Lattices, reciprocal

The reciprocal lattice, which from a mathematical standpoint is equivalent to the dual space of the direct space, is very often used in X-ray diffraction, since it makes it possible to associate each family of planes with its normal. 

The triplet (a, b, c ) constitutes a basis of this space if the vectors in question are such that they verify the following relations  

The volume V of the direct lattice s primitive cell is given by V = a.(b a c), and likewise V = a. (b AC ). These relations can be used to express the vectors of the reciprocal lattice according to those of the direct lattice, resulting in the following equations  

Moreover, if d(hki) is the interplanar distance between two planes of the (h,k,l) family, then it can be shown that  

The interplanar distance is equal to the inverse of the norm of the corresponding vector in the reciprocal lattice. 

Besides the 2D Bravais lattice (2.3) in radius vector space, it is worthwhile to introduce a lattice in the space of wave vectors parallel to the surface. A basis in such a space can be determined as 

The reciprocal lattice concept can be illustrated by constructing a Fourier series in the space of 2D periodic functions. Let /(r) be a periodic function of a radius-vector in the surface plane, i.e., 

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. 

Conception of the reciprocal lattice is of great importance in studies of crystal lattices. In solid state physics, it is convenient in most applications to define another space that is imambiguously related to the real space. This notion, though being a mathematical abstraction, is useful in many applications. In short, the reciprocal lattice is a lattice defined in a particular way based on the real space. 

A number of physical phenomena lead us to the concept of reciprocal lattice  

There are several possible analytical treatments for introduction of the basic concept of the reciprocal lattice and the reciprocal space. 

The formal relation between a real and a reciprocal lattices is as follows. The reciprocal lattice is a set of imaginary points constructed in such a way that the direction of vectors in this lattice from one point to another coincides with the direction of normals to planes in the real crystal lattice. The magnitude of the reciprocal lattice vector g is equal to the reciprocal of the interplanar spacing 1/d in the real lattice multipUed by 2jt. 

The crystal planes hkl in the real crystal lattice define the coordinates of points of the reciprocal lattice space, also called fe-space. A plane in the real-space maps to a point in the reciprocal space and on the contrary, so there is one-to-one correspondence between planes in the real space and points in the reciprocal space. 

A unit cell in the reciprocal lattice is described by the vectors a, If, c, which are defined as follows [2,3,5,6]  

This means that a is perpendicular to both b and c, If is perpendicular to both a and c, and c is perpendicular to both b and a. 

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

since the Miller indices of a plane implies that the plane intercepts the base vectors at the point j-, j,y, a triangular portion of the plane has sides 

Consequently, the vector Gm = Gm is perpendicular to the plane (hkl). Then, it is possible to calculate Gw L that is, the vector modulus. To perform this calculation, we must define the unit vector in the direction of the vector Gm as follows  

The concept of the reciprocal lattice is very useful in discussing the diffraction of x-rays and neutrons from crystalline materials, especially in conjunction with the Ewald sphere construction discussed in Section 1.5.3. The regular arrangement of atoms and atomic groupings in a crystal can be described in terms of the crystal lattice, which is uniquely specified by giving the three unit cell vectors a, b9 and c. It turns out that the diffraction from a crystal is similarly associated with a lattice in reciprocal space. The reciprocal lattice is specified by means of the three unit cell vectors a, b, and c in the same way as the crystal lattice is based on a9 b, and c. In fact, the crystal lattice and the reciprocal lattice are related to each other by the Fourier transform relationship. 

It is often easier and more elegant to express the conditions for diffraction in terms of a mathematical transformation known as the reciprocal lattice. The reciprocal lattice vectors a, b, c are defined in terms of the real lattice vectors a, b, c by the relations  

Two properties of the reciprocal lattice make it of value in diffraction theory. We state these without proof. 

1 The vector r (hkl) from the origin to the point h, k, I of the reciprocal lattice is normal to the plane (hkl) of the real lattice. 

Most readers are probably familiar with the Bragg formula for x-ray diffraction (XRD), 2d sin 6 = nX, and the selection rules that tell us that the sum of the Miller indices for body-centered cubic (bcc) crystals must be even and the indices for face-centered cubic (fee) crystals must be all even or all odd in order to produce a diffraction peak. The intent of this chapter is to give a more general derivation of the Laue conditions leading to Bragg reflections of electrons and phonons that take place in three-dimensional (3D) crystals. These reflections from lattice planes within the crystal are fimdamental to the imderstand-ing of the thermal and electronic properties of materials. In the process, the reciprocal lattice will be introduced which will be widely used in subsequent chapters to describe how materials behave thermally, electronically, magnetically, and photonically. 

Much of solid-state physics is carried out in reciprocal space, sometimes called Fourier space. The reasons for this will become more apparent as we progress. One of the more obvious reasons is that it is easier to represent the momentum of photons and phonons as well as particles such as electrons and neutrons and thus their interactions in reciprocal space. Recall that the momentum of either a massless particle such as a photon or a particle with mass can be written as 

All products in Eqs. 1.12 and 1.13 are scalar (or dot) products. The dot product of the two vectors, Vj and Vz is defined as a scalar quantity, which is equal to the product of the absolute values of the two vectors and the cosine of the angle a between them  

Considering Eqs. 1.12 to 1.15, it is possible to show that the elementary translations in the reciprocal lattice are defined as 

In Eqs. 1.16 and 1.17, the two scalar quantities V and F are the volumes of the unit cell in the direct and reciprocal lattices, respectively. Hence, a is perpendicular to both b and c b is perpendicular to both a and c and c is perpendicular to both a and b. In terms of the interplanar distances, d is perpendicular to the corresponding crystallographic planes, and its length is inversely proportional to d, i.e. 

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 

Now we consider periodic crystalline structures. The simplest case is one-dimensional structure realized, for instance, in the smectic A phase, see Fig. 5.8a the density is periodic along x with period a, and wavevector q = q = h(2nla), /t is an integer. Then the density function can be written as 

As t iqjta) = 1 only for q = 2n a (otherwise it equals 0) the same equation may be rewritten as 

Therefore, f) is a set of the 5-like peaks on the (y-scale separated by distances 2%ja These peaks form a one-dimensional reciprocal lattice with basic vector 2tt/a, shown in Fig. 5.8b. 

In the three-dimensional-lattice, there are three basic vectors a, b, and c, Fig. 5.9a, and we can introduce a concept of the reciprocal three-dimensional lattice. It is a lattice in the wavevector space having the dimension of inverse length for each coordinate in the inverse space. Such a lattice may be built by translations of the elementary cell shown in Fig. 5.9b. The basic vectors of the reciprocal lattice are a, b, c and the vector of the reciprocal lattice is given by 

When the crystal is irradiated by an X-ray beam, its lattice scatters the radiation selectively. A strong diffraction is observed when the wavevector of scattering for a particular angle (i.e. q) coincides with the vector of reciprocal lattice, as shown in the Ewald sphere. Fig. 5.10. The condition 

The main characteristic of a crystalline arrangement consists in the existence of crystallographic planes, which, implicitly collect de periodicity of engagement of structural dots (atoms and assemblies of atoms) in all the directions in which the crystal expands the present discussion follows (Putz, 2006). 

Miller notations gave the indices of equivalent crystallographic planes (hkl), belonging to the same family of planes, that have in common the 

the characterization of crystallographic planes and of the families of such planes by the normal directions to them generated the possibility of representation of all the planes families by the projections of the normal by stereographic projection method. 

stereographic projection, also efficient for establish the orientations of families of planes, can not give indications towards the interplanar specific distance to that family. 

This way, a new type of crystallographic projection should be introduced in relation to crystallographic planes, families of planes, normal directions but also to the interplanar distance. 

Let us consider two parallel crystallographic planes defined by the Miller indices hkl having their planes perpendicular to the diagram (Fig. 6.6) and X-ray having definite wavelength, A is incident at a particular angle. 

The concept and the applicability of reciprocal lattice are very important in structural analysis of crystaUine materials. The importance of this hypothetical lattice can be best understood as we move through the following chapters. 

It can be rightly said that The reciprocal lattice is as important in crystal structure analysis as the walking stick of a blind man moving in a narrow lane having frequent turns. It is extremely difficult if not impossible to picture the different intersecting crystal planes satisfying the Bragg s reflection in three-dimensional lattice from the two-dimensional array of spots or lines. 

as the dot product of a with b and c vanishes, therefore, a is perpendicular to both of them, i.e., perpendicular to the plane containing b and c in the direct lattice. This gives the direction of a and from the relation a a = 1, we can have the magnitude of it. Similarly, b is perpendicular to the plane containing c, and a and c are perpendicular to the plane containing a and b. So, as a, b, and c the three vectors define the actual lattice (called direct lattice ), the three vectors introduced as a, b, and c will also define another lattice. The relations between these vectors can be derived as [4] 

as a b X c is the volmne, V of the unit cell of the direct lattice, the above relations can be written as 

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Expressing (k) is complicated by the fact that k is not unique. In the Kronig-Penney model, if one replaced k by k + lTil a + b), the energy remained unchanged. In tluee dimensions k is known only to within a reciprocal lattice vector, G. One can define a set of reciprocal vectors, given by... 

Reciprocal lattice vectors are usefiil in defining periodic fimctions. For example, the valence charge density, p (r), can be expressed as... 

Figure Bl.8.2. Bragg s law. Wlien X = 2d sin 0, there is strong, constructive interference. (B) THE RECIPROCAL LATTICE...

Figure Bl.8.3. Ewald s reciprocal lattice construction for the solution of the Bragg equation. If Sj-s. is a vector of the reciprocal lattice, Bragg s law is satisfied for the corresponding planes. This occurs if a reciprocal lattice point lies on the surface of a sphere with radius 1/X whose centre is at -s..

The amplitude and therefore the intensity, of the scattered radiation is detennined by extending the Fourier transfomi of equation (B 1.8.11 over the entire crystal and Bragg s law expresses die fact that this transfomi has values significantly different from zero only at the nodes of the reciprocal lattice. The amplitude varies, however, from node to node, depending on the transfomi of the contents of the unit cell. This leads to an expression for the structure amplitude, denoted by F(hld), of the fomi... 

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.

Figure Bl.21.4. Direct lattices (at left) and reciprocal lattices (middle) for the five two-dimensional Bravais lattices. The reciprocal lattice corresponds directly to the diffraction pattern observed on a standard LEED display. Note that other choices of unit cells are possible e.g., for hexagonal lattices, one often chooses vectors a and b that are subtended by an angle y of 120° rather than 60°. Then the reciprocal unit cell vectors also change in the hexagonal case, the angle between a and b becomes 60° rather than 120°.

Alternatively, the electron can exchange parallel momentum with the lattice, but only in well defined amounts given by vectors that belong to the reciprocal lattice of the surface. That is, the vector is a linear combination of two reciprocal lattice vectors a and b, with integer coefficients. Thus, g = ha + kb, with arbitrary integers h and k (note that all the vectors a,b, a, b and g are parallel to the surface). The reciprocal lattice vectors a and are related to tire direct-space lattice vectors a and b through the following non-transparent definitions, which also use a vector n that is perpendicular to the surface plane, as well as vectorial dot and cross products ... 

The reciprocal lattices shown in figure B 1.21.3 and figure B 1.21.4 correspond directly to the diffraction patterns observed in FEED experiments each reciprocal-lattice vector produces one and only one diffraction spot on the FEED display. It is very convenient that the hemispherical geometry of the typical FEED screen images the reciprocal lattice without distortion for instance, for the square lattice one observes a simple square array of spots on the FEED display. 

One of the spots in such a diffraction pattern represents the specularly reflected beam, usually labelled (00). Each other spot corresponds to another reciprocal-lattice vector = ha + kb and is thus labelled (hk), witli integer h and k. 

Anuther concept that is extremely powerful when considering lattice structures is the fi i i/imca/ lattice. X-ray crystallographers use a reciprocal lattice defined by three vectors a, b and c in which a is perpendicular to b and c and is scaled so that the scalar juoduct of a and a equals 1. b and c are similarly defined. In three dimensions this leads to the following definitions ... 

Note that the denominator in each case is equal to the volume of the unit cell. The fact that a, b and c have the units of 1/length gives rise to the terms reciprocal space and reciprocal latlice. It turns out to be convenient for our computations to work with an expanded reciprocal space that is defined by three closely related vectors a , b and c, which are multiples by 2tt. of the X-ray crystallographic reciprocal lattice vectors ... 

Variation in energy for a tour (r-X-M-F) of the reciprocal lattice for a 2D square lattice of hydrogen 2S. (Figure adapted in part from Hoffmann R 1988. Solids and Surfaces A Chemist s View on Bonding in nded Structures. New York, VCH Publishers.)... 

Plane waves are often considered the most obvious basis set to use for calculations on periodic sy stems, not least because this representation is equivalent to a Fourier series, which itself is the natural language of periodic fimctions. Each orbital wavefimction is expressed as a linear combination of plane waves which differ by reciprocal lattice vectors ... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

Pounds per square inch psi Reciprocal lattice vector (cir- G... 

Figure 4 Schematic vector diagrams illustrating the use of coherent inelastic neutron scattering to determine phonon dispersion relationships, (a) Scattering m real space (h) a scattering triangle illustrating the momentum transfer, Q, of the neutrons in relation to the reciprocal lattice vector of the sample t and the phonon wave vector, q. Heavy dots represent Bragg reflections.

The vibrational excitations have a wave vector q that is measured from a Brillouin zone center (Bragg peak) located at t, a reciprocal lattice vector. 

Diffraction is usefiil whenever there is a distinct phase relationship between scattering units. The greater the order, the better defined are the diffraction features. For example, the reciprocal lattice of a 3D crystal is a set of points, because three Laue conditions have to be exactly satisfied. The diffraction pattern is a set of sharp spots. If disorder is introduced into the structure, the spots broaden and weaken. Two-dimensional structures give diffraction rods, because only two Laue conditions have to be satisfied. The diffraction pattern is again a set of sharp spots, because the Ewald sphere cuts these rods at precise places. Disorder in the plane broadens the rods and, hence, the diffraction spots in x and y. The existence of streaks, broad spots, and additional diffuse intensity in the pattern is a common... 

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