William Stratton Ph.D. Thesis - MINDS@UW Home

2.03 Calculation of the Diffracted Intensity
The derivation of V proceeds as follows: first, we consider the intensity from a single bin,
which may or may not contain one crystal. Then we sum over the bins in a column to obtain the
intensity from that column, Ii, which involves the projection through the samples thickness.
Next, we sum over Ii’s to obtain 〈I〉 and 〈I 2 〉. We shall see that these last two sums depend
mainly on the distribution of crystals in the bins, which we will assume is random. 〈I〉 and
〈I 2 〉 are defined as
c 1
1
N
I ≡ ∑ Ii
, and (2.2)
N i=
c
c 1
I ≡ ∑ I . (2.3)
2 2
1
N
N c i=
i
The intensity from a bin can be calculated from a modified intensity equation for the
dark-field TEM image intensity from an arbitrary set of atoms in an monoatomic system derived
previously 74 ,
Natoms Natoms
{ }
( ) = γ ∑∑ jl ( ) exp −2π ( ⋅ jl )
I k A Q i k r , (2.4)
γ λ
j l
( )
2 2
≡ f k .
Ajl is the Airy function evaluated at rjl, Q is the radius of the objective aperture in reciprocal
space, rjl is the distance between atoms j and l, λ is the wavelength of the imaging electron, and k
is the dark field scattering vector. The intensity prefactor γ is a function of the atomic scattering
factor f(k) for a monoatomic system. The intensity expression for a polyatomic system would
have the atomic scattering factors inside the sums, not in γ. I(k) in equation (2.4) is unitless.
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