Diffracted beams, plane crystal

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). 

Figure 18.6 Diffraction of x-rays by a crystal, (a) When a beam of x-rays (red) shines on a crystal all atoms (green) in the crystal scatter x-rays in all directions. Most of these scattered x-rays cancel out, but in certain directions (blue arrow) they reinforce each other and add up to a diffracted beam, (b) Different sets of parallel planes can be arranged through the crystal so that each corner of all unit cells is on one of the planes of the set. The diagram shows in two dimensions three simple sets of parallel lines red, blue, and green. A similar effect is seen when driving past a plantation of regularly spaced trees. One sees the trees arranged in different sets of parallel rows.

Figure 3.4 X-ray beam passing through the Ewald sphere and diffracted by planes in a single crystal produces reflection spots. (Adapted with permission from Figure 1.13 of Drenth, J. Principles of Protein X-ray Crystallography, 2nd ed., Springer-Verlag, New York, 1999. Copyright 1999 Springer-Verlag, New York.)...

X-ray radiation wavelength—that is, 1/ X. When the crystal is rotated, the reciprocal lattice rotates with it and different points within the lattice are brought to diffraction. The diffracted beams are called reflections because each of them can be regarded as a reflection of the primary X-ray beam against planes in the crystal. 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

The XRD pattern of a powdered sample is measured with a stationary X-ray source (usually Cu Ka) and a movable detector, which scans the intensity of the diffracted radiation as a function of the angle 2 6 between the incoming and the diffracted beams. When working with powdered samples, an image of diffraction lines occurs because a small fraction of the powder particles will be oriented such that by chance a certain crystal plane is at the right angle 6 with the incident beam for constructive interference (see Fig. 6.1). 

According to Friedel s law, a diffracted X-ray beam from the (010) side of the R crystal will have the same intensity as that from the opposite (010) side. Moreover, the intensity of this beam will be equal in magnitude to those of the diffracted beams from the (010) and (010) planes of the S crystal. On such a basis one cannot distinguish between the R and S structures. 

The geometrical aspect concerns the position of the diffracted beams on a pattern it only depends on the direct lattice of the crystal through the Bragg law =2dhkisin9B - dhu being the interplanar distance of the diffracted (hkl) lattice planes and 0b the Bragg angle. In other words, it only depends on the lattice parameters of the crystal a, b, c, a, P and y. 

Usually many set of lattice planes can simultaneously be exactly or close to the Bragg orientation and give a multi-beam pattern made of several diffracted beams as shown in the example on figure 2c. A special type of multi-beam pattern concerns Zone-Axis Patterns (ZAP). This type of pattern is observed when a high symmetry [uvw] direction of the crystal is parallel to the incident beam. In this case, the spots on the pattern are arranged along Laue zones (Figure 2d). 

The simplest conditioner is a perfect crystal of the same type as the specimen, using the same reflecting planes, with the deviation of the diffracted beam in the opposite sense to that at the specimen. This is the classic +, - symmetrical double crystal method , as shown inFigrrre 1.5, which gives excellent and easily interpreted resrrlts. Many variations are, however, possible, for example to maximise the sensitivity to strain, or to emphasise the contribution of near-srrrface layers to the diffraction, and we shall treat these in detail in this book. 

The image intensity /(x, y) at the image plane of the objective lens results from two-dimensional Fourier synthesis of the diffracted beams (the square of the FT of the waves at the exit face of the crystal), modified by a phase-contrast transfer function factor (CTF, sin /), given by Scherzer (1949), as... 

The crystal surface is imagined to consist of individual atomic layers parallel to the surface. Whenever convenient, the LEED electrons between these layers are represented by a set of plane waves (to each diffracted beam corresponds one plane wave), as the electron-solid interaction potential is assumed to be a constant between the layers. 

This, therefore, is the origin of the arcs on a powder photograph each arc represents the combined diffracted beams from all the crystals which happen to be suitably oriented for reflecting with one particular set of planes. 

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