56
3 Crystals
(a)
(b)
Fig. 3.33 a Hexagonality index α (3.13) of Zn 1−x Cd x S for various ternary compositions. Dashed line is a guide to the
eye. b Regions of different polytypes in ZnSe x S 1−x . Adapted from [243]
3.6 Reciprocal Lattice
The reciprocal lattice is of utmost importance for the description and investigation of periodic structures,
in particular for X-ray diffraction [245], surface electron diffraction, phonon dispersion or scattering
and the electronic band structure. It is the quasi-Fourier transformation of the crystal lattice. The crystal
lattice is also called the direct lattice, in order to distinguish it from the reciprocal lattice.
3.6.1 Reciprocal Lattice Vectors
When R denotes the set of vectors of the direct lattice, the set G of the reciprocal lattice vectors is
given by the condition
6
exp (i G · R) = 1
(3.14)
for all R ∈ R and G ∈ G. Therefore, for all vectors r and a reciprocal lattice vector G
exp (i G · (r + R)) = exp (i G · r) .
(3.15)
Each Bravais lattice has a certain reciprocal lattice. The reciprocal lattice is also a Bravais lattice, since
when G 1 and G 2 are two reciprocal lattice vectors, then this is obviously true also for G 1 + G 2 . For
the primitive translation vectors a 1 , a 2 and a 3 of the direct lattice, the vectors b 1 , b 2 and b 3 that span
the reciprocal lattice are given directly for any lattice as
b 1 =
2π
V a
(a 2 × a 3 )
(3.16a)
b 2 =
2π
V a
(a 3 × a 1 )
(3.16b)
b 3 =
2π
V a
(a 1 × a 2 ) ,
(3.16c)
where V a = a 1 · (a 2 × a 3 ) is the volume of the unit cell spanned by the vectors a i . The volume of the
unit cell in reciprocal space is V
∗
a = (2π)
3
/V a .
6 The dot product a · b of two vectors shall also be denoted as ab.