Single particle strength in the continuum

Volume 32B, number 4

SINGLE

PHYSICS L E T T E R S

PARTICLE

STRENGTH

6July 1970

IN T H E

CONTINUUM

B. H. J. McKELLAR

Department of Theoretical Physics, Universily of Sydney, Sydney, N.S.W., Australia Received 24 April 1970

The coupling to inelastic channels is shown to transfer about 10~ of the single particle strength to inelastic continuum states, where it is difficult to observe it directly. However it is pointed out that such a depletion of the single particle strength is consistent with recent analyses of stripping reactions on doubly magic nuclei.

In the a n a l y s i s of d i r e c t r e a c t i o n s on a target A it is often demanded that the sum r u l e of Macf a r l a n e and F r e n c h [1] (in our n o r m a l i s a t i o n ) Total single p a r t i c l e strength in A + 1

(1)

+ Total single p a r t i c l e strength in A - 1 = 1 should be s a t u r a t e d by the bound s t a t e s in the A + 1 and A - 1 s y s t e m s (or at l e a s t well defined r e s o n a n c e s ) . It has been noted p r e v i o u s l y that continuum states can contribute to the sum r u l e , but such c o n t r i b u t i o n s were r e g a r d e d as n e g l i gible [2]. The purpose of this note is to point out that, because of the coupling of the ground state channel to the excited state channels, the continuum states should be expected to take up about 10% of the single p a r t i c l e strength. To have an explicit example in mind, consider the r e a c t i o n A(d,p)B. The states of A will be r e p r e s e n t e d by the state v e c t o r s [A;m> and the s t a t e s of B by the state v e c t o r s IB;m 5. In each case m = 0 will r e p r e s e n t the ground state. The probability that a state IB:m } is a n e u t r o n in the single p a r t i c l e state (ba added to the ground state of A i s called the strength of the single p a r t i c l e state ~a in the state IB;m 5, or the s p e c troscopic factor Sam. This n o r m a l i s a t i o n of Sam differs from that of Macfarlane and F r e n c h [1], in that Sam is always l e s s than 1. It is to be e m p h a s i s e d that Sam depends c r u c i ally on the choice of the single p a r t i c l e wave function ~a" In p r i n c i p l e , there is a choice which m a x i m i s e s S ~ for a given m, giving an i n t r i n s i c s p e c t r o s c o p i c factor. (In p r a c t i c e o p t i m i s i n g the choice of the single p a r t i c l e wave function is a topic of c u r r e n t i n t e r e s t [3]. ) Since a n t i s y m m e t r i s a t i o n of the wave functions i n t r o d u c e s comp l i c a t i o n s which a r e i n e s s e n t i a l to our a r g u m e n t , we treat the added n e u t r o n as d i s t i n g u i s h a b l e and write 246

<~,rnlB;k>

= ~
(2)

where } a r e the core co-ordinates and r n the neutron co-ordinates. k 2 Sa = l(qhalk;0) i (3) so if we choose (rn i qha5 =

1

I
= ( r n IF5 we obtain the i n t r i n s i c spectroscopic factor of the state k = 0 S°

= <0;0 !0;05

(5)

This i n t r i n s i c spectroscopic factor suffers depletion into continuum states. To show this notice the spectroscopic factor sum rule (i), which may be derived in this case by i n s e r t i n g the complete set of s t a t e s iB;k5 in the equation ((A;01(Fi) (IFhiA;0>) : 1

(6)

takes the form

with

S~ = i((A;01(Fi) IB;k>12

(8)

(notice that S O is c o r r e c t l y given by eq. (5)). E n u m e r a t i n g the states IB;k>, t h e r e a r e (i) bound s t a t e s IB;k>, k = 0, 1,2 . . . .

, and

(ii) continuum states IB;m,k>, which a r e formed by n e u t r o n s of m o m e n t u m k incident on the n u c l e u s A in the state IA;m>. To simplify the d i s c u s s i o n a s s u m e there is just one bound state I B;0> of the n u c l e u s B, and that the nucleus A has just two states t A;05 and IA;I>. The n e u t r o n - A potential contains t e r m s

Volume 32B.

number

PHYSICS

4

w h i c h c o u p l e t h e g r o u n d s t a t e c h a n n e l to t h e e x cited state channel. Let the parameter p, which we a s s u m e to be s m a l l , c h a r a c t e r i s e t h e s t r e n g t h of t h i s c o u p l i n g . T h e n o n e c a n r e a d i l y c l a s s i f y the single neutron wave functions (rn] k;rn) according to their order of magnitude with regard to p. This classification is presented in table 1. In this way we obtain the order of magnitude of the various contributions to the sum rule (7) s°

= <0;0 i0:0> : I - <0;1i 0;1>

= i

2)

s}. k = b12 ~ Il 2 Since
sO, k : k

:

(9)

= 0 = (0;0 { 0 , k ; 0 ) + ( 0 ; l l 0 , k ; 1 )

! ( 0 : 1 t 0 , k;1>12

(10)

= O ( u 4)

li2 i<0;011,k;0) 12 :

0(#) 2

(11) T h u s we s e e t h a t m o s t of t h e s i n g l e p a r t i c l e s t r e n g t h i s t r a n s f e r r e d to t h e c o n t i n u u m s t a t e i B ; 1 , k > w i t h a n e u t r o n i n c i d e n t on t h e e x c i t e d s t a t e of t h e c o r e *. T h e g e n e r a l a r g u m e n t i s n o t i n v a l i d a t e d by t h e p r e s e n c e of a d d i t i o n a l b o u n d s t a t e s of B o r a d d i t i o n a l e x c i t e d s t a t e s of A. In t h e s p e c i a l c a s e t h a t t h e n e u t r o n - A p o t e n t i a l i s s e p a r a b l e o n e c a n s o l v e t h e two c h a n n e l problem in closed form and verify that the integ r a t i o n of sO 'k a n d S 1,k o v e r k d o e s not a l t e r t h e o r d e r of m a g n i t u d e of t h e s e t e r m s . In t h i s m o d e l , w h i c h we w i l l d i s c u s s in d e t a i l e l s e w h e r e , one obtains the result that

1 -{S ° + f d k s l , k}

: O(~4),

(12)

showing that the sum rule is essentially saturated by the bound state and the excited channel continuum state. The physical importance of this effect depends on the size of ~. One can estimate /~ crudely from the imaginary part of the optical potential, which in the Feshbach formal±sat±on [5] is given by ImVop t

= -VUol }l,k;l>(1,k;llUlo

(13)

w h e r e UIO i s t h e c o u p l i n g p o t e n t i a l b e t w e e n t h e channels. Thus Im

Vopt/Re

Vop t = O ( p 2 ) ,

(14)

* It is i n t e r e s t i n g to note that these e s t i m a t e s show that both the n u m e r a t o r and denominator of the f r a c tion (0;0 [ O . k ; O ) / ( 1 - S 0) appearing in the BHMM theory a r e of o r d e r #~, s~howing that the BHMM theory has a finite limit as sO --'1 [4].

6 July 1970

LETTERS

Table 1 Classification of neutron wave funeLions (rn !k;m> by order of magnitude in ~z. o(1)

O(/~)



<"n!°:l>







Table 2 Spectroscopic factors from stripping reaction on doubly magic nuclei. Final state

S DWBA

BHMM

170 l d 5 / 2

0.9

e0.1

[1O]

0.45 ±0.10 [11]

41Ca l f 7 / 2

0.85 i-0.1

[121

0.60 ± 0.10 [13]

49Ca 2P3/2

1.0

89Sr 2d5/2

-

91Zr 2d5/2 209pb

[14]

0.7 ~ 0.1 [15] 0.80 ± 0.15 [16]

0.78

[13]

0.70 :~ 0.10 [11] 0.65 ~0.10 [11] 0.65 ± 0.10 [17]

suggesting that about 10% of the single particle strength will be transferred to continuum states. Direct observation of this continuum strength would be an extremely difficult experimental problem. One possibility is through a direct (),n) reaction which leaves the residual nucleus in an excited state. The cross section may be estimated by the technique of Breit and Yost [6] to be of the order of 1 mb near threshold, where it may be possible to disentangle the direct process from the competing compound nuclear processes. Indirect evidence for the existence of single particle strength in the continuum may be drawn from the failure to saturate the MacfarlaneFrench sum rule with the bound states. Here one encounters the important question of the reliability with which one can extract spectroscopic factors from the experimental data on stripping r e actions. In table 2 we present the spectroscopic factors extracted by the DWBA and the BHMM theories from the stripping data on doubly magic nuclei. The BHMM values in particular indicate that there is a significant reduction in the spectroscopic factor from the naive value of unity. Three mechanisms have been proposed which can deplete the strength. (a) The hard core correlation of the Brueckner theory, which have been estimated by Brandow [7] to give a depletion of 15% to 20%. (b) The coupling to vibrational states in the core. Bertsch and Kuo [8] find that this gives a depletion of 15% to 25%. (c) The continuum states, which according to the estimate of the present paper take up about 10% of the single particle strength. 247

Volume 32B, number 4

PHYSICS

If t h e s e e f f e c t s c a n be a d d e d w i t h o u t d o u b l e counting, a point which r e q u i r e s f u r t h e r i n v e s t i g a t i o n , t h e n one w o u l d e x p e c t s p e c t r o s c o p i c f a c t o r s f o r s t r i p p i n g on doubly m a g i c n u c l e i to be in t h e r a n g e 0.45 - 0.6, and to v a r y s i g n i f i c a n t l y f r o m n u c l e u s to n u c l e u s . E v e n if w e a s s u m e t h a t a l l of t h e h a r d c o r e c o r r e l a t i o n h a v e a l r e a d y b e e n t a k e n into a c c o u n t by B e r t s c h and Kuo who u s e d t h e G m a t r i c e s of Kuo a n d B r o w n . it i s d i f f i c u l t to s e e how the s p e c t r o s c o p i c f a c t o r c o u l d e x c e e d 0.75 f o r t h e s t a t e s c o n s i d e r e d . A s h a s r e c e n t l y b e e n e m p h a s i s e d by B r o w n [9] t h e DWBA s p e c t r o s c o p i c f a c t o r s a r e c e r t a i n l y too l a r g e . H o w e v e r t h e BHMM v a l u e s a r e c o n s i s tent with t h e s e e s t i m a t e s , p r o v i d i n g c o n f i r m a t i o n of both the e s t i m a t e s and the BHMM t h e o r y . To c o n c l u d e we e m p h a s i s e t h a t c o n t i n u u m s t a t e s m u s t be c o n s i d e r e d when d i s c u s s i n g the s a t u r a t i o n of the M a c f a r l a n e - F r e n c h s u m r u l e s a n d h o p i n g f o r b e t t e r t h a n 10% a c c u r a c y . T h i s w o r k w a s s u p p o r t e d in p a r t by t h e S c i e n c e F o u n d a t i o n w i t h i n the U n i v e r s i t y of Sydney. It i s a p l e a s u r e to a c k n o w l e d g e t h e i n t e r e s t of i t s D i r e c t o r , P r o f e s s o r H. M e s s e l .

248

LETTERS

6 J u l y 1970

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