CHAPTER FOUR
DYNAMICAL THEORY OF ELECTRON DIFFRACTION AND IMAGE SIMULATION
4.2. Theory of Electron Diffraction
4.2.1 The Direct Space Bragg Equation
Although the process of electron scattering within a crystal is in general a complicated process it may be explained in a simple way through the Bragg relation. Consider Fig.
4.1.
Fig. 4.1. (a) A wave incident on a set of hkl planes at an angle θ, (b) the incident and diffracted directions and the plane normal must lie in a planar section through a conical surface with its top in the plane (from De Graef (2003))
A wave is incident on a set of planes with indices (hkl) at an angle θ. Assuming the planes to be semi-transparent mirrors, part of the intensity of the wave will be transmitted through the first plane and part of it reflected. If Snell’s law is applied since an ideal mirror was assumed, the incident and reflected angles must be the same, and both directions of the wave must be coplanar about a normal n to the (hkl) planes.
Next, if the reflected waves from the first and second planes are considered it is seen that there is a difference in path length between the two which relate to the waves interference, the path length difference should be an integral number of wavelengths and thus for constructive interference the relation is,
n
dhklsin
2 (4.2)
which reduces to, 2dhklsin for use in the electron microscope since the n is incorporated into the value of dhkl using the Miller index notation . It should be noted that the Bragg equation only describes the geometric representation of diffraction and does not give any information on the intensity of the diffracted wave. The equation is derived with respect to a specific plane and states that the incident and diffracted waves must travel in directions which lie on a conical surface with top in the diffracting plane and opening angle π/2-θ as illustrated in Fig. 4.1(b).
4.2.2 The Bragg Equation in Reciprocal Space
The direct space Bragg equation however elegant and simple is not useful when the absolute direction of a diffracted wave needs to be determined. To do this the Bragg equation should be reformulated in terms of reciprocal space which in turn leads to the Ewald sphere construction.
From the de Broglie relation it is known that a plane wave may be represented by its momentum vector p = hk where k has units of reciprocal length and that λ = 1/|k|. k fully characterises the direction and wavelength of the plane wave with respect to the crystal reference frame and thus a reciprocal space equivalent of the Bragg equation can be derived since the plane (hkl) may be represented by the vector ghkl in reciprocal space.
Fig. 4.2. (a) The incident and diffracted wave vectors with reciprocal lattice vector g, (b) The redrawn vector construction (from De Graef (2003))
Consider Fig. 4.2 which illustrates the wave vectors of the incident and diffracted
This is equivalent to the direct space Bragg condition and is shown by projecting the above equation onto the vector g,
dhkl same distance from the origin of reciprocal space and the reciprocal lattice point hkl.
This leads to a simple geometric construction for the direction of a diffracted wave when the incident wave vector and crystal orientation are given. This is known as the Ewald sphere and is depicted in Fig 4.3.
Fig. 4.3. The Ewald sphere construction (from De Graef (2003))
The following procedure is followed. First draw the reciprocal lattice with origin O.
Then draw the incident wave vector k such that its end point coincides with O. The point C is then taken as the center of a sphere with radius |k|. Whenever a reciprocal lattice point falls on this sphere, the Bragg condition is satisfied and a diffracted vector k+g may occur. Note also that more than one reciprocal lattice point may lie on the Ewald sphere which indicates that multiple beam diffraction may occur. This is due to a relaxation of the Bragg condition due to the sample geometry causing the reciprocal points to extend into rods (see Hirsch et al. (1965)).
4.2.3 Darwin-Howie-Whelan Equations
Bragg’s law shows that electrons are diffracted in directions k' given by k+g for all reciprocal lattice points g lying on the Ewald sphere. Hence it may be anticipated that the total wave function at the exit surface of the crystal will be a superposition of plane waves, one in each of the directions predicted by the Bragg equation. All that remains is the computation of the complex amplitudes of each of the diffracted waves which are given by the Darwin-Howie-Whelan (DHW) equations:
transform of which is given by,
V0 is the mean inner potential of the crystal and is positive and causes acceleration of the electrons and hence the relativistic acceleration of the electron increases to,
0
which in turn is used to simplify the Schrödinger equation to,
obtain a general solution of the form,
which upon substitution into equation 4.10 and using a high energy approximation the DHW equations are obtained. To follow the complete derivation the reader is referred to de Graef (2003). The symbol θg in equation 4.5 is known as the phase factor, sg is the deviation from the Bragg condition and,
'
with ξg the extinction distance, the absorption length and g' g g' g. The extinction distance and absorption length is given by the following,
cos
1
g
k g
g
U
cos
1 '
' k gg
g
U
The extinction distances as calculated by Electron Diffraction (2003) is given in Table 4.1. for commonly used reflections for 3C-SiC.
Table 4.1. – The extinction distances in 3C-SiC at 200 kV as calculated by Electron Diffraction (2003).
Index ξg (in Å)
111 622
220 825
200 1915
400 1383
113 1431
4.2.4 DHW Equations: Two Beam Case
When the special case in which the intensity of the incident beam is shared only between the transmitted ψ0 and one diffracted beam ψg is considered, the DHW equations are given by two coupled differential equations. They are,
beam left-hand side g' = 0 g' = g
These equations may be written in the form,
q S
by means of the following substitution,
e z procedure the solution to the problem is derived as follows. Equation 4.17 is written as,
Substituted into equation 4.18 and rearranged to give,
This is the differential equation for the harmonic oscillator with general solution,
) found to be 1. The value for B is found by computing dT/dz and substituting into the equation for S above and applying the boundary condition S(z = 0) = 0 which leads to B = -isg/σ and thus,
These are the general solutions to the dynamical two-beam equations including absorption. Note also that σ is complex.
4.2.5 The Two-Beam Scattering Matrix
In Section 4.2.4 the solutions to equations 4.17 and 4.18 was found by using the
In addition another set of independent initial conditions can be used to obtain a special
which shows that it is assumed that all the incident amplitude is in the direction of the scattered beam rather than the transmitted beam. Using this it is found that the sign of the excitation error changed. From equation 4.26 it is also seen that S(-s,z) = S(s,z). The amplitude at the exit plane of the crystal with thickness z0 is then given by,
0 general solution to a system of N first-order differential equations may be written as a linear combination of special solutions, where the special solutions can be obtained by stating N independent boundary conditions. Thus in the case for N = 2 two sets of independent boundary conditions given by equations 4.27 and 4.28 are used to find
solutions which in turn are combined to give a general solution from which the scattering matrix is obtained. The general solution may then be written as follows,
)
It should also be noted that the scattering matrix is a complex matrix of the form,
i r
scat S iS
S (4.33)
4.2.6 Crystal Defects and Displacement Fields
The presence of a defect in a crystal causes a disruption in the positions of the atoms as predicted by the Bravais lattice translation vectors. The atoms are displaced from their positions by a vector R which is a function of position r. This vector field R(r) is known as the displacement field and relates the displaced position r’ of the atom to the ideal position r by,
R(r) r
r' (4.34)
Also the presence of this displacement field causes a deviation in the potential of the crystal from the perfect crystal at the deformed position given by,
)
When this is incorporated into the Darwin-Howie-Whelan equations it modifies the
which only constitutes a change in the excitation error to an effective excitation error given by,