Coherent
and
Incoherent
Spin
Scattering
Ra d i u s
Let
assume
an
operator
for
the
coherent
and
incoherent
spin
as

a = A+ B I . i   
J=I +i

2 2 2
J = I + i + 2I . i ~
~
~
  1 2 2 2
I .i = J − I − i
2
~
~
~  
aζ = A ζ + BI. i ζ
[
]
ζ → spin state eigenvector €
1 2 2 2
aζ = Aζ + B J −I −i ζ
2
~
~
~
[
]
1
a = A + B [ J(J + 1) − I(I + 1) − i(i + 1)]
2
For i = 1/2 ⇒ J = I −1/2 and J = I + 1/2
a) For J = I −1/2
we have
1
−
a = A + B[(I −1/2)(I + 1/4) − I 2 − I − 3/4 ]
2
or
1
−
a = A − B(I + 1)
2
b) For J = I + 1/2
we have
1
+
a = A + B[(I + 1/2)(I + 3/2) − I 2 − I − 3/4 ]
2
or
1
+
a = A + BI
2
−
A and B as a function of a and a
From
€
1
a = A + BI
2
and
1
−
a = A − B(I + 1)
2
+
+
A and B become
I +1 +
I
−
a
A=
a +
2I + 1
2I + 1
and
2
+
−
(a − a )
B=
2I + 1
+
−
If
a = a = a then
B = 0 and
A = a .
This
indicates
that
no
spin
coherent
scattering
exists.
Hence,
A relates
to
the
coherent
and
B to
the
incoherent
scattering.
€
€
€
The
expected
values
for
the
operator
be
derived
as:
€
€
2
~ ~*
< a >= ζ a a ζ
€
~2
a
can
€
 
 
  2
2
< a >= A + AB < I . i > + BA < I . i > +B < (I . i ) >
2
2
The
cross
terms
are
zero
since
no
correlation
exists
between
the
neutron
spin
and
the
nucleus
spin,
i.e.,
< I . i >= 0 .
2
2
2
2
€
2
2
2
2
y y
x x
z z
2
2
2
x
y
z
 
< a >= A + B < (I . i ) >
 
< (I . i ) >=< (I i ) + (I i ) + (I i ) >
< (i ) >=< (i ) >=< (i ) >= 1/4
  2
1 2
< (I . i ) >= < I >
4
or
  2
1
< (I . i ) >= I(I + 1) 4
1
< a >= A + B I(I + 1)
4
2
2
2
With A and B as
I +1 +
I
−
a
A=
a +
2I + 1
2I + 1
and
2
+
−
(a − a )
B=
2I + 1
2
 I +1 +
I(I + 1) +
I
2
−
− 2
a +
< a >= 
a +
(a
−
a
)
2
 2I + 1
2I + 1  (2I + 1)
acoh
€
I +1 +
I
−
=
a +
a
2I + 1
2I + 1
1/ 2
ainch
[I(I + 1)]
=
2I + 1
+
−
(a − a )
MIT OpenCourseWare
http://ocw.mit.edu
22.106 Neutron Interactions and Applications
Spring 2010
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