The isovector monopole state and Coulomb displacement energies

Volume 36B, number 4

P HYSICS L E T T ER S

AND

THE ISOVECTOR MONOPOLE COULOMB DISPLACEMENT

20 September 1971

STATE ENERGIES

N. A U E R B A C H

Department of Physics and Astronomy, Tel-Aviv University, Ramat-Aviv, Israel Received 25 July 1971 A correction to the Coulomb displacement energies arising from the isovector monopole state is c a l culated using a stun rule approach.

A s e r i o u s d i f f i c u l t y a r i s e s w h e n the C o u l o m b d i s p l a c e m e n t e n e r g y i s c a l c u l a t e d in m i r r o r n u c l e i s u c h a s 1 7 0 - 1 7 F , 4 1 C a - 41Sc e t c . [1]. A f t e r isolating several small terms: exchange, electrom a g n e t i c s p i n - o r b i t e t c . [2], one i s l e f t with the m a i n c o n t r i b u t i o n , t h e so c a l l e d d i r e c t t e r m , g i v e n by

E DIR =e2 f I ~ n l j ( r l ) t 2

pP(r2) dr 1 dr 2 1~1=722]

(1)

w h e r e ~nlj i s t h e w a v e f u n c t i o n of the e x c e s s n e u t r o n o u t s i d e t h e N = Z c a s e , ( f o r e x a m p l e in 4 1 C a t h i s i s the w a v e f u n c t i o n of the l f 7 / 2 n e u t r o n ) and pP the c h a r g e d i s t r i b u t i o n of the c o r e m e a s u r e d in e l a s t i c e l e c t r o n s c a t t e r i n g . W h e n o n e c a l c u l a t e s t h e w a v e f u n c t i o n of the e x c e s s n e u t r o n in a W o o d s - S a x o n p o t e n t i a l w h i c h f i t s t h e e l e c t r o n s c a t t e r i n g c h a r g e d i s t r i b u t i o n one o b t a i n s a C o u l o m b d i s p l a c e m e n t e n e r g y which i s a p p r o x i m a t e l y 0.5 MeV s m a l l e r t h a n t h e m e a s u r e d v a l u e . In o t h e r w o r d s , in o r d e r to r e p r o d u c e the m e a s u r e d C o u l o m b e n e r g y o n e h a s to use for the excess neutron a well radius which is 20% s m a l l e r t h a n t h e W o o d s - S a x o n r a d i u s of t h e c o r e p r o t o n s [1]. T h i s r e s u l t s in a n e u t r o n e x c e s s r o o t m e a n s q u a r e r a d i u s ( r . m . s . ) ( r 2 ) 1/2 of about t h e s a m e v a l u e a s c o r e r . m . s , r a d i u s . Such a r e s u l t in 1 7 0 o r 4 1 C a i s d i s t u r b i n g f r o m t h e p o i n t of v i e w of t h e s h e l l m o d e l . In o r d e r to s o l v e t h i s p r o b l e m it was s u g g e s t e d [3] that a s m a l l a d m i x t u r e of the T = 1, J = 0 + i s o v e c t o r m o n o p o l e s t a t e [4] into the T = 0, J = 0 + core will resolve the discrepancy. This admixt u r e p r o d u c e s a d i f f e r e n c e in t h e p r o t o n and n e u t r o n d e n s i t i e s in t h e c o r e w h i c h in t u r n i n d u c e s a small symmetry potential. This potential acts d i f f e r e n t l y on p r o t o n s and n e u t r o n s and t h e r e f o r e g i v e s r i s e to a C o u l o m b e n e r g y c o r r e c t i o n . H o w e v e r , in o r d e r to c a l c u l a t e t h i s c o r r e c t i o n t h e a u t h o r s of r e f . [3] h a v e m a d e two m a j o r a s s u m p -

t i o n s . I) T h e y a s s u m e d an a r b i t r a r y v a l u e f o r the T = I a d m i x t u r e to the T = 0 c o r e . 2) T h e y a d j u s t e d the s h a p e of the d i s t r i b u t i o n P l = PP - p n to g i v e the p r o p e r m a g n i t u d e and s i g n f o r the correction. In the p r e s e n t w o r k we e v a l u a t e the c o r r e c t i o n t e r m a v o i d i n g the a s s u m p t i o n s m a d e in r e f . [3]. In p a r t i c u l a r we a r e a b l e to study in a l e s s m o d e l d e p e n d e n t way the s h a p e of the P l d e n s i t y by u s i n g an i s o v e c t o r s u m r u l e . We w i l l show that t h e c o n t r i b u t i o n f r o m the p o l a r i z a t i o n of the c o r e to the C o u l o m b e n e r g y i s m u c h s m a l l e r and of o p p o s i t e s i g n t h a n in r e f . [3]. We f i r s t w r i t e an e x p r e s s i o n f o r the T = l c o r e p o l a r i z a t i o n c o r r e c t i o n , v a l i d f o r a n u c l e u s with any n u m b e r of e x c e s s n e u t r o n s . In the p a r e n t n u c l e u s (i.e. n u c l e u s with TO = ½(N - Z) the i s o v e c t o r m o n o p o l e s t r e n g t h will be s p l i t into two g r o u p s of s t a t e s ; t h o s e with i s o s p i n T = T o + I and t h o s e with T = T o *. In a n a l o g y to t h e g i a n t d i p o l e s t a t e one a s s u m e s that m o s t of t h e m o n o p o l e s t r e n g t h i s e x h a u s t e d by a s i n g l e s t a t e f o r e a c h i s o s p i n T o + 1 and To, by the s o - c a l l e d i a n t i s o v e c t o r m o n o p o l e s t a t e ] M ) To + 1 and M)To. T h i s a s s u m p t i o n h a s b e e n b o r n e out by c a l c u l a t i o n s [5, 6]. T h e s e c a l c u l a t i o n s a l s o i n d i c a t e that due to the r e s i d u a l p a r t i c l e - h o l e i n t e r a c t i o n the e x c i t a t i o n e n e r g y of the i s o v e c t o r monopole state is moved upwards from its unperturbed position, 2/~. The hydrodynamical model a c t u a l l y p u t s the T = I m o n o p o l e s t a t e at about 4~w [4]. T h e t M ~ T o + 1 c o m p o n e n t of t h e i s o v e c t o r m o n o p o l e i s the m a i n s o u r c e of i s o s p i n m i x i n g in the g r o u n d s t a t e [4] w h i l e the T = T o c o m p o n e n t I M)To p r o d u c e s the C o u l o m b e n e r g y c o r r e c t i o n considered here. S i n c e the g r o u n d s t a t e h a s t h e s a m e i s o s o i n a s

~

* In the analog nucleus there is an additional component with T = T o - 1. 293

PHYSICS

Volume 36B, number 4

the state IM)To t h e r e i s a strong i n t e r a c t i o n m a t r i x e l e m e n t which a d m i x e s the two. If we denote the u n p e r t u r b e d p a r e n t ground state by 17r> and the admixed one by I ~ ' ) then in f i r s t o r d e r p e r t u r b a t i o n theory:

1~') = [~) + ElM)To with

<~1vNUClM>To E --

(2)

EMT ° - E o

where E o and EMT e a r e the u n p e r t u r b e d e n e r g i e s of the ground state and the [M>T° state. If we neglect i s o s p i n mixing i n the ground state the exp r e s s i o n for the total Coulomb d i s p l a c e m e n t e n ergy b e c o m e s [2, 7]: E TOT = 2T1° <,'

][T+, [M, T_]] [='>

-

1 2T ° <~1 [T+, [H,

T_]] In)

+ AE d

(4)

£ AEd : To ~1 [T+, [H,

T-]]]M)To

AE d iS the c o r r e c t i o n t e r m we a r e dealing with in the p r e s e n t work. F o r the c o m m u t a t o r [H, T_] we take only the one-body Coulomb i n t e r a c t i o n , neglecting the two=body Coulomb i n t e r a c t i o n and other s m a l l i s o s p i n violating p a r t s of the h a m i l t o n i a n [2]. E v a l u a t i n g the double c o m m u t a t o r we obtain for AE d the e x p r e s s i o n :

1 Ze2<~lVlolM}ToX To R3

(Tr[vNUC[M)T 0 (5) (EMTo EO)

where A V1 = ~ r 2 , t z ( i ) i=1 i s the ~ = 0 component of the i s o v e c t o r monopole o p e r a t o r V~ and R = 1.4 A1/3 the charge r a d i u s of the nucleus. It is p o s s i b l e to e s t i m a t e the m a t r i x e l e m e n t s (g [ V~1 [M) without invoking s~ecific models, if one a s s u m e s that the state [ M ) exhaust the e n t i r e i s o v e c t o r monopole s t r e n g t h sum r u i e . It i s these m a t r i x e l e m e n t s which 294

e n t e r the c a l c u l a t i o n of i s o s p i n mixing in the ground states and other Cdulomb mixing p r o c e s s e s [6]. We now t u r n to the evaluation of the n u c l e a r m a t r i x element, eq. (5), for m i r r o r nuclei. (The g e n e r a l i z a t i o n to nuclei with m o r e than one exc e s s n e u t r o n i s s t r a i g h t forward). We r e p r e s e n t the p a r e n t state I~} as a n e u t r o n in a ~nlj(r) orbit coupled to the J= 0+, T = 0 core - iI~o. The state IM) To=½ may be d e s c r i b e d to a good a p p r o x i m a t i o n as the s a m e n e u t r o n in state ~nlj(r) coupled to the J = 0+, T = 1 m o n o pole state of the c o r e -~1. The n u c l e a r m a t r i x element becomes:

<~[vNUC[

=

M) TO = ½

(6) 1

-~f~ The - 1 / 4 - 3 f a c t o r is the C l e b s c h - G o r d o n coefficient r e s u l t i n g f r o m the coupling of the t = ½ i s o spin of the n e u t r o n to the T = 1 i s o s p i n of the core to give total i s o s p i n To = ½. If we a s s u m e a zero range and density independent n u c l e a r i n t e r a c t i o n , we can write the m a t r i x e l e m e n t of eq. (6) as:

<~I vNUCIM}To =½

with

Ed -

20 September 1971

(3)

where T+ and T - a r e the i s o s p i n r a i s i n g and lowering o p e r a t o r s . Using eq. (2) and (3) and neglecting the t e r m p r o p o r t i o n a l to c 2 we obtain E TOT

LETTERS

(7)

=_~_(svT 4_ vS) f iW~0(r)l 2 P0M Vo ( r ) r2dr where VT and VS a r e the v o l u m e i n t e g r a l s of the the t r i p l e t and singlet components of the n u c l e a r Vo force. P0M is the /~=0 component of the i s o v e c t o r monopole t r a n s i t i o n density f r o m the ground state to the T= 1 monopole state, The i n t e g r a l f P 0 M r 2 d r = 0, and t h e r e f o r e

Vo

P0M (r) has a node. The value and sign of the m a t r i x e l e m e n t of eq. (7) depend in a s e n s i t i v e m a n n e r on the p o s i t i o n of the node of P0M V° (r). The basic difficulty in c a l c u l a t i n g the m a t r i x e l e m e n t of eq(8) lies in a model independent de-

Vo

r i v a t i o n of P0M (r).

Vo

Our evaluation of P0M i s b a s e d on a sum r u l e approach. In a r e c e n t p a p e r [8] a s u m r u l e for the i s o s c a l a r monopole f o r m f a c t o r has b e e n d e r i v e d and used to study the excitation of the f i r s t excited J= 0+, T= 0 state i n 160 by e l e c t r o n s c a t t e r i n g . Under c e r t a i n conditions s i m i l a r sum r u l e s can be d e r i v e d for the c o m p o n e n t s

Volume 36B, number 4

PHYSICS

of the i s o v e c t o r monopole f o r m f a c t o r s [9]. In the c a s e of a z e r o r a n g e n u c l e a r ' i n t e r a c t i o n the f o l lowing r e l a t i o n holds f o r the F o u r i e r t r a n s f o r m o f the i s o v e c t o r monopole o p e r a t o r V~z= 0:

~2

(8)

[ v l ( - q ' ) , [ H , V l o ( q ) ] ] = ~-~ q " q A ( q - q')

~ h e r e H i s the total hamilton.tan and A(q) i s the F o u r i e r t r a n s f o r m of the i s o s c a l a r d e n s i t y o p e r a t o r . Taking the c o r e g r o u n d s t a t e e x p e c t a t i o n v a l u e of eq. (8) and i n t r o d u c i n g i n t e r m e d i a t e s t a t e s in the u s u a l m a n n e r one can obtain [9]:

(En - E g . s . ) ( ~ o Ivl[n> F o n ( q ) =

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20 September 1971

into eq. (7) and (5) we obtain for the Coulomb e n ergy correction

1 Ze 2 ~2 1 3 V T - VS A E d - ~f3 R 3 2m (EMT ° _ Eo)2 4

fl~nlJ (r)12

3 o ( r ) + r d pd(rr ) J] r 2 d r

(13)

Note that the m a t r i x e l e m e n t (nl Vlo [M>T o d o e s not a p p e a r in this final e x p r e s s i o n . The e x p r e s sion

1 Ze 2 ~2 Pl = 2 4"3 R3 2m

1 ( E M T ° - Eo) 2

V~ POM(r)

/

n

/

/~2q2

2m-

d

~ - F° (q)

(9)

where Fo(q) is the Fourier transform of the ground state isoscalar density and Vo r2 Fon(q ) = 4n fPon (r) jo(qr) dr

(10)

i s the F o u r i e r t r a n s f o r m of the p = 0 component TF

of the i s o v e c t o r d e n s i t y PoVn g bet~,een the ground s t a t e [*o> and a s t a t e In>. Eg.s" is the c o r e ground s t a t e e n e r g y . We a s s u m e that the sum on the left side of eq. (9) i s e x h a u s t e d by the giant i s o v e c t o r state. Then eq. (9) b e c o m e s :

(11)

FOM(q 2) =

n2

Vo

P0M (r) =

2m <*oI VoJ

IC

E

2

E 1 i s the e n e r g y of the i s o v e c t o r monopole s t a t e in the c o r e . T r a n s f o r m i n g this equation into s p a c e c o o r d i n a t e s we get:

~2

Vo we s u b s t i t u t e f o r p(r) the function ( A / Z ) P0M p P ( r ) , w h e r e pP i s the c h a r g e d i s t r i b u t i o n m e a s u r e d in e l e c t r o n s c a t t e r i n g . T h i s p r o c e d u r e i s s o m e w h a t i n c o n s i s t e n t s i n c e pP i s s l i g h t l y d i f f e r -

q2 ~ d F o (q2)

i

2m <*o IVlo 14)1> ( E l - E g . s . )

---

r e p r e s e n t s the d i f f e r e n c e between the neutron and p r o t o n d i s t r i b u t i o n s pn _ pp in the ground s t a t e of the c o r e . Multiplying it by (3VT - VS)/4 , we get the c o r r e s p o n d i n g s y m m e t r y p o t e n t i a l . AE d t h e r e f o r e may be i n t e r p r e t e d a s the s y m m e t r y e n e r g y of the e x t r a nucleon. In o r d e r to c a l c u l a t e the t r a n s i t i o n d e n s i t y

(12)

½F3p (r) + r

1

-

L

dp(r) l

dr J

w h e r e p i s the g r o u n d s t a t e n u c l e a r density. If we n e g l e c t the slight d i f f e r e n c e between the <~1 and <,olVlol¢l> matrix e l e m e n t s a s

Vlo IM>To

well a s the d i f f e r e n c e between the s p a c i n g s E M T ° - E o and E 1 - E g . s . then substituting eq.(12)

r

-4

/

-6 /,%~o~M(r) -8 -10 -12 -14 -16L

Vo Fig. 1. The isovector monopole transition density p 0l~i(r) and ~1f7/2 (r) wave function 41Ca. The two graphs have different and arbitrary ordinate units. 295

Volume 36B, number 4

PHYSICS

Table 1 The calculated direct term of the Coulomb displacement energy and the correction t e r m for orbits in 41Ca and 170. Both E ~ I R and A E d are given in MeV. Orbit

E ~ IR

AE d

41~a

if7/2 ld3/2

7.0 7.6

-0.011 -0.060

170

ld5/2

3.4

-0.014

lPl/2

3.7

-0.033

ent f r o m pn. H o w e v e r , t h i s i n c o n s i s t e n c y a f f e c t s t h e c a l c u l a t i o n of AE d only in s e c o n d o r d e r . T h e

Vo i s shown in fig.1. c a l c u l a t e d t r a n s i t o n d e n s i t y P0M T h e w a v e f u n c t i o n of the l f T / 2 n e u t r o n was c a l c u l a t e d in a W o o d s - S a x o n p o t e n t i a l with p a r a m e t e r s c h o s e n to r e p r o d u c e the d i s t r i b u t i o n p P ( r ) . T h e f u n c t i o n I ~ f 7/2 (r)[ 2 × r 2 is shown a l s o in

Vo

fig. 1. The integral fP0M[ Cf 7/2[ 2 r2 dr is a r e sult Of a d e l i c a t e c a n c e l l a t i o n of two c o n t r i b u t i o n s , a t t r a c t i v e in the i n t e r i o r and r e p u l s i v e at the e x t e r i o r . T h e f i n a l r e s u l t f o r a l l c a s e s we c o n s i d e r e d i s a t t r a c t i v e and the c o r r e c t i o n A E d i s of t h e w r o n g s i g n to r e s o l v e the d i s c r e p a n c y in the C o u l o m b d i s p l a c e m e n t e n e r g i e s . To c a l c u l a t e the a b s o l u t e v a l u e s of A Ed we h a v e to know the v a l u e s of (3V T - VS),/4 and E M T - E o. The f i r s t c a n be e s t i m a t e d f r o m the s y m m ° e t r y p o t e n t i a l , and i s about - 300 MeV fm 3. F o r t h e e n e r g y d i f f e r e n c e we u s e d the v a l u e 170A - 1 / 3 (about 4~w) d e r i v e d in t h e h y d r o d y n a m i c a l m o d e l [4]. T h e r e s u l t s of the c a l c u l a t i o n of the d i r e c t t e r m , eq. (1), and c o r r e c t i o n t e r m , eq. (13), f o r 1 7 0 and 41Ca a r e shown in t a b l e 1. We s e e that the c o r r e c t i o n s a r e v e r y s m a l l f o r the o u t e r o r b i t s and of the w r o n g s i g n to h e l p i m p r o v e the s i t u a t i o n . T h e inner orbits have somewhat larger negative ~E d corrections because their wave functions are p e a k e d c l o s e r to the o r i g i n and the e f f e c t of the c a n c e l l a t i o n is s m a l l e r . One of the u n d e r l y i n g a s s u m p t i o n s in the p r e s e n t t r e a t m e n t of A E d was the d e n s i t y i n d e p e n d e n c e of the e f f e c t i v e n u c l e a r f o r c e . A s t r o n g d e n s i t y d e p e n d e n t f o r c e which s u p p r e s s e s the i n t e r a c t i o n in the i n t e r i o r and a c t s m a i n l y at the s u r f a c e m a y p r o d u c e a t e r m A E d of the r i g h t s i g n and v a l u e to r e m o v e the d i s c r e p a n c y . I n d e e d in a r e c e n t p a p e r [10] an a t t e m p t w a s m a d e to c a l c u l a t e the P l = PP - p n d e n s i t y f r o m a v a r i a t i o n a l p r i n c i p l e *. U s i n g t h e n a n u c l e a r i n t e r a c t i o n with a 1/p d e n s i t y d e p e n d e n c e the a u t h o r s h a v e 296

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20 September 1971

found t h e c o r r e c t i o n A E d to b e r e p u l s i v e and l a r g e , Such a s t r o n g d e n s i t y d e p e n d e n c e s e e m s however unrealistic. Introducing a weaker density d e p e n d e n c e , n a m e l y of the f o r m (1 - (~pk) with k = 1 o r 2 and ot -.< 0.5 we c o u l d not get the d e s i r e d v a l u e f o r the c o r r e c t i o n . C o n c l u s i o n s s i m i l a r to o u r s h a v e b e e n r e a c h e d r e f . [11] and [12] w h e r e a d i f f e r e n t a p p r o a c h h a s b e e n u s e d . T h e a d d e n t i o n a l s y m m e t r y p o t e n t i a l of the c o r e h a s b e e n c a l c u l a t e d in the f r a m e w o r k of a H a r t r e e - F o c k t h e o r y with d e n s i t y d e p e n d e n t f o r c e s . T h e d e n s i t y d e p e n d e n c e was not a s s t r o n g a s in r e f . [10] and the r e s u l t i n g c o r r e c t i o n t e r m was s m a l l . It w a s a l s o n o t e d s e v e r a l y e a r s ago that one o b t a i n s the w r o n g s i g n f o r the i s o t o p e shift when u s i n g d e n s i t y - i n d e p e n d e n t f o r c e s [13, 14]. We m i g h t c o n c l u d e that u n l e s s the n u c l e a r f o r c e i s v e r y s t r o n g l y d e n s i t y d e p e n d e n t the c o n t r i b u t i o n of the i s o v e c t o r m o n o p o l e s t a t e c a n n o t r e s o l v e the e x i s t i n g d i s c r e p a n c y b e t w e e n the c a l c u l a t e d and e x p e r i m e n t a l C o u l o m b d i s p l a c e m e n t energies. We would l i k e to thank P r o f e s s o r A. K. K e r m a n f o r helpful d i s c u s s i o n s and P r o f e s s o r G. E. B r o w n f o r u s e f u l c o r r e s p o n d e n c e c o n c e r n i n g the i s o v e c t o r m o n o p o l e s t a t e and the i s o t o p e shift. * It is worth mentioning that the shape of the density P l derived in ref. [10] might be obtained from eq. (12) by using for p(r) a Gaussian distribution.

References [1] J . A . Nolen and J. P. Schiffer, Ann. Rev. Nucl. Sci. 19 (1969) 471. [2] N. Auerbach, J. HUfner, A.K. Kerman and C.M. Shakin, Phys. Rev. Letters 23 {1969) 484. [3] E.H. Auerbach, S. Kahana and J. Weneser, Phys. Rev. Letters 23 (1969) 1253. [4] A. Bohr, J. Damgaard and B. R. Mottelean, Nuclear Structure, p.1, eds. Hossain, Haren-ar-Rashid, M. Islam (North-Holland, Amsterdam 1967). [5] J. Damgaard, V.V. Gortchakov, G.M. Vagradov and A. Molinari, Nucl. Phys. A121 (1968) 625. [6] N. Auerbach, to be published. [7] N. Auerbach, J. Hilfner, A.K. Kerman and C.M. Shakin, submitted to Rev. Mod. Phys. [8] E. I. Kao and S. Fallerios, Phys. Rev. Letters 25 (1970) 827. [9] J. Noble, Phys. Letters 35B {1971) 140, and University of Pennsylvania preprint (1970). [i0] J. Damgaard, C.K. Scott and E. Osnes, Nucl. Phys. A154 (1970) 12. ]11] J . W . Negele, Nucl. Phys. A165 (1971) 305. (12] N. V. Giai, D. Vautherin, M. Veneroni and D.M. Brink, Phys. Letters 35B (1971) 135. [13] R. C. Barret, Nucl. Phys. 88 (1966) 128. [14] A. Lande, A. Molinari and G. E. Brown, Nucl. Phys. A l l 5 (1968) 241.