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Second order effective charges with realistic forces
Volume 34B, n u m b e r 3
SECOND
PHYSICS
ORDER
EFFECTIVE
LETTERS
CHARGES
WITH
15 F e b r u a r y 1971
REALISTIC
FORCES
*
P. J. ELLIS
Nuclear Physics Laboratory, Oxford, UK and S. S I E G E L
NOAA Computer Division, Room 1328, F.O.B. No. 4, Suitland, Maryland 20023, USA Received 1 December 1970
With various r e a l i s t i c m a t r i x e l e m e n t s the second o r d e r contributions to the effective charge are s m a l l and positive for neutrons. F o r protons they are s m a l l and negative and cancel a good p a r t of the f i r s t o r d e r values; the cancellation among the n u m b e r conserving sets is poor. We r e m a r k that c.m. effects do not need to be considered in evaluating effective c h a r g e s or valence energies.
W e b e g i n b y d i s c u s s i n g c . m . e f f e c t s in p e r turbation theory. Splitting the Hamiltonian into h a r m o n i c o s c i l l a t o r a n d p e r t u r b i n g p a r t s , we have H = H H O + V; <~j ~~1
V =i " V i j - ~ m w
22
~
= i
where R is the c.m. coordinate and V is a funct i o n of r e l a t i v e c o o r d i n a t e s only. S i n c e H H O c a n also be separated into relative and c.m. parts the perturbed wave function can be written ~ = ~2(rel) ~2(R)~D, w h e r e ~ D i s t h e m o d e l w a v e f u n c t i o n t h e c . m . p a r t of w h i c h we a s s u m e to b e in a 0s o s c i l l a t o r s t a t e . Now t h e u s u a l E 2 o p e r a t o r c a n b e w r i t t e n a s a f u n c t i o n of r e l a t i v e c o ordinates only plus terms which are vector and t e n s o r in t h e c . m . c o o r d i n a t e [1]. T h e s e l a t t e r t e r m s c a n g i v e no c o n t r i b u t i o n a s t h e c . m . m u s t r e m a i n i n a n s s t a t e s i n c e R 2 i s s c a l a r in t h e o r b i t a l a n g u l a r m o m e n t u m . T h u s in c a l c u l a t i n g E 2 r a t e s o n l y t h e r e l a t i v e p a r t of t h e w a v e f u n c tion will contribute. Consider the total energy, E = E ( r e l ) + E(R). Now, s i n c e t h e m a t r i x e l e m e n t s of A M w 2 R 2 a r e i n d e p e n d e n t of A M ÷'or oscillator c.m. wave functions, the contribution E(R) w i l l b e i n d e p e n d e n t of A. C o n s e q u e n t l y if the total energy is separated into core and val* P a r t of this work was c a r r i e d out at Rutgers Univ e r s i t y and supported by the National Science Foundation.
e n c e c o n t r i b u t i o n s a n d if E(R) i s r e m o v e d f r o m t h e c o r e p a r t w i t h t h e c o r e v a l u e of A, t h e v a l e n c e e n e r g i e s s h o u l d b e c a l c u l a t e d w i t h V. W e c o n c l u d e t h a t in c a l c u l a t i n g E2 r a t e s o r v a l e n c e energies, V can be used without modification for c.m. effects. W e u s e t h e l i n k e d v a l e n c e f o r m a l i s m of B r a n d o w [2] to c a l c u l a t e t h e e f f e c t i v e c h a r g e f o r t h e E2 t r a n s i t i o n s in m a s s 17 t h r o u g h s e c o n d order using realistic matrix elements. We res t r i c t o u r s p a c e to t h e fp s h e l l a n d b e l o w w h i c h eliminates four diagrams with single particle intermediate states; these diagrams are exp e c t e d to g i v e a v e r y s m a l l c o n t r i b u t i o n [3]. W e assume that harmonic oscillator wave functions a r e s u f f i c i e n t l y c l o s e to s e l f - c o n s i s t e n c y t h a t t h e single particle energy insertions are cancelled by the oscillator potential insertions (although t h i s a s s u m p t i o n i s p r o b a b l y not a d e q u a t e [4] in e v a l u a t i n g t h e z e r o t h o r d e r d i a g r a m 1 of fig. 1). T h e r e m a i n i n g d i a g r a m s a r e s h o w n in fig. 1; e x change and topologically equivalent diagrams are implied. The effective charge is given by the r a t i o of t h e s e d i a g r a m s to t h e z e r o t h o r d e r d i a g r a m in i s o s c a l a r a n d i s o v e c t o r n o t a t i o n , f r o m which the neutron and proton values may be ded u c e d ( e . g . , s e e r e f . [3]). E x p r e s s i o n s f o r t h e d i a g r a m s in fig. 1 h a v e b e e n g i v e n b y S i e g e l a n d Z a m i c k [3], in t h e i r c a l c u l a t i o n of t h e s e c o n d order effective charge with the semi-realistic K a l l i o - K o l l t v e i t (KK) f o r c e . (In c a r e f u l l y c h e c k i n g t h e i r f o r m u l a t i o n we u n f o r t u n a t e l y f o u n d a 1 f a c t o r of ~ m i s s i n g f r o m t h e e x p r e s s i o n s f o r
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Volume 34B, number 3
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15 February
1971
diagrams 5, 9, 18, 19, 28 and 29; the c o r r e c t e d KK totals a r e given in table 2.) In ref. [3] the wave functions were explicitly normalised by factors of (1 + x)- I / 2 , whereas following Brandow [2] we expand this as (1 -½x) and write the diagrams in folded form. The factor of ½ is shown on the diagrams since it does not follow from the usual rules. We have used two s e t s of the essentially exact G-matrix elements of B a r r e t t , Hewitt and McCarthy [5]. Therefore diagrams 28-31 of fig. 1 should not be included since the p a r t i c l e p a r t i c l e ladders have been properly taken into account. Set BHM1 corresponds to a starting energy of -3 MeV, which introduces a gap b e tween occupied and unoccupied levels (note that s t a t e s with one and two p a r t i c l e s in the unoccupied space a r e shifted equally). These matrix elements a r e fairly s i m i l a r to those calculated r a t h e r differently by Kuo and Brown (KB). These authors also used the Hamada-Johnston potential and introduced a gap [6]. The set BHM2 c o r r e sponds to a starting energy of 82 MeV which means essentially an unshifted harmonic o s c i l l a tor spectrum. The contributions of the second o r d e r diagrams 4-27 of fig. 1 a r e listed in table 1 for the BHMI case. We have included the 0s through the
7
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LETTERS
51
Fig. i. Perturbation theory diagrams discussed in the evaluation of the effective charge. Here E is the E2 electromagnetic operator.
Table i Contributions of the secondorder diagrams to the effective charge for neutrons and protons using the BHM1 matrix elements.
Diagram number
d~-
ds~.
d~ -
s½
d~-2
d3~
s½ -
dl2
N
P
N
P
N
P
N
P
4 5 6 7 8 9 10 11 12 13
0.026 -0.013 0 0 0.009 -0.013 0 0 0 0
Subtotal
0.008
14 15 16 17
18 19 20 21 22 23 24 25 26 27
178
0 0 0 0
0.005 0.005 -0.010 -0.010 -0.014 -0.014 0.033 0.033 0.010 0.010
d~-
d~
N
P
0.027 0.027 -0.003 -0.010 -0.041 0 -0.041 0 0.012 0.002 -0.004 -0.009 -0.037 0 -0.037 0 0.019 0 0.019 0
0.030 -0.003 -0.041 -0.050 0.003 -0.003 -0.043 -0.037 0.019 0.022
0.038 -0.013 0 0 0.008 -0.012 0 0 0 0
0.040 -0.003 -0.041 -0.053 0.004 -0.003 -0.071 -0.037 0.019 0.035
0.036 -0.012 0 0 0.011 -0.022 0 0 0 0
0.033 -0.004 -0.050 -0.053 0.015 -0.003 -0.071 -0.043 0.022 0.035
0.040 -0.019 0 0 0.021 -0.026 0 0 0 0
0.037 -0.003 -0.053 -0.053 0.020 -0.005 -0.071 -0.071 0.035 0.035
-0.087
0.010
-0.105
0.021
-0.111
0.013
-0.120
0.016
-0.129
0 0 0 0
0.005 0 -0.024 0
0 0 0 0
0 0 0 0
0 0.005 0 -0.024
0 0 0 0
0 0 0 0
0.002 0.002 -0.020 -0.020 -0.026 -0.026 0.045 0.045 0.014 0.014
0 0 0 0
0 0 0 0
0.004 0.001 0.005 0.002 0.002 -0.001 0.003 0.001 - 0 . 0 0 4 - 0 . 0 0 1 -0.005 0.002 0.004 0.001 0.003 0.001 - 0 . 0 0 7 - 0 . 0 0 4 - 0 . 0 2 1 - 0 . 0 1 1 - 0 . 0 1 8 - 0 . 0 2 1 - 0 . 0 2 0 -0.023 - 0 . 0 1 2 - 0 . 0 1 5 - 0 . 0 1 2 - 0 . 0 1 8 - 0 . 0 1 1 - 0 . 0 1 8 - 0 . 0 2 0 -0.023 - 0 . 0 1 1 - 0 . 0 1 7 - 0 . 0 1 8 - 0 . 0 1 9 - 0 . 0 1 3 - 0 . 0 2 6 - 0 . 0 1 9 -0.028 - 0 . 0 1 6 - 0 . 0 2 1 - 0 . 0 2 1 - 0 . 0 2 2 - 0 . 0 0 8 - 0 . 0 1 8 - 0 . 0 1 9 -0.028 0.016 0.023 0.027 0.045 0.012 0.028 0.025 0.040 0.035 0.043 0.053 0.057 0.020 0.024 0.025 0.040 0.010 0.012 0.020 0.022 0.005 0.007 0.010 0.014 0.007 0.009 0.011 0.014 0.003 0.006 0.010 0.014
Volume 34B, number 3
PHYSICS LETTERS
(0flp) s h e l l s and allowed all p o s s i b l e i n t e r m e diate s t a t e s . The e n e r g y d e n o m i n a t o r s w e r e taken to be m u l t i p l e s of/~¢o = 14 MeY. The c o n t r i b u t i o n s l i s t e d under, e . g . , d ~ - s½ r e f e r to an i nit i a l 0d~ and a final l s ~ state. The d i a g r a m s 4-7 and ~-13 f o r m two n ~ m b e r c o n s e r v i n g s e t s , i . e . , if the E2 o p e r a t o r is r e p l a c e d by the n u m b e r o p e r a t o r t h es e graphs give i d e n t i c a l l y z e r o . Brandow [2] has s u g g e s t e d that such s e t s might c a n c e l to a c o n s i d e r a b l e d e g r e e and f o r the a n a l ogous d i a g r a m s in the e f f e c t i v e i n t e r a c t i o n (where the E o p e r a t o r is r e p l a c e d by a v a l e n c e line) B a r r e t t and K i r s o n found this to be t r u e [7]. In the p r e s e n t c a s e the neutron subtotals f o r d i a g r a m s 4-13 (see table 1) a r e quite s m a l l . Howe v e r , f o r the p r o t o n s the totals a r e s i z e a b l e and the c a n c e l l a t i o n is quite poor. (Note that f o r our i n t e r m e d i a t e s t a t e s d i a g r a m s (10+ 11) = - 2 × d i a g r a m s (12 + 13).) D i a g r a m s 14 and 15 give v e r y s m a l l co n t ri b u t i o n s as do the h o l e - h o l e i n t e r a c t i o n d i a g r a m s 18 and 19. One m a y wish to include the i n s e r t i o n s of d i a g r a m s 16 and 17 in the single p a r t i c l e p o t e n ti a l definition [2]; we have c a l c u l a t e d the d i a g r a m s anyway and found them to be s m a l l . D i a g r a m s 20-23 which in volve a particle-hole interaction are sizeable and n e g a t i v e and tend to c a n c e l the TDA and RPA d i a g r a m s 24-27. The effect is not so pronounced as it is f o r the analogous d i a g r a m s of the e f f e c t i v e i n t e r a c t i o n w h e r e t h e r e is g e n e r a l l y an o v e r c a n c e l l a t i o n [7]. C o m p a r i s o n of our r e s u l t s with those of B a r r e t t and K i r s o n [7] f o r the e f fective interaction indicates a strong correlation in sign b et ween analogous graphs. A l m o s t w it h out exception, a graph which i n c r e a s e s the e f f e c t i v e c h a r g e ( m o r e p o s i ti v e ) i n c r e a s e s the e f f e c t i v e i n t e r a c t i o n ( m o r e negative), and v i c e v e r s a . D i e p e r i n k and B r u s s a a r d [8] have c a r r i e d out a second o r d e r ca l c u la ti o n in the c a l c i u m r e gion. C o m p a r i s o n with our r e s u l t s i n d ic a t e s that the s i z e s of the d i a g r a m s a r e quite s i m i l a r ,
15 February 1971
a p a r t f r o m the TDA and RPA d i a g r a m s (24-27) which a r e a f a c t o r of t h r e e l a r g e r than in the o x ygen region. They thus obtain a l a r g e r p o s i t i v e second o r d e r contribution. The z e r o t h o r d e r e f f e c t i v e c h a r g e is t r i v i a l l y z e r o f o r n e u t r o n s and one f o r protons. The total f i r s t and second o r d e r contributions a r e l i s t e d in table 2. They a r e i n v a r i a n t under i n t e r c h a n g e of initial and final s t a t e s s i n c e the d i a g r a m s a r e e i t h e r s y m m e t r i c under i n t e r c h a n g e o r o c c u r in p a i r s which t r a n s f o r m into one a n o t h e r under i n t e r c h a n g e of initial and final quantum n u m b e r s . In addition, the KK totals depend only on the s i n gle p a r t i c l e o r b i t a l an g u l ar m o m e n t a s i n c e the f o r c e is c e n t r a l [9]. The f i r s t o r d e r r e s u l t s ( d i a g r a m s 2 and 3) a r e s i m i l a r f o r the BHM1 and KB c a s e s . The BHM2 m a t r i x e l e m e n t s yield a much i n c r e a s e d value f o r n e u t r o n s and a s m a l l e r i n c r e a s e f o r p r o t o n s si n ce the p r o t o n v a l u e s d e pend only on the T = 1 m a t r i x e l e m e n t s which a r e known to be l e s s s e n s i t i v e to the s t a r t i n g e n e r g y than those with T = 0 [5]. As r e g a r d s the KK i n t e r a c t i o n , we r e m a r k only that in f i r s t and second o r d e r it g i v es r e s u l t s in q u a l i t a t i v e a g r e e m e n t with the o t h er c a s e s . As can be seen , the r e a l i s t i c m a t r i x e l e m e n t s y i el d total second o r d e r proton c h a r g e s which a r e f a i r l y s m a l l and a l m o s t always negative. They tend to c a n c e l the f i r s t o r d e r v a l u e s indeed f o r the s , -d~ c a s e f i r s t plus second o r d e r is n e g a t i v e 2. Th~ neutron v a l u e s a r e s m a l l and g e n e r a l l y p o s i t i v e , being roughly half the TDA plus RPA v a l u e s f o r the BHM1 and KB c a s e s . We have c o m p a r e d the BHM1 c a s e to the r e s u l t s obtained by allowing the e n e r g y d e n o m i n a t o r s to be 2/~w only (including d i a g r a m s 26 and 27). The r e s u l t s a r e quite s i m i l a r and c o m p a r i s o n of the v a r i o u s d i a g r a m s i n d i c a t e s that the 2~w contributions dominate. The KB v a l u e s a g r e e well with the BHM1 c a s e ; this is t r u e f o r the individual d i a g r a m s as well as the totals. Note that we have not included the
Table 2 First and second order effective charges calculated with various matrix elements.
1st Order
2nd Order
d~-
ds_
d~-
sl
ds-
d½
sl-
d3
d3-
d3
N
P
N
P
N
P
N
P
N
P
BHM1 KB BHM2 KK
0.258 0.329 0.615 0.406
0.099 0.103 0.145 0.144
0.219 0.269 0.512 0.285
0.115 0.119 0.147 0.101
0.310 0.371 0.678 0.406
0.191 0.187 0.238 0.144
0.192 0.239 0.490 0.285
0.057 0.053 0.088 0.101
0.269 0.331 0.640 0.406
0.104 0.099 0.150 0.144
BHM1 B H M 1 (2 h~¢0} KB BHM2 KK
0.056 0.066 0.058 0.068 0.070
-0.057 -0.036 -0.038 -0.031 0.012
0.035 0.043 0.046 0.028 0.026
-0.095 -0.049 -0.069 -0.176 -0.059
0.058 0.081 0.072 0.104 0.070
-0.039 -0.019 -0.019 0.001 0.012
0.008 0.027 0.010 -0.037 0.026
-0.156 -0.089 -0.132 -0.276 -0.059
0.011 0.037 0.020 0.008 0.070
-0.124 -0.083 -0.093 -0.121 0.012
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Volume 34B, n u m b e r 3
PHYSICS
l a d d e r d i a g r a m s 2 8 - 3 1 in c o m p u t i n g t a b l e 2. D i a g r a m s 28 a n d 29 g i v e a s i g n i f i c a n t c o n t r i b u t i o n f o r n e u t r o n s ( ~ + 0.06 w i t h t h e KB m a t r i x e l e m e n t s ) . T h u s it w o u l d a p p e a r t h a t t h e l a d d e r s h a v e b e e n a d e q u a t e l y t a k e n i n t o a c c o u n t in t h e K u o - B r o w n m a t r i x e l e m e n t s , in a g r e e m e n t w i t h t h e r e c e n t a r g u m e n t s of K i r s o n [10]. T h e B H M 2 t o t a l s a r e in q u i t e good a g r e e m e n t w i t h t h e BHM1 a n d KB v a l u e s . T h i s r e s u l t i s s o m e w h a t s u r prising since the individual diagram contribut i o n s a r e m u c h l a r g e r b y t y p i c a l l y f a c t o r s of 2-4. B a r r e t t a n d K i r s o n [7] s u g g e s t e d a s l a s h a b i l i ty c l a s s i f i c a t i o n w h e r e f o r a g i v e n d e g r e e of s l a s h a b i l i t y t h e d i a g r a m s w o u l d c o n v e r g e in t e r m s of t h e n u m b e r of G - m a t r i x i n t e r a c t i o n s . Thus, for the zero-times slashable diagrams o n l y d i a g r a m 1 w h i c h d o e s not c o n t a i n a G - i n t e r a c t i o n w o u l d b e i n c l u d e d . T h e r e a r e no f i r s t o r d e r z e r o - t i m e s s l a s h a b l e d i a g r a m s , b u t in s e c ond o r d e r d i a g r a m s 4 - 1 7 of fig. 1 c o n t r i b u t e . These include the number-conserving sets (diag r a m s 4 - 1 3 ) w h i c h a r e l a r g e f o r p r o t o n s ( ~ - 0.1 for BHM1) so that this classification would not a p p e a r to b e a p p l i c a b l e h e r e . E x p e r i m e n t a l l y , one r e q u i r e s e f f e c t i v e c h a r g e s ~ 0.5 a n d b y s u i t a b l e c h o i c e of t h e s t a r t ing e n e r g y n e u t r o n v a l u e s of t h i s o r d e r of m a g nitude can be obtained. However, the second ord e r p r o t o n v a l u e c a n c e l s a good p a r t of t h e f i r s t o r d e r v a l u e , a t l e a s t if t h e H a r t r e e - F o c k d i a g r a m s a r e n e g l e c t e d . T h u s t h e c o n v e r g e n c e of t h e p e r t u r b a t i o n e x p a n s i o n in t e r m s of t h e n u m b e r of G - i n t e r a c t i o n s a p p e a r s to b e p o o r h e r e a s
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15 F e b r u a r y 1971
w a s t h e c a s e f o r t h e e f f e c t i v e i n t e r a c t i o n [7]. W e a r e g r a t e f u l f o r t h e u s e of t h e m a t r i x e l e m e n t s of T. T. S. Kuo a n d of B. R. B a r r e t t , R . G . L . H e w i t t a n d R. J . M c C a r t h y . W e t h a n k B. R. B a r r e t t , H . A . M a v r o m a t i s a n d L. Z a m i c k f o r useful discussions.
Reference s [1] A. Bohr and B. R. Mottelson, N~elear s t r u c t u r e I (Benjamin, New York, 1969) p. 342. [2] B.H.Brandow, Revs. Mod. Phys. 39 (1967) 771; B. H. Brandow, L e c t u r e s in theoretical physics XI-B, eds. K. T. Mahanthappa and W. E. Brittin (Gordon and Breach, New York, 1969) p. 55. [3] S. Siegel and L. Zamick, Nucl. Phys. A145 (1970) 89. [4] H. A. M a v r o m a t i s and B. Singh, Nucl. Phys. A139 (1969) 451; P . J . E l l i s and S.Siegel, Nucl. Phys. A152 (1970) 547. [5] B . R . B a r r e t t , R . G . L . H e w i t t and R . J . M c C a r t h y , Phys. Rev., to be published; B. R. B a r r e t t , R . G . L . Hewitt and R . J . McCarthy, preprint. [6] T . T . S . K u o and G . E . Brown, Nucl. Phys. 85 (1966) 4O; T . T . S . K u o , Nucl. Phys. A103 (1967) 71. [7] B.R. B a r r e t t and M. W. Kirson, Phys. L e t t e r s 30B (1969) 8; B. R. B a r r e t t and M. W. Kirson, Nucl. Phys. A148 (1970) 145. [8] A. E. L. Dieperink and P. J. B r u s s a a r d , P r o c . Int. Conf. on P r o p e r t i e s of nucl. states (Montreal, 1969) p. 715. [9] P. F e d e r m a n and L. Zamick, Phys. Rev. 177 (1969) 1534. [lO] M . W . K i r s o n , Phys. L e t t e r s 32B (1970) 33.
SECOND
PHYSICS
ORDER
EFFECTIVE
LETTERS
CHARGES
WITH
15 F e b r u a r y 1971
REALISTIC
FORCES
*
P. J. ELLIS
Nuclear Physics Laboratory, Oxford, UK and S. S I E G E L
NOAA Computer Division, Room 1328, F.O.B. No. 4, Suitland, Maryland 20023, USA Received 1 December 1970
With various r e a l i s t i c m a t r i x e l e m e n t s the second o r d e r contributions to the effective charge are s m a l l and positive for neutrons. F o r protons they are s m a l l and negative and cancel a good p a r t of the f i r s t o r d e r values; the cancellation among the n u m b e r conserving sets is poor. We r e m a r k that c.m. effects do not need to be considered in evaluating effective c h a r g e s or valence energies.
W e b e g i n b y d i s c u s s i n g c . m . e f f e c t s in p e r turbation theory. Splitting the Hamiltonian into h a r m o n i c o s c i l l a t o r a n d p e r t u r b i n g p a r t s , we have H = H H O + V; <~j ~~1
V =i " V i j - ~ m w
22
~
= i
where R is the c.m. coordinate and V is a funct i o n of r e l a t i v e c o o r d i n a t e s only. S i n c e H H O c a n also be separated into relative and c.m. parts the perturbed wave function can be written ~ = ~2(rel) ~2(R)~D, w h e r e ~ D i s t h e m o d e l w a v e f u n c t i o n t h e c . m . p a r t of w h i c h we a s s u m e to b e in a 0s o s c i l l a t o r s t a t e . Now t h e u s u a l E 2 o p e r a t o r c a n b e w r i t t e n a s a f u n c t i o n of r e l a t i v e c o ordinates only plus terms which are vector and t e n s o r in t h e c . m . c o o r d i n a t e [1]. T h e s e l a t t e r t e r m s c a n g i v e no c o n t r i b u t i o n a s t h e c . m . m u s t r e m a i n i n a n s s t a t e s i n c e R 2 i s s c a l a r in t h e o r b i t a l a n g u l a r m o m e n t u m . T h u s in c a l c u l a t i n g E 2 r a t e s o n l y t h e r e l a t i v e p a r t of t h e w a v e f u n c tion will contribute. Consider the total energy, E = E ( r e l ) + E(R). Now, s i n c e t h e m a t r i x e l e m e n t s of A M w 2 R 2 a r e i n d e p e n d e n t of A M ÷'or oscillator c.m. wave functions, the contribution E(R) w i l l b e i n d e p e n d e n t of A. C o n s e q u e n t l y if the total energy is separated into core and val* P a r t of this work was c a r r i e d out at Rutgers Univ e r s i t y and supported by the National Science Foundation.
e n c e c o n t r i b u t i o n s a n d if E(R) i s r e m o v e d f r o m t h e c o r e p a r t w i t h t h e c o r e v a l u e of A, t h e v a l e n c e e n e r g i e s s h o u l d b e c a l c u l a t e d w i t h V. W e c o n c l u d e t h a t in c a l c u l a t i n g E2 r a t e s o r v a l e n c e energies, V can be used without modification for c.m. effects. W e u s e t h e l i n k e d v a l e n c e f o r m a l i s m of B r a n d o w [2] to c a l c u l a t e t h e e f f e c t i v e c h a r g e f o r t h e E2 t r a n s i t i o n s in m a s s 17 t h r o u g h s e c o n d order using realistic matrix elements. We res t r i c t o u r s p a c e to t h e fp s h e l l a n d b e l o w w h i c h eliminates four diagrams with single particle intermediate states; these diagrams are exp e c t e d to g i v e a v e r y s m a l l c o n t r i b u t i o n [3]. W e assume that harmonic oscillator wave functions a r e s u f f i c i e n t l y c l o s e to s e l f - c o n s i s t e n c y t h a t t h e single particle energy insertions are cancelled by the oscillator potential insertions (although t h i s a s s u m p t i o n i s p r o b a b l y not a d e q u a t e [4] in e v a l u a t i n g t h e z e r o t h o r d e r d i a g r a m 1 of fig. 1). T h e r e m a i n i n g d i a g r a m s a r e s h o w n in fig. 1; e x change and topologically equivalent diagrams are implied. The effective charge is given by the r a t i o of t h e s e d i a g r a m s to t h e z e r o t h o r d e r d i a g r a m in i s o s c a l a r a n d i s o v e c t o r n o t a t i o n , f r o m which the neutron and proton values may be ded u c e d ( e . g . , s e e r e f . [3]). E x p r e s s i o n s f o r t h e d i a g r a m s in fig. 1 h a v e b e e n g i v e n b y S i e g e l a n d Z a m i c k [3], in t h e i r c a l c u l a t i o n of t h e s e c o n d order effective charge with the semi-realistic K a l l i o - K o l l t v e i t (KK) f o r c e . (In c a r e f u l l y c h e c k i n g t h e i r f o r m u l a t i o n we u n f o r t u n a t e l y f o u n d a 1 f a c t o r of ~ m i s s i n g f r o m t h e e x p r e s s i o n s f o r
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15 February
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diagrams 5, 9, 18, 19, 28 and 29; the c o r r e c t e d KK totals a r e given in table 2.) In ref. [3] the wave functions were explicitly normalised by factors of (1 + x)- I / 2 , whereas following Brandow [2] we expand this as (1 -½x) and write the diagrams in folded form. The factor of ½ is shown on the diagrams since it does not follow from the usual rules. We have used two s e t s of the essentially exact G-matrix elements of B a r r e t t , Hewitt and McCarthy [5]. Therefore diagrams 28-31 of fig. 1 should not be included since the p a r t i c l e p a r t i c l e ladders have been properly taken into account. Set BHM1 corresponds to a starting energy of -3 MeV, which introduces a gap b e tween occupied and unoccupied levels (note that s t a t e s with one and two p a r t i c l e s in the unoccupied space a r e shifted equally). These matrix elements a r e fairly s i m i l a r to those calculated r a t h e r differently by Kuo and Brown (KB). These authors also used the Hamada-Johnston potential and introduced a gap [6]. The set BHM2 c o r r e sponds to a starting energy of 82 MeV which means essentially an unshifted harmonic o s c i l l a tor spectrum. The contributions of the second o r d e r diagrams 4-27 of fig. 1 a r e listed in table 1 for the BHMI case. We have included the 0s through the
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51
Fig. i. Perturbation theory diagrams discussed in the evaluation of the effective charge. Here E is the E2 electromagnetic operator.
Table i Contributions of the secondorder diagrams to the effective charge for neutrons and protons using the BHM1 matrix elements.
Diagram number
d~-
ds~.
d~ -
s½
d~-2
d3~
s½ -
dl2
N
P
N
P
N
P
N
P
4 5 6 7 8 9 10 11 12 13
0.026 -0.013 0 0 0.009 -0.013 0 0 0 0
Subtotal
0.008
14 15 16 17
18 19 20 21 22 23 24 25 26 27
178
0 0 0 0
0.005 0.005 -0.010 -0.010 -0.014 -0.014 0.033 0.033 0.010 0.010
d~-
d~
N
P
0.027 0.027 -0.003 -0.010 -0.041 0 -0.041 0 0.012 0.002 -0.004 -0.009 -0.037 0 -0.037 0 0.019 0 0.019 0
0.030 -0.003 -0.041 -0.050 0.003 -0.003 -0.043 -0.037 0.019 0.022
0.038 -0.013 0 0 0.008 -0.012 0 0 0 0
0.040 -0.003 -0.041 -0.053 0.004 -0.003 -0.071 -0.037 0.019 0.035
0.036 -0.012 0 0 0.011 -0.022 0 0 0 0
0.033 -0.004 -0.050 -0.053 0.015 -0.003 -0.071 -0.043 0.022 0.035
0.040 -0.019 0 0 0.021 -0.026 0 0 0 0
0.037 -0.003 -0.053 -0.053 0.020 -0.005 -0.071 -0.071 0.035 0.035
-0.087
0.010
-0.105
0.021
-0.111
0.013
-0.120
0.016
-0.129
0 0 0 0
0.005 0 -0.024 0
0 0 0 0
0 0 0 0
0 0.005 0 -0.024
0 0 0 0
0 0 0 0
0.002 0.002 -0.020 -0.020 -0.026 -0.026 0.045 0.045 0.014 0.014
0 0 0 0
0 0 0 0
0.004 0.001 0.005 0.002 0.002 -0.001 0.003 0.001 - 0 . 0 0 4 - 0 . 0 0 1 -0.005 0.002 0.004 0.001 0.003 0.001 - 0 . 0 0 7 - 0 . 0 0 4 - 0 . 0 2 1 - 0 . 0 1 1 - 0 . 0 1 8 - 0 . 0 2 1 - 0 . 0 2 0 -0.023 - 0 . 0 1 2 - 0 . 0 1 5 - 0 . 0 1 2 - 0 . 0 1 8 - 0 . 0 1 1 - 0 . 0 1 8 - 0 . 0 2 0 -0.023 - 0 . 0 1 1 - 0 . 0 1 7 - 0 . 0 1 8 - 0 . 0 1 9 - 0 . 0 1 3 - 0 . 0 2 6 - 0 . 0 1 9 -0.028 - 0 . 0 1 6 - 0 . 0 2 1 - 0 . 0 2 1 - 0 . 0 2 2 - 0 . 0 0 8 - 0 . 0 1 8 - 0 . 0 1 9 -0.028 0.016 0.023 0.027 0.045 0.012 0.028 0.025 0.040 0.035 0.043 0.053 0.057 0.020 0.024 0.025 0.040 0.010 0.012 0.020 0.022 0.005 0.007 0.010 0.014 0.007 0.009 0.011 0.014 0.003 0.006 0.010 0.014
Volume 34B, number 3
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(0flp) s h e l l s and allowed all p o s s i b l e i n t e r m e diate s t a t e s . The e n e r g y d e n o m i n a t o r s w e r e taken to be m u l t i p l e s of/~¢o = 14 MeY. The c o n t r i b u t i o n s l i s t e d under, e . g . , d ~ - s½ r e f e r to an i nit i a l 0d~ and a final l s ~ state. The d i a g r a m s 4-7 and ~-13 f o r m two n ~ m b e r c o n s e r v i n g s e t s , i . e . , if the E2 o p e r a t o r is r e p l a c e d by the n u m b e r o p e r a t o r t h es e graphs give i d e n t i c a l l y z e r o . Brandow [2] has s u g g e s t e d that such s e t s might c a n c e l to a c o n s i d e r a b l e d e g r e e and f o r the a n a l ogous d i a g r a m s in the e f f e c t i v e i n t e r a c t i o n (where the E o p e r a t o r is r e p l a c e d by a v a l e n c e line) B a r r e t t and K i r s o n found this to be t r u e [7]. In the p r e s e n t c a s e the neutron subtotals f o r d i a g r a m s 4-13 (see table 1) a r e quite s m a l l . Howe v e r , f o r the p r o t o n s the totals a r e s i z e a b l e and the c a n c e l l a t i o n is quite poor. (Note that f o r our i n t e r m e d i a t e s t a t e s d i a g r a m s (10+ 11) = - 2 × d i a g r a m s (12 + 13).) D i a g r a m s 14 and 15 give v e r y s m a l l co n t ri b u t i o n s as do the h o l e - h o l e i n t e r a c t i o n d i a g r a m s 18 and 19. One m a y wish to include the i n s e r t i o n s of d i a g r a m s 16 and 17 in the single p a r t i c l e p o t e n ti a l definition [2]; we have c a l c u l a t e d the d i a g r a m s anyway and found them to be s m a l l . D i a g r a m s 20-23 which in volve a particle-hole interaction are sizeable and n e g a t i v e and tend to c a n c e l the TDA and RPA d i a g r a m s 24-27. The effect is not so pronounced as it is f o r the analogous d i a g r a m s of the e f f e c t i v e i n t e r a c t i o n w h e r e t h e r e is g e n e r a l l y an o v e r c a n c e l l a t i o n [7]. C o m p a r i s o n of our r e s u l t s with those of B a r r e t t and K i r s o n [7] f o r the e f fective interaction indicates a strong correlation in sign b et ween analogous graphs. A l m o s t w it h out exception, a graph which i n c r e a s e s the e f f e c t i v e c h a r g e ( m o r e p o s i ti v e ) i n c r e a s e s the e f f e c t i v e i n t e r a c t i o n ( m o r e negative), and v i c e v e r s a . D i e p e r i n k and B r u s s a a r d [8] have c a r r i e d out a second o r d e r ca l c u la ti o n in the c a l c i u m r e gion. C o m p a r i s o n with our r e s u l t s i n d ic a t e s that the s i z e s of the d i a g r a m s a r e quite s i m i l a r ,
15 February 1971
a p a r t f r o m the TDA and RPA d i a g r a m s (24-27) which a r e a f a c t o r of t h r e e l a r g e r than in the o x ygen region. They thus obtain a l a r g e r p o s i t i v e second o r d e r contribution. The z e r o t h o r d e r e f f e c t i v e c h a r g e is t r i v i a l l y z e r o f o r n e u t r o n s and one f o r protons. The total f i r s t and second o r d e r contributions a r e l i s t e d in table 2. They a r e i n v a r i a n t under i n t e r c h a n g e of initial and final s t a t e s s i n c e the d i a g r a m s a r e e i t h e r s y m m e t r i c under i n t e r c h a n g e o r o c c u r in p a i r s which t r a n s f o r m into one a n o t h e r under i n t e r c h a n g e of initial and final quantum n u m b e r s . In addition, the KK totals depend only on the s i n gle p a r t i c l e o r b i t a l an g u l ar m o m e n t a s i n c e the f o r c e is c e n t r a l [9]. The f i r s t o r d e r r e s u l t s ( d i a g r a m s 2 and 3) a r e s i m i l a r f o r the BHM1 and KB c a s e s . The BHM2 m a t r i x e l e m e n t s yield a much i n c r e a s e d value f o r n e u t r o n s and a s m a l l e r i n c r e a s e f o r p r o t o n s si n ce the p r o t o n v a l u e s d e pend only on the T = 1 m a t r i x e l e m e n t s which a r e known to be l e s s s e n s i t i v e to the s t a r t i n g e n e r g y than those with T = 0 [5]. As r e g a r d s the KK i n t e r a c t i o n , we r e m a r k only that in f i r s t and second o r d e r it g i v es r e s u l t s in q u a l i t a t i v e a g r e e m e n t with the o t h er c a s e s . As can be seen , the r e a l i s t i c m a t r i x e l e m e n t s y i el d total second o r d e r proton c h a r g e s which a r e f a i r l y s m a l l and a l m o s t always negative. They tend to c a n c e l the f i r s t o r d e r v a l u e s indeed f o r the s , -d~ c a s e f i r s t plus second o r d e r is n e g a t i v e 2. Th~ neutron v a l u e s a r e s m a l l and g e n e r a l l y p o s i t i v e , being roughly half the TDA plus RPA v a l u e s f o r the BHM1 and KB c a s e s . We have c o m p a r e d the BHM1 c a s e to the r e s u l t s obtained by allowing the e n e r g y d e n o m i n a t o r s to be 2/~w only (including d i a g r a m s 26 and 27). The r e s u l t s a r e quite s i m i l a r and c o m p a r i s o n of the v a r i o u s d i a g r a m s i n d i c a t e s that the 2~w contributions dominate. The KB v a l u e s a g r e e well with the BHM1 c a s e ; this is t r u e f o r the individual d i a g r a m s as well as the totals. Note that we have not included the
Table 2 First and second order effective charges calculated with various matrix elements.
1st Order
2nd Order
d~-
ds_
d~-
sl
ds-
d½
sl-
d3
d3-
d3
N
P
N
P
N
P
N
P
N
P
BHM1 KB BHM2 KK
0.258 0.329 0.615 0.406
0.099 0.103 0.145 0.144
0.219 0.269 0.512 0.285
0.115 0.119 0.147 0.101
0.310 0.371 0.678 0.406
0.191 0.187 0.238 0.144
0.192 0.239 0.490 0.285
0.057 0.053 0.088 0.101
0.269 0.331 0.640 0.406
0.104 0.099 0.150 0.144
BHM1 B H M 1 (2 h~¢0} KB BHM2 KK
0.056 0.066 0.058 0.068 0.070
-0.057 -0.036 -0.038 -0.031 0.012
0.035 0.043 0.046 0.028 0.026
-0.095 -0.049 -0.069 -0.176 -0.059
0.058 0.081 0.072 0.104 0.070
-0.039 -0.019 -0.019 0.001 0.012
0.008 0.027 0.010 -0.037 0.026
-0.156 -0.089 -0.132 -0.276 -0.059
0.011 0.037 0.020 0.008 0.070
-0.124 -0.083 -0.093 -0.121 0.012
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l a d d e r d i a g r a m s 2 8 - 3 1 in c o m p u t i n g t a b l e 2. D i a g r a m s 28 a n d 29 g i v e a s i g n i f i c a n t c o n t r i b u t i o n f o r n e u t r o n s ( ~ + 0.06 w i t h t h e KB m a t r i x e l e m e n t s ) . T h u s it w o u l d a p p e a r t h a t t h e l a d d e r s h a v e b e e n a d e q u a t e l y t a k e n i n t o a c c o u n t in t h e K u o - B r o w n m a t r i x e l e m e n t s , in a g r e e m e n t w i t h t h e r e c e n t a r g u m e n t s of K i r s o n [10]. T h e B H M 2 t o t a l s a r e in q u i t e good a g r e e m e n t w i t h t h e BHM1 a n d KB v a l u e s . T h i s r e s u l t i s s o m e w h a t s u r prising since the individual diagram contribut i o n s a r e m u c h l a r g e r b y t y p i c a l l y f a c t o r s of 2-4. B a r r e t t a n d K i r s o n [7] s u g g e s t e d a s l a s h a b i l i ty c l a s s i f i c a t i o n w h e r e f o r a g i v e n d e g r e e of s l a s h a b i l i t y t h e d i a g r a m s w o u l d c o n v e r g e in t e r m s of t h e n u m b e r of G - m a t r i x i n t e r a c t i o n s . Thus, for the zero-times slashable diagrams o n l y d i a g r a m 1 w h i c h d o e s not c o n t a i n a G - i n t e r a c t i o n w o u l d b e i n c l u d e d . T h e r e a r e no f i r s t o r d e r z e r o - t i m e s s l a s h a b l e d i a g r a m s , b u t in s e c ond o r d e r d i a g r a m s 4 - 1 7 of fig. 1 c o n t r i b u t e . These include the number-conserving sets (diag r a m s 4 - 1 3 ) w h i c h a r e l a r g e f o r p r o t o n s ( ~ - 0.1 for BHM1) so that this classification would not a p p e a r to b e a p p l i c a b l e h e r e . E x p e r i m e n t a l l y , one r e q u i r e s e f f e c t i v e c h a r g e s ~ 0.5 a n d b y s u i t a b l e c h o i c e of t h e s t a r t ing e n e r g y n e u t r o n v a l u e s of t h i s o r d e r of m a g nitude can be obtained. However, the second ord e r p r o t o n v a l u e c a n c e l s a good p a r t of t h e f i r s t o r d e r v a l u e , a t l e a s t if t h e H a r t r e e - F o c k d i a g r a m s a r e n e g l e c t e d . T h u s t h e c o n v e r g e n c e of t h e p e r t u r b a t i o n e x p a n s i o n in t e r m s of t h e n u m b e r of G - i n t e r a c t i o n s a p p e a r s to b e p o o r h e r e a s
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w a s t h e c a s e f o r t h e e f f e c t i v e i n t e r a c t i o n [7]. W e a r e g r a t e f u l f o r t h e u s e of t h e m a t r i x e l e m e n t s of T. T. S. Kuo a n d of B. R. B a r r e t t , R . G . L . H e w i t t a n d R. J . M c C a r t h y . W e t h a n k B. R. B a r r e t t , H . A . M a v r o m a t i s a n d L. Z a m i c k f o r useful discussions.
Reference s [1] A. Bohr and B. R. Mottelson, N~elear s t r u c t u r e I (Benjamin, New York, 1969) p. 342. [2] B.H.Brandow, Revs. Mod. Phys. 39 (1967) 771; B. H. Brandow, L e c t u r e s in theoretical physics XI-B, eds. K. T. Mahanthappa and W. E. Brittin (Gordon and Breach, New York, 1969) p. 55. [3] S. Siegel and L. Zamick, Nucl. Phys. A145 (1970) 89. [4] H. A. M a v r o m a t i s and B. Singh, Nucl. Phys. A139 (1969) 451; P . J . E l l i s and S.Siegel, Nucl. Phys. A152 (1970) 547. [5] B . R . B a r r e t t , R . G . L . H e w i t t and R . J . M c C a r t h y , Phys. Rev., to be published; B. R. B a r r e t t , R . G . L . Hewitt and R . J . McCarthy, preprint. [6] T . T . S . K u o and G . E . Brown, Nucl. Phys. 85 (1966) 4O; T . T . S . K u o , Nucl. Phys. A103 (1967) 71. [7] B.R. B a r r e t t and M. W. Kirson, Phys. L e t t e r s 30B (1969) 8; B. R. B a r r e t t and M. W. Kirson, Nucl. Phys. A148 (1970) 145. [8] A. E. L. Dieperink and P. J. B r u s s a a r d , P r o c . Int. Conf. on P r o p e r t i e s of nucl. states (Montreal, 1969) p. 715. [9] P. F e d e r m a n and L. Zamick, Phys. Rev. 177 (1969) 1534. [lO] M . W . K i r s o n , Phys. L e t t e r s 32B (1970) 33.