Reciprocal lattices, Bragg equation

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... 

The two-dimensional Bragg condition leads to the definition of reciprocal lattice vectors at and aj which fulfil the set of equations ... 

The notion of a reciprocal lattice cirose from E vald who used a sphere to represent how the x-rays interact with any given lattice plane in three dimensioned space. He employed what is now called the Ewald Sphere to show how reciprocal space could be utilized to represent diffractions of x-rays by lattice planes. E vald originally rewrote the Bragg equation as ... 

We should first correct the wavevector inside the crystal for the mean refractive index, by multiplying the wavevectors by the mean refractive index (1 + IT). This expression is derived from classical dispersion theory. Equation (4. 18) shows us that is negative, so the wavevector inside the crystal is shorter than that in vacuum (by a few parts in 10 ), in contrast to the behaviom of electrons or optical light. The locus of wavevectors that have this corrected value of k lie on spheres centred on the origin of the reciprocal lattice and at the end of the vector h, as shown in Figure 4.11 (only the circular sections of the spheres are seen in two dimensions). The spheres are in effect the kinematic dispersion surface, and indeed are perfectly correct when the wavevectors are far from the Bragg condition, since if D 0 then the deviation parameter y, 0 from... 

The process of reflection by the real lattice cannot be visualized in terms of the reciprocal lattice but the condition for reflection by the real lattice (the Bragg equation) naturally has its jjrecise geometrical equivalent in terms of the reciprocal lattice. This is illustrated in Fig. 81, in which X Y represents the orientation of a set of crystal planes which we will suppose is in a reflecting position. Along the normal to this... 

If, as in Fig. 108 a, the c axis of the crystal is displaced from the axis of rotation in the plane normal to the beam (for the mean position of the crystal), the zero layer (hk0) of the reciprocal lattice is tilted in this same direction, and its plane cuts the sphere of reflection in the circle AD. During the 15° oscillation a number of hkO points pass through the surface of the sphere, and thus X-rays reflected by these hkO planes of the crystal strike the film at corresponding points on the flattened-out film (Fig. 108 b) the spots fall on a curve BAD, whose distance from the equator is a maximum at a Bragg angle 0 = 45° and zero at 6 = 90°. If, on the other hand, the displacement of the c axis is in the plane containing the beam (Fig. 108 c), the spots on the film fall on a curve whose maximum distance from the equator is at 6 = 90° (Fig. 108 d). When the displacement of the c axis has components in both directions,... 

If one examines the series in Equation (2.9) it will be seen that the terms will tend to reinforce whenever Q=2vl/d, where / denotes an integer. These values of Q define a one-dimensional reciprocal lattice and whenever Q takes on one of these values the diffracted waves will reinforce. These values of Q correspond to the familiar Bragg reflections given by 2d sin 0 =/A. It should be noted that for finite values of N the reciprocal lattice points have a finite width. 

The direction of a HOLZ line is normal to the reciprocal lattice vector and its position is decided by the Bragg condition. In diffraction analysis, it is useful to express HOLZ lines using line equations in an orthogonal zone-axis coordinate system (x, y, z), with z parallel to the zone-axis direction. The x direction can be taken along the horizontal direction of the experimental pattern and y is normal to x. The Bragg diffraction equation (2) expressed in this coordinate is given by... 

The simple geometrical arrangement of the reciprocal lattice, Ewald s sphere, and three vectors (ko, ki, and d hki) in a straightforward and elegant fashion yields Braggs equation. From both Figure 2.27 and Figure 2.28, it is clear that vector ki is a sum of two vectors, ko and d hki ... 

Since every Kai/Ka2 double peak is caused by scattering from a single reciprocal lattice point, the rf-spacing remains constant and the scattered intensity is proportional to the intensities of the two components in the characteristic spectrum. Using Braggs law the following equation reflects the relationship between the positions of the diffraction peaks in the doublet ... 

Remember from Chapter 4 that the periods and frequencies of waves are reciprocally related.) Exactly those properties are expressed by their reciprocal lattice vectors h. The amplitudes of these electron density waves vary according to the distribution of atoms about the planes. Although the electron density waves in the crystal cannot be observed directly, radiation diffracted by the planes (the Fourier transforms of the electron density waves) can. Thus, while we cannot recombine directly the spectral components of the electron density in real space, the Bragg planes, we can Fourier transform the scattering functions of the planes, the Fhki, and simultaneously combine them in such a way that the end result is the same, the electron density in the unit cell. In other words, each Fhki in reciprocal, or diffraction space is the Fourier transform of one family of planes, hkl. With the electron density equation, we both add these individual Fourier transforms together in reciprocal space, and simultaneously Fourier transform the result of that summation back into real space to create the electron density. 

Let us return, for a moment, to Figure 10, the Bragg s law description of X-ray diffraction. X-rays are reflected by planes of lattice points, uniquely described by the three indices h, k, l. These three indices form the basis of another lattice, which we called the reciprocal lattice, where the distance from the origin to each point hkl was 1 /dhu, where dm was the distance between the Bragg planes. Each Bragg plane can be defined by its normal, which turns out to be a multiple of the reciprocal space basis vectors a, b, c. We can then refer to this plane, as well as to the Fourier term associated with it, by a reciprocal lattice vector d / = (ha + kb + lc ). Rewriting Equation (9) in terms of electron density, we get... 

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). 

To derive the Bragg equation, we used an assumption of specular reflection, which is borne out by experiment. For crystalline materials, destructive interference completely destroys intensity in all directions except where Equation (5) holds. This is no longer true for disordered materials where diffracted intensity can be observed in all directions away from reciprocal lattice points, known as diffuse scattering, as discussed in Chapter 16. 

Comparing Equation (32) with Equation (13) proves the identity of Ahu and the reciprocal lattice vectorh Bragg s equation, Equation (26), can be restated as ... 

The Bragg equation shows that diffraction occurs when the scattering vector equals a reciprocal lattice vector. The scattering vector depends on the geometry of the experiment whereas the reciprocal lattice is determined by the orientation and the lattice parameters of the crystalline sample. Ewald s construction combines these two concepts in an intuitive way. A sphere of radius 1//1 is constructed and positioned in such a way that the Bragg equation is satisfied, and diffraction occurs, whenever a reciprocal lattice point coincides with the surface of the sphere (Figure 1.8). 

As was shown in Equation (34) of Chapter 1, Bragg s law dictates that the scattering vectors for a Bragg peak, h, correspond to these reciprocal lattice vectors. The three-dimensionality of the diffraction pattern makes the identification of the three vectors a, b, c, straightforward, from which the direct space unit cell vectors ... 

Diffraction occurs when a reciprocal lattice point passes through the Ewald sphere and satisfies the Bragg law (Equation 1). The measured intensity, l(hkl), of a diffracted X-ray beam can be calculated for a crystal rotating with a uniform angular velocity, to, through a reflecting position... 

GEOMETRY BRAGG S LAW, THE LAUE EQUATIONS THE RECIPROCAL LATTICE AND THE EWALD SPHERE... 

The wave-vector Bragg equation describes diffraction phenomenon and can be derived using the reciprocal lattice approach as shown in the next section. 

A construction due to Ewald illustrates the importance of the reciprocal lattice in X-ray crystallography. As Figure 4 shows, the Bragg equation is satisfied where the reflection sphere is cut by a lattice point of the reciprocal lattice constructed around the center of the cry.stal. Rotating the crystal together with the reciprocal lattice around a few different directions in the crystal fulfills the reflection condition for all points of the reciprocal space within the limiting sphere. The reciprocal lattice vector S is perpendicular to the set of net planes, and has the absolute magnitude 1/d. In vector notation ... 

In the case that the incident direction is chosen such as only a single eigen wave vector of the reciprocal lattice can be found on the Ewald s reflection sphere (Figure 5.1), so that the Bragg s equation (5.1) to be satisfied, the system (5.78) is reduced at two equations, speeific for the approximations made the present discussion follows (Azaroflf et al., 1974 Putz and Lacrama, 2005). 

That is to say, peaks are observed in the diffraction data when the momentum transfer vector is equal to a vector of the reciprocal lattice. Such peaks are known as Bragg peaks and Equation [34] is Bragg s law, for diffraction from a single crystal. Figure 6 shows a typical single crystal diffraction pattern for every peak in the observed diffraction pattern the momentum transfer vector, Q, satisfies Bragg s law. Each and every Bragg peak may be identified by a unique combination of indices h, k, /). 

The effect can be better understood if the concept of the reciprocal lattice is introduced from the Bragg equation it is clear that sin 0, i.e., the measure of the deflection of the diffracted beam from the nondiffracted, is invCTsely proportional to the plane separation d. However, if we defined a reciprocal lattice in which corresponding distances were l/d, then the deflection of X-rays (sin 0) would be directly proportional to distances. The reciprocal lattice is constructed thus for a plane with index (hkl) we define a lattice point as the origin and constract a line perpendicular... 

For simplicity, we assume a 2D-periodic structure ( 2D-crystallinity ) in the monolayer film floating on the subphase The molecules are arranged in identical unit cells which form a regular lattice. Then, in GID, Bragg diffraction occurs when the lateral scattering vector Qhor, c.f. Fig. 1.1b, coincides with a reciprocal lattice vector Ghk The scattering is concentrated in so-called Bragg Rods (parallel to the Qz axis), defined by two Laue conditions or, in vector notation, by the equation Qhor=Ghk- By constrast, XR... 

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