Bragg reflection, crystallography

Extensive use of three-dimensional electron-density or difference electron-density maps in crystallography became possible only with the advent of high-speed computers. The magnitude of the problem can be illustrated" for a compound crystallizing in an orthorhombic unit cell with dimensions a = 11.98, 6 = 15.82, c = 11.49 A, Z = 4, for which 4397 (independent) Bragg reflections were measured at a resolution of... 

P. Suortti, Bragg reflection profile shape in x-ray powder diffraction patterns, in The Rietveld method. lUCr monographs on crystallography 5, R.A. Young, Ed., Oxford University Press, Oxford, New York (1993). 

The refractive index n in this formula is an average between the ordinary and extraordinary indices defined by = and = ex, and the wavelength 2 corresponds to the center of the reflection peak. In reality, as shown in Annex II where Maxwell equations are solved exactly, the Bragg reflection occurs in quite a large band of wavelengths (analog to the Darwin band in X-ray crystallography) defined by... 

How is the diffraction pattern obtained in an x-ray experiment such as that shown in Figure 18.5b related to the crystal that caused the diffraction This question was addressed in the early days of x-ray crystallography by Sir Lawrence Bragg of Cambridge University, who showed that diffraction by a crystal can be regarded as the reflection of the primary beam by sets of parallel planes, rather like a set of mirrors, through the unit cells of the crystal (see Figure 18.6b and c). 

The work was picked up immediately by the Braggs, father and son, who, by interpreting the scattering of X-rays as reflection from the different planes in the crystal were able to deduce the structures of some very simple crystals in the years 1913 and 1914. And for this they got the Nobel Prize in 1915. That was the beginning of X-ray crystallography. 

The phenomenon of anomalous scattering is extensively used in modem macromolecular crystallography to solve the phase problem. To understand how this is done, we need to return to the simple picture of X-rays reflecting from Bragg planes, where it makes no difference which side of the plane is the reflecting surface . This leads to two structure factors Fhki and F h differing only in the sign of their phase. The phase — a complex number - drops out because we measure intensities (/= F2 see above) and I k,i and are equal. 

The intensity of the beam diffracted by all the atoms of the unit cell in a direction predicted by the Bragg law is proportional simply to the square of the amplitude of the resultant beam, and [Fp is obtained by multiplying the expression given for F in Eq. (4-11) by its complex conjugate. Equation (4-11) is therefore a very important relation in x-ray crystallography, since it permits a calculation of the intensity of any hkl reflection from a knowledge of the atomic positions. 

One final irony in this whole situation is that crystallographic data are referred to as "reflections" thus reinforcing an incorrect model. Even the cif dictionary begins all items involving the raw data as reflections. It is no wonder that nonspecialists are confused by Bragg s Law and its place in understanding crystallography. 

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 ... 

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