Vector Bragg Equation

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

Equation 4.70 is referred as the Vector Bragg Equation which expresses the relationship between H, a vector characterizing a crystal plane, and s, a vector characterizing the scattering geometry, for constructive interference to occur. Such a vector equation implies two conditions ... 

This defines a set of 0 s corresponding to the set of s.We may now interpret the vector Bragg equation... 

Figure 4.13. Relationships used in deriving the Vector 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 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 ... 

Setting the magnitude of s to 1/2, we get the Bragg equation in terms of the magnitude of the scattering vector h ... 

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). 

In terms of the wave vector, k, the Bragg equation can be written as ... 

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 ... 

The Laue condition for diffraction (including x-ray, electron, and neutron diffraction) can simply be stated as Ghke = where Ak is the vector difference between the incident and scattered wave vector. For elastic scattering, the Laue condition can also be written as 2k G = G which is shown to be equivalent to the more familiar Bragg equation,... 

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

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... 

Equations (1) are the von Laue conditions, which apply to the reflection of a plane wave in a crystal. Because of eqs. (1), the momentum normal to the surface changes abruptly from hk to the negative of this value when k terminates on a face of the BZ (Bragg reflection). At a general point in the BZ the wave vector k + bm cannot be distinguished from the equivalent wave vector k, and consequently... 

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... 

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... 

Both the Laue equations and the Bragg law can be derived from Eq. (7). The former are obtained by forming the dot product of each side of the equation and the three crystal-lattice vectors aj, aj, sl successively. For example,... 

By definition, h, k, and I are divided by their largest common integer to be Miller indices. The vector from Bragg s Equation (26) points in the plane normal direction parallel to d but with length Ijd. We can now write in terms of the vector d ... 

A further simplification of the texture correction can be made if the form of the ODF is assumed to be cylindrically symmetric (e.g. from spirming the sample) and ellipsoidal. If the unique ellipsoid axis is parallel to the diffraction vector (s) and perpendicular to a flat sample surface as it is in a Bragg-Brentano experiment, the general axis equation (commonly referred to as the March-Dollase equation) is ... 

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 ... 

As mentioned previously, CTRs arise as a result of the abrupt termination of a crystal lattice, and the diffuse diffracted intensity connects Bragg points in reciprocal space. In this case, the scattering vector is normal to the surface, and as a result, this technique is very sensitive to surface and interface roughness but not to in-plane atomic correlations. Thus, it yields information that is complementary to that obtained by grazing incidence diffraction. The most important feature of CTR is the characteristic decay of the scattered intensity described by Eq. (38). For surfaces that are not perfectly terminated (i.e., rough) the intensity will decay faster than predicted by this equation, and this can be used as a measure of root-mean-square surface roughness. 

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