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Figure 1.15 Graphical interrelation between wavevectors of the incident (k_i) and scattered (k_f) 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

(1.38)

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, Θ_B) 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, G^qc, 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 (k_i) and diffracted (k_f) waves. The point of degeneracy of quantum states is marked by the letter D.

where G₀ is some constant and are unit vectors expressed via the golden ratio (Eq. (1.6)), as:

      (1.42)

Using this terminology, regular crystals constitute the largest class, for which n = 3 (see Скачать книгу