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ELECTRO-EXCITATION OF THE 1+ STATE AT 3,486 MeV IN 88Sr+
L. van der Bijl, H. Blok, R. Ent, J. Heisenberg, O. Schwentker, A. Richter, P.K.A. de Witt Huberts
To cite this version:
L. van der Bijl, H. Blok, R. Ent, J. Heisenberg, O. Schwentker, et al.. ELECTRO-EXCITATION OF
THE 1+ STATE AT 3,486 MeV IN 88Sr+. Journal de Physique Colloques, 1984, 45 (C4), pp.C4-465-
C4-469. �10.1051/jphyscol:1984436�. �jpa-00224102�
JOURNAL DE PHYSIQUE
Colloque C4, suppl&rnent a u n03, Tome 45, m a r s 1984 page C4-465
8 8
t
ELECTRO-EXCITATION OF THE 1' STATE AT 3 , 4 8 6 MeV I N S rL.T. v a n d e r B i j l , H. B l o k , H.P. Blok, R. E n t , J. ~ e i s e n b e r g * , 0 . Schwentker*, A. R i c h t e r * * and P.K.A. de W i t t H u b e r t s
***
V r i j e Universiteit, Amsterdam, The Netherlands
*University of New Hampshire, Durham, U. S. A . :~;echnische HochschuZe, Darmstade, F.R. G.
NIkZEF-K, Amsterdam, The Netherlands
RdsumB:
Le facteur de forme, pour l ' e x c i t a t i o n de 1 ' B t a t 1' 5 3,846 MeV dans " ~ r p a r d i f f u s i o n in61astique d ' d l e c t r o n s , a d t d mesure pour q = 0,22-2,57 fm-'.
La p o l a r i s a t i o n n u c l d a i r e du coeur e t l a p o l a r i s a t i o n A-trousemblent toutes deux necessairesi l a d e s c r i p t i o n : de l a diminution observee de l a valeur B ( M ~ ) e t des donnees en q f a i b l e ; de I 'a1 l u r e p a r t i c u l i 6 r e du facteur de forme pour les valeurs intermddiaires de q.
Abstract: The form f a c t o r f o r e x c i t a t i o n o f 1' s t a t e a t 3.846 MeV i n ' ' ~ r by
~ n 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 has been measured f o r q=0.22-2.57 fm-'. Both nuclear core p o l a r i z a t i o n and A-hole p o l a r i z a t i o n seem t o be necessary t o describe the observed reduction of the B(M1)-value and data a t low q and the p e c u l i a r shape o f the form f a c t o r a t intermediate values o f q.
l NTRODUCT l ON
I t has been known f o r some t i m e t h a t n u c l e a r t r a n s i t i o n s i n v o l v i n g t h e s p i n - i s o - s p i n o p e r a t o r seem t o be quenched. Well known examples a r e Gamow-Teller t r a n s i t i o n s
i n a-decay (1) and M I - t r a n s i t i o n s and m a g n e t i c moments ( 2 ) . The l a t t e r r e q u i r e a r e - d u c t i o n o f t h e s p i n g - f a c t o r t o about 70 % o f t h e f r e e n u c l e o n v a l u e , a l t h o u g h a r e c e n t a n a l y s i s o f t r a n s i t i o n s i n ( s , d ) - s h e l l n u c l e i i n d i c a t e s t h a t a l m o s t no r e n o r - m a l i z a t i o n i s n e c e s s a r y , i f wave f u n c t i o n s f r o m l a r g e - b a s i s s h e l l model c a l c u l a t i o n s a r e used ( 3 ) . R e c e n t l y a s i m i l a r quenching has been observed i n t h e e x c i t a t i o n o f t h e Gamow-Tel l e r resonance i n ( p ,n) r e a c t i o n s ( 4 ) . From t h e o r e t i c a l s i d e t h e s e phenomena have r e c e i v e d g r e a t a t t e n t i o n l a s t y e a r s , because i t has been suggested (2,s-8) t h a t t h e o b s e r v e d quenching o f such t r a n s i t i o n s i s p a r t l y t h e r e s u l t o f a r e n o r - m a l i z a t i o n o f t h e t r a n s i t i o n o p e r a t o r due t o p o l a r i z a t i o n phenomena i n v o l v i n g t h e A(1232) i s o b a r . I n 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 i s one o f t h e b e s t t o o l s t o t e s t t h i s i n t e r e s t i n g p o s s i b i l i t y , as i t a l l o w s t o d e t e r m i n e t h e f o r m f a c t o r , i.e. t h e q-dependence o f a c e r t a i n t r a n s i t i o n , w h i c h i s a much more s e n s i t i v e t e s t o f t h e o r e t i c a l p r e d i c t i o n s than, f o r i n s t a n c e , a B(MI) - v a l ue, w h i c h c o r r e s - ponds t o o n l y one p o i n t o f t h e f o r m f a c t o r , v i z . t h e v a l u e n e a r q = 0. E s p e c i a l l y M I - t r a n s i t i o n s s h o u l d be i n v e s t i g a t e d as h e r e t h e e f f e c t s a r e e x p e c t e d t o be
l a r g e s t .
Few form f a c t o r s o f 1' s t a t e s have been measured. A w e l l known example i s t h e I +
s t a t e a t 15.1 MeV e x c i t a t i o n energy i n
"c.
The f o r m f a c t o r o f t h i s s t a t e d i f f e r s c o n s i d e r a b l y f r o m t h e one c a l c u l a t e d w i t h t h e Cohen-Kurath wave f u n c t i o n s ( 9 ) . T h i s d i s c r e p a n c y has s t i m d l a t e d v a r i o u s t h e o r e t i c a l c a l c u l a t i o n s ( 9 - 1 2 ) . The c a l c u l a t i o n by T o k i and Weise (12) i n d i c a t e s t h a t a f a i r d e s c r i p t i o n o f t h e f o r m f a c t o r can be o b t a i n e d , i f one i n c l u d e s a p a r t f r o m T - and p-exchange terms i n t h e i n t e r a c t i o n a l s o A - h o l e e x c i t a t i o n s .A s t r o n g M1 t r a n s i t i o n has been o b s e r v e d (13) i n " ~ a t o a s t a t e a t 10.23 MeV e x c i t a t i o n e n e r g y . T h i s t r a n s i t i o n has been i n t e r p r e t e d as b e i n g p r e d o m i n a n t l y due t o a n e u t r o n I f 7,2-+1f5,2 s p i n - f l i p e x c i t a t i o n . I n t h i s case a l m o s t no change i n t h e shape o f t h e f o r m f a c t o r has been observed, b u t m a i n l y an o v e r a l l r e d u c t i o n o f about a f a c t o r o f two compared t o a s h e l l - m o d e l c a l c u l a t i o n i n t h e f u l l f p - s h e l l . T h i s r e - d u c t i o n has been e x p l a i n e d i n terms o f t h e A - h o l e model (14).
t T h i s work was p a r t o f the research program o f the S t i c h t i n g voor Fundamenteel Onderzoek der Materie (FOM) which i s f i n a n c i a l l y supported by the Nederlandse Organ i s a t i e voor Zuiver-Wetenschappel i j k Onderzoek (ZWO)
.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1984436
C4-466 JOURNAL DE PHYSIQUE
I n h e a v i e r n u c l e i t h e r e a r e few we1 I known 1+ s t a t e s . The 1' s t a t e a t 3.486 MeV e x c i t a t i o n e n e r g y i n "Sr s h o u l d p r o v i d e an e x c e l l e n t t e s t case, as i t i s w e l l i s o - l a t e d f r o m n e i g h b o u r i n g s t a t e s and i t s n u c l e a r s t r u c t u r e , i n c o n t r a s t t o t h a t o f t h e 1 2 c case, i s r a t h e r s i m p l e . Model c a l c u l a t i o n s i n d i c a t e t h a t t h e e x c i t a t i o n o f t h i s s t a t e i s a l m o s t c o m p l e t e l y due t o a p r o t o n 2p3/2+2p1/2 t r a n s i t i o n w i t h a s m a l l n e u t r o n l g 9/2+1g7/2 t r a n s i t i o n as t h e main a d m i x t u r e . The ~ ( M l ) - v a I u e 0.92 0.15 pi, as measured b y y - r a y f l u o r e s c e n c e ( I S ) , i s a f a c t o r o f t w o smal l e r t h a n t h e c a l c u l a - t e d v a l u e . I n o r d e r t o s t u d y t h e q-dependence o f t h i s r e d u c t i o n t h e f o r m f a c t o r f o r e x c i t a t i o n o f t h i s s t a t e i n t h e ( e , e l ) r e a c t i o n was measured.
EXPERIMENTAL DETAILS AND RESULTS
Data were o b t a i n e d a t t h r e e l a b o r a t o r i e s : a t Darmstadt f o r q = 0.22 t o 0.60 fm-', a t t h e Bates l i n e a r a c c e l e r a t o r o f H I T f o r q = 1.19 t o 2.57 frn-' and w i t h t h e new e l e c t r o n a c c e l e r a t o r o f NIKHEF-K f r o m q = 0.67 t o 1.03 fm-'. D e t a i I s o f t h e experimen- t a l s e t - u p a t t h e d i f f e r e n t l a b o r a t o r i e s can be f o u n d i n r e f s . 16-18. T a r g e t s were s e l f - s u p p o r t i n g f o i l s o f "St- m e t a l , e n r i c h e d t o 99.84 %, w i t h a t h i c k n e s s o f 10-25 mg/cm2. A t a l l t h r e e l a b o r a t o r i e s s p e c t r a were t a k e n a t a backward a n g l e w i t h a r e s o -
l u t i o n o f a b o u t 50 keV. Examples a r e shown i n f i g . 1. E s p e c i a l l y f o r t h e s p e c t r a i n t h e r e g i o n q = 0.6-1.0 f m - l , where t h e peak due t o t h e e x c i t a t i o n o f t h e lf s t a t e i s
E = 209.5 MeV 8 = 160'
EXCITATION ENERGY ( M e V ) EXCITATION ENERGY ( M e V )
F i g . 1 . Spectra for the 8 8 ~ r ( e , e ' ) reaction measured at different laboratories:
a) NIKHEF-K, b ) Bates
v e r y s m a l l and r i d e s on a l a r g e r a d i a t i o n t a i l , s p e c i a l c a r e was t a k e n t o d e t e r m i n e r e l i a b l e c r o s s s e c t i o n s by measuring t h e peak a t v a r i o u s p l a c e s o f t h e d e t e c t o r t h e r e b y e l i m i n a t i n g p o s s i b l e s m a l l d i f f e r e n c e s i n channel e f f i c i e n c i e s . The s p e c t r a were a n a l y s e d i n a s t a n d a r d way. A b s o l u t e c r o s s s e c t i o n s were d e t e r m i n e d by n o r m a l i - z i n g t o t h e e l a s t i c peak o f w h i c h t h e c r o s s s e c t i o n was c a l c u l a t e d w i t h a phase s h i f t code f r o m t h e b e s t f i t t o a l l a v a i l a b l e (e,e) and muonic X-ray d a t a . The r e s u l - t a n t f o r m f a c t o r i s shown i n f i g . 2, where we have p l o t t e d a/o/(++ tg2;) w i t h o t h e
P t h e M o t t c r o s s s e c t i o n f o r u n i t charge t i m e s t h e r e c o i l f a c t o r (see r e f . 1 9 ) . THEORETICAL ANALYSIS
The measured f o r m f a c t o r has been compared t o t h e o r e t i c a l p r e d i c t i o n s i n w h i c h d i f f e r e n t assumptions a b o u t t h e n u c l e a r s t r u c t u r e were made. We w i l l m a i n l y use t h e r e s u l t s o f Broken-Pai r model c a l c u l a t i o n s ( 2 0 ) , i n w h i c h a G - m a t r i x d e r i v e d f r o m a meson-exchange p o t e n t i a l (21) was used as i n t e r a c t i o n . A l l c a l c u l a t i o n s were done i n DWBA. The main e f f e c t o f t h e d i s t o r t i o n s i s t o s h i f t t h e f o r m f a c t o r t o l a r g e r v a l u e s o f q. T h e r e f o r e t h e d a t a and c a l c u l a t i o n s have been p l o t t e d a g a i n s t q
e f f '
(e,el ) 1+ 3486 MeV
. .
-
VALENCE - - * 1 ---- VAL. + CP:+'
...
VAL.+CP+A '+++++ VAL.+ C P + A + MEC ?
Fig. 2. Measured form f a c t o r compared w i t h the r e s u l t s o f d i f f e r e n t c a l c u l a t i o n s . The corresponding B(M1)-values are: valence: 2.07, Val. + CP: 2.09, v a l . + CP + A: 1.06, v a l . + CP
+
A + MEC: 1.24 u i . The experimental value i s 0.92*0.15u;.
B a s i c a l l y t h e c a l c u l a t i o n s i n v o l v e t h e f o l l o w i n g i n g r e d i e n t s (see r e f . 19 f o r d e t a i - l e d f o r m u l a e ) . The f o r m f a c t o r can be expressed i n terms o f a t r a n s i t i o n d e n s i t y , w h i c h can be w r i t t e n as a sum o v e r s i n g l e - p a r t i c l e d e n s i t i e s w e i g h t e d b y s p e c t r o s c o - p i c a m p l i t u d e s S, w h i c h s h o u l d be g i v e n b y t h e n u c l e a r - s t r u c t u r e c a l c u l a t i o n s . The t r a n s i t i o n d e n s i t y was c a l c u l a t e d w i t h t h e program WSAXM u s i n g s i n g l e - p a r t i c l e wave f u n c t i o n s c a l c u l a t e d i n a Woods-Saxon w e l l wiCh parameters r,= 1.29 f m , a = 0.70 f m , A = 20 and n o n - l o c a l i t y parameter B = 0.85 fm ( w i t h t h e s e parameters measured rms r a d i i and t r a n s i t i o n c h a r g e d e n s i t i e s i n " ~ r were r e p r o d u c e d ) . T h i s d e n s i t y was t h e n used
i n t h e DWBA program HEIMAG t o c a l c u l a t e c r o s s s e c t i o n s .
I n a s i m p l e p i c t u r e t h e e x c i t a t i o n o f t h e 1+ s t a t e i s due t o a p r o t o n 2 ~ ~ , ~ + 2 p 1 /2 t r a n s i t i o n . Because o f t h e p r e s e n c e o f a p a i r i n g d i s t r i b u t i o n i n t h e g r o u n d s t a t e o f
" ~ r t h e amp1 i t u d e a o f t h i s t r a n s i t i o n i s smal i e r t h a n one ( a = S(p3/2+p112)- S ( P ~ / ~ - + P ~ / ~ ) ) . The v a l u e o f a i s c r u c i a l f o r t h e q u e s t i o n i f t h e B(MI)-value i s quenched. One B r o k e n - P a i r model c a l c u l a t i o n s u s i n g d i f f e r e n t s i n g l e - p a r t i c l e e n e r g i e s and i n t e r a c t i o n s y i e l d a = 0.55-0.65. Two-BP c a l c u l a t i o n s (22) g i v e v a l u e s t h a t a r e a few p e r c e n t s m a l l e r . C a l c u l a t i o n s by Hughes ( 2 3 ) g i v e a = 0.51.
The combined a n a l y s i s o f t h e r e s u l t s o f ( d , 3 ~ e ) and ( He,d) r e a c t i o n s (24) y i e l d s a=0.50-0.60. The main a d m i x t u r e i n t h e wave f u n c t i o n o f t h e 1+ s t a t e i s a n e u t r o n 1 g j j 2 g 7 / 2 component. V a r i o u s c a l c u l a t i o n s y i e l d
B =
0.01-0.04 f o r t h e a m p l i t u d e o f t h t s component. The f u l l c u r v e i n f i g . 2 was c a l c u l a t e d w i t h a = 0.616 and8 =
0.040 as g i v e n b y t h e 1 BP c a l c u l a t i o n s w i t h t h e G-matrix. The e x p e r i m e n t a l d a t a a r e o v e r e s t i m a t e d a t s m a l l v a l u e s o f q ( a l s o t h e c a l c u l a t e d B(MI)-value i s a f a c t o r o f t w o h i g h e r t h a n t h e measured one) and t h e shape i s n o t c o r r e c t i n t h e r e g i o nq = 0.6-1.3 I f one i n c r e a s e s t h e model space, many components i n t h e wave f u n c t i o n o f t h e 1 fm-'.
+
s t a t e a r e o b t a i n e d . As a l m o s t a l l o f t h e s e g i v e r i s e t o n - o r 1 - f o r b i d d e n t r a n s i - t i o n s t h e B(M1)-value i s h a r d l y i n f l u e n c e d , b u t t h e f o r m f a c t o r a t l a r g e r v a l u e s o f q can change c o n s i d e r a b l y . The r e s u l t s o f a c a l c u l a t i o n w h i c h i n c l u d e s t h i s n u c l e a r c o r e p o l a r i z a t i o n (CP) c a l c u l a t e d i n a ( ~ d ) - ( ~ f h ) space a r e shown as t h e dashed c u r v e i n f i g . 2. There i s a l m o s t no change i n t h e c a l c u l a t e d B(M1)-value and t h e f o r m f a c - t o r a t low q, w h i c h a r e s t i l l t o o h i g h , b u t by d e s t r u c t i v e i n t e r f e r e n c e w i t h t h e va- l a n c e p a r t o f t h e t r a n s i t i o n t h e f o r m f a c t o r i s changed q u i t e a b i t i n t h e r e g i o n q = 0.7-1.2 fm-'. I t seems t h a t i n t h i s r e g i o n even a l i t t l e more d e s t r u c t i v e
C4-468 JOURNAL DE PHYSIQUE
i n t e r f e r e n c e i s needed, as i t l o o k s as i f t h e f o r m f a c t o r ( F ~ ) ' has an e x t r a maximum a t q = 0 . 8 5 f m - ' compared t o t h e c a l c u l a t i o n s . T h i s c o u l d be e x p l a i n e d i f FT would be reduced s o much as t o become n e g a t i v e i n t h a t r e g i o n ( t h e f a c t t h a t t h e r e I S n o t a r e a l minimum a t q = 0 . 6 fm-' i s due t o D\!B/? e f f e c t s , as i t was c a l c u l a t e d t h a t i f t h e PWBA f o r m f a c t o r has a minimum t h e r e , t h e DWBA r e s u l t s o n l y show a s h o u l d e r . T h i s i s i l l u s t r a t e d i n f i g . 3 t o be d i s c u s s e d l a t e r ) .
O f t e n m a g n e t i c t r a n s i t i o n s and moments a r e d e s c r i b e d by u s i n g an e f f e c t i v e s p i n g - f a c t o r o f about 70 % o f t h e f r e e g - v a l u e i n t h e t r a n s i t i o n o p e r a t o r . We i n v e s t i - g a t e d i f such a r e d u c t i o n m i g h t d e s c ? i b e o u r d a t a . I t t u r n s o u t t h a t t h e B ( M ~ ) - v a l u e and t h e f i r s t maximum o f t h e f o r m f a c t o r can be d e s c r i b e d by c h o o s i n g a r e d u c t i o n f a c t o r o f 0.75, b u t t h e shape d i s c r e p a n c y a t h i g h e r q remains, so a q-independent r e n o r m a l i z a t i o n i s n o t a b l e t o d e s c r i b e t h e d a t a . I t has been shown a l r e a d y (25) i n a s e m i q u a n t i t a t i v e way t h a t i n c l u d i n g a p a r t f r o m t h e u s u a l n u c l e a r CP, w h i c h c o n s i s t s o f n u c l e o n p a r t i c l e - h o l e c o n f i g u r a t i o n s , a l s o A-hole p o l a r i z a t i o n , i . e . components i n t h e wave f u n c t i o n i n w h i c h a n u c l e o n i s e x c i t e d i n t o a A(1232) i s o b a r , t h e r e b y c r e a t i n g a A-hole s t a t e , leads t o a q-dependent r e n o r m a l i z a t i o n o f t h e i s o v e c t o r p a r t o f g which reduces t h e B(M1)-value and changes t h e shape o f t h e f o r m f a c t o r .
5 '
The change i n shape, w h i c h i s a l m o s t absent i n t h e 4 8 ~ a case, i s due t o t h e f a c t t h a t t h e t r a n s i t i o n i n i s m a i n l y a p r o t o n t r a n s i t i o n , i n w h i c h t h e o r b i t a l and s p i n p a r t s o f t h e MI o p e r a t o r i n t e r f e r e . By t h e q-dependent r e n o r m a l i z a t i o n o f g t h i s
i n t e r f e r e n c e and t h e r e b y t h e shape o f t h e f o r m f a c t o r i s changed.
The d o t t e d c u r v e i n f i g . 2 was c a l c u l a t e d (26) by i n c l u d i n g b o t h n u c l e a r CP and A - h o l e p o l a r i z a t i o n ( u s i n g q ' = 0 . 6 ) . The v a l e n c e t r a n s i t i o n used was
+
0 . 2 ~ l g ~ / ~ + l g ~ / ~ , w h i c h g i v e s B ( M ~ ) = 1.63u i .
F u r t h e r d e t a i I s o f t h e s e c a l c u l a t i o n s can be found i n r e f . 27. I t i s seen t h a t t h e s e c a l c u l a t i o n s g i v e a f a i r d e s c r i p t i o n o f t h e data. The d i s c r e p a n c i e s n e a r q = 0.9 f m - ' s h o u l d n o t be t a k e n t o o s e r i o u s l y because t h e s m a l l f o r m f a c t o r i n t h a t r e g i o n i s t h e r e s u l t o f a d e s t r u c - t i v e i n t e r f e r e n c e between t h e v a l e n c e p a r t o f t h e t r a n s i t i o n and t h e p o l a r i z a t i o n e f f e c t s (see f i g . 3 f o r an i l l u s t r a t i o n o f t h i s ) and t h e r e f o r e r a t h e r s e n s i t i v e t o d e t a i l s o f t h e c a l c u l a t i o n s .The i n f l u e n c e o f meson exchange c u r r e n t s has a l s o been c a l c u l a t e d by H 5 r t i n g e t a l . ( 2 7 ) . As can be seen i n f i g . 2 t h e main e f f e c t i s t o r a i s e t h e B ( M ~ ) - v a l u e and low q p o i n t s a g a i n , and a l s o t h e h i g h q p o i n t s , t h e r e b y i n c r e a s i n g t h e d i s c r e p a n c y w i t h t h e e x p e r i m e n t a l data.
1 +
P W B A
Fig. 3. Plane-wave form f a c t o r FT f o r d i f f e r e n t c a l c u l a t i o n s showing the d e s t r u c t i v e i n t e r - ference of both A and CP i n the region q = O . G - l . 3 f m - ' . The negative a r t of F T i n theaval. + CP
+
b c a l c u l a t i o n gives r i s e t o an e x t r a maximum i n ( F Tr2.
CONCLUSIONS
The m e a s u r e d f o r m f a c t o r f o r e x c i t a t i o n i n i n 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 o f t h e I' s t a t e a t 3.486 MeV i n d e v i a t e s c o n s i d e r a b l y f r o m t h e o n e c a l c u l a t e d w i t h v a l e n c e s p a c e wave f u n c t i o n s . The i n c l u s i o n o f t h e e f f e c t s o f A - h o l e s t a t e s seems t o b e n e c e s s a r y t o b e a b l e t o d e s c r i b e t h e m e a s u r e d B ( M ~ ) - v a l u e a n d t h e f o r m f a c t o r d a t a a t l o w q. T h i s c o n c l u s i o n depends o n t h e f i l l i n g o f t h e p r o t o n 2 p 3 / 2 a n d 2pIl2 o r b i - t a l s i n t h e g r o u n d s t a t e o f 8 8 ~ r . M o r e e x p e r i m e n t a l i n f o r m a t i o n o n t h i s w o u l d b e o f
i m p o r t a n c e . However, a m o r e d i f f u s e p a i r i n g d i s t r i b u t i o n t h e n h a s been assumed i n t h e p r e s e n t c a l c u l a t i o n s d o e s n o t seem v e r y l i k e l y i n v i e w o f s e v e r a l m o d e l c a l c u l a t i o n s a n d w o u l d c o n t r a d i c t t h e d a t a f r o m s t r i p p i n g a n d p i c k - u p r e a c t i o n s . N u c l e a r c o r e p o l a r i z a t i o n p l u s t h e e f f e c t o f t h e A seems t o b e r e s p o n s i b l e f o r t h e p e c u l i a r s h a p e o f t h e t r a n s i t i o n i n t h e r e g i o n q = 0 . 6 - 1 . 3 fm-'. MEC c o r r e c t i o n s i n c r e a s e t h e f o r m f a c t o r b o t h a t l o w a n d a t h i g h q , i n c r e a s i n g t h e d i s c r e p a n c y w i t h t h e d a t a i n t h e
.
l a t t e r r e g i o n .
We t h a n k d r s . P. H e n g e v e l d a n d K. A l l a a r t f o r p e r f o r m i n g t h e B P - c a l c u l a t i o n s a n d d r . H B r t i n g f o r s e n d i n g us t h e r e s u l t s o f h i s c a l c u l a t i o n s .
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