Pram~nx, Vol, 19, No. 3, September 1982, pp. 249-254. © Printed in India.
The ( ~ - d ) cluster model of 6Li and muon capture
I A H M A D a n d S K S I N G H
Department of Physics, Aligarh Muslim University, Aligarh 202 001, India MS received 3 October 1981 ; revised 5 June 1982
Abstract. The ( ~ - d) cluster model with parameters determined from electron scattering and pion photoproduction processes is used to calculate the muon capture rate in eLi. "Ihe result is found to be better than the results calculated in other models and is in agreement with the exlzerimental data.
Key~vords. ,dvluon capture; cluster model; form factors; impulse approximation.
1. Introduction
The process o f m u o n c a p t u r e in light nuceli has been widely studied b y using either the impulse a p p r o x i m a t i o n 0A) or the elementary particle m o d e l a p p r o a c h ( P r i m a k o f f 1975; W a l e c k a 1975; M u k h o p a d h y a y 1977). While b o t h these models give similar results for the partial m u o n capture rate in 6Li none o f these can be said to be in g o o d agreement with the experimental result ( H w a n g 1978; C a m m a r a t a a n d D o n n e l l y 1976; B e r g s t r o m e t al 1975; Donnetly and Walecka 1973; D e l o r m e 1970: K i m a n d M i n t z 1970). W h e n the question o f the existence o f second class currents seems to have been settled n o w ( W u 1977; G a r v e y 1977), the disagreement is in general attri- buted to the uncertainties in the knowledge o f the nuclear wave function in 6Li a n d other light nuclei ( P r i m a k o f f 1977). In the case o f 6Li no a t t e m p t s have been m a d e to calculate the effect o f m e s o n exchange currents - - a n effect suspected to p l a y a n i m p o r t a n t role in w e a k a n d electromagnetic interactions o f nuclei even at low energies a n d m o m e n t u m transfers (Parthasarathy and W a g h m a r e 1979; I v a n o v a n d T r u b l i k 1978, 1979, G u i c h o n et al 1978, 1977, P o r r m a n n 1981, H e n l e y a n d H w a n g 1980, J a u s and W o o l c o c k 1981). In nuclei, the study o f m e s o n exchange currents is generally related with the description o f basic nuclear potential which is used to calculate the nucleon wave functions. A better wave function f o r the nucleus m a y therefore simu- late to some extent the effects o f meson exchange currents. M a n y a u t h o r s h a v e recently explored various wave functions o f 6Li while studying the threshold p i o n p h o t o p r o d u e t i o n a n d have used these wave functions to analyse other w e a k a n d electromagnetic processes in this nucleus ( C a m m a r a t a a n d Donnelly 1976; B e r g s t r o m et al 1975, C a n n a t a et al 1974; K o c h and Donnelly 1974, 1973). These studies h a v e shown t h a t n o w a v e function o f GLi in the shell model using h a r m o n i c oscillator basis is c a p a b l e o f giving a consistent description o f electron scattering, m u o n capture, p i o n p h o t o p r o d u c t i o n a n d radiative pion capture processes.*
*In an improved analysis by Mukhopadhyay (1977) it is shown that the experimental capture rate can be reproduced in shell model with harmonic oscillator basis with the oscillator parameter (b) in the range of 1-33 < b < 1-75. The values of this parameter b from electron scattering and threshold pion photoproduction data are higher than 1"75 fro.
249
250 I A h m a d and S K Singh
The ( a - - d ) cluster model for the 6Li nucleus has attracted considerable attention during the last several years. A number o f studies show that the model works quite well for the description o f low lying states o f this nucleus. In particular it is very successful in accounting for the various electric a n d magnetic from factors o f ~Li.
It has also explained quite successfully the data o n p-6Li scattering from the Saclay group (Ahmad a n d K h a n 1979; Bruge 1978, Aslanides et al 1975) especially at low m o m e n t u m transfer range which concerns us here. This model has however not been used so far to analyse the present data on weak processes in 6Li (Ahmad and K h a n 1979; Noble 1974; Raphael 1973; Kudeyarov 1971 ; Neudatchin and Smirmov 1964).
We have earlier shown that this model gives a better description of the threshold pion photoproduction process in 6Li (Singh and A h m a d 1977). In this paper we calculate the partial m u o n capture rate in e l i using the (a -- d) cluster model wave function o f e l i with the parameters determined from the electron scattering experi- ments and show t h a t our result for the capture ~ate is better than those calculated in the various shell models and is in fair agreement with the experimental result.
2. Capture rate
The transition probability for muon capture is expressed as:
d w = 2~r3(E,, i- Ei -- m,, -- Er) dPv [m]-',
" (2 ~)a (l)
where the matrix element m is given by (Commins 1973) G cos 0
m--- ,.. f dx ~,,(x) [ft(q 2) v t' -~- if2(q2)cfl 'v
x' 2 , qr
--!- gt(q ~) 7 '~ --i- gz(q ~-) 75 q~j Jr, ~p(.v)
where Jr, ---= -~v(x) 7 t, (1 -- 75) ~b (x). (2)
Using the standard methods for nonrelativistic reduction of the covariant single nucleon operator applicable to nuclei the matrix element m is derived to be (Commins 1973, Singh 1972. 1974. 1975).
'
m = dx <f[ X,, 2 r i [Gv(q"-) -',- GA(q °') a.~ri i
+ Gp(q"-) ai "flY] Xt, exp ( - - i p,,-x,.) ~bt,(xi) 3 (x--xz) [i), (3) where Gv(q ~-) -= G cos 0 .fl(q 2) ( l - - Et,'2M).
GA(q"- ) = G cos 0 [_g~(q2) _ [f~(q2) +A(q~)] (E,,/2M)].
Gp(q 2) = G cos 0 - - m g3(q 2) .-~- gl(q z) -- [./i(¢) + f2(q2)] ~ (4)
(~ - d) cluster model o f % i and muon capture 251 gv(qZ)=me, ga(q=) a n d m o m e n t u m transfer q---Pv { i ) a n d
If )
are the initial a n d final nuclear wave functions described in (¢ -- d) cluster model through a function ~JM given by~jM(X1, X2,...,X6)___ A[~( j I J)
rnj rn, mj~o,
o (Xl, "", X,)~7 J
(x+, x J 4,7' (xl, . , x4; xs,as), (5)
where A stands for antisymmetrisation,
i b°,
0 and ff;J are the wave functions o f and dinucleon cluster in the initial and final nuclei; 4'7' is the wave function which describes the relative motion o f the ~ and d cluster inside eLi.Neglecting the effect o f antisymmetrisation* and D state o f the deuteroa, in d cluster, the matrix element (3) is calculated to be
m_-- f X+(1--''~v)
v 2 (G A (q2) + Gp (q2) Pv) X ~ ° o(x -[- 2X/3)
• (¢ (5) -- o" (6)) 4'~J (x + 2X/3) ]~ o (X) 12 exp ( - - i Pv" x)
~bz (x) dX dx, (6)
4
where X = (1/4) ~ xi -- (1/2) ~ xi is the relative coordinate between a a n d i = 1 i = 5 , 6
dinucleon clusters. Taking the rouen wave function to be Z8/2
~ (x) -- (= ao)Xl--- z exp (-- Zr/ao) U~, (7)
the capture rate is calculated to be
W = (G = cos 2 0/~r
~) R (m~ Evil
= -1- E=,/3) a 3 Z 3 F~j (Pv) F2=- a(2pv/3) {G~I (p~)
+ 1/3[G~ (p~
- - 2G A(p2) Gp (p~]}, (8)
where m is the reduced mass of the m u o n and R is the reduction factor needed to properly take into account the effect o f the charge distribution o f e l i nucleus a n d is taken to be 0-95 following Walecka (1975, 1976). This corresponds to an effective charge Zcff = 2-94 extrapolated by Eckhouse et al (1963) following F o r d and Willis (1962).
*It has been shown by Kudeyarov et al (1971) in a similar transition that because of small over- lap of the two clusters the intercluster nucleon exchange effects and antisymmetrisation effects are not significant at low momentum transfers (q = 1"3 fm -1) if we lake ~b0°('R) to have no modes.
P.---4
252 I Ahmad and S K Singh
The form factors Fd (q) and F~--d (q) in (8) are given by
F,, (q) = f ,~*~ (x) exp ( - - i q . x) ff,~ (x) dx, (9) F ~ _ , (q) = J [ff~ (x) 12 exp
(i
q - X) dX, (10) 4,, (x) and ~d (X) being the radial part offf ° (x) and ff~'J (x) respectively.Evaluation of Fd (q) requires a knowledge of the wave function ~a for the n-p cluster within eLi as well as for the wave function ~nn of the nn cluster within SHe.
The wave function '~a, although it describes a system with the same quantum number as the deuterons, need not necessarily be taken asthe free deuteron wave function. As a matter of fact, the commonly used gaussian wave function for the deuteron within the nucleus eLi departs greatly from the realistic free deuteron wave function. This is understandable because the free deuteron being a loosely bound system is more amenable to polalization effects than a tightly bound cluster like the a-particle.
This is the main reason why in almost all the studies of eLi nucleus in the a-d duster model, the deuteron wave function parameter is treated as adjustable. In what follows, we also work in the same spirit and take ffd (a) as suggested from the elec- tron scattering experiments.
As regards the nn cluster wave function ~,,, unfortunately, we do not have suflieient information as yet. Therefore in the absence of anything to the contrary it does not appear to be a bad approximation to assume the bound nn cluster wave function in SHe to be the same as the bound np cluster wave-function in eLi. Under this approximation which sounds reasonable for the present study, the form factor Fa (q) is found to be
F,, ( q ) ----
exp
(-- q'lS) " (q),(11)
where F~, (q) is the proton form factor (Raphael 1973). In order to calculate the relative form factor F,,_ 4 (q) the L = 0 intercluster wave function ~b ° (R) is taken to be
fit ° (R) = NX ~ exp (-- 2X'2/3/~).
(12)
This form of ~bo° (R) is specially suitable at low q~ as the antisymmetrisation effects with this wave-function are shown to be small (Raphael 1973; Kudeyarev et al
1971). With this wave function F~--d is calculated to be
F,,_,, (2pv/3) =- (1/15) e', ~' (4y 't -- 20 y2 -F. 15); y~ ---- ( l l l 2 ) p v ~ b 2. (13)
3. Results and discussion
Results for the partial muon capture rate is calculated from (8), (11) and (13) for Ev = 100.7 MeV and q~ = 1.072 × 10 -2 GeV ~ applicable to t h e m u o n capture in 8Li.
The following values have been used for the various form factor used in (4)
(O)
gx" ° = (1 --t-
and fz, fz' (O)
(1 +
(a - d) cluster model o f eLi and muon capture 253 w i t h g l ( 0 ) = 1-23; g , ( 0 ) = 7g](0); f ~ ( 0 ) + f ~ ( 0 ) = 4.706, M A = 0.95 GeV and Mo = 0.92 GeV. The results for the capture rate are shown in figure 1 for various values of by and b 2 suggested by electron scattering and pion photoproduction experiments (Singh and Abroad 1977). In order to compare our results with other calculations in literature we have also shown in this figure the muon capture rates calculated by various authors. It is clear that the results obtained in this model are better than the results in all other models when compared with the experimental value (Deutsch et al 1968). For fixed b 2, a lower value of b 2 (i.e b~/b~< I) as suggested by inelastic electron scattering experiments on eLi gives even better results. If we compare these calculations with our earlier calculation (Singh and Ahmad 1977) of threshold pion photoproduction which also measures the matrix elements of the isovector axial current between the same nuclear states but in a different kinematic region, then we find that in muon capture the matrix element is higher than the corresponding results in the shell model while in the threshold pion photoproduction, the matrix element in the cluster model is smaller than the corresponding results in the shell model. The difference is not much (at least for b~d/b2,,~ 1) but it is in the right direction of better agreement with experimental results. This is because of the different dependence of the matrix element in these models. The comparison with the experimental value suggests that the (¢ -- d) cluster model wave function provides a better description of the e l i nucleus than the shell model with the harmonic oscillator basis. An analysis of various weak interaction processes in eLi using improved cluster model wave func- tions including the effects of antisymmetrisation and internucleon cluster exchanges, etc is presently under consideration.
' I - - I 1950
e×pe;i=i~s,O, 1750 (o)_ b 2= 5-56 fm z 1
(b) z ~
EPM
calculations ~ ~ . ~
• Delorme (1970)
• Hwang (1978)
-...
I(..~
o Cammarata & Donnel~y (1976) Shell model calculations
® Cammarata & Donnelly (1976)
® Walecka
e Bergstrorn etal ( 1 9 7 5 )
0/6
Figure 1.
® @
-8
' t
®l I
08 10
x
Muon capture rates calculated in various models.
1550~
1350
1150
254 I A h m a d and S K Singh References
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