Diffraction intensity planes

In Figures 6.17, 6.18, and 6.19 are presented pairs of diffraction intensity planes for three different protein crystals. From two photographs such as these, which are usually (e.g., Figure 6.19) orthogonal to one another, the symmetry of the entire three-dimensional diffraction pattern may be deduced. In some cases, however, additional diffraction images... 

If the detection system is an electronic, area detector, the crystal may be mounted with a convenient crystal direction parallel to an axis about which it may be rotated under tlie control of a computer that also records the diffracted intensities. Because tlie orientation of the crystal is known at the time an x-ray photon or neutron is detected at a particular point on the detector, the indices of the crystal planes causing the diffraction are uniquely detemiined. If... 

In the concepts developed above, we have used the kinematic approximation, which is valid for weak diffraction intensities arising from imperfect crystals. For perfect crystals (available thanks to the semiconductor industry), the diffraction intensities are large, and this approximation becomes inadequate. Thus, the dynamical theory must be used. In perfect crystals the incident X rays undergo multiple reflections from atomic planes and the dynamical theory accounts for the interference between these reflections. The attenuation in the crystal is no longer given by absorption (e.g., p) but is determined by the way in which the multiple reflections interfere. When the diffraction conditions are satisfied, the diffracted intensity ft-om perfect crystals is essentially the same as the incident intensity. The diffraction peak widths depend on 26 m and Fjjj and are extremely small (less than... 

FIGURE 27.9 (a) Voltammetry curve for the UPD of TI on Au(l 11) in 0.1 M HCIO4 containing ImMTlBr. Sweep rate 20mV/s. The in-plane and surface normal structural models are deduced from the surface X-ray diffraction measurements and X-ray reflectance. The empty circles are Br and the filled circles are Tl. (b) Potential-dependent diffraction intensities at the indicated positions for the three coadsorbed phases. (From Wang et al., 1998, with permission from Elsevier.)... 

Figure 14 Angles y and iji defining the position of the normal to a given hkl crystal plane in the sample reference system. Ix, incident X-ray beam lhkl, diffracted intensity 6B, Bragg angle. Adapted from Lafrance et al. [82]. Reprinted with permission of John Wiley 8t Sons, Inc.

Figure 2.22 A three-dimensional plot of the diffracted intensity as a function of specimen rotations both parallel ( ) and perpendicular ( j to the dispersion plane. The larger peak is that from the substrate GaAs and the smaller from the GaAIAs layer. CuK i, slit-limited from a single GaAs beam conditioner...

Fig. 8. Generation of the form of the helical diffraction pattern. (A) shows that a continuous helical wire can be considered as a convolution of one turn of the helix and a set of points (actually three-dimensional delta-functions) aligned along the helix axis and separated axially by the pitch P. (B) shows that a discontinuous helix (i.e., a helical array of subunits) can be thought of as a product of the continuous helix in (A) and a set of horizontal density planes spaced h apart, where h is the subunit axial translation as in Fig. 7. This discontinuous set of points can then be convoluted with an atom (or a more complicated motif) to give a helical polymer. (C)-(F) represent helical objects and their computed diffraction patterns. (C) is half a turn of a helical wire. Its transform is a cross of intensity (high intensity is shown as white). (D) A full turn gives a similar cross with some substructure. A continuous helical wire has the transform of a complete helical turn, multiplied by the transform of the array of points in the middle of (A), namely, a set of planes of intensity a distance n/P apart (see Fig. 7). This means that in the transform in (E) the helix cross in (D) is only seen on the intensity planes, which are n/P apart. (F) shows the effect of making the helix in (E) discontinuous. The broken helix cross in (E) is now convoluted with the transform of the set of planes in (B), which are h apart. This transform is a set of points along the meridian of the diffraction pattern and separated by m/h. The resulting transform in (F) is therefore a series of helix crosses as in (E) but placed with their centers at the positions m/h from the pattern center. (Transforms calculated using MusLabel or FIELIX.)...

Definition rocking curve (RC) is a function of the total intensity of X-rays reflected by a sample versus its angular position in rotation around the axis perpendicular to the diffraction plane. The sample is adjusted to have diffracting crystallographic planes perpendicular to the diffraction plane. 

Powder X-ray diffraction (XRD) is a fundamental technique for the structural characterization of condensed matter. It provides evidence of bulk structures in various dimensions. By coherent scattering, the translational symmetry of a lattice is represented in a diffraction pattern, and the atomic species with their average site occupations are reflected in intensities. In powder diffraction, a full structure analysis has become possible as a result of advances in modeling strategies (Langford and Louer, 1996 McCusker et al., 1999). If the diffracting lattice planes are comparable in their dimensions to the wavelength of the X-rays (i.e., they are nanosized), or if the lattice plane distance is not constant but described by a... 

Figure 15 Neutron time-of-flight study of 0-(BEDT-TTF)2I3 performed at the IPNS pulsed source (Argonne). (a) Neutron diffraction intensity distribution in the h = 4.92 reciprocal lattice plane at 20 K and ambient pressure. Satellite peaks are observed (arrows) (b) the same reciprocal plane after applying a pressure of 0.14 GPa, warming to room temperature, and cooling back down to 20 K. (From Ref. 64.)...

The kinematic approximation breaks down at a certain crystal thickness when the diffracted intensity approaches that of the incident beam. A usefiil criterion for kinematic approximation is r < fg/4, where fg is the extinction distance of the strongest reflection in the diffraction pattern. The extinction distance is orientation dependent. In case only one set of lattice planes is strongly diffracting (the two-beam condition), the extinction distance is given by = h lmeX Vg. 

In addition to this information, however, the diffraction pattern also provides information on the quality of the crystal lattice and the thermal motion of the atoms in the unit cell. Figure 3 shows examples of the diffraction patterns that would be observed under various circumstances for the lattice shown in Fig. 2. The examples assume diffraction from the (010) and (100) planes of the lattice that are, respectively, planes parallel to the jcz-plane (Fig. 2A) and yz-plane (not shown). A diffraction pattern for a crystalline sample is recorded by rotating a crystal in the x-ray beam to record systematically the reflections from the various lattice planes. To see the diffracted intensities for the example under consideration, the crystal would be mounted with its z-axis parallel to the x-ray beam and then rotated about the jr-axis to obtain reflections from the (010) planes with spacings of d(0k0) and about the y-axis to obtain reflections from the (100) planes with spacings of d h00). All of the reflections are recorded on a single frame of a two-dimensional detector or a single piece of photographic film. 

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