Crystals Bragg planes

As one may infer from the quotation, W. L. Bragg realized that a crystal can act as an x-ray grating made up of equidistant parallel planes (Bragg planes) of atoms or ions from which unmodified scattering of x-rays can occur in such fashion that the waves from different planes are in phase and reinforce each other. When this happens, the x-rays are said to undergo Bragg reflection by the crystal and a diffraction pattern results. 

Although onty two Bragg planes have been considered, it is clear that planes below BBf will also contribute to the reflected beam, though to an exponentially decreasing extent. The relationship between X and sin will evidently be important in determining the useful wavelength range of a particular crystal... 

Mile. Cauchois20 used in transmission with a convergent beam a thin crystal bent cylindrically with the Bragg planes about as shown in Figure 4-11. Mica, gypsum, and quartz proved suitable materials. The action of the crystal is shown graphically in Figure 4-11, and it can be demonstrated analytically as well.22... 

With x-rays, however, one can have his cake and eat it too That the two conditions given above can both be met for a curved crystal was - appreciated first by Du Mond and Kirkpatrick18 and put into practice first by Johansson.21 This situation exists because-the crystal is a three-dimensional lattice of exceedingly small spacing. It is therefore possible to bend the crystal until the Bragg planes have the radius R,... 

The theory of this aberration was worked out in the 1920s by Schwarzchild. For simphcity we shall discuss the case of a beam conditioner comprising a single crystal and an aperture as in the classic double-crystal arrangement. If the Bragg planes are tilted about an axis contained in the incidence plane and the... 

Bragg planes, then rays which are not contained in the incidence plane will not see equal angles with respect to the specimen and the reference. If we set the crystals so that the median ray (in the incidence plane) makes equal angles, then an inclined ray may make the Bragg angle for the reference crystal but will not be diffracted from the specimen (Figure 2.21). The result is that only a band of rays satisfies the Bragg conditions for both crystals. The band moves up (or down) as the crystals are rotated. The consequences are ... 

Figure 2.21 Bragg plane tilt aberration, (a) Diffracting planes parallel, diffraction occurs simultaneously over the whole height of the beam, (b) Diffracting planes skewed, diffraction only takes place over a narrow band, (c) As the crystal is rotated to measure the rocking curve, the band moves up or down the crystal. The integrated intensity remains approximately the same as in case (a) but the peak intensity decreases and the width increases...

Two peaks should be visible, the K 2 being half of the intensity of the K 1 peak. If this intensity ratio is not 2 1 it implies that the Bragg planes are not vertical and the orientation of the crystal in the plane normal to the incident beam should be adjusted. 

The width of the image can be deduced using this simple idea of contrast being formed when the misorientation around the defect exceeds the perfect crystal reflecting range. We consider the case of a screw dislocation nmning normal to the Bragg planes, where the line direction / coincides with the diffraction vector g. The effective misorientation at distance r from the core is =bH r (8.41)... 

Fig. 4. Schematic illustration of Bragg diffraction from photonic crystal lattice planes.

The lattice wave numbers and Bragg planes for the system. shown in Fig. 16-3. We imagine, though, a much larger crystal, so the mesh of wave numbers allowed by periodic boundary conditions becomes very fine and is not shown. 

Bragg s law relates the diffraction angle 4 in a crystal with planes a distance d apart to the wavelength X and a... 

FIGURE 3.17. The construction of an Ewald sphere of reflection, illustrated in two dimensions (the Ewald circle), (a) Bragg s Law and the formation of a Bragg reflection hkl. The crystal lattice planes hkl are shown, (b) Construction of an Ewald circle, radius 1/A, with the crystal at the center C and Q-C-0 as the incident beam direction. 

As the crystal is rotated, so is its crystal lattice and its reciprocal lattice. If, during the rotation of the crystal a reciprocal lattice point touches the circumference of the Ewald circle (the surface of the Ewald sphere), Bragg s Law and the Laue conditions are satisfied. The resnlt js a Bragg reflection in the direction CP, with values of h, k, and 1 corresponding both to hkl values for the reciprocal lattice point and for the crystal lattice planes. 

Bragg s Law, the Bragg equation In diffraction of X rays by crystals, each diffracted beam can be considered to be reflected from a set of parallel lattice planes. If the angle between the diffracted X-ray beam (wavelength X) and the normal (perpendicular) to a set of crystal lattice planes is 90° - Ohki, and if the perpendicular spacing of the lattice planes is dhti, then ... 

The channel-cut monochromator is the simplest type employed experimentally. A channel is cut in a perfect crystal (e.g.. Si) to provide two parallel reflecting surfaces that have a particular crystal plane [e.g., the Si (220)] parallel to the surface. The Bragg condition is used to select a particular wavelength and the reflected beam emerges parallel to the incident beam but is vertically displaced by 2D cos d, where D is the distance between the two faces and 0 is the angle between the beam and the Bragg planes. The accuracy of data collected using channel-cut crystal monochromators may be limited due to harmonic con-... 

In this equation f is the atomic scattering factor or atomic form factor, h,k,l are the Miller indices of the reflecting plane, and x ,y ,z are the coordinates of the scattering atom in decimal fractions of unit cell parameters, a,b,c, respectively. For simple structures the structure factor indicates what types of Bragg planes in a given kind of structure can produce a diffraction peak, i.e., have non-cancelled, coherent scattering, and also indicates the relative intensity of the allowed peaks (Warren 1969). A few structure factors for simple crystal structures are shown below. 

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. 

What is the resolution of the diffraction pattern, namely what is the family of Bragg planes in the crystal of smallest interplanar spacing that is represented by a measurable intensity in the diffraction pattern Definition of measurable intensity may here become a contentious question. 

Generating X rays and directing them on to the macromolecular crystal is only half of the data collection process. Equally important is recording the diffraction intensities produced by the interaction of the X rays with the atoms distributed about the Bragg planes in the... 

The crystal is a three-dimensional object and the different Bragg planes have unique orientations in space. Since the reflection angle 9 is defined with respect to the plane, the orientation of that plane (and hence the orientation of the crystal) with respect to the incoming X-ray beam will determine the actual direction to which the X-rays are diffracted. The X-rays diffracted in a given direction give rise to one diffraction spot (Figure 8), also known as a reflection. 

Figure 3.15 Electron deflection by Bragg diffraction of a crystalline specimen (a) image formation in crystalline samples and (b) diffraction at crystal lattice planes and at the contours of inclusions. (Reproduced with permission from M. von Heimandahl, Electron Microscopy of Materials, Academic Press, New York. 1980 Elsevier B. V.)...

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