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Exact calculations of nucleon-deuteron scattering with internuclear forces including a tensor component
Volume 37B, number 5
PHYSICS
LETTERS
27 December 1971
EXACT CALCULATIONS OF NUCLEON-DEUTERON SCATTERING INTERNUCLEAR FORCES INCLUDING A TENSOR COMPONENT
WITH
Y. AVISHAI A r g o n n e National L a b o r a t o r y , A ~gonne , Illinois, USA
and A. S. RINAT ( R E I N E R ) W e i z m a n n Institute of Science, Rehoz,ot, I s r a e l
Received 20 October 1971
The F a d d e ' e v - M i t r a - A m a d o - L o v e l a c e equations have been solved, with a separable tensor component added to a rank one triplet nucleon-nucleon interaction. C r o s s - s e c t i o n s and nucleon polarizations for elastic nucleon-deuteron s c a t t e r i n g have been computed for 31 MeV > E > 7.8 MeV. F o r lower energies both functions agree reasonably well with experiment for a D-wave contribution PD = 4-5~. D i s c r e pancies grow with i n c r e a s i n g energies.
There exist, roughly speaking, three practical a p p r o a c h e s to the n u c l e a r t h r e e - b o d y p r o b l e m . T h e s e a r e the c o u p l e d - c h a n n e l potential a p p r o a c h of F a d d e ' e v - M i t r a - A m a d o - L o v e l a c e ( F M A L ) []], d i s p e r s i o n t h e o r i e s [2] a n d v a r i a t i o n a l c a l c u l a t i o n s [3]. V i r t u a l l y a l l c a l c u l a t i o n s in t h e f i r s t two c a t e g o r i e s h a v e t i l l now b e e n b a s e d on c e n t r a l n u c l e o n - n u c l e o n f o r c e s w i t h p a r a m e t e r s f i t t e d to S - w a v e p h a s e s (in p a r t i c u l a r to z e r o - e n e r g y s c a t t e r i n g data) and bound state p r o p e r t i e s .
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These forces usually contain or have been exp a n d e d into a r e s t r i c t e d n u m b e r of s e p a r a b l e central components. The theory then allows for c o m p u t a t i o n of, a m o n g s t o t h e r s , b i n d i n g e n e r gies, electromagnetic form factors, nucleond e u t e r o n e l a s t i c and b r e a k - u p c r o s s s e c t i o n s e t c [e.g. 4]. The o b s e r b e d a g r e e m e n t with e x p e r i m e n t is
200l-
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........... ,~ o
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i 90
,=,~, ." I 120
Ep =:.8 Me', 150 180
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8corn
Fig. 1. Differential c r o s s - s e c t i o n zor nucleon-deuteron s c a t t e r i n g at ED = 7.8 and E n = 14.4 MeV [14]. Calculated a r e n=d c r o s s - s e c t i o n s , t h e r e f o r e forward Coulomb i n t e r f e r e n c e regions should be ignored.
'
I
I
I
I
I
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60
90
120
150
180
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Fig. 2. Same as fig. 1 for Ep = 22.7 and 31.0 MeV [14].
487
Volume 37B, number 5
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PHYSICS
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I
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LETTERS
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27 December 1971
I
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90
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Fig. 3. Nucleonpolarizations in nucleon-deuteron scattering for Ep, n = 7.8 MeV and Ep = 30 MeV [17]. often satisfactory and actually embarassing in view of the meagre information that went into the effective central forces. A whole body of additional information like polarization data can only be approached if noncentral components are included in the internucleon force• "Realistic" forces containing a tensor part like the Hamada-Johnston potential have previously been used in variational calculations [3] whose scope, however, till today appears limited to boundstate properties and threshold scattering data. The same held true for FMAL calculations [5]. There is at this state an at least two-fold incentive to allow non-central components in the nucleon-nucleon interactions to be used as input for calculations of general properties of the t h r e e nucleon system. Users of purely central forces have always been aware that these are effeclive interactions
488
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Fig• 4• Same as fig. 3 for Ep = 14.5 and 30.8 MeV [17]• and not the best known or "realistic" ones• Does then the recorded agreement for elastic c r o s s sections p e r s i s t if a wider class of (stil effective'.) forces including a tensor part is chosen? And if so, how well do the eratically varying polarization data, when fitted at all, determine the ratio of tensor to central forces? We present in the following the first results of a FMAL calculation of elastic cross-sections and nucleon polarizations. The simplest possible effective nucleon-nucleon interaction has been employed, namely a rank-two potential with tensor component of the Hulth6n-Yamaguchi type for the spin-triplet interaction in momentum space:
3v(p, p,) = _Xt[()2 +fl21-12 - u ,2 • [(P
2-1 +~2)
u8-1/2 S -
8-1/2S12(p)p2~2 +/3~)-2~
(,)p,2. 12 P
(P
,2+ 2 ) - 2 . ~t J"
(1)
In (1) t h e r e a p p e a r two r a n g e p a r a m e t e r s ~2, fit a n d f u r t h e r v w h i c h is r e l a t e d to the D - s t a t e p r o b a b i l i t y in the d e u t e r o n . S12 is the t e n s o r
PHYSICS
Volume 37B, number 5 operator S12(P ) = 3 ( a 1 . p ) (a 2 . p ) - a 1 . a 2.
(2)
F o r the s p i n - s i n g l e t i n t e r a c t i o n w e c h o s e • ,2
1 v ( p , P ') = - X s ( p 2 + ~ 2 s )-1 (P
2.-1
+fis)
•
(3)
A l l p a r a m e t e r s but P D h a v e b e e n a d a p t e d to z e r o e n e r g y s c a t t e r i n g d a t a by P h i l l i p s [6] *. One of the m a j o r o b s t a c l e s to s u c h a c o m p l e t e c a l c u l a t i o n a p p e a r e d to be the i n v o l v e d a n g u l a r m o m e n t u m a l g e b r a , but e v e n a f t e r g r a d u a l s o l u t i o n of t h e s e g e o m e t r i c a l a s p e c t s of the p r o b l e m [8,9] the r e m a i n i n g n u m e r i c a l c o m p l i cations are still formidable**. The resulting coupled channel equations are of the s t a n d a r d type [1]
Z= B + Z'r B
(4)
w i t h Z e l a s t i c a m p l i t u d e s and ~- the p r o p a g a t o r f o r i n t e r m e d i a t e s t a t e s (bound d e u t e r o n and v i r t u a l s p i n - s i n g l e t " s t a t e s " ) [1]. B is the d r i v i n g f o r c e w h i c h is j u s t n u c l e o n e x c h a n g e . D r i v i n g f o r c e s and a m p l i t u d e s c o n s e r v e only jTr, the t o t a l a n g u l a r m o m e n t u m and p a r i t y and a r e o t h e r w i s e m a t r i c e s in e i t h e r the c h a n n e l s p i n S and o r b i t a l a n g u l a r m o m e n t u m L o r a l t e r n a t i v e l y in the h e l i c i t i e s . The g e n e r a l f o r m of B is g i v e n in r e f . [9]. Eq. (4) h a s b e e n s o l v e d by s t a n d a r d m a t r i x a p p r o x i m a t i o n u s i n g a c o n t o u r d e f o r m a t i o n [13]. T h e s i z e of the m a t r i x is 4 N x 4N, 4 b e i n g the n u m b e r of c h a n n e l s c o n t r i b u t i n g to e a c h J ~r w h e n a l l o w i n g the a b o v e m e n t i o n e d i n t e r m e d i a t e s t a t e s . ( F o r J~r = 1+ the n u m b e r is 3). By N w e d e n o t e the n u m b e r of g r i d p o i n t s w h i c h r e p l a c e the c o n t i n u o u s v a r i a b l e in the i n t e g r a l eq. (4) ( c h o s e n to be 12). P a r t i a l w a v e s w i t h L , L' < 5 (i.e. J m a x "< ~ ) h a v e b e e n i n c l u d e d t h r o u g h o u t . We • Most calculations had been finished before publication of the newest nucleon-nucleon z e r o - e n e r g y data by Houk [7] which would entail slight modifications of Phillips' p a r a m e t e r s . In particular the singlet effective range used was r = 2.7 fm, as compared to Houk's value r s = 2.74 fSm. • * At this stage, we mention Sloan's simplifying Kmatrix approach [10]. Subsequent calculations by Kraus and Kowalski showed remarkable agreement for both c r o s s - s e c t i o n s and polarizations [11]. The agreement appeared to be totally fortuitous and has been obtained accidentally by inclusion of an insufficient number of partial waves [12]. This unfortunate outcome incidentally does not condemn the K-matrix approach, but only shows that when K is approximated by nucleon exchange and (properly projected) double nucleon exchange, that input is insufficient.
LETTERS
27 December 1971
h a v e a s s e m b l e d in figs. 1,2 e l a s t i c c r o s s - s e c t i o n s f o r s o m e e n e r g i e s high enough to g i v e s i g n i f i c a n t p o l a r i z a t i o n s . We c h o o s e the lab e n e r g i e s E = = 7.8, 14.4, 22.7 and 30.8 MeV, and P D = 4 and 7%. T h e r e s u l t s f o r p u r e l y c e n t r a l f o r c e s (PD = 0) w h e n e v e r c a l c u l a t e d b e f o r e a r e a l s o i n c l u d e d [14]. C o m p a r i s o n w i t h e x p e r i m e n t a l d a t a s h o w s f o r the l o w e r e n e r g i e s m u t u a l t h e o r e t i c a l d e v i a t i o n s of the s a m e o r d e r as the m o d e s t d i s c r e p a n c i e s w i t h e x p e r i m e n t . C o n s e q u e n t l y the a n s w e r to the f i r s t q u e s t i o n a b o v e is that, j u d g e d from cross-section measurements, mixed as w e l l as p u r e l y c e n t r a l f o r c e s a r e about e q u a l l y e f f e c t i v e and can h a r d l y be d i s c r i m i n a t e d b e t w e e n , w h e n t e s t e d on n - d c r o s s - s e c t i o n s . F o r h i g h e r e n e r g i e s both p o s i t i o n and m a g n i t u d e of the m i n i m u m a r e not too w e l l r e p r o d u c e d , w i t h P D ~ 0 g i v i n g the b e s t fit. T h e s e n s i t i v i t y of the s a m e a r e a has b e e n s t r e s s e d b e f o r e in ND -1 c a l c u l a t i o n [2] (see a l s o r e f . [15]). W e now t u r n to p o l a r i z a t i o n s , w h i c h w e r e c a l c u l a t e d u s i n g S e y l e r ' s e x p r e s s i o n s [16]. We show in figs. 3 and 4 c a l c u l a t e d and o b s e r v e d p o l a r i z a t i o n s [17] of the o u t g o i n g n u c l e o n s in s c a t t e r i n g b e t w e e n u n p o l a r i z e d n u c l e o n s and deuterons. A g a i n one has to d e s c r i m i n a t e b e t w e e n l o w e r and h i g h e r e n e r g i e s . P o l a r i z a t i o n f o r the f o r m e r a r e w e l l r e p r o d u c e d and d e f i n i t e l y f a v o r a r a t h e r low P D ~ 4%. The s a m e p e r c e n t a g e fits the a n g u l a r r e g i o n s 0 < 60 o, 0 > 140 °, but the r e l a t i v e ly l a r g e n e g a t i v e p o l a r i z a t i o n s f o r m e d i u m a n g les remain unexplained. W e f i n a l l y c o n f i r m P h i l l i p s ' r e s u l t s f o r the t r i t o n b i n d i n g e n e r g y and d o u b l e t nd s c a t t e r i n g l e n g t h [6]. If the l a t e s t r e p o r t e d e x p e r i m e n t a l r e s u l t s a r e c o r r e c t [18], t h e r e w i l l be no d i s c r e p a n c y left f o r t h e s e q u a n t i t i e s . We c o n c l u d e and note that e n i g m a t i c a l l y e n o u g h c e n t r a l and m i x e d f o r c e s a r e about e q u a l ly e f f e c t i v e in d e s c r i b i n g d i f f e r e n t i a l c r o s s - s e c t i o n s . A p p a r e n t l y not too m a n y d e t a i l s of the t e n s o r f o r c e , a r e r e q u i r e d to r e p r o d u c e p o l a r i z a t i o n s up to 15 MeV, w h e r e a v a l u e PD - 4% is d e f i n i t e l y f a v o u r e d . It d o e s not c o m e a s a s u r p r i s e to s e e f a i l the s i m p l e s t p o s s i b l e i n t e r a c t i o n s (1) - (3) in a d e s c r i p t i o n of s e n s i t i v e i n t e r f e r e n c e r e g i o n s both in c r o s s - s e c t i o n s and p o l a r i z a t i o n s . C a l c u l a t i o n s b a s e d on a m o r e s o p h i s t i c a t e d input a r e c l e a r l y c a l l e d f o r . A m o r e e x t e n s i v e r e p o r t w i l l be p u b l i s h e d elsewhere.
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References [1] L. F a d d e ' e v , J E T P 12 (1961) 1014; A.N. M i t r a . Nuel. P h y s . 32 (1962) 529: R. Aaron et al., P h y s . Rev. 140B (1965) 1291; C. L o v e l a c e , P h y s . Rev. 135B (1964) 1225; A. C. P h i l l i p s , P h y s . Rev. 142 (1966) 984. [2] Y. Avishai, W. EbenhSh and A. S. Rinat (Re[her), Ann. of P h y s . 55 (1969) 341: P h y s . L e t t e r s 28B (1969) 387; G. B a r t o n and A. C. P h i l l i p s , Nucl. P h y s . A132 (1969) 97; M . P . L o c h e r , Nucl. P h y s . B23 (1970) 116; R . H . J . B o w e r , P r i n c e t o n Ph. D. t h e s i s , unpublished (1971). [3] L . M . D e l v e s et al., P h y s . L e t t e r s 20B (1969) 472; J. W. H u m b e r s t o n and M. A. Lennell, P h y s . L e t t e r s 31B (1970) 423; L. M . D e l v e s and M . A . HenneIl, Nucl. P h y s . A160 (1971) 347. [4] L. M. D e l v e s and A. C. P h i l l i p s , Rev. of Mod. P h y s . 41 (1969) 497. [5] A. N. M i t r a , G . L . Schrenk and V. S. Bhasin. Ann. of P h y s . 40 (1966) 357; G. L. Schrenk and A. N. M i t r a , P h y s . Rev. L e t t e r s 19 (1967) 530. [6] A.C. P h i l l i p s , Nucl. P h y s . A107 (1968) 209. [7] T . L . Houk, P h y s . Rev. C3 (1971) 1886.
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2 7 D e c e m b e r 1971
[8] I. S[oan, Nucl. P h y s . A139 (1969) 373. [9] M. Stingl and A. S. Rinat (Reiner), Nucl. P h y s . A154 (1970) 613; Ei Baz e t a [ . , Nuovo Cimento 64A (1969) 13. [10] I. Sloan, Phys. Rev. 165 {1968) 1587. [11] J . K r a u s and K . L . Kowalski, P h y s . L e t t e r s 31B {1970) 263; P h y s . Rev. C1 {1970) 1319. [12] J. C. A a r o n s and I. Sloan, to be published. [13] J. H. H e t h e r i n g t o n and L. H. Schick, P h y s . Rev, B137 (1965) 935. [14] Exp. data taken f r o m Ep = 7.8 MeV: J. B r o l l e y , P h y s . Rev. 117 (1960) 1307; En = 14.4 MeV: P h y s . Rev. 91 {1953) 90; 97 (1955) 757; 174 {1968) 1105; E ~ = 22.7 MeV: Nucl. P h v s . Al13 (1968) 461; E~o = 30.75 MeV: Nucl. PhYs. 50 {1964) 32. [15] I.'Sloan, Nucl. Phys. A168 (1971) 211. [16] Seyler, Nucl. Phys. A124 (1969) 253. [17] E = 8 MeV:Nuel. Phys. A95 (1967) 608; ~._ = 7.8 M e V : B A P S 14 (1969) 21: L~_~ = 14.5 MeV: Nucl. P h y s . A127 {1969) 169; ~ = 22.7 MeV: P h y s . Rev. L e t t e r s 21 (1968) 1393; 1~6 = 30 MeV: R u t h e r f o r d High E n e r g y Lab. Rep. (66). [18] W. Dilg, L. K o e s t e r and W. N i s t l e r , P h y s . L e t t e r s 36B (1971) 208.
PHYSICS
LETTERS
27 December 1971
EXACT CALCULATIONS OF NUCLEON-DEUTERON SCATTERING INTERNUCLEAR FORCES INCLUDING A TENSOR COMPONENT
WITH
Y. AVISHAI A r g o n n e National L a b o r a t o r y , A ~gonne , Illinois, USA
and A. S. RINAT ( R E I N E R ) W e i z m a n n Institute of Science, Rehoz,ot, I s r a e l
Received 20 October 1971
The F a d d e ' e v - M i t r a - A m a d o - L o v e l a c e equations have been solved, with a separable tensor component added to a rank one triplet nucleon-nucleon interaction. C r o s s - s e c t i o n s and nucleon polarizations for elastic nucleon-deuteron s c a t t e r i n g have been computed for 31 MeV > E > 7.8 MeV. F o r lower energies both functions agree reasonably well with experiment for a D-wave contribution PD = 4-5~. D i s c r e pancies grow with i n c r e a s i n g energies.
There exist, roughly speaking, three practical a p p r o a c h e s to the n u c l e a r t h r e e - b o d y p r o b l e m . T h e s e a r e the c o u p l e d - c h a n n e l potential a p p r o a c h of F a d d e ' e v - M i t r a - A m a d o - L o v e l a c e ( F M A L ) []], d i s p e r s i o n t h e o r i e s [2] a n d v a r i a t i o n a l c a l c u l a t i o n s [3]. V i r t u a l l y a l l c a l c u l a t i o n s in t h e f i r s t two c a t e g o r i e s h a v e t i l l now b e e n b a s e d on c e n t r a l n u c l e o n - n u c l e o n f o r c e s w i t h p a r a m e t e r s f i t t e d to S - w a v e p h a s e s (in p a r t i c u l a r to z e r o - e n e r g y s c a t t e r i n g data) and bound state p r o p e r t i e s .
2
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I
/o
,oar
_7.
I
oo,~ "~
a
[ PDin O/Io
---
o
......
4
_
i
I | En =14.4 MeV -~
/I
/,-4 ,/.' I 7/ / //..:"
,
...
bT~zoo
~:.
//
"~',.
A
These forces usually contain or have been exp a n d e d into a r e s t r i c t e d n u m b e r of s e p a r a b l e central components. The theory then allows for c o m p u t a t i o n of, a m o n g s t o t h e r s , b i n d i n g e n e r gies, electromagnetic form factors, nucleond e u t e r o n e l a s t i c and b r e a k - u p c r o s s s e c t i o n s e t c [e.g. 4]. The o b s e r b e d a g r e e m e n t with e x p e r i m e n t is
200l-
I /
I
~
,ooi-
\
oo
i _ Ep =31.OMeV
'
........
~o~- ' ~ . : \ ~oo
IP o in %1 O
---
.........
\
i
.'"
\
"~,.~,'~
''<'::~\
.." '~";
/,L i
-
'1
J
\,
,/
~... 'I~N
/--'11
"<,":k,
50
i
./'4
........... ,~ o
I 30
I 60
i 90
,=,~, ." I 120
Ep =:.8 Me', 150 180
lO
Ep =22.7Me
8corn
Fig. 1. Differential c r o s s - s e c t i o n zor nucleon-deuteron s c a t t e r i n g at ED = 7.8 and E n = 14.4 MeV [14]. Calculated a r e n=d c r o s s - s e c t i o n s , t h e r e f o r e forward Coulomb i n t e r f e r e n c e regions should be ignored.
'
I
I
I
I
I
30
60
90
120
150
180
~com
Fig. 2. Same as fig. 1 for Ep = 22.7 and 31.0 MeV [14].
487
Volume 37B, number 5
I
OJO --
o I
PHYSICS
I
- .....
I
I
LETTERS
I
27 December 1971
I
0.20
I
[
I ~ 1
Po ~ %
/ E=14~4 MeV
"'"
0.10
0 0,10 a_
0
...."
/ ...... . . ' " ' "
'I ...... t
t
-0.10
l
-0.10
t t
-0.20
1i
I
0 0
30
60
90
120
150
30
I
I
60
90
180
e tom
Fig. 3. Nucleonpolarizations in nucleon-deuteron scattering for Ep, n = 7.8 MeV and Ep = 30 MeV [17]. often satisfactory and actually embarassing in view of the meagre information that went into the effective central forces. A whole body of additional information like polarization data can only be approached if noncentral components are included in the internucleon force• "Realistic" forces containing a tensor part like the Hamada-Johnston potential have previously been used in variational calculations [3] whose scope, however, till today appears limited to boundstate properties and threshold scattering data. The same held true for FMAL calculations [5]. There is at this state an at least two-fold incentive to allow non-central components in the nucleon-nucleon interactions to be used as input for calculations of general properties of the t h r e e nucleon system. Users of purely central forces have always been aware that these are effeclive interactions
488
.:'
t
l
i
f20
I
150
180
con',
Fig• 4• Same as fig. 3 for Ep = 14.5 and 30.8 MeV [17]• and not the best known or "realistic" ones• Does then the recorded agreement for elastic c r o s s sections p e r s i s t if a wider class of (stil effective'.) forces including a tensor part is chosen? And if so, how well do the eratically varying polarization data, when fitted at all, determine the ratio of tensor to central forces? We present in the following the first results of a FMAL calculation of elastic cross-sections and nucleon polarizations. The simplest possible effective nucleon-nucleon interaction has been employed, namely a rank-two potential with tensor component of the Hulth6n-Yamaguchi type for the spin-triplet interaction in momentum space:
3v(p, p,) = _Xt[()2 +fl21-12 - u ,2 • [(P
2-1 +~2)
u8-1/2 S -
8-1/2S12(p)p2~2 +/3~)-2~
(,)p,2. 12 P
(P
,2+ 2 ) - 2 . ~t J"
(1)
In (1) t h e r e a p p e a r two r a n g e p a r a m e t e r s ~2, fit a n d f u r t h e r v w h i c h is r e l a t e d to the D - s t a t e p r o b a b i l i t y in the d e u t e r o n . S12 is the t e n s o r
PHYSICS
Volume 37B, number 5 operator S12(P ) = 3 ( a 1 . p ) (a 2 . p ) - a 1 . a 2.
(2)
F o r the s p i n - s i n g l e t i n t e r a c t i o n w e c h o s e • ,2
1 v ( p , P ') = - X s ( p 2 + ~ 2 s )-1 (P
2.-1
+fis)
•
(3)
A l l p a r a m e t e r s but P D h a v e b e e n a d a p t e d to z e r o e n e r g y s c a t t e r i n g d a t a by P h i l l i p s [6] *. One of the m a j o r o b s t a c l e s to s u c h a c o m p l e t e c a l c u l a t i o n a p p e a r e d to be the i n v o l v e d a n g u l a r m o m e n t u m a l g e b r a , but e v e n a f t e r g r a d u a l s o l u t i o n of t h e s e g e o m e t r i c a l a s p e c t s of the p r o b l e m [8,9] the r e m a i n i n g n u m e r i c a l c o m p l i cations are still formidable**. The resulting coupled channel equations are of the s t a n d a r d type [1]
Z= B + Z'r B
(4)
w i t h Z e l a s t i c a m p l i t u d e s and ~- the p r o p a g a t o r f o r i n t e r m e d i a t e s t a t e s (bound d e u t e r o n and v i r t u a l s p i n - s i n g l e t " s t a t e s " ) [1]. B is the d r i v i n g f o r c e w h i c h is j u s t n u c l e o n e x c h a n g e . D r i v i n g f o r c e s and a m p l i t u d e s c o n s e r v e only jTr, the t o t a l a n g u l a r m o m e n t u m and p a r i t y and a r e o t h e r w i s e m a t r i c e s in e i t h e r the c h a n n e l s p i n S and o r b i t a l a n g u l a r m o m e n t u m L o r a l t e r n a t i v e l y in the h e l i c i t i e s . The g e n e r a l f o r m of B is g i v e n in r e f . [9]. Eq. (4) h a s b e e n s o l v e d by s t a n d a r d m a t r i x a p p r o x i m a t i o n u s i n g a c o n t o u r d e f o r m a t i o n [13]. T h e s i z e of the m a t r i x is 4 N x 4N, 4 b e i n g the n u m b e r of c h a n n e l s c o n t r i b u t i n g to e a c h J ~r w h e n a l l o w i n g the a b o v e m e n t i o n e d i n t e r m e d i a t e s t a t e s . ( F o r J~r = 1+ the n u m b e r is 3). By N w e d e n o t e the n u m b e r of g r i d p o i n t s w h i c h r e p l a c e the c o n t i n u o u s v a r i a b l e in the i n t e g r a l eq. (4) ( c h o s e n to be 12). P a r t i a l w a v e s w i t h L , L' < 5 (i.e. J m a x "< ~ ) h a v e b e e n i n c l u d e d t h r o u g h o u t . We • Most calculations had been finished before publication of the newest nucleon-nucleon z e r o - e n e r g y data by Houk [7] which would entail slight modifications of Phillips' p a r a m e t e r s . In particular the singlet effective range used was r = 2.7 fm, as compared to Houk's value r s = 2.74 fSm. • * At this stage, we mention Sloan's simplifying Kmatrix approach [10]. Subsequent calculations by Kraus and Kowalski showed remarkable agreement for both c r o s s - s e c t i o n s and polarizations [11]. The agreement appeared to be totally fortuitous and has been obtained accidentally by inclusion of an insufficient number of partial waves [12]. This unfortunate outcome incidentally does not condemn the K-matrix approach, but only shows that when K is approximated by nucleon exchange and (properly projected) double nucleon exchange, that input is insufficient.
LETTERS
27 December 1971
h a v e a s s e m b l e d in figs. 1,2 e l a s t i c c r o s s - s e c t i o n s f o r s o m e e n e r g i e s high enough to g i v e s i g n i f i c a n t p o l a r i z a t i o n s . We c h o o s e the lab e n e r g i e s E = = 7.8, 14.4, 22.7 and 30.8 MeV, and P D = 4 and 7%. T h e r e s u l t s f o r p u r e l y c e n t r a l f o r c e s (PD = 0) w h e n e v e r c a l c u l a t e d b e f o r e a r e a l s o i n c l u d e d [14]. C o m p a r i s o n w i t h e x p e r i m e n t a l d a t a s h o w s f o r the l o w e r e n e r g i e s m u t u a l t h e o r e t i c a l d e v i a t i o n s of the s a m e o r d e r as the m o d e s t d i s c r e p a n c i e s w i t h e x p e r i m e n t . C o n s e q u e n t l y the a n s w e r to the f i r s t q u e s t i o n a b o v e is that, j u d g e d from cross-section measurements, mixed as w e l l as p u r e l y c e n t r a l f o r c e s a r e about e q u a l l y e f f e c t i v e and can h a r d l y be d i s c r i m i n a t e d b e t w e e n , w h e n t e s t e d on n - d c r o s s - s e c t i o n s . F o r h i g h e r e n e r g i e s both p o s i t i o n and m a g n i t u d e of the m i n i m u m a r e not too w e l l r e p r o d u c e d , w i t h P D ~ 0 g i v i n g the b e s t fit. T h e s e n s i t i v i t y of the s a m e a r e a has b e e n s t r e s s e d b e f o r e in ND -1 c a l c u l a t i o n [2] (see a l s o r e f . [15]). W e now t u r n to p o l a r i z a t i o n s , w h i c h w e r e c a l c u l a t e d u s i n g S e y l e r ' s e x p r e s s i o n s [16]. We show in figs. 3 and 4 c a l c u l a t e d and o b s e r v e d p o l a r i z a t i o n s [17] of the o u t g o i n g n u c l e o n s in s c a t t e r i n g b e t w e e n u n p o l a r i z e d n u c l e o n s and deuterons. A g a i n one has to d e s c r i m i n a t e b e t w e e n l o w e r and h i g h e r e n e r g i e s . P o l a r i z a t i o n f o r the f o r m e r a r e w e l l r e p r o d u c e d and d e f i n i t e l y f a v o r a r a t h e r low P D ~ 4%. The s a m e p e r c e n t a g e fits the a n g u l a r r e g i o n s 0 < 60 o, 0 > 140 °, but the r e l a t i v e ly l a r g e n e g a t i v e p o l a r i z a t i o n s f o r m e d i u m a n g les remain unexplained. W e f i n a l l y c o n f i r m P h i l l i p s ' r e s u l t s f o r the t r i t o n b i n d i n g e n e r g y and d o u b l e t nd s c a t t e r i n g l e n g t h [6]. If the l a t e s t r e p o r t e d e x p e r i m e n t a l r e s u l t s a r e c o r r e c t [18], t h e r e w i l l be no d i s c r e p a n c y left f o r t h e s e q u a n t i t i e s . We c o n c l u d e and note that e n i g m a t i c a l l y e n o u g h c e n t r a l and m i x e d f o r c e s a r e about e q u a l ly e f f e c t i v e in d e s c r i b i n g d i f f e r e n t i a l c r o s s - s e c t i o n s . A p p a r e n t l y not too m a n y d e t a i l s of the t e n s o r f o r c e , a r e r e q u i r e d to r e p r o d u c e p o l a r i z a t i o n s up to 15 MeV, w h e r e a v a l u e PD - 4% is d e f i n i t e l y f a v o u r e d . It d o e s not c o m e a s a s u r p r i s e to s e e f a i l the s i m p l e s t p o s s i b l e i n t e r a c t i o n s (1) - (3) in a d e s c r i p t i o n of s e n s i t i v e i n t e r f e r e n c e r e g i o n s both in c r o s s - s e c t i o n s and p o l a r i z a t i o n s . C a l c u l a t i o n s b a s e d on a m o r e s o p h i s t i c a t e d input a r e c l e a r l y c a l l e d f o r . A m o r e e x t e n s i v e r e p o r t w i l l be p u b l i s h e d elsewhere.
489
Volume 37B, n u m b e r 5
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References [1] L. F a d d e ' e v , J E T P 12 (1961) 1014; A.N. M i t r a . Nuel. P h y s . 32 (1962) 529: R. Aaron et al., P h y s . Rev. 140B (1965) 1291; C. L o v e l a c e , P h y s . Rev. 135B (1964) 1225; A. C. P h i l l i p s , P h y s . Rev. 142 (1966) 984. [2] Y. Avishai, W. EbenhSh and A. S. Rinat (Re[her), Ann. of P h y s . 55 (1969) 341: P h y s . L e t t e r s 28B (1969) 387; G. B a r t o n and A. C. P h i l l i p s , Nucl. P h y s . A132 (1969) 97; M . P . L o c h e r , Nucl. P h y s . B23 (1970) 116; R . H . J . B o w e r , P r i n c e t o n Ph. D. t h e s i s , unpublished (1971). [3] L . M . D e l v e s et al., P h y s . L e t t e r s 20B (1969) 472; J. W. H u m b e r s t o n and M. A. Lennell, P h y s . L e t t e r s 31B (1970) 423; L. M . D e l v e s and M . A . HenneIl, Nucl. P h y s . A160 (1971) 347. [4] L. M. D e l v e s and A. C. P h i l l i p s , Rev. of Mod. P h y s . 41 (1969) 497. [5] A. N. M i t r a , G . L . Schrenk and V. S. Bhasin. Ann. of P h y s . 40 (1966) 357; G. L. Schrenk and A. N. M i t r a , P h y s . Rev. L e t t e r s 19 (1967) 530. [6] A.C. P h i l l i p s , Nucl. P h y s . A107 (1968) 209. [7] T . L . Houk, P h y s . Rev. C3 (1971) 1886.
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[8] I. S[oan, Nucl. P h y s . A139 (1969) 373. [9] M. Stingl and A. S. Rinat (Reiner), Nucl. P h y s . A154 (1970) 613; Ei Baz e t a [ . , Nuovo Cimento 64A (1969) 13. [10] I. Sloan, Phys. Rev. 165 {1968) 1587. [11] J . K r a u s and K . L . Kowalski, P h y s . L e t t e r s 31B {1970) 263; P h y s . Rev. C1 {1970) 1319. [12] J. C. A a r o n s and I. Sloan, to be published. [13] J. H. H e t h e r i n g t o n and L. H. Schick, P h y s . Rev, B137 (1965) 935. [14] Exp. data taken f r o m Ep = 7.8 MeV: J. B r o l l e y , P h y s . Rev. 117 (1960) 1307; En = 14.4 MeV: P h y s . Rev. 91 {1953) 90; 97 (1955) 757; 174 {1968) 1105; E ~ = 22.7 MeV: Nucl. P h v s . Al13 (1968) 461; E~o = 30.75 MeV: Nucl. PhYs. 50 {1964) 32. [15] I.'Sloan, Nucl. Phys. A168 (1971) 211. [16] Seyler, Nucl. Phys. A124 (1969) 253. [17] E = 8 MeV:Nuel. Phys. A95 (1967) 608; ~._ = 7.8 M e V : B A P S 14 (1969) 21: L~_~ = 14.5 MeV: Nucl. P h y s . A127 {1969) 169; ~ = 22.7 MeV: P h y s . Rev. L e t t e r s 21 (1968) 1393; 1~6 = 30 MeV: R u t h e r f o r d High E n e r g y Lab. Rep. (66). [18] W. Dilg, L. K o e s t e r and W. N i s t l e r , P h y s . L e t t e r s 36B (1971) 208.