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Nucleon-Nucleon T-matrix factorization for realistic potentials
Volume 37B, number 3
PHYSICS
LETTERS
29 November 1971
NUCLEON-NUCLEON T-MATRIX FACTORIZATION REALISTIC POTENTIALS
FOR
V. N. E F I M O V , Yu. I. F E N I N
Joint Institute for Nuclear Research, Dubna, USSR and E. G. T K A C H E N K O
V. G. Hlopin Radium Institute, Leningrad, USSR Received 25 September 1971 The nucleon-nucleon T-matrix off energy shell is factorized with respect to momentum variables for potentials with a hard core. The expressions derived give a practical possibility of solving numerically the three-nucleon Faddeev equations. In s o l v i n g the F a d d e e v e q u a t i o n s f o r a t h r e e - n u c l e o n s y s t e m an a p p r o x i m a t i o n is w i d e l y e m p l o y e d to r e p r e s e n t the t w o - b o d y o f f - s h e l l T - m a t r i x in the f o r m of a f i n i t e s u m of t e r m s e a c h b e i n g f a c t o r i z e d ~¢ith r e s p e c t to m o m e n t u m v a r i a b l e s . Such a p r o c e d u r e (with the a c c o u n t of only a few l o w e s t a n g u l a r m o m e n t u m c o m p o n e n t s of the t w o - b o d y T - m a t r i x ) m a k e s it p o s s i b l e to r e d u c e the t w o - d i m e n s i o n a l F a d d e e v e q u a t i o n s to f i n i t e s y s t e m s of o n e - d i m e n s i o n a l i n t e g r a l e q u a t i o n s , the n u m e r i c a l s o l v i n g of which i s q u i t e f e a s i b l e with m o d e r n c o m p u t e r p o s s i b i l i t i e s . In t h i s c o n n e c t i o n t h e r e a r i s e s the p r o b l e m of d e v e l o p i n g m e t h o d s f o r a p p r o x i m a t e f a c t o r i z a t i o n of the n u c l e o n - n u c l e o n T - m a t r i x . T h e s e s h o u l d be r e g u l a r m a t h e m a t i c a l p r o c e d u r e s c h a r a c t e r i z e d by a s u f f i c i e n t l y r a p i d c o n v e r g e n c e . In a d d i t i o n , t h e s e m e t h o d s should be s u i t a b l e f o r the c a s e s of r e a l i s t i c n u c l e o n - n u c l e o n p o t e n t i a l s p o s s e s s i n g a r e p u l s i o n at s m a l l d i s t a n c e s . Our way of c o n s t r u c t i n g the f a c t o r i z e d t w o - b o d y T - m a t r i x i s b a s e d upon a p p r o x i m a t e s o l v i n g of i n t e g r a l e q u a t i o n s by m e a n s of the m o m e n t u m m e t h o d [1]. In the p r o b l e m of two n u c l e o n s i n t e r a c t i n g with e a c h o t h e r by a c e n t r a l p o t e n t i a l we h a v e to r e s t r i c t o u r s e l v e s h e r e to the S - w a v e c a s e . T h e g e n e r a l i z a t i o n to the s t a t e s With o t h e r a n g u l a r m o m e n t a i s t r i v i a l . L e t the p o t e n t i a l h a v e a h a r d r e p u l s i v e c o r e with a r a d i u s r c. T h e i n t e g r a l e q u a t i o n [2] f o r the o f f s h e l l w a v e f u n c t i o n of the S - s t a t e
~hk(r,z) : Jo(kZ) + 2 y r ' 2 d r ' K ( r , r ' , z ) V ( r ' ) ~ k ( r ' , z ) 0
,
oO
f p2dp jo(Pr) Jo(Pr')(p2+r2)
-1, z : _~2 < 0
0 K(r,
Y', z) = oo
~p2dpjo(Pr)jo(Pr')(p2-s2)
-1,
z=S2
> 0
,
(1)
0 h a s a s i n g u l a r i t y , w h i c h c o m e s f r o m the h a r d c o r e of the p o t e n t i a l . T h e B r u e c k n e r a p p r o x i m a t i o n [4] h a s b e e n u s e d in the p r e s e n t w o r k to e l i m i n a t e t h i s s i m g u l a r i t y :
V(r')~Vk(r',z ) = w6(r'-re)
, r'<
re; ~k(r',z)
= 0, r ' = r e
(2)
If an a t t r a c t i v e p a r t of t h e p o t e n t i a l i s of c o n s t a n t s i g n at r > r e , t h e n the f u n c t i o n q k ( r , z ) = = r 4 V - ( ~ ~Vk ( r , z ) c a n b e i n t r o d u c e d . By t a k i n g (2) into a c c o u n t the m o d i f i e d n o n - s i n g u l a r e q u a t i o n f o r t h e q k ( r ,z) h a s the f o r m : 269
Volume 37B, number 3
PHYSICS
LETTERS
29 N o v e m b e r 1971
Table 1
Comparison of the separable (TS) and precise (TP) on-shell two-body matrices (P=K=S). ng is a number of basic functions in the momentum method. Energy is posit.lye and equal to S squared. P(i/f)
0.001
0.5
TS Re TS Im TS
-5.4056E 00 2.9222E-02
1.4817E-01 1.9890E 00
-5.3663E 00 2.8798E-02
1.4080E-01 1.9900E 00
-5.3662E00 2.8797E-02
1.4078E-01 1.9900E00
-5.3662E 00 2.8797E-02
1.4078E-01 1.9900E 00
TS Re TS Im TS
4.9363E-01 5.7956E-01
-1.2950E-01 6.1744E-02
4.9323E-01 5.8199E-01
-1.2934E-01 6.1555E-02
ng= 4
TP Re TP Im TP
3.0
ng= 3
TS Re TS ]m TS
1.0
ng = 2
4.9323E-01 5.8200E-01
-1.2934E-01 6.1554E-02
precise matrix 4.9323E-01 5.8200E-01
.-1.2934E-01 6.1554E-02
oo
qk(r,z)
= Jo(r,k,z)+
f dr'So(r,r',z)#k(r',z)
,
~'C J o ( r , k, z)
= [jo(kr) - jo(krc)K(r, re, z)/K(rc, rc, z ) ] r 4 - W ~
,
(3)
So(r , r', z) = [K ( r, r', z) - K (r, rc, z)K( rc, r', z)/K (r c, rc, z)] 2 r r ' 4-W(~ V(r ') T h e a p p r o x i m a t e - in the s'ense of (2)- T - m a t r i x (for z < 0) and A - m a t r i x w r i t t e n down in the f o l l o w i n g f o r m :
T(p,k , - 7 2) = - -~ J°(Prc)J°(kr c) + f dr Jo(r, p, - 7 2 ) ' I ' k ( r , - 7 2 ) 2 K(rc.,:rc ' _72) r c
[3] (for z > 0) c a n be
,
(4)
n Jo(Prc)Jo(krc) f d r d o ( r , P , S 2 ) ~ k ( r , S 2) A(p, k, $2)=-~ K ( r c , r c , $2) + r c
(5)
T h e f i r s t t e r m s in e q s . (4) and (5) a r e due to the a c c o u n t of the h a r d c o r e in t h e h a r d - s h e l l a p p r o x i m a t i o n [4]; the s e c o n d t e r m s of (4) and (5) d e s c r i b e c o n t r i b u t i o n s of the s m o o t h a t t r a c t i v e p a r t of the p o t e n t i a l (at r > r c ) in t h e T - m a t t r i x and t h e A - m a t r i x r e s p e c t i v e l y . A c c o r d i n g to the g e n e r a l s c h e m e of the m o m e n t u m m e t h o d [1] the e x p r e s s i o n N-1 •
,
(6)
),=0 where
YX (r ,z) = J dr ' So(r, r ' , z ) Y~_l(r ', z);
fo(r, z)= do(r, 4 - ~ ,
z)
,
(7)
~C s h o u l d be i n s e r t e d in eq. (3) i n s t e a d of the ,I,k ( r , z ) - f u n c t i o n to be found. T h e n an o r t h o g o n a l i t y f o r the d i f f e r e n c e b e t w e e n the l e f t - and r i g h t - h a n d p a r t s of the o b t a i n e d e q u a t i o n to N b a s i c f u n c t i o n s Y;t(r, z ) s h o u l d be r e q u i r e d . Such an o r t h o g o n a l i t y r e q u i r e m e n t p r o d u c e s a s y s t e m of n o n - h o m o g e n e o u s l i n e a r e q u a t i o n s f o r the c o e f f i c i e n t s C~2J)(""k, z ) [2]: 270
Volume 37B, number 3
PHYSICS
N-1 [ax, ~t' (z) - a ~t, ~.+1 (z)] C(XN ) (k, z
LETTERS
=M~(k,z)
29 November 1971
(8)
;
~,=0 oo
a~,~t, = ~ d r Y~t(r,z) Y~,(r,z);
M~t(k ,z) = f dr J o ( r , k , z ) Yx(r, z)
(9)
r c
r E
T h e c o e f f i c i e n t s d e t e r m i n e the a p p r o x i m a t e N - o r d e r s o l u t i o n (6) f o r the i n t e g r a l eq. (3). T h e a b o v e s c h e m e of c o n s t r u c t i n g an a p p r o x i m a t e o f f - s h e l l w a v e - f u n c t i o n l e a d s to a s e p a r a b l e r e p r e s e n t a t i o n of the A - m a t r i x (z > 0) and the T - m a t r i x (z < 0):
A ( g ) ( p , k , S 2 ) : - - ~jo(Prc) Jo(krc) + N-1 c(N ) ( k , S 2) Uk(p ,S 2) ~ 2 K(rc ' rc ' S 2) ~=0 T (N) (p, k, -7 2) = - ~- J°(Prc) J ° ( k r c ) + ~ 1 2 K ( r c , r c , - y 2)
,
(10)
c(xN ) (k, -y2)Mk(p, - y 2)
(ii)
k=0
A t z > 0 t h e r e l a t i o n b e t w e e n t h e A - and T - m a t r i c e s
[3].
off the e n e r g y s h e l l d o e s not v i o l a t e s e p a r a b i l i t y
F o r the p o t e n t i a l
V(r) = - 2rn
-co
r)
r
~< r e
,
=
Vn e x p ( - ~ n ( r - rc) ,
r > r c
,
(12)
n
a l l the i n t e g r a l s (7), (9) h a v e b e e n t a k e n in the c l o s e d f o r m , b a s i c f u n c t i o n s (7) c o n n e c t e d with e a c h o t h e r by s i m p l e r e c u r r e n c e a l g e b r a i c e q u a t i o n s , and f a c t o r i z e d o f f - s h e l l T - m a t r i x h a s e a s i l y b e e n c a l c u l a t e d by the C D C - 1 6 0 4 A . T a b l e 1 i l l u s t r a t e s the c o n v e r g e n c e of o u r m e t h o d f o r a p o t e n t i a l with h a r d c o r e and a t t r a c t i v e p a r t in the f o r m of p l a i n e x p o n e n t i a l b e y o n d r c ( V = 1 0 . 4 7 f m - 2 , ~ = 2 . 4 f m - 1 , r c = 0 . 3 5 f r o [5]). T h e c a s e e x h i b i t e d i s of p = k = s w h e n the t w o - b o d y T - m a t r i x c o i n c i d e s with the s c a t t e r i n g a m p l i t u d e . T h e r e s u l t s of a p p l i c a t i o n of t h e m e t h o d to t h e t h r e e - n u c l e o n p r o b l e m w i l l be g i v e n in d e t a i l e l s e where.
References [1] Yu.V. Vorobjev, Method momentov v prikladnoj matematike, GIFML, Moscou, 1958. [2] V. N. Efimov, Preprint oiji p-2546, Dubna, 1966; Compt. Rend. Cong. Intern. de Physique nuc[eaire, V. 2, Paris, 1964. [3] C. Lovelace, Phys. Rev. 135 (1964) B1225. [4] K.A. Brueckner, Theory of nuclear structure, (London, Methuen; N.Y., Wiley; Paris, Dunod, 1959). [5] Y. C. Tang, E.W. Schmid and R. C. Herndon, Nucl. Phys. 65 (1965) 203; I. R. Afnan and Y. C. Tang, Phys. Rev. 175 (1968) 1337.
271
PHYSICS
LETTERS
29 November 1971
NUCLEON-NUCLEON T-MATRIX FACTORIZATION REALISTIC POTENTIALS
FOR
V. N. E F I M O V , Yu. I. F E N I N
Joint Institute for Nuclear Research, Dubna, USSR and E. G. T K A C H E N K O
V. G. Hlopin Radium Institute, Leningrad, USSR Received 25 September 1971 The nucleon-nucleon T-matrix off energy shell is factorized with respect to momentum variables for potentials with a hard core. The expressions derived give a practical possibility of solving numerically the three-nucleon Faddeev equations. In s o l v i n g the F a d d e e v e q u a t i o n s f o r a t h r e e - n u c l e o n s y s t e m an a p p r o x i m a t i o n is w i d e l y e m p l o y e d to r e p r e s e n t the t w o - b o d y o f f - s h e l l T - m a t r i x in the f o r m of a f i n i t e s u m of t e r m s e a c h b e i n g f a c t o r i z e d ~¢ith r e s p e c t to m o m e n t u m v a r i a b l e s . Such a p r o c e d u r e (with the a c c o u n t of only a few l o w e s t a n g u l a r m o m e n t u m c o m p o n e n t s of the t w o - b o d y T - m a t r i x ) m a k e s it p o s s i b l e to r e d u c e the t w o - d i m e n s i o n a l F a d d e e v e q u a t i o n s to f i n i t e s y s t e m s of o n e - d i m e n s i o n a l i n t e g r a l e q u a t i o n s , the n u m e r i c a l s o l v i n g of which i s q u i t e f e a s i b l e with m o d e r n c o m p u t e r p o s s i b i l i t i e s . In t h i s c o n n e c t i o n t h e r e a r i s e s the p r o b l e m of d e v e l o p i n g m e t h o d s f o r a p p r o x i m a t e f a c t o r i z a t i o n of the n u c l e o n - n u c l e o n T - m a t r i x . T h e s e s h o u l d be r e g u l a r m a t h e m a t i c a l p r o c e d u r e s c h a r a c t e r i z e d by a s u f f i c i e n t l y r a p i d c o n v e r g e n c e . In a d d i t i o n , t h e s e m e t h o d s should be s u i t a b l e f o r the c a s e s of r e a l i s t i c n u c l e o n - n u c l e o n p o t e n t i a l s p o s s e s s i n g a r e p u l s i o n at s m a l l d i s t a n c e s . Our way of c o n s t r u c t i n g the f a c t o r i z e d t w o - b o d y T - m a t r i x i s b a s e d upon a p p r o x i m a t e s o l v i n g of i n t e g r a l e q u a t i o n s by m e a n s of the m o m e n t u m m e t h o d [1]. In the p r o b l e m of two n u c l e o n s i n t e r a c t i n g with e a c h o t h e r by a c e n t r a l p o t e n t i a l we h a v e to r e s t r i c t o u r s e l v e s h e r e to the S - w a v e c a s e . T h e g e n e r a l i z a t i o n to the s t a t e s With o t h e r a n g u l a r m o m e n t a i s t r i v i a l . L e t the p o t e n t i a l h a v e a h a r d r e p u l s i v e c o r e with a r a d i u s r c. T h e i n t e g r a l e q u a t i o n [2] f o r the o f f s h e l l w a v e f u n c t i o n of the S - s t a t e
~hk(r,z) : Jo(kZ) + 2 y r ' 2 d r ' K ( r , r ' , z ) V ( r ' ) ~ k ( r ' , z ) 0
,
oO
f p2dp jo(Pr) Jo(Pr')(p2+r2)
-1, z : _~2 < 0
0 K(r,
Y', z) = oo
~p2dpjo(Pr)jo(Pr')(p2-s2)
-1,
z=S2
> 0
,
(1)
0 h a s a s i n g u l a r i t y , w h i c h c o m e s f r o m the h a r d c o r e of the p o t e n t i a l . T h e B r u e c k n e r a p p r o x i m a t i o n [4] h a s b e e n u s e d in the p r e s e n t w o r k to e l i m i n a t e t h i s s i m g u l a r i t y :
V(r')~Vk(r',z ) = w6(r'-re)
, r'<
re; ~k(r',z)
= 0, r ' = r e
(2)
If an a t t r a c t i v e p a r t of t h e p o t e n t i a l i s of c o n s t a n t s i g n at r > r e , t h e n the f u n c t i o n q k ( r , z ) = = r 4 V - ( ~ ~Vk ( r , z ) c a n b e i n t r o d u c e d . By t a k i n g (2) into a c c o u n t the m o d i f i e d n o n - s i n g u l a r e q u a t i o n f o r t h e q k ( r ,z) h a s the f o r m : 269
Volume 37B, number 3
PHYSICS
LETTERS
29 N o v e m b e r 1971
Table 1
Comparison of the separable (TS) and precise (TP) on-shell two-body matrices (P=K=S). ng is a number of basic functions in the momentum method. Energy is posit.lye and equal to S squared. P(i/f)
0.001
0.5
TS Re TS Im TS
-5.4056E 00 2.9222E-02
1.4817E-01 1.9890E 00
-5.3663E 00 2.8798E-02
1.4080E-01 1.9900E 00
-5.3662E00 2.8797E-02
1.4078E-01 1.9900E00
-5.3662E 00 2.8797E-02
1.4078E-01 1.9900E 00
TS Re TS Im TS
4.9363E-01 5.7956E-01
-1.2950E-01 6.1744E-02
4.9323E-01 5.8199E-01
-1.2934E-01 6.1555E-02
ng= 4
TP Re TP Im TP
3.0
ng= 3
TS Re TS ]m TS
1.0
ng = 2
4.9323E-01 5.8200E-01
-1.2934E-01 6.1554E-02
precise matrix 4.9323E-01 5.8200E-01
.-1.2934E-01 6.1554E-02
oo
qk(r,z)
= Jo(r,k,z)+
f dr'So(r,r',z)#k(r',z)
,
~'C J o ( r , k, z)
= [jo(kr) - jo(krc)K(r, re, z)/K(rc, rc, z ) ] r 4 - W ~
,
(3)
So(r , r', z) = [K ( r, r', z) - K (r, rc, z)K( rc, r', z)/K (r c, rc, z)] 2 r r ' 4-W(~ V(r ') T h e a p p r o x i m a t e - in the s'ense of (2)- T - m a t r i x (for z < 0) and A - m a t r i x w r i t t e n down in the f o l l o w i n g f o r m :
T(p,k , - 7 2) = - -~ J°(Prc)J°(kr c) + f dr Jo(r, p, - 7 2 ) ' I ' k ( r , - 7 2 ) 2 K(rc.,:rc ' _72) r c
[3] (for z > 0) c a n be
,
(4)
n Jo(Prc)Jo(krc) f d r d o ( r , P , S 2 ) ~ k ( r , S 2) A(p, k, $2)=-~ K ( r c , r c , $2) + r c
(5)
T h e f i r s t t e r m s in e q s . (4) and (5) a r e due to the a c c o u n t of the h a r d c o r e in t h e h a r d - s h e l l a p p r o x i m a t i o n [4]; the s e c o n d t e r m s of (4) and (5) d e s c r i b e c o n t r i b u t i o n s of the s m o o t h a t t r a c t i v e p a r t of the p o t e n t i a l (at r > r c ) in t h e T - m a t t r i x and t h e A - m a t r i x r e s p e c t i v e l y . A c c o r d i n g to the g e n e r a l s c h e m e of the m o m e n t u m m e t h o d [1] the e x p r e s s i o n N-1 •
,
(6)
),=0 where
YX (r ,z) = J dr ' So(r, r ' , z ) Y~_l(r ', z);
fo(r, z)= do(r, 4 - ~ ,
z)
,
(7)
~C s h o u l d be i n s e r t e d in eq. (3) i n s t e a d of the ,I,k ( r , z ) - f u n c t i o n to be found. T h e n an o r t h o g o n a l i t y f o r the d i f f e r e n c e b e t w e e n the l e f t - and r i g h t - h a n d p a r t s of the o b t a i n e d e q u a t i o n to N b a s i c f u n c t i o n s Y;t(r, z ) s h o u l d be r e q u i r e d . Such an o r t h o g o n a l i t y r e q u i r e m e n t p r o d u c e s a s y s t e m of n o n - h o m o g e n e o u s l i n e a r e q u a t i o n s f o r the c o e f f i c i e n t s C~2J)(""k, z ) [2]: 270
Volume 37B, number 3
PHYSICS
N-1 [ax, ~t' (z) - a ~t, ~.+1 (z)] C(XN ) (k, z
LETTERS
=M~(k,z)
29 November 1971
(8)
;
~,=0 oo
a~,~t, = ~ d r Y~t(r,z) Y~,(r,z);
M~t(k ,z) = f dr J o ( r , k , z ) Yx(r, z)
(9)
r c
r E
T h e c o e f f i c i e n t s d e t e r m i n e the a p p r o x i m a t e N - o r d e r s o l u t i o n (6) f o r the i n t e g r a l eq. (3). T h e a b o v e s c h e m e of c o n s t r u c t i n g an a p p r o x i m a t e o f f - s h e l l w a v e - f u n c t i o n l e a d s to a s e p a r a b l e r e p r e s e n t a t i o n of the A - m a t r i x (z > 0) and the T - m a t r i x (z < 0):
A ( g ) ( p , k , S 2 ) : - - ~jo(Prc) Jo(krc) + N-1 c(N ) ( k , S 2) Uk(p ,S 2) ~ 2 K(rc ' rc ' S 2) ~=0 T (N) (p, k, -7 2) = - ~- J°(Prc) J ° ( k r c ) + ~ 1 2 K ( r c , r c , - y 2)
,
(10)
c(xN ) (k, -y2)Mk(p, - y 2)
(ii)
k=0
A t z > 0 t h e r e l a t i o n b e t w e e n t h e A - and T - m a t r i c e s
[3].
off the e n e r g y s h e l l d o e s not v i o l a t e s e p a r a b i l i t y
F o r the p o t e n t i a l
V(r) = - 2rn
-co
r)
r
~< r e
,
=
Vn e x p ( - ~ n ( r - rc) ,
r > r c
,
(12)
n
a l l the i n t e g r a l s (7), (9) h a v e b e e n t a k e n in the c l o s e d f o r m , b a s i c f u n c t i o n s (7) c o n n e c t e d with e a c h o t h e r by s i m p l e r e c u r r e n c e a l g e b r a i c e q u a t i o n s , and f a c t o r i z e d o f f - s h e l l T - m a t r i x h a s e a s i l y b e e n c a l c u l a t e d by the C D C - 1 6 0 4 A . T a b l e 1 i l l u s t r a t e s the c o n v e r g e n c e of o u r m e t h o d f o r a p o t e n t i a l with h a r d c o r e and a t t r a c t i v e p a r t in the f o r m of p l a i n e x p o n e n t i a l b e y o n d r c ( V = 1 0 . 4 7 f m - 2 , ~ = 2 . 4 f m - 1 , r c = 0 . 3 5 f r o [5]). T h e c a s e e x h i b i t e d i s of p = k = s w h e n the t w o - b o d y T - m a t r i x c o i n c i d e s with the s c a t t e r i n g a m p l i t u d e . T h e r e s u l t s of a p p l i c a t i o n of t h e m e t h o d to t h e t h r e e - n u c l e o n p r o b l e m w i l l be g i v e n in d e t a i l e l s e where.
References [1] Yu.V. Vorobjev, Method momentov v prikladnoj matematike, GIFML, Moscou, 1958. [2] V. N. Efimov, Preprint oiji p-2546, Dubna, 1966; Compt. Rend. Cong. Intern. de Physique nuc[eaire, V. 2, Paris, 1964. [3] C. Lovelace, Phys. Rev. 135 (1964) B1225. [4] K.A. Brueckner, Theory of nuclear structure, (London, Methuen; N.Y., Wiley; Paris, Dunod, 1959). [5] Y. C. Tang, E.W. Schmid and R. C. Herndon, Nucl. Phys. 65 (1965) 203; I. R. Afnan and Y. C. Tang, Phys. Rev. 175 (1968) 1337.
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