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Note on the 4-body system
Volume 7, number 4
PHYSICS
LETTERS
of fit. B e t w e e n 5 and 10 p h o t o g r a p h s h a v e b e e n m a d e at e a c h of 11 p r e s s u r e s . T h e e r r o r b a r s on the p o i n t s shown in fig. 1 a r e e q u a l to one s t a n d a r d d ev i a t i o n f r o m the m e a n . U n d e r the a s s u m p t i o n that the only d e p o l a r i z i n g c o l l i s i o n s that c a n o c c u r a r e c o l l i s i o n s of o r i e n t e d c e s i u m a t o m s with the w a l l s o r with b u f f e r g a s a t o m s , the f o l l o w i n g a p p r o x i m a t e e x p r e s s i o n f o r the r e l a x a t i o n t i m e ~- h a s b e e n d e r i v e d 3, 4) n- = [ ( n / R ) 2 D o ( P o / p ) + NO~ t , r e l ( P / p o ) ] -1 , w h e r e R is the r a d i u s of the v a p o u r c e l l , D o i s the d i f f u s i o n c o e f f i c i e n t of c e s i u m in n i t r o g e n at a t m o s p h e r i c p r e s s u r e and 52°C, Po i s a t m o s p h e r i c p r e s s u r e , P i s the a c t u a l n i t r o g e n p r e s s u r e , N o i s the d e n s i t y of n i t r o g e n at a t m o s p h e r i c p r e s s u r e and 52°C, and Vrel is the m e a n r e l a t i v e v e l o c i t y of a c e s i u m a t o m and n i t r o g e n m o l e c u l e at 52°C. T h e s o l i d c u r v e in fig. 1 r e p r e s e n t s the b e s t fit of t h i s e q u a t i o n to the e x p e r i m e n t a l data. F r o m t h i s c u r v e we obtain v a l u e s of 0.29 c m 2 / s e c f o r Do, and 7 . 0 × 10 -23 c m 2 f o r ~. T h e s e a r e q u a l i t a t i v e l y c o n -
NOTE
ON
THE
1 December 1963
s i s t e n t with t h e v a l u e s of 0.31 c m 2 / s e c f o r Do and 5.2)< 10 .2 3 c m 2 f o r ~ that F r a n z e n found f o r r u b i dium in neon. It r e m a i n s of i n t e r e s t to c o m p a r e the r e s u l t s r e p o r t e d in t h i s p a p e r with d i f f u s i o n c o e f f i c i e n t s and d i s o r i e n t a t i o n c r o s s s e c t i o n s f o r c e s i u m in the v a r i o u s i n e r t g a s e s . M e a s u r e m e n t s of t h e s e p a r a m e t e r s a r e now in p r o c e s s . We w i s h to e x t e n d thanks to Mr . G. S a n d o r f i f o r h i s help in the c o n s t r u c t i o n of the e x p e r i m e n t a l a p p a r a t u s and in the c o l l e c t i o n and a n a l y s i s of data. We a l s o w i s h to thank D r . G. Kane f o r d e v i s i n g a s u i t a b l e c o m p u t e r p r o g r a m , and P r o f e s s o r H. F r a u e n f e l d e r and D r . J. R. E h r m a n f o r s e v e r a l helpful discussions. 1) 2) 3) 4) 5) 6)
H.G.Dehmelt, Phys. Rev. 105 (1957) 1487. R . L . d e Zafra, A m . J . P h y s . 2 8 (1960) 646. W. Franzen: Phys. Rev. 115 (1959) 850. R.A. Bernheim, J. Chem. Phys. 36 (1962) 35. R. G. B r e w e r J. Chem.Phys. 37 (1962) 2504. W.W. Holloway J r . , E. LGscher and R. Novick, Phys. Rev. 126 (1962) 2109. 7) F.A. Franz, Rev. Sei. Instr. 34 (1963) 589.
4-BODY
SYSTEM
T. STOVALL and M. DANOS National Bureau of Standards, Washington, D. C. Received 28 October 1963
R e c e n t e x p e r i m e n t s on the s c a t t e r i n g of d e u t e r o n s on t r i t i u m 1) h a v e b e e n i n t e r p r e t e d by W e r n t z 2) a s a m a n i f e s t a t i o n of an e x c i t e d s t a t e in He 4 lying within 1 MeV of the c o n t i n u u m . He c o n c l u d e d that t h i s s t a t e is J = 0 +, T= O. W e s e t o u r s e l v e s the t a s k of f i n d in g out w h e t h e r a T = 0 s t a t e c a n be u n d e r s t o o d with r e a s o n a b l e p a r a m e t e r s . In a d d i t i o n , we hope to o b t a i n an i n d i c a t i o n of w h e t h e r it i s l i k e l y that a T = 1 s t a t e a l s o e x i s t s , t h i s b e i n g n e c e s s a r y f o r p a r t i c l e s t a b i l i t y of H 4 and Li4, a q u e s t i o n about w h i c h t h e r e h a s b e e n r e n e w e d i n t e r e s t . We u s e the t e r m " s t a t e " to m e a n that the a p p r o p r i a t e s c a t t e r i n g p h a s e s h i f t g o e s t h r o u g h 90 ° . U s i n g a v e r y s i m p l e m o d e l w h i c h w i l l be d e s c r i b e d b e l o w , we find that i n d e e d a T = 0 s t a t e c a n b e a c c o m m o d a t e d e a s i l y with v e r y r e a s o n a b l e p a r a m e t e r s , w h i l e t h i s is not t r u e f o r a T = 1 s t a t e . N a t u r a l l y , a T = 1 c o m p o n e n t i s a d m i x e d to the T = 0 s t a t e b e c a u s e of the C o u l o m b f o r c e s . T h i s a d m i x t u r e h a s , h o w e v e r , a v e r y s m a l l a m p l i t u d e and we do not s e e t h e p o s s i b i l i t y of a p r e d o m i n a n t l y T = 1 r e s o n a n c e in the He 4 s y s t e m .
278
We consider the following model: We replace the 4-body problem of the He 4 nucleus by a onebody problem in which one nucleon moves in a potential which is supposed to represent all the other energies and the interactions of the particle with the other three particles, i.e., we consider the motion of, say, a neutron with respect to the recoiling He 3 system. Such a treatment would be valid if the adiabatic approximation were applicable. We do not expect this to be true really ; the numerical values have to be considered, therefore, as having only a qualitative significance. Our program is to find a potential which reproduces the bound state characteristics of He'*, namely the binding energy and the rms charge radius. According to Levinson's theorem 3) there exists a scattering state. As we will see, the scattering state occurs at an energy which, considering the accuracy of the model, is somewhat too high. In the bound state the potential well is of very short range (and thus quite deep because of the well-known approximate parameter VR 2
Volume 7, number 4
PHYSICS
w h i c h d e t e r m i n e s t h e b i n d i n g e n e r g y ) owing to t h e c o m p a c t n e s s of He4. In the s c a t t e r i n g s t a t e , on the o t h e r hand, the s i z e of the p o t e n t i a l w e l l i s g i v e n b y the l a r g e r s i z e of He3. S i n c e t h e s c a t t e r i n g s t a t e i s q u i t e w i d e a n d s i n c e the i n c o m i n g p a r t i c l e m o v e s f a s t e r than t h e p a r t i c l e s b o u n d in the He 3 s y s t e m , one m u s t e x p e c t t h a t upon t h e a p p r o a c h of the i n c o m i n g p a r t i c l e the He3 s y s t e m h a s no t i m e to a d j u s t i t s e l f f r o m t h e He 3 d e n s i t y to the He 4 d e n s i t y . We t h u s t a k e the p o t e n t i a l f o r the s c a t t e r i n g s t a t e to h a v e a p p r o p r i a t e l y l a r g e r r a d i u s arid s m a l l e r depth. W i t h t h i s s o - c a l l e d t r a n s f o r m e d p o t e n t i a l we o b t a i n a r e a s o n a b l e r a n g e f o r the s c a t t e r i n g e n e r gies. We c o n s i d e r f o r t h e g r o u n d s t a t e of He 4 only the c o n f i g u r a t i o n (S~.)4. T h e n t h e o p t i c a l m o d e l S c h r S d i n g e r e q u a t i o n } o r one p a r t i c l e (we n e g l e c t the Coulomb force) is
{- h 2 ~2 ÷ V(x)- E} u(x) = 0
(1)
w h e r e / z = ~3 M i s t h e r e d u c e d m a s s , x i s the d i s t a n c e of the p a r t i c l e f r o m the c e n t r e of m a s s of the r e c o i l i n g He 3 o r H 3 s y s t e m , E i s the e n e r g y , and we a s s u m e t h a t t h e p o t e n t i a l V(x) h a s the f o r m V
Y(x) - 1+ e ( x - R ) / a
XM:
2 dx =-fx2co(x) d3x ;
fw(x) d3x: 1.
(3)
If t h e p r o t o n a n d the r e c o i l i n g H 3, o r a l t e r n a t e l y the r e c o i l i n g He 3, w e r e p o i n t c h a r g e s the m e a n s q u a r e r a d i u s , RC2 , of the c h a r g e d i s t r i b u t i o n w o u l d b e g i v e n by 2 R c 2 = ½[(¼XM)2+(¼XM) 2] + ½12(¼XM)2]; (4) t h e f a c t o r 2 on the l e f t h a n d s i d e c o m e s f r o m t h e c h a r g e of He, a n d the two b r a c k e t s c o n t a i n t h e c o n t r i b u t i o n s f r o m the two p o s s i b i l i t i e s : T h e odd p a r ticle can be a proton and then the recoiling system i s H 3 ( f i r s t b r a c k e t ) o r the odd p a r t i c l e c a n b e a n e u t r o n a n d the r e c o i l i n g s y s t e m then i s He 3 ( s e c ond b r a c k e t ) . B o t h t h e s e p o s s i b i l i t i e s a r e g i v e n the 1 w e i g h t ~ and thus RC 2
3
2
- 16 X M
•
(5)
In o r d e r to o b t a i n the a c t u a l r m s r a d i u s one h a s to c o n s i d e r the f i n i t e s i z e of the p a r t i c l e a n d of the r e -
1 D e c e m b e r 1963
c o i l i n g t h r e e - b o d y s y s t e m . F r o m the r e s u l t s of S l a g g i e a n d W i c h m a n n 4) c o n c e r n i n g the a s y m p t o t i c b e h a v i o u r of the t h r e e - b o d y w a v e f u n c t i o n we s e e t h a t upon s e p a r a t i o n of a b o u n d s y s t e m into f r a g m e n t s , a t l a r g e s e p a r a t i o n the f r a g m e n t s a c q u i r e the s i z e t h e y would h a v e if t h e y w e r e b y t h e m s e l v e s , i . e . , the f a c t t h a t t h e y h a v e a n e g a t i v e k i n e t i c e n e r g y d o e s not i n f l u e n c e t h e i r i n t e r n a l s t r u c t u r e . T h u s f o r i n c r e a s i n g x the r e c o i l i n g three-body system expands from a density approp r i a t e to the He 4 s y s t e m to a d e n s i t y a p p r o p r i a t e to the H 3 and He 3 s y s t e m . We f u r t h e r a s s u m e t h a t the c h a r g e d i s t r i b u t i o n s of the p r o t o n , H 3 and He 3 h a v e a G a u s s i a n f o r m , the l a t t e r two b e i n g the s a m e and e q u a l to
~(y) = .~e -(y/D(x))2,
f ~,,(.y) d 3 y = l ,
(6)
w h e r e y i s m e a s u r e d f r o m t h e c e n t r e of m a s s of the t h r e e - p a r t i c l e s y s t e m a n d 2~ i s t h e n o r m a l i s a tion f a c t o r . We t a k e the " d e n s i t y a p p r o p r i a t e to the He 4 s y s t e m " to be c h a r a c t e r i z e d by X M and the " d e n s i t y a p p r o p r i a t e to the r e c o i l i n g He 3 o r H 3 s y s t e m " to b e c h a r a c t e r i z e d b y R H. In (6) we c h o o s e the f u n c t i o n D(x) s o that i t i n t e r p o l a t e s s m o o t h l y b e t w e e n t h e s e two v a l u e s :
(2)
w h i c h h a s t h r e e a d j u s t a b l e p a r a m e t e r s , V, R a n d a. We want to find s e t s of t h e s e p a r a m e t e r s w h i c h y i e l d the c o r r e c t b i n d i n g e n e r g y and r m s c h a r g e r a d i u s . Solving (1) w e i m m e d i a t e l y o b t a i n the b i n d i n g e n e r g y B = - E . To f i n d t h e r m s c h a r g e d i s t r i b u t i o n we e m p l o y t h e f o l l o w i n g p r o c e d u r e : W i t h the s o l u t i o n of (1) _we c o n s t r u c t t h e m e a n s q u a r e of x w h i c h we c a l l
x2:jx2u
LETTERS
D(x) = D o = X M J 2 f o r x < X M D(x) = D O + (R H - Do)(1 - e - ( X - X M / ( R H - D o ) )
(Ta) (Tb)
for x >1X M . We w e r e g u i d e d in the c h o i c e (Ta) f o r the r e l a t i o n b e t w e e n D o a n d R M b y the g e o m e t r y of the t e t r a h e d r o n . H e r e R H i s the r m s v a l u e of the c h a r g e d i s t r i b u t i o n s in H 3 o r He 3 a f t e r c o r r e c t i o n f o r the f i n i t e s i z e of the p r o t o n . Now we c a n w r i t e down the c h a r g e d i s t r i b u t i o n of He 4. F i r s t of a l l we e l i m i n a t e the e f f e c t of the f i n i t e s i z e of the p r o t o n b y t h e r e l a t i o n 2 = R2 2 RC rms - r P
(8)
L e t r b e m e a s u r e d f r o m the c e n t r e of m a s s of the f o u r - p a r t i c l e s y s t e m . T h e n the m e a n s q u a r e r a d i u s of the c h a r g e d i s t r i b u t i o n i s RC2 = ½/r2D(~. ) d3 r ,
(9)
w h e r e the c h a r g e d e n s i t y
P(r) = 32/co(x) co(y) 6 ({x + y - r) d3x d3y
(10)
+ ½/co(x) 6(~x + r ) d 3 x . We s e t of would MeV,
u s e d the NBS IBM 7094 c o m p u t e r to find a p a r a m e t e r s of the p o t e n t i a l , eq. (2), which r e p r o d u c e t h e b i n d i n g e n e r g y , B = 20.57 and the r m s r a d i u s of the p o i n t c h a r g e d i s 279
Volume 7, n u m b e r 4
PHYSICS
t r i b u t i o n , R e 2 = 1 . 9 5 2 1 x 1 0 -26 c m 2 * 6). T h e s e a r c h was done by a s s u m i n g the d i f f u s e n e s s p a r a m e t e r a a n d c h a n g i n g V a n d R u n t i l both B and R e 2 w e r e r e p r o d u c e d to b e t t e r than 1%. We f i n d that in a t y p i c a l c a s e R e 2 c o n t r i b u t e s 21% to RC 2, the f i n i t e s i z e of the t h r e e - p a r t i c l e s y s t e m c o n t r i b u t e s 62%, a n d the e x p a n s i o n of the t h r e e - p a r t i c l e s y s t e m 17%. T h e p a r a m e t e r s o b t a i n e d in this way a r e p l o t t e d i n fig. 1. IOOC --
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03 0.4 PARAMETER,
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0.5 o, x l ( ) 1 3 c m
Fig. 1. P a r a m e t e r s of the potential which yield the correct bound state properties of the four-particle system. I00
_
b
i
. . . . .
7- ....
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LETTERS
1 December 1963
With each of t h e s e p o t e n t i a l s we then s e a r c h e d for the e n e r g y of the s c a t t e r i n g s t a t e . T h e s e e n e r g i e s a r e p l o t t e d i n fig. 2 a s open c i r c l e s . T h e y l i e r a t h e r high, at 20 to 40 MeV i n s t e a d of the few MeV obs e r v e d e x p e r i m e n t a l l y . The r e a s o n s for t h i s have b e e n d i s c u s s e d above. We now t r a n s f o r m the pot e n t i a l by m u l t i p l y i n g a a n d R by ~ and, c o n s i d e r i n g that V is a p p r o x i m a t e l y p r o p o r t i o n a l to the d e n s i t y , d i v i d e V b y ~3. We c h o o s e ~ s u c h that R a f t e r the t r a n s f o r m a t i o n e q u a l s RH, i . e . , ~ = R H / R . Again we s e a r c h e d for the s c a t t e r i n g s t a t e . The e n e r g i e s that we o b t a i n e d a r e p l o t t e d a s dots on fig. 2. It i s s e e n that t h e r e does e x i s t a t r a n s f o r m e d p o t e n t i a l which g i v e s the c o r r e c t s c a t t e r i n g s t a t e e n e r g y . T h u s f r o m the a r g u m e n t s g i v e n h e r e a T = 0 s c a t t e r i n g s t a t e j u s t above the t h r e s h o l d for p a r t i c l e e m i s s i o a i s e x p e c t e d to exist. A s t u d y of the p a r a m e t e r s of the t r a n s f o r m e d p o t e n t i a l i n d i c a t e s to us that a n y r e a s o n a b l e t r a n s f o r m a t i o n would y i e l d a t r a n s f o r m e d p o t e n t i a l which g i v e s the c o r rect scattering state energy. In o r d e r to have a T = 1 s t a t e in the f o u r - p a r t i c l e s y s t e m one m u s t h a v e at l e a s t one p a r t i c l e in a p - s t a t e . W e r n t z a n d B r e n n a n 7) have g i v e n r e a s o n s for b e l i e v i n g that the s t a t e s of the four nucleon system are pure isospin states. Moreover, this would b e e x p e c t e d to be t r u e e v e n if we do not c o n s i d e r t h e s e a r g u m e n t s s i n c e the C o u l o m b i n t e r a c t i o n e n e r g y in the few p a r t i c l e s y s t e m is s m a l l . We s e a r c h e d for p - w a v e s t a t e s in both the t r a n s f o r m e d and u n t r a n s f o r m e d p o t e n t i a l s . No r e s o n a n c e s e x i s t in t h e s e p o t e n t i a l s . It s h o u l d be r e m e m b e r e d that the i n t e r a c t i o n of n u c l e o n s in r e l a t i v e p - s t a t e s i s w e a k e r than i n r e l a t i v e s s t a t e s . By u s i n g the s a m e p o t e n t i a l we have in effect a s s u m e d that the p- and s - s t a t e i n t e r a c t i o n s w e r e the s a m e . We thus f e e l that we have o v e r e s t i m a t e d the i n t e r a c t i o n s t r e n g t h a n d that it is r a t h e r u n l i k e l y that t h e r e should e x i s t p - s t a t e s in n a t u r e in the f o u r - p a r t i c l e s y s t e m . The i n t e r a c t i o n of h i g h e r l s t a t e s with the p o t e n t i a l w e l l i s s t i l l w e a k e r than the p - s t a t e i n t e r a c t i o n , a n d we do not think that they e x i s t i n n a t u r e e i t h e r . T h u s we do not expect that a T = 1 s t a t e of the a - 2 a r t i c l e e x i s t s . T h e r e f o r e p r o b a b l y Li 4 and H'~ a r e not s t a b l e u n d e r p a r t i c l e e m i s s i o n and following Gol' d a n s k y 8) H 5 is a l s o not s t a b l e u n d e r p a r t i c l e emission.
.E
References
.4
1) H.W. Lafevre, R.R. Borchers and C. H. Poppe, Phys. Rev. 128 (1962) 1328. 2) Carl Werntz, Phys.Rev. 128 (1962) 1336. 3) N. Levinson, Dan. Mat. Fys. Medd. Bind 25 (1949) 0
I
0.1
[
I
I
I
Fig. 2. Energies of the T= 0 scattering state. 280
J
0.2 0.3 0.4 0.5 DIFFUSENESS P A R A M E T E R , o, xlO"'3era
* The code was based on ref. 6). We are grateful to Dr. R. S. Caswell for modifying and supplementing the code of ref. 6) to suit our needs.
Volume 7, n u m b e r 4
PHYSICS
4) E. Leo Slaggie and Eyvind H.Wichmann, J. Math. Phys. 3 (1962) 946. 5) H. Collard et al., private communication. 6) R.S. Caswell, National Bureau of Standards Technical Note No. 159 (1962).
LETTERS
1December 1963
7) Carl Werntz and J.G. Brennan, Physics Letters 6 (1963) 113. 8) V.I. Go1'dansky, J. Exptl. Theoret. Phys. (USSR) 38 (1960) 1637 ; translation : Soviet Phys. JETP 11 (1960) 1179.
ABSENCE OF TENSOR TERM IN THE OPTICAL MODEL FOR THE d-Ca ELASTIC SCATTERING A T 22 M e V J. R A Y N A L Service de Physique ThSorique, Centre d'Etudes Nucl~aires de Saclay B. P. no. 2, Gif - s u r - Yvette (S &O) , France Received 26 October 1963
O p t i c a l m o d e l c a l c u l a t i o n s for the d i f f e r e n t i a l c r o s s s e c t i o n of e l a s t i c s c a t t e r i n g of d e u t e r o n s on n u c l e i a g r e e v e r y w e l l with e x p e r i m e n t a l data, c h i e f l y at not too high e n e r g y 1) a n d for h e a v y t a r g e t s 2), taking into a c c o u n t only a c e n t r a l p o t e n t i a l with s l o w l y v a r y i n g p a r a m e t e r s ; h o w e v e r m u l t i p l e s o l u t i o n s c a n b e found. Some p o l a r i s a t i o n m e a s u r e m e n t s have r e c e n t l y b e e n p e r f o r m e d at S a c l a y on the Ca n u c l e u s n e a r 22 MeV l e a d i n g to the v e c t o r p o l a r i s a t i o n p a r a m e t e r 3) a n d to two of the t h r e e t e n s o r p o l a r i s a t i o n p a r a m e t e r s 4). C r o s s s e c t i o n m e a s u r e m e n t s with the s a m e t a r g e t w e r e m a d e by Y n t e m a at 21.4 MeV 5). C r o s s s e c t i o n data c a n n o t be f i t t e d v e r y w e l l by a c e n t r a l p o t e n t i a l a l o n e 2). Some a t t e m p t s w e r e m a d e to fit t h e s e data with L S coupling a n d t e n s o r p o t e n t i a l s b y a n a u t o m a t i c s e a r c h p r o g r a m 6) with the IBM 7090 at S a c l a y a s well a s for o t h e r t a r g e t s 7). The i m a g i n a r y p a r t of the c e n t r a l p o t e n t i a l a l w a y s t u r n e d out to be s m a l l e r and p l a c e d at g r e a t e r r a d i u s than for o t h e r t a r g e t s . Good r e s u l t s w e r e o b t a i n e d with a s o m e w h a t c o m p l i c a t e d p o t e n t i a l i n c l u d i n g the u s u a l L S coupling t e r m t a k e n with a c o m p l e x c o e f f i c i e n t a n d the t e n s o r t e r m 1 {(S~)2 - -~}dfir2 (1 + e ( r - R ) / a ) " The l a t t e r was about twice g r e a t e r than the one exp e c t e d f r o m c o n s i d e r a t i o n s on the D - w a v e of the d e u t e r o n 8) and had the o p p o s i t e sign. The v e c t o r p o l a r i s a t i o n g i v e n by this p o t e n t i a l was in r e a s o n a b l e a g r e e m e n t witht he e x p e r i m e n t ; w h e r e a s the t e n s o r p o l a r i s a t i o n w a s m u c h g r e a t e r than the one o b t a i n e d in the e x p e r i m e n t s 4). If one i n t r o d u c e s the p o l a r i s a t i o n s into the s e a r c h , the t e n s o r p o t e n t i a l b e c o m e s a s m a l l f r a c t i o n of that e x p e c t e d ; h o w e v e r , n o r e a l l y s a t i s f a c t o r y fit was o b t a i n e d : in p a r t i c u l a r , the fit for the c r o s s s e c t i o n w a s q u i t e poor.
T h i s was t a k e n a s an i n d i c a t i o n that the t e n s o r t e r m was of no g r e a t help in this s i t u a t i o n , and we d e c i d e d to p u r s u e the s e a r c h with the L S coupling only. T h e r e s u l t s a r e a s f o l l o w s : 1. We take a Saxon c e n t r a l p o t e n t i a l , the u s u a l L S coupling a n d a s u r f a c e a b s o r p t i o n and we t r y to fit the c r o s s s e c t i o n r e s u l t s . A g r e e m e n t h a s b e e n o b t a i n e d only in the 100 MeV r e g i o n of depth for the c e n t r a l p a r t . S e a r c h e s w e r e m a d e with v a r i o u s v a l u e s of the r e d u c e d r a d i u s of the c e n t r a l p o t e n t i a l r a n g i n g f r o m 1 to 1.25 fm. T h e c o r r e s p o n d i n g par a m e t e r s a r e l i s t e d in t a b l e 1. The m i n i m u m of ×2 i s r e a c h e d for the v a l u e s 1 . 1 0 - 1.15. The v e c t o r p o l a r i s a t i o n g i v e n by this p o t e n t i a l is in r e a s o n a b l e a g r e e m e n t with e x p e r i m e n t but the z e r o at 54 ° has a shift of s o m e d e g r e e s . 2. We then p e r f o r m e d the s a m e c a l c u l a t i o n s v a r y i n g the r a d i u s and the d i f f u s e n e s s p a r a m e t e r of the L S coupling i n s t e a d of taking i t s u s u a l f o r m . The fit was i m p r o v e d a s is e v i d e n t f r o m t a b l e 2. Two kind of s o l u t i o n s c a n b e s e p a r a t e d h e r e dep e n d i n g on w h e t h e r the d i f f u s e n e s s p a r a m e t e r of L S coupling i s g r e a t e r or s m a l l e r than 0.45 fm. In the l a s t c a s e t h e r e is an i m p r o v e m e n t of the fit of the f o r w a r d s c a t t e r i n g c r o s s s e c t i o n , but the e x p e r i m e n t a l e r r o r s a r e m o r e i m p o r t a n t in this r e g i o n 5); at the s a m e a n g l e s , the v e c t o r p o l a r i s a t i o n and T22 a r e too s m a l l . H o w e v e r , s i n c e the z e r o s of the v e c t o r p o l a r i s a t i o n a r e well p l a c e d the o v e r a l l a g r e e m e n t is b e t t e r than in the f i r s t case. The l a s t line of each t a b l e g i v e s the r e s u l t of s e a r c h e s including vector polarisation and tensorial effect at ~ = 0, ~ and ~ with s t a t i s t i c a l e r r o r s (36 v a l u e s of each kind). The c o r r e s p o n d i n g c r o s s s e c t i o n s and p o l a r i s a t i o n s a r e plotted in figs. 1-4 with e x p e r i m e n t a l p o i n t s . The e f f i c i e n c y in the p r o d u c t i o n of the p o l a r i z e d b e a m is s u p p o s e d to b e 281
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of fit. B e t w e e n 5 and 10 p h o t o g r a p h s h a v e b e e n m a d e at e a c h of 11 p r e s s u r e s . T h e e r r o r b a r s on the p o i n t s shown in fig. 1 a r e e q u a l to one s t a n d a r d d ev i a t i o n f r o m the m e a n . U n d e r the a s s u m p t i o n that the only d e p o l a r i z i n g c o l l i s i o n s that c a n o c c u r a r e c o l l i s i o n s of o r i e n t e d c e s i u m a t o m s with the w a l l s o r with b u f f e r g a s a t o m s , the f o l l o w i n g a p p r o x i m a t e e x p r e s s i o n f o r the r e l a x a t i o n t i m e ~- h a s b e e n d e r i v e d 3, 4) n- = [ ( n / R ) 2 D o ( P o / p ) + NO~ t , r e l ( P / p o ) ] -1 , w h e r e R is the r a d i u s of the v a p o u r c e l l , D o i s the d i f f u s i o n c o e f f i c i e n t of c e s i u m in n i t r o g e n at a t m o s p h e r i c p r e s s u r e and 52°C, Po i s a t m o s p h e r i c p r e s s u r e , P i s the a c t u a l n i t r o g e n p r e s s u r e , N o i s the d e n s i t y of n i t r o g e n at a t m o s p h e r i c p r e s s u r e and 52°C, and Vrel is the m e a n r e l a t i v e v e l o c i t y of a c e s i u m a t o m and n i t r o g e n m o l e c u l e at 52°C. T h e s o l i d c u r v e in fig. 1 r e p r e s e n t s the b e s t fit of t h i s e q u a t i o n to the e x p e r i m e n t a l data. F r o m t h i s c u r v e we obtain v a l u e s of 0.29 c m 2 / s e c f o r Do, and 7 . 0 × 10 -23 c m 2 f o r ~. T h e s e a r e q u a l i t a t i v e l y c o n -
NOTE
ON
THE
1 December 1963
s i s t e n t with t h e v a l u e s of 0.31 c m 2 / s e c f o r Do and 5.2)< 10 .2 3 c m 2 f o r ~ that F r a n z e n found f o r r u b i dium in neon. It r e m a i n s of i n t e r e s t to c o m p a r e the r e s u l t s r e p o r t e d in t h i s p a p e r with d i f f u s i o n c o e f f i c i e n t s and d i s o r i e n t a t i o n c r o s s s e c t i o n s f o r c e s i u m in the v a r i o u s i n e r t g a s e s . M e a s u r e m e n t s of t h e s e p a r a m e t e r s a r e now in p r o c e s s . We w i s h to e x t e n d thanks to Mr . G. S a n d o r f i f o r h i s help in the c o n s t r u c t i o n of the e x p e r i m e n t a l a p p a r a t u s and in the c o l l e c t i o n and a n a l y s i s of data. We a l s o w i s h to thank D r . G. Kane f o r d e v i s i n g a s u i t a b l e c o m p u t e r p r o g r a m , and P r o f e s s o r H. F r a u e n f e l d e r and D r . J. R. E h r m a n f o r s e v e r a l helpful discussions. 1) 2) 3) 4) 5) 6)
H.G.Dehmelt, Phys. Rev. 105 (1957) 1487. R . L . d e Zafra, A m . J . P h y s . 2 8 (1960) 646. W. Franzen: Phys. Rev. 115 (1959) 850. R.A. Bernheim, J. Chem. Phys. 36 (1962) 35. R. G. B r e w e r J. Chem.Phys. 37 (1962) 2504. W.W. Holloway J r . , E. LGscher and R. Novick, Phys. Rev. 126 (1962) 2109. 7) F.A. Franz, Rev. Sei. Instr. 34 (1963) 589.
4-BODY
SYSTEM
T. STOVALL and M. DANOS National Bureau of Standards, Washington, D. C. Received 28 October 1963
R e c e n t e x p e r i m e n t s on the s c a t t e r i n g of d e u t e r o n s on t r i t i u m 1) h a v e b e e n i n t e r p r e t e d by W e r n t z 2) a s a m a n i f e s t a t i o n of an e x c i t e d s t a t e in He 4 lying within 1 MeV of the c o n t i n u u m . He c o n c l u d e d that t h i s s t a t e is J = 0 +, T= O. W e s e t o u r s e l v e s the t a s k of f i n d in g out w h e t h e r a T = 0 s t a t e c a n be u n d e r s t o o d with r e a s o n a b l e p a r a m e t e r s . In a d d i t i o n , we hope to o b t a i n an i n d i c a t i o n of w h e t h e r it i s l i k e l y that a T = 1 s t a t e a l s o e x i s t s , t h i s b e i n g n e c e s s a r y f o r p a r t i c l e s t a b i l i t y of H 4 and Li4, a q u e s t i o n about w h i c h t h e r e h a s b e e n r e n e w e d i n t e r e s t . We u s e the t e r m " s t a t e " to m e a n that the a p p r o p r i a t e s c a t t e r i n g p h a s e s h i f t g o e s t h r o u g h 90 ° . U s i n g a v e r y s i m p l e m o d e l w h i c h w i l l be d e s c r i b e d b e l o w , we find that i n d e e d a T = 0 s t a t e c a n b e a c c o m m o d a t e d e a s i l y with v e r y r e a s o n a b l e p a r a m e t e r s , w h i l e t h i s is not t r u e f o r a T = 1 s t a t e . N a t u r a l l y , a T = 1 c o m p o n e n t i s a d m i x e d to the T = 0 s t a t e b e c a u s e of the C o u l o m b f o r c e s . T h i s a d m i x t u r e h a s , h o w e v e r , a v e r y s m a l l a m p l i t u d e and we do not s e e t h e p o s s i b i l i t y of a p r e d o m i n a n t l y T = 1 r e s o n a n c e in the He 4 s y s t e m .
278
We consider the following model: We replace the 4-body problem of the He 4 nucleus by a onebody problem in which one nucleon moves in a potential which is supposed to represent all the other energies and the interactions of the particle with the other three particles, i.e., we consider the motion of, say, a neutron with respect to the recoiling He 3 system. Such a treatment would be valid if the adiabatic approximation were applicable. We do not expect this to be true really ; the numerical values have to be considered, therefore, as having only a qualitative significance. Our program is to find a potential which reproduces the bound state characteristics of He'*, namely the binding energy and the rms charge radius. According to Levinson's theorem 3) there exists a scattering state. As we will see, the scattering state occurs at an energy which, considering the accuracy of the model, is somewhat too high. In the bound state the potential well is of very short range (and thus quite deep because of the well-known approximate parameter VR 2
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w h i c h d e t e r m i n e s t h e b i n d i n g e n e r g y ) owing to t h e c o m p a c t n e s s of He4. In the s c a t t e r i n g s t a t e , on the o t h e r hand, the s i z e of the p o t e n t i a l w e l l i s g i v e n b y the l a r g e r s i z e of He3. S i n c e t h e s c a t t e r i n g s t a t e i s q u i t e w i d e a n d s i n c e the i n c o m i n g p a r t i c l e m o v e s f a s t e r than t h e p a r t i c l e s b o u n d in the He 3 s y s t e m , one m u s t e x p e c t t h a t upon t h e a p p r o a c h of the i n c o m i n g p a r t i c l e the He3 s y s t e m h a s no t i m e to a d j u s t i t s e l f f r o m t h e He 3 d e n s i t y to the He 4 d e n s i t y . We t h u s t a k e the p o t e n t i a l f o r the s c a t t e r i n g s t a t e to h a v e a p p r o p r i a t e l y l a r g e r r a d i u s arid s m a l l e r depth. W i t h t h i s s o - c a l l e d t r a n s f o r m e d p o t e n t i a l we o b t a i n a r e a s o n a b l e r a n g e f o r the s c a t t e r i n g e n e r gies. We c o n s i d e r f o r t h e g r o u n d s t a t e of He 4 only the c o n f i g u r a t i o n (S~.)4. T h e n t h e o p t i c a l m o d e l S c h r S d i n g e r e q u a t i o n } o r one p a r t i c l e (we n e g l e c t the Coulomb force) is
{- h 2 ~2 ÷ V(x)- E} u(x) = 0
(1)
w h e r e / z = ~3 M i s t h e r e d u c e d m a s s , x i s the d i s t a n c e of the p a r t i c l e f r o m the c e n t r e of m a s s of the r e c o i l i n g He 3 o r H 3 s y s t e m , E i s the e n e r g y , and we a s s u m e t h a t t h e p o t e n t i a l V(x) h a s the f o r m V
Y(x) - 1+ e ( x - R ) / a
XM:
2 dx =-fx2co(x) d3x ;
fw(x) d3x: 1.
(3)
If t h e p r o t o n a n d the r e c o i l i n g H 3, o r a l t e r n a t e l y the r e c o i l i n g He 3, w e r e p o i n t c h a r g e s the m e a n s q u a r e r a d i u s , RC2 , of the c h a r g e d i s t r i b u t i o n w o u l d b e g i v e n by 2 R c 2 = ½[(¼XM)2+(¼XM) 2] + ½12(¼XM)2]; (4) t h e f a c t o r 2 on the l e f t h a n d s i d e c o m e s f r o m t h e c h a r g e of He, a n d the two b r a c k e t s c o n t a i n t h e c o n t r i b u t i o n s f r o m the two p o s s i b i l i t i e s : T h e odd p a r ticle can be a proton and then the recoiling system i s H 3 ( f i r s t b r a c k e t ) o r the odd p a r t i c l e c a n b e a n e u t r o n a n d the r e c o i l i n g s y s t e m then i s He 3 ( s e c ond b r a c k e t ) . B o t h t h e s e p o s s i b i l i t i e s a r e g i v e n the 1 w e i g h t ~ and thus RC 2
3
2
- 16 X M
•
(5)
In o r d e r to o b t a i n the a c t u a l r m s r a d i u s one h a s to c o n s i d e r the f i n i t e s i z e of the p a r t i c l e a n d of the r e -
1 D e c e m b e r 1963
c o i l i n g t h r e e - b o d y s y s t e m . F r o m the r e s u l t s of S l a g g i e a n d W i c h m a n n 4) c o n c e r n i n g the a s y m p t o t i c b e h a v i o u r of the t h r e e - b o d y w a v e f u n c t i o n we s e e t h a t upon s e p a r a t i o n of a b o u n d s y s t e m into f r a g m e n t s , a t l a r g e s e p a r a t i o n the f r a g m e n t s a c q u i r e the s i z e t h e y would h a v e if t h e y w e r e b y t h e m s e l v e s , i . e . , the f a c t t h a t t h e y h a v e a n e g a t i v e k i n e t i c e n e r g y d o e s not i n f l u e n c e t h e i r i n t e r n a l s t r u c t u r e . T h u s f o r i n c r e a s i n g x the r e c o i l i n g three-body system expands from a density approp r i a t e to the He 4 s y s t e m to a d e n s i t y a p p r o p r i a t e to the H 3 and He 3 s y s t e m . We f u r t h e r a s s u m e t h a t the c h a r g e d i s t r i b u t i o n s of the p r o t o n , H 3 and He 3 h a v e a G a u s s i a n f o r m , the l a t t e r two b e i n g the s a m e and e q u a l to
~(y) = .~e -(y/D(x))2,
f ~,,(.y) d 3 y = l ,
(6)
w h e r e y i s m e a s u r e d f r o m t h e c e n t r e of m a s s of the t h r e e - p a r t i c l e s y s t e m a n d 2~ i s t h e n o r m a l i s a tion f a c t o r . We t a k e the " d e n s i t y a p p r o p r i a t e to the He 4 s y s t e m " to be c h a r a c t e r i z e d by X M and the " d e n s i t y a p p r o p r i a t e to the r e c o i l i n g He 3 o r H 3 s y s t e m " to b e c h a r a c t e r i z e d b y R H. In (6) we c h o o s e the f u n c t i o n D(x) s o that i t i n t e r p o l a t e s s m o o t h l y b e t w e e n t h e s e two v a l u e s :
(2)
w h i c h h a s t h r e e a d j u s t a b l e p a r a m e t e r s , V, R a n d a. We want to find s e t s of t h e s e p a r a m e t e r s w h i c h y i e l d the c o r r e c t b i n d i n g e n e r g y and r m s c h a r g e r a d i u s . Solving (1) w e i m m e d i a t e l y o b t a i n the b i n d i n g e n e r g y B = - E . To f i n d t h e r m s c h a r g e d i s t r i b u t i o n we e m p l o y t h e f o l l o w i n g p r o c e d u r e : W i t h the s o l u t i o n of (1) _we c o n s t r u c t t h e m e a n s q u a r e of x w h i c h we c a l l
x2:jx2u
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D(x) = D o = X M J 2 f o r x < X M D(x) = D O + (R H - Do)(1 - e - ( X - X M / ( R H - D o ) )
(Ta) (Tb)
for x >1X M . We w e r e g u i d e d in the c h o i c e (Ta) f o r the r e l a t i o n b e t w e e n D o a n d R M b y the g e o m e t r y of the t e t r a h e d r o n . H e r e R H i s the r m s v a l u e of the c h a r g e d i s t r i b u t i o n s in H 3 o r He 3 a f t e r c o r r e c t i o n f o r the f i n i t e s i z e of the p r o t o n . Now we c a n w r i t e down the c h a r g e d i s t r i b u t i o n of He 4. F i r s t of a l l we e l i m i n a t e the e f f e c t of the f i n i t e s i z e of the p r o t o n b y t h e r e l a t i o n 2 = R2 2 RC rms - r P
(8)
L e t r b e m e a s u r e d f r o m the c e n t r e of m a s s of the f o u r - p a r t i c l e s y s t e m . T h e n the m e a n s q u a r e r a d i u s of the c h a r g e d i s t r i b u t i o n i s RC2 = ½/r2D(~. ) d3 r ,
(9)
w h e r e the c h a r g e d e n s i t y
P(r) = 32/co(x) co(y) 6 ({x + y - r) d3x d3y
(10)
+ ½/co(x) 6(~x + r ) d 3 x . We s e t of would MeV,
u s e d the NBS IBM 7094 c o m p u t e r to find a p a r a m e t e r s of the p o t e n t i a l , eq. (2), which r e p r o d u c e t h e b i n d i n g e n e r g y , B = 20.57 and the r m s r a d i u s of the p o i n t c h a r g e d i s 279
Volume 7, n u m b e r 4
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t r i b u t i o n , R e 2 = 1 . 9 5 2 1 x 1 0 -26 c m 2 * 6). T h e s e a r c h was done by a s s u m i n g the d i f f u s e n e s s p a r a m e t e r a a n d c h a n g i n g V a n d R u n t i l both B and R e 2 w e r e r e p r o d u c e d to b e t t e r than 1%. We f i n d that in a t y p i c a l c a s e R e 2 c o n t r i b u t e s 21% to RC 2, the f i n i t e s i z e of the t h r e e - p a r t i c l e s y s t e m c o n t r i b u t e s 62%, a n d the e x p a n s i o n of the t h r e e - p a r t i c l e s y s t e m 17%. T h e p a r a m e t e r s o b t a i n e d in this way a r e p l o t t e d i n fig. 1. IOOC --
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LETTERS
1 December 1963
With each of t h e s e p o t e n t i a l s we then s e a r c h e d for the e n e r g y of the s c a t t e r i n g s t a t e . T h e s e e n e r g i e s a r e p l o t t e d i n fig. 2 a s open c i r c l e s . T h e y l i e r a t h e r high, at 20 to 40 MeV i n s t e a d of the few MeV obs e r v e d e x p e r i m e n t a l l y . The r e a s o n s for t h i s have b e e n d i s c u s s e d above. We now t r a n s f o r m the pot e n t i a l by m u l t i p l y i n g a a n d R by ~ and, c o n s i d e r i n g that V is a p p r o x i m a t e l y p r o p o r t i o n a l to the d e n s i t y , d i v i d e V b y ~3. We c h o o s e ~ s u c h that R a f t e r the t r a n s f o r m a t i o n e q u a l s RH, i . e . , ~ = R H / R . Again we s e a r c h e d for the s c a t t e r i n g s t a t e . The e n e r g i e s that we o b t a i n e d a r e p l o t t e d a s dots on fig. 2. It i s s e e n that t h e r e does e x i s t a t r a n s f o r m e d p o t e n t i a l which g i v e s the c o r r e c t s c a t t e r i n g s t a t e e n e r g y . T h u s f r o m the a r g u m e n t s g i v e n h e r e a T = 0 s c a t t e r i n g s t a t e j u s t above the t h r e s h o l d for p a r t i c l e e m i s s i o a i s e x p e c t e d to exist. A s t u d y of the p a r a m e t e r s of the t r a n s f o r m e d p o t e n t i a l i n d i c a t e s to us that a n y r e a s o n a b l e t r a n s f o r m a t i o n would y i e l d a t r a n s f o r m e d p o t e n t i a l which g i v e s the c o r rect scattering state energy. In o r d e r to have a T = 1 s t a t e in the f o u r - p a r t i c l e s y s t e m one m u s t h a v e at l e a s t one p a r t i c l e in a p - s t a t e . W e r n t z a n d B r e n n a n 7) have g i v e n r e a s o n s for b e l i e v i n g that the s t a t e s of the four nucleon system are pure isospin states. Moreover, this would b e e x p e c t e d to be t r u e e v e n if we do not c o n s i d e r t h e s e a r g u m e n t s s i n c e the C o u l o m b i n t e r a c t i o n e n e r g y in the few p a r t i c l e s y s t e m is s m a l l . We s e a r c h e d for p - w a v e s t a t e s in both the t r a n s f o r m e d and u n t r a n s f o r m e d p o t e n t i a l s . No r e s o n a n c e s e x i s t in t h e s e p o t e n t i a l s . It s h o u l d be r e m e m b e r e d that the i n t e r a c t i o n of n u c l e o n s in r e l a t i v e p - s t a t e s i s w e a k e r than i n r e l a t i v e s s t a t e s . By u s i n g the s a m e p o t e n t i a l we have in effect a s s u m e d that the p- and s - s t a t e i n t e r a c t i o n s w e r e the s a m e . We thus f e e l that we have o v e r e s t i m a t e d the i n t e r a c t i o n s t r e n g t h a n d that it is r a t h e r u n l i k e l y that t h e r e should e x i s t p - s t a t e s in n a t u r e in the f o u r - p a r t i c l e s y s t e m . The i n t e r a c t i o n of h i g h e r l s t a t e s with the p o t e n t i a l w e l l i s s t i l l w e a k e r than the p - s t a t e i n t e r a c t i o n , a n d we do not think that they e x i s t i n n a t u r e e i t h e r . T h u s we do not expect that a T = 1 s t a t e of the a - 2 a r t i c l e e x i s t s . T h e r e f o r e p r o b a b l y Li 4 and H'~ a r e not s t a b l e u n d e r p a r t i c l e e m i s s i o n and following Gol' d a n s k y 8) H 5 is a l s o not s t a b l e u n d e r p a r t i c l e emission.
.E
References
.4
1) H.W. Lafevre, R.R. Borchers and C. H. Poppe, Phys. Rev. 128 (1962) 1328. 2) Carl Werntz, Phys.Rev. 128 (1962) 1336. 3) N. Levinson, Dan. Mat. Fys. Medd. Bind 25 (1949) 0
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* The code was based on ref. 6). We are grateful to Dr. R. S. Caswell for modifying and supplementing the code of ref. 6) to suit our needs.
Volume 7, n u m b e r 4
PHYSICS
4) E. Leo Slaggie and Eyvind H.Wichmann, J. Math. Phys. 3 (1962) 946. 5) H. Collard et al., private communication. 6) R.S. Caswell, National Bureau of Standards Technical Note No. 159 (1962).
LETTERS
1December 1963
7) Carl Werntz and J.G. Brennan, Physics Letters 6 (1963) 113. 8) V.I. Go1'dansky, J. Exptl. Theoret. Phys. (USSR) 38 (1960) 1637 ; translation : Soviet Phys. JETP 11 (1960) 1179.
ABSENCE OF TENSOR TERM IN THE OPTICAL MODEL FOR THE d-Ca ELASTIC SCATTERING A T 22 M e V J. R A Y N A L Service de Physique ThSorique, Centre d'Etudes Nucl~aires de Saclay B. P. no. 2, Gif - s u r - Yvette (S &O) , France Received 26 October 1963
O p t i c a l m o d e l c a l c u l a t i o n s for the d i f f e r e n t i a l c r o s s s e c t i o n of e l a s t i c s c a t t e r i n g of d e u t e r o n s on n u c l e i a g r e e v e r y w e l l with e x p e r i m e n t a l data, c h i e f l y at not too high e n e r g y 1) a n d for h e a v y t a r g e t s 2), taking into a c c o u n t only a c e n t r a l p o t e n t i a l with s l o w l y v a r y i n g p a r a m e t e r s ; h o w e v e r m u l t i p l e s o l u t i o n s c a n b e found. Some p o l a r i s a t i o n m e a s u r e m e n t s have r e c e n t l y b e e n p e r f o r m e d at S a c l a y on the Ca n u c l e u s n e a r 22 MeV l e a d i n g to the v e c t o r p o l a r i s a t i o n p a r a m e t e r 3) a n d to two of the t h r e e t e n s o r p o l a r i s a t i o n p a r a m e t e r s 4). C r o s s s e c t i o n m e a s u r e m e n t s with the s a m e t a r g e t w e r e m a d e by Y n t e m a at 21.4 MeV 5). C r o s s s e c t i o n data c a n n o t be f i t t e d v e r y w e l l by a c e n t r a l p o t e n t i a l a l o n e 2). Some a t t e m p t s w e r e m a d e to fit t h e s e data with L S coupling a n d t e n s o r p o t e n t i a l s b y a n a u t o m a t i c s e a r c h p r o g r a m 6) with the IBM 7090 at S a c l a y a s well a s for o t h e r t a r g e t s 7). The i m a g i n a r y p a r t of the c e n t r a l p o t e n t i a l a l w a y s t u r n e d out to be s m a l l e r and p l a c e d at g r e a t e r r a d i u s than for o t h e r t a r g e t s . Good r e s u l t s w e r e o b t a i n e d with a s o m e w h a t c o m p l i c a t e d p o t e n t i a l i n c l u d i n g the u s u a l L S coupling t e r m t a k e n with a c o m p l e x c o e f f i c i e n t a n d the t e n s o r t e r m 1 {(S~)2 - -~}dfir2 (1 + e ( r - R ) / a ) " The l a t t e r was about twice g r e a t e r than the one exp e c t e d f r o m c o n s i d e r a t i o n s on the D - w a v e of the d e u t e r o n 8) and had the o p p o s i t e sign. The v e c t o r p o l a r i s a t i o n g i v e n by this p o t e n t i a l was in r e a s o n a b l e a g r e e m e n t witht he e x p e r i m e n t ; w h e r e a s the t e n s o r p o l a r i s a t i o n w a s m u c h g r e a t e r than the one o b t a i n e d in the e x p e r i m e n t s 4). If one i n t r o d u c e s the p o l a r i s a t i o n s into the s e a r c h , the t e n s o r p o t e n t i a l b e c o m e s a s m a l l f r a c t i o n of that e x p e c t e d ; h o w e v e r , n o r e a l l y s a t i s f a c t o r y fit was o b t a i n e d : in p a r t i c u l a r , the fit for the c r o s s s e c t i o n w a s q u i t e poor.
T h i s was t a k e n a s an i n d i c a t i o n that the t e n s o r t e r m was of no g r e a t help in this s i t u a t i o n , and we d e c i d e d to p u r s u e the s e a r c h with the L S coupling only. T h e r e s u l t s a r e a s f o l l o w s : 1. We take a Saxon c e n t r a l p o t e n t i a l , the u s u a l L S coupling a n d a s u r f a c e a b s o r p t i o n and we t r y to fit the c r o s s s e c t i o n r e s u l t s . A g r e e m e n t h a s b e e n o b t a i n e d only in the 100 MeV r e g i o n of depth for the c e n t r a l p a r t . S e a r c h e s w e r e m a d e with v a r i o u s v a l u e s of the r e d u c e d r a d i u s of the c e n t r a l p o t e n t i a l r a n g i n g f r o m 1 to 1.25 fm. T h e c o r r e s p o n d i n g par a m e t e r s a r e l i s t e d in t a b l e 1. The m i n i m u m of ×2 i s r e a c h e d for the v a l u e s 1 . 1 0 - 1.15. The v e c t o r p o l a r i s a t i o n g i v e n by this p o t e n t i a l is in r e a s o n a b l e a g r e e m e n t with e x p e r i m e n t but the z e r o at 54 ° has a shift of s o m e d e g r e e s . 2. We then p e r f o r m e d the s a m e c a l c u l a t i o n s v a r y i n g the r a d i u s and the d i f f u s e n e s s p a r a m e t e r of the L S coupling i n s t e a d of taking i t s u s u a l f o r m . The fit was i m p r o v e d a s is e v i d e n t f r o m t a b l e 2. Two kind of s o l u t i o n s c a n b e s e p a r a t e d h e r e dep e n d i n g on w h e t h e r the d i f f u s e n e s s p a r a m e t e r of L S coupling i s g r e a t e r or s m a l l e r than 0.45 fm. In the l a s t c a s e t h e r e is an i m p r o v e m e n t of the fit of the f o r w a r d s c a t t e r i n g c r o s s s e c t i o n , but the e x p e r i m e n t a l e r r o r s a r e m o r e i m p o r t a n t in this r e g i o n 5); at the s a m e a n g l e s , the v e c t o r p o l a r i s a t i o n and T22 a r e too s m a l l . H o w e v e r , s i n c e the z e r o s of the v e c t o r p o l a r i s a t i o n a r e well p l a c e d the o v e r a l l a g r e e m e n t is b e t t e r than in the f i r s t case. The l a s t line of each t a b l e g i v e s the r e s u l t of s e a r c h e s including vector polarisation and tensorial effect at ~ = 0, ~ and ~ with s t a t i s t i c a l e r r o r s (36 v a l u e s of each kind). The c o r r e s p o n d i n g c r o s s s e c t i o n s and p o l a r i s a t i o n s a r e plotted in figs. 1-4 with e x p e r i m e n t a l p o i n t s . The e f f i c i e n c y in the p r o d u c t i o n of the p o l a r i z e d b e a m is s u p p o s e d to b e 281