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Asymmetry parameter of neutrons and tritons from the capture of polarised μ- in 4He
Volume 4, number 1
PHYSICS
ASYMMETRY FROM
THE
PARAMETER CAPTURE
LETTERS
1 March 1963
OF NEUTRONS OF POLARISED
AND TRITONS #- I N 4 H e
A. B I E T T I Iustituto di Fisiea dell'UniversitA di Roma, and Instituto Nazionale di Fisica Nucleare, Sezione di Roma Received 29 January 1963
t
~(En)
Weak magnetism term inclu4ed
It is well known that a m e a s u r e m e n t of the a s y m - oAm e t r y p a r a m e t e r of the r e c o i l n u c l e i o r of the 0.3 e m i t t e d n e u t r o n s in the c a p t u r e of p o l a r i s e d m u o n s by s p i n z e r o n u c l e i , c a n give i m p o r t a n t i n f o r m a t i o n OJ about the coupling c o n s t a n t s of the p r o c e s s , and m a i n l y the p s e u d o s c a l a r t e r m and the weak m a g 0 netism term. -03 It s e e m s to be of s o m e i n t e r e s t to p e r f o r m such a n e x p e r i m e n t in the n u c l e i of the low m a s s n u m -0 b e r r e g i o n , f o r i n s t a n c e , He 4 and C 12. In effect, at the p r e s e n t m o m e n t , the c o m p a r i s o n of the theo r y with the experiments done in the nuclei of the o~ =(En)
go'lsgA~~__ gp=SgA 30
0
~
~
5p
4,0
6PEnM~V
%=-~6%
~ -
.48f
Fig. 1. Weak magnetism term omitted
medium-high m a s s number region, is very difficult Q3 b e c a u s e of o u r p o o r knowledge of the n u c l e a r wave functions, On the o t h e r hand, in a r e c e n t p a p e r 1) it w a s shown that the d e p e n d e n c e , in the r e a c t i o n ~- + He 4 - H3 + n + u of the c a p t u r e r a t e and the n e u t r o n and t r i t o n s p e c t r a , on a f i n a l s t a t e i n t e r a c t i o n b e t w e e n the n e u t r o n and the t r i t o n i s v e r y s m a l l . The n u c l e a r d e p e n d e n c e is t h e r e f o r e r e s t r i c t e d only to the " s i z e " of the a - p a r t i c l e and the t r i t o n 1). The d i s t r i b u t i o n of the n e u t r o n s e m i t t e d [rom the c a p t u r e of p o l a r i s e d m u o n s i s g i v e n by
-
1
o.2-
o.1
gp-egA gp=O
o -o., -Q2
gp=-8~. gp=-16g A r (He)=l.49f
P (T)-l.4Bf
Fig, 2. Weak magnetism term included
04~ ~(ET) 0.3
dW
d E n = W(En) [I - P # ~(En) cos 8sn],
(I)
ghere esn is the angle between the m u o n spin direc:ion and the m o m e n t u m of the neutron, P/~ is the |egree of polarisation of the muons and W(En) is he neutron spectrum given in ref. I) c~(En)' the Lsymmetry parameter, is given by
~
Q
, y
~
-o.2]h/
5
lo
r(H~=,.~, ~T,.,.~,
15
20 ET MeV
=-16gA
Fig. 3.
/ ~(E'n)d R y !(En)
R u cos Onu
(]1-I2)2
0.41#((ET) / 0' j
,
= p
(2)
? ~(En) d r y r y (/1-/2) 2
Weak magnetism term omitted
o.?
o.,t/",,
gP=16gA _ _
/J
%.o
k+(En)
' h e r e R u i s the n e u t r i n o m o m e n t u m , env i s the ngle b e t w e e n the n e u t r i n o and n e u t r o n m o m e n t a , ~(En) and the i n t e g r a l s o v e r the n u c l e a r wave unctions I1 and I2, a r e g i v e n in ref. 1). F r o m the ) n c l u s i o n s of ref. 1), t h e s e i n t e g r a l s a r e e v a l u -
-0.1
Fig. 4. 63
Volume 4, number I
PHYSICS
LETTERS
gpIBgA Weak magetism term included
0.4- =((En)
t 0.4 o((ET) 1 0.3!
0.3 r~
(~2-
gp.BgA Weak magnetism term include0 ~ ~a~
r (~4B
f
(T)=1.48f
0.10
1 March 1963
~
~
E
n MeV
°.11 0
1L0
t5
20 El NIeV
-0.1-
Fig 5
-0.2-
a t e d in the a p p r o x i m a t i o n of f r e e p a r t i c l e s in the f i n a l s t a t e , o i s a r a t i o b e t w e e n the c o n s t a n t s 2) 2 -
2 +
2
-
2C(")C(")A P
w h e r e the Gu~) , GA(~) , G u ~ ) a r e g i v e n in t e r m s of the c o u p l i n g c o n s t a n t s g v ~ ' ) , g A ~ ) , g p ~ ) in the e q s . ( l b ) and ( l c ) of r e f . 2). F o r the t r i t o n d i s t r i b u t i o n , we h a v e a f o r m u l a a n a l o g o u s to eq. (1), with the s u b s t i t u t i o n E T f o r e n e r g y and 0ST f o r a n g l e . (~(ET) i s g i v e n b y a f o r m u l a s i m i l a r to eq. (2), with the s u b s t i t u t i o n E T f o r e n e r g y and 0Tv f o r a n g l e . The n u m e r i c a l c a l c u l a t i o n s w e r e c a r r i e d out by the 1620 IBM e l e c t r o n i c c o m p u t e r a t the U n i v e r s i t y of R o m e . T a k i n g f o r the r . m. s. r a d i u s of the ( ~ - p a r t i c l e a n d the t r i t o n r e p e c t i v e l y the v a l u e s 1.49 f m 3) * a n d 1.48 f m 4), the c u r v e s f o r a ( E n) w i t h g p ( P ) / g A ( g ) = 0 , ± 8 , J:16, a r e g i v e n in f i g s . l and 2 with the weak magnetism term included and omitted respectively. U n d e r the s a m e c o n d i t i o n s , the c u r v e s f o r a ( E T ) a r e g i v e n in f i g s . 3 and 4. T h e c u r v e s f o r the c a s e * This number comes from the measured value 1.68 fm c o r r e c t e d for the charge distribution of the proton.
gp~)/gA ~)
= - 8 a r e not shown in f i g s . 1 and 3 b e c a u s e the m a x i m u m v a l u e of ~(En) and of ~(E T) d o e s not e x c e e d 10-2. In f i g s . 5 and 6 ~{En) and ~(ET) a r e g i v e n r e s p e c t i v e l y f o r two d i f f e r e n t v a l u e s of the ~ - p a r t i c l e r . m. s. r a d i u s : 1.48 f m and 1.68 f m 3). If the v a l ue of the t r i t o n r a d i u s i s c h a n g e d f r o m 1.48 f m to 1.78 f m 5), f o r n e u t r o n s and t r i t o n s with k i n e t i c e n e r g y b i g g e r r e s p e c t i v e l y than 15 MeV and 5 MeV the a s y m m e t r y p a r a m a t e r s h o w s a d i f f e r e n c e of l e s s than 4%. In c o n c l u s i o n we o b t a i n the i n t e r e s t i n g r e s u l t : the d e p e n d e n c e of a ( E n) and a ( E T ) on the n u c l e a r p a r a m e t e r s , i s s m a l l c o m p a r e d with the big d e p e n d e n c e on the v a r i a t i o n s of g p , and the i n c l u s i o r o r the o m i s s i o n of the w e a k m a g n e t i s m t e r m . In t h i s m a n n e r , i t s e e m s to b e c l e a r that an a s y m m e t r y e x p e r i m e n t done in He4, h a s a r a t h e r secure theoretical interpretation.
References 1) 2) 3) 4)
A.Bietti and P.Di Porto, Nuovo Cimento, in p r e s s . H.Primakoff, Revs. Modern Phys. 31 (1959) 802. G.R.Burleson, Phys. Rev. 121 (1961) 624. A.K. Kaninsky and M. Chirokov, Compt. Rend. du Congr~s Intern. de Physique Nuel~aire, P a r i s , p. 940. 5) A. Fujii and H. Primakoff, Nuovo Cimento 12 (1959) 3~
ERRATUM S.K.Bhattacherjee, E.Giberman, H.J.Lipkin, A.Nir and M. S c h m o r a k , T r a n s v e r s e p o l a r i s a t i o n of c o n v e r s i o n e l e c t r o n s f o l l o w i n g b e t a d e c a y of Hg203, P h y s i c s L e t t e r s 2 (1962) p. 347 Due to a n u n f o r t u n a t e m i s u n d e r s t a n d i n g t h e n a m e of M. S c h m o r a k w a s o m i t t e d f r o m the a u t h o r s of this paper.
64
PHYSICS
ASYMMETRY FROM
THE
PARAMETER CAPTURE
LETTERS
1 March 1963
OF NEUTRONS OF POLARISED
AND TRITONS #- I N 4 H e
A. B I E T T I Iustituto di Fisiea dell'UniversitA di Roma, and Instituto Nazionale di Fisica Nucleare, Sezione di Roma Received 29 January 1963
t
~(En)
Weak magnetism term inclu4ed
It is well known that a m e a s u r e m e n t of the a s y m - oAm e t r y p a r a m e t e r of the r e c o i l n u c l e i o r of the 0.3 e m i t t e d n e u t r o n s in the c a p t u r e of p o l a r i s e d m u o n s by s p i n z e r o n u c l e i , c a n give i m p o r t a n t i n f o r m a t i o n OJ about the coupling c o n s t a n t s of the p r o c e s s , and m a i n l y the p s e u d o s c a l a r t e r m and the weak m a g 0 netism term. -03 It s e e m s to be of s o m e i n t e r e s t to p e r f o r m such a n e x p e r i m e n t in the n u c l e i of the low m a s s n u m -0 b e r r e g i o n , f o r i n s t a n c e , He 4 and C 12. In effect, at the p r e s e n t m o m e n t , the c o m p a r i s o n of the theo r y with the experiments done in the nuclei of the o~ =(En)
go'lsgA~~__ gp=SgA 30
0
~
~
5p
4,0
6PEnM~V
%=-~6%
~ -
.48f
Fig. 1. Weak magnetism term omitted
medium-high m a s s number region, is very difficult Q3 b e c a u s e of o u r p o o r knowledge of the n u c l e a r wave functions, On the o t h e r hand, in a r e c e n t p a p e r 1) it w a s shown that the d e p e n d e n c e , in the r e a c t i o n ~- + He 4 - H3 + n + u of the c a p t u r e r a t e and the n e u t r o n and t r i t o n s p e c t r a , on a f i n a l s t a t e i n t e r a c t i o n b e t w e e n the n e u t r o n and the t r i t o n i s v e r y s m a l l . The n u c l e a r d e p e n d e n c e is t h e r e f o r e r e s t r i c t e d only to the " s i z e " of the a - p a r t i c l e and the t r i t o n 1). The d i s t r i b u t i o n of the n e u t r o n s e m i t t e d [rom the c a p t u r e of p o l a r i s e d m u o n s i s g i v e n by
-
1
o.2-
o.1
gp-egA gp=O
o -o., -Q2
gp=-8~. gp=-16g A r (He)=l.49f
P (T)-l.4Bf
Fig, 2. Weak magnetism term included
04~ ~(ET) 0.3
dW
d E n = W(En) [I - P # ~(En) cos 8sn],
(I)
ghere esn is the angle between the m u o n spin direc:ion and the m o m e n t u m of the neutron, P/~ is the |egree of polarisation of the muons and W(En) is he neutron spectrum given in ref. I) c~(En)' the Lsymmetry parameter, is given by
~
Q
, y
~
-o.2]h/
5
lo
r(H~=,.~, ~T,.,.~,
15
20 ET MeV
=-16gA
Fig. 3.
/ ~(E'n)d R y !(En)
R u cos Onu
(]1-I2)2
0.41#((ET) / 0' j
,
= p
(2)
? ~(En) d r y r y (/1-/2) 2
Weak magnetism term omitted
o.?
o.,t/",,
gP=16gA _ _
/J
%.o
k+(En)
' h e r e R u i s the n e u t r i n o m o m e n t u m , env i s the ngle b e t w e e n the n e u t r i n o and n e u t r o n m o m e n t a , ~(En) and the i n t e g r a l s o v e r the n u c l e a r wave unctions I1 and I2, a r e g i v e n in ref. 1). F r o m the ) n c l u s i o n s of ref. 1), t h e s e i n t e g r a l s a r e e v a l u -
-0.1
Fig. 4. 63
Volume 4, number I
PHYSICS
LETTERS
gpIBgA Weak magetism term included
0.4- =((En)
t 0.4 o((ET) 1 0.3!
0.3 r~
(~2-
gp.BgA Weak magnetism term include0 ~ ~a~
r (~4B
f
(T)=1.48f
0.10
1 March 1963
~
~
E
n MeV
°.11 0
1L0
t5
20 El NIeV
-0.1-
Fig 5
-0.2-
a t e d in the a p p r o x i m a t i o n of f r e e p a r t i c l e s in the f i n a l s t a t e , o i s a r a t i o b e t w e e n the c o n s t a n t s 2) 2 -
2 +
2
-
2C(")C(")A P
w h e r e the Gu~) , GA(~) , G u ~ ) a r e g i v e n in t e r m s of the c o u p l i n g c o n s t a n t s g v ~ ' ) , g A ~ ) , g p ~ ) in the e q s . ( l b ) and ( l c ) of r e f . 2). F o r the t r i t o n d i s t r i b u t i o n , we h a v e a f o r m u l a a n a l o g o u s to eq. (1), with the s u b s t i t u t i o n E T f o r e n e r g y and 0ST f o r a n g l e . (~(ET) i s g i v e n b y a f o r m u l a s i m i l a r to eq. (2), with the s u b s t i t u t i o n E T f o r e n e r g y and 0Tv f o r a n g l e . The n u m e r i c a l c a l c u l a t i o n s w e r e c a r r i e d out by the 1620 IBM e l e c t r o n i c c o m p u t e r a t the U n i v e r s i t y of R o m e . T a k i n g f o r the r . m. s. r a d i u s of the ( ~ - p a r t i c l e a n d the t r i t o n r e p e c t i v e l y the v a l u e s 1.49 f m 3) * a n d 1.48 f m 4), the c u r v e s f o r a ( E n) w i t h g p ( P ) / g A ( g ) = 0 , ± 8 , J:16, a r e g i v e n in f i g s . l and 2 with the weak magnetism term included and omitted respectively. U n d e r the s a m e c o n d i t i o n s , the c u r v e s f o r a ( E T ) a r e g i v e n in f i g s . 3 and 4. T h e c u r v e s f o r the c a s e * This number comes from the measured value 1.68 fm c o r r e c t e d for the charge distribution of the proton.
gp~)/gA ~)
= - 8 a r e not shown in f i g s . 1 and 3 b e c a u s e the m a x i m u m v a l u e of ~(En) and of ~(E T) d o e s not e x c e e d 10-2. In f i g s . 5 and 6 ~{En) and ~(ET) a r e g i v e n r e s p e c t i v e l y f o r two d i f f e r e n t v a l u e s of the ~ - p a r t i c l e r . m. s. r a d i u s : 1.48 f m and 1.68 f m 3). If the v a l ue of the t r i t o n r a d i u s i s c h a n g e d f r o m 1.48 f m to 1.78 f m 5), f o r n e u t r o n s and t r i t o n s with k i n e t i c e n e r g y b i g g e r r e s p e c t i v e l y than 15 MeV and 5 MeV the a s y m m e t r y p a r a m a t e r s h o w s a d i f f e r e n c e of l e s s than 4%. In c o n c l u s i o n we o b t a i n the i n t e r e s t i n g r e s u l t : the d e p e n d e n c e of a ( E n) and a ( E T ) on the n u c l e a r p a r a m e t e r s , i s s m a l l c o m p a r e d with the big d e p e n d e n c e on the v a r i a t i o n s of g p , and the i n c l u s i o r o r the o m i s s i o n of the w e a k m a g n e t i s m t e r m . In t h i s m a n n e r , i t s e e m s to b e c l e a r that an a s y m m e t r y e x p e r i m e n t done in He4, h a s a r a t h e r secure theoretical interpretation.
References 1) 2) 3) 4)
A.Bietti and P.Di Porto, Nuovo Cimento, in p r e s s . H.Primakoff, Revs. Modern Phys. 31 (1959) 802. G.R.Burleson, Phys. Rev. 121 (1961) 624. A.K. Kaninsky and M. Chirokov, Compt. Rend. du Congr~s Intern. de Physique Nuel~aire, P a r i s , p. 940. 5) A. Fujii and H. Primakoff, Nuovo Cimento 12 (1959) 3~
ERRATUM S.K.Bhattacherjee, E.Giberman, H.J.Lipkin, A.Nir and M. S c h m o r a k , T r a n s v e r s e p o l a r i s a t i o n of c o n v e r s i o n e l e c t r o n s f o l l o w i n g b e t a d e c a y of Hg203, P h y s i c s L e t t e r s 2 (1962) p. 347 Due to a n u n f o r t u n a t e m i s u n d e r s t a n d i n g t h e n a m e of M. S c h m o r a k w a s o m i t t e d f r o m the a u t h o r s of this paper.
64