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Quadrupole moment and effective E2 transition moments in 51V
Volume 25B, n u m b e r 5
PHYSICS
LETTERS
18 S e p t e m b e r 1967
W e w o u l d l i k e to t h a n k P r o f e s s o r s J . D{}browski a n d J . P n i e w s k i a n d D r . J . Z a k r z e w s k i f o r t h e i r k i n d interest and discussion. We thank also Dr. J. Zakrzewski for supplying us with the experimental data.
References 1. N. B y e r s and W. N. Cottingham. Nuovo Cimento 11 (1959) 554. 2. Y . C . T a n g . Nuovo Cimento 10 (1958) 780. 3. V . A . L y u l ' k a , Zh. Exp. i T e o r . Fiz. 42 (1962) 1692. 4. J . S z y m a f s k i . Nuovo Cimento 10 (1958) 830. 5. D.Chlebowska and J.Szymafiski, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 7 (1959) 643. 6. R . H . D a l i t z , Nucl. Phys. 41 (1963) 78. 7. J. Zakrzewski, private communication. 8. K.S.Suh. Phys. Rev. 111 {1958) 941. 9. J.Sacton. K - C o l l a b o r a t i o n , p r e p r i n t (1965).
QUADRUPOLE
MOMENT AND MOMENTS
EFFECTIVE IN 51V*
E2
TRANSITION
I. TALMI** Palmer Physical Laboratory. Princeton University. Princeton, New Jersey. USA Received 14 August 1967
Recently m e a s u r e d E2 r a t e s for Coulomb excitation from the ground state of 51V to excited s t a t e s a r e consistent with the use of effective single particle E2 o p e r a t o r s and f3 wave functions. F r o m these o p e r a t o r s the quadrupole m o m e n t of 51V is calculated and found to be in v e r } good a g r e e m e n t with the new e x p e r i m e n tal (negative) value. The enhancement of the single proton E2 o p e r a t o r is equivalent to an effective charge eef f = 1.61e.
T h e e n e r g y l e v e l s of t h e 51V n u c l e u s h a v e b e e n k n o w n f o r m a n y y e a r s to b e w e l l d e s c r i b e d i n t e r m s of t h e l f 3 p r o t o n c o n f i g u r a t i o n [1] (and a c l o s e d lf~ n e u t r o n s h e l l ) . T h e low l y i n g odd p a r i t y l e v e l s h a v e the spin~J=½ (g.s.), j = 5 (0.32MeV), j = 3 (0.93)/J=~ (1.61), j = 9 ( 1 . 8 1 ) a n d p o s s i b l y J = ~ (2.70). T h e s e s t a t e s a r e t h e o n l y o n e s a l l o w e d by t h e P a u l i p r i n c i p l e f o r t h e f3 c o n f i g u r a t i o n of i d e n t i c a l n u c l e o n s . T h e i r e n e r g i e s c a n b e v e r y w e l l o b t a i n e d f r o m t h e f2 m a t r i x e ~ e m e n t s of t h e e f f e c t i v e n u c l e a r Y i n t e r a c t i o n t a k e n f r o m n e l• g h b o u r i n g n u c l e i , e . g . 5 0 T i [1]. L i k e m o s t , if n o t a l l , n u c l e i , t h e E 2 t r a n s i t i o n r a t e s i n 51V a r e m u c h b i g g e r t h a n t h e s i n g l e p r o t o n e s t i m a t e s . T h i s i s n o t a s t r o n g o b j e c t i o n t o t h e f3 a s s i g n m e n t s i n c e t r a n s i t i o n p r o b a b i l i t i e s , u n l i k e energies, are rather sensitive to small admixtures in the wave functions. Moreover, it is clear that the r e a l w a v e f u n c t i o n s , w i t h t h e s t r o n g t w o - b o d y c o r r e l a t i o n s , a r e v e r y d i f f e r e n t f r o m t h e s h e l l m o d e l f3 w a v e f u n c t i o n s . W e c a n c a l c u l a t e n u c l e a r e n e r g i e s in t h e s h e l l m o d e l b e c a u s e we u s e r e n o r m a l i z e d o~ e f f e c t i v e t w o - b o d y i n t e r a c t i o n s . T h e s e a r e s u p p o s e d t o i n c l u d e t h e e f f e c t of t w o p a r t i c l e s h o r t r a n g e correlations. Similarly, it is expected that the electromagnetic moments to be used in the shell model w o u l d a l s o b e r e n o r m a l i z e d i n s o m e way. F o r e x a m p l e , a n i m p o r t a n t c o n t r i b u t i o n t o E 2 o p e r a t o r s w o u l d c o m e f r o m t h e p o l a r i z a t i o n of t h e c o r e b y t h e o u t s i d e n u c l e o n s . It m a y a l s o h a p p e n t h a t t h e c h a n g e d e l e c t r o m a g n e t i c m o m e n t s w i l l h a v e t h e f o r m of r e n o r m a l i z e d s i n g l e n u c l e o n o p e r a t o r s ( o p e r a t o r s w i t h " e f f e c t i v e c h a r g e s " ) . T h i s w i l l b e t h e c a s e if t h e c o r e p o l a r i z a t i o n of t h e i n d i v i d u a l n u c l e o n s a r e i n d e * This work was supported by the U . S . A t o m i c Energy Commission and the Higgins Scientific T r u s t Fund. This work made use of the P r i n c e t o n Computer F a c i l i t i e s supported in part by the National Science Foundation Grant NSF GP-579. ** National Science Foundation Senior F o r e i g n Scientist Fellow. On leave of absence from the Weizmann Institute of Science, Rehovoth, Israel.
313
Volume 25B, number 5
PHYSICS
LETTERS
18 September 1967
Table 1 Experimental and calculated B(E2) transition strengths in 51V (in units of e 2 × 10 -50 cm4). j
_7 2
Reduced matrix element squared
1 ~
£ 2
0.11
5 2
12
3-~ : 0.34
9_ 2
22
15 1.47
26
~5 = 0.41
11
-24
3 - 1.33
B(E2. ½ --*J) experimental [3]
0.27
0.92
0.22
0.90
B(E2, ~ ~ J )
0.22
0.96
0.27
0.86
calculated
p e n d e n t of e a c h o t h e r . If t h i s i s so, it is p o s s i b l e to r e l a t e t r a n s i t i o n p r o b a b i l i t i e s within the s a m e c o n f i g u r a t i o n by u s i n g s h e l l m o d e l w a v e f u n c t i o n s and e f f e c t i v e s i n g l e p a r t i c l e e l e c t r o m a g n e t i c m o m e n t s . Such a c a l c u l a t i o n y i e l d s c o n d i t i o n s that should be s a t i s f i e d by the o b s e r v e d t r a n s i t i o n strengths. The E2 m o m e n t s of the t r a n s i t i o n s f r o m the g r o u n d s t a t e to the e x c i t e d s t a t e s of 51V w e r e c o n s i d e r e d in t h i s a p p r o a c h [2]. U n f o r t u n a t e l y , the v a l u e s of the t r a n s i t i o n m o m e n t s d e t e r m i n e d by e l e c t r o n s c a t t e r i n g w e r e not v e r y a c c u r a t e [2] and it s e e m e d i m p o s s i b l e to r e l a t e t h e s e t r a n s i t i o n s to the f3 w a v e f u n c t i o n s . H o w e v e r , a c c u r a t e m e a s u r e m e n t s of the E2 t r a n s i t i o n m o m e n t s w e r e r e c e n t l y earr~ied out u s i n g C o u l o m b e x c i t a t i o n s of 51V with 12C i o n s [3]. The r e s u l t s , a s n o t e d by L e m b e r g et al. [3], a r e in v e r y good a g r e e m e n t with the c a l c u l a t e d v a l u e s and s u p p o r t the i d e a t h a t e n h a n c e d E 2 t r a n s i t i o n s can fit into the s h e l l m o d e l d e s c r i p t i o n of n u c l e a r s t a t e s . T h e r a t e unit t i m e of the E2 t r a n s i t i o n Ji ~ J f i s d i r e c t l y p r o p o r t i o n a l to [2,4]
consislency
1
B ( E 2 Ji --~Jf) - 2 j i + l
I(Ji I I ~ T~2)It J f )
12
(1)
3 w h e r e the r a n k 2 i r r e d u c i b l e t e n s o r --.i~Jl T!2)" is the e f f e c t i v e s i n g l e p a r t i c l e o p e r a t o r . T h e r e d u c e d m a t r i x e l e m e n t s in eq. (1) can be c a l c u l a t e d f r o m the w a v e f u n c t i o n s . U s i n g c o e f f i c i e n t s of f r a c t i o n a l p a r e n t a g e we obtain [2,4]
j3(j=½)
(J3J'H
2) iJ j3j) =
= 3(jltT(2)llJ) ~ [J2(J1)JJ'l >J3J'][J2(J1)J JI } j 3 j ] (_l)Jl+J+J'(2J,+l)(2J+l)~'t jJ'J1} J1 Jj 2 •
.
,
.
7
3
.
(2)
5
Using the values of c.f.p, and Racah coe[hclents for J =J =~ and J=~, ~, ~, ~, we obtain the B(E2 ½~ J) in terms of one parameter {~ II T(2){I~){z The coefficients which should be multiplied by this single proton reduced matrix element squared in order to obtain the absolute value squared of (2) are listed in table 1. The method makes sense if a value of this parameter can be foundwhich will accurately reproduce the rates of the observed transitions (also listed in table 1). We make use of the observed four transitions to determine the best value which turns out to be 1(½ l}T(2){I½){2 =
5.19 e 2 ×
I0 -5° cm 4 .
(3)
The B(E2) values calculated with this effective operator are given in table I. They are in good agreement with the observed values. We can use the value (3) in order to calculate another matrix element of the effective E2 operator, namely the static quadrupole moment of the 51V ground state. The experimental value of this quantity was not very well knownfor many years. Eventhe sign seemed to be wrong (positive) for three protons in the f~_ shell (it should be exactly ½of the single proton quadrupole moment which should be negative). 2 More recent measurements gave a practically vanishing quadrupole moment which added much to the confusion. Recently, Childs [5] has accurately determined from experiment the quadrupole moment and obtained the result 51V(g.s.) quadrupole moment (-0.052 • 0.010) e x 10-24 cm2 . 314
(4)
Volume 25B, number 5
PHYSICS L E T T E R S
18 September 1967
We can see whether this value i s c o n s i s t e n t with the r e d u c e d m a t r i x e l e m e n t (3). The sign cannot be d e t e r m i n e d f r o m (3) but the sign of the e x p e r i m e n t a l m o m e n t is c o n s i s t e n t with the shell model d e s c r i p tion. The o p e r a t o r to be used with real wave functions for E2 t r a n s i t i o n s is e r 2 Y2n(0, (p). The q u a d r u pole m o m e n t is defined a s the expectation value in the state with Mj = J of the o p e r a t o r e (3z 2 - r 2) = = v~6~-~5r 2 Y20(0,~). The p r o p o r t i o n a l i t y factor of these two o p e r a t o r s is thus l~f~n. In o r d e r to c a l c u late the quadrupole m o m e n t in the J = ½ state of the f3 configuration we have to calculate 2
O = W16~ ~
=
.3
7 _£ J=~M-21i~_l
T (2)
(i)] j 3 J=i7M=~) =
( ~ 2~ ) (~3 j=~ tl2 T~2) It j3 J--N) : \-72 0~
(5)
=V 5 \--~0~ The last equality is due to the reduced m a t r i x e l e m e n t , given by eq. (2), being equal to ~ (table 1) m u l tiplied by (-~tl T(2) tl ~)' T h u s , we take the negative square root of the value (3) and multiply it by -~ /V/2~/30. The value we obtain this way is ]
- -
..
Q = -0.058 e x 10 -24 cm 2.
(6)
As we see, this value c a l c u l a t e d from E2 t r a n s i t i o n r a t e s a g r e e s v e r y well with the static quadrupole m o m e n t of the 51V ground state. It is c u s t o m a r y to give the e n h a n c e m e n t of v a r i o u s t r a n s i t i o n s in t e r m s of an effective charge. This is the charge to be u s e d with the r e a l (not effective) t r a n s i t i o n o p e r a t o r in o r d e r to obtain the o b s e r v e d t r a n s i t i o n r a t e . It should be r e m e m b e r e d , however, that t r a n s i t i o n s due to different e l e c t r o m a g n e t i c m u l t i p o l e s will probably have different "effective charges" even if a given multipole r a d i a t i o n can be c a l culated with effective single n u c l e o n o p e r a t o r s . In o r d e r to find the effective charge for the E2 o p e r a t o r s in 51V let us c a l c u l a t e the r e d u c e d m a t r i x e l e m e n t of the r e a l E2 o p e r a t o r . We use the e x p r e s s i o n [2,4]
I(Jll 2y(2)(O,dP)tlJ)'2 7
.
.
:
5(2j+1)2(2l+1)2 l l 2 2 1 / (000t j .
.
.
J ½212~ °°
]2
Jo
.
for j = ~, l = 3. F o r h a r m o m c oscxllator wave functions the r a d i a l i n t e g r a l xs equal to (4n + 2/+ 3 J 4 v = 9/4~ which for the o s c i l l a t o r c o n s t a n t ~ given by [6] e 2 ~ = 0.3 MeV, is equal to 16.4 x 10-26 cm 2. A v e r y close value to this i s obtained by using a Saxon-Wood potential. Using this value we obtain 1 (lf~ It r 2 y ( 2 ) I t lf~) 12 = 0.02 x 10 -48 cm 4.
(8)
C o m p a r i n g the value (87 with (37 we obtain the value of the lf~ proton effective charge for the E2 o p e r a tors 2 = 2.59 e 2, (9) eeff eeff = 1.61 e . T h i s value for the proton effective charge is in very good a g r e e m e n t with the v a l u e s obtained from other E2 t r a n s i t i o n s in n e i g h b o u r i n g f~ shell nuclei [7]. The author would like to e x p r e s s his thanks to The Academy of Science of the USSR for the invitation to p a r t i c i p a t e in the Kharkov Meeting and to the p a r t i c i p a n t s of that s t i m u l a t i n g m e e t i n g for t h e i r kind hospitality.
315
Volume25B, number 5
PHYSICS
References 1. R.D. L a w s o n a n d J . L . U r e t s k y . Phys. Rcv. 106 (1957) 1369; I . T a l m i . Proe. 1957 Rehovoth Conf. on nuclear s t r u c t u r e (North-Holland Publishing Co., A m s t e r dam 1958) p. 31; J . E . S c h w i i g e r , Phys. Rev. 121 (1961) 569. 2. H.W. KendalI a n d I . T a l m i . Phys. Rev. 128 (1962) 792. 3. O.F.Afonin, A . P . G r e e n b e r g . I. Kh. L e m b e r g and I. N. Chugunov, XVII Annual National Meeting on nuclear. ' s t r u c t u r e and nuclear spectroscopy, K h a r kov. 1967.
SEMISCHEMATICAL MODEL AND 2 PARTICLE-2
LETTERS
18 September 1967
4. A. de-Shalit and I. Talmi, Nuclear shell theory (Academic P r e s s , New York 1963). 5. W.J. Childs, Phys. Rcv. 156 (1967) 71. 6. B . C . C a r l s o n a n d I . T a l m i . Phys. Rev. 96 (1954) 436. 7. J . V e r v i e r , Phys. L e t t e r s 13 (1964) 47.
OF THE COUPLING HOLE STATES IN
OF THE
THE PARTICLE-HOLE CONTINUUM
V . V . B A L A S H O V a n d N. M. K A B A C H N I K Institute of Nuclear P h y s i c s of the Moscow University. Moscow, USSR
Received 8 August 1967
Two-particle, two-hole s t a t e s are introduced to d e s c r i b e the splitting of the dipole resonance.
The particle-hole (lp-lh) states are doorway s t a t e s in t h e c o l l e c t i v e e x c i t a t i o n of n u c l e i by e x t e r n a l f i e l d . T o e x p l a i n t h e s t r o n g d a m p i n g of s u c h e x c i t a t i o n s i t i s n o t s u f f i c i e n t to i n c l u d e t h e p o s s i b i l i t y of t h e i r d i r e c t d e c a y a c c o m p a n i e d w i t h n u c l e o n e m i s s i o n , It i s n e c e s s a r y to t a k e i n t o a c c o u n t t h e c o u p l i n g of t h e s e e x c i t a t i o n s w i t h t h e s t a t e s of a m o r e c o m p l i c a t e d n a t u r e which may occur in the same energy region. The t y p i c a l e x a m p l e of s u c h a s i t u a t i o n i s t h e g i a n t r e s o n a n c e of p h o t o d i s i n t e g r a t i o n of n u c l e i . A n u m b e r of i m p o r t a n t f e a t u r e s of t h e g i a n t r e s o n a n c e (the s t r u c t u r e a n d a g r e a t w i d t h ) c a n n o t b e e x p l a i n e d w i t h i n t h e f r a m e w o r k of t h e u s u a l ( l p - l h ) t r e a t m e n t . In a l a r g e n u m b e r of w o r k s t h e e n r i c h m e n t of t h e d i p o l e e x c i t a t i o n s p e c t r u m of t h e g i a n t r e s o n a n c e w a s o b t a i n e d by i n c l u d i n g the 2 particle-2 hole (2p-2h) configurations. Such an inclusion was effected by diagonalizing the H a m i l t o n i a n of e i t h e r s h e l l m o d e l o r c o l l e c t i v e m o d e l . T h e s e w o r k s do n o t i n c l u d e t h e s p e c i f i c f e a t u r e s of t h e f o r m a t i o n a n d d i s i n t e g r a t i o n of t h e c o l l e c t i v e d i p o l e s t a t e s w h i c h a r e due t o t h e f a c t t h a t t h e s e s t a t e s a r e in t h e c o n t i n u u m , a b o v e the threshold for nucleon emission. In t h i s p a p e r t h e a p p r o x i m a t e v e r s i o n of t h e 316
tzo 1oo
40
°lf I 19
20
21
22
23
2~
2Y
26
27
f Meg
Fig. 1. The total photoabsorption c r o s s section for 160, including (solid curve) and omitting (dashed c u r ve) the (2p-2h) states. u n i f i e d t h e o r y of n u c l e a r r e a c t i o n s [1] e l a b o r a t e d i n [2,3] f o r t h e p r o b l e m of ( l p - l h ) e x c i t a t i o n i n the continuum is generalized for the case when (2p-2h) configurations are included. These config u r a t i o n s m a y b e c o n s i d e r e d to b e t h e d o o r w a y s t a t e s i n t h e p r o c e s s of d a m p i n g of p a r t i c l e - h o l e excitation.
PHYSICS
LETTERS
18 S e p t e m b e r 1967
W e w o u l d l i k e to t h a n k P r o f e s s o r s J . D{}browski a n d J . P n i e w s k i a n d D r . J . Z a k r z e w s k i f o r t h e i r k i n d interest and discussion. We thank also Dr. J. Zakrzewski for supplying us with the experimental data.
References 1. N. B y e r s and W. N. Cottingham. Nuovo Cimento 11 (1959) 554. 2. Y . C . T a n g . Nuovo Cimento 10 (1958) 780. 3. V . A . L y u l ' k a , Zh. Exp. i T e o r . Fiz. 42 (1962) 1692. 4. J . S z y m a f s k i . Nuovo Cimento 10 (1958) 830. 5. D.Chlebowska and J.Szymafiski, Bull. Acad. Polon. Sci. Ser. Sci. Math. Astron. Phys. 7 (1959) 643. 6. R . H . D a l i t z , Nucl. Phys. 41 (1963) 78. 7. J. Zakrzewski, private communication. 8. K.S.Suh. Phys. Rev. 111 {1958) 941. 9. J.Sacton. K - C o l l a b o r a t i o n , p r e p r i n t (1965).
QUADRUPOLE
MOMENT AND MOMENTS
EFFECTIVE IN 51V*
E2
TRANSITION
I. TALMI** Palmer Physical Laboratory. Princeton University. Princeton, New Jersey. USA Received 14 August 1967
Recently m e a s u r e d E2 r a t e s for Coulomb excitation from the ground state of 51V to excited s t a t e s a r e consistent with the use of effective single particle E2 o p e r a t o r s and f3 wave functions. F r o m these o p e r a t o r s the quadrupole m o m e n t of 51V is calculated and found to be in v e r } good a g r e e m e n t with the new e x p e r i m e n tal (negative) value. The enhancement of the single proton E2 o p e r a t o r is equivalent to an effective charge eef f = 1.61e.
T h e e n e r g y l e v e l s of t h e 51V n u c l e u s h a v e b e e n k n o w n f o r m a n y y e a r s to b e w e l l d e s c r i b e d i n t e r m s of t h e l f 3 p r o t o n c o n f i g u r a t i o n [1] (and a c l o s e d lf~ n e u t r o n s h e l l ) . T h e low l y i n g odd p a r i t y l e v e l s h a v e the spin~J=½ (g.s.), j = 5 (0.32MeV), j = 3 (0.93)/J=~ (1.61), j = 9 ( 1 . 8 1 ) a n d p o s s i b l y J = ~ (2.70). T h e s e s t a t e s a r e t h e o n l y o n e s a l l o w e d by t h e P a u l i p r i n c i p l e f o r t h e f3 c o n f i g u r a t i o n of i d e n t i c a l n u c l e o n s . T h e i r e n e r g i e s c a n b e v e r y w e l l o b t a i n e d f r o m t h e f2 m a t r i x e ~ e m e n t s of t h e e f f e c t i v e n u c l e a r Y i n t e r a c t i o n t a k e n f r o m n e l• g h b o u r i n g n u c l e i , e . g . 5 0 T i [1]. L i k e m o s t , if n o t a l l , n u c l e i , t h e E 2 t r a n s i t i o n r a t e s i n 51V a r e m u c h b i g g e r t h a n t h e s i n g l e p r o t o n e s t i m a t e s . T h i s i s n o t a s t r o n g o b j e c t i o n t o t h e f3 a s s i g n m e n t s i n c e t r a n s i t i o n p r o b a b i l i t i e s , u n l i k e energies, are rather sensitive to small admixtures in the wave functions. Moreover, it is clear that the r e a l w a v e f u n c t i o n s , w i t h t h e s t r o n g t w o - b o d y c o r r e l a t i o n s , a r e v e r y d i f f e r e n t f r o m t h e s h e l l m o d e l f3 w a v e f u n c t i o n s . W e c a n c a l c u l a t e n u c l e a r e n e r g i e s in t h e s h e l l m o d e l b e c a u s e we u s e r e n o r m a l i z e d o~ e f f e c t i v e t w o - b o d y i n t e r a c t i o n s . T h e s e a r e s u p p o s e d t o i n c l u d e t h e e f f e c t of t w o p a r t i c l e s h o r t r a n g e correlations. Similarly, it is expected that the electromagnetic moments to be used in the shell model w o u l d a l s o b e r e n o r m a l i z e d i n s o m e way. F o r e x a m p l e , a n i m p o r t a n t c o n t r i b u t i o n t o E 2 o p e r a t o r s w o u l d c o m e f r o m t h e p o l a r i z a t i o n of t h e c o r e b y t h e o u t s i d e n u c l e o n s . It m a y a l s o h a p p e n t h a t t h e c h a n g e d e l e c t r o m a g n e t i c m o m e n t s w i l l h a v e t h e f o r m of r e n o r m a l i z e d s i n g l e n u c l e o n o p e r a t o r s ( o p e r a t o r s w i t h " e f f e c t i v e c h a r g e s " ) . T h i s w i l l b e t h e c a s e if t h e c o r e p o l a r i z a t i o n of t h e i n d i v i d u a l n u c l e o n s a r e i n d e * This work was supported by the U . S . A t o m i c Energy Commission and the Higgins Scientific T r u s t Fund. This work made use of the P r i n c e t o n Computer F a c i l i t i e s supported in part by the National Science Foundation Grant NSF GP-579. ** National Science Foundation Senior F o r e i g n Scientist Fellow. On leave of absence from the Weizmann Institute of Science, Rehovoth, Israel.
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LETTERS
18 September 1967
Table 1 Experimental and calculated B(E2) transition strengths in 51V (in units of e 2 × 10 -50 cm4). j
_7 2
Reduced matrix element squared
1 ~
£ 2
0.11
5 2
12
3-~ : 0.34
9_ 2
22
15 1.47
26
~5 = 0.41
11
-24
3 - 1.33
B(E2. ½ --*J) experimental [3]
0.27
0.92
0.22
0.90
B(E2, ~ ~ J )
0.22
0.96
0.27
0.86
calculated
p e n d e n t of e a c h o t h e r . If t h i s i s so, it is p o s s i b l e to r e l a t e t r a n s i t i o n p r o b a b i l i t i e s within the s a m e c o n f i g u r a t i o n by u s i n g s h e l l m o d e l w a v e f u n c t i o n s and e f f e c t i v e s i n g l e p a r t i c l e e l e c t r o m a g n e t i c m o m e n t s . Such a c a l c u l a t i o n y i e l d s c o n d i t i o n s that should be s a t i s f i e d by the o b s e r v e d t r a n s i t i o n strengths. The E2 m o m e n t s of the t r a n s i t i o n s f r o m the g r o u n d s t a t e to the e x c i t e d s t a t e s of 51V w e r e c o n s i d e r e d in t h i s a p p r o a c h [2]. U n f o r t u n a t e l y , the v a l u e s of the t r a n s i t i o n m o m e n t s d e t e r m i n e d by e l e c t r o n s c a t t e r i n g w e r e not v e r y a c c u r a t e [2] and it s e e m e d i m p o s s i b l e to r e l a t e t h e s e t r a n s i t i o n s to the f3 w a v e f u n c t i o n s . H o w e v e r , a c c u r a t e m e a s u r e m e n t s of the E2 t r a n s i t i o n m o m e n t s w e r e r e c e n t l y earr~ied out u s i n g C o u l o m b e x c i t a t i o n s of 51V with 12C i o n s [3]. The r e s u l t s , a s n o t e d by L e m b e r g et al. [3], a r e in v e r y good a g r e e m e n t with the c a l c u l a t e d v a l u e s and s u p p o r t the i d e a t h a t e n h a n c e d E 2 t r a n s i t i o n s can fit into the s h e l l m o d e l d e s c r i p t i o n of n u c l e a r s t a t e s . T h e r a t e unit t i m e of the E2 t r a n s i t i o n Ji ~ J f i s d i r e c t l y p r o p o r t i o n a l to [2,4]
consislency
1
B ( E 2 Ji --~Jf) - 2 j i + l
I(Ji I I ~ T~2)It J f )
12
(1)
3 w h e r e the r a n k 2 i r r e d u c i b l e t e n s o r --.i~Jl T!2)" is the e f f e c t i v e s i n g l e p a r t i c l e o p e r a t o r . T h e r e d u c e d m a t r i x e l e m e n t s in eq. (1) can be c a l c u l a t e d f r o m the w a v e f u n c t i o n s . U s i n g c o e f f i c i e n t s of f r a c t i o n a l p a r e n t a g e we obtain [2,4]
j3(j=½)
(J3J'H
2) iJ j3j) =
= 3(jltT(2)llJ) ~ [J2(J1)JJ'l >J3J'][J2(J1)J JI } j 3 j ] (_l)Jl+J+J'(2J,+l)(2J+l)~'t jJ'J1} J1 Jj 2 •
.
,
.
7
3
.
(2)
5
Using the values of c.f.p, and Racah coe[hclents for J =J =~ and J=~, ~, ~, ~, we obtain the B(E2 ½~ J) in terms of one parameter {~ II T(2){I~){z The coefficients which should be multiplied by this single proton reduced matrix element squared in order to obtain the absolute value squared of (2) are listed in table 1. The method makes sense if a value of this parameter can be foundwhich will accurately reproduce the rates of the observed transitions (also listed in table 1). We make use of the observed four transitions to determine the best value which turns out to be 1(½ l}T(2){I½){2 =
5.19 e 2 ×
I0 -5° cm 4 .
(3)
The B(E2) values calculated with this effective operator are given in table I. They are in good agreement with the observed values. We can use the value (3) in order to calculate another matrix element of the effective E2 operator, namely the static quadrupole moment of the 51V ground state. The experimental value of this quantity was not very well knownfor many years. Eventhe sign seemed to be wrong (positive) for three protons in the f~_ shell (it should be exactly ½of the single proton quadrupole moment which should be negative). 2 More recent measurements gave a practically vanishing quadrupole moment which added much to the confusion. Recently, Childs [5] has accurately determined from experiment the quadrupole moment and obtained the result 51V(g.s.) quadrupole moment (-0.052 • 0.010) e x 10-24 cm2 . 314
(4)
Volume 25B, number 5
PHYSICS L E T T E R S
18 September 1967
We can see whether this value i s c o n s i s t e n t with the r e d u c e d m a t r i x e l e m e n t (3). The sign cannot be d e t e r m i n e d f r o m (3) but the sign of the e x p e r i m e n t a l m o m e n t is c o n s i s t e n t with the shell model d e s c r i p tion. The o p e r a t o r to be used with real wave functions for E2 t r a n s i t i o n s is e r 2 Y2n(0, (p). The q u a d r u pole m o m e n t is defined a s the expectation value in the state with Mj = J of the o p e r a t o r e (3z 2 - r 2) = = v~6~-~5r 2 Y20(0,~). The p r o p o r t i o n a l i t y factor of these two o p e r a t o r s is thus l~f~n. In o r d e r to c a l c u late the quadrupole m o m e n t in the J = ½ state of the f3 configuration we have to calculate 2
O = W16~ ~
=
.3
7 _£ J=~M-21i~_l
T (2)
(i)] j 3 J=i7M=~) =
( ~ 2~ ) (~3 j=~ tl2 T~2) It j3 J--N) : \-72 0~
(5)
=V 5 \--~0~ The last equality is due to the reduced m a t r i x e l e m e n t , given by eq. (2), being equal to ~ (table 1) m u l tiplied by (-~tl T(2) tl ~)' T h u s , we take the negative square root of the value (3) and multiply it by -~ /V/2~/30. The value we obtain this way is ]
- -
..
Q = -0.058 e x 10 -24 cm 2.
(6)
As we see, this value c a l c u l a t e d from E2 t r a n s i t i o n r a t e s a g r e e s v e r y well with the static quadrupole m o m e n t of the 51V ground state. It is c u s t o m a r y to give the e n h a n c e m e n t of v a r i o u s t r a n s i t i o n s in t e r m s of an effective charge. This is the charge to be u s e d with the r e a l (not effective) t r a n s i t i o n o p e r a t o r in o r d e r to obtain the o b s e r v e d t r a n s i t i o n r a t e . It should be r e m e m b e r e d , however, that t r a n s i t i o n s due to different e l e c t r o m a g n e t i c m u l t i p o l e s will probably have different "effective charges" even if a given multipole r a d i a t i o n can be c a l culated with effective single n u c l e o n o p e r a t o r s . In o r d e r to find the effective charge for the E2 o p e r a t o r s in 51V let us c a l c u l a t e the r e d u c e d m a t r i x e l e m e n t of the r e a l E2 o p e r a t o r . We use the e x p r e s s i o n [2,4]
I(Jll 2y(2)(O,dP)tlJ)'2 7
.
.
:
5(2j+1)2(2l+1)2 l l 2 2 1 / (000t j .
.
.
J ½212~ °°
]2
Jo
.
for j = ~, l = 3. F o r h a r m o m c oscxllator wave functions the r a d i a l i n t e g r a l xs equal to (4n + 2/+ 3 J 4 v = 9/4~ which for the o s c i l l a t o r c o n s t a n t ~ given by [6] e 2 ~ = 0.3 MeV, is equal to 16.4 x 10-26 cm 2. A v e r y close value to this i s obtained by using a Saxon-Wood potential. Using this value we obtain 1 (lf~ It r 2 y ( 2 ) I t lf~) 12 = 0.02 x 10 -48 cm 4.
(8)
C o m p a r i n g the value (87 with (37 we obtain the value of the lf~ proton effective charge for the E2 o p e r a tors 2 = 2.59 e 2, (9) eeff eeff = 1.61 e . T h i s value for the proton effective charge is in very good a g r e e m e n t with the v a l u e s obtained from other E2 t r a n s i t i o n s in n e i g h b o u r i n g f~ shell nuclei [7]. The author would like to e x p r e s s his thanks to The Academy of Science of the USSR for the invitation to p a r t i c i p a t e in the Kharkov Meeting and to the p a r t i c i p a n t s of that s t i m u l a t i n g m e e t i n g for t h e i r kind hospitality.
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References 1. R.D. L a w s o n a n d J . L . U r e t s k y . Phys. Rcv. 106 (1957) 1369; I . T a l m i . Proe. 1957 Rehovoth Conf. on nuclear s t r u c t u r e (North-Holland Publishing Co., A m s t e r dam 1958) p. 31; J . E . S c h w i i g e r , Phys. Rev. 121 (1961) 569. 2. H.W. KendalI a n d I . T a l m i . Phys. Rev. 128 (1962) 792. 3. O.F.Afonin, A . P . G r e e n b e r g . I. Kh. L e m b e r g and I. N. Chugunov, XVII Annual National Meeting on nuclear. ' s t r u c t u r e and nuclear spectroscopy, K h a r kov. 1967.
SEMISCHEMATICAL MODEL AND 2 PARTICLE-2
LETTERS
18 September 1967
4. A. de-Shalit and I. Talmi, Nuclear shell theory (Academic P r e s s , New York 1963). 5. W.J. Childs, Phys. Rcv. 156 (1967) 71. 6. B . C . C a r l s o n a n d I . T a l m i . Phys. Rev. 96 (1954) 436. 7. J . V e r v i e r , Phys. L e t t e r s 13 (1964) 47.
OF THE COUPLING HOLE STATES IN
OF THE
THE PARTICLE-HOLE CONTINUUM
V . V . B A L A S H O V a n d N. M. K A B A C H N I K Institute of Nuclear P h y s i c s of the Moscow University. Moscow, USSR
Received 8 August 1967
Two-particle, two-hole s t a t e s are introduced to d e s c r i b e the splitting of the dipole resonance.
The particle-hole (lp-lh) states are doorway s t a t e s in t h e c o l l e c t i v e e x c i t a t i o n of n u c l e i by e x t e r n a l f i e l d . T o e x p l a i n t h e s t r o n g d a m p i n g of s u c h e x c i t a t i o n s i t i s n o t s u f f i c i e n t to i n c l u d e t h e p o s s i b i l i t y of t h e i r d i r e c t d e c a y a c c o m p a n i e d w i t h n u c l e o n e m i s s i o n , It i s n e c e s s a r y to t a k e i n t o a c c o u n t t h e c o u p l i n g of t h e s e e x c i t a t i o n s w i t h t h e s t a t e s of a m o r e c o m p l i c a t e d n a t u r e which may occur in the same energy region. The t y p i c a l e x a m p l e of s u c h a s i t u a t i o n i s t h e g i a n t r e s o n a n c e of p h o t o d i s i n t e g r a t i o n of n u c l e i . A n u m b e r of i m p o r t a n t f e a t u r e s of t h e g i a n t r e s o n a n c e (the s t r u c t u r e a n d a g r e a t w i d t h ) c a n n o t b e e x p l a i n e d w i t h i n t h e f r a m e w o r k of t h e u s u a l ( l p - l h ) t r e a t m e n t . In a l a r g e n u m b e r of w o r k s t h e e n r i c h m e n t of t h e d i p o l e e x c i t a t i o n s p e c t r u m of t h e g i a n t r e s o n a n c e w a s o b t a i n e d by i n c l u d i n g the 2 particle-2 hole (2p-2h) configurations. Such an inclusion was effected by diagonalizing the H a m i l t o n i a n of e i t h e r s h e l l m o d e l o r c o l l e c t i v e m o d e l . T h e s e w o r k s do n o t i n c l u d e t h e s p e c i f i c f e a t u r e s of t h e f o r m a t i o n a n d d i s i n t e g r a t i o n of t h e c o l l e c t i v e d i p o l e s t a t e s w h i c h a r e due t o t h e f a c t t h a t t h e s e s t a t e s a r e in t h e c o n t i n u u m , a b o v e the threshold for nucleon emission. In t h i s p a p e r t h e a p p r o x i m a t e v e r s i o n of t h e 316
tzo 1oo
40
°lf I 19
20
21
22
23
2~
2Y
26
27
f Meg
Fig. 1. The total photoabsorption c r o s s section for 160, including (solid curve) and omitting (dashed c u r ve) the (2p-2h) states. u n i f i e d t h e o r y of n u c l e a r r e a c t i o n s [1] e l a b o r a t e d i n [2,3] f o r t h e p r o b l e m of ( l p - l h ) e x c i t a t i o n i n the continuum is generalized for the case when (2p-2h) configurations are included. These config u r a t i o n s m a y b e c o n s i d e r e d to b e t h e d o o r w a y s t a t e s i n t h e p r o c e s s of d a m p i n g of p a r t i c l e - h o l e excitation.