UNITARY POLE APPROXIMATION FOR THE COULOMB-PLUS-YAMAGUCHI POTENTIAL AND APPLICATION TO A 3-BODY BOUND-STATE CALCULATION

PHYSICAL REVIEW C VOLUME 43, NUMBER 6 JUNE 1991

Unitary pole approximation for the Coulomb-plus-Yamaguchi potential

and application to a three-body bound-state calculation K.

Ueta

Insti tuto deFI'sicada Universidade de SaoPaulo, Caixa Postal 20516, 01498 SaoPaulo, Brazil

G.

W.Bund

Instituto deFisica Teorica, Universidade Estadual Paulista, Rua Pamplona, 145,01405SaoPaulo, Brasil (Received 28December 1990)

The unitary pole approximation is used to construct a separable representation for a potential U which consists ofa Coulomb repulsion plus an attractive potential ofthe Yamaguchi type. The ex- actbound-state wave function isemployed. U is chosen as the potential which binds the proton in

the 1d5&& single-particle orbit in

'"F.

_Using the separable representation derived for U, and assum-

ing a separable Yamaguchi potential to describe the 1d,zz neutron in

"0,

the energies and wave functions ofthe ground state

(1+)

and the lowest 0+ _state of

"F

are calculated in the core-plus- two-nucleons model solving the Faddeev equations.

I.

INTRODUCTION

In some three-body processes, the

T

^matrix associated to apair

of

particles is dominated by the bound-state pole

of

the pair. In this case the interaction Ubetween these particles can be approximated by the separable potential

Ul^qyB

)

^&qIBslUlqlBs)—_i

&qIBsl U

)

being the bound-state wave function. This approx- imation is known as unitary pole approximation (UPA).

Our purpose is to construct, based on the UPA, a se- parable approximation for a potential Uwhich consists

of

a short-range attractive part Vand

of

a Coulomb repul- sion

Vc=(ZZ'e )fr.

As is well known, the inclusion

of

the Coulomb interaction in the three-body problem presents many difficulties. Although it has been possible to extend the Faddeev formalism to include the Coulomb force,' the numerical applications have been restricted to low values (

S4) of

the product

ZZ'.

^' The usual re- placement

of

the Coulomb

T

matrix by VC (Born approx- imation) becomes questionable as

ZZ'

_increases.

In a recent calculation

of

the p-d breakup reaction, the long-range tail

of

the Coulomb interaction isreplaced by a short-range potential and the Ernst-Shakin-Thaler (EST)method is used to obtain a separable approxima- tion for this cutoA' Coulomb potential. The

EST

method is more general than the UPA. However, for the UPA it is not necessary to make any screening

of

the Coulomb tail.

II.

CONSTRUCTION OF THEUPA FORM FACTOR

Having in mind applications to three-body systems consisting

of

two nucleons outside an inert and massive core, we consider Uas being the single-particle potential

of

the proton.

For

simplicity, it will be assumed that the

short-range part ^{V is} already separable and acts in a specific

(lj)

orbit

of

the shell model:

&plvip ^&

= —,

~lj

^{" g„(p)g„(p}

⁾

y

^&

^pl~„„)

^&

W„„lp &,

where p^isthe momentum

of

the proton (mass m )and

&

pl%i „) ⁼ g

^(Imi

^,

^{' ml Jd)u—Yi}

^'(p)l

^—,

^',

^m,

⁾

^.

ml

I

(2)

l

giiJ

(p)=

( ²

+

^{p2)l+1 (4)}

With this choice, the two-body problem corresponding to the potential

V+

Vc can be solved exactly. ^' The ener- gy ^E'Ij

of

the bound state isdetermined by the equation

2l

+1

a-'=

_Ij

— ^4-' _(2P)

^'

_'(P+K)

l

+1 — _s/~

X2F)

1,

—

_l

— —;l

^S

+2 — —;

^s

^p

^K

P+K

2

where s

= —

_{(2mZZ'e )/2 and}

K=(2mlel l)'

. The cor- responding wave function is given by

qlBi,

s(p) =X. . . ^,

^[gi)(p) Ui&(p)]

p(, „(p), —

P

+K

where

We assume also that the error made in considering the core as a point charge, which leads to an excess Coulomb repulsion in the region corresponding to the interior

of

the core, is compensated for by making the potential V more attractive.

It

is convenient to choose a form factor

of

the Yamaguchi type,

43 2887

1991

The American Physical Society

BRIEFREPQRTS

vi~(p)

= ——

^{1 4pa}

p (pz+~z)(/3z ^z) 'n

s/Ic /3 Ic

i+

^t p ^/c

(+i

^{n s/K /3+K} p

+K

(7) and _Xi~ is a normalization factor. The

C„(z)

^{are the}

Gegenbauer polynomials.

From expression (6)we see that the separable potential

U„,

^which generates the same bound state as the poten- tial

V+

V&,is given by

~~

0. 050

iE

0.025

—. —

V s^C_2pxlO

AI

&pl

&„, ^Ip'&= — '

g(, (p)g(,

(p') &

&pl&(,

„)

^&

^&(, „p'),

2m

2 3 4

momentum (fm

")

with

gi,(p)

=gi,

(p) vi,(p)—_,

(8)

FIG.

2. Form factors _g2&/2 ofUPA, gz5/2 ofthe Yamaguchi potential, and the difterence v25/, between g,5/2 and g⁷&/2.

and A& is determined _by requiring that the bound state

has the energy eigenvalue ei,

.

Expression (8) is the UPA [Eq. (1)]when the degeneracy introduced by the quantum number

p

^isincluded.

As an example, we consider U as the interaction which describes the 1d5&& single-particle bound state

of

the pro- ton in '

F.

The energy

of

the bound state (ezs&z=

— _0.

_{596 MeV) is reproduced if we take}

A,z5&z=945 fm and

/3=1.

464 fm ' in Eqs. (2) and (4).

The value

of

/3 is the same as the one appropriate for the neutron _lds&z single-particle state in '

0

^(seeSec.

^III).

^In

Fig. 1we show the radial function

~2 EP

2+E ^K

u(r)=Br

/e

" x 'r 1+ x

0 2K

Xe

^'~

'"dx

where

3

isa normalization constant and y

=is/ir

(Ref. 7, p. 432). The corresponding mean-square radius is

3.

74 fm. This value is very close to the value

3.

69fm given in Ref. 8. In Fig. 2 we plot the functions _{gz spaz(p), v zspaz(p),} and gz s~z(p) [Eqs. (4), (7), and

(9)]. It

can be shown analytically that vz spaz(p) behaves as p ^when

p~0

and asp for p

~ ~.

^{Finally, we mention} that the coupling

constant Az spaz which appears ⁱⁿ

Eq.

(8) results in being equal to 1103fm

III.

APPLICATION

TOATHREE-BODY BOUND-STATE CALCULATION We consider the nucleus ^'

F

as a three-body system composed

of

an '

0

^core plus a proton (particle 1)and a neutron (particle 2). We restrict ourselves to bound states dominated by the (ldszz, ids&z) configuration and, in fact, consider _only the ground state

(1+)

and the lowest

0+

state (excitation energy

1.

042MeV).

For

the proton-'

0

interaction U&

=

_V&+-Vc, we use the UPA potential described in the previous section.

For

the neutron-core interaction V2, a potential

of

the same form as the short-range part V&

of

the proton interaction [Eqs. (2)—_{(4)]is used.} The parameters are chosen in such a way to reproduce the energy (

—

_{4. 146}MeV) and the ra- dius (3.464 fm)

of

the Idszz single-particle state in ' O.

Thus the values ^A,~zz&z=924 fm and ^/3~z z&z=1.464^fm are obtained. We here make the remark that since kz'~&&

(=945

fm ) is larger than Az'5'&z, V, ^is more attractive than Vz. This is expected since, as was pointed out in

Sec. II,

V, has an additional attractive part to compen- sate forthe excess Coulomb repulsion.

For

the neutron-proton interaction V,z, we use the se- parable s-wave Yamaguchi potential

&s

&ql I']z_lq&

= — g gs(q)gs(q')

S=o& m

I

15 20

gsq=

⁽

^)=-

s FID.

1. Exact radial wave function for the 1d&/2 state in the

Coulomb-plus- Yamaguchi potential.

In Eq.

(11),

q is the relative momentum ^—,'

(P, —

_Pz),

_P,

-be- ing the momentum

of

particle

i,

and ~SMs ) is the spin

43

BRIEF

REPORTS

0.5

C3

(2}

Q)~5 Zy., -H (2)_g

Qs H

1

V)

~Q5

C)

0

,2+;0_2'

I I

2 3

momentum (frn ^1}

I + ^I

2

3 4

momentum (fm-^~}

FIG.

3. Spectator functions for the ground state of

"F. _FIG.

4. Spectator functions for the lowest 0+_state of

"F.

wave function for the proton-neutron system. The values used for the parameters are AO=0. 149

fm, ^PO=1.

¹⁶⁵

fm

',

_A,

=0.

382

fm,

^and

^P1=1.

^{406 fm}

^',

^{which are}

determined from the values

a, = — _23.

_{71 fm, ro,}

:2.

70 fm,

a, = _5.

_{42 fm, and} Io,

= _1.

_{76 fm for} the scattering

I

length and effective range

of

the neutron-proton scatter- ing.

Performing the Faddeev decomposition

of

the total wave function,

4=+"'+4' ^)+'0'

^', we obtain the fol- lowing expressions forthe components

4I":

(&)

(&)

_(P,

_,P2)

=

2m

g

^g2

c

^{5/2( 1) 5/2, l'j';}

^J

^(~2)

⁺²

^5/2,!'j';JM (P1~P2)

2mE

— _P' — P',

₂ 1'j'^, P2 ^J~

(2)

(2) 2m ₍₂₎ H2 5/2,l'j',

J(

1)

(P„P2) =

_2mE

_{— Pi}

_P2

g

^{g2 5/2( 2)}

^+1'j',

^{25/2;JM (P1~P2)}

&'j

P)

^{J& J}

(3) 2m HLS^.

J

⁽

P

)

X

^~gS(P) +LO(L)S;JM (Pt~p)

2mE

P

^J

2

(13)

(14)

(15)

In Eqs. (13)—(15),

E

isthe energy

of

the three-body bound state,

(J, MJ)

denotes the total angular momentum,

P

is the center-of-mass momentum

P, +P2,

and the 'Y's are the usual total angular momentum eigenfunctions.

The spectator functions

H

satisfy the homogeneous in- tegral equations given in Refs. 10 and

11. For

the 1

state, ^we have

(l', j')=(2,

^—,

'),

(2,^=,

'), (4,

^—', ),

L=O

_{2, and}

S=1

in expansions (13)—(15) and, for the

0+

_state,

(l', j')=(2,

^—',),

L=O,

and

S=O.

The coupled integral equations are transformed into a system

of

algebraic equations by applying the X-point Gauss quadrature method for the integrals. The vanishing

of

the associated determinant gives the energy eigenvalue. In this way we get

E,

⁺

= — _10.

_{3 M V a d}

_E

₊

= — _7.

_{93 MeV. These}

numbers are close to the experimental values'

E;"+' = — _9.

_{75 MeV and}

E'"+' = —

_{8. 71 MeV,} despite the fact that only the _1d5y2 interaction isconsidered.

In order to evaluate the contribution

of

the Coulomb force to the three-body bound-state energy, ^we replace the valence proton by a neutron and calculate the energy

of

the

0

ground state

of

^'

0

^obtaining

E

+

(' 0) = — _11.

_{38 MeV. Therefore,} in our model, the

switching on

of

the proton-core Coulomb interaction raises the energy by an amount

E +(' F)

E+(' 0)= — ^— 7.

93

MeV+11.

38

MeV=3.

45 MeV. Ex- perimentally, one has E'"+~'('

F) E'"+'(' 0)—

=

^—

—

_8.71

_MeV+12.

₁₉

_MeV=3.

_{48 MeV. This shows that}

the UPA is able to yield a correct value

of

the Coulomb energy.

In Figs. 3and 4,^we p1ot the spectator functions versus momentum. From the closeness

of II"'

and

H'

', _we conclude that the asymmetry introduced in the total wave function [Eqs. (13)—(15)]by the Coulomb force lies mainly in the difference U2 5&2 between the form factors g2 5/2 and g2 5/2 (g25/2 in Fig. 2) and is about

20%. ~e

make here the remark that in the numerical calculations we found it convenient to multiply

gz»z

and gz ^5&2by a factor

of 15.

50 and, accordingly, divide

A2»2

^{and A,}z ~&2

by

(15.

50)

.

Therefore, the actual values

of H'"

_and

H'

^' are 15^~50times those shown in Figs. 3and

4.

It

is our purpose to extend the present calculation to other levels

of

^'

F,

^{and further, we expect to be able to}

apply the UPA to describe the (d,n) stripping reaction on 16O

2890

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REPORTS 43

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