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On the impulse-approximation treatment of allowed muon capture in 6Li
ON T H E
IMPULSE-APROXIMATION
OF ALLOWED
MUON CAPTURE
TREATMENT IN 6Li
*
N. C. M U K H O P A D H Y A Y * *
Department of Physics, University of Chicago, Chicago, Ill., 60637, USA and Physics Division, Argonne National Laboratory, Argonne, Ill., 60439, USA Received 24 May 1972 The rate of the reaction 6Li + ~-~6He(g.s.) + ug is ~aicu[ated with Cohen-Kurath shell-mode[ wave functions. Calculated without the tensor term, it agrees with experiment if the oscillator root-mean-square rzdius is within the range 2.10 fm ~
Despite considerable theoretical attention, it is still not clear [I] whether an impulse-approximation analysis of the process 6Li + ~-(1S)-*6He(g.s.) + u~
(1)
y i e l d s a t r a n s i t i o n r a t e in a g r e e m e n t with e x p e r i ment. T h i s l e t t e r r e - e x a m i n e s this q u e s t i o n with the aid of the C o h e n - K u r a t h [2] i n t e r m e d i a t e coupling wave functions for the n u c l e a r states involved. This study has t h r e e p r i m a r y a i m s : (a) to find out whether the i m p u l s e - a p p r o x i m a t i o n a n a l y s i s for p r o c e s s (1) can account for the m e a s u r e d [3] t r a n s i t i o n rate W~xp
+330 -1 = (1600_129)s ,
(2)
(b) to study the c o n t r i b u t i o n s of v a r i o u s induced t e r m s in the weak hadronic c u r r e n t (including the p o s s i b l e p r e s e n c e of a G - p a r i t y - v i o l a t i n g t e n sor t e r m ) , and (c) to e x a m i n e the r e l a t i v e i m p o r t a n c e of v a r i o u s n u c l e a r multipole o p e r a t o r s . The method of a n a l y s i s is that d e s c r i b e d in a study [4] of muon c a p t u r e in 12C. The b a s i c idea is to use the i m p u l s e a p p r o x i m a t i o n to r e l a t e the t r a n s i t i o n amplitude for n u c l e a r c a p t u r e to that for the e l e m e n t a r y c a p t u r e p r o c e s s Ix- + p ,
•
.
.
-
n : :t~ar } h : o i c n t : : a : t 'th : [ ~sPtte:lble t ~°[ : h : n ~ : d • Research done under the auspices of the U. s. Atomic Energy Commission, in partial fulfillment of requirements of a Ph. D.degree at the University of Chicago. • * Fulbright Grantee, the University of Chicago, and Resident Student Associate (Thesis), Argonne National Laboratory. Supported in part by the University of Chicago and the U. S. Atomic Energy Commission.
c u r r e n t - c u r r e n t form. The f o r m f a c t o r s in the weak hadron c u r r e n t and the m a n n e r of t h e i r d e t e r m i n a t i o n a r e given in ref. [4]; in addition to the dominant a x i a l - v e c t o r (A) t e r m , the p r o c e s s (1) involves vector (V), weak m a g n e t i s m (WM), induced p s e u d o s c a l a r (PS), and induced t e n s o r (T) t e r m s . All but the t e n s o r f o r m factor a r e w e l l - d e t e r m i n e d for p r e s e n t p u r p o s e s . The v e r y p r e s e n c e of the G - p a r i t y - v i o l a t i n g t e n s o r t e r m is u n c e r t a i n . , Here we compute the c a p t u r e r a t e with both zero and the W i l k i n s o n - A l b u r g e r [5] upper l i m i t for T. The i m p u l s e - a p p r o x i m a t i o n a n a l y s i s in [4] yields an e x p r e s s i o n of the f o r m
:s(Irl/212 + 1T3/2[ 2)
(3)
for the t r a n s i t i o n r a t e W~ of the p r o c e s s (1), where S is the s t a t i s t i c a l factor and Tj is the a m p litude for the e m i s s i o n of a n e u t r i n o in a state of total a n g u l a r m o m e n t u m j(j=½ o r 23-for the n e u t r i n o in an S l / 2 0 r d3/2 state, r e s p e c t i v e l y ) . The t r a n s i t i o n a m p l i t u d e s have the f o r m 6 Tol= 6He(g.s.)[ [i=~l{aa a +ba[ai>(Y2(t'i)] 1 + C a [ r / × p / ] l + d a r i (a i " p i ) } r i- 116Li(g.s.) ,
(4) where a n , ba, etc. a r e functions of the nucleon f o r m f a c t o r s and lepton v a r i a b l e s and r~ c o n v e r t s the ith p r o t o n into a nuetron. The wave functions for the i n i t i a l and the final n u c l e a r s t a t e s a r e the i n t e r m e d i a t e - c o u p l i n g s h e l l - m o d e l wave functions of Cohen and Kurath [2]. In the m a s s - 6 s y s t e m , these a r e v e r y c l o s e to the ones obtained in the LS l i m i t . The Cohen-
157
Volume 40B, number 2
PHYSICS
Kurath wave functions give a reasonable account of m a n y of t h e l o w - e n e r g y p r o p e r t i e s of t h e mass-6 system; in particular, these reproduce t h e m e a s u r e d 6 H e - ~ 6 L i . f l - d e c a y r a t e to w i t h i n a b o u t 5%. With the weak form factors and muon wave f u n c t i o n s g i v e n i n r e f . [4] a n d h a r m o n i c - o s c i l l a t o r nuclear radial wave functions with an rms radius of 2.43 f m [ 6 ] , w e o b t a i n f o r t h e c a p t u r e r a t e t h e v a l u e s W ~ = 1813 s -1 if t h e t e n s o r t e r m i s n e g l e c t e d o r W~ = 1872 s -1 if t h e W i l k i n s o n - A l b u r g e r u p p e r l i m i t i s a s s u m e d f o r T. T h e s e v a l u e s , w h i c h i n c l u d e n u c l e a r f i n i t e - s i z e e f f e c t s [7] i n the mu~n spinors, are both well inside the error l i m i t s of t h e m e a s u r e d t r a n s i t i o n r a t e . C o n t r i b u t ~ n s to t h e r a t e of t h e p r o c e s s (1) arise from various weak hadron form factors ( t a b l e 1) a n d v a r i o u s n u c l e a r m u l t i p o l e o p e r a t o r s ( t a b l e 2). E a c h c o n t r i b u t i o n h a s b e e n c a l c u l a t e d both for a zero tensor term (upper entry) and for a tensor term equal to the Wilkinson-Alburger upper limit (lower entry). It is clear that the tensor interaction at the present upper limit has only a marginally significant effect on both the coupling-constant and the muir[pole decompositions. Table 1 shows that, as expected, the axialvector term is by f a r the most important of the t e r m s in the weak hadron current. However, the weak-magnetism and the pseudoscalar t e r m s give sizable contributions through interference with the axial-vector t e r m ; these two interference contributions are almost equal, but opposite in sign. The pseudoscalar contribution in the present problem is about 30~o more than that in the 12C ~- 12B(g.s.) problem. * The vector t e r m contributes about 6~0 of the rate through interference with the axial-vector term. Table 2 reflects the dominance of the GamowTeller operator o in the process (I). The m a r ginal contribution of other nuclear multipole ope r a t o r s , either individually or through i n t e r f e r ence with a , is an important result. The contribution from the interference of a with r(o" p) is much smaller than that for capture in the 12C ground state. • The percentage contributions quoted in tables 1 and 2 of [4] involve a numerical error. The correct decomposition, using theCohen-Kurath(6-16) 2BME model and the tensor term at the Wiikinson-Alburgerlimit is: in table ], A=82.0, WM=I.4, PS=3.1, A-V'=5.2, A-WM=19.2, A-PS=-15.1, A-T=4.5, T-PS=-I.4, other terms individually less than 1%; in table 2, AA=I04.3, AB=3.7, AD=-10.2, DD=I.6, others individually less than 1%. The primary conclusions of [4] viz., dominance of the axial-vector term and of the Gamow-Teller operator, remain unchanged. 158
26 June 1972
LETTERS
Table 1 Contributions (%) to the muon capture rate in 6Li. The w e a k - i n t e r a c t i o n coupling t e r m s that contribute a r e axial vector (A), vector (V), weak m a g n e t i s m (WM), p s e u d o s c a l a r (PS) and T e n s o r (T); they are defined in [4]. The two e n t r i e s for each contribution due to a coupling t e r m or i n t e r f e r e n c e between two t e r m s were calculated without the t e n s o r t e r m (upper entry, row a) and with it (lower entry, row b). In computing the e n t r i e s in both table 1 and table 2, the value used for the r e duced m a t r i x e l e m e n t of the G a m o w - T e l l e r o p e r a t o r is the " e x p e r i m e n t a l value" deduced from the m e a s u r e d fit by use of eq. (5).
A
a b
A
V
WM
PS
89.2 86.4
4.95 4.'~9
20.0 19.3
-20.2 -19.6
0.II 0.ii
0.84 C.81
0.08 0.07
0 -0.01
1.60 1.55
-0.09 -0..09
0 0.05
3.56 3.44
0 -1.49
a b
V WM I~S
T 0 4.44
0 0.16
a
T
b
T h e o v e r w h e l m i n g d o m i n a n c e of t h e G a m o w T e l l e r o p e r a t o r a i n p r o c e s s (1) a l l o w s u s to o b t a i n t h e r a t e of p r o c e s s (1) i n a n e a r l y m o d e l independent manner (from the nuclear-coupling p o i n t of v i e w ) by d e t e r m i n i n g t h e r e d u c e d m a t r i x e l e m e n t [ l a ~ [I of t h i s o p e r a t o r f r o m t h e e x p e r i m e n t a l f ~ v a l u e of t h e ~ - d e c a y f r o m t h e g r o u n d s t a t e of Vile to t h a t of L i b y u s e of t h e e q u a t i o n lexp = [Ggl(O)]-l[(2u31n2)/(ft)exp]l/2
, (5)
Table 2 Direct and i n t e r f e r e n c e t e r m s (both e x p r e s s e d as p e r centages of the rate) in th~ muitipole decomposition of the muon capture rate in Li. The notation for the mulitpoles is-4=a, D=[OxY2(~)] 1, C=[]'x p ]1 and D = ~ ( t / ' p ) . As in table 1, e n t r i e s a and b for each contribution were calculated r e s p e c t i v e l y without and with the the tensor term.
A
a b
B
a b
C
a b
D
a b
A
B
C
D
100.4 100.8
1.91 1.69
-0.39 -0..40
-2.10 -2.27
0.03 0.03
~ 0 ~ 0
0.06 0.06
~ 0 ~ 0
0.02 0.01
0.08 0.08
Volume 40B, number 2
PHYSICS LETTERS
where g l ( 0 ) is the a x i a l - v e c t o r coupling c o n s t a n t at zero m o m e n t u m t r a n s f e r [4]. The r e s u l t is a G a m o w - ~ e l l e r m a t r i x e l e m e n t about 3% s m a l l e r than that given by the C o h e n - K u r a t h n u c l e a r model. The c o r r e s p o n d i n g v a l u e s for the muon c a p t u r e r a t e a r e : W~z=1705.s -1 if the t e n s o r t e r m is i g n o r e d and W =1761 s -1 with the W i l k i n s o n A l b u r g e r upper l i m i t for T. Both v a l u e s a r e n e a r the middle of the r a n g e of m e a s u r e d values. With h a r m o n i c - o s c i l l a t o r r a d i a l wave functions for the two " a c t i v e " n u c l e o n s outside the " i n e r t " s - p a r t i c l e c o r e , the muon c a p t u r e r a t e is found to be a n e a r l y - l i n e a r function of the o s c i l l a t o r r m s r a d i u s r 0. In the a b s e n c e of the t e n s o r t e r m and with the a s s u m p t i o n that the G o l d b e r g e r T r e i m a n r e l a t i o n [4] is c o r r e c t , the range of r 0 that y i e l d s a fit to the e x p e r i m e n t a l l y o b s e r v e d rate (2) is found to be 2.10 f m < r 0 --<2.76 fm; this is in excellent a g r e e m e n t with the n u c l e a r r m s r a d i u s obtained f r o m a n a l y s e s of the r e c e n t e l a s tic 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 on 6Li[8-11]. In this connection, one r e c a l l s that tn the a n a l y s i s of the charge f o r m f a c t o r of 6Li, c o r r e l a t i o n effects a r e taken into account by multiplying the h a r m o n i c - o s c i l l a t o r r a d i a l functionSRnl (r) by a c o r r e c t i o n f a c t o r f ( r ) , which a p p r o a c h e s unity in the vicinity of the m o m e n t u m - t r a n s f e r c h a r a c t e r istic of the p r o c e s s (1)[8]. The a p p l i c a b i l i t y of the h a r m o n i c - o s c i l l a t o r model to the i n t e r p r e t a t i o n of the process: (3) is not s u r p r i s i n g : the r e c e n t a n a l y s i s [12] of the 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 f o r m factor for the 3.56 MeV state of 6Li shows that the s i m p l e h a r monic o s c i l l a t o r and two other different f o r m s of effective potential all give the same r e s u l t at low m o m e n t u m t r a n s f e r (£ 100 MeV/c) and a r e in good a g r e e m e n t with the e x p e r i m e n t . In t h e i r a n a l y s i s of the i n e l a s t i c - s c a t t e r i n g f o r m factor for the 3.56 MeV l e v e l s in 6Li, B e r n h e i m and Bishop [13] obtained 2.80 fm for the r m s r a d i u s of the h a r m o n i c - o s c i l l a t o r well. This is r e a s o n ably close to the r a n g e of v a l u e s c o m p a t i b l e with the r a t e (2). However, in a m o r e r e c e n t a n a l y s i s Hutcheon et al. [9] find that the Saskatoon data at higher m o m e n t u m t r a n s f e r (q >0.5 fm -1) d e m a n d s an r m s r a d i u s of 3.15 fm for the o s c i l l a t o r well. The values of W_ deduced with this r m s r a d i u s a r e 1188 s -1 fore'T=0 and 1225 s -1 for T at the c u r r e n t upper l i m i t . Both of these v a l u e s of W~t a r e i n c o n s i s t e n t with the e x p e r i m e n t a l l y m e a s u r e d r a t e (2). However, one should keep in mind that the l o w - m o m e n t u m - t r a n s f e 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 data cannot d i s t i n g u i s h between well r a d i i of 2.81 fm and 3.54 fm[9]. To see how the c a l c u l a t e d value of the muon c a p t u r e rate is affected by a b e t t e r t r e a t m e n t of
26 June 1972
the n u c l e a r r a d i a l wave functions, a study in which the o s c i l l a t o r wave functions a r e r e p l a c e d by suitably bound Woods-Saxon wave functions has been initiated. Although the absolute t r a n s i tion r a t e s a r e v e r y s e n s i t i v e to the r a d i a l s t r u c ture of the "active" nucleons, the p e r c e n t a g e decomposition in t e r m s of n u c l e a r multipoles and of v a r i o u s weak hadron f o r m f a c t o r s a r e r e l tively i n s e n s i t i v e . To a good a p p r o x i m a t i o n , changing the r a d i a l wave functions within '~sane" l i m i t s s i m p l y s c a l e s the c o n t r i b u t i o n s of the v a r i o u s n u c l e a r m u l t i p o l e s and of the v a r i o u s t e r m s in the hadronic weak c u r r e n t . Thus, the p e r c e n t a g e decompositions given in tables 1 and 2 a r e n e a r l y independent of the u n c e r t a i n t i e s in the r a d i a l wave functions. The m a i n c o n c l u s i o n s of this study of a p p l i c a bility of the i m p u l s e a p p r o x i m a t i o n to the a n a l y s i s of the r e a c t i o n (1) a r e s u m m a r i z e d in the following p a r a g r a p h s . (1) In the a b s e n c e of a t e n s o r t e r m in the weak hadron c u r r e n t , the r e s u l t s of the p r e s e n t s h e l l model a n a l y s i s of the 6Li muon c a p t u r e t r a n s i t i o n to the ground state of 6He agree with e x p e r i m e n t if the o s c i l l a t o r r m s r a d i u s is r e s t r i c t e d to the range 2.10 fm--
159
Volume 40B, n u m b e r 2
PHYSICS
w i t h t h e a x i a l - v e c t o r t e r m . T h e c o n t r i b u t i o n s of the vector term and the induced tensor term at the Wilkinson-Alburger limit are marginal. (4) R e l a t i v e d e c o m p o s i t i o n s of t h e t r a n s i t i o n r a t e in t e r m s of v a r i o u s w e a k h a d r o n f o r m f a c tors and nuclear multipole operators are nearly i n d e p e n d e n t of t h e o s c i l l a t o r r a d i u s i n t h e r a n g e 1.0 fm--
References [1] c. w. Kim and S. L. Mintz, Nuc[. Phys. B27 (1971) 621. [2] S. Cohen and D. Kurath, Nuc[. Phys. 73(1965) 1. $
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LETTERS
26 June 1972
[3] J . P . D e u t s e h , L. G r e n a c s , P. Igo-Kemenes, P. Lipnik and P. C. Macq, Phys. L e t t e r s 26B, (1968) 315. [4] N. C. Mukhopadhyay and M. H. Macfarlane, Phys. Rev. L e t t e r s 27 (1971) 1823; E r r a t u m : Phys. Rev. L e t t e r s , to be published. [5] D. H. Wilkinson and D. E. Aiburger, Phys. Rev. L e t t e r s 26 (1971) 1127. [6] J. D e l o r m e , p r e p r i n t , Institut de Physique Nue[6aire de Lyon, LYCEN-7005 (1970), and r e f e r e n c e s therein. [7] C. W. Kim and H. Primakoff, Phys. Rev. ] 40B (1965) 566. [8] S. S. M. Wong and D. L. Lin, Nuc[. Phys. A101 (1967) 663. [9] R. M. Hutcheon, T . E . Drake, V.W. Stobie, G.A. B e e r and H. S. Cap[an, Nuc[. Phys. A107 (1968) 266. [10] G. C. Li, I. Sick, R. R. Whitney and M. R. Year[an, NucL Phys. A162 (1971) 583. [11] F..a. Bum[[let, F. R. Buskirk, J. N. Dyer and W. A. Monson, Phys. Rev. C5 (1972) 391. [12] R. Neuhausen and R. M. Hutcheon, Nuc[. Phys. A164 (1971) 497. [13] M. B e r n h e i m and G. R. Bishop, P h y s . L e t t e r s 5 (1963) 270. [14~ A. Lodder and C.C. Jonker, Phys. L e t t e r s 15 (1965) 245 and r e f e r e n c e s therein. [15] J . K r i i g e r and P. Van Leuven, Phys. L e t t e r s 28B (1969) 623.
IMPULSE-APROXIMATION
OF ALLOWED
MUON CAPTURE
TREATMENT IN 6Li
*
N. C. M U K H O P A D H Y A Y * *
Department of Physics, University of Chicago, Chicago, Ill., 60637, USA and Physics Division, Argonne National Laboratory, Argonne, Ill., 60439, USA Received 24 May 1972 The rate of the reaction 6Li + ~-~6He(g.s.) + ug is ~aicu[ated with Cohen-Kurath shell-mode[ wave functions. Calculated without the tensor term, it agrees with experiment if the oscillator root-mean-square rzdius is within the range 2.10 fm ~
Despite considerable theoretical attention, it is still not clear [I] whether an impulse-approximation analysis of the process 6Li + ~-(1S)-*6He(g.s.) + u~
(1)
y i e l d s a t r a n s i t i o n r a t e in a g r e e m e n t with e x p e r i ment. T h i s l e t t e r r e - e x a m i n e s this q u e s t i o n with the aid of the C o h e n - K u r a t h [2] i n t e r m e d i a t e coupling wave functions for the n u c l e a r states involved. This study has t h r e e p r i m a r y a i m s : (a) to find out whether the i m p u l s e - a p p r o x i m a t i o n a n a l y s i s for p r o c e s s (1) can account for the m e a s u r e d [3] t r a n s i t i o n rate W~xp
+330 -1 = (1600_129)s ,
(2)
(b) to study the c o n t r i b u t i o n s of v a r i o u s induced t e r m s in the weak hadronic c u r r e n t (including the p o s s i b l e p r e s e n c e of a G - p a r i t y - v i o l a t i n g t e n sor t e r m ) , and (c) to e x a m i n e the r e l a t i v e i m p o r t a n c e of v a r i o u s n u c l e a r multipole o p e r a t o r s . The method of a n a l y s i s is that d e s c r i b e d in a study [4] of muon c a p t u r e in 12C. The b a s i c idea is to use the i m p u l s e a p p r o x i m a t i o n to r e l a t e the t r a n s i t i o n amplitude for n u c l e a r c a p t u r e to that for the e l e m e n t a r y c a p t u r e p r o c e s s Ix- + p ,
•
.
.
-
n : :t~ar } h : o i c n t : : a : t 'th : [ ~sPtte:lble t ~°[ : h : n ~ : d • Research done under the auspices of the U. s. Atomic Energy Commission, in partial fulfillment of requirements of a Ph. D.degree at the University of Chicago. • * Fulbright Grantee, the University of Chicago, and Resident Student Associate (Thesis), Argonne National Laboratory. Supported in part by the University of Chicago and the U. S. Atomic Energy Commission.
c u r r e n t - c u r r e n t form. The f o r m f a c t o r s in the weak hadron c u r r e n t and the m a n n e r of t h e i r d e t e r m i n a t i o n a r e given in ref. [4]; in addition to the dominant a x i a l - v e c t o r (A) t e r m , the p r o c e s s (1) involves vector (V), weak m a g n e t i s m (WM), induced p s e u d o s c a l a r (PS), and induced t e n s o r (T) t e r m s . All but the t e n s o r f o r m factor a r e w e l l - d e t e r m i n e d for p r e s e n t p u r p o s e s . The v e r y p r e s e n c e of the G - p a r i t y - v i o l a t i n g t e n s o r t e r m is u n c e r t a i n . , Here we compute the c a p t u r e r a t e with both zero and the W i l k i n s o n - A l b u r g e r [5] upper l i m i t for T. The i m p u l s e - a p p r o x i m a t i o n a n a l y s i s in [4] yields an e x p r e s s i o n of the f o r m
:s(Irl/212 + 1T3/2[ 2)
(3)
for the t r a n s i t i o n r a t e W~ of the p r o c e s s (1), where S is the s t a t i s t i c a l factor and Tj is the a m p litude for the e m i s s i o n of a n e u t r i n o in a state of total a n g u l a r m o m e n t u m j(j=½ o r 23-for the n e u t r i n o in an S l / 2 0 r d3/2 state, r e s p e c t i v e l y ) . The t r a n s i t i o n a m p l i t u d e s have the f o r m 6 Tol= 6He(g.s.)[ [i=~l{aa a +ba[ai>(Y2(t'i)] 1 + C a [ r / × p / ] l + d a r i (a i " p i ) } r i- 116Li(g.s.) ,
(4) where a n , ba, etc. a r e functions of the nucleon f o r m f a c t o r s and lepton v a r i a b l e s and r~ c o n v e r t s the ith p r o t o n into a nuetron. The wave functions for the i n i t i a l and the final n u c l e a r s t a t e s a r e the i n t e r m e d i a t e - c o u p l i n g s h e l l - m o d e l wave functions of Cohen and Kurath [2]. In the m a s s - 6 s y s t e m , these a r e v e r y c l o s e to the ones obtained in the LS l i m i t . The Cohen-
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Volume 40B, number 2
PHYSICS
Kurath wave functions give a reasonable account of m a n y of t h e l o w - e n e r g y p r o p e r t i e s of t h e mass-6 system; in particular, these reproduce t h e m e a s u r e d 6 H e - ~ 6 L i . f l - d e c a y r a t e to w i t h i n a b o u t 5%. With the weak form factors and muon wave f u n c t i o n s g i v e n i n r e f . [4] a n d h a r m o n i c - o s c i l l a t o r nuclear radial wave functions with an rms radius of 2.43 f m [ 6 ] , w e o b t a i n f o r t h e c a p t u r e r a t e t h e v a l u e s W ~ = 1813 s -1 if t h e t e n s o r t e r m i s n e g l e c t e d o r W~ = 1872 s -1 if t h e W i l k i n s o n - A l b u r g e r u p p e r l i m i t i s a s s u m e d f o r T. T h e s e v a l u e s , w h i c h i n c l u d e n u c l e a r f i n i t e - s i z e e f f e c t s [7] i n the mu~n spinors, are both well inside the error l i m i t s of t h e m e a s u r e d t r a n s i t i o n r a t e . C o n t r i b u t ~ n s to t h e r a t e of t h e p r o c e s s (1) arise from various weak hadron form factors ( t a b l e 1) a n d v a r i o u s n u c l e a r m u l t i p o l e o p e r a t o r s ( t a b l e 2). E a c h c o n t r i b u t i o n h a s b e e n c a l c u l a t e d both for a zero tensor term (upper entry) and for a tensor term equal to the Wilkinson-Alburger upper limit (lower entry). It is clear that the tensor interaction at the present upper limit has only a marginally significant effect on both the coupling-constant and the muir[pole decompositions. Table 1 shows that, as expected, the axialvector term is by f a r the most important of the t e r m s in the weak hadron current. However, the weak-magnetism and the pseudoscalar t e r m s give sizable contributions through interference with the axial-vector t e r m ; these two interference contributions are almost equal, but opposite in sign. The pseudoscalar contribution in the present problem is about 30~o more than that in the 12C ~- 12B(g.s.) problem. * The vector t e r m contributes about 6~0 of the rate through interference with the axial-vector term. Table 2 reflects the dominance of the GamowTeller operator o in the process (I). The m a r ginal contribution of other nuclear multipole ope r a t o r s , either individually or through i n t e r f e r ence with a , is an important result. The contribution from the interference of a with r(o" p) is much smaller than that for capture in the 12C ground state. • The percentage contributions quoted in tables 1 and 2 of [4] involve a numerical error. The correct decomposition, using theCohen-Kurath(6-16) 2BME model and the tensor term at the Wiikinson-Alburgerlimit is: in table ], A=82.0, WM=I.4, PS=3.1, A-V'=5.2, A-WM=19.2, A-PS=-15.1, A-T=4.5, T-PS=-I.4, other terms individually less than 1%; in table 2, AA=I04.3, AB=3.7, AD=-10.2, DD=I.6, others individually less than 1%. The primary conclusions of [4] viz., dominance of the axial-vector term and of the Gamow-Teller operator, remain unchanged. 158
26 June 1972
LETTERS
Table 1 Contributions (%) to the muon capture rate in 6Li. The w e a k - i n t e r a c t i o n coupling t e r m s that contribute a r e axial vector (A), vector (V), weak m a g n e t i s m (WM), p s e u d o s c a l a r (PS) and T e n s o r (T); they are defined in [4]. The two e n t r i e s for each contribution due to a coupling t e r m or i n t e r f e r e n c e between two t e r m s were calculated without the t e n s o r t e r m (upper entry, row a) and with it (lower entry, row b). In computing the e n t r i e s in both table 1 and table 2, the value used for the r e duced m a t r i x e l e m e n t of the G a m o w - T e l l e r o p e r a t o r is the " e x p e r i m e n t a l value" deduced from the m e a s u r e d fit by use of eq. (5).
A
a b
A
V
WM
PS
89.2 86.4
4.95 4.'~9
20.0 19.3
-20.2 -19.6
0.II 0.ii
0.84 C.81
0.08 0.07
0 -0.01
1.60 1.55
-0.09 -0..09
0 0.05
3.56 3.44
0 -1.49
a b
V WM I~S
T 0 4.44
0 0.16
a
T
b
T h e o v e r w h e l m i n g d o m i n a n c e of t h e G a m o w T e l l e r o p e r a t o r a i n p r o c e s s (1) a l l o w s u s to o b t a i n t h e r a t e of p r o c e s s (1) i n a n e a r l y m o d e l independent manner (from the nuclear-coupling p o i n t of v i e w ) by d e t e r m i n i n g t h e r e d u c e d m a t r i x e l e m e n t [ l a ~ [I of t h i s o p e r a t o r f r o m t h e e x p e r i m e n t a l f ~ v a l u e of t h e ~ - d e c a y f r o m t h e g r o u n d s t a t e of Vile to t h a t of L i b y u s e of t h e e q u a t i o n lexp = [Ggl(O)]-l[(2u31n2)/(ft)exp]l/2
, (5)
Table 2 Direct and i n t e r f e r e n c e t e r m s (both e x p r e s s e d as p e r centages of the rate) in th~ muitipole decomposition of the muon capture rate in Li. The notation for the mulitpoles is-4=a, D=[OxY2(~)] 1, C=[]'x p ]1 and D = ~ ( t / ' p ) . As in table 1, e n t r i e s a and b for each contribution were calculated r e s p e c t i v e l y without and with the the tensor term.
A
a b
B
a b
C
a b
D
a b
A
B
C
D
100.4 100.8
1.91 1.69
-0.39 -0..40
-2.10 -2.27
0.03 0.03
~ 0 ~ 0
0.06 0.06
~ 0 ~ 0
0.02 0.01
0.08 0.08
Volume 40B, number 2
PHYSICS LETTERS
where g l ( 0 ) is the a x i a l - v e c t o r coupling c o n s t a n t at zero m o m e n t u m t r a n s f e r [4]. The r e s u l t is a G a m o w - ~ e l l e r m a t r i x e l e m e n t about 3% s m a l l e r than that given by the C o h e n - K u r a t h n u c l e a r model. The c o r r e s p o n d i n g v a l u e s for the muon c a p t u r e r a t e a r e : W~z=1705.s -1 if the t e n s o r t e r m is i g n o r e d and W =1761 s -1 with the W i l k i n s o n A l b u r g e r upper l i m i t for T. Both v a l u e s a r e n e a r the middle of the r a n g e of m e a s u r e d values. With h a r m o n i c - o s c i l l a t o r r a d i a l wave functions for the two " a c t i v e " n u c l e o n s outside the " i n e r t " s - p a r t i c l e c o r e , the muon c a p t u r e r a t e is found to be a n e a r l y - l i n e a r function of the o s c i l l a t o r r m s r a d i u s r 0. In the a b s e n c e of the t e n s o r t e r m and with the a s s u m p t i o n that the G o l d b e r g e r T r e i m a n r e l a t i o n [4] is c o r r e c t , the range of r 0 that y i e l d s a fit to the e x p e r i m e n t a l l y o b s e r v e d rate (2) is found to be 2.10 f m < r 0 --<2.76 fm; this is in excellent a g r e e m e n t with the n u c l e a r r m s r a d i u s obtained f r o m a n a l y s e s of the r e c e n t e l a s tic 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 on 6Li[8-11]. In this connection, one r e c a l l s that tn the a n a l y s i s of the charge f o r m f a c t o r of 6Li, c o r r e l a t i o n effects a r e taken into account by multiplying the h a r m o n i c - o s c i l l a t o r r a d i a l functionSRnl (r) by a c o r r e c t i o n f a c t o r f ( r ) , which a p p r o a c h e s unity in the vicinity of the m o m e n t u m - t r a n s f e r c h a r a c t e r istic of the p r o c e s s (1)[8]. The a p p l i c a b i l i t y of the h a r m o n i c - o s c i l l a t o r model to the i n t e r p r e t a t i o n of the process: (3) is not s u r p r i s i n g : the r e c e n t a n a l y s i s [12] of the 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 f o r m factor for the 3.56 MeV state of 6Li shows that the s i m p l e h a r monic o s c i l l a t o r and two other different f o r m s of effective potential all give the same r e s u l t at low m o m e n t u m t r a n s f e r (£ 100 MeV/c) and a r e in good a g r e e m e n t with the e x p e r i m e n t . In t h e i r a n a l y s i s of the i n e l a s t i c - s c a t t e r i n g f o r m factor for the 3.56 MeV l e v e l s in 6Li, B e r n h e i m and Bishop [13] obtained 2.80 fm for the r m s r a d i u s of the h a r m o n i c - o s c i l l a t o r well. This is r e a s o n ably close to the r a n g e of v a l u e s c o m p a t i b l e with the r a t e (2). However, in a m o r e r e c e n t a n a l y s i s Hutcheon et al. [9] find that the Saskatoon data at higher m o m e n t u m t r a n s f e r (q >0.5 fm -1) d e m a n d s an r m s r a d i u s of 3.15 fm for the o s c i l l a t o r well. The values of W_ deduced with this r m s r a d i u s a r e 1188 s -1 fore'T=0 and 1225 s -1 for T at the c u r r e n t upper l i m i t . Both of these v a l u e s of W~t a r e i n c o n s i s t e n t with the e x p e r i m e n t a l l y m e a s u r e d r a t e (2). However, one should keep in mind that the l o w - m o m e n t u m - t r a n s f e 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 data cannot d i s t i n g u i s h between well r a d i i of 2.81 fm and 3.54 fm[9]. To see how the c a l c u l a t e d value of the muon c a p t u r e rate is affected by a b e t t e r t r e a t m e n t of
26 June 1972
the n u c l e a r r a d i a l wave functions, a study in which the o s c i l l a t o r wave functions a r e r e p l a c e d by suitably bound Woods-Saxon wave functions has been initiated. Although the absolute t r a n s i tion r a t e s a r e v e r y s e n s i t i v e to the r a d i a l s t r u c ture of the "active" nucleons, the p e r c e n t a g e decomposition in t e r m s of n u c l e a r multipoles and of v a r i o u s weak hadron f o r m f a c t o r s a r e r e l tively i n s e n s i t i v e . To a good a p p r o x i m a t i o n , changing the r a d i a l wave functions within '~sane" l i m i t s s i m p l y s c a l e s the c o n t r i b u t i o n s of the v a r i o u s n u c l e a r m u l t i p o l e s and of the v a r i o u s t e r m s in the hadronic weak c u r r e n t . Thus, the p e r c e n t a g e decompositions given in tables 1 and 2 a r e n e a r l y independent of the u n c e r t a i n t i e s in the r a d i a l wave functions. The m a i n c o n c l u s i o n s of this study of a p p l i c a bility of the i m p u l s e a p p r o x i m a t i o n to the a n a l y s i s of the r e a c t i o n (1) a r e s u m m a r i z e d in the following p a r a g r a p h s . (1) In the a b s e n c e of a t e n s o r t e r m in the weak hadron c u r r e n t , the r e s u l t s of the p r e s e n t s h e l l model a n a l y s i s of the 6Li muon c a p t u r e t r a n s i t i o n to the ground state of 6He agree with e x p e r i m e n t if the o s c i l l a t o r r m s r a d i u s is r e s t r i c t e d to the range 2.10 fm--
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Volume 40B, n u m b e r 2
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w i t h t h e a x i a l - v e c t o r t e r m . T h e c o n t r i b u t i o n s of the vector term and the induced tensor term at the Wilkinson-Alburger limit are marginal. (4) R e l a t i v e d e c o m p o s i t i o n s of t h e t r a n s i t i o n r a t e in t e r m s of v a r i o u s w e a k h a d r o n f o r m f a c tors and nuclear multipole operators are nearly i n d e p e n d e n t of t h e o s c i l l a t o r r a d i u s i n t h e r a n g e 1.0 fm--
References [1] c. w. Kim and S. L. Mintz, Nuc[. Phys. B27 (1971) 621. [2] S. Cohen and D. Kurath, Nuc[. Phys. 73(1965) 1. $
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[3] J . P . D e u t s e h , L. G r e n a c s , P. Igo-Kemenes, P. Lipnik and P. C. Macq, Phys. L e t t e r s 26B, (1968) 315. [4] N. C. Mukhopadhyay and M. H. Macfarlane, Phys. Rev. L e t t e r s 27 (1971) 1823; E r r a t u m : Phys. Rev. L e t t e r s , to be published. [5] D. H. Wilkinson and D. E. Aiburger, Phys. Rev. L e t t e r s 26 (1971) 1127. [6] J. D e l o r m e , p r e p r i n t , Institut de Physique Nue[6aire de Lyon, LYCEN-7005 (1970), and r e f e r e n c e s therein. [7] C. W. Kim and H. Primakoff, Phys. Rev. ] 40B (1965) 566. [8] S. S. M. Wong and D. L. Lin, Nuc[. Phys. A101 (1967) 663. [9] R. M. Hutcheon, T . E . Drake, V.W. Stobie, G.A. B e e r and H. S. Cap[an, Nuc[. Phys. A107 (1968) 266. [10] G. C. Li, I. Sick, R. R. Whitney and M. R. Year[an, NucL Phys. A162 (1971) 583. [11] F..a. Bum[[let, F. R. Buskirk, J. N. Dyer and W. A. Monson, Phys. Rev. C5 (1972) 391. [12] R. Neuhausen and R. M. Hutcheon, Nuc[. Phys. A164 (1971) 497. [13] M. B e r n h e i m and G. R. Bishop, P h y s . L e t t e r s 5 (1963) 270. [14~ A. Lodder and C.C. Jonker, Phys. L e t t e r s 15 (1965) 245 and r e f e r e n c e s therein. [15] J . K r i i g e r and P. Van Leuven, Phys. L e t t e r s 28B (1969) 623.