Introduction to Solid State Physics for Materials Engineers. Emil Zolotoyabko

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(1.26)

is the phase wave velocity. Substituting Eqs. (1.24—1.26) into Eq. (1.23) yields:

(1.27)

      Introducing a new vector, G, which is called vector of reciprocal lattice,

(1.28)

and combining Eqs. (1.27, 1.28), we find

(1.29)

According to Eqs. (1.28, 1.29), different values of k_f = k_i + 2πG are permitted in a periodic medium, but only those that provide scalar products of certain vectors, G, with all possible vectors, r_s, to be integer numbers, m. By substituting Eq. (1.1) into Eq. (1.29), we finally obtain:

(1.30)

In order to find the set of allowed vectors, G, satisfying Eq. (1.30), the reciprocal space is built, which is based on three non-coplanar vectors b₁, b₂, and b₃. Real (direct) space and reciprocal space are interrelated by the orthogonality (reciprocity) conditions:

(1.31)

      where δ_ij is the Kronecker symbol, equal to 1 for i = j or 0 for i ≠ j (i, j = 1, 2, 3). To produce the reciprocal space from real space, we use the following mathematical procedure:

(1.32)

      where V_c stands for the volume of the parallelepiped (unit cell) built in real space on vectors a₁, a₂, a₃:

(1.33)

By using Eqs. (1.32, 1.33), it is easy to directly check that the procedure (1.32) provides proper orthogonality conditions (1.31). For example, a₁ · b₁ = a₁ · [a₂ × a₃]/V_c = V_c/V_c = 1, whereas a₂ · b₁ = a₂ · [a₂ × a₃]/ V_c = 0. Certainly, the volume of the unit cell, V_rec, in reciprocal space

      (1.34)

      is inverse to V_c. To prove this statement, we use the relationship well-known in vector algebra:

      (1.35)

      In the reciprocal space, the allowed vectors, G, are linear combinations of the basis vectors, b₁, b₂, b₃:

(1.36)

      with integer projections (hkl), known as Miller indices. The ends of vectors, G, being constructed from the common origin (000), produce the nodes of a reciprocal lattice. For all vectors, G, Eq. (1.30) is automatically valid due to the orthogonality conditions (1.31). We repeat that in the medium with translational symmetry, only those wavevectors, k_f, may exist, which are in rigid interrelation with the initial wavevector k_i, satisfying СКАЧАТЬ