _{intrinsic coordinates pin.}

Consequently, given

a trial intrinsic wave function

4(int

not

_yet

_{anti symmetrized,}

the Pauli

_principle

requires

the evaluation of the so-called

exchange

kernels

this is a terrible task.

_Usually,

one

_simplifies

a bit the

problem by choosing

trial wave functions which are

built on definite

_eigenmodes

of

_{subsystems :}

this is the essence of the cluster models

[2].

This allows a drastic

reduction of the Hilbert space while

_keeping

most of the

_physical

_{correlations;}

nevertheless the Pauli

principle

must be fulfilled and the

_problem

of

_exchange

kernels still remains. Even in the cluster

_models,

the sub-wave functions are themselves chosen very

simple

(usually Gaussians)

and this _may lead to inconsisten-cies between the

_{dynamical description}

of the clusters and the total system

[13].

In this _paper, we

_present

a

_general

method

_allowing

an exact calculation of the

_{exchange kernels,}

based on

the

_expansion

of the intrinsic wave function in terms

of

independent

harmonic oscillator wave functions.

The

_{generalized Brody-Moshinsky}

coefficients are

crucial ingredients

in this method In the next

_section,

the

relations between cluster internal coordinates and those obtained

_{by permutation}

of the

_particles

are

discussed in detail. In section

_3,

a number of

_properties

concerning

the

_{generalized Brody-Moshinsky}

coeffi-cients is

_presented

In section 4 the formalism is

_applied

to the

_description

of the nucleon-nucleon

_scattering

in terms of

_quark

structure. Conclusions are drawn

in the last section.

2. Relations between intrinsic cluster coordinates.

.2.1

DEFINITIONS. - Let

us describe a

_system

of n

identical

_particles.

In this _paper,

_only

the coordinate

degrees

of freedom are

_considered;

in fact the

spin

and

_{isospin degrees}

of freedom are treated in a

straight-forward, although

lenghtly,

way with

help

of Racah

algebra.

The

_particles

are thus referred in the

_ordinary

space

by

their coordinates

ri (i

=

1, 2,

...,

n)

relative

to some

_{arbitrary origin.}

Since the total Hamiltonian

is invariant under space translation it is necessary to introduce the centre of mass coordinate

Besides,

there remains n - 1

linearly independent

coordinates

_(called

intrinsic

_{coordinates); they}

can be

chosen in an infinite number _{of ways but the} most

common choice defines the Jacobi

_coordinates,

which

represent the distance of the

_(i

+

₁₎

th

_particle

relative

to the centre of mass of the i

_{previous particles}

a is a scale

_parameter

whose role will be

_emphasized

later. One can write more

formally

where A is a n - 1

by n

matrix

It has the

_properties

It is

_simple

to invert

_(2.1)

and

_{(2. 3)}

to find n-1

To achieve a _quantum mechanical

_description

of the _system we need the canonical momenta of R and pi which are referred to as P _{and ni.}

_Starting

from the canonical

_conjugates

_{pin of the initial}

coor-dinates _{ri, it is} not difficult to show that

and the inverse relations

Exactly

in the same _way we have

For a Hamiltonian invariant under space

_translation,

the

_potential

_{energy does} not

_depend

on the R

coor-dinate,

and with the Jacobi coordinates we are left

with an

_interacting

_system of n - 1

_particles

with

mass m . Thus,

a is a scale mass

_parameter,

which is not

a

relevant for the essential features of the

_problem.

The total orbital wave function is

_separated

into a centre

of mass _part and an intrinsic one.

2.2 ACTION OF A PERMUTATION. -

Since the Pauli

principle imposes

a

complete

antisymmetric

wave-function,

one needs to know the effect of a

_permutation

P on the orbital wave function. If P is the

_permutation

which

_changes

the ith

_particle

into the

_Pith

_particle,

the action of P is

_{particularly simple expressed}

in the

_original

coordinates since _ri -

IiP)

=

r,,.

Thus

The centre of mass coordinate is

_symmetric

for every

permutation

so

R(P)

= R. On the other

hand,

the effect

of the

_permutation

on the intrinsic wave function is not

which is different from :

Hence,

as seen from

_{(2.13), (2.14)}

and

(2.15),

the effect of P on the intrinsic wave function is much more

complicated

than a

_{simple permutation}

of the intrinsic Jacobi coordinates.

However,

it is trivial to show that

_p(P)

are also

intrinsic coordinates in the sense that

_they

do not

depend

on the R coordinate. Hence

and hence the

_permutation

matrix

orthogonal

2. 3 CLUSTERING OF THE SYSTEM. -

Very

often it is

advantageous

to _express the

_eigenmodes

of a _system

in terms of

eigenmodes

of

_sub-systems;

in _many cases

the choice of

_sub-systems

_{(or clusters)}

lies on a definite

physical

situation.

Moreover,

the restriction to few

eigenmodes

of the clusters allows a drastic truncation

in the total Hilbert space, while

retaining

most of the relevant correlations.

Hence,

it is _necessary to have the relation between the

original

Jacobi coordinates for the total

_system

as defined in the

_previous

sub-section and the intrinsic coordinates of the clusters. At the moment let us _suppose that the

_n-particle

system is

_partitioned

in two

_subsystems

_{of si and s2}

particles

(si

> 1

_{and si}

+ _s2 =

n).

It is natural _to

introduce the intrinsic coordinates of the first cluster

and of the second cluster

Since

_{only n -}

2 intrinsic coordinates have been

_defined,

we need one more which is taken as the

_separation

dis-tance of the two

clusters,

namely

By

replacing the ri

in terms _{of pi in these}

_expressions

it is

_just

a matter of calculation to find the new Jacobi

coordinates as function of the old ones : it comes

We feel that it is more convenient to write this

_change

of basis in a matricial

_form;

then the cluster coordinates are _put in the form

which is a vector with n - 1

_components

and

_equations

_(2.22)

can be recast as

From the matrix elements defined

_{by (2.24)}

one can show that F[2] is an

_orthogonal

_matrix,

and

_consequently

this

_property

holds too for the matrix F[c] defined

_{by (2.23)}

The

_{generalization}

to an

_arbitrary

number of clusters is obtained

_by

recurrence. Let us _suppose

_{that sl}

is now

partitioned

into two

_clusters,

then the new first cluster is divided into two further

_clusters,

and so on. Thus one

considers the

_partition

of the

_original

_system

in k

_clusters,

each one

_{having si}

_particles

_(si

>

_1,

_si

_{+ s2}

_{+... + sk} =

n).

The various

_partitions

of the total

_system

_{play a}

_very

_important

role in _any

_manybody

theories

_eliminating

correctly

the centre of mass motion

_[3]

_(see

for instance the Yakubovski

_equations

for the

_four-body

_problem

_[4]).

To

_simplify

the notation it is convenient to define

The intrinsic Jacobi coordinates for the lth cluster are

and the

_separation

coordinates

then these new intrinsic coordinates are related to the old ones

_through

with the matrix

_Fl:tsl

defined

_by

As

usual,

one can write in a more formal _way

then

Since each sub-matrix FE’3 is

_orthogonal,

this is also true for the total transformation matrix FEc3

Now,

we are able to demonstrate the

_general

theorem

_concerning

the intrinsic Jacobi coordinates.

Theorem :

_Any

_arbitrary

_permuted

intrinsic coordinate of an

_arbitrary

cluster

_system

is related to another

Proof :

_Starting

from the

_original

coordinates

_(p),

one builts the first

_permuted

coordinates 1

and then the cluster

_permuted

coordinates

Consequently

Since each matrix

_appearing

in

_{(2 . 32)}

is an

_orthogonal

_matrix,

their

_product

is also an

orthogonal

matrix

Q.E.D.

As is well

_known,

_any

_orthogonal

matrix in n dimensions can be

expressed

as the

_product

of

ortho-gonal

matrices which leaves an n - 2 dimensional

subspace

invariant

_(in

other

words,

any « _rotation »

in an n dimensional space is obtained

by

successive « rotations » in fixed

_planes).

Each infinitesimal generator of

_0(n)

is a rotation in two

_dimensions,

and

n(n -

1)

2

generators are

_enough

to generate any

general

rotation. Thus

n(n - 1)

is the maximum number of «

_plane

_{rotations » necessary} to built the

original

orthogonal

matrix.

_However,

it is

_interesting

to search for the « _{best way »} to do the

_{decomposition}

- that is the

one which minimizes the number of

operations.

The

_procedure

to do that is not

_unique,

and each

_particular

case

_requires

a

_special

treatment to find « the best _{way ».} The cluster wave function

being decomposed

into

_{independent particle}

wave-functions,

the

_problem

of

_evaluating

the

_exchange

kernels reduces to

_expressing

two

_independent

func-tions of rotated coordinates in terms of the functions in the

_original

coordinates. This last

problem

can be solved if the cluster function is

_expanded

into a

harmonic oscillator basis.

3. Generalized

_{Brody-Moshinsky}

coefficients.

3.1 DEFINITIONS. - The harmonic oscillator

(HO)

wave functions

Onl.(r)

are defined with the

norma-lization and

_phase

conventions of

_{Moshinsky [5]}

or

Erdelyi et

al.

_[6]

where b is the size

_parameter

b =

_(Ii/Mro)1/2,

and the radial

_part

is written

and

_{Ll n +112(X)}

is a

_generalized

_Laguerre

_polynomial.

Let us consider the Hamiltonian of two

_independent

harmonic oscillators

An

_{eigenfunction}

with

_angular

momentum A is

_readily

The Hamiltonian

_(3.3)

is invariant under the rotation of the coordinates

Consequently

[0.1(r)

l/JNL(R)]).1l is

also an

eigen-vector _{with energy}

_{núJ(2 n}

+ 1 + 2 N +

_L).

Hence one can write

It is

_important

to note that the summation contains

a

_finite

number of _terms,

_namely

all the

_(n,

_I,

_N,

L)

multiplets compatible

with the conservation _{of energy}

2 nl + II + 2 n2 + 12 = 2 n + I + 2 N + Land

constraint of the resultant

_angular

momentum

II - L A 1 + Landparity(-

l)lt +12

=

(-I)’+L.

The transformation

_{coefficients nlN L; À. nl}

_II

_n2

_12,

À. >fJ

are called

Brody-Moshinsky

(BM)

coefficients

with

_{angle p.}

The standard BM

_coefficient,

cor-responding

to

play a

very

important

role in numerous fields in

physics, especially

in the

expression

of matrix elements of

_two-body

effective interactions

_[7].

Their

impor-tance was first

_{recognized by}

Talmi

_{[ 15]}

who calculated

some of them

_by

hand. Later on,

Moshinsky [5]

studied their

_properties

in more detail and in collaboration

with

_{Brody [7]}

made extensive tables.

Then,

a number

of works has been devoted to find new relations

between them and to refine and

simplify

their cal-culation

_(see

_Appendix

6 of Lawson’s book

_[7]).

Curiously,

since their introduction

_by

_{Moshinsky [8],}

the BM coefficients with an

_{angle fl #}

_n/2,

have

as we are aware

of,

were _very

scarcely

used in

practical

cases. In this

_section,

we

give

the main

properties

of

these BM coefficients and show how useful

they

are

in the cluster

theory.

3.2 CALCULATIONS. - The standard BM

were

cal-culated in several different ways : recurrence relations

[5,

9],

more or less

sophisticated analytical

formulae .

[10], diagonalization

of

_special

matrices

_[11].

In

_fact,

all these methods can be as well

applied

to the BM with

_{angle P.}

This

_property

was not realized first

since,

in his

original

paper

[8], Moshinsky

proposed

a

complicated

formula

including

the standard

coeffi-cients ;

readily

it is not a _very convenient one. Which

method to choose for

_computing

them

_depends

on

what one wants. If one is interested in

particular,

unrelated

coefficients,

it is

_preferable

to use

_analytical

formulae;

if one wants to evaluate entire lines of the

BM matrices the method

_proposed

_{by Raynal [12]}

is

_certainly

_better;

if on the other

hand,

one is interested

in the whole

_matrix,

or all the BM _up to a

_given

number of _quanta the

_{diagonalization}

or the recur-rence methods are

_preferable.

The latter was used

for the

_application

to the nucleon-nucleon interaction

presented

in the next section.

_{Thus, here,}

the BM coefficients are calculated

_by

recurrence relations in the same _way as the

_original

standard

coefficients.

In that sense, all the BM coefficients are treated on an

_{equal footing.}

Since the method is not new and

follows

exactly

the same _steps as in the

original

work

of

_{Moshinsky [5],}

we do not insist on its _However, the final results never

_appeared

in the

_literature,

as

far as we

know,

and we think that it is

interesting

to

report

them. In a first

_step,

the

_particular

_{nlNL ;}

A

I n,

h

n2

l2 ;

_{A >,}

coefficients

with n1

= n2 = 0 _are calculated

with

Then,

other coefficients with

_{n, 0}

0 are obtained

by

recurrence relations

3.3 SOME PROPERTIES. -

Directly

from their definition

the

_{following phase}

relations can be

proved

Applying

this

_twice,

and

_(due

to the definition

_{of pin}

_(3.4))

taking

care of

one _gets

Owing

to the fact that the BM are

real,

by replacing

the variables of

integration

r, R

by

rl, r2 in

(3 . 8)

and then

making

r2

-+ - r2, the following

relations can be obtained

Now,

when a rotation of

_{angle fl}

is followed

by

another rotation of

angle

_y, one obtains a _very

_important

com-position

rule

In

_{particular, putting}

_y

_{= - p}

in this

_equation,

the

_{orthogonality}

relation is recovered

It is

_possible

to derive also a _very

_important

« closure relation ». Let us start with the

_{following integral}

where

_R1

and

_R2

are the coordinates rotated with an

_{angle fl}

(Ri

and

_{R2 correspond}

to r and R

_respectively

in definition

_(3.4)).

Noting

that

a first

_expression

for

calculating h

is

readily

obtained

Now

_changing

the

_integration

variables _{rl, r2} into

_R1

and _r2, some

_lengthy

calculations allow us to write

a similar

_expression

holds if in the left hand

_side

_{OOnI ;}

_{1 nl00;}

_{1 >,}

is

_replaced

_{by (}

_n100,

_{1 nl00;}

1

_{> fJ}

and in the

_right

hand side C - S and S --+ - C. This

_expression

with 1 = 0 appears in the

exchange

normalization

kernel of the nucleon-nucleon

_scattering

discussed in the next

_chapter,

and in reference

_[13]

it was used as a

convergence test of relation

(3.17).

4.

_Application

to the nucleon-nucleon interaction. 4.1 THE GENERAL PROBLEM. - We consider the

pro-, blem of the

_scattering

of two

_baryon

_clusters,

each

one

_being

formed

_by

three

quarks

of

_type

u and d. In

principle

each

_quark

is characterized

_by

its

_colour,

isospin, spin

and space

degree

of freedom.

However,

this _paper is concerned with the action of the Pauli

principle

on _{intrinsic space} coordinates and thus we

shall

_neglect

the

_{colour, isospin,}

and

_spin

coordinates

as well as orbital

_coupling

considerations

_(all

these can be treated in a

_{straightforward}

_although

_lengthy

way with the

help

of Racah

algebra).

Since the

physical

situation

_imposes

that the 6

_{particles (with}

coordinates

ri)

are

separated

into two clusters of 3

_particles,

the natural coordinates for this

_problem

can be chosen

(following

the recommendations of

_chapter

₂₎

as

Taking fully

into account the Pauli

_principle,

the trial wave function reads

where fc.m.,

95N,

x are the centre

_{of mass,}

intrinsic

cluster,

relative wave

function,

_respectively

and A the total

antisymmetrizer.

The centre of mass

function,

_{being symmetric}

in the

_6-q

_coordinates,

can be

_kept

out of the

antisymmetrizer

and is

_easily

dealt with. It will be

_ignored

in the

following.

Let T be _any

_operator

_symmetric

in the

_exchange

of the

_particles.

The main

_problem

is the calculation of

_{quantity ql}

I T I ql

>.

Since

A + = A,

A2

= A and AT A = T it is well known that it is sufficient to

keep

the

antisymmetrizer

_on the bra _vector

only.

Moreover,

the intrinsic functions

_q5N

are built

antisymmetric

for all

_permutations

of the

subgroup

_S3

x

_S3

c

_S6.

This

_property

allows the number of different contributions to the

_{average ql}

_{I T I ql >}

to be reduced from 720

to

_2;

one can choose for instance 4

_(identity)

and

_{P 14 (transposition}

of

_particles

1 and

₄₎

and the

_{corresponding}

contributions are denoted as direct and

exchange

contributions. In

_general

the direct

_part

_{is easy} to calculate and the main

_problem

in cluster

_theory

lies in the evaluation of the

_exchange

_parts.

If it is

_possible

to

_expand

the trial intrinsic wave function in terms of the harmonic oscillator basis

_0(r),

this

_problem

reduces to the

com-putation

of

In this

_expression

the

_permuted

_{coordinates xp,} yp,

xp, yp, Rp

are those obtained in definition

_(4.1)

_by

permuting

r. H _r4. From the conclusions of sections 2 and

_3,

one sees that this

_quantity

is obtained with a

summation of a

finite

number of terms

_including

generalized Brody-Moshinsky

coefficients. In the

following

we

_apply

these

_general

ideas for T = ’11

(exchange

normalization

contribution)

and for

T =

V;nt (exchange potential

contribution).

In

fact,

among the 9 terms

coming

from the cluster

interaction,

only

three

_give

different contributions and one can

write

_formally

4.2 EXCHANGE NORMALIZATION CONTRIBUTION.

it is trivial that

_{quantity (4.3)}

with T = ’1J

_{(called JBT)}

immediately

reduces to

_(the

_integration

on x and x’

variables

_{gives 6 symbols}

which are omitted in the

following)

Keeping

in a first _step the coordinates R and

_Rp

fixed

and

_making

a « standard » rotation in the

_{(y, y’)}

plane

and in the

_{(yp, yp)}

_plane

one introduces the new

coordinates

This

_{allows, through}

the introduction of two standard

BM,

expression (4.6)

to be

_changed

into

or, after trivial

_integration

on the u

_variable,

into

It is _easy to show that

then

_Rp

_{and vp} are derived from v and R

_by

a rotation

of

_{angle PI}

such that C

_{= /8-/3}

and S

_{= 1/3;}

con-sequently (4. 8)

is

_nothing

else that a

_generalized

BM

with

_angle

_Pl.

Thus

_{expression (4.6)}

can be written

very

symbolically

meaning

that it is a summation of a finite number of

terms

_including

3 different

_{Brody-Moshinsky}

coef-ficients

_(two

standard coefficients and one with

_angle

PI).

We think that this is the most economical pro-cedure for

_{calculating (4.6).}

4.3 EXCHANGE

_V14

CONTRIBUTION. - The

problem

here is the evaluation of

Due to the fact that

_{V 14 depends}

on the coordinate r =

ri - r4

only,

it is natural to

_impose

that this variable must be one of the final

_integration

variables.

For instance one can choose the

following

coordinates

Since x p = x and

_Xp

=

x’,

the

integration

over the x

and x’ variables reduces

_{’BJ 14}

to

Keeping

the R and

_Rp

coordinates fixed and

_making

a « standard » rotation on the

(y, y’) plane

and

(yp, yP)

plane,

one can define the new variables

Hence with the

_help

of two standard BM and

inte-gration

on u

variable,

_{’lJ 14}

now reduces to

Since,

on the other

hand,

it is evident that

_{(R, v)}

is deduced from

_{(r, z)}

_{by a}

rotation

_through

an

_angle

_P2

such that C =

Vl/3,

S

_{= 2/3 ;}

_similarly

_{vp and -}

_Rp

are obtained from z and r

_through

the same rotation.

Then,

using

two

BMp2,

the scalar

_{product (4.15)}

is transformed into

which is almost trivial. All this can be summarized

_by

4.4 EXCHANGE

_{Vl 5}

CONTRIBUTION. - This

requires

the evaluation of

As

_already

stated,

one must

_keep

r =

rl - rs as an

_integration

_variable;

it is _easy to show that x can be chosen as well. In a first choice we _propose to take

as

integration

variables. Since xp = _{x, we are} left with

One verifies

_easily

that

Introducing

new variables as

and,

with the

_angle

_P4

such that with the

_help

of

_BMfJ4’

_{0(v) O(R)}

is transformed into

_{0(s) 0(t).}

Now let us _put

it is trivial to show that

and,

with the

_angle

_P4

such that i with the

_help

of

_BMp4,Ø(V)

_Ø(R)

is transformed into

_{0(s) 0(t).}

At this

_stage

we are left with the evaluation of

Lastly

from

(4.23)

and

_(4.24)

one sees that a rotation with

_angle

_P3

transforms

_{0(u) 0(s)}

into

_{Ø(yp) O(Rp)}

and

/3

1

the rotation with

angle

Ps

such that C =

/3- 2 ,

S

= 2

transforms

_0(t)

_O(x’)

into

_{0(r) q6(z)}

which achieves the

reduction.

_Symbolically

one has

alternative

_procedure

which more or less exhibits the same levels of

_difficulty.

We

_choose,

as

_integration

variables

It follows that a « standard rotation » transforms

_(y,

_R)

into

_(-

_v,

_u)

with

the same rotation transforms

_{4J(Yp) 4J(Rp)}

into

_{O(z) 4J( -}

_v).

Moreover, a rotation with

_angle

_P1

as defined

_{by (4 . 9)}

allows us to _go from

_{4J( -}

_u)

_§(y’)

to

_¢(w)

_§(z)

where

Lastly

a rotation with

_angle

_P5

=

n/3

transforms

§(w) §(x’)

into

§(r) 4> (s).

The same rotation also transforms

0(-

y’)

§(x’)

into

_{0(r) io(s). Consequently}

the calculation

_{of V15}

can be summarized

symbolically by

If we _compare

(4.31)

to

_(4.27),

it _appears that both

_computations

are of

_equal

_difficulty.

This

_example

_clearly

exhibits that the

_{decomposition}

of the

original quantity

into

_{successive-planar}

rotations _may be obtained

by

several

_equivalent

_ways. The

_procedure

to be chosen

_depends

on each

particular

case considered.

4. 5 EXCHANGE

_V25

CONTRIBUTION. - In that case one must calculate

As

_already

_stated,

it is _necessary to

_keep

_{r2 - r5} = r as an

_integration

variable. In this _case, which is the most

complicated (because

one cannot

_keep

x as

_integration

variable),

we found convenient to define the

_following

integration

variables

They

are related to

_{the original}

ones

_by

Noting

that z and

defining

we obtain

_{0(u) 0(v)}

_(and

_0(u)

_0(vp))

from

_{0(y) 0(y’)}

_(and

_§(yp)

_0(y,P))

_by

a standard BM. After

integration

on

the u and z variables we are left with

It is not difficult to see that a rotation with

_angle

_P2

transforms v and R

_(and

_{vp and}

_Rp)

into

and _r14

_(s

and -

_{r14) ;}

hence with a

_BMp2

we _go from

_0(v)

_§(R)

_(and

_0(vp)

_§(Rp))

to

_0(s)

_{o(rl 4).}

_Finally

a standard

BM transforms

_§(w)

_0(s)

into

_0(r)

_cp(r36).

_Symbolically

this

_{V 25}

contribution is written

The various

_{decompositions presented}

here were the

underlying

technical

ingredients

of reference

_[13].

_They

allow the introduction of radial excitations of the nucleon in the

_description

of nucleon-nucleon

_scattering.

More

complete

formulae

_including

_spin-isospin

_components can be found in the internal

_report

_[14].

5. Conclusions.

In this _paper, we studied the

properties

of the

gene-ralized

_{Brody-Moshinsky}

coefficients in

_detail;

in the literature _up to now,

only

the standard coefficients have deserved some work but

_here,

all the coefficients are treated on

_{equal footing.}

A new closure

_relation,

which

_appeared

to be

_important

in the

_microscopic

description

of the nucleon-nucleon

_interaction,

was

derived. On the other

_hand,

it is shown that _every set

of

_permuted

intrinsic coordinates is related to another

one

_by

an

_orthogonal

transformation. This

_property

is _very

_important

for the cluster models which

rigo-rously

eliminate the centre of mass motion. If it is

possible

to

_expand

the kernels

_coming

from the

Schrodinger equation (in

most cases, this

equation

is of the Hill-Wheeler

_type)

in terms of a harmonic

oscil-lator basis with

_{sufficiently rapid}

_{convergence, then}

our work shows that it is

_possible

to calculate

_exactly

the kernels -

specially

the

_{complicated exchange}

kernels - with

a

_finite

number of terms which include

generalized Brody-Moshinsky

coefficients. The

practi-cal _{way -} that is the most economical one -

to do that

_depends

on each

_particular

_{system which} is

considered. This method should be convenient for the

_study

of

_light

_systems

which could not be solved

exactly

up to now

_(let

_say with a

particule

number

between 4 and

_8).

The

_application

to the nucleon-nucleon interaction described in terms of the

consti-tuent

_quarks

₍₆

_particles)

has been

_abundantly

dis-cussed. This method _opens new

_{possibilities}

in the cluster

_theory;

in

_particular

it is

_possible

to retain the radial excitations of the nucleon in the

_description

of the nucleon-nucleon

_scattering.

Acknowledgments.

We are _very much indebted to Prof. C.

Gignoux

for

constant interest in this work and numerous

illuminat-ing

discussions. We are also

grateful

to Dr. A. K. Jain for

_suggesting

the second

_possibility

of

_evaluating

_{V 15}

and to Drs. J. Carbonell and S. Shlomo for a careful

reading

of the

_manuscript.

References

[1] PEIERLS, R. E. and _Yoccoz, J. Proc.

_Phys.

Soc. A 70

(1957)

381.

PEIERLS, R. E. and _{THOULESS, D.,} J. Nucl.

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38

(1962) 154.

[2]

WILDERMUTH, K. and _TANG, Y. C., A

unified theory

of the

nucleus

_(Vieweg)

1977.

WILDERMUTH, K. and McCLURE, W., Cluster represen-tations

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nuclei

_{(Springer-Verlag)}

1966.

KUKULIN, V. I., NEUDATCHIN, V. _G., OBUKHOVSKI, I. T. and SMIRNOV, Y. _F., Cluster as

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nuclei

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BENCZE,

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[7]

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Brody-Moshinsky

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R. _D.,

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1980,

Appendix

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[14]

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