Revista Brasileira de Física, V 01. 11, nP 1, 1981
Three-Alpha Forces in the "O Nucleus*
D. A. AGRELLO**, V. C. AGUILERA NAVARRO and J. N. MAKI**
lnsfiruto de Fgca Tedrica, RLBPamplona 145, CEPO1405 - São Paulo, Brasil
Recebido um 22 de Agosto de 1980
The e f f e c t o f r e c e n t l y determined t h r e e - a l p h a f o r c e s
i n the
g r o u n d - s t a t e energy and e l a s t i c f o r m f a c t o r o f 160 i s discussed, i n t h e
a l p h a - p a r t i c l e model.
Discutem-se os e f e i t o s de f o r ç a s de t r ê s a l f a s , recentemente
propostas na 1 i t e r a t u r a , na e n e r g i a e no f a t o r de forma e l á s t i c o do estado fundamental do 160, no modelo de p a r t í c u l a a l f a .
1. INTRODUCTION
I n t h i s paper, we s h a l l d i s c u s s the e f f e c t o f
recently
de-
termined three- alpha f o r c e s l y 2 i n t h e ground s t a t e o f t h e 160 nucleus.
Due t o t h e n a t u r e o f those f o r c e s , and o f t h e two-body i n t e r a c t i o n her e considered, t h e o x i g e n nucleus i s p r o p e r l y d e s c r i b e d i n the alphap a r t i c l e rnodel. I n t h i s v e r y simpl i f i e d d e s c r i p t i o n , t h e f o u r alpha part i c l e s a r e considered as s t r u c t u r e l e s s and
each w i t h e l e c t r i c charge equal t o 22.
indestructible
Although these
entities,
partirles
can
be taken as p o i n t ones f o r t h e d i s c u s s i o n o f t h e s p e c t r u m , a c o r r e c t i o n
due t o i t s e x t e n s i o n i n space i s taken i n t o account i n t h e study o f t h e
form f a c t o r o f t h e nucleus.
Previous c a l c u l a t i o n s u s i n g t h e aboue
a l p h a - p a r t i c l e model
and two- alpha f o r c e s have f a i l e d i n s a t i s f a z t o r i l y d e s c r i b i n g t h e g r o u n d
-
*
P a r t i a l l y supported by FINEP, B r a s i l , under c o n t r a c t 522/CT.
**
CNPq f e l lowship, B r a s i 1.
- s t a t e energy o f the 160 nucleus,
i n s p i t e a f succeeding i n
the
des-
c r i p t i o n o f o t h e r n u c l e a r p r o p e r t i es as t h e r a d i u s and main aspects o f
t h e charge form f a c t o r , f o r exampl e.
I f one wants t o e x p l o i t
a
bit
more the model, i t i s c ! e a r l y o f i n t e r e s t t o i n v a s t i g a t e the e f f e c t o f
a t h r e e - a l p h a f o r c e a c t i n g among t he nucleus c o n s t i t u e n t s .
I n the f o l -
lowing, we develop t h i s program.
The mathematical method used i s a v a r i a t i o n a l a n a l y s i s o f t h e
hamiltonian.
body f o r c e s
F i r s t we c o n s i d e r a h a m i l t o n i a n cor;structed o f o n l y two-
-
p l u s t h e t o t a l k i n e t i c energy, o b v i o u s l y .
Next, we t u r n
on t h e t h r e e - a l p h a i n t e r a c t i o n arid repeat t h e c a l c u l a r i o n s
resul ts.
t o compare
Indeed, we take advantage o f some r e s u l t s by other a u t h 0 t - 5 ~ ' ~
concerning t h e c a l c u l a t i o n s w i t h two- alpha f o r c e s o n l y .
The t r i a l i u n c t i o n i s expanded i n terms
of
invariant four- particle harmonic- oscillator functions.
translatíonally
It i s construc-
t e d i n such iyay as t o have d e f i n i t e t o t a l o r b i t a l a n g u l a r momentum
arid p o s i t i v e p a r i t y .
L-O
The v a r i a t i o n a l f u n c t i o n i s c o m p l e t e l y 'symmetric
as i t should be i f we want i t t o d e s c r i b e a boson system. The e x p l i c i t
c o n s t r u c t i o n o f t h i s f u n c t i o n i s discussed i n the n e x t s e c t i o n .
T h e p r c s e n t a t i o n and d i s c u s s i o n o f t h e three- alpha f o r c e s cons i d e r e d a r e made i r i s e c t i o n 3, where t h e m a t r i x elements o f t h e h a m i l t o n i a n a r e a l s o discussed.
The form f a c t o r i s c a l c u l a t e d i n s e c t i o n 4
and a l l t h e r e s u l t s o f t h i s paper a r e discussed i n t h e l a s t s e c t i o n .
2. THE VARIATIONAL FUNCTION
The ground s t a t e o f t h e 160 nucleus, d e s c r i b e d i n
f o u r s t r u c t u r e l e s s and i n d e s t r u c t i b l e alpha p a r t i c l e s ,
is
terms o f
represented
by a c o m p l e t e l y symmetric wavefunction w i t h t o t a l o r b i t a l a n g u l a r momentum L=O and p o s i t i v e p a r i t y .
independent,
Since t h e f o r c e s considered
t h e v a r i a t i o n a l a n a i y s i s we i n t e n d t o c a r r y
out
are spin
can
be
done e n t i r e l y i n t h e c o n f i g u r a t i o n space so t h a t t h e above c h a r a c t e r i z a t i o n o f the wavefunct i o n wi I 1 be enough.
energies,
ant.
I n o r d e r t o a v o i d spurious
t h e wavefunction must be i n a d d i t i o n t r a n s l a t i o n a l l y i n v a r i -
I n the following,
we s h a l l c o n s t r u c t f o u r - p a r t i c l e harmonic- o s -
c i l l a t o r s t a t e s w i t h the above p r o p e r t i e s and then expand t h e v a r i a t i onal f u n c t i o n i n terms o f such s t a t e s .
Since t h i s was a l r e a d y done i n
~ , now r e s t r i c o u r s e l v e s t o a b r i e f presensome d e t a i l s e l s e ~ h e r e ~ 'we
t a t i o n o f that matter.
The t r i a l f u n c t i o n i s expressed as t h e expansion
where a, a r e v a r i a t i o n a l parameters t o be determined and $v a r e
p a r t i c l e h a r m o n i c - o s c i l l a t o r wavefunctions (FPHO).
f o r t h e FPHO quantum numbers,
four-
The index v stands
The c u t i n t h e sum (2.1)
is
determined
by t h e degree o f t h e a p p r o x i m a t i o n desi r e d ( a r a l lowed by t h e c o s t s i n volved).
T h i s i s d e f i n e d by t h e maximum quantum number
of
the
FPHO
components.
I t i s v e r y c o n v e n i e n t t o i n t r o d u c e t h e Jacobi c o o r d i n a t e s f o r
a system o f f o u r p a r t i c l e s , namely,
3
where xa, a = a,b,c,d,
are the coordinates o f the four p a r t i c l e s refer-
r e d t o an a r b i t r a r y frame.
These c o o r d i n a t e s a r e g i v e n
in inities of
( R / V ~ ) ' / ~ where
,
m i s t h e a l p h a - p a r t i c l e mass and w t h e osc i I ! a t o r f r e -
quence.
I t must be n o t e d t h a t
coordinate.
The o t h e r
$
-+
xd
i s e s s e n t i a l l y the
are-translational l y invariants.
+
be shown l a t e r on, t h e c o o r d i n a t e x
two-body m a t r i x elements, w h i l e
three- body m a t r i x elements.
3
x
C
.
center
-+ a
xb
of
mass
As i t wi l l
i s u s e f u i l i n the c a l c u l a t i o n o f
i s adequate f o r t h e
evaluation o f
The form f a c t o r depends on t h e c o o r d i n a t e
These comments l a r g e l y j u s t i f y t h e c h o i c e o f t h e c o o r d i n a t e s (2.2).
Let
ln L m > stand f o r t h e s t a t e o f a harmonic o s c i l l a t o r i n
a a a
3
the coordinate xa. If we allow only zero-quantum excitation in the center oP mass coordinate, the total number of quanta N
associated to
four oscillators in the coordinates (2.2) is given by
N = 2n +R +2n +R +2n +R
a a b b c c '
fornd=R = O .
d
L
Thc FPHO state with definite total orbital angular momentum
i s obtained from the In .t nr > by ordinary coupling. The particular
a a a
case L=O will 8e denoted simply by
jnR n i n R >
aaJ b b ' e c
I t can be easily shown that the state (2.4) is ipdependentof the order
3
of coupling the various angular momenta R
a
.
The richness of properties of the Jacobi coordinates
is not yet enough to facilitate the discussion of
the
(2.2)
permutational
symmetry of the FPHO states. For this purpose it is convenient to introduce the Kramer-kshinsky coordinates which are defined by
We shall denote the FPHO states, with L=O, in the Kramer-Moshinsky coordinates by the ket
This state has n q = RI+ = 0, allowing again no excitation of the center
of mass, and is represented by a round ket ) to distinguish betweenthe
angular ket
> in (2.4).
I t must be emphasised t h a t t h e quantum numbers niti
a f f e c t the coordinates
-f
y - i n (2.5) and
not
i n (2.6)
the laboratory c o o r d i n a t e s
-f
2..
Z
The r e l a t i v e Kramer-Moshinsky c o o r d i n a t e s and
the
relative
Jacobi ones a r e r e l a t e d between themselves through an o r t h o g o n a l transf o r m a t i o n , so t h e t o t a l number o f quanta (2.3)
i s a l s o g i v e n by
E x p l i c i t l y , we have t h e f o l l o w i n g r e l a t i o n
The p a r i t y o f s t a t e (2.6)
(2.7).
parity
of
N
in
As t h e p a r i t y o f t h e ground s t a t e o f t h e 160, i s even, we must
have t1+i12+t3 even.
be odd.
i s g i v e n by t h e
So, t h e t h r e e 9,'s must be even o r two o f
them caii
T h i s l a s t p o s s i b i l i t y i s t o be d i s r e g a r d e d i f we want ( 2 . 6 ) t o
be c o m p l e t e l y symmetric 5 . Thus, a l l t h e a n g u l a r momenta i n
(2.6)
are
.
even
Finally,
t h e s t a t e $v w i t h t h e p r o p e r t i e s mentioned
beginning . o f t h i s s e c t i o n i s g i v e n by .
where t h e n o r m a l i z a t i o n c o e f f i c i e n t A
v
i s such t h a t
in
the
I
&/6
A
=
v
c
1/6
i F a11 p a i r s (niL.j
are different
i f a l i p a i r s ( n . 1 . ) a r e equal
J1/12 o t h e r w i s e
z
Z
.
3. MATRIX ELEMEMTS
I n t h e l a s t s e c t i o n , we have discussed t h e
ternis o f which we expanded o u r v a r i a t i o n a l f u n c t i o n .
FPHO
states
in
l t was mentioned
t h a t t h e Kramer-Moshinsky coord i r i a t e s ( 2 . 5 ) a r e c o n v e n i e n t t o c o n s t r u c t
FPHO s t a t e s w i t h d e f i n e d p e r m u t a t i o n a l symmetry and p a r i t y .
Neverthe-
lesá, t h e JacoSi c o o r d i n a t e s ( 2 . 2 ) a r e s u i t a b l e f o r t h e c a l c u l a t i o n o f
the m a t r i x elements o f two-body and three- body o p e r a t o r s
f o r the c a l c u l a t i o n o f t h e form f a c t o r s .
from t h e s t a t e s ( 2 . 6 ) t o the s t a t e s
So a
linear
as
well
as
transformation
( 2 . 4 ) i s r a t h e r convenient.
The
c o e f f i c i e n t s a s s o c i a t e d w i t h t h i s t r a n s f o r m a t i o n a r e those i n t h e f o l l o w i n g e x p r e s s i o n ( i , j , k = 1,2,3)
Z l naRa ' nb Rb ' nc Rc > c na Ra ' nbRb ' nc Rc I ni Ri 'n j Rj , kn k ~ )
(3.1)'
where t h e sum i s extended over a11 non n e g a t i v e i n t e g e r v a l u e s o f
ia
nb,,!Lb,
nc and Rc s u b j e c t e d t o t h e c o n d i t i o n (2.7)
n
u'
and t o t h e con-
di tion
n e r a l 5'6.
The b r a c k e t ln R n R n R ln R n .a n a ) i s known
a a ' b b ' c c
i i ' j j ' k k
I n t h e case o f L=O i t reduces t o
i n ge-
The b r a c k e t s appear i n g i n t h e RHS o f (3
c o e f f icie:nts5.
. 3)
are
Moshinsky
The general i z e d Moshinsky c o e f f i c i e n t s a r e a s s o c i a t e d
w i t h t h e a n g l e f3 d e f i n e d by
The p r o p e r t i e s of these c c e f f i c i e n t s a r e discussed i n Ref . I .
I n t h e f o l l o w i n g , we s h a l l d i s c u s s t h e h a m i l t o n i a n whose mat r i x elements between s t a t e s (3.1) we a r e i n t e r e s t e d i n .
L e t us c o n s i d e r t h e f o u r - p a r t i c l e h a m i l t o n i a n
4
H = (Ew/2)
1
s=
The f i r s t term i s t h e k i n e t i c energy, Ve t h e Coulomb p o t e n t i a l , Vaa t h a
aa
i n t e r a c t i o i i , and V
The a-cr
3a
the 3cr-potential.
p o t e n t i a l we s h a l l use i n t h i s p a p e r w a s c o n s t r u c t e d
hy A1 i and ~ o d m e r ~ .I t i s an !L-dependence phenomenol o g i c a l p o t e n t i a l
whose parnrrieters were f i t t e d as t o reproduce t h e phase s h i f t s
and 6 4 a s s o c i a t e d w i t h a-a
scattering.
&o>
E x p l i c i t l y i t i s given
by
&2
a
s u p e r p o s i t i o n o f a t t r a c t i v e and r e p u l s i v e gaussians as
A l i and Bodmer a d j u s t e d severa1 s e t s o f parameters.
The ona we
shall
use here i s shown i n Table 1 .
The % - p o t e n t i a l
i s d e s c r i b e d by an a t t r a c t i v e gaussian
For t h i s component o f t h e p o t e n t i a l energy, we considered two r e c e n t l y
c o n s t r u c t e d p o t e n t ia1 s1 '
2
whose parameters were a d j u s t e d
in distinct
Table 1
- Parameters o f
t h e Ali-Bodmer p o t e n t i a l conside-
r e d i n t h i s paper.
forms.
60th p o t e n t i a l s were a d j u s t e d i n o r d e r t o reproduce t h e ground
- s t a t e energy o f t h e
l2c
nucleus.
A d d i t i o n a l l y , P o r t i l h o and Com r e -
+
q u i r e d i t t o d e s c r i b e t h e energy gap o f t h e lower O
w h i l e Ogasawara and H i u r a r e q u i r e d t h e d e s c r i p t i o n
t h a t nucleus.
and
of
2'
states,
the radius o f
These d i f f e r e n t c r i t e r i a l e d t o p o t e n t i a l s o f v e r y d i s -
t i n c t c h a r a c t e r i s t i c s as i t i s shown i n t a b l e 2 .
As the b a s i c s t a t e s a r e c o m p l e t e l y symmetric, f o r
effect o f
m a t r i x elements e v a l u a t i o n we may t a k e f o r t h e p o t e n t i a l energy t h e e x press i o n
where t h e f a c t o r s 6 and
4 account
f o r t h e number o f p a i r s and t r i o s o f
particles, respectively.
We a r e i n t e r e s t e d i n t h e i n t r i n s i c h a m i l t o n i a n
HI
1
Portilho
1
i
and
-
7
-
200
5.06
w3
Coon
Oga sawa r a
and
0.25
Hiura
Table 2
-
Parameters o f t h e 3-a
potentials.
which i s
obtained from (3.5)
by s u b t r a c t i n g the center-of-rnass k i n e t i c
energy,
name 1 y
Now, i t i s convenient t o add and s u b t r a c t from (3.9)
the term
so t h a t (3.9) can be expressed e n t i r e l y i n terms o f Jacobi coordinates
as
The f i r s t : term represents the hamiltonian o f
3 harmonic o s c i 1 l a t o r s w i t h
e i genva 1 ue
.
where N i s given by (2.7).
I f we represent
s i c s t a t e s (2.6),
by N the s e t o f quantum numbers o f the ba-
i.e.,
we have t o evaluate the m a t r i x elemehts
E s s e n t i a l l y , we have t o e v a l u a t e two types o f m a t r i x elements, namely
( n ; a l , ~ : ~ , n i ~ b I v(xa>
, , +Ve(xa>- (fico"/)
$Iniri,n
,
.a .,nk\)
3 3
(3.15)
and
To e v a i u a t e these m a t r i x elements i t i s o b v i o u s l y
t o express b r a s and k e t s i n Jacobi c o o r d i n a t e s . T h i s can
be
convenient
accompli-
shed by t h e use o f t h e t r a n s f o r m a t i o n (3.1) and It i s easy t o see
that
the two-body c o n t r i b u t i o n s are g i v e n by
where the summations a r e r e s t r i c t by t h e p r o p e r t i e s
coefficients.
of
the
Moshinsky
The "reduced" m a t r i x ~ l e m e n t scan be expressed i n
o f Talmi i n t e g r a l s 5
resul t i n g
terms
I n (3.19),
m o c 2 = 0.511 MeV i s an a r b i t r a r y s c a l e f a c t o r .
We n o t e t h a t t h e m a t r i x elements (3.17) a r e i n v a r i a n t
under
t h e exchange o f t h e p a i r s ( n . t .)
due
++ ( n .R .) and/or ( n ! ~ ! ) +-+ (nl.&!)
Z Z
3 3
Z Z
3 3
t o synmetry p r o p e r t i e s o f t h e Moshinsky c o e f f i c i e n t s . T h i s f a c t redu-
ces d r a s t i c a l l y t h e number o f m a t r i x elements t o be e f f e c t i v e l y evaluated.
The m a t r i x elements (3.16)
a r e w r i t t e n as
where t h e summations a r e r e s t r i c t e d by p r o p e r t i e s o f t h e s e v e r a 1
shinsky c o e f f i c i e n t s i n v o l v e d .
Since we can f a c t o r i z e t h e 3-body o p e r a t o r as
-3A (xa+x2)
vh
,xb ) =
R
z
V, f(xa)f(zb)
,
t h e correspond i ng reduced m a t r i x elements can be w r i t t e n as
Mo-
ta S p
in l + n
a
a
%
-+ R and
a
5 q I nj
+
nb
Again, we note t h a t the m a t r i x elements (3.20)
+
kb
are
.
(3.23)
invariant.
"
(n.R .) and/or (nR
; );
(nj").
In spite
J J
o f the economy t h i s symmetry provides, the presence o f generalized Moi )i
under the exchanges (nR
++
shinsky c o e f f i c i e n t s requires too much cornputer time.
I n order t o save
computer time, the f o l l o w i n g development was c a r r i e d out.
Let us introduce the f o l l o w i n g orthogonal transformation
-+
where yi
a r e the Kramer-Moshinsky coordinates (2.5).
I t i s easy t o see
that
so t h a t (3.21)
reads
g
The o s c i l l a t o r s i n the coordinates
wi t h those i n the coordinates
and
?
-+
y z and
-+
y 3 are
connected
through o r d i n a r y Moshinsky trans-
formation brackets, so t h a t we can w r i t e
where (72%) and (NL) a r e associated w i t h the coordinates
pectively.
$ and
-+
Y,
res-
Now a l i t t l e Racah a l g e b r a leads us t o t h e f i n a l e x p r e s s i o n
(x ,r )
(n;~I,nÍa:,n'e'lV
z z J J k k
3a
5+R1/2+3/2
,
x
F
2 1
a
b
Ini a i,n j a j
q+R '/2+3/2
,n L? ) =
k k
; R '+3/2 ; 2A2/(~+1) ( 2 ~ + 1 ) ]
.
(3.28)
Taking i n t o account t h e p r o p e r t i e s o f t h e Clebsch-Gordan and
Moshinsky
c o e f f i c i e n t s i t can be shown t h a t a l l t h e i ' s i n v o l v e d i n t h e
t i o n s a r e even.
changes (n.R .)
3 3
F i n a l l y , we n o t e t h a t (3.28)
*
(nkRk) a n d b r (n!ii.)
3 3
The above e x p r e s s i o n (3.28)
*
summa-
i s i n v a r i a n t u n d e r t h e ex-
(nkik).
i s much more compl i c a t e d t o w r i t e
dokn than t h e e q u i v a l e n t one g i v e n i n ( 3 . 2 2 ) .
Nevertheless,
its
use
a l l o w e d us t o reduce computer t i m e t o l e s s than h a l f t h e p r e v i o u s t i m e !
The d i s c u s s i o n o f t h e m a t r i x elements o f t h e
complete.
hamiltonian
I n t h e n e x t s e c t i o n , we d i s c u s s t h e charge form f a c t o r .
is
4. THE ÇHARGE FORM FACTOR OF"O
It i s w e l l kncwn t h a t t h e charge f o r w f a c t o r o f t h e
c i e u s f a c t o r i z e s as the p r o d u c t o f i t s body forrn f a c t o r by
p a r t i c l e charge form f a c t o r ,
the
160 nualpha
i.e,
As we a r e h e r e i n t e r e s t e d o n l y i n t h e e f f e c t o f t h r e e - a l p h a
forces i n
t h e c h a i g e form f a c t o r and s i n c e t h e a l p h a - p a r t i c l e charge form f a c t o r
i s o b v i o u s l y independent o f t h e i n t e r a c t i o n s between
the
alphas,
we
s h a l l r e s t r i c t o u r s e l v e s t o t h e d i s c u s s i o n o f t h e 150 body form f a c t o r .
I f t h e ground s t a t e i s d e s c r i b e d by t h e v a r i a t i o n a l
then t h e body form f a c t o r i s g i v e n by
func t iori
(2.1),
5
whete v stands f o r t h e s e t (nltl ,n21i2 , n 3 R 3 ) ,
and 8,
E
are defined i n
(3.15).
On t h e c o e f f i c i e n t s a,
were deterrnined through t h e v a r i a t i o -
na1 a n a l y s i s o f t h e h a m i l t o n i a n , i t rernains o n l y t o e v a l u a t e t h e m t r i x
elements t h a t appear i n ( 4 . 2 ) .
A specirnen o f them i s g i v e n by
where use was made o f t h e t r a n s f o r m a t i o n (3. I ) .
in
(4.4)
t i o n runs o v e r a11 t h e quantum number t h a t do n o t appear i n
There ai-e 16 summat i o n s .
the
summa-
the
LHS.
The presence o f t h e severa1 Kronecker symbol s
and the p r o p e r t i e s o f t h e Moshinsky c o e f f i c i e n t s l e a v e us w i t h o n l y
7
independent summations.
The e x p r e s s i o n
so easy t o be w r i t t e n down, a l s o con-
(4.4),
Thus, we had t o f i n d an a l t e r n a t i v e ex-
sumes t o n much computer t i m e .
p r e s s i o n which does n o t i n v o l v e g e n e r a l i z e d Moshinsky b r a c k e t s .
procedure i s somewhat s i m i l a r t o t h a t developed f o r t h e
evaluation
The
of
t h e m a t r i x elements o f t h e 3-body o p e r a t o r s i n l a s t s e c t i o n .
From ( 2 . 8 ) we see t h a t
Now, u s i n g t h e expansion
jo
(a
+
+ +
w i t h p = y ~ + y 3and
/:-;I
= 4n
a =
-
t o mean t h a t t h e term X=O
r
j X ( a ~ ) j X ( a ~ ) Y (Q$YhU(p)
hp
Av
k/S
.
The prime i n t h e summation
symbol i s
i s excluded from t h e sum.
Once t h e i n d i c e s 1,2 and 3 a r e separated i n SlandS2the
responding c o n t r i b u t i o n s can be e v a l u a t e d e a s i l y a f t e r some Racah
gebra g i v i n g
coral-
and
(even)
To e v a l u a t e t h e m a t r i x elements o f S 3 we use a g a í n t h e t r a n s f o r m a t i o n (3.24)
E3
5
and someilacahalgebra t o g e t
(n!R;,n',Q1 n ' . L ' l ~jn R n R n R
z
j 3 ' k k 3 i i ' j j ' k k '
'
(even)
Because a11 Ri, L . and Lk a r e even,
3
and
the
Gaunt c o e f f i -
c i e n t s (a0b01c0) r e q u i r e t h a t a+b+c = even, t h e A i n (4.9)
and
(4.10)
i s even.
An a l t e r n a t i v e e q u i v a l e n t e e x p r e s s i o n f o r (4.4)
i s then g i -
ven by
E1+E2+E3. A i 1
i t remains t o e v a l u a t e a r e
the
reduced
matrix
e l e m e n t ~o f j o . They a r e e a s i l y o b t a i n e d throiigh t h e use o f Talmi
in-
t e g r a l ~and a r e g i v e n fiy
where M(c:,b,z) i s a Kummer f u n c t i o n and p runs o v e r t h e values
Again, we o b t a i n e d an e x p r e s s i o n much more c o m p l i c a t e d
when
we t r i e d t o a v o i d t h e g e n e r a l i z e d Moshinsky c o e f f i c i e n t s . However, ex-
-
p r e s s i o n s (4.8)
(4.10) h e l p u s t o save a l o t o f computer time.
5. RESUL'TS AND DISCUSSION
Using o n l y the two- alpha i n t e r a c t i o n (3.6) w i t h t h e
t e r s o f Table I, we found a ground- state energy o f - 4.8
160 nucleus.
I n t h i s case, t h e numerical c a l c u l a t i o n s
o u t i n t h e 10-quantum a p p r o x i m a t i o n what means
f u n c t i o n (2.1)
has 40 components.
that
MeV
paramefor
wsre
the
carried
variational
The above energy agrees with t h e one
The experimental
p r e v i o u s l y o b t a i n e d by Mendez-Morerio e t d 4 .
is
- 14.5 MeV.
the
So, t h e u-a p o t e n t i a l a l o n e p r o v i d e s c n l y
30% o f
value
the
experimental v a l u e i n t h a t a p p r o x i m a t i o n .
I'urning on the 3-a
potential
(3.7),
we l e a r n t h a t t h e P o r t i -
lho-Coon p o t e n t i a l w i t h t h e parameters o f Table 2 o v e r b i n d s t h e system
e a r l y i n t h e 6-quantum approximation, s u r e l y on account
range c h a r a c t e r .
of
it s
long
On t h e o t h e r hand, t h e p o t e n t i a l o f Ogasawara-Hiurc,
whose parameters a r e l i s t e d i n Toble 2, g i v e s -4.74 NeV i n t h e 8-quantum approximation.
I t must be stre-ssed t h a t w i t h t h e h e l p uf t h i s 3 3
f o r c e we r e q u i r e d a subspace rnuch srnal l e r ( h a l f t h e d i m e n s i o n )
t h a t r e q u i r e d by u s i n g o n l y 2-u
than
p o t e n t i a l t o o b t a i n t h e sanie energy.
T h i s i s s i g n i f i c a n t and g i v e s us a c l e a r d e m o n s t r a t i o n o f t h e e f f e c t i veness o f t h e
3-a f o r c e i n t h e 160 nucleus.
Numerical c a l c u l a t i o n s up t o a g r e a t e r number o f
-
the approximation
quanta
in
t h a t would a l l o w a r e l i a b l e d i s c u s s i o n on e x c i t e d
s t a t e s and i n e l a s t i c form f a c t o r , f o r i n s t a n c e
-
were n o t c a r r i e d o u t
because o f t h e v e r y expensive coaputer c o s t s i n s p i t e o f t h e r e d u c t i o n
i n process t i m e as mentioned b e f o r e .
Concerning t h e 160 f o r m f a c t o r , o n l y t h e body form f a c t o r i s
presented because we p r e s e n t l y a r e i n t e r e s t e d o n l y i n v e r i f y t h e e f f e c t
of a
3-a f o r c e .
T h i s i s shown i n Fig.1.
The dashed l i n e i s o b t a i n e d
w i t h o u t t h e presence o f a t h r e e - a l p h a f o r c e .
When we t u r n t h i s f o r c e
on we o b t a i n t h e body form f a c t o r represented by t h e continuous
line.
We see from t h e f i g u r e t h a t the form f a c t o r i s v e r y i n s e n s i t i v e t o t h e
presence o f t h e
3-a f o r c e f o r small momentum t r a n s f e r zq. A s l i g h t e f -
f e c t can be noted a c c o r d i n g as Rq increases.
l60
BODY FORM FACTOR
Fig.1
-
Body form f a c t a r of 150 w i t h Ali-Badmer + Ogasawara-Hiura po-
tentials.
Dashed l i n e : 3-0 force turned o f f .
force t u r n e d on.
Continuous
line:
3-a
The a u t h o r s thanks v e r y much P r o f . O.
Portilho for his
help
a t the Computational Center o f t h e Universidade de ~ r a s i l i a .
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