Construction of a reciprocal lattice

The steps in the construction of a reciprocal lattice are given below and are illustrated for a section of a monoclinic unit cell. 

The Miller indices, (h kl), of plane in a lattice or a crystal define the positions where the plane cuts the unit cell edges, oq, and cq. A set of parallel planes has the same Miller indices. Negative intersections are represented by bar h, (h), bar k, (k), or bar l(J). 

The intensity of radiation diffracted from a set of hkl planes depends on the relative phases of the waves from adjacent planes, which in turn depends on the symmetry of the structure and the types of atoms that are in the unit cell. When these waves are completely in step the diffracted beam is intense. When they are completely out of step, the diffracted intensity is zero (see also Section 14.7). The relative intensity of the diffracted radiation from a set of planes is portrayed by the weighted reciprocal lattice. 

The reflections not present because of the symmetry of the structure are termed systematic absences. These are  

---

Figure 2.6 The construction of a reciprocal lattice (a) draw the plane lattice and mark the unit cell (b) draw lines perpendicular to the two sides of the unit cell to give the axial directions of the reciprocal lattice basis vectors (c) determine the perpendicular distances from the origin of the direct lattice to the end faces of the unit cell, dv> and dm, and take the inverse of these distances, 1 /d o and 1 /dm, as the reciprocal lattice axial lengths, a and b (d) mark the lattice points at the appropriate reciprocal distances, and complete the lattice (e) the vector joining the origin of the reciprocal lattice to a lattice point hk is perpendicular to the (hk) planes in the real lattice and of length 1 /dhk...

Figure 2.10 The construction of a reciprocal lattice (a) the a-c section of the unit cell in a monoclinic (mP) direct lattice (b) reciprocal lattice axes lie perpendicular to the end faces of the direct cell (c) reciprocal lattice points are spaced a = 1 /do o and c = 1 /V/0oi (d) the lattice plane is completed by extending the lattice (e) the reciprocal lattice is completed by adding layers above and below the first plane...

Figure SI.3 The construction of a reciprocal lattice draw the unit cell (part a) draw hnes perpendicular to the end faces of the unit cell (part h) to give the axes of the reciprocal lattice (part c) mark lattice points at l/dioo, HdQ 10, and 1/iioo i and fill in the lattice by extending these over the required region (part d)...

Because of the inverse relationship between interatomic distances and the directions in which constructive interference between the scattered electrons occurs, the separation between LEED spots is large when interatomic distances are small and vice versa the LEED pattern has the same form as the so-called reciprocal lattice. This concept plays an important role in the interpretation of diffraction experiments as well as in understanding the electronic or vibrational band structure of solids. In two dimensions the construction of the reciprocal lattice is simple. If a surface lattice is characterized by two base vectors a and a2, the reciprocal lattice follows from the definition of the reciprocal lattice vectors a and a2 ... 

Because P is a reciprocal lattice point, the length of the line OP is ldhkl, where h, k, and 1 are the indices of the set of planes represented by P. (Recall from the construction of the reciprocal lattice that the length of a line from O to a reciprocal-lattice point hkl is Vdhkl). So HOP = dhkl and... 

A crystallographic plane (hkl) is represented as a light spot of constructive interference when the Bragg conditions (Equation 2.3) are satisfied. Such diffraction spots of various crystallographic planes in a crystal form a three-dimensional array that is the reciprocal lattice of crystal. The reciprocal lattice is particularly useful for understanding a diffraction pattern of crystalline solids. Figure 2.7 shows a plane of a reciprocal lattice in which an individual spot (a lattice point) represents crystallographic planes with Miller indices (hkl). 

This construction places a reciprocal lattice point at one end of h. By definition, the other end of h lies on the surface of the sphere. Thus, Bragg s law is only satisfied, when another reciprocal lattice point coincides with the surface of the sphere. Diffraction is emanating from the sample in these directions. To detect the diffracted intensity, one simply moves the detector to the right position. Any vector between two reciprocal lattice points has the potential to produce a Bragg peak. The Ewald sphere construction additionally indicates which of these possible reflections satisfy experimental constraints and are therefore experimentally accessible. 

Sometimes it can be advantageous to construct the reciprocal lattice of the centred rectangular oc-lattice using the primitive unit cell. In this way it will be found that the primitive reciprocal lattice so formed can also be described as a centred rectangular lattice. This is a general feature of reciprocal lattices. Each direct lattice generates a reciprocal lattice of the same type, i.e. mp — mp, oc —> oc, etc. In addition, the reciprocal lattice of a reciprocal lattice is the direct lattice. 

Figure 2.7 The first Brillouin zone of a reciprocal lattice (a) the real lattice and Wigner-Seitz cell (b) the reciprocal lattice and first Brillouin zone. The zone is constructed by drawing the perpendicular bisectors of the lines connecting the origin, 00, to the nearest neighbouring lattice points, in an identical fashion to that used to obtain the Wigner-Seitz cell in real space...

Figure 6.4 Ewald Sphere Construction that demonstrates how the complete X-ray scattering pattern (all vertices of reciprocal lattice) can be visualised by adjusting i) the wavelength (X) of X-ray beam So incident upon a crystal mounted at position M ii) the orientation of the crystal relative to beam Sq. Each vertex of the reciprocal lattice corresponds with a different hW-reflection. A given hW-reflection may be visualised only when scattered beam s cuts the surface of the Ewald sphere at a position P coincident with a corresponding reciprocal lattice vertex. In principle, X and crystal orientation at M may be adjusted to visualise the vast majority of vertices of a reciprocal lattice and hence the vast majority if not all of the hW-reflections possible from a given mounted crystal (Laue Condition).

Figure 4.9 The construction of the reciprocal lattice. ai, U2, as are the vectors of a real crystal lattice, foi, fo2i 3 are the vectors of the reciprocal lattice. Units of the vector magnitudes are m and rad respectively.

The calculation of the diffraction pattern for a periodic system revolves around the construction of the reciprocal lattice and subsequent placement of the first Brillouin zone however, in this case the aperiodicity of the pentagonal array requires a different approach due to the lack of translational symmetry. The reciprocal lattice of such an array is densely filled with reciprocal lattice vectors, with the consequence that the wave vector of a transmitted/reflected light beam encounters many diffraction paths. The resultant replay fields can be accurately calculated by taking the FT of the holograms. To perform the 2D fast Fourier transform (FFT) of the quasi-crystalline nanotube array, a normal scanning electron micrograph was taken, as shown in Fig. 1.13. 

In the case of a periodic solid the vibrational modes become phonons and the dynamical matrix becomes a function of a reciprocal lattice vector k chosen from the Brillouin zone. This means that in constructing D(k) all interactions are multiplied by the phase factor exp(ikrjj), where rp is the interatomic vector. A more detailed discussion of the theory of phonons can be found elsewhere (Dove 1993 Chapter 13 by Kubicki). 

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..

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the... 

In other words, the condition for reflection, in terms of the reciprocal lattice, is this construct a sphere of unit radius having the primary beam along its diameter. Place the origin of the reciprocal lattice at... 

A Bernal chart for a cylindrical camera of any radius may be constructed graphically by drawing the plan and elevation of this model. Thus, if the height of any reciprocal lattice point above the origin is r and its distance from the axis of rotation is r(, the position of the reflection on the film is obtained in the following way. Draw a circle of radius r (Fig. 85 e), and then a chord NUT at a distance r from the centre (this is the circle of contact seen edgewise) UT is the radius of the circle of contact for this reciprocal lattice point. Join OT and produce to W on the line XC which is parallel to OU. WX is then the ordinate y of the spot on the film. Now draw the plan, that is, draw another circle of radius r (Fig. 85/ ) and in it describe a circle of radius UT. On this circle NT mark off the points Pv P2 which are at a distance r from X, and produce UPX to Yx and UP2 to Y2. The arcs XJ and XY2 are the abscissae x of the two reflections on the film produced by this plane. By doing this for a number of different values of r and rg, the complete chart is obtained. 

If a triclinic crystal is rotated round any axis of the real cell (Fig. 93), the photograph exhibits layer lines (since the various levels of the reciprocal lattice are normal to the axis of rotation), but not row lines, since none of the points on upper or lower levels are at the same distance from the axis of rotation as corresponding points on the zero level. The indices for points on the zero level are found in the same way as for photographs of monoclinic crystals rotated round the 6 axis for the zero level of a triclinic crystal rotated round c, a net with elements a, 6, and y is constructed (Fig. 94), and distances of points from the origin are measured. The other levels, projected on to the equator, are displaced with regard to the zero level in a direction which does not lie along an equatorial reciprocal axis the simplest way of measuring values is, as before, to use the zero level network,... 

The whole zero layer of the reciprocal lattice can thus be plotted directly, using the polar coordinates and /. Cartesian coordinates and f are usually more convenient, however these are given by e cos y and / = sin y. It is a simple matter to construct a chart giviug Cartesian coordinates for all positions on the film such a chart )s illustrated ra Plate IX. Transparent copies of such charts can be obtained from the l isfcitute of Physics (47 Beigrave Square, London, S.W. 1). 

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... 

The formalism of the reciprocal lattice and the Ewald construction can be applied to the diffraction at surfaces. As an example, we consider how the diffraction pattern of a LEED experiment (see Fig. 8.21) results from the surface structure. The most simple case is an experiment where the electron beam hits the crystal surface perpendicularly as shown in Fig. A.5. Since we do not have a Laue condition to fulfill in the direction normal to the surface, we get rods vertical to the surface instead of single points. All intersecting points between these rods and the Ewald sphere will lead to diffraction peaks. Therefore, we always observe diffraction... 

Figure A.5 Ewald construction for surface diffraction, a) a side view of the reciprocal lattice at the surface. Constructive interference occurs for all intersection points of the vertical rods with the Ewald sphere. This is equivalent to the condition when the component qj of the scattering vector parallel to the surface is identical to a reciprocal lattice vector of the surface lattice, b) the top view of the reciprocal surface lattice. The circle is the projection of the Ewald sphere. If we disregarding the radiation scattered into the crystal, the number of lattice points within the circle (corresponding to the intersections of the rods with the Ewald sphere) is identical to the maximum number of observed diffraction peaks.

Another way of stating the result is that constructive interference, or diffraction of X rays by electrons, will occur if and only if the two wavevectors k and k (equal in magnitude) differ by a reciprocal lattice vector G so that k + G = k r whence (k + G)2 = fc 2 = fc2, or... 

In order to understand the information contained within the diffraction pattern of a crystal lattice, it is necessary to construct a secondary lattice known as a reciprocal lattice. This lattice is related to the real crystalline array by the following... 

Ewald sphere, sphere of reflection A geometrical construction used for predicting conditions for diffraction by a crystal in terms of its reciprocal lattice rather than its crystal lattice. It is a sphere, of radius 1/A (for a reciprocal lattice with dimensions d = X/d). The diameter of this Ewald sphere lies in the direction of the incident beam. The reciprocal lattice is placed with its origin at the point where the incident beam emerges from the sphere. Whenever a reciprocal lattice point touches the surface of the Ewald sphere, a Bragg reflection with the indices of that reciprocal lattice point will result. Thus, if we know the orientation of the crystal, and hence of its reciprocal lattice, with respect to the incident beam, it is possible to predict which reciprocal lattice points are in the surface of this sphere and hence which planes in the crystal are in a reflecting position. 

Figure 12 The Ewald construction drawn for the reflection (-2 2 0). The crystal is located at the origin O and the endpoint of the vector s lies at a lattice point of the reciprocal lattice (gray). The radius of the circle is A 1.

---