Analysis of the extended defects in 3C-SiC.pdf - Nelson Mandela ...

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Consider Fig. 4.2 which illustrates the wave vectors of the incident and diffracted
waves in Fig. 4.1 with the reciprocal lattice vector g. By redrawing the vector
construction such that k' starts at point C Fig. 4.2 (b) is obtained which shows the
relation,
k' k
g
(4.3)
This is equivalent to the direct space Bragg condition and is shown by projecting the
above equation onto the vector g,
k'
g k
g
g
g;
k g sin k g sin g
sin sin 1
d
hkl
2
(4.4)
from which the Bragg equation follows. The Bragg condition is thus satisfied
whenever the point C falls on the perpendicular bisector plane of the vector g. The
perpendicular bisector plane to the vector ghkl consists of all points that are at the
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))