International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 1.1, pp. 8-9   | 1 | 2 |

Section 1.1.6. Some analytical aspects of the reciprocal space

U. Shmuelia*

aSchool of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel
Correspondence e-mail: ushmueli@post.tau.ac.il

1.1.6. Some analytical aspects of the reciprocal space

| top | pdf |

1.1.6.1. Continuous Fourier transform

| top | pdf |

Of great interest in crystallographic analyses are Fourier transforms and these are closely associated with the dual bases examined in this chapter. Thus, e.g., the inverse Fourier transform of the electron-density function of the crystal [F({\bf h}) = \textstyle\int\limits_{\rm cell} \rho({\bf r}) \exp (2 \pi i{\bf h} \cdot {\bf r})\; \hbox{d}^{3}{\bf r}, \eqno(1.1.6.1)] where [\rho({\bf r})] is the electron-density function at the point r and the integration extends over the volume of a unit cell, is the fundamental model of the contribution of the distribution of crystalline matter to the intensity of the scattered radiation. For the conventional Bragg scattering, the function given by (1.1.6.1)[link], and known as the structure factor, may assume nonzero values only if h can be represented as a reciprocal-lattice vector. Chapter 1.2[link] is devoted to a discussion of the structure factor of the Bragg reflection, while Chapters 4.1[link] , 4.2[link] and 4.3[link] discuss circumstances under which the scattering need not be confined to the points of the reciprocal lattice only, and may be represented by reciprocal-space vectors with non-integral components.

1.1.6.2. Discrete Fourier transform

| top | pdf |

The electron density [\rho({\bf r})] in (1.1.6.1)[link] is one of the most common examples of a function which has the periodicity of the crystal. Thus, for an ideal (infinite) crystal the electron density [\rho({\bf r})] can be written as [\rho({\bf r}) = \rho({\bf r} + u{\bf a} + v{\bf b} + w{\bf c}), \eqno(1.1.6.2)] and, as such, it can be represented by a three-dimensional Fourier series of the form [\rho({\bf r}) = {\textstyle\sum\limits_{\bf g}} C({\bf g}) \exp (-2 \pi i{\bf g} \cdot {\bf r}), \eqno(1.1.6.3)] where the periodicity requirement (1.1.6.2)[link] enables one to represent all the g vectors in (1.1.6.3)[link] as vectors in the reciprocal lattice (see also Section 1.1.2[link] above). If we insert the series (1.1.6.3)[link] in the integrand of (1.1.6.1)[link], interchange the order of summation and integration and make use of the fact that an integral of a periodic function taken over the entire period must vanish unless the integrand is a constant, equation (1.1.6.3)[link] reduces to the conventional form [\rho({\bf r}) = {1\over V} {\sum\limits_{\bf h}} F({\bf h}) \exp (-2 \pi i{\bf h} \cdot {\bf r}), \eqno(1.1.6.4)] where V is the volume of the unit cell in the direct lattice and the summation ranges over all the reciprocal lattice.

Fourier transforms, discrete as well as continuous, are among the most important mathematical tools of crystallography. The discussion of their mathematical principles, the modern algorithms for their computation and their numerous applications in crystallography form the subject matter of Chapter 1.3[link] . Many more examples of applications of Fourier methods in crystallography are scattered throughout this volume and the crystallographic literature in general.

1.1.6.3. Bloch's theorem

| top | pdf |

It is in order to mention briefly the important role of reciprocal space and the reciprocal lattice in the field of the theory of solids. At the basis of these applications is the periodicity of the crystal structure and the effect it has on the dynamics (cf. Chapter 4.1[link] ) and electronic structure of the crystal. One of the earliest, and still most important, theorems of solid-state physics is due to Bloch (1928[link]) and deals with the representation of the wavefunction of an electron which moves in a periodic potential. Bloch's theorem states that:

The eigenstates [\psi] of the one-electron Hamiltonian [ {\scr h} = (-\hbar^{2}/2m) \nabla^{2} + U({\bf r})], where U(r) is the crystal potential and [{\bf U}({\bf r} + {\bf r}_{L}) = U({\bf r})] for all [{\bf r}_{L}] in the Bravais lattice, can be chosen to have the form of a plane wave times a function with the periodicity of the Bravais lattice.

Thus [\psi({\bf r}) = \exp (i{\bf k} \cdot {\bf r})u({\bf r}), \eqno(1.1.6.5)] where [u({\bf r} + {\bf r}_{L}) = u({\bf r}) \eqno(1.1.6.6)] and k is the wavevector. The proof of Bloch's theorem can be found in most modern texts on solid-state physics (e.g. Ashcroft & Mermin, 1975[link]). If we combine (1.1.6.5)[link] with (1.1.6.6)[link], an alternative form of the Bloch theorem results: [\psi({\bf r} + {\bf r}_{L}) = \exp (i{\bf k} \cdot {\bf r}_{L}) \psi ({\bf r}). \eqno(1.1.6.7)] In the important case where the wavefunction [\psi] is itself periodic, i.e. [\psi({\bf r} + {\bf r}_{L}) = \psi({\bf r}),] we must have [\exp (i{\bf k} \cdot {\bf r}_{L}) = 1]. Of course, this can be so only if the wavevector k equals [2\pi] times a vector in the reciprocal lattice. It is also seen from equation (1.1.6.7)[link] that the wavevector appearing in the phase factor can be reduced to a unit cell in the reciprocal lattice (the basis vectors of which contain the [2\pi] factor), or to the equivalent polyhedron known as the Brillouin zone (e.g. Ziman, 1969[link]). This periodicity in reciprocal space is of prime importance in the theory of solids. Some Brillouin zones are discussed in detail in Chapter 1.5.[link]

References

Ashcroft, N. W. & Mermin, N. D. (1975). Solid state physics. Philadelphia: Saunders College.
Bloch, F. (1928). Über die Quantenmechanik der Elektronen in Kristallgittern. Z. Phys. 52, 555–600.
Ziman, J. M. (1969). Principles of the theory of solids. Cambridge University Press.








































to end of page
to top of page