Diffraction intensities

X-ray diffraction by a crystal arises from X-ray scattering by individual atoms in the crystal. The diffraction intensity relies on collective scattering by all the atoms in the crystal. In an atom, the X-ray is scattered by electrons, not nuclei of atom. An electron scatters the incident X-ray beam to all directions in space. The scattering intensity is a function of the angle between the incident beam and scattering direction (26). The X-ray intensity of electron scattering can be calculated by the following equation. 

I0 is the intensity of the incident beam, r is the distance from the electron to the detector and K is a constant related to atom properties. The last term in the equation shows the angular effect on intensity, called the polarization factor. The incident X-ray is unpolarized, but the scattering process polarizes it. 

The structure extinction of intensity can be calculated by the structure factor (F). Suppose that there are N atoms per unit cell, and the location of atom n is known as un, vn, wn and its atomic structure factor is/ . The structure factor for (hkl) plane (F w) can be calculated. 

For example, a body-centered cubic (BCC) united cell has two atoms located at [0,0,0] and [0.5,0.5,0.5], We can calculate the diffraction intensity of (001) and (002) using Equation 2.11. 

This structure factor calculation confirms the geometric argument of (001) extinction in a body-centered crystal. Practically, we do not have to calculate the structure extinction using Equation 2.11 for simple crystal structures such as BCC and face-centered cubic (FCC). Their structure extinction rules are given in Table 2.2, which tells us the detectable crystallographic planes for BCC and FCC crystals. From the lowest Miller indices, the planes are given as following. 

A primitive structure that contains only one atom per unit cell will produce a diffraction peak for every plane that satisfies the Laue condition. However, the presence or absence as well as the intensity of the reflections from a structure containing multiple atoms per unit cell will be determined by the contents of the unit cell. 

Geometry for analyzing scattering of x-rays from electron density located at p relative to the /th atom in the unit cell translated by from the origin. 

The electron density in the vicinity if the th atom is given by y(p) = J The electron concentration n(r) in the crystal may be written as 

The scattering amphtude is proportional to the electron concentration times the appropriate phase factor  

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As noted earlier, most electron diffraction studies are perfonned in a mode of operation of a transmission electron microscope. The electrons are emitted themiionically from a hot cathode and accelerated by the electric field of a conventional electron gun. Because of the very strong interactions between electrons and matter, significant diffracted intensities can also be observed from the molecules of a gas. Again, the source of electrons is a conventional electron gun. 

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... 

Similar models for the crystal stmcture of Fortisan Cellulose II came from two separate studies despite quite different measured values of the diffraction intensities (66,70). Both studies concluded that the two chains in the unit cell were packed antiparallel. Hydrogen bonding between chains at the corners and the centers of the unit cells, not found in Cellulose I, was proposed to account for the increased stabiUty of Cellulose II. The same model, with... 

When a ledge is formed on an atomically smooth monolayer during tire formation of a thin film the intensity of the diffraction pattern is reduced due to the reduction in the beatrr intensity by inelastic scattering of electrons at the ledge-monolayer junction. The diffraction intensity catr thus be used during deposition of several monolayers to indicate the completion of a monolayer through the relative increase in intensity at tlris time. Observation of this effect of intensity oscillation is used in practice to count the number of monolayers which are laid down during a deposition process. 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

XRD offers unparalleled accuracy in the measurement of atomic spacings and is the technique of choice for determining strain states in thin films. XRD is noncontact and nondestructive, which makes it ideal for in situ studies. The intensities measured with XRD can provide quantitative, accurate information on the atomic arrangements at interfaces (e.g., in multilayers). Materials composed of any element can be successfully studied with XRD, but XRD is most sensitive to high-Z elements, since the diffracted intensity from these is much lar r than from low-Z elements. As a consequence, the sensitivity of XRD depends on the material of interest. With lab-based equipment, surface sensitivities down to a thickness of -50 A are achievable, but synchrotron radiation (because of its higher intensity)... 

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... 

Interdiffusion of bilayered thin films also can be measured with XRD. The diffraction pattern initially consists of two peaks from the pure layers and after annealing, the diffracted intensity between these peaks grows because of interdiffusion of the layers. An analysis of this intensity yields the concentration profile, which enables a calculation of diffusion coefficients, and diffusion coefficients cm /s are readily measured. With the use of multilayered specimens, extremely small diffusion coefficients (-10 cm /s) can be measured with XRD. Alternative methods of measuring concentration profiles and diffusion coefficients include depth profiling (which suffers from artifacts), RBS (which can not resolve adjacent elements in the periodic table), and radiotracer methods (which are difficult). For XRD (except for multilayered specimens), there must be a unique relationship between composition and the d-spacings in the initial films and any solid solutions or compounds that form this permits calculation of the compo-... 

S. Clake, D. D. Vvedensky. Origin of reflection high-energy electron-diffraction intensity oscillations during molecular-beam epitaxy a computational modeling approach. Phys Rev Lett 55 2235, 1987. 

In the powder diffraction technique, a monochromatic (single-frequency) beam of x-rays is directed at a powdered sample spread on a support, and the diffraction intensity is measured as the detector is moved to different angles (Fig. 1). The pattern obtained is characteristic of the material in the sample, and it can be identified by comparison with a database of patterns. In effect, powder x-ray diffraction takes a fingerprint of the sample. It can also be used to identify the size and shape of the unit cell by measuring the spacing of the lines in the diffraction pattern. The central equation for analyzing the results of a powder diffraction experiment is the Bragg equation... 

Figure 2. (a) Blaze condition for a reflection grating, (b) diffracted intensity for unblazed... 

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 3. Example of XRPD on small Au clusters supported on silica. Total diffraction intensity has been measured with area detector (IP) on BM08-GILDA beamline at the ESRF with A = 0.6211 A and 2min exposure time. Diffraction patterns were collected on Au-supported sample (Exp) and on silica support (Support). Difference patterns, corrected for fluorescence, IP efficiency, etc., are shown (n-Au).

For instance, with the introduction of SR sources, particles with a radius of a few nanometers can be studied with conventional methods. This has also stimulated a new kind of microscopy, named diffraction microscopy, where the Fraunhofer diffraction intensity patterns are measured at fine intervals in reciprocal space. By means of this oversampling a computer assisted solution of the... 

When low-temperature studies are performed, the maximum resolution is imposed by data collection geometry and fall-off of the scattered intensities below the noise level, rather than by negligible high-resolution structure factor amplitudes. Use of Ag Ka radiation would for example allow measurement of diffracted intensities up to 0.35 A for amino-acid crystals below 30 K [40]. Similarly, model calculations show that noise-free structure factors computed from atomic core electrons would be still non-zero up to O.lA. 

Blessing, R.H. (1987) Data reduction and error analysis for accurate single crystal diffraction intensities, Cryst. Rev., 1, 3-58. 

Since the to angle tracks the 29 value, we need specify only 2 9, tp and x in order to correctly label the intensities read by the detector. In orientation texture studies the diffracted intensity is mapped as a function of a and x for a fixed 29. This provides (under appropriate assumptions) a measure of the probability... 

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