International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.1, pp. 29
https://doi.org/10.1107/97809553602060000758 Chapter 1.1. Reciprocal space in crystallography ^{a}School of Chemistry, Tel Aviv University, 69 978 Tel Aviv, Israel After a brief introduction in Section 1.1.1, Section 1.1.2 of this chapter presents a formal definition of dual (reciprocal) bases and a brief overview of applications of the reciprocal lattice basis to lattice geometry, diffraction conditions and Fourier synthesis of functions with the periodicity of the crystal. In Section 1.1.3, the fundamental relationships between direct and reciprocal bases are derived and summarized. Section 1.1.4 introduces the basics of tensor notation and the representation of the above relationships in this notation, which is particularly well adapted to analytical considerations as well as to computer programming. Several examples from various areas of crystallographic computing follow this introduction and a detailed derivation of the finite rotation operator is presented in this context. This is followed by a section on transformation of basis vectors (Section 1.1.5), of importance in many areas of crystallography. The chapter is concluded with brief mentions of analytical aspects of the concept of reciprocal space in crystallography (Section 1.1.6). The purpose of this chapter is to introduce the reader to or remind them of the fundamentals of concepts which are employed throughout this volume. 
The purpose of this chapter is to provide an introduction to several aspects of reciprocal space, which are of general importance in crystallography and which appear in the various chapters and sections to follow. We first summarize the basic definitions and briefly inspect some fundamental aspects of crystallography, while recalling that they can be usefully and simply discussed in terms of the concept of the reciprocal lattice. This introductory section is followed by a summary of the basic relationships between the direct and associated reciprocal lattices. We then introduce the elements of tensoralgebraic formulation of such dual relationships, with emphasis on those that are important in many applications of reciprocal space to crystallographic algorithms. We proceed with a section that demonstrates the role of mutually reciprocal bases in transformations of coordinates and conclude with a brief outline of some important analytical aspects of reciprocal space, most of which are further developed in other parts of this volume.
The notion of mutually reciprocal triads of vectors dates back to the introduction of vector calculus by J. Willard Gibbs in the 1880s (e.g. Wilson, 1901). This concept appeared to be useful in the early interpretations of diffraction from single crystals (Ewald, 1913; Laue, 1914) and its first detailed exposition and the recognition of its importance in crystallography can be found in Ewald's (1921) article. The following free translation of Ewald's (1921) introduction, presented in a somewhat different notation, may serve the purpose of this section:
To the set of , there corresponds in the vector calculus a set of `reciprocal vectors' , which are defined (by Gibbs) by the following properties:andwhere i and k may each equal 1, 2 or 3. The first equation, (1.1.2.1), says that each vector is perpendicular to two vectors , as follows from the vanishing scalar products. Equation (1.1.2.2) provides the norm of the vector : the length of this vector must be chosen such that the projection of on the direction of has the length , where is the magnitude of the vector ….
The consequences of equations (1.1.2.1) and (1.1.2.2) were elaborated by Ewald (1921) and are very well documented in the subsequent literature, crystallographic as well as other.
As is well known, the reciprocal lattice occupies a rather prominent position in crystallography and there are nearly as many accounts of its importance as there are crystallographic texts. It is not intended to review its applications, in any detail, in the present section; this is done in the remaining chapters and sections of the present volume. It seems desirable, however, to mention by way of an introduction some fundamental geometrical, physical and mathematical aspects of crystallography, and try to give a unified demonstration of the usefulness of mutually reciprocal bases as an interpretive tool.
Let us assume that the coordinates of all the (direct) lattice points are integers. This can only be true for Ptype lattices. Consider the equation of a lattice plane in the direct lattice. It can be shown (e.g. Buerger, 1941; also Shmueli, 2007) that this equation is given bywhere h, k and l, known as Miller indices of the (hkl) lattice plane, are (under the above assumption) relatively prime integers (i.e. do not have a common factor other than or ). In this equation, x, y and z are the coordinates of any point lying in the plane and are expressed as fractions of the magnitudes of the basis vectors a, b and c of the direct lattice, and n is an integer denoting the serial number of the lattice plane within the family of parallel and equidistant planes. The interplanar spacing is denoted by , the value corresponding to the plane passing through the origin.
Let and , where u, v, w are any integers, denote the position vectors of the point xyz and a lattice point uvw lying in the plane (1.1.2.3), respectively, and assume that r and are different vectors. If the plane normal is denoted by N, where N is proportional to the vector product of two inplane lattice vectors, the vector form of the equation of the lattice plane becomesFor equations (1.1.2.3) and (1.1.2.4) to be identical, the plane normal N must satisfy the requirement that , where n is an (unrestricted) integer.
While the Miller indices of lattice planes in Ptype lattices must be relatively prime, if the direct lattice is based on a nonprimitive unit cell (any centring type) the Miller indices of some lattice planes are no longer relatively prime (e.g. Nespolo, 2015).
Let us now consider the basic diffraction relations (e.g. Lipson & Cochran, 1966). Suppose a parallel beam of monochromatic radiation, of wavelength , falls on a lattice of identical point scatterers. If it is assumed that the scattering is elastic, i.e. there is no change of the wavelength during this process, the wavevectors of the incident and scattered radiation have the same magnitude, which can conveniently be taken as . A consideration of path and phase differences between the waves outgoing from two point scatterers separated by the lattice vector (defined as above) shows that the condition for their maximum constructive interference is given bywhere and s are the wavevectors of the incident and scattered beams, respectively, and n is an arbitrary integer.
Since , where u, v and w are unrestricted integers, equation (1.1.2.5) is equivalent to the equations of Laue:where is the diffraction vector, and h, k and l are integers corresponding to orders of diffraction from the threedimensional lattice (Lipson & Cochran, 1966). The diffraction vector thus has to satisfy a condition that is analogous to that imposed on the normal to a lattice plane.
The next relevant aspect to be commented on is the Fourier expansion of a function having the periodicity of the crystal lattice. Such functions are e.g. the electron density, the density of nuclear matter and the electrostatic potential in the crystal, which are the operative definitions of crystal structure in Xray, neutron and electrondiffraction methods of crystal structure determination. A Fourier expansion of such a periodic function may be thought of as a superposition of waves (e.g. Buerger, 1959), with wavevectors related to the interplanar spacings , in the crystal lattice. Denoting the wavevector of a Fourier wave by g (a function of hkl), the phase of the Fourier wave at the point r in the crystal is given by , and the triple Fourier series corresponding to the expansion of the periodic function, say G(r), can be written as where C(g) are the amplitudes of the Fourier waves, or Fourier coefficients, which are related to the experimental data. Numerous examples of such expansions appear throughout this volume.
The permissible wavevectors in the above expansion are restricted by the periodicity of the function G(r). Since, by definition, , where is a directlattice vector, the righthand side of (1.1.2.7) must remain unchanged when r is replaced by . This, however, can be true only if the scalar product is an integer.
Each of the above three aspects of crystallography may lead, independently, to a useful introduction of the reciprocal vectors, and there are many examples of this in the literature. It is interesting, however, to consider the representation of the equation which is common to all three, in its most convenient form. Obviously, the vector v which stands for the plane normal, the diffraction vector, and the wavevector in a Fourier expansion, may still be referred to any permissible basis and so may , by an appropriate transformation.
Let , where A, B and C are linearly independent vectors. Equation (1.1.2.8) can then be written as or, in matrix notation, or The simplest representation of equation (1.1.2.8) results when the matrix of scalar products in (1.1.2.11) reduces to a unit matrix. This can be achieved (i) by choosing the basis vectors to be orthonormal to the basis vectors , while requiring that the components of be integers, or (ii) by requiring that the bases and coincide with the same orthonormal basis, i.e. expressing both v and , in (1.1.2.8), in the same Cartesian system. If we choose the first alternative, it is seen that:
It follows that, at least in the present case, algebraic simplicity goes together with ease of interpretation, which certainly accounts for much of the importance of the reciprocal lattice in crystallography. The second alternative of reducing the matrix in (1.1.2.11) to a unit matrix, a transformation of (1.1.2.8) to a Cartesian system, leads to nonintegral components of the vectors, which makes any interpretation of v or much less transparent. However, transformations to Cartesian systems are often very useful in crystallographic computing and will be discussed below (see also Chapters 2.3 and 3.3 in this volume).
We shall, in what follows, abandon all the temporary notation used above and write the reciprocallattice vector as or and denote the directlattice vectors by , as above, or by The representations (1.1.2.13) and (1.1.2.14) are used in the tensoralgebraic formulation of the relationships between mutually reciprocal bases (see Section 1.1.4 below).
We now present a brief derivation and a summary of the most important relationships between the direct and the reciprocal bases. The usual conventions of vector algebra are observed and the results are presented in the conventional crystallographic notation. Equations (1.1.2.1) and (1.1.2.2) now become and respectively, and the relationships are obtained as follows.
It is seen from (1.1.3.1) that must be proportional to the vector product of b and c, and, since , the proportionality constant K equals . The mixed product can be interpreted as the positive volume of the unit cell in the direct lattice only if a, b and c form a righthanded set. If the above condition is fulfilled, we obtain and analogously where V and are the volumes of the unit cells in the associated direct and reciprocal lattices, respectively. Use has been made of the fact that the mixed product, say , remains unchanged under cyclic rearrangement of the vectors that appear in it.
The reciprocal relationship of V and follows readily. We have from equations (1.1.3.2), (1.1.3.3) and (1.1.3.4) If we make use of the vector identity and equations (1.1.3.1) and (1.1.3.2), it is seen that .
The relationships of the angles between the pairs of vectors (b, c), (c, a) and (a, b), respectively, and the angles between the corresponding pairs of reciprocal basis vectors, can be obtained by simple vector algebra. For example, we have from (1.1.3.3):
If we make use of the identity (1.1.3.5), and compare the two expressions for , we readily obtain Similarly, and The expressions for the cosines of the direct angles in terms of those of the reciprocal ones are analogous to (1.1.3.6)–(1.1.3.8). For example,
Various computational and algebraic aspects of mutually reciprocal bases are most conveniently expressed in terms of the metric tensors of these bases. The tensors will be treated in some detail in the next section, and only the definitions of their matrices are given and interpreted below.
Consider the length of the vector . This is given by and can be written in matrix form as where and This is the matrix of the metric tensor of the direct basis, or briefly the direct metric. The corresponding reciprocal metric is given by The matrices G and are of fundamental importance in crystallographic computations and transformations of basis vectors and coordinates from direct to reciprocal space and vice versa. Examples of applications are presented in Part 3 of this volume and in the remaining sections of this chapter.
It can be shown (e.g. Buerger, 1941) that the determinants of G and equal the squared volumes of the direct and reciprocal unit cells, respectively. Thus, and and a direct expansion of the determinants, from (1.1.3.12) and (1.1.3.14), leads to and The following algorithm has been found useful in computational applications of the above relationships to calculations in reciprocal space (e.g. data reduction) and in direct space (e.g. crystal geometry).
The direct and reciprocal sets of unitcell parameters, as well as the corresponding metric tensors, are now available for further calculations.
Explicit relations between direct and reciprocallattice parameters, valid for the various crystal systems, are given in most textbooks on crystallography [see also Chapters 1.1 and 1.2 of Volume C (Koch, 2004)].
The present section summarizes the tensoralgebraic properties of mutually reciprocal sets of basis vectors, which are of importance in the various aspects of crystallography. This is not intended to be a systematic treatment of tensor algebra; for more thorough expositions of the subject the reader is referred to relevant crystallographic texts (e.g. Patterson, 1967; Sands, 1982), and other texts in the physical and mathematical literature that deal with tensor algebra and analysis.
Let us first recall that symbolic vector and matrix notations, in which basis vectors and coordinates do not appear explicitly, are often helpful in qualitative considerations. If, however, an expression has to be evaluated, the various quantities appearing in it must be presented in component form. One of the best ways to achieve a concise presentation of geometrical expressions in component form, while retaining much of their `transparent' symbolic character, is their tensoralgebraic formulation.
We shall adhere to the following conventions:
A familiar concept but a fundamental one in tensor algebra is the transformation of coordinates. For example, suppose that an atomic position vector is referred to two unitcell settings as follows: and Let us multiply both sides of (1.1.4.1) and (1.1.4.2), on the right, by the vectors , m = 1, 2, or 3, i.e. by the reciprocal vectors to the basis . We obtain from (1.1.4.1) where is the Kronecker symbol which equals 1 when and equals zero if , and by comparison with (1.1.4.2) we have where is an element of the required transformation matrix. Of course, the same transformation could have been written as where .
A tensor is a quantity that transforms as the product of coordinates, and the rank of a tensor is the number of transformations involved (Patterson, 1967; Sands, 1982). E.g. the product of two coordinates, as in the above example, transforms from the a′ basis to the a basis as the same transformation law applies to the components of a contravariant tensor of rank two, the components of which are referred to the primed basis and are to be transformed to the unprimed one:
The expression for the scalar product of two vectors, say u and v, depends on the bases to which the vectors are referred. If we admit only the covariant and contravariant bases defined above, we have four possible types of expression:
There are numerous applications of tensor notation in crystallographic calculations, and many of them appear in the various chapters of this volume. We shall therefore present only a few examples.
It happens rather frequently that a vector referred to a given basis has to be reexpressed in terms of another basis, and it is then required to find the relationship between the components (coordinates) of the vector in the two bases. Such situations have already been indicated in the previous section. The purpose of the present section is to give a general method of finding such relationships (transformations), and discuss some simplifications brought about by the use of mutually reciprocal and Cartesian bases. We do not assume anything about the bases, in the general treatment, and hence the tensor formulation of Section 1.1.4 is not appropriate at this stage.
Let and be the given and required representations of the vector r, respectively. Upon the formation of scalar products of equations (1.1.5.1) and (1.1.5.2) with the vectors of the second basis, and employing again the summation convention, we obtain or where and . Similarly, if we choose the basis vectors , l = 1, 2, 3, as the multipliers of (1.1.5.1) and (1.1.5.2), we obtain where and . Rewriting (1.1.5.4) and (1.1.5.5) in symbolic matrix notation, we have leading to and and leading to and
Equations (1.1.5.7) and (1.1.5.9) are symbolic general expressions for the transformation of the coordinates of r from one representation to the other.
In the general case, therefore, we require the matrices of scalar products of the basis vectors, G(12) and G(22) or G(11) and G(21) – depending on whether the basis or , k = 1, 2, 3, was chosen to multiply scalarly equations (1.1.5.1) and (1.1.5.2). Note, however, the following simplifications.

It should be noted that the above transformations do not involve any shift of the origin. Transformations involving such shifts, notably the symmetry transformations of the space group, are treated rather extensively in Volume A of International Tables for Crystallography (2005) [see e.g. Part 5 there (Arnold, 2005)].
This example deals with the construction of a Cartesian system in a crystal with given basis vectors of its direct lattice. We shall also require that the Cartesian system bear a clear relationship to at least one direction in each of the direct and reciprocal lattices of the crystal; this may be useful in interpreting a physical property which has been measured along a given lattice vector or which is associated with a given lattice plane. For a better consistency of notation, the Cartesian components will be denoted as contravariant.
The appropriate version of equations (1.1.5.1) and (1.1.5.2) is now and where the Cartesian basis vectors are: , and , and the vectors and are given by where and , i, k = 1, 2, 3, are arbitrary integers. The vectors and must of course be chosen to be mutually perpendicular, . The axis of the Cartesian system thus coincides with a directlattice vector, and the axis is parallel to a vector in the reciprocal lattice.
Since the basis in (1.1.5.12) is a Cartesian one, the required transformations are given by equations (1.1.5.10) as where , k, i = 1, 2, 3, form the matrix of the scalar products. If we make use of the relationships between covariant and contravariant basis vectors, and the tensor formulation of the vector product, given in Section 1.1.4 above (see also Chapter 3.1 ), we obtain
Note that the other convenient choice, and , interchanges the first two columns of the matrix T in (1.1.5.14) and leads to a change of the signs of the elements in the third column. This can be done by writing instead of , while leaving the rest of unchanged.
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 electrondensity function of the crystal where is the electrondensity 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), and known as the structure factor, may assume nonzero values only if h can be represented as a reciprocallattice vector. Chapter 1.2 is devoted to a discussion of the structure factor of the Bragg reflection, while Chapters 4.1 , 4.2 and 4.3 discuss circumstances under which the scattering need not be confined to the points of the reciprocal lattice only, and may be represented by reciprocalspace vectors with nonintegral components.
The electron density in (1.1.6.1) 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 can be written as and, as such, it can be represented by a threedimensional Fourier series of the form where the periodicity requirement (1.1.6.2) enables one to represent all the g vectors in (1.1.6.3) as vectors in the reciprocal lattice (see also Section 1.1.2 above). If we insert the series (1.1.6.3) in the integrand of (1.1.6.1), 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) reduces to the conventional form 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 . Many more examples of applications of Fourier methods in crystallography are scattered throughout this volume and the crystallographic literature in general.
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 ) and electronic structure of the crystal. One of the earliest, and still most important, theorems of solidstate physics is due to Bloch (1928) and deals with the representation of the wavefunction of an electron which moves in a periodic potential. Bloch's theorem states that:
The eigenstates of the oneelectron Hamiltonian , where U(r) is the crystal potential and for all 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 where and k is the wavevector. The proof of Bloch's theorem can be found in most modern texts on solidstate physics (e.g. Ashcroft & Mermin, 1975). If we combine (1.1.6.5) with (1.1.6.6), an alternative form of the Bloch theorem results: In the important case where the wavefunction is itself periodic, i.e. we must have . Of course, this can be so only if the wavevector k equals times a vector in the reciprocal lattice. It is also seen from equation (1.1.6.7) 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 factor), or to the equivalent polyhedron known as the Brillouin zone (e.g. Ziman, 1969). 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.
Acknowledgements
I wish to thank Professor D. W. J. Cruickshank for bringing to my attention the contribution of M. von Laue (Laue, 1914), who was the first to introduce general reciprocal bases to crystallography.
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