Derivation of the Bragg Equation

The easiest access to the structural information in powder diffraction is via the well-known Bragg equation (W. L. Bragg, 1912), which describes the principle 

Another equivalent, and highly useful, expression of the Bragg equation is  

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. 

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Several important relationships in crystallography directly follow from a derivative of the Bragg equation [Equation (5)]. First we rewrite Bragg s law making the /-spacing the subject of the equation ... 

A) Derivation of the Bragg equation for X-ray diffraction B) schematic diagram of X-ray diffraction apparatus... 

Fig. 1.3-17 Geometrical derivation of the Bragg equation nk = 2d sin 9. n can be set to 1 when it is included in a higher-order hkl...

Diagram illustrating the derivation of the Bragg equation for the diffraction of x-rays by crystals. 

Q.28.1 Show that Equation (29.12) is a general form of the Bragg equation as derived from the von Laue equations. 

This Chapter is concerned with methods for obtaining the relative phase angles for each Bragg reflection so that the correct electron-density map can be calculated and, from it, the correct molecular structure determined. When scattered light is recombined by a lens, as described in Chapters 3 and 6, the relationships between the phases of the various diffracted beams are preserved. In X-ray diffraction experiments, however, only the intensities of the Bragg reflections are measured, and information on the relative phases is lost. An attempt is maxle to remedy this situation by deriving relative phases by one of the methods to be described in this Chapter. Then Equation 6.3 (Chapter 6) is used to obtain the electron-density map. Peaks in this map represent atomic positions. 

In any crystalline material an infinite set of planes exists with different Miller indices, and each set of planes has a particular separation. For any set of planes there will be a diffraction maximum at a particular angle, 9. By combining the equation relating to the lattice parameter for a particular system with the Bragg equation, a direct relationship between the diffraction angle and the lattice parameters can be derived. 

In the above treatment, which is analogous to the Bragg treatment of x-ray diffraction, the assumption of specular reflection is made. This can be avoided by a treatment similar to Laue s derivation of his diffraction equations. 

As we derived in Section 8.1.3, nX = 2d sin d. This is the Bragg equation, which states that coherence occurs when nX = 2d sin 9. It can be used to measure d, the distance between planes of electron density in crystals, and is the basis of X-ray crystallography, the determination of the crystal structure of solid crystalline materials. Liquids, gases, and solids such as glasses and amorphous polymers have no well-ordered structure therefore they do not exhibit diffraction of X-rays. 

In a recent experiment the same invariance was also foimd for the lateral size of the blocks in the lamellar crystals [16]. Crystal block diameters can generally be derived from the linewidth of Bragg reflections in WAXS patterns, by application of the Scherrer equation... 

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... 

Unit cell dimensions are obtained from measurements of 29 values of several Bragg reflections for which the indices h, k, and I are known. Values of 26 are measured as accurately as possible, and, since the wavelength A. of the radiation used is known, a value of dhu may be found by Bragg s Law, Equation 3.2 (Chapter 3). The value of is related to the unit cell dimensions and, if 26 values are measured for several reflections, values of the unit cell dimensions may be derived. The selected group of reflections chosen to do these calculations should contain a distribution of Miller indices and they should have relatively high 29 values ... 

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,... 

Similar equations can be derived for species of /-types. An expression for the average interaction energy follows from the Bragg-Williams approximation... 

In the typical practice of the XSW technique, however, only a limited set of hkl measurements is taken, and the analysis resorts to comparing the measured / and values to those predicted by various competing structural models. The procedures of structural analysis using fH and will be described in more detail in a later section of this chapter. It should be stressed that the Bragg XSW positional information acquired is in the same absolute coordinate system as used for describing the substrate unit cell. This unit cell and its origin were previously chosen when the structure factors FH and Fs where calculated and used in Equations (9), (10), and (12). As previously derived and experimentally proven (Bedzyk and Materlik 1985), the phase of the XSW is directly linked to the phase of the structure factor. This is an essential feature of the XSW method that makes it unique namely, it does not suffer from the well known phase problem of X-ray diffraction. 

In this equation d represents the distance between Bragg Planes, 6 the angle of incidence and reflection of the X-ray beam, k the wavelength of the X-rays, and n the order of diffraction. The law was derived by Sir W. H. Bragg and his son Sir W. L. Bragg and first reported in 1912 [3]. The Braggs were awarded the Noble Prize in physics in 1915 for their use of this equation to solve the structures of simple salts. 

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