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First principles calculations
of the dichroic effect in the transmission electron microscope
Within the first Born approximation the differential
scattering crossection is given by the formula
where A are given in terms of Bloch wave coefficients, T is determined
by the sample thickness and S(q,q',E) is the mixed dynamic formfactor
defined as
The problem of the calculation of the scattering
crosssection can be split into several tasks. The first one is
the description of the splitting of the incoming electron beam
into Bloch waves. This is calculated according to the dynamical
diffraction theory. Equivalent problem is solved for the projection
of diffracted beams into the direction defined by the detector
position. These tasks can be calculated by knowing the incoming
beam energy and the geometry of the experiment (sample orientation,
detector orientation). The crucial task, where the input from
firstprinciples codes is needed is the evaluation of the mixed
dynamic formfactor S(q,q',E).
In our approach S(q,q',E) is evaluated from
the knowledge of bandstructure calculated within the density functional
theory. Knowledge of eigenfunctions and eigenenergies is necessary
for this task. This calculation is implemented using the WIEN2k
code.
Similarly as in the XMCD the determining features
of the electronic structure for observing the dichroism in 3d
transition metals are 1) spinorbit splitting of the initial (core)
state 2p into 2p1/2 and 2p3/2; and 2) the exchange splitting of
the valence 3d band, i.e. different density of states for spinup
and spindown electrons (as shown in the figure above for the
case of bcc iron).
The shape of the 3d DOS to a large extent determines
the shape of the L2 and L3 peaks of the scattering crossections
(and the dichroism). This is demonstrated on the 2nd figure, where
normalized scattering crosssections and the resulting dichroic
effect are displayed (for clarity, these are raw results without
any smearings, which of course are present in the real experiment).
We have studied the influence of various parameters
on the strength of the dichroic effect:
1. The thickness of the sample seems to be a
very delicate parameter, beacuse small variations of the thickness
can strongly influence the strength of the dichroic effect. Luckily,
for iron it seems, that there is a relatively wide interval of
thicknesses with a rather strong dichroic signal.
2. The precision of the detector position seems
to be a less sensitive parameter. Relatively large displacements
along the direction perpendicular to the systematic row of reflections
influence the dichroic signal only weakly.
3. All calculations above have been performed
assuming a parallel illumination and pointlike detection. In
the last figure we have tested the sensitivity of the dichroic
effect on the size of the convergence and collection angles (set
to be equal in the figure). As one could guess from the previous
figure, where small detector displacements did not degrade the
dichroic signal, the effect of finite angles is modest.
