Diploma Thesis - Erich Schmid Institute

3. Basic X-ray diffraction expressions
3.1. Reciprocal lattice
Basic X-ray diffraction
A mathematical formalism will be defined to describe scattering phenomena on
crystal structures with translation symmetry [1]. Supposing the crystal periodicity, the
selection of available functions is reduced. A periodical function alone, like sinus or
cosines, is unable to describe the electron density distribution and phase
phenomenon, the basic attributes of X-ray diffraction effect. Fourier series can be
applied to describe the scattering effect.
n
G
= ∫ N(
r)
exp( −i
G . r)
dV
(equ. 3.1)
cell
∑
n ( r ) = nG
exp( i G . r)
(equ. 3.2)
G
The imaginary unit, square root of minus one, is expressed by i. N(r) defines the
electron density within one unit cell while, on the other hand, n(r) denotes the
electron density of the whole crystal. The two vectors r and G in equation 3.1 and 3.2
that appear in the exponential term can be expressed as:
r = x a + y a + z a
(equ. 3.3)
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1
2
2
3
G = h b + k b + l b
(equ. 3.4)
The summation over G in equation 3.2 should be understood as a summations over
h, k and l from minus infinity to plus infinity. The relation between ai and bi vectors
can be found by calculating the scalar product of G and r. According to the Fourier
series the scalar product must have the following form.
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G . r = 2π
(h x + k y + l z)
(equ. 3.5)
Considering the equation 3.5, the scalar product of a1 and b2 or a1 and b3 is zero, so
a system of linear equations can be written as:
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