- Email: [email protected]
Even and odd parity states in 9Be
Volume27B, number 2
PHYSICS LETTERS
EVEN
AND ODD
PARITY
10June 1968
STATES
IN
9Be
M. BOUTEN, M.C. BOUTEN, H. DEPUYDT * and L. SCI-IOTSMANS * International Atomic Energy Agency, International Centre f o r Theoretical Physics, Trieste Received 26 April 1968
Results of a projected Hartree-Fock calculation are presented both for the even and odd parity states of 9Be. It is found that the even parity states come down very low in the energy spectrum and explain the occurrence of non-normal parity states at low energies.
T h e r e is now c o n s i d e r a b l e e x p e r i m e n t a l e v i dence [1] for the e x i s t e n c e of v e r y l o w - l y i n g non= n o r m a l p a r i t y states in 9Be, the lowest one being a ~1+ s t a t e at 1.7 MeV. It was suggested a l r e a d y s e v e r a l y e a r s ago by Lane [2] that t h e s e l e v e l s could be d e s c r i b e d by weakly coupling a 0s o r 0 d - n e u t r o n to the lowest states of 8Be. This idea was developed m o r e quantitatively by B a r k e r [3] and r e c e n t l y also by A d l e r et al. [4] in a s o m e what modified f o r m , u s i n g E l l i o t t ' s S U 3 - c l a s s i f i cation. Although they obtain v e r y s a t i s f a c t o r y a g r e e m e n t with the r e l a t i v e e x p e r i m e n t a l energy s p l i t t i n g s between the even p a r i t y s t a t e s and with s o m e other s p e c t r o s c o p i c data, the position of t h e s e states in r e l a t i o n to the odd p a r i t y l e v e l s is not calculated. The question whether the occ u r r e n c e of n o n - n o r m a l p a r i t y s t a t e s a t v e r y low e n e r g i e s can be u n d e r s t o o d r e m a i n s open. Recent H a r t r e e - F o c k c a l c u l a t i o n s for 8Be give s o m e i n f o r m a t i o n about the e x i s t e n c e of l o w - l y i n g even p a r i t y s t a t e s in 9Be. B a s s i c h i s et al. [5] have used T a b a k i n ' s i n t e r a c t i o n in an u n r e s t r i c t e d H a r t r e e - F o c k calculation for 8Be. F o r the lowest unoccupied H a r t r e e - F o c k o r b i t a l s with negative and positive p a r i t y they obtain, r e s p e c t i v e l y , the following s i n g l e - p a r t i c l e e n e r gies: ~ ( m = ~, I T = - ) = 3.95 MeV ~(m=~, ~ = +) = 10.68MeV . Both these single-particle energies are positive, which would mean that the 9Be nucleus is not bound, in contradiction to the experimental situation. The a n s w e r to this difficulty i s probably that the c o r r e l a t i o n energy will be much l a r g e r
* Applied Mathematics Department, S.C.K. Mol, Belgium.
for 9Be than for 8Be. This can be u n d e r s t o o d f r o m the fact that the energy gap between o c c u pied and unoccupied o r b i t a l s i s l a r g e r for even • Z = N n u c l e i than for n e i g h b o u r i n g odd n u c l e i . The r e l a t i v e position of positive and n e g a t i v e p a r i t y l e v e l s in 9Be can roughly b e e s t i m a t e d a s the d i f f e r e n c e of t h e s e s i n g l e - p a r t i c l e e n e r g i e s . This gives a v a l u e of 6.7 MeV which i s a l r e a d y much s m a l l e r than the value ~w ~ 15 MeV given by the s i n g l e - p a r t i c l e s h e l l - m o d e l . A m o r e a c c u r a t e way to obtain this energy d i f f e r e n c e would r e q u i r e an a n g u l a r m o m e n t u m p r o j e c t i o n f r o m the different i n t r i n s i c H a r t r e e - F o c k s t a t e s . In this l e t t e r , we p r e s e n t the r e s u l t s obtained f r o m a p r o j e c t e d H a r t r e e - F o c k calculation for the lowest even and odd p a r i t y s t a t e s of 9Be. We s t a r t f r o m the H a m i l t o n i a n for the i n t e r n a l energy
H= ~i Ti
-Tcm
+ ½i~j " " Vij
(1)
and take as n u c l e o n nucleon i n t e r a c t i o n the p o t e n tial of Volkov VI2
=
{-83.34
exp[-(r12/1.6) 2]
+
(2)
+ 144.86 exp [-(r12/0.82)2]}× (0.4 +0.6 P x ) , w h e r e P x is the M a j o r a n a space exchange o p e r a tor. It was shown in p r e v i o u s p r o j e c t e d H a r t r e e Fock c a l c u l a t i o n s [6] that this f o r c e gives r e a s o n able e n e r g i e s and wave functions for the l o w - l y ing l e v e l s of the Ugther p - s h e l l n u c l e i . As we a r e p r i m a r i l y i n t e r e s t e d h e r e i n the e n e r g y diff e r e n c e between even and odd p a r i t y states and not in the f i n e r d e t a i l s of the wave functions, we b e l i e v e that the n e g l e c t of n o n - c e n t r a l f o r c e s in the H a m i l t o n i a n can be justified. Following the pr(~cedure d e s c r i b e d p r e v i o u s l y [6] we take as t r i a l function ~ L = ~ L ~ where 61
Volume27B, number 2
PHYSICS LETTERS
10 June 1968
~0 -- 10s0) + 810d0) E (MeV)
7
¢0 -- 10p0> ÷ ~10~0> ~1 = lOp1) + ~]Ofl)
-35
X0 : [ ls0} - ~2[0d0) . The b a s i c states [nlm) a r e h a r m o n i c o s c i l l a t o r functions with phases and quantum n u m b e r s as in ref. 7. Putting 8 = 6 = ~ = 0, the S l a t e r d e t e r m i n a n t s @ r e d u c e to the leading s t a t e s of the lowest r e p r e s e n t a t i o n s with negative and positive parity of E l l i o t t ' s S U 3 - c l a s s i f i c a t i o n . With the t r i a l function ~L, we c a l c u l a t e
(~,LIHI q,L>
EL~/
-40
~
EL(f35~bb') = (@L[@L >
F./,, S'f/ ~,j SF.,
-45
-50
//"
A,,, ~''" /
a
I
2
I
8
12
L (L+ l)
20
Fig. 1. Plot of the energies EO and E L for the oddparity states as a function of L(L+ 1). ~L is a p r o j e c t i o n o p e r a t o r for the total o r b i t a l a n g u l a r m o m e n t u m and @ i s a single Slater d e t e r m i n a n t belonging to the m o s t s y m m e t r i c p a r tition [441]. F o r the negative parity states ,:I, = detl~P04 ~4,~n+
~-0,Yl I
(3)
and for the positive parity states @
=
detlq]4~ ~_4 n+ 0 Xo
H e r e ~04 m e a n s that function ~ 0 with the l a b e l s , and ×~+ is a and spatial function chosen as
62
"
(4)
four o r b i t a l s have the s p a t i a l four different spin i s o s p i n n e u t r o n o r b i t a l with spin + ½ X0. The spatial functions a r e
(6)
u s i n g the method d e s c r i b e d in ref. 6, and m i n i m i z e the energy with r e s p e c t to the v a r i a t i o n a l p a r a m e t e r s ~ , 6 , ~ , b and b'. H e r e b and b' a r e the h a r m o n i c o s c i l l a t o r length p a r a m e t e r s , r e s pectively, of the even p a r i t y and odd parity o r b i t a l s . Taking t h e m as v a r i a t i o n a l p a r a m e t e r s , we allow the different o r b i t a l s s o m e f r e e d o m in choosing t h e i r s p a t i a l extension. The other par a m e t e r s / 3 , 6 and ~ d e t e r m i n e the d e f o r m a t i o n of the o r b i t a l s . It i s seen that the o r b i t a l X0 cont a i n s no independent v a r i a t i o n a l p a r a m e t e r , either for its size or for its deformation. The r e a s o n for this i s that the c o m p u t e r on which this c a l c u l a t i o n was p r o g r a m m e d is not sufficiently l a r g e to i n t r o d u c e m o r e v a r i a t i o n a l p a r a m e t e r s . F o r the s a m e r e a s o n , the choice of the o r b i t a l s (5) i s somewhat m o r e r e s t r i c t e d than those in our p r e v i o u s work [6]. F o r this calculation we a l s o had to t r u n c a t e the p o l y n o m i a l s , called FAB in ref. 6, a f t e r t e r m s of d e g r e e 6 in 8, 5 and ~. F r o m our p r e v i o u s work, we e s t i m a t e the u n c e r tainty on the energy i n t r o d u c e d by this t r u n c a t i o n to be l e s s than 0.3 MeV. The r e s u l t s obtained for the negative parity l e v e l s with L = 1, 2, 3, 4 and for the positive p a r i ty l e v e l s with L = 0 and 2 a r e p r e s e n t e d in table 1 H e r e , E ~ is the energy obtained for fl = 6 = ~ = 0 and for the v a l u e s of b and b' given in the table. In the l a s t ~olumn we also give the d e f o r m a t i o n e n e r g i e s ELU _ E L , i.e., the energy gained by i n t r o d u c i n g the p a r a m e t e r s 8, 6 and ~. This d e f o r mation energy i s c o n s i d e r a b l y l a r g e r for the two lower L - v a l u e s of the negative parity band than for the two upper L - v a l u e s . As a result,^ the exact r o t a t i o n a l band s p e c t r u m of E ~ = A + + B L (L + 1) is d i s t o r t e d c o n s i d e r a b l y (fig. 1). As in our p r e v i o u s c a l c u l a t i o n s [6], we again find that the deformation of the i n t r i n s i c state d e c r e a s e s a s we go to higher L - v a l u e s . This b e -
Volume27B, number 2
PHYSICS LETTERS
10June 1968
Table 1 Energies and wave functions for the low-lying negative and positive parity states of 9Be and for the ground state of 8Be.
9Be
8Be
L~
~L
~
~
~
b
b,
~o
~ o _ EL
I-
-49.7
0.357
0.233
0.205
1.62
1.69
-38.89
10.81
2-
-47.7
0.336
0.220
0.201
1.62
1.69
-37.47
10.23
3-
-42.6
0.258
0.166
0.182
1.61
1.68
-35.35
7.25
4-
-39.6
0.229
0.153
0.181
1.61
1.67
-32.55
7.05
0+
-45.4
0.420
0.315
-
1.74
1.75
-28.89
16.51
2+
-43.0
0.395
0.296
-
1.74
1.75
-28.82
14.18
0+
-53.35
0.375
0.239
-
1.61
1.65
-
-
h a v i o u r is just the opposite of what one might expect f r o m the c e n t r i f u g a l f o r c e in a r o t a t i o n a l picture. The m o s t s t r i k i n g r e s u l t of table 1 i s the v e r y l a r g e d e f o r m a t i o n energy for the positive p a r i t y l e v e l s . W h e r e a s t h e s e l e v e l s lie well above the four odd p a r i t y s t a t e s when the d e f o r m a t i o n is neglected, they drop down v e r y low between the negative p a r i t y l e v e l s when the d e f o r m a t i o n a t t a i n s its s t a b l e value. The energy d i f f e r e n c e b e tween the lowest negative and lowest p o s i t i v e p a r i t y level c o m e s down f r o m 10 MeV to 4.3 MeV. Since the d e f o r m a t i o n for the L = 0 state is v e r y l a r g e , it i s to be expected that an additional f r e e dom for the o r b i t a l ×0 to a d j u s t its d e f o r m a t i o n to that of the other o r b i t a l s will give a n a d d i t i o nal energy gain, so that the e x p e r i m e n t a l value 1.7 MeV may well be reached. In the l a s t row of table 1, we have also given, for c o m p a r i s o n , the energy and wave function obtained for the ground s t a t e of 8Be, u s i n g the t r i a l function ~L=O-- ~L=O~, where
= detl 04%41 and ~ 0 , ~°0 a r e given by. eq. (5). A l a r g e r b i n d ing e n e r g y is found for 8Be than for 9Be, j u s t as in the H a r t r e e - F o c k c a l c u l a t i o n of B a s s i c h i s et al. m e n t i o n e d above. The s a m e r e m a r k about the c o r r e l a t i o n energy i s valid here. F r o m t a b l e 1, it i s seen that the s i z e and d e f o r m a t i o n of the n e g a t i v e p a r i t y s t a t e s of 9Be i s quite s i m i l a r to those of the ground s t a t e of 8Be. The p o s i t i v e p a r i t y o r b i t a l s have, however, both a l a r g e r s i z e and d e f o r m a t i o n . In view of the l a r g e d e f o r m a t i o n s obtained in this c a l c u l a t i o n it i s a l s o i n t e r e s t i n g to c a l c u l a t e the quadrupole m o m e n t of the ground s t a t e of 9Be. The e x p e r i m e n t a l value i s not well known, the different d e t e r m i n a t i o n s [8] y i e l d i n g v a l u e s r a n g -
ing between 2.9 and 6.9 fm 2. The most r e l i a b l e v a l u e s e e m s to be obtained f r o m r e c e n t highe n e r g y e l e c t r o n s c a t t e r i n g e x p e r i m e n t s [8], giving the v a l u e Q = 3.8 f m 2. The s a m e m e a s u r e m e n t s a l s o d e t e r m i n e the r . m . s , r a d i u s of the charge d i s t r i b u t i o n to be R = 2.5 fro. On the s i m p l e s h e l l - m o d e l in LS-coupling, one gets Q = ~b 2. Choosing b so that the e x p e r i m e n t a l r a d i u s is r e p r o d u c e d (b = 1.77 fm) gives a quadrupole m o m e n t Q = 1.88 f m z. The p r o j e c ted H a r t r e e - F o c k function gives R = 2.41 fm and Q = 2.73 fm 2. F r o m i n t e r m e d i a t e coupling c a l c u l a t i o n s [9,10], it is well known that even a weak spin o r b i t f o r c e b r i n g s about an i m p o r t a n t i n c r e a s e of the quadrupole moment. T h i s i s due to the fact that the lowest p a r t i t i o n [441] of 9Be contains L = 1, 2, 3, 4 and a weak spin o r b i t f o r c e will mix s o m e L = 2 state with the L = 1 ground state. In a detailed c a l c u l a t i o n for 9Be, it is consequently not allowed to n e g l e c t the spin f o r c e s . T h i s n e g l e c t was p o s s i b l e in our p r e vious c a l c u l a t i o n s [6] for 4 --
63
Volume27B, number 2
PHYSICS
LETTERS
References
6. M. Bouten, P. Van Leuven, H. Depuydt and L. S c h o t s m a n s , Nucl. P h y s . A100 (1967) 90. 7. T. Brody and M. Moshinsky, T a b l e s of t r a n s f o r m a tion b r a c k e t s , (Mexico, 1960). 8. M. B e r n h e i m , T. Stovall and D. V i n c i g u e r r a , Nucl. P h y s , A97 (1967) 488. 9. J . B . F r e n c h , E . C . Halbert and S. Pandya, P h y s . Rev. 99 (1955) 1387. 10. F . C . B a r k e r , Nucl. P h y s . 83 (1966) 418.
1. J . J . K r o e p e l and C . P . B r o w n e , Nucl. P h y s . A108 (1968) 289. 2. A . M . L a n e , Rev. Mod. P h y s . 32 (1960) 619. 3. F . C . B a r k e r , Huel. P h y s . 28 (1961) 96. 4. C . A d l e r , T . C o r c o r a n and C . M a s t , Nucl. P h y s . 88 (1966) 145. 5. W.H. B a s s i c h i s , A . K . K e m a n and J. P. Svenne, P h y s . R e v . 160 (1967) 746. *****
64
10 June 1968
PHYSICS LETTERS
EVEN
AND ODD
PARITY
10June 1968
STATES
IN
9Be
M. BOUTEN, M.C. BOUTEN, H. DEPUYDT * and L. SCI-IOTSMANS * International Atomic Energy Agency, International Centre f o r Theoretical Physics, Trieste Received 26 April 1968
Results of a projected Hartree-Fock calculation are presented both for the even and odd parity states of 9Be. It is found that the even parity states come down very low in the energy spectrum and explain the occurrence of non-normal parity states at low energies.
T h e r e is now c o n s i d e r a b l e e x p e r i m e n t a l e v i dence [1] for the e x i s t e n c e of v e r y l o w - l y i n g non= n o r m a l p a r i t y states in 9Be, the lowest one being a ~1+ s t a t e at 1.7 MeV. It was suggested a l r e a d y s e v e r a l y e a r s ago by Lane [2] that t h e s e l e v e l s could be d e s c r i b e d by weakly coupling a 0s o r 0 d - n e u t r o n to the lowest states of 8Be. This idea was developed m o r e quantitatively by B a r k e r [3] and r e c e n t l y also by A d l e r et al. [4] in a s o m e what modified f o r m , u s i n g E l l i o t t ' s S U 3 - c l a s s i f i cation. Although they obtain v e r y s a t i s f a c t o r y a g r e e m e n t with the r e l a t i v e e x p e r i m e n t a l energy s p l i t t i n g s between the even p a r i t y s t a t e s and with s o m e other s p e c t r o s c o p i c data, the position of t h e s e states in r e l a t i o n to the odd p a r i t y l e v e l s is not calculated. The question whether the occ u r r e n c e of n o n - n o r m a l p a r i t y s t a t e s a t v e r y low e n e r g i e s can be u n d e r s t o o d r e m a i n s open. Recent H a r t r e e - F o c k c a l c u l a t i o n s for 8Be give s o m e i n f o r m a t i o n about the e x i s t e n c e of l o w - l y i n g even p a r i t y s t a t e s in 9Be. B a s s i c h i s et al. [5] have used T a b a k i n ' s i n t e r a c t i o n in an u n r e s t r i c t e d H a r t r e e - F o c k calculation for 8Be. F o r the lowest unoccupied H a r t r e e - F o c k o r b i t a l s with negative and positive p a r i t y they obtain, r e s p e c t i v e l y , the following s i n g l e - p a r t i c l e e n e r gies: ~ ( m = ~, I T = - ) = 3.95 MeV ~(m=~, ~ = +) = 10.68MeV . Both these single-particle energies are positive, which would mean that the 9Be nucleus is not bound, in contradiction to the experimental situation. The a n s w e r to this difficulty i s probably that the c o r r e l a t i o n energy will be much l a r g e r
* Applied Mathematics Department, S.C.K. Mol, Belgium.
for 9Be than for 8Be. This can be u n d e r s t o o d f r o m the fact that the energy gap between o c c u pied and unoccupied o r b i t a l s i s l a r g e r for even • Z = N n u c l e i than for n e i g h b o u r i n g odd n u c l e i . The r e l a t i v e position of positive and n e g a t i v e p a r i t y l e v e l s in 9Be can roughly b e e s t i m a t e d a s the d i f f e r e n c e of t h e s e s i n g l e - p a r t i c l e e n e r g i e s . This gives a v a l u e of 6.7 MeV which i s a l r e a d y much s m a l l e r than the value ~w ~ 15 MeV given by the s i n g l e - p a r t i c l e s h e l l - m o d e l . A m o r e a c c u r a t e way to obtain this energy d i f f e r e n c e would r e q u i r e an a n g u l a r m o m e n t u m p r o j e c t i o n f r o m the different i n t r i n s i c H a r t r e e - F o c k s t a t e s . In this l e t t e r , we p r e s e n t the r e s u l t s obtained f r o m a p r o j e c t e d H a r t r e e - F o c k calculation for the lowest even and odd p a r i t y s t a t e s of 9Be. We s t a r t f r o m the H a m i l t o n i a n for the i n t e r n a l energy
H= ~i Ti
-Tcm
+ ½i~j " " Vij
(1)
and take as n u c l e o n nucleon i n t e r a c t i o n the p o t e n tial of Volkov VI2
=
{-83.34
exp[-(r12/1.6) 2]
+
(2)
+ 144.86 exp [-(r12/0.82)2]}× (0.4 +0.6 P x ) , w h e r e P x is the M a j o r a n a space exchange o p e r a tor. It was shown in p r e v i o u s p r o j e c t e d H a r t r e e Fock c a l c u l a t i o n s [6] that this f o r c e gives r e a s o n able e n e r g i e s and wave functions for the l o w - l y ing l e v e l s of the Ugther p - s h e l l n u c l e i . As we a r e p r i m a r i l y i n t e r e s t e d h e r e i n the e n e r g y diff e r e n c e between even and odd p a r i t y states and not in the f i n e r d e t a i l s of the wave functions, we b e l i e v e that the n e g l e c t of n o n - c e n t r a l f o r c e s in the H a m i l t o n i a n can be justified. Following the pr(~cedure d e s c r i b e d p r e v i o u s l y [6] we take as t r i a l function ~ L = ~ L ~ where 61
Volume27B, number 2
PHYSICS LETTERS
10 June 1968
~0 -- 10s0) + 810d0) E (MeV)
7
¢0 -- 10p0> ÷ ~10~0> ~1 = lOp1) + ~]Ofl)
-35
X0 : [ ls0} - ~2[0d0) . The b a s i c states [nlm) a r e h a r m o n i c o s c i l l a t o r functions with phases and quantum n u m b e r s as in ref. 7. Putting 8 = 6 = ~ = 0, the S l a t e r d e t e r m i n a n t s @ r e d u c e to the leading s t a t e s of the lowest r e p r e s e n t a t i o n s with negative and positive parity of E l l i o t t ' s S U 3 - c l a s s i f i c a t i o n . With the t r i a l function ~L, we c a l c u l a t e
(~,LIHI q,L>
EL~/
-40
~
EL(f35~bb') = (@L[@L >
F./,, S'f/ ~,j SF.,
-45
-50
//"
A,,, ~''" /
a
I
2
I
8
12
L (L+ l)
20
Fig. 1. Plot of the energies EO and E L for the oddparity states as a function of L(L+ 1). ~L is a p r o j e c t i o n o p e r a t o r for the total o r b i t a l a n g u l a r m o m e n t u m and @ i s a single Slater d e t e r m i n a n t belonging to the m o s t s y m m e t r i c p a r tition [441]. F o r the negative parity states ,:I, = detl~P04 ~4,~n+
~-0,Yl I
(3)
and for the positive parity states @
=
detlq]4~ ~_4 n+ 0 Xo
H e r e ~04 m e a n s that function ~ 0 with the l a b e l s , and ×~+ is a and spatial function chosen as
62
"
(4)
four o r b i t a l s have the s p a t i a l four different spin i s o s p i n n e u t r o n o r b i t a l with spin + ½ X0. The spatial functions a r e
(6)
u s i n g the method d e s c r i b e d in ref. 6, and m i n i m i z e the energy with r e s p e c t to the v a r i a t i o n a l p a r a m e t e r s ~ , 6 , ~ , b and b'. H e r e b and b' a r e the h a r m o n i c o s c i l l a t o r length p a r a m e t e r s , r e s pectively, of the even p a r i t y and odd parity o r b i t a l s . Taking t h e m as v a r i a t i o n a l p a r a m e t e r s , we allow the different o r b i t a l s s o m e f r e e d o m in choosing t h e i r s p a t i a l extension. The other par a m e t e r s / 3 , 6 and ~ d e t e r m i n e the d e f o r m a t i o n of the o r b i t a l s . It i s seen that the o r b i t a l X0 cont a i n s no independent v a r i a t i o n a l p a r a m e t e r , either for its size or for its deformation. The r e a s o n for this i s that the c o m p u t e r on which this c a l c u l a t i o n was p r o g r a m m e d is not sufficiently l a r g e to i n t r o d u c e m o r e v a r i a t i o n a l p a r a m e t e r s . F o r the s a m e r e a s o n , the choice of the o r b i t a l s (5) i s somewhat m o r e r e s t r i c t e d than those in our p r e v i o u s work [6]. F o r this calculation we a l s o had to t r u n c a t e the p o l y n o m i a l s , called FAB in ref. 6, a f t e r t e r m s of d e g r e e 6 in 8, 5 and ~. F r o m our p r e v i o u s work, we e s t i m a t e the u n c e r tainty on the energy i n t r o d u c e d by this t r u n c a t i o n to be l e s s than 0.3 MeV. The r e s u l t s obtained for the negative parity l e v e l s with L = 1, 2, 3, 4 and for the positive p a r i ty l e v e l s with L = 0 and 2 a r e p r e s e n t e d in table 1 H e r e , E ~ is the energy obtained for fl = 6 = ~ = 0 and for the v a l u e s of b and b' given in the table. In the l a s t ~olumn we also give the d e f o r m a t i o n e n e r g i e s ELU _ E L , i.e., the energy gained by i n t r o d u c i n g the p a r a m e t e r s 8, 6 and ~. This d e f o r mation energy i s c o n s i d e r a b l y l a r g e r for the two lower L - v a l u e s of the negative parity band than for the two upper L - v a l u e s . As a result,^ the exact r o t a t i o n a l band s p e c t r u m of E ~ = A + + B L (L + 1) is d i s t o r t e d c o n s i d e r a b l y (fig. 1). As in our p r e v i o u s c a l c u l a t i o n s [6], we again find that the deformation of the i n t r i n s i c state d e c r e a s e s a s we go to higher L - v a l u e s . This b e -
Volume27B, number 2
PHYSICS LETTERS
10June 1968
Table 1 Energies and wave functions for the low-lying negative and positive parity states of 9Be and for the ground state of 8Be.
9Be
8Be
L~
~L
~
~
~
b
b,
~o
~ o _ EL
I-
-49.7
0.357
0.233
0.205
1.62
1.69
-38.89
10.81
2-
-47.7
0.336
0.220
0.201
1.62
1.69
-37.47
10.23
3-
-42.6
0.258
0.166
0.182
1.61
1.68
-35.35
7.25
4-
-39.6
0.229
0.153
0.181
1.61
1.67
-32.55
7.05
0+
-45.4
0.420
0.315
-
1.74
1.75
-28.89
16.51
2+
-43.0
0.395
0.296
-
1.74
1.75
-28.82
14.18
0+
-53.35
0.375
0.239
-
1.61
1.65
-
-
h a v i o u r is just the opposite of what one might expect f r o m the c e n t r i f u g a l f o r c e in a r o t a t i o n a l picture. The m o s t s t r i k i n g r e s u l t of table 1 i s the v e r y l a r g e d e f o r m a t i o n energy for the positive p a r i t y l e v e l s . W h e r e a s t h e s e l e v e l s lie well above the four odd p a r i t y s t a t e s when the d e f o r m a t i o n is neglected, they drop down v e r y low between the negative p a r i t y l e v e l s when the d e f o r m a t i o n a t t a i n s its s t a b l e value. The energy d i f f e r e n c e b e tween the lowest negative and lowest p o s i t i v e p a r i t y level c o m e s down f r o m 10 MeV to 4.3 MeV. Since the d e f o r m a t i o n for the L = 0 state is v e r y l a r g e , it i s to be expected that an additional f r e e dom for the o r b i t a l ×0 to a d j u s t its d e f o r m a t i o n to that of the other o r b i t a l s will give a n a d d i t i o nal energy gain, so that the e x p e r i m e n t a l value 1.7 MeV may well be reached. In the l a s t row of table 1, we have also given, for c o m p a r i s o n , the energy and wave function obtained for the ground s t a t e of 8Be, u s i n g the t r i a l function ~L=O-- ~L=O~, where
= detl 04%41 and ~ 0 , ~°0 a r e given by. eq. (5). A l a r g e r b i n d ing e n e r g y is found for 8Be than for 9Be, j u s t as in the H a r t r e e - F o c k c a l c u l a t i o n of B a s s i c h i s et al. m e n t i o n e d above. The s a m e r e m a r k about the c o r r e l a t i o n energy i s valid here. F r o m t a b l e 1, it i s seen that the s i z e and d e f o r m a t i o n of the n e g a t i v e p a r i t y s t a t e s of 9Be i s quite s i m i l a r to those of the ground s t a t e of 8Be. The p o s i t i v e p a r i t y o r b i t a l s have, however, both a l a r g e r s i z e and d e f o r m a t i o n . In view of the l a r g e d e f o r m a t i o n s obtained in this c a l c u l a t i o n it i s a l s o i n t e r e s t i n g to c a l c u l a t e the quadrupole m o m e n t of the ground s t a t e of 9Be. The e x p e r i m e n t a l value i s not well known, the different d e t e r m i n a t i o n s [8] y i e l d i n g v a l u e s r a n g -
ing between 2.9 and 6.9 fm 2. The most r e l i a b l e v a l u e s e e m s to be obtained f r o m r e c e n t highe n e r g y e l e c t r o n s c a t t e r i n g e x p e r i m e n t s [8], giving the v a l u e Q = 3.8 f m 2. The s a m e m e a s u r e m e n t s a l s o d e t e r m i n e the r . m . s , r a d i u s of the charge d i s t r i b u t i o n to be R = 2.5 fro. On the s i m p l e s h e l l - m o d e l in LS-coupling, one gets Q = ~b 2. Choosing b so that the e x p e r i m e n t a l r a d i u s is r e p r o d u c e d (b = 1.77 fm) gives a quadrupole m o m e n t Q = 1.88 f m z. The p r o j e c ted H a r t r e e - F o c k function gives R = 2.41 fm and Q = 2.73 fm 2. F r o m i n t e r m e d i a t e coupling c a l c u l a t i o n s [9,10], it is well known that even a weak spin o r b i t f o r c e b r i n g s about an i m p o r t a n t i n c r e a s e of the quadrupole moment. T h i s i s due to the fact that the lowest p a r t i t i o n [441] of 9Be contains L = 1, 2, 3, 4 and a weak spin o r b i t f o r c e will mix s o m e L = 2 state with the L = 1 ground state. In a detailed c a l c u l a t i o n for 9Be, it is consequently not allowed to n e g l e c t the spin f o r c e s . T h i s n e g l e c t was p o s s i b l e in our p r e vious c a l c u l a t i o n s [6] for 4 --
63
Volume27B, number 2
PHYSICS
LETTERS
References
6. M. Bouten, P. Van Leuven, H. Depuydt and L. S c h o t s m a n s , Nucl. P h y s . A100 (1967) 90. 7. T. Brody and M. Moshinsky, T a b l e s of t r a n s f o r m a tion b r a c k e t s , (Mexico, 1960). 8. M. B e r n h e i m , T. Stovall and D. V i n c i g u e r r a , Nucl. P h y s , A97 (1967) 488. 9. J . B . F r e n c h , E . C . Halbert and S. Pandya, P h y s . Rev. 99 (1955) 1387. 10. F . C . B a r k e r , Nucl. P h y s . 83 (1966) 418.
1. J . J . K r o e p e l and C . P . B r o w n e , Nucl. P h y s . A108 (1968) 289. 2. A . M . L a n e , Rev. Mod. P h y s . 32 (1960) 619. 3. F . C . B a r k e r , Huel. P h y s . 28 (1961) 96. 4. C . A d l e r , T . C o r c o r a n and C . M a s t , Nucl. P h y s . 88 (1966) 145. 5. W.H. B a s s i c h i s , A . K . K e m a n and J. P. Svenne, P h y s . R e v . 160 (1967) 746. *****
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10 June 1968