γ-Neutrino correlations in allowed nuclear μ - meson capture

γ-Neutrino correlations in allowed nuclear μ - meson capture

Volume 24B, number 10 PHYSICS LETTERS 15 May 1967 v-NEUTRINO CORRELATIONS IN ALLOWED NUCLEAR P-MESON CAPTURE A. P. BUKHVOSTOV and N. P. POPOV A...

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Volume 24B, number 10

PHYSICS

LETTERS

15 May 1967

v-NEUTRINO CORRELATIONS IN ALLOWED NUCLEAR P-MESON CAPTURE

A. P. BUKHVOSTOV and N. P. POPOV

A.F.Ioffe Physzco-Techmcal Instztute Lenzngrad USSR Received 20 March 1967

Angular yu c o r r e l a h o n and circular polar~zatmn of y - r a y quanta are calculated for allowed capture of unpolamzed ~z-meson with a r b i t r a r y populahon of # - m e m o atom hyperfme levels.

In a previous paper [i] concerning angular >,v correlation in nuclear g-meson capture the correlation formula was obtained for partial transitions of any order of forbiddenness, without taking into consideration the hyperfine splitting of #-mesic atom levels. This formula holds for capture by spinless nuclei; in the case of an unpolarized ~-meson it also holds for capture by nonzero spin nuclei if the population of the hyperflne structure levels is statistical. As the available data [2] give evidence for possible transitions between these levels, we give here the result of a calculation of angular yv correlation and circular polarization of V's for capture of unpolarized #-mesons in the case of arbitrary population of the hyperfine structure levels. The angular correlation of the neutrino and unpolarlzed y-ray quantum in an allowed Jo ~)I~32 transition has the form

W~ 1 + b2P2(c~s o~) where a is the angle between the directions of emissidn of the neutrino and y, Ps (cos a) is the Legendre polinomial. Gamov-Teller transitions are of the most practical interest. For j I =J o + 1

b2 Wo :

r,L,,,(Jo + 2)(2jo + 5) Fp2BC(2jo+ 1) + C2(3o - 1) (1 - P)C2~ ] ' ~ 2 ~/5(Jo + I)(2.7o + l) t 3(Joo+ I) -

Wo = P B2(2Jo + 1) + C2()o + 2) 3Uo + 1) + (1 - p)C 2 w h e r e p i s t h e p o p u l a t i o n of the F+ h y p e r f i n e s t r u c t u r e s t a t e w i t h t o t a l a n g u l a r m o m e n t u m j o + ½, it can be m e a s u r e d in e x p e r i m e n t b y o b s e r v i n g t h e t i m e d e p e n d e n c e of r a t e of c a p t u r e . F o r s t a t i s t i c a l p o p u l a t i o n P = (Jo + 1)/(22o + 1). B and C a r e d e f i n e d by cV

B= - CA[lO1]+~-~Cp-CA)([101-]+ f2[121+]) + 3 q - ~ c v ( l + p p

-Un)([101-]- f½(121+]) + ~/~-~[111p]

c-- ,;½ ?all 21]

-

-CA)C101- ] + ¢-21121+]) +

v0 +

n)([I01-] -

cA

+

- cA[011p] c

V

The r e d u c e d n u c l e a r m a t r i x e l e m e n t s [101], etc. a r e g i v e n in t a b l e 2 t a k e n f r o m r e f . 3. c i is the w e a k i n t e r a c t i o n c o n s t a n t ; # p and # n a r e t h e a n o m a l o u s m a g n e t i c m o m e n t s of the p r o t o n and n e u t r o n , r e s p e c t i v e l y ; q i s the e n e r g y of the n e u t r i n o ; M i s the m a s s of the n u c l e o n . F o r v - r a d i a t i o n of c h a r a c t e r 2 L

QL = ~-~S+ I)(2L + I)(2JI+ I) CLIS 0LI W(J2LJlSiJlL) C: :. is the Clebsch-Gordan coefficient, W(abcd;ef) is the Racah function. For the c a s e j I =Jo - I the coefficient b2 can be obtained from the above formula by substition of p ~1

- p a n d j o --' - (Jo + 1).

497

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PHYSICS

LETTERS

15 May 1967

Ot

O~

i 02.

O4

Oi

02

/

Fig. 1. The c u r v e s of /)2 as a function of C J c a for the r e a c t i o n 10B(3+) /~ 10Be*(2 +) Y-, 10Be(0 +)" Ey = a.a6s MeV for the c a s e s 1) Capture from the lower h y p e r free s t r u c t u r e state ( F ) . 2) Capture from the htgher hyperfme s t r u c t u r e state (F+). 3) Statlstmal population of levels.

Fig. 2. The curves of /~2 as a function of Cp/Ca for the r e a c t i o n I4N(1 +) t:~ 14C~(2+) ~ 14C(0+), Ey = 7.01 MeV [5] for the cases 1) CaptuI e from the hlghel hyperfme s t r u c t u r e state (F ~). 2) Capture from the lower h y p e r tme s t r u c t u r e state (F_). 3) Statmtmal populatmn of levels.

O n e c a n s e e t h a t b2 i s n o n z e r o o n l y if t h e n e u t r i n o s p h e r i c a l w a v e w i t h t o t a l a n g u l a r m o m e n t u m I = 3 i s p r e s e n t (C ¢ 0); t h e l a t t e r d e p e n d s s t r o n g l y o n Cp. F i ~ s . 1 a n d 2 s h o w t h e d e p e n d e n c e of t h e c o r r e l a t i o n c o e f f i c i e n t on Cp/C a f o r ~ - m e s o n c a p t u r e b y 10B a n d I ~ N r e s p e c h v e l y . T h e c u r v e s 1, 3 o n b o t h f i g u r e s a r e c a l c u l a t e d n e g l e c t i n g t h e c o r r e c t i o n n u c l e a r m a t r i x e l e m e n t s w h o s e c o n t r i b u t i o n t o t h e c o r r e l a h o n c o e f f i m e n t s d o e s n o t e x c e e d 20% [4]. F o r c a p t u r e b y 10B f r o m t h e l o w e s t h y p e r f i n e s t r u c t u r e s t a t e ( F . ) t h e v a l u e of b 2 d e p e n d s o n Cp s t r o n g ly a n d m o n o t o n i c a l l y * . H o w e v e r , f o r c a p t u r e b y 14N f r o m t h e l o w e s t s t a t e b 2 i s i n d e p e n d e n t b o t h of w e a k i n t e r a c t i o n c o n s t a n t s a n d of n u c l e a r m a t r i x e l e m e n t s . T h i s i s a c o n s e q u e n c e of c o n s e r v a t i o n of t o t a l a n g u l a r m o m e n t u m . In t h e c a p t u r e f r o m t h e s t a t e F - = Jo - ½ w i t h t h e t r m ~ s i h o n t o j 1 =.1o + 1 a s w e l l a s m 3. t h e c a p t u r e f r o m F + = Jo + ½- w i t h t h e t r a n s i t i o n to J l = Jo - 1 t h e n e u t r i n o c a n b e e m i t t e d o n l y w i t h I ~ ~, t h e f o r b i d d e n c o n t r i b u t i o n s b e i n g n e g l e c t e d , t h e o n l y p o s m b l e w a v e i s t h a t w i t h I = ~, ~ so t h a t i t c a n c e l s out from the correlation**. T h e d e g r e e of c i r c u l a r p o l a r i z a t i o n of t h e y - r a y q u a n t u m i s d e f i n e d b y t h e f o r m u l a : {h

cos

+ b3P3(cos

+ b2e2(eos

* In the case of s t a t i s t i c a l population of hyperfme s t r u c t u r e levels the values of b2 a g r e e wtth the corresponding curve from ref. 4. ** It is worth notmmg that the capture r a t e from t h e s e states should be e o n m d e r a b l e only for a l a r g e p s e u d o s e a l a r constant. 498

Volume 24B, n u m b e r 10

PHYSICS

LETTERS

15 May 1967

F°rTl :Jo + 1 ,f)0

blWo

=

+ 2 r (270 + I ) ( B +

C) 2 - { j o C

2

3(jo + 1)

Q1L [13(3-~~l)LP

+3( I -p)C2}j

L 3 y'Oo + 2)0o + 3)(2:0 + 1)(24 + 5) b3Wo --- Q3 5(Jo + ~ 7Jo(J--~+ I-) (P

Jo + 1 ~o +-~)Cz

F°r71 --Jo - 1 the coefficients bI and b3 can be obtained from these formulae by substitution ofp ~1 -p andJo -~-(Jo + I). In the case of ~-meson capture by 10B the y-ray quanta forming the angle ~ < 90 ° with respect to the neutrino m o m e n t u m have m M n l y left-handed circular polarization; for capture by 14N they have mostly right-handed polarIzatmn if o~ < 45 o. Thls is true for any population parameter p. For capture by 10B from F+ state and by 14N from F- state the polarization is independent of Cp, the dependence being rather strong for capture from the other state and from a mixture of the two states. Thus, the study of the angular yv correlation and of the ?' circular polarization can give information about the value of Cp and about the forbidden contributions to the allowed p-meson capture. T h e a u t h o r s a r e d e e p l y g r a t e f u l to A. I. M u k h i n f o r h e l p f u l d i s c u s s i o n .

R e f e~'cnces 1. 2. 3. 4.

N , P . P o p o v , Zh. Eksp. 1 T e o r . Flz. 44 (1963) 1679 (Soviet Phvs J E T P 17 (1963) 1130). R.Wlnston Phys. Rev. 129 (1963) 2766. M . M o m t a a n d A . F u j l l , Phys. Rev. 118 (1960) 606. G , M . B u k a t and N . P . P o p o v , Zh. Eksp. 1 T e o r . Flz. 46 (1964) 1782 (Soviet Phys, J E T P 19 (1964) 1200). Z . O m e w l c z and A. Plkulskl, m p r e s s 5. R . R . C a r l s o n , Phys. Rev. 148 (1966) 99l.

THE

EFFECT OF HYPERFINE

NUCLEAR DEFORMATION STRUCTURE ANOMALY I.

ON

UNNA

Departn~ent of Theoretical Physzcs, Hebrew Universdy, Jerusalem

Isreal

Received 3 April 1967

A r e c e n t m e a s u r e m e n t of the hyperfme s t r u c t u r e anomaly of deformed 171 173yb isotopes is i n t e r p r e t e d m t e r m s of the Nllsson model. It t u r n s out that the anomaly ~s best reproduced with d e f o r m a t m n s wh,ch a r e consistent with those obtained by other, m o r e direct, methods.

Recently a high precision measurement has b e e n c a r r i e d o u t , f o r t h e f i r s t t i m e , of t h e h y p e r f i n e s t r u c t u r e a n o m a l y c a u s e d b y two s t r o n g l y d e formed nuclei. We find that the measured anomaly c a n b e a c c o u n t e d f o r b y t h e N i l s s o n m o d e l f o r t h e s e n u c l e i . O n l y d e f o r m a t i o n s in t h e r a n g e of those measured by other experiments on these nuclei reproduce correctly the observed result. Budik and Snir obtained for the hyperfine struct u r e a n o m a l y of t h e 3 P 1 s t a t e i n t h e i s o t o p e s 1 7 1 y b a n d 1 7 3 y b t h e v a l u e [1]

A ( 3 P 1 ) = ( - 0 . 3 7 6 ± 0.020)% The anomaly,

A, i s d e f i n e d b y

A ( 3 P 1 ) =-A171(3P1)g173/A173(3P1)g171 - 1 Here, A are the magnetic hyperfine structure constants and g are the nuclear ground state gf a c t o r s [2]. The atomic state considered here, as well as t h e o t h e r two m e m b e r s of t h e t r i p l e t , b e l o n g to t h e two e l e c t r o n c o n f i g u r a t i o n p s . U s i n g t h e a t o m i c 499