Coulomb disintegration of μ-mesonic atoms

Volume 3, number 1

PHYSICS

COULOMB

DISINTEGRATION

LETTERS

OF

15 November 196~

A~OMS

~-MESONIC

Y. N. KIM Institute of Theoretical Physics, University of Alberta, Edmonton, Alberta, Canada Received 22 October 1962

T h e c a s c a d e s c h e m e of t h e ~ - m e s o n i c a t o m s h a s b e e n i n v e s t i g a t e d b y v a r i o u s r e s e a r c h e r s 1-4). H o w e v e r , t h e p a r a d o x of m i s s i n g X - r a y s and s o m e related problems remain unexplained 5,6). The p u r p o s e of t h i s l e t t e r i s n o t to a t t e m p t to s o l v e t h i s p a r a d o x but m e r e l y to p o i n t out t h a t t h e l i f e h i s t o r y of the n e g a t i v e ~ - m e s o n s , e v e n a f t e r t h e i n i t i a l f o r m a t i o n of t h e m e s o n i c a t o m s , m a y be m o r e c o m plicated than hitherto assumed. T h e m u o n s , a f t e r b e i n g s l o w e d down t h r o u g h i o n i z a t i o n and s c a t t e r i n g in p a s s i n g t h r o u g h d e n s e m a t t e r , a r e c a p t u r e d into bound s t a t e s b y t h e n u c l e i to f o r m m e s o n i c a t o m s . It h a s b e e n u s u a l l y a s s u m e d t h a t t h e m u o n s a r e c a p t u r e d in s t a t e s of p r i n c i p a l q u a n t u m n u m b e r n ~ 15 and then c a s c a d e down to t h e g r o u n d s t a t e , in a t i m e m u c h s h o r t e r (r~ 10-13 s e c ) t h a n the l i f e t i m e of m u o n s , m a i n l y t h r o u g h r a d i a t i v e a n d A u g e r t r a n s i t i o n s 7). To f i n d the p o p u l a t i o n of e a c h s t a t e of l o w e r n, one h a s to a s s u m e a c e r t a i n d i s t r i b u t i o n of m u o n s a m o n g the s u b - s t a t e s of n. T h e u s u a l a s s u m p t i o n h a s b e e n t h e d i s t r i b u t i o n with s t a t i s t i c a l w e i g h t ( 2 / + 1), although c a l c u l a t i o n s h a v e a l s o b e e n p e r f o r m ed with o t h e r i n i t i a l d i s t r i b u t i o n s 2). R e c e n t l y M a n n and R o s e m a d e a d e t a i l e d c a l c u l a t i o n of the i n i t i a l d i s t r i b u t i o n of m u o n s in bound s t a t e s by c o n s i d e r i n g the c a p t u r e r a t e a s a f u n c t i o n of the i n c i d e n t m e s o n e n e r g y 8). T h e i r c a l c u l a t i o n s h o w s t h a t , f o r m e s o n i c c a r b o n a t o m s , the m u o n s a r e c a p t u r e d when t h e i r k i n e t i c e n e r g y i s a r o u n d 8 keV and a l s o t h a t the s t a t e s with ;z ~ 7, i n s t e a d of n - 15 a s h a s been usually conjectured, capture the most muons. P r e v i o u s l y it h a s b e e n thought that the m u o n s a r e c a p t u r e d a f t e r t h e y a r e s l o w e d down and have c o m e e s s e n t i a l l y to r e s t . A c c o r d i n g to the c a l c u l a t i o n of Mann and R o s e , the p i c k u p of n e g a t i v e m u o n s in f l i g h t should be t h e m a i n m o d e of f o r m a t i o n of t h e ~ - m e s o n i c a t o m s . H o w e v e r , it i s u n l i k e l y that t h e e j e c t e d e l e c t r o n s a l w a y s c a r r y off a l l of t h i s e n e r g y a s w e l l a s t h a t c o m i n g f r o m the l a r g e d i f f e r e n c e b e t w e e n the b i n d i n g e n e r g i e s of t h e o r d i n a r y and m e s o n i c a t o m s , a l t h o u g h t h i s h a s often b e e n a s s u m e d . T h e i m m e d i a t e e f f e c t of t h i s c o n s i d e r a t i o n i s f o r the t h u s f o r m e d m e s o n i c a t o m s to a c q u i r e a c o n s i d e r a b l e

p a r t of t h e k i n e t i c e n e r g y of the i n c i d e n t m u o n s . T h e e n e r g y i m p a r t e d to the m e s o n i c a t o m s i s too s m a l l to i n d u c e n u c l e a r e x c i t a t i o n , e x c e p t p o s s i b l y f o r the v e r y h e a v y a t o m s , y e t i t i s t o o l a r g e f o r collective electronic phenomena. Instead, this e n e r g y w i l l t e n d to b e e x p e n d e d a s t h e k i n e t i c e n e r gy of m o t i o n of the c e n t e r of m a s s of t h e m e s o n i c a t o m s . A s a r e s u l t , the m e s o n i c a t o m s , w h i c h a r e f o r m e d f r o m the p i c k u p of m u o n s in f l i g h t by l i g h t n u c l e i , w i l l b e m o v i n g with k i n e t i c e n e r g i e s of up to a few keV. It i s to b e noted that t h i s i d e a of m o v i n g 9 - m e s o n i c a t o m s i s e n t i r e l y d i f f e r e n t , in s o f a r a s the o r i g i n and m a g n i t u d e of e n e r g y i s c o n c e r n e d , f r o m t h a t c o n j e c t u r e d by D a y and M o r r i s o n 9). T h e m o t i o n of t h e m e s o n i c a t o m s w i l l i n t e n s i f y t h e S t a r k e f f e c t , a s h a s b e e n s u g g e s t e d by D a y et a l . 10). H o w e v e r , if the v e l o c i t y of the m e s o n i c a t o m s i s s u f f i c i e n t l y high, t h e n t h e r e w i l l a l s o ap~ p e a r a new p r o c e s s , n a m e l y t h e e l e c t r i c b r e a k - u p of the m e s o n i c a t o m s in the C o u l o m b f i e l d due to the n u c l e i of n e i g h b o u r i n g a t o m s. To i n v e s t i g a t e the p o s s i b i l i t y of t h i s p r o c e s s , we h a v e c a l c u l a t e d t h e p r o b a b i l i t y f o r a t y p i c a l c a s e . In a c e r t a i n a p p r o x i m a t i o n t h i s p r o b a b i l i t y m a y be w r i t t e n 2~ r = ~ f d k I(~fxf]HI ~iXi) 12 N(Kf) z(k) ,

(1)

w h e r e $ d e s c r i b e s t h e r e l a t i v e m o t i o n of t h e n u c l e u s of the n e i g h b o u r i n g a t o m a n d the c e n t e r of m a s s of the m e s o n i c a t o m , and × r e f e r s to the muon. In eq. (1) t h e i n t e g r a t i o n o v e r a l l u n m e a s u r e d q u a n t i t i e s i s i n c l u d e d . N(Kf) and z(k) a r e t h e d e n s i t i e s of s t a t e s f o r ~f and ×f r e s p e c t i v e l y . In c a r r y i n g out the c a l c u l a t i o n , w e have a s s u m e d t h a t the c e n t e r of m a s s of the ~ - m e s o n i c a t o m c o i n c i d e s with i t s n u c l e u s . In t h i s a p p r o x i m a t i o n the i n t e r a c tion e n e r g y i s Z 2 e2

H-

IRi

Z e2

IR-

,'I '

(2)

w h e r e t h e p o s i t i o n v e c t o r s R(R, ®, ~ ) and r ( r , 8, ~) r e f e r to t h e n u c l e u s of t h e n e i g h b o u r i n g a t o m and the m u o n with r e s p e c t to t h e n u c l e u s of t h e m e s o n i c a t o m . We d e s c r i b e t h e bound and e j e c t e d m u o n s b y h y d r o g e n l i k e w a v e f u n c t i o n s and p l a n e w a v e s , r e 33

Volume 3, number 1

PHYSICS

s p e c t i v e l y . The i n i t i a l (final) state of the m o t i o n of the c e n t e r of m a s s of the m e s o n i c atom is d e s c r i b e d by d i s t o r t e d C o u l o m b w a v e s a s y m p t o t i c to p l a n e w a v e s m o v i n g in the d i r e c t i o n K i (Kf) and outgoing (incoming) s p h e r i c a l waves. We have

Rnli(r) Ylimi(8,~) ,

IXi) =

I×f) : V-½ 4 ,

I:)i) =

-~

~. i/f If mf

(3)

jlf(ter) Ylfmf(k)

x YLiMi(O,¢ ) (/~R)-I FLi(KiR) l~f} = ~'. 4rr (-1) Mf i l l e-i~Lf(rg)

Lf Mf

,

(6)

In these equations R(r) is the radial part of the hydrogenic wave functions, V is the volume of the box in which the system is included and F is the regular solution to the Coulomb radial wave equation. We might note here that the phase shifts OLO?) drop out in the final stage of the calculation. For the densities of states, we have

mf~k (21rh)3

(7)

(21r)3 //.

(8)

and 11)

If we restrict If = l i than the

c o n s i d e r dipole t r a n s i t i o n s only and f u r t h e r the c a l c u l a t i o n to the c a s e of Lf = L i + 1, 1 (this g i v e s a m u c h l a r g e r c o n t r i b u t i o n c a s e Lf = L i - 1, If = /i + 1), we obtain

r:96z2~2kdk~(2~+1)

1

0

J t

× {J(R)! 2 ~gK(r)]2 , (9) 1

'0f) i s W i g n e r ' s 3j syrflbol,

J(~) = ~--~.!'dR R-2 FL(KiR) FL+I(KfR) L and

K(,') = ~¢ d,"

R.,di(Y ) jlf(k,') ~3

(10)

(11)

Eq. (11) can be c a l c u l a t e d a n a l y t i c a l l y for li = n - 1. Using eq. (9) we e s t i m a t e d n u m e r i c a l l y the p r o b a b i l i t i e s of C o u l o m b d i s i n t e g r a t i o n of p - m e s o n i c Li a t o m s w h e n the m u o n s a r e in 0z = 4, ! = 3) and 0z = 3,1 = 2) s t a t e s , r e s p e c t i v e l y . An i n c i d e n t e n e r g y of 5 keV for the r e l a t i v e m o t i o n of the c e n t e r of m a s s of the m e s o n i c atom and the n u c l e u s of 34

type of process

quantum numbers of initial and final states

probability (sec-1)

radiative

4

3

3

2

2.4 x 1011

Auger

4

3

3

2

2 × 1013

Coulomb disintegration

4

3

2

2 × 1015

radiative

3

2

2

1

1.1 × 1012

Auger

3

2

2

1

5 × 1012

Coulomb dis integration

3

2

1

4 × 109

(5)

YLf-Mf~f)

× YLfMf(~a)) (/~R)-1 FLf(I~R) ,

N(Kf) -

Table 1 Probabilities of various processes for the p-mesonic Li atom. The incident energy for the Coulomb disintegration is 5 keV.

YLi_Mi(Ki)

LiMi

z(k) = V

15 November 1962

the n e i g h b o u r i n g atom i s a s s u m e d . T h e s e a r e c o m p a r e d in t a b l e 1 with the p r o b a b i l i t i e s of t y p i c a l r a d i a t i v e and A u g e r t r a n s i t i o n s f o r the s a m e s t a t e s .

Y/fmf(0,~) ,

(4)

4~r (-1) Mi i L i e i(~Li(rii)

LETTERS

The C o u l o m b d i s i n t e g r a t i o n p r o b a b i l i t y i s e n t i r e l y n e g l i g i b l e even for the (n = 7, / = 6) s t a t e if the i n c i d e n t e n e r g y i s 1 keV. Under C o u l o m b d i s i n t e g r a t i o n the m u o n s a r e e j e c t e d with s m a l l k i n e t i c e n e r g y . F o r e x a m p l e , in the above c a s e s , the m a j o r c o n t r i b u t i o n to the C o u l o m b d i s i n t e g r a t i o n c o m e s f r o m s m a l l v a l u e s of k , c o r r e s p o n d i n g to a kinetic energy of about 0.I keV. The Coulombdisintegration probability varies sensitively as a function of the incident energy. This striking sensitivity can be understood from the facy that J(R) decreases rapidly with increasing v a l u e of Ki/K f. T h e r e f o r e , a p - m e s o n i c atom with k i n e t i c e n e r g y l a r g e r than a c e r t a i n v a l u e will i n e v i t a b l y suffer C o u l o m b d i s i n t e g r a t i o n when it c o m e s into c l o s e c o n t a c t with the n u c l e i of o t h e r a t o m s , p r o v i d e d its n value is not too s m a l l . In our e x a m p l e , the p - m e s o n i c Li atom will p a s s t h r o u g h m a t t e r with v e l o c i t y v - 107 c m / s e c . We a s s u m e that it will be s u b j e c t e d to one i n t e n s e C o u l o m b field due to the n u c l e u s of a n e i g h b o u r i n g atom in, r o u g h l y s p e a k i n g , about ten t r a v e r s a l s of a unit cell of the m a t t e r 10), so that the t i m e b e tween C o u l o m b e x c i t a t i o n s is about 10-14 sec. T h i s t i m e is b a r e l y enough to allow a few t r a n s i t i o n s , r a d i a t i v e and A u g e r , to take p l a c e . Since, in a c t u a l i t y , the k i n e t i c e n e r g y of the p - m e s o n i c a t o m s when they a r e f o r m e d i s s p r e a d b e t w e e n z e r o and a few keV, a c e r t a i n f r a c t i o n of the p - m e s o n i c a t o m s will d i s i n t e g r a t e b e f o r e they r e a c h the ground

Volume 3, number 1

PHYSICS

state. The e j e c t e d m u o n s have a low v e l o c i t y and will e i t h e r d e c a y o r be r e c a p t u r e d by o t h e r n u c l e i , p r o b a b l y in o r b i t s of l a r g e n. T h e c a s c a d e will r e p e a t but t h i s t i m e C o u l o m b d i s i n t e g r a t i o n will not play a s i g n i f i c a n t r o l e . R e p e a t e d c a s c a d e would i n c r e a s e the d e p o l a r i z a t i o n of the n e g a t i v e m u o n s . H o w e v e r , it would be p o i n t l e s s to c a r r y out m o r e a c c u r a t e c a l c u l a t i o n s u n t i l we have m o r e q u a n t i t a t i v e i n f o r m a t i o n about the f o r m a t i o n of the m e s o n i c a t o m s . The e x p e r i m e n t a l c o m p a r i s o n of the f a l l - o f f with i n c i d e n t e n e r g y of the e l a s t i c s c a t t e r i n g c r o s s s e c t i o n s for p o s i t i v e a n d n e g a t i v e m u o n s on n u c l e i would help to e s t i m a t e the a m o u n t of the f o r m a t i o n of m e s o n i c a t o m s t h r o u g h the pickup of m u o n s in flight. The a u t h o r w i s h e s to e x p r e s s h i s t h a n k s to Prof. L. E. H. T r a i n o r f o r e n c o u r a g e m e n t and u s e f u l d i s cussions.

A NOTE

ON

LETTERS

15 November 1962

References 1) J.A.Wheeler, Revs. Modern Phys. 21 (1949) 133. 2) Y. Eisenberg and D. Kessler, Nuovo Cimento 19 (1961) 1195. 3) G.R. Burbidge and A.H.De Borde, Phys. Rev. 89 (1953) 189. A.H.DeBorde, Proe. Phys. Soe. (London) A 67 (1954) 57. 4) M.Demuer, Nuclear Phys. 1 (1956) 516. 5) M.B.Stearns and M.Stearns, Phys. Rev. 105 (1957) 1573. 6) M.A.Ruderman, Phys. Rev. 118 (1960) 1632. 7) E.Fermi and E.Teller, Phys. Rev. 72 (1947) 399. 8) R.A.Mann and M.E.Rose, Phys. Rev. 121 (1961) 293. 9) T.B.Day and P.Morrison, Phys. Rev. 107 (1957) 912. 10) T.B.Day, G.A.Snow and J.Sucher, Phys. Rev. Letters 3 (1959) 61. 11) A. Sommerfeld, Atombau und Spektrallinien, Vol. If, Chap. VII (F. Vieweg und Solm, Braunschweig, 1939).

CASTAGNOLI'S

FORMULA

A. A. KAMAL and G. K. RAO Physics Department, Osmania University, Hyderabad 7, A.P., India Received 23 October 1962

Of all the i n d i r e c t m e t h o d s for the e n e r g y d e t e r m i n a t i o n of j e t s , C a s t a g n o l i ' s method has b e e n m o s t e x t e n s i v e l y used. The well known f o r m u l a i s log ~c = (log cot O) ,

(I)

where Vc is the c.m. system Lorentz factor for the colliding particles and O is the space angle of the shower tracks. The basic assumptions on which the derivation of this equation is based are 1. the fore and aft angular distribution of the m e sons in the c.m. system, 2. ~ ~- ~c -~ 1, w h e r e fls is the v e l o c i t y of the shower p a r t i c l e s in the c.m. s y s t e m and ~c i s the c.m. s y s t e m v e l o c i t y i t s e l f . The f i r s t c o n d i tion i s expected to be s a t i s f i e d s p e c i a l l y in the N-N c o l l i s i o n s , w h e r e a s , the second m a y not be a l w a y s s a t i s f i e d . It i s now g e n e r a l l y a g r e e d , a s it was a n t i c i p a t e d in C a s t a g n o l i ' s o r i g i n a l p a pe:- 1), that the a p p l i c a t i o n of this e q u a t i o n , apart from introducing statistical fluctuations, will r e s u l t in s e r i o u s o v e r e s t i m a t e s of the p r i m a r y e n e r g y . We m a y then r e c a s t C a s t a g n o l i ' s equation a s log ~ ~c -
(2)

where x is a numerical correction factor. Clearly × will a l w a y s be l a r g e r than unity, except in the l i m i t i n g c a s e in which r e q u i r e m e n t 2 i s s a t i s f i e d . U s i n g d i f f e r e n t p r o c e d u r e s s e v e r a l a u t h o r s have e v a l u a t e d the v a l u e of x, d e t a i l s of which a r e cont a i n e d in t a b l e 1. While v a r i o u s p r o c e d u r e s yield v a l u e s of x b e t w e e n 1.3 and 2, a l l of them suffer f r o m the c o m m o n defect of u s i n g a r b i t r a r y a s s u m p t i o n s in the c a l c u l a t i o n s . It should be e m p h a s i s e d that x i s v e r y s e n s i t i v e to the shape of the e n e r g y s p e c t r u m at lower e n e r g i e s and that it i s n e i t h e r a c c u r a t e l y known or can it be expected to be c o n s t a n t t h r o u g h o u t the e n e r g y r a n g e . In our opinion a r e a s o n a b l e value of × can c o n v e n i e n t l y be c o m p u t e d by u s i n g the e x p e r i m e n t a l v a l u e s of P/, the t r a n s v e r s e m o m e n t a of s h o w e r p a r t i c l e s . Since P/ a p p e a r s to be i n d e p e n d e n t of the angle of e m i s s i o n and i t s d i s t r i b u t i o n is p r a c t i c a l l y i n v a r i a n t at h i g h e r e n e r g i e s , t h i s a p p r o a c h i s f r e e f r o m any i n h e r e n t a s s u m p t i o n s . The exp r e s s i o n u s e d i s 10) log ~ =
(3)

where ,~I is the r e s t mass of the shower particle. The source of data used for this calculation is in-

35