Reciprocal lattice point

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..

In crystallography, the difiiraction of the individual atoms within the crystal interacts with the diffracted waves from the crystal, or reciprocal lattice. This lattice represents all the points in the crystal (x,y,z) as points in the reciprocal lattice (h,k,l). The result is that a crystal gives a diffraction pattern only at the lattice points of the crystal (actually the reciprocal lattice points) (O Figure 22-2). The positions of the spots or reflections on the image are determined hy the dimensions of the crystal lattice. The intensity of each spot is determined hy the nature and arrangement of the atoms with the smallest unit, the unit cell. Every diffracted beam that results in a reflection is made up of beams diffracted from all the atoms within the unit cell, and the intensity of each spot can be calculated from the sum of all the waves diffracted from all the atoms. Therefore, the intensity of each reflection contains information about the entire atomic structure within the unit cell. 

Figure 10. Distribution of reciprocal lattice points of a plate texture along straight lines parallel to the texture axis and perpendicular to the face lying on the support (a) and distribution of circular scattering regions of the reciprocal lattice of a texture on coaxial cylinders.

Since the variation of any physical property in a three dimensional crystal is a periodic function of the three space coordinates, it can be expanded into a Fourier series and the determination of the structure is equivalent to the determination of the complex Fourier coefficients. The coefficients are indexed with the vectors of the reciprocal lattice (one-to-one relationship). In principle the expansion contains an infinite number of coefficients. However, the series is convergent and determination of more and more coefficients (corresponding to all reciprocal lattice points within a sphere, whose radius is given by the length of a reciprocal lattice vector) results in a determination of the stmcture with better and better spatial resolution. Both the amplitude and the phase of the complex number must be determined for any Fourier coefficient. The amplitudes are determined from diffraction... 

The end of each such vector, starting from the origin, is a reciprocal lattice point (sometimes abbreviated relp ). 

The number of units in an MQW will be much more limited than the number of atomic planes sampled by the X-ray beam in a standard reflection. The intensity will be low, but also the MQW will behave as a thin crystal —the reciprocal lattice points will be extended into rods perpendicular to the crystal surface. This will broaden the reflection, and thus the width of each satellite peak is determined by the number of units in the MQW. It has even been possible, by... 

Figure 7.8 A scattering map in reciprocal space. Equal intensity contours are shown schematically, and the Ewald sphere is represented as a plane near reciprocal lattice points 0 and h. The dynamical diffraction from the specimen is displaced slightly from the relp and from the centre of the diffuse scatter by the refractive index effect...

Figure 7.15 Reciprocal space maps of GaAs with graded InGaAs buffer layer around the 224 reciprocal lattice point, (a) [110] direction, (b) [ 10] direction. The 004 maps in those directions are shown in the insets...

As will become apparent, it is important to place the photographic plate as close to the specimen as possible. With rotating anode generators, care should be taken not to allow the full power of the beam to fall on the plate when stationary as this leads to an unsightly overexposed vertical line on the topograph. The presence of a horizontal stripe on the recorded topograph is often due to the presence of a second reciprocal lattice point lying on the Ewald sphere. It can be removed by a small rotation of the crystal about the diffraction vector as if to take a stereo pair. 

We see that very close to the Bragg condition, where the dispersion strrface is highly cttrved, R K and the crystal acts as a powerful angrtlar amplifier. A reaches 3.5xl0 in the centre of the dispersion surface for sihcon in the 220 reflection with MoK radiation. Far from the centre, the dispersion strrface becomes asymptotic to the spheres about the reciprocal lattice points and A approaches unity. Thus when the whole of the dispersion strrface is excited by a spherical wave, owing to the amplification close to the Bragg condition, the density of wavelields will be veiy low in the centre of the Borrmann fan and... 

It is clear from Eq. (2.2b) that the frequency to in Eq. (2.7) is a function of q, because q governs the relative displacement of two interacting atoms. The co(q) dependence on q (the dispersion relationships) is illustrated in Fig. 2.1 for the rock-salt structure. It can be shown that all normal modes can be represented in the first Brillouin zone, which extends from 0 to nja in the a direction of the rock-salt structure, or, more generally, is bounded by faces located halfway between the reciprocal lattice points in the space defined by1 = 27r<5fJ-. The... 

We consider the behavior of the origin term when the continuous variable S (= H at the reciprocal lattice points) approaches zero ... 

Fig. 80. Formation of reciprocal lattice rotation diagram, set of planes is the corresponding reciprocal lattice point V, the distance of which from the reciprocal lattice origin X is inversely proportional to...

Fig. 81. The condition for reflection in terms of the reciprocal lattice. Keflection occurs when a reciprocal lattice point P touches the surface of the sphere of reflection.

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. 

---