Introduction to Solid State Physics for Materials Engineers. Emil Zolotoyabko
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translational symmetry, which should be used instead of the momentum conservation law in a homogeneous medium. We remind that the latter law means 2πG = kf − ki = 0, i.e. kf = ki.
Note that each vector of reciprocal lattice, G = hb1 + kb2 + lb3, is perpendicular to a specific crystallographic plane in real space. This statement directly follows from Eq. (1.29), which defines the geometric plane for the ends of certain vectors, rs, the plane being perpendicular to the vector G (Figure 1.14). Bearing in mind possible wave diffraction when propagating through a periodic medium, it is worth to introduce a set of parallel planes of this type (i.e. those given by Eq. (1.29)), which are separated by the d-spacing
Figure 1.14 Sketch of a crystal plane, normal to the vector of reciprocal lattice, G, which contains the ends of vectors, rs, satisfying Eq. (1.29).
Figure 1.15 Graphical interrelation between wavevectors of the incident (ki) and scattered (kf) waves and the vector of reciprocal lattice, G.
In fact, using graphical representation of Eq. (1.28) (Figure 1.15) and solving the wavevector triangle, we find (with the aid of Eq. (1.24)) that
Substituting Eq. (1.37) into Eq. (1.38), we finally obtain the so-called Bragg law:
(1.39)
which provides the relationship between the possible directions for the diffracted wave propagation (via Bragg angles, ΘB) and inter-planar spacings (d-spacings), d, in crystals. We stress that if λ > 2d, Bragg diffraction is not possible.
Note that for quasicrystals, the diffraction conditions (like Eq. (1.28)) can be deduced from the quasi-momentum (quasi-wavevector) conservation law in the n-dimensional space (hyperspace, n > 3), in which the vectors of reciprocal lattice, Gqc, are:
(1.40)
In case of icosahedral symmetry, n = 6, and the set of basis vectors has the following form:
(1.41)
Figure 1.16 The traces of isoenergetic surfaces (red curves) in reciprocal space for the incident (ki) and diffracted (kf) waves. The point of degeneracy of quantum states is marked by the letter D.
where G0 is some constant and
(1.42)
Using this terminology, regular crystals constitute the largest class, for which n = 3 (see