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On the parity nonconserving alpha-decay from the 3+, T=0 11.06 MeV state in 16O
Volume 38B, number 4
ON
THE
P HYSI C S L E T T ER S
PARITY NONCONSERVING T H E 3 " , T =0 1 1 . 0 6 MeV
21 February 1972
ALPHA-DECAY STATE IN 160*
FROM
M. FINK
Inst~tut fur Theoret~sche Phys~k Ruhr-Umvers~tat Bochum, W-Germany M. GARI **
Cahforma Institute of Technology, Pasadena, Cahforma and J. G. Z A B O L I T Z K Y
Institut.fur Theoret~sche Physzk Ruhr-Univers~tilt Bochum, W-Germany Received 21 December 1972
The irregular alpha-decay from the 3+ T = 0 11.06 MeV state in 160 is calculated in the Cabblbomodel of weak interactions. Including short range correlations via the generalized Bethe Goldstone equation th,s process seems to be a very favorable one: F i r ~ 3 × 10-6 eV.
E f f e c t s of p a r i t y n o n c o n s e r v i n g n u c l e o n - n u c l e o n i n t e r a c t i o n s (NNI) a r e now e s t a b l i s h e d in e l e c t r o m a g n e t i c t r a n s i t i o n s [1] a s w e l l a s in a l p h a d e c a y [2]. U n f o r t u n a t e l y , m o s t of the e x p e r i m e n t s h a v e b e e n p e r f o r m e d on h e a v y n u c l e i , w h e r e a r e l i a b l e t h e o r e t i c a l c a l c u l a t i o n i s e x t r e m e l y d i f f i c u l t to p e r f o r m if one t r i e s to u s e a d e s c r i p t i o n of the n u c l e a r s t r u c t u r e which is at l e a s t in s o m e way r e a l i s t i c . T h e only p o s i t i v e r e s u l t in a l p h a - d e c a y m e a s u r e m e n t s has b e e n o b t a i n e d by the g r o u p of W ~ f l e r [2] on the e x a m p l e of the 2 - , 0; 8.8 MeV s t a t e in 160. T h e y found a d e c a y width of r ai r = (9 + 3) × 10 -11 eV. T h i s r e s u l t is in good a g r e e m e n t with t h e o r e t i c a l p r e d i c t i o n s [3,4] u s i n g the C a b b i b o m o d e l of w e a k i n t e r a c t i o n s , i n d i c a t i n g t h a t no c a n c e l l a t i o n of S e a g u l l and S c h w i n g e r t e r m s should o c c u r in the w e a k N - ~ N + v e c t o r m e s o n a m p l i t u d e [17]. H o w e v e r , t h i s is b a s e d on o n l y one e x p e r i m e n t and one has to be v e r y c a r e ful with final s t a t e m e n t s . In t h i s p a p e r we s h a l l p r e s e n t c a l c u l a t i o n s on the a - d e c a y of the 3 +, 0; 11.06 MeV s t a t e in 1 6 0 ( s e e fig. 1). T h e d e c a y width of t h i s l e v e l h a s b e e n a l r e a d y e s t i m a t e d to b e F3÷> 10-7 eV [3]. B e c a u s e in t h i s s p e c i a l s i t u * This work was supported in part by Deutsche F o r schungsgemeinschaft, and in part by the U.S. Atomic Energy Commission. Prepared under Contract AT(04-3)-63 for the San Francisco Operations Office, U. S. Atomic Energy Commission. ** On leave of absence from Ruhr-Umversittlt Boehum, W-Germany.
ation the c a l c u l a t i o n of the i r r e g u l a r a l p h a - d e c a y width can be r e d u c e d m a v e r y good a p p r o x i m a tion to m e a s u r e d a l p h a - d e c a y r a t e s , without any a s s u m p t i o n on the n u c l e a r w a v e f u n c t i o n s , t h i s d e c a y could be of c r u c i a l i m p o r t a n c e f o r the a b o v e m e n t i o n e d q u e s t i o n s c o n c e r n i n g the w e a k N-~ N + v e c t o r - m e s o n a m p l i t u d e . In f i r s t o r d e r p e r t u r b a t i o n t h e o r y the p a r i t y m i x e d s t a t e at 11.06 MeV can be w r i t t e n a s : 13; E o = 11.06 MeV> = 13+, T = 0)+
+ ~i Fi]3-' T=O, Ei>+~j Fj]3-, T = I ; E j ) (1) with ( 3 - , T Ivweak 13+, T:O> Fk =
Eo - ~ k
(2)
w h e r e V w e a k d e n o t e s the p a r i t y n o n c o n s e r v i n g NN p o t e n t i a l : V w e a k = V AT=I + V AT=O .
(3)
U s i n g the s t a n d a r d a - d e c a y t h e o r y [5], the d e c a y r a t e F i r of the E o = 11.06 MeV s t a t e can be w r i t t e n as:
Pz:a
(Eo,Rol
2MR °
+ .~Fj(12C Y
I
Fi(12C ÷ a 137 r :0>a °
+ at3 i T=I)Ro [2 ,
(4) 189
PHYSICS
Volume 38B, number 4
wlth R o equal to the sum of daughter and a-particle radu and PL(Eo, Ro) the penetrabthty for angular momentum L. Here the admixtures of opposite pamty rata the groundstates of 12C and a-particle have been neglected, since the first possible states to admix are in both cases about 20 MeV above the groundstate. The second term in eq. (4) can only contribute through isospin mlxture and can therefore be neglected. The decay rate is thus given in a good approximation by F ~r _
PL =3 (Eo,Ro) I 2MRo ~
FZ<12C
+
21 February 1972
LETTERS 1326 1513
1144
riO8
ra
×
-
PL=3 (Eo,Ro) 2MRo
0009 0 09
3-,0 3+,0
083
/
/ 7¢4 f / 12C +eO
615
3",0
a{3;,T=O)Ro{2.
S i n c e the a m p l i t u d e s F z and the r e g u l a r r e (5) d u c e d a - w i d t h s a r e d o m i n a t e d by the 3 - , 0 s t a t e at 11.44 MeV t h i s e q u a t i o n r e d u c e s to the f o l l o w trig s i m p l e e x p r e s s i o n : ir
/ / 0"0
/~/
/ ~
3"~ I 3-,0
o
0% o
MeV
d', T
If{2
Fao (MeV)
~s0
}<12C +a{3-,0; E = 11.44>Ro{2 .
(6)
Fig. 1. Partial level scheme of 160 as gtven by Ajzenberg-Selove [16].
Multiplying by:
x = pr;
PL =3 (E,Ro)/PL=3 (Eo,Ro)
f 2 = 0.08;
and u s i n g the d e f i n i t i o n of F a , we h a v e the f o l l o w ing r e l a t i o n f o r
VLxT=ap =Gp
F i r _ P5:3
a
riar [6]: (E°'RO)IFI2rreg
PL=3 (E, R o)
+
(Ol -
Y(x) = exp-x/x
(Sa)
[(1 + ~ v ) ( l a l × a 2 ) . [ P 1 2 , f ( r ) ]
"2)'{"12,
f(r)}+
T~'~)
(8b)
(~)
E
w h e r e F r o g d e n o t e s the r e g u l a r m e a s u r e d a - d e c a y r a t e of the 3 +, 0; E = 11.44 MeV s t a t e . T h e r a t i o of p e n e t r a b d t h e s to the s a m e L and R o d o e s not d e p e n d v e r y s e n s i t i v e l y on the a l p h a - p a r t i c l e p o t e n t i a l and the c h a n n e l r a d i u s R o. T h e c a l c u l a t i o n of the t r r e g u l a r a l p h a - d e c a y width t h u s is p r a c t i c a l l y r e d u c e d to the c a l c u l a t i o n of the m t x ing a m p l i t u d e F, and the s e v e r e p r o b l e m of h a v ing a r e l i a b l e a - d e c a y t h e o r y is a v o i d e d without any a s s u m p t i o n s on the n u c l e a r w a v e f u n c t m n s . As we h a v e s e e n f r o m eq. (4), b e c a u s e of i s o s y m m e t r y only the A T = 0 p a r t of the w e a k NN p o t e n t i a l can c o n t r i b u t e m o u r c a s e . In the p r e s ent p a p e r we s h a l l i n v e s t i g a t e m d e t a i l the c o n t r i b u t m n s b a s e d on t h e C a b b i b o m o d e l of w e a k i n t e r a c t i o n s . In t h i s m o d e l the c o n t r i b u t i o n s f r o m p and 2~ e x c h a n g e (fig. 2) a r e g i v e n t h r o u g h [7,8]:
with
ll/
lr2
+z
;+ z 2 ; f(r)
=exp(-mpr)/r)
and
G cos20 Gp -
A
8 ~-2~ M
One of the r e a s o n s for d i s c u s s i n g the 27r c o n t r i b u t i o n in t h i s p a p e r is the r e c e n t s t a t e m e n t [9] that t h i s p o t e n t i a l s h o u l d c o n t r i b u t e a s l a r g e a s the p e x c h a n g e p a r t . A p a r t f r o m the q u e s t i o n of double c o u n t i n g , a d d i n g V2~ to Vp would c h a n g e
AT=0, 2 2~
=
(a)
N with 190
x
(b)
Fig. 2. Nonvanishing p r o c e s s e s in the Cabblbo-model of weak mteractions, leading to weak NN potentials with T(+,) mospin dependence. The cross denotes the weak Interactmn.
Volume 38B, number 4
PHYSICS
Table 1 Comparison of the matrkx elements M p and M 2~ for different stron N N interactmns. weak
V 2~
Vp
~c
8.3
strong NNI uncorrelated ORSAY (soft-core) 0.27
2.8
Reid (soft-core) (itS)
0.20
2.3
Reid (hard-core) (RH)
0.17
2.2
Hamada-Johnston (HJ)
0.16
2.0
t h e r e c e n t c a l c u l a t i o n s on a l p h a - d e c a y f r o m the 2 - , 8.8 MeV s t a t e d r a s t i c a l l y . B e c a u s e the c a l c u l a t i o n s in ref. [9] h a v e b e e n p e r f o r m e d without i n t r o d u c i n g s h o r t - r a n g e c o r r e l a h o n s in the n u c l e o n w a v e f u n c t i o n s , we s h a l l d i s c u s s t h i s c a s e too f o r the s a k e of c o m p a r i s o n . F o r the n u c l e a r s t r u c t u r e d e s c r i p t i o n we u s e the w a v e f u n c t m n s of Z u k e r et al. [11], c l a s s i f i e d in the s e n i o r i t y s c h e m e and Jj - c o u p l i n g . 12C is t r e a t e d a s an i n n e r c o r e in the u s u a l way, a l l o w i n g the r e s t of the p a r h c l e s to o c c u p y the 2 S 1 / 2 , l p 1/2 and l d 5 / 2 s h e l l s . T h e s e w a v e f u n c t i o n s a r e g i v e n in an i n d e p e n d e n t p a r t i c l e p i c t u r e , which m e a n s that t h e n u c l e o n - n u c l e o n c o r r e l a t i o n s at s m a l l r e l a t i v e d i s t a n c e s h a s to be i n t r o d u c e d s e p a r a t e l y by s o l u t i o n s of the B e t h e - G o l d s t o n e [12] (BG) o r g e n e r a l i z e d B e t h e - G o l d s t o n e (GBG) [13] e q u a t i o n f o r f i n i t e n u c l e i . In o u r c l a c u l a t i o n s [3,6,10] we s o l v e d the GBG e q u a t i o n with the a s s u m p t i o n that t h e c o r r e l a t e d 2 p a r t i c l e s t a t e I ~ ; A ) can be e x p a n d e d in e s s e n t i a l l y the s a m e way into r e l a t i v e and CM m o t i o n a s the u n c o r r e l a t e d 2 p a r t i c l e s t a t e l Ul l 1 n 2 l 2 ; A ) in the o s c i l l a t o r m o d e l : Inllln2/2;A)=
~
nlNL
(nlNL/n l lln212)AlnlNL;A) (9a)
namely:
I~;A) =
21 February 1972
Table 2 Resulting alpha-decay width in the Cabblbo model of weak interactions calculated with different strong NN interactions. HamadaJohnston
Reid hard-core
Reid soft-core
BGT (r c = 0.4 fm)
2.4 × 10 -6
2.7 × 10 -6
3.1 × 10 -6
4.4 × 10 -6
5.0
BGT (r c = 0.4 fro)
I
LETTERS
~ ( n l N L / n l l l n 2 12) A {IC)nlNLAINL)}A nlNL (9b)
T h i s m e a n s that we i n t r o d u c e d the e f f e c t of the N N - c o r r e l a t i o n s only in the r e l a t i v e m o t i o n . T h e CM m o t i o n is a s s u m e d to be not a f f e c t e d ; t h i s i s not e x a c t l y t r u e , h o w e v e r , it should be a good apprommation. The relative wavefunction(rtC) w i l l , owing to the s p e c i a l A n s a t z (9b), depend on
t h e q u a n t u m n u m b e r s nlNL A a s i n d i c a t e d . With the e x p a n s i o n (9b) one can r e d u c e the BG o r GBG e q u a t i o n to an e q u a h o n f o r the r e l a t i v e m o t i o n IC > only. T h e a d v a n t a g e of thts p r o c e d u r e [14] is that the c a l c u l a h o n of the t w o - b o d y m a t r i x e l e m e n t s r e m a i n s e s s e n t i a l l y the s a m e a s in the u n c o r r e l a t e d c a s e - the r e l a t i v e w a v e f u n c h o n (rlnl) h a s to be r e p l a c e d by (rlc~nlNLA. T h e s o l u t i o n of t h e c o r r e l a t e d w a v e f u n c h o n t h r o u g h eq. (9b) e n s u r e s that the m t r o d u c t m n of the c o r r e l a t i o n s into the m a t r i x e l e m e n t s will be c o n s i s t e n t . To g e t a f e e h n g f o r the r e l a t i v e d e p e n d e n c e of the A - p a r t i c l e m a t r i x e l e m e n t s of V 2v and V p on the s t r o n g NN c o r r e l a t i o n s we s o l v e d the GBG e q u a tion f o r d i f f e r e n t s t r o n g NN i n t e r a c t i o n s : H a m a d a J o h n s t o n (HJ), R e i d h a r d - c o r e (RH), R e i d s o f t c o r e (RS), B r u c k n e r - G a m m e l - T h a l e r (BGT) and O r s a y soft c o r e . T h e u n c o r r e l a t e d c a s e c o r r e s ponds j u s t to the t w o - p a r t i c l e s t a t e eq. (9a). In t a b l e 1 we c o m p a r e the m a t r l x e l e m e n t s (3 +, 01Vweak [ 3 - , 0 ) f o r the w e a k p and 2~ p o t e n t i a l ( a r b i t r a r y units). We s e e that the r e d u c t i o n in Vp owing to s t r o n g i n t e r a c t i o n s is about a f a c t o r 3 - 4 , in c o m p l e t e a g r e e m e n t with o u r e a r l i e r c a l c u l a h o n s [10]. Owing to the v e r y s o f t - c o r e in the O r s a y NN p o t e n t i a l the r e d u c t i o n in t h i s c a s e i s not v e r y s t r o n g . The m a t r i x - e l e m e n t s f o r the 2n p o t e n t i a l a r e i n f i n i t e f o r c a l c u l a t i o n s with u n c o r r e l a t e d w a v e f u n c t l o n s , which is due to the f a c t , that V 2~ is p r o p o r t i o n a l to r -5 at the o r i g i n if no cutoff is u s e d m the m o m e n t u m s p a c e (eq. (8a)). We should, h o w e v e r , k e e p in m i n d that a cutoff in m o m e n t u m s p a c e is not n e c e s s a r y if realistic wavefunchons are used, because this r e g i o n is c o v e r e d c o m p l e t e l y t h r o u g h the h a r d c o r e e f f e c t m the w a v e f u n c t i o n s . C o r r e l a t i o n s due to the O r s a y p o t e n t i a l cannot be u s e d without a cutoff in the weak p o t e n h a l , b e c a u s e of the v e r y soft c o r e . C o m p a r i n g the a m p l i t u d e M p with M 2n f o r h a r d - c o r e NN p o t e n t i a l s , we s e e that M2~ is n e a r l y a f a c t o r 10 s m a l l e r than M P, showing that the 2~ c o n t r i b u t i o n should be n e g l i g i b l e a l s o in F i r In v i e w of p o s s i b l e double c o u n t i n g we inc l u d e d only the p - e x c h a n g e p a r t m F i r g i v e n in t a b l e 2. T h e d e c a y r a t e of t h i s 11.06 MeV l e v e l is c o n s i d e r a b l y l a r g e r ( N 104) than the one of the 191
Volume 38B, n u m b e r 4
PHYSICS
8.8 M e V l e v e l in 1 6 0 w h i c h i s a l r e a d y m e a s u r e d . T h i s s h o u l d l e a v e a good c h a n c e f o r i t s m e a s u r e m e n t in s p h t e of t h e d i f f i c u l t i e s c o n c e r n i n g t h e b a c k g r o u n d of a l l o w e d a l p h a - p a r t i c l e s . Measurem e n t s of t h i s d e c a y r a t e a r e p l a n n e d b y W ~ f l e r [15]. M a n y t h a n k s a r e due to H. Kt~mmel a n d A. H. Huffman for helpful discusslons and G E Brown for valuable correspondence. The numerical c a l c u l a t i o n s h a v e b e e n p e r f o r m e d on t h e T R 440 of t h e " R e c h e n z e n t r u m d e r R u h r - U n i v e r s i t ~ t t Bochum".
R e f e r e n ce s [1] E. D. L,pson, F. Boehm and F. C. Vanderleeden, to be published (CALT-63-163); F. Boehm, Intern. Conf. m Delft, 1969 (CALT-63-142). [2] H. Hgttig, K. Hfinchen and H. W~lffler, Phys. Rev. L e t t e r s 25 (1970) 941.
192
LETT ERS
21 F e b r u a r y 1972
[31 M. Gari and H. Kgmmel, Phys. Rev. L e t t e r s 23 (1969) 26. [4] E. M. Henley, T. E. Keliher and D. U. L. Yu, Phys. Rev. L e t t e r s 23 (1969) 941. [51 H. F. Mang, Ann. Rev. Nucl. Scl. 14 (1964) 1. [61 M. Garl, Phys. L e t t e r s 31B (1970) 627. [71 E. F m c h b a c h , D. Tadlc and K. T r a b e r t , Phys. Rev. 186 (1969) 1688. [81 R. J. B h n - S t o y l e , Phys. Rev. 118 (1960) 1605. [9] N. Vmh Mau, Syrup. on Nucleons and weak i n t e r actions, Z a g r e b - Y u g o s l a v i a , July 1971. [10] M. Gain, H. Ktimmel and J. G. Zabohtzky, Nucl. Phys. A161 (1971) 625. [11] A. P. Zuker, B. Buck and F. B. M e G r o r y , Phys. Rev. L e t t e r s 21 (1968) 39. [12] A. Kalho and B. D. Day, Phys. L e t t e r s 25B (1967) 72. [13] H. Kflmmel, Nucl. Phys. A176 (1971) 205. [14l M. Gari, Doktorarbeit Mamz 1969. [15] H. Wgffler, M a x - P l a n c k Institut Mamz, Germany (private cummunieation). [16] F. Ajzenberg-Selove, Nucl. Phys. A166 (1971) 1. [17] P. Olesen and F.S. Rao, Phys. L e t t e r s 29B (1969) 23',
ON
THE
P HYSI C S L E T T ER S
PARITY NONCONSERVING T H E 3 " , T =0 1 1 . 0 6 MeV
21 February 1972
ALPHA-DECAY STATE IN 160*
FROM
M. FINK
Inst~tut fur Theoret~sche Phys~k Ruhr-Umvers~tat Bochum, W-Germany M. GARI **
Cahforma Institute of Technology, Pasadena, Cahforma and J. G. Z A B O L I T Z K Y
Institut.fur Theoret~sche Physzk Ruhr-Univers~tilt Bochum, W-Germany Received 21 December 1972
The irregular alpha-decay from the 3+ T = 0 11.06 MeV state in 160 is calculated in the Cabblbomodel of weak interactions. Including short range correlations via the generalized Bethe Goldstone equation th,s process seems to be a very favorable one: F i r ~ 3 × 10-6 eV.
E f f e c t s of p a r i t y n o n c o n s e r v i n g n u c l e o n - n u c l e o n i n t e r a c t i o n s (NNI) a r e now e s t a b l i s h e d in e l e c t r o m a g n e t i c t r a n s i t i o n s [1] a s w e l l a s in a l p h a d e c a y [2]. U n f o r t u n a t e l y , m o s t of the e x p e r i m e n t s h a v e b e e n p e r f o r m e d on h e a v y n u c l e i , w h e r e a r e l i a b l e t h e o r e t i c a l c a l c u l a t i o n i s e x t r e m e l y d i f f i c u l t to p e r f o r m if one t r i e s to u s e a d e s c r i p t i o n of the n u c l e a r s t r u c t u r e which is at l e a s t in s o m e way r e a l i s t i c . T h e only p o s i t i v e r e s u l t in a l p h a - d e c a y m e a s u r e m e n t s has b e e n o b t a i n e d by the g r o u p of W ~ f l e r [2] on the e x a m p l e of the 2 - , 0; 8.8 MeV s t a t e in 160. T h e y found a d e c a y width of r ai r = (9 + 3) × 10 -11 eV. T h i s r e s u l t is in good a g r e e m e n t with t h e o r e t i c a l p r e d i c t i o n s [3,4] u s i n g the C a b b i b o m o d e l of w e a k i n t e r a c t i o n s , i n d i c a t i n g t h a t no c a n c e l l a t i o n of S e a g u l l and S c h w i n g e r t e r m s should o c c u r in the w e a k N - ~ N + v e c t o r m e s o n a m p l i t u d e [17]. H o w e v e r , t h i s is b a s e d on o n l y one e x p e r i m e n t and one has to be v e r y c a r e ful with final s t a t e m e n t s . In t h i s p a p e r we s h a l l p r e s e n t c a l c u l a t i o n s on the a - d e c a y of the 3 +, 0; 11.06 MeV s t a t e in 1 6 0 ( s e e fig. 1). T h e d e c a y width of t h i s l e v e l h a s b e e n a l r e a d y e s t i m a t e d to b e F3÷> 10-7 eV [3]. B e c a u s e in t h i s s p e c i a l s i t u * This work was supported in part by Deutsche F o r schungsgemeinschaft, and in part by the U.S. Atomic Energy Commission. Prepared under Contract AT(04-3)-63 for the San Francisco Operations Office, U. S. Atomic Energy Commission. ** On leave of absence from Ruhr-Umversittlt Boehum, W-Germany.
ation the c a l c u l a t i o n of the i r r e g u l a r a l p h a - d e c a y width can be r e d u c e d m a v e r y good a p p r o x i m a tion to m e a s u r e d a l p h a - d e c a y r a t e s , without any a s s u m p t i o n on the n u c l e a r w a v e f u n c t i o n s , t h i s d e c a y could be of c r u c i a l i m p o r t a n c e f o r the a b o v e m e n t i o n e d q u e s t i o n s c o n c e r n i n g the w e a k N-~ N + v e c t o r - m e s o n a m p l i t u d e . In f i r s t o r d e r p e r t u r b a t i o n t h e o r y the p a r i t y m i x e d s t a t e at 11.06 MeV can be w r i t t e n a s : 13; E o = 11.06 MeV> = 13+, T = 0)+
+ ~i Fi]3-' T=O, Ei>+~j Fj]3-, T = I ; E j ) (1) with ( 3 - , T Ivweak 13+, T:O> Fk =
Eo - ~ k
(2)
w h e r e V w e a k d e n o t e s the p a r i t y n o n c o n s e r v i n g NN p o t e n t i a l : V w e a k = V AT=I + V AT=O .
(3)
U s i n g the s t a n d a r d a - d e c a y t h e o r y [5], the d e c a y r a t e F i r of the E o = 11.06 MeV s t a t e can be w r i t t e n as:
Pz:a
(Eo,Rol
2MR °
+ .~Fj(12C Y
I
Fi(12C ÷ a 137 r :0>a °
+ at3 i T=I)Ro [2 ,
(4) 189
PHYSICS
Volume 38B, number 4
wlth R o equal to the sum of daughter and a-particle radu and PL(Eo, Ro) the penetrabthty for angular momentum L. Here the admixtures of opposite pamty rata the groundstates of 12C and a-particle have been neglected, since the first possible states to admix are in both cases about 20 MeV above the groundstate. The second term in eq. (4) can only contribute through isospin mlxture and can therefore be neglected. The decay rate is thus given in a good approximation by F ~r _
PL =3 (Eo,Ro) I 2MRo ~
FZ<12C
+
21 February 1972
LETTERS 1326 1513
1144
riO8
ra
×
-
PL=3 (Eo,Ro) 2MRo
0009 0 09
3-,0 3+,0
083
/
/ 7¢4 f / 12C +eO
615
3",0
a{3;,T=O)Ro{2.
S i n c e the a m p l i t u d e s F z and the r e g u l a r r e (5) d u c e d a - w i d t h s a r e d o m i n a t e d by the 3 - , 0 s t a t e at 11.44 MeV t h i s e q u a t i o n r e d u c e s to the f o l l o w trig s i m p l e e x p r e s s i o n : ir
/ / 0"0
/~/
/ ~
3"~ I 3-,0
o
0% o
MeV
d', T
If{2
Fao (MeV)
~s0
}<12C +a{3-,0; E = 11.44>Ro{2 .
(6)
Fig. 1. Partial level scheme of 160 as gtven by Ajzenberg-Selove [16].
Multiplying by:
x = pr;
PL =3 (E,Ro)/PL=3 (Eo,Ro)
f 2 = 0.08;
and u s i n g the d e f i n i t i o n of F a , we h a v e the f o l l o w ing r e l a t i o n f o r
VLxT=ap =Gp
F i r _ P5:3
a
riar [6]: (E°'RO)IFI2rreg
PL=3 (E, R o)
+
(Ol -
Y(x) = exp-x/x
(Sa)
[(1 + ~ v ) ( l a l × a 2 ) . [ P 1 2 , f ( r ) ]
"2)'{"12,
f(r)}+
T~'~)
(8b)
(~)
E
w h e r e F r o g d e n o t e s the r e g u l a r m e a s u r e d a - d e c a y r a t e of the 3 +, 0; E = 11.44 MeV s t a t e . T h e r a t i o of p e n e t r a b d t h e s to the s a m e L and R o d o e s not d e p e n d v e r y s e n s i t i v e l y on the a l p h a - p a r t i c l e p o t e n t i a l and the c h a n n e l r a d i u s R o. T h e c a l c u l a t i o n of the t r r e g u l a r a l p h a - d e c a y width t h u s is p r a c t i c a l l y r e d u c e d to the c a l c u l a t i o n of the m t x ing a m p l i t u d e F, and the s e v e r e p r o b l e m of h a v ing a r e l i a b l e a - d e c a y t h e o r y is a v o i d e d without any a s s u m p t i o n s on the n u c l e a r w a v e f u n c t m n s . As we h a v e s e e n f r o m eq. (4), b e c a u s e of i s o s y m m e t r y only the A T = 0 p a r t of the w e a k NN p o t e n t i a l can c o n t r i b u t e m o u r c a s e . In the p r e s ent p a p e r we s h a l l i n v e s t i g a t e m d e t a i l the c o n t r i b u t m n s b a s e d on t h e C a b b i b o m o d e l of w e a k i n t e r a c t i o n s . In t h i s m o d e l the c o n t r i b u t i o n s f r o m p and 2~ e x c h a n g e (fig. 2) a r e g i v e n t h r o u g h [7,8]:
with
ll/
lr2
+z
;+ z 2 ; f(r)
=exp(-mpr)/r)
and
G cos20 Gp -
A
8 ~-2~ M
One of the r e a s o n s for d i s c u s s i n g the 27r c o n t r i b u t i o n in t h i s p a p e r is the r e c e n t s t a t e m e n t [9] that t h i s p o t e n t i a l s h o u l d c o n t r i b u t e a s l a r g e a s the p e x c h a n g e p a r t . A p a r t f r o m the q u e s t i o n of double c o u n t i n g , a d d i n g V2~ to Vp would c h a n g e
AT=0, 2 2~
=
(a)
N with 190
x
(b)
Fig. 2. Nonvanishing p r o c e s s e s in the Cabblbo-model of weak mteractions, leading to weak NN potentials with T(+,) mospin dependence. The cross denotes the weak Interactmn.
Volume 38B, number 4
PHYSICS
Table 1 Comparison of the matrkx elements M p and M 2~ for different stron N N interactmns. weak
V 2~
Vp
~c
8.3
strong NNI uncorrelated ORSAY (soft-core) 0.27
2.8
Reid (soft-core) (itS)
0.20
2.3
Reid (hard-core) (RH)
0.17
2.2
Hamada-Johnston (HJ)
0.16
2.0
t h e r e c e n t c a l c u l a t i o n s on a l p h a - d e c a y f r o m the 2 - , 8.8 MeV s t a t e d r a s t i c a l l y . B e c a u s e the c a l c u l a t i o n s in ref. [9] h a v e b e e n p e r f o r m e d without i n t r o d u c i n g s h o r t - r a n g e c o r r e l a h o n s in the n u c l e o n w a v e f u n c t i o n s , we s h a l l d i s c u s s t h i s c a s e too f o r the s a k e of c o m p a r i s o n . F o r the n u c l e a r s t r u c t u r e d e s c r i p t i o n we u s e the w a v e f u n c t m n s of Z u k e r et al. [11], c l a s s i f i e d in the s e n i o r i t y s c h e m e and Jj - c o u p l i n g . 12C is t r e a t e d a s an i n n e r c o r e in the u s u a l way, a l l o w i n g the r e s t of the p a r h c l e s to o c c u p y the 2 S 1 / 2 , l p 1/2 and l d 5 / 2 s h e l l s . T h e s e w a v e f u n c t i o n s a r e g i v e n in an i n d e p e n d e n t p a r t i c l e p i c t u r e , which m e a n s that t h e n u c l e o n - n u c l e o n c o r r e l a t i o n s at s m a l l r e l a t i v e d i s t a n c e s h a s to be i n t r o d u c e d s e p a r a t e l y by s o l u t i o n s of the B e t h e - G o l d s t o n e [12] (BG) o r g e n e r a l i z e d B e t h e - G o l d s t o n e (GBG) [13] e q u a t i o n f o r f i n i t e n u c l e i . In o u r c l a c u l a t i o n s [3,6,10] we s o l v e d the GBG e q u a t i o n with the a s s u m p t i o n that t h e c o r r e l a t e d 2 p a r t i c l e s t a t e I ~ ; A ) can be e x p a n d e d in e s s e n t i a l l y the s a m e way into r e l a t i v e and CM m o t i o n a s the u n c o r r e l a t e d 2 p a r t i c l e s t a t e l Ul l 1 n 2 l 2 ; A ) in the o s c i l l a t o r m o d e l : Inllln2/2;A)=
~
nlNL
(nlNL/n l lln212)AlnlNL;A) (9a)
namely:
I~;A) =
21 February 1972
Table 2 Resulting alpha-decay width in the Cabblbo model of weak interactions calculated with different strong NN interactions. HamadaJohnston
Reid hard-core
Reid soft-core
BGT (r c = 0.4 fm)
2.4 × 10 -6
2.7 × 10 -6
3.1 × 10 -6
4.4 × 10 -6
5.0
BGT (r c = 0.4 fro)
I
LETTERS
~ ( n l N L / n l l l n 2 12) A {IC)nlNLAINL)}A nlNL (9b)
T h i s m e a n s that we i n t r o d u c e d the e f f e c t of the N N - c o r r e l a t i o n s only in the r e l a t i v e m o t i o n . T h e CM m o t i o n is a s s u m e d to be not a f f e c t e d ; t h i s i s not e x a c t l y t r u e , h o w e v e r , it should be a good apprommation. The relative wavefunction(rtC) w i l l , owing to the s p e c i a l A n s a t z (9b), depend on
t h e q u a n t u m n u m b e r s nlNL A a s i n d i c a t e d . With the e x p a n s i o n (9b) one can r e d u c e the BG o r GBG e q u a t i o n to an e q u a h o n f o r the r e l a t i v e m o t i o n IC > only. T h e a d v a n t a g e of thts p r o c e d u r e [14] is that the c a l c u l a h o n of the t w o - b o d y m a t r i x e l e m e n t s r e m a i n s e s s e n t i a l l y the s a m e a s in the u n c o r r e l a t e d c a s e - the r e l a t i v e w a v e f u n c h o n (rlnl) h a s to be r e p l a c e d by (rlc~nlNLA. T h e s o l u t i o n of t h e c o r r e l a t e d w a v e f u n c h o n t h r o u g h eq. (9b) e n s u r e s that the m t r o d u c t m n of the c o r r e l a t i o n s into the m a t r i x e l e m e n t s will be c o n s i s t e n t . To g e t a f e e h n g f o r the r e l a t i v e d e p e n d e n c e of the A - p a r t i c l e m a t r i x e l e m e n t s of V 2v and V p on the s t r o n g NN c o r r e l a t i o n s we s o l v e d the GBG e q u a tion f o r d i f f e r e n t s t r o n g NN i n t e r a c t i o n s : H a m a d a J o h n s t o n (HJ), R e i d h a r d - c o r e (RH), R e i d s o f t c o r e (RS), B r u c k n e r - G a m m e l - T h a l e r (BGT) and O r s a y soft c o r e . T h e u n c o r r e l a t e d c a s e c o r r e s ponds j u s t to the t w o - p a r t i c l e s t a t e eq. (9a). In t a b l e 1 we c o m p a r e the m a t r l x e l e m e n t s (3 +, 01Vweak [ 3 - , 0 ) f o r the w e a k p and 2~ p o t e n t i a l ( a r b i t r a r y units). We s e e that the r e d u c t i o n in Vp owing to s t r o n g i n t e r a c t i o n s is about a f a c t o r 3 - 4 , in c o m p l e t e a g r e e m e n t with o u r e a r l i e r c a l c u l a h o n s [10]. Owing to the v e r y s o f t - c o r e in the O r s a y NN p o t e n t i a l the r e d u c t i o n in t h i s c a s e i s not v e r y s t r o n g . The m a t r i x - e l e m e n t s f o r the 2n p o t e n t i a l a r e i n f i n i t e f o r c a l c u l a t i o n s with u n c o r r e l a t e d w a v e f u n c t l o n s , which is due to the f a c t , that V 2~ is p r o p o r t i o n a l to r -5 at the o r i g i n if no cutoff is u s e d m the m o m e n t u m s p a c e (eq. (8a)). We should, h o w e v e r , k e e p in m i n d that a cutoff in m o m e n t u m s p a c e is not n e c e s s a r y if realistic wavefunchons are used, because this r e g i o n is c o v e r e d c o m p l e t e l y t h r o u g h the h a r d c o r e e f f e c t m the w a v e f u n c t i o n s . C o r r e l a t i o n s due to the O r s a y p o t e n t i a l cannot be u s e d without a cutoff in the weak p o t e n h a l , b e c a u s e of the v e r y soft c o r e . C o m p a r i n g the a m p l i t u d e M p with M 2n f o r h a r d - c o r e NN p o t e n t i a l s , we s e e that M2~ is n e a r l y a f a c t o r 10 s m a l l e r than M P, showing that the 2~ c o n t r i b u t i o n should be n e g l i g i b l e a l s o in F i r In v i e w of p o s s i b l e double c o u n t i n g we inc l u d e d only the p - e x c h a n g e p a r t m F i r g i v e n in t a b l e 2. T h e d e c a y r a t e of t h i s 11.06 MeV l e v e l is c o n s i d e r a b l y l a r g e r ( N 104) than the one of the 191
Volume 38B, n u m b e r 4
PHYSICS
8.8 M e V l e v e l in 1 6 0 w h i c h i s a l r e a d y m e a s u r e d . T h i s s h o u l d l e a v e a good c h a n c e f o r i t s m e a s u r e m e n t in s p h t e of t h e d i f f i c u l t i e s c o n c e r n i n g t h e b a c k g r o u n d of a l l o w e d a l p h a - p a r t i c l e s . Measurem e n t s of t h i s d e c a y r a t e a r e p l a n n e d b y W ~ f l e r [15]. M a n y t h a n k s a r e due to H. Kt~mmel a n d A. H. Huffman for helpful discusslons and G E Brown for valuable correspondence. The numerical c a l c u l a t i o n s h a v e b e e n p e r f o r m e d on t h e T R 440 of t h e " R e c h e n z e n t r u m d e r R u h r - U n i v e r s i t ~ t t Bochum".
R e f e r e n ce s [1] E. D. L,pson, F. Boehm and F. C. Vanderleeden, to be published (CALT-63-163); F. Boehm, Intern. Conf. m Delft, 1969 (CALT-63-142). [2] H. Hgttig, K. Hfinchen and H. W~lffler, Phys. Rev. L e t t e r s 25 (1970) 941.
192
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21 F e b r u a r y 1972
[31 M. Gari and H. Kgmmel, Phys. Rev. L e t t e r s 23 (1969) 26. [4] E. M. Henley, T. E. Keliher and D. U. L. Yu, Phys. Rev. L e t t e r s 23 (1969) 941. [51 H. F. Mang, Ann. Rev. Nucl. Scl. 14 (1964) 1. [61 M. Garl, Phys. L e t t e r s 31B (1970) 627. [71 E. F m c h b a c h , D. Tadlc and K. T r a b e r t , Phys. Rev. 186 (1969) 1688. [81 R. J. B h n - S t o y l e , Phys. Rev. 118 (1960) 1605. [9] N. Vmh Mau, Syrup. on Nucleons and weak i n t e r actions, Z a g r e b - Y u g o s l a v i a , July 1971. [10] M. Gain, H. Ktimmel and J. G. Zabohtzky, Nucl. Phys. A161 (1971) 625. [11] A. P. Zuker, B. Buck and F. B. M e G r o r y , Phys. Rev. L e t t e r s 21 (1968) 39. [12] A. Kalho and B. D. Day, Phys. L e t t e r s 25B (1967) 72. [13] H. Kflmmel, Nucl. Phys. A176 (1971) 205. [14l M. Gari, Doktorarbeit Mamz 1969. [15] H. Wgffler, M a x - P l a n c k Institut Mamz, Germany (private cummunieation). [16] F. Ajzenberg-Selove, Nucl. Phys. A166 (1971) 1. [17] P. Olesen and F.S. Rao, Phys. L e t t e r s 29B (1969) 23',