C o h e r e n t a n d I n c o h e r e n t S p i n Scattering R a d i u s
L e t
a s s u m e a n
o p e r a t o r
f o r
t h e
c o h e r e n t a n d
i n c o h e r e n t s p i n a s
~ ~ ~
a A B I . i
J I i
J 2 I 2 i 2 2 I . i
1 2
i
I . i 2 J I
2 2
~ ~ ~
a A B I . i
spin state eigenvector
~ ~ ~ 1 2
a A B 2 J I
2 2
i
a A B 1
2
J ( J 1 ) I ( I 1 ) i ( i 1 )
For i 1 / 2 J I 1 / 2 and J I 1 / 2
a ) For J I 1 / 2
we have
1 2
a A 2 B ( I 1 / 2)( I 1 / 4 ) I
or
I 3 / 4
a A
1 B ( I 1 )
2
b ) For J I 1 / 2
we have
1 2
a A 2 B ( I 1 / 2)( I 3 / 2 ) I
or
I 3 / 4
a A
1 B I
2
A and B as a function of a and a
From
1
a A B I
2
and
a A
1 B ( I 1 )
2
A and B become
I 1
A
2 I 1
a
I
a
2 I 1
and
B
2 ( a a ) 2 I 1
I f a
a a
t h e n
B 0
a n d
A a .
T h i s
i n d i c a t e s
t h a t n o
s p i n
c o h e r e n t
s c a t t e r i n g
e x i s t s . H e n c e , A r e l a t e s t o t h e c o h e r e n t a n d B t o
t h e i n c o h e r e n t s c a t t e r i n g .
~ 2
T h e e x p e c t e d v a l u e s f o r t h e o p e r a t o r
a c a n
b e d e r i v e d a s :
a 2
~ ~ *
a a
a 2
A 2
AB
I . i BA I . i B 2 ( I . i ) 2
T h e c r o s s t e r m s
a r e
z e r o
s i n c e n o
c o r r e l a t i o n
e x i s t s b e t w e e n
t h e n e u t r o n s p i n a n d
t h e
n u c l e u s
s p i n , i . e . ,
I . i
0 .
a 2 A 2 B 2 ( I . i ) 2
x
x
y
y
( I . i
) 2 ( I i
) 2 ( I i ) 2
( I z i z
) 2
x y
( i ) 2 ( i ) 2
( i z )
1 / 4
2
( I .
or
( I .
) 2 1
i
4
i
) 2 1
4
I 2
I ( I 1 )
a 2
A 2 B 2 1
4
I ( I 1 )
With A and B as
I 1
A
2 I 1
a
I
a
2 I 1
and
B
2 ( a a ) 2 I 1
I 1
I 2 I ( I 1 )
a 2
a
a
( a a ) 2
2 I 1
2 I 1
(2 I 1 ) 2
a coh
I 1
a
2 I 1
I
a
2 I 1
a [ I ( I
1 ) ] 1 / 2
( a a )
inch
2 I 1
M IT OpenCourseWare http://ocw.mit.edu
22.106 Neutron Interactions and Applications
Spring 20 10
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