The Direct Lattice

The crystallographer s view of a crystal starts from the definition of a lattice A lattice is a collection of points repeated at intervals of length a, az and 3 along three non-coplanar directions, indefinitely. The three constants ai, az, and a are called lattice parameters, and the vectors ai, ai, and as, oriented in the same three non-coplanar directions with the lattice parameters 

A vector g joining any two lattice points is a lattice vector. Every lattice vector can always be expressed by the basis vectors and three integer coefficients i, ni, and n  

Basis vectors ai, and 33 define a parallelepiped called the unit cell, which is primitive, because it contains one lattice point. All cells that are obtained by translation of this unit cell, the origin cell, through the application of all vectors g in Eq. [1], fill the space completely. Then, the entire lattice can be subdivided into cells and every vector g can be used to label a cell with respect to the origin cell, or 0-cell. Actually, the definition of a unit cell is arbitrary, and many (an infinite number) different possible choices exist, because all cells containing the same number of lattice points are equivalent. The actual shape of a unit cell depends on the lattice type. 

In summary, specifying the geometry of a crystal requires the following information  

The position r of an atom in the unit cell is usually not expressed in terms of Cartesian coordinates, but in terms of fractional coordinates x, X2, x such that 

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Figure Bl.21.3. Direct lattices (at left) and corresponding reciprocal lattices (at right) of a series of connnonly occurring two-dimensional superlattices. Black circles correspond to the ideal (1 x 1) surface structure, while grey circles represent adatoms in the direct lattice (arbitrarily placed in hollow positions) and open diamonds represent fractional-order beams m the reciprocal space. Unit cells in direct space and in reciprocal space are outlined.

The traditional eharaeterisation of an electron density in a crystal amounts to a statement that the density is invariant under all operations of the space group of the crystal. The standard notation for sueh an operation is (Rim), where R stands for the point group part (rotations, reflections, inversion and combinations of these) and the direct lattice vector m denotes the translational part. When such an operation works on a vector r we get... 

A Bloch function for a crystal has two characteristics. It is labeled by a wave vector k in the first Brillouin zone, and it can be written as a product of a plane wave with that particular wave vector and a function with the "little" period of the direct lattice. Its counterpart in momentum space vanishes except when the argument p equals k plus a reciprocal lattice vector. For quasicrystals and incommensurately modulated crystals the reciprocal lattice is in a certain sense replaced by the D-dimensional lattice L spanned by the vectors It is conceivable that what corresponds to Bloch functions in momentum space will be non vanishing only when the momentum p equals k plus a vector of the lattice L. 

A is the volume of the unit cell in the direct lattice of the crystal The range of integration is restricted to the first Brillouin zone of the crystal, and the volume of the zone is (27t)3/A. 

In this expression SA is summed over the direct lattice, R( SB is summed over the reciprocal lattice, b,. The parameter r is chosen so as to obtain equally rapid convergence in the sums over the direct and reciprocal lattices. The 6 functions are defined by... 

The geometrical aspect concerns the position of the diffracted beams on a pattern it only depends on the direct lattice of the crystal through the Bragg law =2dhkisin9B - dhu being the interplanar distance of the diffracted (hkl) lattice planes and 0b the Bragg angle. In other words, it only depends on the lattice parameters of the crystal a, b, c, a, P and y. 

The direct lattice and reciprocal lattice unit cells are marked on the crystal pattern of a planar hexagonal net in Figure 16.10, using eqs. (13), (14), and (18). The scales chosen for... 

We now describe a general method for the construction of the BZ. It is a consequence of the SP relation eqs. (7)—(9) that every reciprocal lattice vector b , is normal to a set of planes in the direct lattice. In Figure 16.11(a), bm is a reciprocal lattice vector that connects lattice point O to some other lattice point P. Let 1 be the plane through Pi that is normal to b , and let 0 be the plane parallel to 1 through O. Let a be the lattice vector from O to some... 

Similar to the direct lattice, all the possible points that lie at the reciprocal lattice can be represented as follows ... 

The Direct Lattice Sum. Dispersion forces between two atoms can be described by a potential function expressed in terms containing inverse powers of the internuclear separations, s. The simplest function of this sort includes a potential energy of attraction proportional to the inverse sixth power of the separation and a repulsion that is zero at distances of separation greater than a particular value se and infinite at separations less than sc. This is the so-called hard sphere or van der Waals model. Such an approximate potential function can be improved in two respects. Investigations of the second virial coefficient have revealed that the potential energy of repulsion is best described as proportional to the inverse twelfth power of the separation and the term in sr9, which accounts for the greater part of the total attraction potential, due to the attraction of mutually induced dipoles, should have added to it the dipole-quadrupole and quadrupole-quadru-pole attractions, expressed as terms in sr8 and s-10, respectively. The complete potential function for the forces between two atoms is, therefore ... 

The concept, incorporated in this approximation, that matter is distributed continuously in each layer plane, while it might not be permissible for a localized adsorbed film, is actually close to the truth for a mobile adsorbed film, in which the rapid translation of the molecules along the surface prevents their responding to its fine structure. This approximatioi i produces just the sort of average that is desired and is so difficult to obtain from the direct lattice sum. 

Note also that in a triclinic crystal a and a are not collinear in a monoclinic crystal (b unique setting) b is parallel to b, but a and c form the obtuse angle [>, while a and c form a smaller acute angle /T given by fJ = 180 — fi. The reciprocal lattice vectors and the direct lattice vectors are a ying-yang duo of concepts, as are position space and momentum space, or space domain and time domain. Fourier transformation helps us walk across from one space to other, as convenience dictates Some problems are easy in one space, others in the space dual to it this amphoterism is frequent in physics. The directions of the direct and reciprocal lattice vectors are shown as face normals in Fig. 7.22. 

In calculating the interplanar spacing, or perpendicular distance between adjacent planes of given indices, dku, in the direct lattice (whether or not these planes coincide with lattice points), it is helpful to consider the reciprocal lattice, which defines a crystal in terms of the vectors that are the normals to sets of planes in the direct lattice and whose lengths are the inverse of dku- The relationship between the interplanar spacing and the magnitude of the reciprocal lattice vectors, a, b, c, is given by ... 

Derive the expression for in terms of the direct lattice, for each of the crystal systems with orthogonal axes. 

These and other calcnlations can be greatly simplified, using the concept of the reciprocal space (which has the dimension of reciprocal length, A ) and of reciprocal crystal lattice in this space. The reciprocal lattice vectors, a, b, and c, are related to those of the direct lattice (a, b, and c) by the relationships... 

A set of these vectors is usually called the direct lattice. For a given lattice, its reciprocal lattice is defined as a set of vectors K satisfying the relationship... 

Use of the reciprocal lattice unites and simplifies crystallographic calcnlations. The motivation for the reciprocal lattice is that the x-ray pattern can be interpreted as the reciprocal lattice with the x-ray diffraction intensities superimposed on it. See Section 14.2 for the definition of the reciprocal lattice vectors a b and c in terms of the direct basis vectors a, b, and c. Table 14.2 shows the parallel between the properties of the direct lattice and the reciprocal lattice, and Table 14.3 relates the direct and reciprocal lattices. 

Rows of points (zone axes) in the direct lattice are perpendicular to nets (planes) of the reciprocal lattice, and vice versa. The repeat distance between points in a particular row of the reciprocal lattice is inversely proportional to the interplanar spacing between the nets of the crystal lattice that are normal to this row of points (d = A/d). 

If the electron density is known correctly, then structure factors and their relative phases can be computed by Fourier transform techniques. The calculation of X-ray scattering factors from the computed orbital electron densities as a function of distance from the nucleus, shown in Figure 6.19, provides an example of this. In a crystal structure analysis it is possible, from the measured diffraction pattern (structure factors and their phases) to compute the Fourier transform and thereby obtain an image of the entire crystal structure. In practice, only the contents of one unit cell are computed because the reciprocal lattice is the Fourier transform of the direct lattice and vice versa, so that the two transforms can be multiplied (Figure 6.17). 

The concept of a reciprocal lattice was first introduced by Ewald and it quickly became an important tool in the illustrating and understanding of both the diffraction geometry and relevant mathematical relationships. Let a, b and c be the elementary translations in a three-dimensional lattice (called here a direct lattice), as shown for example in Figure 1.4. A second lattice, reciprocal to the direct lattice, is defined by three elementary translations a"", b and c, which simultaneously satisfy the following two conditions ... 

An important consequence of Eq. 1.18 is that a set, which consists of an infinite number of crystallographic planes in the direct lattice, is replaced by a single vector or by a point at the end of the vector in the reciprocal lattice. Furthermore, Eqs. 1.16 and 1.17 can be simplified in the orthogonal crystal systems to... 

The two-dimensional example illustrating the relationships between the direct and reciprocal lattices (or spaces), which are used to represent crystal structures and diffraction patterns, respectively, is shown in Figure 1.40. Pin important property of the reciprocal lattice is that its symmetry is the same as the symmetry of the direct lattice. However, in the direct space atoms can be located anywhere in the unit cell, whereas diffraction peaks are represented only by the points of the reciprocal lattice, and the unit cells themselves are "empty" in the reciprocal space. Furthermore, the contents of every unit cell in the direct space is the same, but the intensity of diffraction peaks, which are conveniently represented using points in the reciprocal space, varies. 

Figure 1.40. Example of converting crystallographic planes in the direct lattice into points in the reciprocal lattice. The corresponding Miller indices are shown near the points in the reciprocal lattice.

For example, in one-dimension the direct lattice is na and the reciprocal lattice is (2tz/a)n (n = 0, 1, 2,...). The P irst Brillouin zone is a cell in the reciprocal lattice that encloses points closer to the origin (zii, n2,nj, = 0) than to any other lattice point. Obviously, for a one-dimensional lattice the first Brilloin zone is —(n/a .. (tt/a). 

Problem 4.3. Show that in a three-dimensional lattice the number of distinctly different k vectors is N1N2N3. Since these vectors can all be mapped into the first Brillouin zone whose volume is bi (b2 x bj) = (27r)fyvv where vv = ai (32 X 33) is the volume of the primitive unit cell of the direct lattice, we can infer that per unit volume of the reciprocal lattice there are M N2N3 / [ (2n ) /w] = wNiN2N3/(2n ) = Q/(27r) states, where Q = L L2L3 is the system volume. Show that this implies that the density (in Z -space) of allowed k states is 1 / (27r ) per unit system volume, same result as for free particle. 

A crystalline solid can be described by three vectors a, b and c, so that the crystal structure remains invariant under translation through any vector that is the sum of integral multiples of these vectors. Accordingly, the direct lattice sites can be defined by the set... 

According to the definitions given by Eqs. (1.1) to (1.3), the product G R = 2jrx integer. Therefore each vector of the reciprocal lattice is normal to a set of planes in the direct lattice, and the volume of a unit cell of the reciprocal lattice is related to the volume of the direct lattice I4 by... 

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