An Introduction to Neutron Scattering - Spallation Neutron Source

Kinematic (Born) Approximation
• We defined the scattering cross section in terms of an incident plane wave & a
weakly scattered spherical wave (called the Born Approximation)
• This picture is not correct for surface reflection, except at large values of Q z
• For large Q z , one may use the definition of the scattering cross section to
calculate R for a flat surface (in the Born Approximation) as follows:
R
=
=
L
because
number of
number of
sin α
From the definition
dσ
dΩ
x
L
=
σ
y
ρ
2
k
x
=
=
k
0
neutrons reflected by a sample of
neutrons incident
L
x
L
y
1
cosα
r
r
∫ dr∫
dr
'e
sin α
of
so
iQ.(
r −r ')
r v r
∫
dk
2
x
4π
Q
on sample ( = ΦL
L
2
2
z
It is easy to show that this is the same as the Fresnel form at large Q
0
1
sin α
sin α
dα.
a cross section we get for a smooth substrate :
=
dσ
dΩ
=
dΩ
ρ
= −k
L
L
x
x
L
y
y
L δ ( Q
x
∫
) δ ( Q
size
x
dσ
dΩ
y
)
y
L
2
0
so
x
L
y
sinα
)
dk
k
x
dk
y
sin α
R = 16π
2
ρ
z
2
/ Q
4
z