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Further remarks on the extended Siegert theorem
Volume 24B, n u m b e r 3
PHYSICS
LETTERS
6 F e b r u a r y 1967
7. G.J.Stephenson and J.B.Marion, in Isobaric spin in nuclear physics eds. Fox and Robson. (Academic P r e s s New York and London 1966), p. 766. 8. Y.C.Tang, E.Sehmid and K.Wildermuth. Phys. I{ev. 131 (1963) 2631.
5. B . J . D a l t o n a n d D.Robson, Phys. Letters 20 (1966) 405. 6. J . B . M a r i o n , Phys. Letters 14 (1965) 315.
* * ~ * *
FURTHER
REMARKS
ON
THE
EXTENDED
SIEGERT
THEOREM
J, I. F U J I T A
Institute f or Nuclear Study, University of Tokyo, Tanashi-Machi, Tokyo Received 6 December 1966
T h e validity of the s o - c a l l e d A h r e n s - F e e n b e r g a p p r o x i m a t i o n is r e i n v e s t i g a t e d , and e x p e r i m e n t a l t e s t s a r e
proposed.
R e c e n t l y the v a l i d i t y of the A h r e n s - F e e n b e r g a p p r o x i m a t i o n [1] was d i s c u s s e d [2], a n d s o m e doubt was c a s t on the o r i g i n a l e s t i m a t e [1] of 10% for the a c c u r a c y . T h e p u r p o s e of t h i s s h o r t note is to r e i n v e s t i g a t e i t s v a l i d i t y f r o m a s o m e w h a t d i f f e r e n t v i e w p o i n t , s i n c e the A h r e n s - F e e n b e r g a p p r o x i m a t i o n is quite u s e f u l in t r e a t i n g b e t a [3,4] or m u o n c a p t u r e [5] m a t r i x e l e m e n t s , a l though it i s not e s s e n t i a l . F i r s t , let u s c o n s i d e r the a s s u m p t i o n s un the b a s i s of which the e x t e n d e d S i e g e r t t h e o r e m can be d e r i v e d . A) T h e c o n s e r v e d v e c t o r c u r r e n t t h e o r y for n u c l e i is g i v e n by [ A s s u m p t i o n I]
j(V p ,)¥
cc [J(Y) ' T$] ,
(1)
which l e a d s to the r e l a t i o n [3,5] of p a r t i a l c o n s e r v a t i o n of the weak v e c t o r c u r r e n t j_(V) , + /~j(V) ~,$
= i[H', j(V) ]. 0,;
(2)
In (2), H ' is the c h a r g e d e p e n d e n t p a r t of the Hamiltonian. [ A s s u m p t i o n II] F o r s i m p l i c i t y , H ' is a s s u m e d to be the s u m of the C o u l o m b p o t e n t i a l s V c and the n e u t r o n - p r o t o n m a s s d i f f e r e n c e . ( A s s u m p t i o n I i s e x p e c t e d to be v a l i d in the l o w e s t o r d e r of e; if the h i g h e r - o r d e r r a d i a t i v e c o r r e c t i o n s a r e c o n s i d e r e d , the i s o t e n s o r p a r t , for i n s t a n c e , m a y a p p e a r in J/x, ¥(V) ). B) F r o m ( 2 ) w e c a n d e r i v e v a r i o u s u s e f u l r e l a tions between forbidden nuclear matrix elements. F o r the f i r s t - f o r b i d d e n t r a n s i t i o n s we o b t a i n [3]
( a ) C V C = - ( W o X 2.5 m e ) ( i r )
-([Vc, it])
(3)
a s an a l m o s t d i r e c t c o n s e q u e n c e of the c o n s e r v e d v e c t o r c u r r e n t t h e o r y . It s h o u l d be n o t e d that, in the n o n - r e l a t i v i s t i c a p p r o x i m a t i o n , we have [6] (a)CVC = (p/M)+ ([ V, ir]} = (p/M*}
(4)
i n s t e a d of ( a ) = (p/M). In ref. 6, the m a g n i t u d e of ([V, i r ] ) was shown to be g e n e r a l l y 40% or m o r e of
(p/M ). C) Next, let u s i n t r o d u c e knowledge of the i s o b a r i c s t a t e s [7], which w e r e e x p e r i m e n t a l l y d i s c o v e r e d [8] for h e a v i e r n u c l e i in (p, n) r e a c t i o n s . Let u s c o n s i d e r the e l e c t r o m a g n e t i c E1 t r a n s t i o n f r o m T_[ i ) t o [ f ) : ( f i l l (y) T _ [ i ) = i ( E T _ i - E l ) 2 ( f l . ~ g
T 3 j zj T _ l i ) , ( 5 ) *
a c c o r d i n g to the u s u a l S i e g e r t t h e o r e m . F r o m (5) and (1), we c a n e a s i l y p r o v e that, for the c o r r e s p o n d i n g b e t a t r a n s i t i o n f r o m I i) to ] f ) ;
(fIH(_V) l i ) : i ( E r _ i _ E f ) g(V){(f 2 T_jZj[ i ) + 5 } , (6) w h e r e 5 = ( f [ T _ 2 "r3jzjli}. If the c o r r e c t i o n due to 5, is n e g l e c t e d , we o b t a i n a c o n v e n i e n t r e l a * In fact the isobaric state T_[ i} is only an approximate eigenstate of the Hamiltonian: T li) = N c s q)s Therefore, ET_ i in (5) should be understood to be an average value of the energies q)s123
Volume24B, number 3
PHYSICS
"f".l 4;>
LETTERS
It is c l e a r t h a t t h e r i g h t - h a n d s i d e b e c o m e s (F~ Z/R ) (fl~ "r_jzj It) and A h r e n s - F e e n b e r g - a p p r o x i m a t i o n i s v a l i d * in the l i m i t w h e r e a l l the F e r m i s t r e n g t h s a r e c o n c e n t r a t e d at the t r a n s i t i o n s to the i s o b a r i c s t a t e s : ( n i t It)-- 0 u n l e s s
In) o: T_li};(fJT_ Ira) = 0
Y(E1)
/ (~/,
d
z-O
%.z
?J
,I
IZ~>
<
It>
Fig. 1. tion f o r ( a ) C V C / ( i t - ) by r e p l a c i n g ( [Vc, it'] ) in (3) by F~Z/R(ir), the e x p e r i m e n t a l v a l u e of F b e i n g about 1.1 f o r r o = 1.2 [8]. It is e a s i l y n o t i c e d that, if t h e i s o s p i n T ~ T z = T o i s and a p p r o x i m a t e l y good q u a n t u m n u m b e r [9], T+I f ) ~ 0 and 5 ~ 0. F u r t h e r m o r e , e v e n if the i s o s p i n a d m i x t u r e i s s i g n i f i c a n t [7], 5 s h o u l d be quite s m a l l in so f a r a s the F e r m i s t r e n g t h i s m o s t l y c o n c e n t r a t e d at the t r a n s i t i o n f r o m Ira) to T_ Ira). T h e r e f o r e , we f e e l that it i s a d i f f i c u l t p r o b l e m to e s t i m a t e the deviation from Ahrens-Feenberg approximation with good a c c u r a c y , s i n c e we n e e d a r e a s o n a b l e n u c l e a r m o d e l at l e a s t with p r o p e r h i g h e r - i s o spin admixture. In the a b o v e d e r i v a t i o n we c a t i o u s l y a v o i d e d m a k i n g u s e of the A h r e n s - F e e n b e r g a p p r o x i m a tion. H o w e v e r , the v a l i d i t y of A h r e n s - F e e n b e r g a p p r o x i m a t i o n i s c l o s e l y r e l a t e d to the e x i s t e n c e of the w e l l - d e f i n e d i s o b a r i c s t a t e s :
6 February 1967
unlessJi)o~
T i m ).
P r e s e n t e x p e r i m e n t a l k n o w l e d g e of the F e r m i t r a n s i t i o n s s u g g e s t s that s u c h an a p p r o x i m a t i o n i s q u i t e r e a s o n a b l e [10]. In t h i s c o n n e c t i o n it is v e r y i n t e r e s t i n g to c a r r y out t e s t e x p e r i m e n t s . T h o u g h s u c h an e x p e r i m e n t [11] l o o k e d h a r d l y p r a c t i c a l in 1963, r e c e n t l y M. S a k a i p o i n t e d out [12] that it i s f e a s i b l e if we u s e c o m p o u n d i s o b a r i c s t a t e s . T h e p r i n c i p l e of s u c h an e x p e r i m e n t is shown in fig. 1; t h e c o r r e s p o n d i n g b e t a and g a m m a E1 t r a n s i t i o n m a t r i x e l e m e n t s a r e to be c o m p a r e d . T h e a u t h o r would l i k e to thank P r o f e s s o r J . P. D e u t s c h f o r c a l l i n g h i s a t t e n t i o n to t h i s p r o b l e m . He a l s o w i s h e s to t h a n k P r o f e s s o r s H. P r i m a k o f f , M. Sakai and Y. Y a m a g u c h i f o r v a l u a b l e d i s c u s s i o n s , and P r o f e s s o r D. C . W o r t h f o r r e a d i n g t h i s manuscript. 1. T . A h r e n s a n d E . F e e n b e r g , Phys. Rev. 86 (1952) 64. 2. J.Damgaard andA.Winther, Phys. Letters 23 (1966) 345. 3. J . I . Fujita, Phys. Rev. 126 (1962)202; Prog. Theor. Phys. 28 (1962) 338; J . E i c h l e r , Z. Phys. 171 (1963)463. 4. J . I . F u j i t a and K. Ikeda, Nucl. Phys. 67 (1965) 145. 5. A.Ft~jii, J . I . F u j i t a a n d M . M o r i t a , Prog. Theor. Phys. 32 (1964) 438. 6. Y. F n j i i a n d J . I . F u j i t a , Phys. Rev. 140 (1965)B239. 7. A . M . L a n e a n d J . M . S o p e r , Phys. Letters 1 (1962) 33; Nucl. Phys. 37 (1962) 506; Phys. Rev. Letters 7 (1961) 420; Phys. Letters 1 (1962) 28. 8. J . D . A n d e r s o n , C.Wongand McClure, Phys. Rev. 126 (1962) 2170; Phys. Rev. 129 (1963) 2718. 9. L . A . S l i v and Yu.I.Kharitonov, Phys. Letters 16 (1965) 176. 10. S.D.Bloom, Nuovo Cimento 32 (1964) 1023; H.Daniel and H.Schmidt, Nucl. Phys. 65 (1965)481; S. K. Bhattaeherjee, S.K. Mitra and H. C. Padhi, Nucl. Phys. 72 (1965) 145; J . I . F u j i t a and K.Ikeda, Prog. Theor. Phys. 36 (1966) 530. 11. J . I . F u j i t a , Brookhaven National Laboratory Report No. BNL 837 C-39, 1963, p. 340. 12. M.Sakai, to be published.
: ½~rl[~ ~-~zj, [Vc, T_]] li>=
=~[~ (E~ -S~i +2.S me)trl~ ~ j
ln> +
n
-
~ ( E f - E m +2.5me)< / IT_ Ira) ( m l ~ ~-3jzjli)].(7) m
124
* It should be r e m a r k e d that, rigorously speaking, we are discussing a modified A h r e n s - F e e b e r g approximation ~e [ Vc rn] It) ~ Ac{flm It) where A c = =
PHYSICS
LETTERS
6 F e b r u a r y 1967
7. G.J.Stephenson and J.B.Marion, in Isobaric spin in nuclear physics eds. Fox and Robson. (Academic P r e s s New York and London 1966), p. 766. 8. Y.C.Tang, E.Sehmid and K.Wildermuth. Phys. I{ev. 131 (1963) 2631.
5. B . J . D a l t o n a n d D.Robson, Phys. Letters 20 (1966) 405. 6. J . B . M a r i o n , Phys. Letters 14 (1965) 315.
* * ~ * *
FURTHER
REMARKS
ON
THE
EXTENDED
SIEGERT
THEOREM
J, I. F U J I T A
Institute f or Nuclear Study, University of Tokyo, Tanashi-Machi, Tokyo Received 6 December 1966
T h e validity of the s o - c a l l e d A h r e n s - F e e n b e r g a p p r o x i m a t i o n is r e i n v e s t i g a t e d , and e x p e r i m e n t a l t e s t s a r e
proposed.
R e c e n t l y the v a l i d i t y of the A h r e n s - F e e n b e r g a p p r o x i m a t i o n [1] was d i s c u s s e d [2], a n d s o m e doubt was c a s t on the o r i g i n a l e s t i m a t e [1] of 10% for the a c c u r a c y . T h e p u r p o s e of t h i s s h o r t note is to r e i n v e s t i g a t e i t s v a l i d i t y f r o m a s o m e w h a t d i f f e r e n t v i e w p o i n t , s i n c e the A h r e n s - F e e n b e r g a p p r o x i m a t i o n is quite u s e f u l in t r e a t i n g b e t a [3,4] or m u o n c a p t u r e [5] m a t r i x e l e m e n t s , a l though it i s not e s s e n t i a l . F i r s t , let u s c o n s i d e r the a s s u m p t i o n s un the b a s i s of which the e x t e n d e d S i e g e r t t h e o r e m can be d e r i v e d . A) T h e c o n s e r v e d v e c t o r c u r r e n t t h e o r y for n u c l e i is g i v e n by [ A s s u m p t i o n I]
j(V p ,)¥
cc [J(Y) ' T$] ,
(1)
which l e a d s to the r e l a t i o n [3,5] of p a r t i a l c o n s e r v a t i o n of the weak v e c t o r c u r r e n t j_(V) , + /~j(V) ~,$
= i[H', j(V) ]. 0,;
(2)
In (2), H ' is the c h a r g e d e p e n d e n t p a r t of the Hamiltonian. [ A s s u m p t i o n II] F o r s i m p l i c i t y , H ' is a s s u m e d to be the s u m of the C o u l o m b p o t e n t i a l s V c and the n e u t r o n - p r o t o n m a s s d i f f e r e n c e . ( A s s u m p t i o n I i s e x p e c t e d to be v a l i d in the l o w e s t o r d e r of e; if the h i g h e r - o r d e r r a d i a t i v e c o r r e c t i o n s a r e c o n s i d e r e d , the i s o t e n s o r p a r t , for i n s t a n c e , m a y a p p e a r in J/x, ¥(V) ). B) F r o m ( 2 ) w e c a n d e r i v e v a r i o u s u s e f u l r e l a tions between forbidden nuclear matrix elements. F o r the f i r s t - f o r b i d d e n t r a n s i t i o n s we o b t a i n [3]
( a ) C V C = - ( W o X 2.5 m e ) ( i r )
-([Vc, it])
(3)
a s an a l m o s t d i r e c t c o n s e q u e n c e of the c o n s e r v e d v e c t o r c u r r e n t t h e o r y . It s h o u l d be n o t e d that, in the n o n - r e l a t i v i s t i c a p p r o x i m a t i o n , we have [6] (a)CVC = (p/M)+ ([ V, ir]} = (p/M*}
(4)
i n s t e a d of ( a ) = (p/M). In ref. 6, the m a g n i t u d e of ([V, i r ] ) was shown to be g e n e r a l l y 40% or m o r e of
(p/M ). C) Next, let u s i n t r o d u c e knowledge of the i s o b a r i c s t a t e s [7], which w e r e e x p e r i m e n t a l l y d i s c o v e r e d [8] for h e a v i e r n u c l e i in (p, n) r e a c t i o n s . Let u s c o n s i d e r the e l e c t r o m a g n e t i c E1 t r a n s t i o n f r o m T_[ i ) t o [ f ) : ( f i l l (y) T _ [ i ) = i ( E T _ i - E l ) 2 ( f l . ~ g
T 3 j zj T _ l i ) , ( 5 ) *
a c c o r d i n g to the u s u a l S i e g e r t t h e o r e m . F r o m (5) and (1), we c a n e a s i l y p r o v e that, for the c o r r e s p o n d i n g b e t a t r a n s i t i o n f r o m I i) to ] f ) ;
(fIH(_V) l i ) : i ( E r _ i _ E f ) g(V){(f 2 T_jZj[ i ) + 5 } , (6) w h e r e 5 = ( f [ T _ 2 "r3jzjli}. If the c o r r e c t i o n due to 5, is n e g l e c t e d , we o b t a i n a c o n v e n i e n t r e l a * In fact the isobaric state T_[ i} is only an approximate eigenstate of the Hamiltonian: T li) = N c s q)s Therefore, ET_ i in (5) should be understood to be an average value of the energies q)s123
Volume24B, number 3
PHYSICS
"f".l 4;>
LETTERS
It is c l e a r t h a t t h e r i g h t - h a n d s i d e b e c o m e s (F~ Z/R ) (fl~ "r_jzj It) and A h r e n s - F e e n b e r g - a p p r o x i m a t i o n i s v a l i d * in the l i m i t w h e r e a l l the F e r m i s t r e n g t h s a r e c o n c e n t r a t e d at the t r a n s i t i o n s to the i s o b a r i c s t a t e s : ( n i t It)-- 0 u n l e s s
In) o: T_li};(fJT_ Ira) = 0
Y(E1)
/ (~/,
d
z-O
%.z
?J
,I
IZ~>
<
It>
Fig. 1. tion f o r ( a ) C V C / ( i t - ) by r e p l a c i n g ( [Vc, it'] ) in (3) by F~Z/R(ir), the e x p e r i m e n t a l v a l u e of F b e i n g about 1.1 f o r r o = 1.2 [8]. It is e a s i l y n o t i c e d that, if t h e i s o s p i n T ~ T z = T o i s and a p p r o x i m a t e l y good q u a n t u m n u m b e r [9], T+I f ) ~ 0 and 5 ~ 0. F u r t h e r m o r e , e v e n if the i s o s p i n a d m i x t u r e i s s i g n i f i c a n t [7], 5 s h o u l d be quite s m a l l in so f a r a s the F e r m i s t r e n g t h i s m o s t l y c o n c e n t r a t e d at the t r a n s i t i o n f r o m Ira) to T_ Ira). T h e r e f o r e , we f e e l that it i s a d i f f i c u l t p r o b l e m to e s t i m a t e the deviation from Ahrens-Feenberg approximation with good a c c u r a c y , s i n c e we n e e d a r e a s o n a b l e n u c l e a r m o d e l at l e a s t with p r o p e r h i g h e r - i s o spin admixture. In the a b o v e d e r i v a t i o n we c a t i o u s l y a v o i d e d m a k i n g u s e of the A h r e n s - F e e n b e r g a p p r o x i m a tion. H o w e v e r , the v a l i d i t y of A h r e n s - F e e n b e r g a p p r o x i m a t i o n i s c l o s e l y r e l a t e d to the e x i s t e n c e of the w e l l - d e f i n e d i s o b a r i c s t a t e s :
6 February 1967
unlessJi)o~
T i m ).
P r e s e n t e x p e r i m e n t a l k n o w l e d g e of the F e r m i t r a n s i t i o n s s u g g e s t s that s u c h an a p p r o x i m a t i o n i s q u i t e r e a s o n a b l e [10]. In t h i s c o n n e c t i o n it is v e r y i n t e r e s t i n g to c a r r y out t e s t e x p e r i m e n t s . T h o u g h s u c h an e x p e r i m e n t [11] l o o k e d h a r d l y p r a c t i c a l in 1963, r e c e n t l y M. S a k a i p o i n t e d out [12] that it i s f e a s i b l e if we u s e c o m p o u n d i s o b a r i c s t a t e s . T h e p r i n c i p l e of s u c h an e x p e r i m e n t is shown in fig. 1; t h e c o r r e s p o n d i n g b e t a and g a m m a E1 t r a n s i t i o n m a t r i x e l e m e n t s a r e to be c o m p a r e d . T h e a u t h o r would l i k e to thank P r o f e s s o r J . P. D e u t s c h f o r c a l l i n g h i s a t t e n t i o n to t h i s p r o b l e m . He a l s o w i s h e s to t h a n k P r o f e s s o r s H. P r i m a k o f f , M. Sakai and Y. Y a m a g u c h i f o r v a l u a b l e d i s c u s s i o n s , and P r o f e s s o r D. C . W o r t h f o r r e a d i n g t h i s manuscript. 1. T . A h r e n s a n d E . F e e n b e r g , Phys. Rev. 86 (1952) 64. 2. J.Damgaard andA.Winther, Phys. Letters 23 (1966) 345. 3. J . I . Fujita, Phys. Rev. 126 (1962)202; Prog. Theor. Phys. 28 (1962) 338; J . E i c h l e r , Z. Phys. 171 (1963)463. 4. J . I . F u j i t a and K. Ikeda, Nucl. Phys. 67 (1965) 145. 5. A.Ft~jii, J . I . F u j i t a a n d M . M o r i t a , Prog. Theor. Phys. 32 (1964) 438. 6. Y. F n j i i a n d J . I . F u j i t a , Phys. Rev. 140 (1965)B239. 7. A . M . L a n e a n d J . M . S o p e r , Phys. Letters 1 (1962) 33; Nucl. Phys. 37 (1962) 506; Phys. Rev. Letters 7 (1961) 420; Phys. Letters 1 (1962) 28. 8. J . D . A n d e r s o n , C.Wongand McClure, Phys. Rev. 126 (1962) 2170; Phys. Rev. 129 (1963) 2718. 9. L . A . S l i v and Yu.I.Kharitonov, Phys. Letters 16 (1965) 176. 10. S.D.Bloom, Nuovo Cimento 32 (1964) 1023; H.Daniel and H.Schmidt, Nucl. Phys. 65 (1965)481; S. K. Bhattaeherjee, S.K. Mitra and H. C. Padhi, Nucl. Phys. 72 (1965) 145; J . I . F u j i t a and K.Ikeda, Prog. Theor. Phys. 36 (1966) 530. 11. J . I . F u j i t a , Brookhaven National Laboratory Report No. BNL 837 C-39, 1963, p. 340. 12. M.Sakai, to be published.
=~[~ (E~ -S~i +2.S me)trl~ ~ j
ln>
n
-
~ ( E f - E m +2.5me)< / IT_ Ira) ( m l ~ ~-3jzjli)].(7) m
124
* It should be r e m a r k e d that, rigorously speaking, we are discussing a modified A h r e n s - F e e b e r g approximation ~e [ Vc rn] It) ~ Ac{flm It) where A c = =