Geometry, diffraction

Early powder diffraction experiments relied mostly on the Debye-Scherrer experiment to record a diffractogram. A broad film strip set into a cylindrical chamber produced the first known two-dimensional powder diffraction data. In contrast to modern methods the thin equatorial strip was the only part of interest and intensities merely optically and qualitatively analysed. This changed drastically with the use of electronic scintillation counters. Intensities were no longer a matter of quality but quantity. Inevitably the introduction of intensity correction functions long known to the single-crystal metier, i.e. Lorentz and polarization corrections (see Section 14.3), made their way into the field of powder diffraction. 

Generally the detectors are set up perpendicular to the primary beam, with the intersection of the primary beam at the detector centre. This setting has some advantages the entire Bragg cones are detected and the deviation of the cone projection from an ideal circle is usually small. Sometimes a detector can be placed off-centre and non-orthogonally to the primary beam. This can enlarge the detectable q-space in a very cost effective manner. The downsides are the strongly elliptical conical projections and the loss of the entire azimuthal information of a diffraction cone. 

The Miller indices (hkl) represent a series of parallel planes in a crystal with spacing of d i-Combining Equations 2.3 and 2.4, we obtain the following relationship between diffraction data and crystal parameters for a cubic crystal system. 

Equation 2.5 does not directly provide the Miller indices of crystallographic planes. We need to convert (h2 + k2 +12) to (hkl) or hkl. This should not be difficult for low-index planes of cubic systems. For example, when (h2 + k2 +12) is equal to 1, the plane index must be 001 when it is equal to 2, the index must be 110. This type of calculation is often not necessary 

A reciprocal lattice is in an imaginary reciprocal space that relates to the corresponding crystal lattice in real space. A direction in the crystal lattice is defined by a vector ruvw with unit vectors a, b and c in real space. 

A dimension in reciprocal space is a reciprocal of the dimension in real space (with a factor of unity). The magnitude of a vector d hkl in a reciprocal lattice equals the reciprocal of plane spacing ( 4h) in real space. 

The relationship between a reciprocal lattice and its real crystal structure is highlighted as follows  

For a stationary crystal and white radiation with 2.max A Amin, the RLPs whose reflections can be recorded lie between the Ewald spheres of radii 1/Amax and 1/Amin. These spheres touch at the origin of the reciprocal lattice (figure 7.2(a)), and the wavelength at which any individual RLP diffracts is determined by the reciprocal radius of the Ewald sphere passing through it. 

The spot coordinate on the film is related to the RLP coordinate as shown in figure 7.3. There is also a sample resolution limit d max (=1/ dmin) so that no reflections are recorded from RLPs outside a sphere centred at the origin with radius (shown in figure 7.2). The acces- 

Spindle Add. fract symm. uniq. captured Fract. symm. uniques not recorded Fractional overlap with previous spindles  

If we consider a given reflecting plane of spacing d there is an associated set of Miller indices (h,k, ). Now 2h,2k,2 may be beyond the resolution limit and h,k, may also have no common integer divisor. Hence, the Laue spot that results will contain h,k, only. 

It can be shown (Cruickshank et al 1987) that the probability that a randomly chosen RLP has no common integer divisor is 

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The simplest diffraction measurement is the determination of the surface or overlayer unit mesh size and shape. This can be performed by inspection of the diffraction pattern at any energy of the incident beam (see Figure 4). The determination is simplest if the electron beam is incident normal to the surface, because the symmetry of the pattern is then preserved. The diffraction pattern determines only the size and shape of the unit mesh. The positions of atoms in the surface cannot be determined from visual inspection of the diffraction pattern, but must be obtained from an analysis of the intensities of the diffracted beams. Generally, the intensity in a diffracted beam is measured as a fimction of the incident-beam energy at several diffraction geometries. These intensity-versus-energy curves are then compared to model calculations. ... 

In this section we will discuss perturbation methods suitable for high-energy electron diffraction. For simplicity, in this section we will be concerned with only periodic structures and a transmission diffraction geometry. In the context of electron diffraction theory, the perturbation method has been extensively used and developed. Applications have been made to take into account the effects of weak beams [44, 45] inelastic scattering [46] higher-order Laue zone diffraction [47] crystal structure determination [48] and crystal structure factors refinement [38, 49]. A formal mathematical expression for the first order partial derivatives of the scattering matrix has been derived by Speer et al. [50], and a formal second order perturbation theory has been developed by Peng [22,34],... 

Electrons diffract from a crystal under the Laue condition k — kg=G, with G = ha +kb +lc. Each diffracted beam is defined by a reciprocal lattice vector. Diffracted beams seen in an electron diffraction pattern are these close to the intersection of the Ewald sphere and the reciprocal lattice. A quantitative understanding of electron diffraction geometry can be obtained based on these two principles. 

For convergent electron beams with a sufficiently small probe, the diffraction geometry can be approximated by a parallel crystal slab whose surface normal direction is n. To satisfy the boundary condition, letting... 

The zone axis coordinate system can be used for specifying the diffraction geometry the incident beam direction and crystal orientation. In this coordinate, an incident beam of wavevector K is specified by its tangential component on x-y plan = k x + k y, and its diffracted beam at Kt+gt, for small angle scatterings. For each point inside the CBED disk of g, the intensity is given by... 

Figure 10 Diffraction geometry with diamond anvils (a) Merrill-Bassett-like geometry (b) Bridgman-like geometry but with both transparent anvils and gasket. (From Ref. 96.)...

Figure 13. Effect of diffraction geometry on XRD powder patterns.

There are several electron diffraction techniques based on (i) electron energy, (ii) the diffraction geometry, and (hi) the electron probe (Figure 1). Each has its own unique applications. High-energy electron diffraction (HEED) uses electrons with energy from tens of kilo electron volts (keV)... 

The surface diffraction geometry of RHEED and FEED is determined by the intersection of the Ewald sphere with the 2-D reciprocal rods of the smface. The incidence for FEED is normal to the smface the diffraction pattern is a projection of the positions of the rods. In RHEED, the incident beam is nearly normal to the smface reciprocal rods. The diffraction spots fall onto a semicircle, which is defined by the Ewald sphere. For more details on RHEED and FEED, refer to Ref 3 and 2 respectively. 

Figure 9 An example of the best fit obtained from an electron structure factor refinement. The experimental intensities are from a few selected line scans across the experimental pattern shown in (a). The fit was obtained using Si(lll) and (222) strucmre factors as adjustable parameters together with parameters describing the electron diffraction geometry...

Thomas, D. J. Modern equations of diffractometry. Diffraction geometry. Acta Cryst. A48, 134-158 (1992). 

The concept of a reciprocal lattice was first introduced by Ewald and it quickly became an important tool in the illustrating and understanding of both the diffraction geometry and relevant mathematical relationships. Let a, b and c be the elementary translations in a three-dimensional lattice (called here a direct lattice), as shown for example in Figure 1.4. A second lattice, reciprocal to the direct lattice, is defined by three elementary translations a"", b and c, which simultaneously satisfy the following two conditions ... 

Diffraction geometry - reflection or transmission - when scattered intensity is registered after the reflection from or after the transmission through the sample, respectively. 

First, the intensities of all the predicted spots on the detector are measured and the errors in the intensities are estimated. Owing to the diffraction geometry and other experimental factors, different measurements of the same reflection are not directly comparable to each other, and a computational procedure known as relative scaling must be used to bring them on a common scale. In all cases some reflections are related by symmetry, therefore, we have multiple observations of the same reflection. This gives us additional information on the experimental errors, because even though the intensities of symmetry-related reflections should be equal, they differ due to experimental error. The differences in the related intensities can be quantified by the residual... 

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