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Single-particle resonances in the unified theory of nuclear reactions
Volume 32B, number 8
PHYSICS LETTERS
17 August 1970
SINGLE-PARTICLE RESONANCES IN THE UNIFIED THEORY OF NUCLEAR REACTIONS
*
W. L. WANG ** and C. M. SHAKIN
Laboratory for Nuclear Science and Physics Department Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Received 30 June 1970
A method for the treatment of single-particle resonances in the theory of nucleon-nucleus scattering is introduced which finds application in the projection operator theory of Feshbaeh. The example of the d3/2 neutron resonance in the scattering from 160 is discussed.
S e v e r a l a u t h o r s have a t t e m p t e d to extend the t e c h n i q u e s of the u s ua l s h e l l - m o d e l bound state c a l c u l a t i o n s to d e s c r i b e n u c l e a r r e a c t i o n s . Most of the t e c h n i q u e s have been d e s c r i b e d in the m o n o g r a p h of Mahaux and W e i d e n m i i l l e r [1] w h e r e e x t e n s i v e r e f e r e n c e s may be found to t h e i r w o r k and that of F e s h b a c h , MacDonald, and o t h e r s . A c o m m o n difficulty in t h e s e f o r m u l a t i o n s is the t r e a t m e n t of s i n g l e - p a r t i c l e r e s o n a n c e s in the continuum. Some t e c h n i q u e s of handling the r e s o n a n c e s t at e have been i n v e s t i g a t e d . The H e i d e l b e r g group has applied a method [2], b a s e d upon the w o r k of W e i n b e r g [3], to modify the c o n t i n u u m - c o n t i n u u m i n t e r a c t i o n in such a way that it may be t r e a t e d in p e r t u r b a t i o n theory. More recently a "generalized innerproduct" method has b e e n d ev e l o p e d by Romo [4]. F u l l e r has a l s o d e v e l o p e d an a p p r o a c h analogous to W e i n b e r g ' s t r e a t m e n t to r e m e d y the d i v e r g e n c e of the Born s e r i e s [5]. The o r t h o g o n a l i z a t i o n method we p r o p o s e h e r e has the advantage of b e i n g conceptually s i m p l e and e a s y to apply. We s h a l l outline its b a s i c concept and its r o l e in the unified t h e o r y de ve lo p ed by F e s h b a c h [6]. We r e p o r t an i l l u s t r a t i v e c a l c u l a t i o n f o r the d3/2 continuum neut r o n r e s o n a n c e in s c a t t e r i n g f r o m 160. In our ap p l i cat i o n of the unified t h e o r y , a p r o j e c t i o n o p e r a t o r P is defined to p r o j e c t on to the s u b s p a c e w h e r e t h e r e is one p a r t i c l e in a s c a t t e r i n g state coupled to the ground s t a t e or * The work was supported in part through funds provided by the Atomic Energy Commission under Contract No. AT(30-1) 2098. ** Department of Nuclear Engineering.
v a r i o u s e x c i t e d s t a t e s of the t a r g e t . P a r t i c l e s in bound o r b i t s coupled to the ground state or the v a r i o u s e x c i t e d s t a t e s a r e in Q = 1 - P. The Q s p a c e a l s o contains all other channels not i n cluded in P . We have in mind the e x a m p l e of neutron s c a t t e r i n g f r o m 1 5 0 d e s c r i b e d as a P l / 2 n eu t r o n hole state. The e x c i t e d st at e of 150 is taken a s a P3/2 hole and P s p a c e contains continuum o r b i t s coupled to e i t h e r of t h ese hole s t a t e s . All o t h e r m o d e s a r e in the Q - s p a c e , p a r t i c u l a r l y s t a t e s of a bound ld5/2 or 2Sl/2 p a r t i c l e coupled to the hole s t a t e s . The d3/2 o r b i t a p p e a r s as a s i n g l e p a r t i c l e r e s o n a n c e in the P space. With the two orthogonal p r o j e c t i o n o p e r a t o r s , P and Q, a solution f o r the s c a t t e r i n g state can be obtained f r o m the S ch r S d i n g er equation, (E - H ) @ : 0, as [6],
P@(+) = ~ + ) + E + -Hpp H p o E -HQQ HQp (17 where
(E -Hpp )~+) = 0 ,
(2)
and
H p Q = PHQ ,
HQQ = QHQ , etc.
We w i s h to t r e a t the c o n t i n u u m - c o n t i n u u m coupling in p e r t u r b a t i o n theory. We m i g h t e x p e c t such a p e r t u r b a t i o n t h e o r y to c o n v e r g e if t h e r e w e r e no r e s o n a n c e in the P s p a c e . O t h e r w i s e we i n t r o d u c e a technique w h e r e b y we r e d e f i n e the P s p a c e , r e m o v i n g to the Q s p a c e any P-space single-particle resonances***. The m a n y - b o d y S c h r b d i n g e r equation (2) can 421
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be reduced to a single-particle Schrbdinger equation [7-8]. In the limit of no direct channel coupling in the P space, we obtain uncoupled singleparticle SchrBdinger equations with a single-particle Hamiltonianha, (E-ha)~:+)
=0
(3)
w h e r e ~b(~!a d e s c r i b e s the s c a t t e r i n g of a p a r t i c l e with ener~'y E in c h a n n e l a . We a s s u m e that t h e r e i s a s i n g l e - p a r t i c l e r e s o n a n c e in one of the c h a n n e l s at the e n e r g y E R. In t h i s c a s e we define a " r e s o n a n c e wave p a c k e t " , q~R(r), w h i c h is o b t a i n e d f r o m the c o n +) t i n u u m wave f u n c t i o n ~ b ~ ) a ( r ) . We choose ~ R ( r ) p r o p o r t i o n a l to ~+R) a ( r ) ~ b r r l e s s t h a n s o m e c u t off r a d i u s R c and eh'uaI to z e r o for r > R c. We p r o c e e d to c o n s t r u c t m o d i f i e d c o n t i n u u m wave f u n c t i o n s @(E!a (r) o r t h o g o n a l to ~n(r):
I #E ,)a >:
E,ol
>'
with
G(a+)(E)[ dpR) (C~RI R : I - (d)R I G(+)(E) dPR)
(5)
LETTERS
17 August 1970
the g r o u n d state or v a r i o u s e x c i t e d s t a t e s of the t a r g e t . C o n f i g u r a t i o n s s u c h a s (ld3/2, lp~J 2) and (ld3/2, lp3~2) a r e t h e r e f o r e in the new Q s p a c e . It can e a s i l y be shown that the ~ ) ~ a r e o r t h o gonal to ~bn and obey the s a m e o r t h o ~ i o r m a l i t y c o n d i t i o n s ~ a s ~b(~a, i.e.
(@(+) E,~ [~R) = 0
(9)
and (+) (+) <¢e',~'lCe,~>
=
(~b(E+),o~,!~(+) ): E,a
%'a
6(E' - E) (10)
In c a s e of a l o c a l p o t e n t i a l , the o p e r a t o r R c a n be o b t a i n e d bv~ e x p l i c i t y c o n s t r u c t i n g the G r e e n f u n c t i o n G ~ ) in t e r m s of the r e g u l a r and i r r e g u l a r s o l u t i o n s of eq. (3). An a l t e r n a t i v e a p p r o a c h [9] is to s o l v e the S c h r h d i n g e r e q u a t i o n , eq. (6), with the p r o j e c t e d H a m i l t o n i a n , eq. (7), and p r o p e r o r t h o g o n a l i t y c o n d i t i o n of eq. (9). We c a n w r i t e eq. (6) a s a n i n h o m o g e n e o u s e q u a t i o n
(ha - e)~(E+)a = CPR)7 . RE,a
(11)
w h e r e the a m p l i t u d e 7 i s d e f i n e d a s , H e r e I i s the i d e n t i t y o p e r a t o r and G(~)(E) is the G r e e n ' s f u n c t i o n , (E - h a +i~) -1. In our e x a m p l e , h a was the H a m i l t o n i a n for a S a x o n - W o o d s p o t e n t i a l and dPR(r) w a s t a k e n p r o p o r t i o n a l to the d3/2 c o n t i n u u m wave at r e s o n a n c e . The c u t - o f f , R c , w a s c h o s e n at about 16 fro. The new P space i s t h e r e f o r e s p a n n e d by the + s t a t e s ~P(E)a which c a n be shown to be the s o l u t i o n s of a ' s i n g l e - p a r t i c l e S c h r h d i n g e r e q u a t i o n with a m o d i f i e d H a m i l t o n i a n , hp,a,
(E - hp, a)g/(E+)a : 0 ,
(6)
E,a We w r i t e the s o l u t i o n in the f o l l o w i n g f o r m ~h(+E)ot = a w,,I) E , a + b ~(E2)o~
w h e r e a, b a r e c o n s t a n t s to be d e t e r m i n e d . The wave f u n c t i o n s ~(El!a and ,,,(2) 'e"E , a s a t i s f y the e q u a tions
(1)
(ha-E)~/E,a
=
~bR>V(1) (14)
w h e r e hp, a i s given a s , hp,a=
(1-ICbR)(dpR1)ha(1-I~R)<~RI).
(13)
(7)
The G r e e n f u n c t i o n (E - hp, a ) - I is g i v e n by
Now the Q s p a c e i s e x t e n d e d to i n c l u d e the s i n g l e - p a r t i c l e r e s o n a n c e " s t a t e " ~bR coupled to *** It might be worthwhile to point out that the method we propose is applicable to situations where there is a particular state to be removed for any reason. The properties of this state are not contained in the framework of our formulation but only appear in the discussion of the specific application. The method presented here was first used to orthogonalize the continuum space to the space of analog states as defined in ref. [8]. 422
( h a - E)~(E21a = ¢ R ) y (2) with ~(i) a r b i t r a r y c o n s t a n t s . If we e n f o r c e the c o n d i t i o n g i v e n in eq. (9) on e q u a t i o n (13), we obtain
(15)
A s i d e f r o m a n o r m a l i z a t i o n c o n s t a n t a, the new wave f u n c t i o n is d e t e r m i n e d s i n c e the a m p l i t u d e s (~R I ~P{~a) a r e j u s t l i n e a r f u n c t i o n s of ~(ff c h o s e n in eq. (1'4). We r e l a t e the S - m a t r i c e s of s i n g l e p a r t i c l e and hp f r o m e q u a t i o n s (3) and (5). We find that
Volume 32B, number 6
PHYSICS LETTERS
they a r e r e l a t e d by a s i m p l e p h a s e shift m o d i f i cation, ~ a , exp (2i 6~) = exp 2i(6 a +A a )
1.0
(16)
w h e r e the 6~ denote the modified phase shifts and 6a denote the o r i g i n a l phase shifts. The m o d i f i cation is given by exp ( 2 i ~ )
17 August 1970
= ( ~ + i p ) / ( ~ - ip)
0.5
(17)
with + ip = ((bRIG ~ ) Iq~R)
(18)
The o r i g i n a l 6 a is a s t r o n g function of e n e r g y at r e s o n a n c e . The m o d i f i c a t i o n phase A~ can a l s o be made to be s t r o n g function of e n e r g y at the r e s o n a n c e and thus p r o d u c e s m o o t h modified pha se shifts 6'~. This a n t i r e s o n a n c e b e h a v i o r can be obtained by a p r o p e r choice of the s t a t e
i
1.0
0.5
oJ
oo
o
1.6
-0.5
1.4
-Lo
12
~t°~(r )
I 0
2
I
I 4
I
i
L
6
i 8
i
.l I0
I 12
E (MeV) -I"
t.O
Fig. 2. The neutron phase shifts in 160. The solid line is the potential phase shift, while the dashed line is the modified phase shift given by eq. (16).
"~ 0.8
•~
0,6
2 o
0.2
~,~°~(r ) 0.0
-0.2
~
~'~,,
5
0
I0
15
r(fm)
Fig. 1. The d3/2 continuum neutron wave function at
resonance and 'its modification by the operator R. A Saxon-Woods potential is used in the calculations and the resonance energy iS taken to be 0.93 MeV. The wave functions are defined to have the asymptotic forms:
~(0) ~ j - 2 ~ sin(kr-½~z+~) lrfi~ k
We have c a l c u l a t e d the wave functions and the c o r r e s p o n d i n g p h a s e shifts - s e e fig. 1 and fig. 2. We notice that the r e s o n a n c e v a n i s h e s a f t e r the m o d i f i cat i o n , and that the effect of the m o d i f i c a tion is l o c a l i z e d n e a r the r e s o n a n c e . We have a l s o found that t h e s e f e a t u r e s a r e not s e n s i t i v e to cut-off r a d i u s c h o s e n f o r the function ~R" It is w o r t h w h i l e to see the p a r t i c u l a r r o l e of the s i n g l e - p a r t i c l e r e s o n a n c e in our f o r m u l a t i o n . We have extended the u su al definition of the Q s p a c e to a c c o m m o d a t e a s i n g l e - p a r t i c l e r e s o n ance in the continuum and t r e a t it e x a c t l y a s a bound state. H o w e v e r this s t a t e , when coupled to the newly defined P s p a c e , g i v e s r i s e to the single p a r t i c l e r e s o n a n c e . F i n a l l y we note that the m o d i f i c a t i o n i n t r o d u c e d by the o p e r a t o r R is just a r e d e f i n i t i o n of r e a c t i o n s p a c e s r a t h e r than a change in p h y s i c a l situation. The c a l c u l a t i o n c l e a r l y shows that this o r t h o g o n a l i z a t i o n method can be v e r y c o n v e n i e n t l y inc o r p o r a t e d with the unified t h e o r y . An a n a l y s i s employing these techniques for nucleon-nucleus s c a t t e r i n g c r o s s s e c t i o n s is b ei n g i n v e s t i g a t e d and w i l l be r e p o r t e d l a t e r . 423
Volume 32B, n u m b e r 6
PHYSICS
References [1] c . Mahaux and H. A. Weidenmilller, Shell-model a p p r o a c h to n u c l e a r reactions (North-Holland A m s t e r dam, 1969). [2] W. Gl~cke, J. H£Ifner and H. A. Weidenm~/ller, Nucl. Phys. A90 (1967) 481. [3] S. Weinberg, Phys. Rev. 130 (1963) 776. [4] W. J. Romo, Nucl. Phys. Al16 (1968) 618.
424
LETTERS
17 August 1970
[5] R. C. F u l l e r , P r e p r i n t , School of Physics and Astronomy, University of Minnesota (1969). [6] H. F e s h b a c h , Ann. Phys. 5 (1958) 357; 19 (1962) 287. [7] A. K. K e r m a n , L e c t u r e s in t h e o r e t i c a l physics VIH-C, University of Colorado P r e s s (1966). [8] N. Auerbach, J. H(lfner, A. K. K e r m a n and C. M. Shakin, to be submitted for publication. [9] A. F. R. De Toledo Piza, private communication.
PHYSICS LETTERS
17 August 1970
SINGLE-PARTICLE RESONANCES IN THE UNIFIED THEORY OF NUCLEAR REACTIONS
*
W. L. WANG ** and C. M. SHAKIN
Laboratory for Nuclear Science and Physics Department Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Received 30 June 1970
A method for the treatment of single-particle resonances in the theory of nucleon-nucleus scattering is introduced which finds application in the projection operator theory of Feshbaeh. The example of the d3/2 neutron resonance in the scattering from 160 is discussed.
S e v e r a l a u t h o r s have a t t e m p t e d to extend the t e c h n i q u e s of the u s ua l s h e l l - m o d e l bound state c a l c u l a t i o n s to d e s c r i b e n u c l e a r r e a c t i o n s . Most of the t e c h n i q u e s have been d e s c r i b e d in the m o n o g r a p h of Mahaux and W e i d e n m i i l l e r [1] w h e r e e x t e n s i v e r e f e r e n c e s may be found to t h e i r w o r k and that of F e s h b a c h , MacDonald, and o t h e r s . A c o m m o n difficulty in t h e s e f o r m u l a t i o n s is the t r e a t m e n t of s i n g l e - p a r t i c l e r e s o n a n c e s in the continuum. Some t e c h n i q u e s of handling the r e s o n a n c e s t at e have been i n v e s t i g a t e d . The H e i d e l b e r g group has applied a method [2], b a s e d upon the w o r k of W e i n b e r g [3], to modify the c o n t i n u u m - c o n t i n u u m i n t e r a c t i o n in such a way that it may be t r e a t e d in p e r t u r b a t i o n theory. More recently a "generalized innerproduct" method has b e e n d ev e l o p e d by Romo [4]. F u l l e r has a l s o d e v e l o p e d an a p p r o a c h analogous to W e i n b e r g ' s t r e a t m e n t to r e m e d y the d i v e r g e n c e of the Born s e r i e s [5]. The o r t h o g o n a l i z a t i o n method we p r o p o s e h e r e has the advantage of b e i n g conceptually s i m p l e and e a s y to apply. We s h a l l outline its b a s i c concept and its r o l e in the unified t h e o r y de ve lo p ed by F e s h b a c h [6]. We r e p o r t an i l l u s t r a t i v e c a l c u l a t i o n f o r the d3/2 continuum neut r o n r e s o n a n c e in s c a t t e r i n g f r o m 160. In our ap p l i cat i o n of the unified t h e o r y , a p r o j e c t i o n o p e r a t o r P is defined to p r o j e c t on to the s u b s p a c e w h e r e t h e r e is one p a r t i c l e in a s c a t t e r i n g state coupled to the ground s t a t e or * The work was supported in part through funds provided by the Atomic Energy Commission under Contract No. AT(30-1) 2098. ** Department of Nuclear Engineering.
v a r i o u s e x c i t e d s t a t e s of the t a r g e t . P a r t i c l e s in bound o r b i t s coupled to the ground state or the v a r i o u s e x c i t e d s t a t e s a r e in Q = 1 - P. The Q s p a c e a l s o contains all other channels not i n cluded in P . We have in mind the e x a m p l e of neutron s c a t t e r i n g f r o m 1 5 0 d e s c r i b e d as a P l / 2 n eu t r o n hole state. The e x c i t e d st at e of 150 is taken a s a P3/2 hole and P s p a c e contains continuum o r b i t s coupled to e i t h e r of t h ese hole s t a t e s . All o t h e r m o d e s a r e in the Q - s p a c e , p a r t i c u l a r l y s t a t e s of a bound ld5/2 or 2Sl/2 p a r t i c l e coupled to the hole s t a t e s . The d3/2 o r b i t a p p e a r s as a s i n g l e p a r t i c l e r e s o n a n c e in the P space. With the two orthogonal p r o j e c t i o n o p e r a t o r s , P and Q, a solution f o r the s c a t t e r i n g state can be obtained f r o m the S ch r S d i n g er equation, (E - H ) @ : 0, as [6],
P@(+) = ~ + ) + E + -Hpp H p o E -HQQ HQp (17 where
(E -Hpp )~+) = 0 ,
(2)
and
H p Q = PHQ ,
HQQ = QHQ , etc.
We w i s h to t r e a t the c o n t i n u u m - c o n t i n u u m coupling in p e r t u r b a t i o n theory. We m i g h t e x p e c t such a p e r t u r b a t i o n t h e o r y to c o n v e r g e if t h e r e w e r e no r e s o n a n c e in the P s p a c e . O t h e r w i s e we i n t r o d u c e a technique w h e r e b y we r e d e f i n e the P s p a c e , r e m o v i n g to the Q s p a c e any P-space single-particle resonances***. The m a n y - b o d y S c h r b d i n g e r equation (2) can 421
V o l u m e 32B, n u m b e r 6
PHYSICS
be reduced to a single-particle Schrbdinger equation [7-8]. In the limit of no direct channel coupling in the P space, we obtain uncoupled singleparticle SchrBdinger equations with a single-particle Hamiltonianha, (E-ha)~:+)
=0
(3)
w h e r e ~b(~!a d e s c r i b e s the s c a t t e r i n g of a p a r t i c l e with ener~'y E in c h a n n e l a . We a s s u m e that t h e r e i s a s i n g l e - p a r t i c l e r e s o n a n c e in one of the c h a n n e l s at the e n e r g y E R. In t h i s c a s e we define a " r e s o n a n c e wave p a c k e t " , q~R(r), w h i c h is o b t a i n e d f r o m the c o n +) t i n u u m wave f u n c t i o n ~ b ~ ) a ( r ) . We choose ~ R ( r ) p r o p o r t i o n a l to ~+R) a ( r ) ~ b r r l e s s t h a n s o m e c u t off r a d i u s R c and eh'uaI to z e r o for r > R c. We p r o c e e d to c o n s t r u c t m o d i f i e d c o n t i n u u m wave f u n c t i o n s @(E!a (r) o r t h o g o n a l to ~n(r):
I #E ,)a >:
E,ol
>'
with
G(a+)(E)[ dpR) (C~RI R : I - (d)R I G(+)(E) dPR)
(5)
LETTERS
17 August 1970
the g r o u n d state or v a r i o u s e x c i t e d s t a t e s of the t a r g e t . C o n f i g u r a t i o n s s u c h a s (ld3/2, lp~J 2) and (ld3/2, lp3~2) a r e t h e r e f o r e in the new Q s p a c e . It can e a s i l y be shown that the ~ ) ~ a r e o r t h o gonal to ~bn and obey the s a m e o r t h o ~ i o r m a l i t y c o n d i t i o n s ~ a s ~b(~a, i.e.
(@(+) E,~ [~R) = 0
(9)
and (+) (+) <¢e',~'lCe,~>
=
(~b(E+),o~,!~(+) ): E,a
%'a
6(E' - E) (10)
In c a s e of a l o c a l p o t e n t i a l , the o p e r a t o r R c a n be o b t a i n e d bv~ e x p l i c i t y c o n s t r u c t i n g the G r e e n f u n c t i o n G ~ ) in t e r m s of the r e g u l a r and i r r e g u l a r s o l u t i o n s of eq. (3). An a l t e r n a t i v e a p p r o a c h [9] is to s o l v e the S c h r h d i n g e r e q u a t i o n , eq. (6), with the p r o j e c t e d H a m i l t o n i a n , eq. (7), and p r o p e r o r t h o g o n a l i t y c o n d i t i o n of eq. (9). We c a n w r i t e eq. (6) a s a n i n h o m o g e n e o u s e q u a t i o n
(ha - e)~(E+)a = CPR)7 . RE,a
(11)
w h e r e the a m p l i t u d e 7 i s d e f i n e d a s , H e r e I i s the i d e n t i t y o p e r a t o r and G(~)(E) is the G r e e n ' s f u n c t i o n , (E - h a +i~) -1. In our e x a m p l e , h a was the H a m i l t o n i a n for a S a x o n - W o o d s p o t e n t i a l and dPR(r) w a s t a k e n p r o p o r t i o n a l to the d3/2 c o n t i n u u m wave at r e s o n a n c e . The c u t - o f f , R c , w a s c h o s e n at about 16 fro. The new P space i s t h e r e f o r e s p a n n e d by the + s t a t e s ~P(E)a which c a n be shown to be the s o l u t i o n s of a ' s i n g l e - p a r t i c l e S c h r h d i n g e r e q u a t i o n with a m o d i f i e d H a m i l t o n i a n , hp,a,
(E - hp, a)g/(E+)a : 0 ,
(6)
E,a We w r i t e the s o l u t i o n in the f o l l o w i n g f o r m ~h(+E)ot = a w,,I) E , a + b ~(E2)o~
w h e r e a, b a r e c o n s t a n t s to be d e t e r m i n e d . The wave f u n c t i o n s ~(El!a and ,,,(2) 'e"E , a s a t i s f y the e q u a tions
(1)
(ha-E)~/E,a
=
~bR>V(1) (14)
w h e r e hp, a i s given a s , hp,a=
(1-ICbR)(dpR1)ha(1-I~R)<~RI).
(13)
(7)
The G r e e n f u n c t i o n (E - hp, a ) - I is g i v e n by
Now the Q s p a c e i s e x t e n d e d to i n c l u d e the s i n g l e - p a r t i c l e r e s o n a n c e " s t a t e " ~bR coupled to *** It might be worthwhile to point out that the method we propose is applicable to situations where there is a particular state to be removed for any reason. The properties of this state are not contained in the framework of our formulation but only appear in the discussion of the specific application. The method presented here was first used to orthogonalize the continuum space to the space of analog states as defined in ref. [8]. 422
( h a - E)~(E21a = ¢ R ) y (2) with ~(i) a r b i t r a r y c o n s t a n t s . If we e n f o r c e the c o n d i t i o n g i v e n in eq. (9) on e q u a t i o n (13), we obtain
(15)
A s i d e f r o m a n o r m a l i z a t i o n c o n s t a n t a, the new wave f u n c t i o n is d e t e r m i n e d s i n c e the a m p l i t u d e s (~R I ~P{~a) a r e j u s t l i n e a r f u n c t i o n s of ~(ff c h o s e n in eq. (1'4). We r e l a t e the S - m a t r i c e s of s i n g l e p a r t i c l e and hp f r o m e q u a t i o n s (3) and (5). We find that
Volume 32B, number 6
PHYSICS LETTERS
they a r e r e l a t e d by a s i m p l e p h a s e shift m o d i f i cation, ~ a , exp (2i 6~) = exp 2i(6 a +A a )
1.0
(16)
w h e r e the 6~ denote the modified phase shifts and 6a denote the o r i g i n a l phase shifts. The m o d i f i cation is given by exp ( 2 i ~ )
17 August 1970
= ( ~ + i p ) / ( ~ - ip)
0.5
(17)
with + ip = ((bRIG ~ ) Iq~R)
(18)
The o r i g i n a l 6 a is a s t r o n g function of e n e r g y at r e s o n a n c e . The m o d i f i c a t i o n phase A~ can a l s o be made to be s t r o n g function of e n e r g y at the r e s o n a n c e and thus p r o d u c e s m o o t h modified pha se shifts 6'~. This a n t i r e s o n a n c e b e h a v i o r can be obtained by a p r o p e r choice of the s t a t e
i
1.0
0.5
oJ
oo
o
1.6
-0.5
1.4
-Lo
12
~t°~(r )
I 0
2
I
I 4
I
i
L
6
i 8
i
.l I0
I 12
E (MeV) -I"
t.O
Fig. 2. The neutron phase shifts in 160. The solid line is the potential phase shift, while the dashed line is the modified phase shift given by eq. (16).
"~ 0.8
•~
0,6
2 o
0.2
~,~°~(r ) 0.0
-0.2
~
~'~,,
5
0
I0
15
r(fm)
Fig. 1. The d3/2 continuum neutron wave function at
resonance and 'its modification by the operator R. A Saxon-Woods potential is used in the calculations and the resonance energy iS taken to be 0.93 MeV. The wave functions are defined to have the asymptotic forms:
~(0) ~ j - 2 ~ sin(kr-½~z+~) lrfi~ k
We have c a l c u l a t e d the wave functions and the c o r r e s p o n d i n g p h a s e shifts - s e e fig. 1 and fig. 2. We notice that the r e s o n a n c e v a n i s h e s a f t e r the m o d i f i cat i o n , and that the effect of the m o d i f i c a tion is l o c a l i z e d n e a r the r e s o n a n c e . We have a l s o found that t h e s e f e a t u r e s a r e not s e n s i t i v e to cut-off r a d i u s c h o s e n f o r the function ~R" It is w o r t h w h i l e to see the p a r t i c u l a r r o l e of the s i n g l e - p a r t i c l e r e s o n a n c e in our f o r m u l a t i o n . We have extended the u su al definition of the Q s p a c e to a c c o m m o d a t e a s i n g l e - p a r t i c l e r e s o n ance in the continuum and t r e a t it e x a c t l y a s a bound state. H o w e v e r this s t a t e , when coupled to the newly defined P s p a c e , g i v e s r i s e to the single p a r t i c l e r e s o n a n c e . F i n a l l y we note that the m o d i f i c a t i o n i n t r o d u c e d by the o p e r a t o r R is just a r e d e f i n i t i o n of r e a c t i o n s p a c e s r a t h e r than a change in p h y s i c a l situation. The c a l c u l a t i o n c l e a r l y shows that this o r t h o g o n a l i z a t i o n method can be v e r y c o n v e n i e n t l y inc o r p o r a t e d with the unified t h e o r y . An a n a l y s i s employing these techniques for nucleon-nucleus s c a t t e r i n g c r o s s s e c t i o n s is b ei n g i n v e s t i g a t e d and w i l l be r e p o r t e d l a t e r . 423
Volume 32B, n u m b e r 6
PHYSICS
References [1] c . Mahaux and H. A. Weidenmilller, Shell-model a p p r o a c h to n u c l e a r reactions (North-Holland A m s t e r dam, 1969). [2] W. Gl~cke, J. H£Ifner and H. A. Weidenm~/ller, Nucl. Phys. A90 (1967) 481. [3] S. Weinberg, Phys. Rev. 130 (1963) 776. [4] W. J. Romo, Nucl. Phys. Al16 (1968) 618.
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17 August 1970
[5] R. C. F u l l e r , P r e p r i n t , School of Physics and Astronomy, University of Minnesota (1969). [6] H. F e s h b a c h , Ann. Phys. 5 (1958) 357; 19 (1962) 287. [7] A. K. K e r m a n , L e c t u r e s in t h e o r e t i c a l physics VIH-C, University of Colorado P r e s s (1966). [8] N. Auerbach, J. H(lfner, A. K. K e r m a n and C. M. Shakin, to be submitted for publication. [9] A. F. R. De Toledo Piza, private communication.