Miller Indices, hkl

The position of any reflection with given Miller indices hkl can be calculated once all unit cell parameters have been determined. This allows starting the last but very important procedure in the analysis of a texture diffraction pattern - estimating the integrated intensities of the diffraction... 

It is sometimes useful to be able to calculate the perpendicular distance dhu between parallel planes (Miller indices hkl). When the axes are at right angles to one another (orthogonal) the geometry is simple and for an orthorhombic system where a b i c and... 

It is often necessary to find the angle between two crystallographic poles of known Miller indices hkl) and (h k V). For the cubic lattice, it is given by... 

The X-ray diffraction pattern obtained from the dry, annealed fibre (see Materials and Methods) is shown in Figure 3. The reflections index on a hexagonal unit cell with dimensions a = b = 1.438 nm, c (fibre axis) = 0.582 nm with meridional reflections occurring only on layer lines with 1 = 2 n, where n is an integer. These constraints on the Miller indices hkl are in accordance with... 

An alternative grid, shown in red, defines an alternative set of planes that also contains all equivalent atoms, and under different conditions, also satisfies Bragg s equation. Such planes, which may be constructed in an endless number of ways, all have one property in common - they make rational intercepts on the axes of the unit cell. The fractional intercepts are defined as a/h, h/k, c/l in terms of the unit cell constants and the integer Miller indices, hkl. 

By comparison with a film strip with a relative intensity scale for some strong hkl reflection exposed to X-rays for known increments of time, the intensity of each spot was read by hand (later by optical photometers), and a set of Miller indices hkl and intensities Ihm were collected (this would take a... 

If the coordirrate system is not chosen in corresporrdence to a primitive unit cell, not all the lattice points have coordinates that are integers and all lattice points will lie on planes in the set orrly if certain restrictiorrs are placed on the combinations of values that the Miller indices hkl may asstrme. 

The interplanar distance d is determined by the Miller indices hkl. For the cubic system it is easy to show by analytical geometry that... 

Find a factor that will reduce the Hdf values to relatively prime integers within experimental error. Refer to Table 1 and identify the lattice type. Also from Table 1, obtain the Miller indices hkl for each fine. 

The sums may be carried out with respect to the atomic positions in direct (real) space or to lattice planes in reciprocal space, an approach introduced in 1913 by Paul Peter Ewald (1888-1985), a doctoral student under Arnold Sommerfeld (Ewald, 1913). In reciprocal space, the structures of crystals are described using vectors that are defined as the reciprocals of the interplanar perpendicular distances between sets of lattice planes with Miller indices (hkl). In 1918, Erwin Rudolf Madelung (1881-1972) invoked both types of summations for calculating the electrostatic energy of NaCl (Madelung, 1918). 

Thus each diffracted beam is regarded as a reflection from a plane with Miller indices (hkl) and is named by the same indices. Since dhu depends on the unit cell parameters, as dhki = 2f/T, where... 

Miller indices, hkl Three relatively prime integers, hkl, that are reciprocals of the fractional intercepts that the crystallographic plane makes with the crystallographic axes. The crystallographic plane hkl) is described by its Miller indices. 

Electron-density waves hkl lie perpendicular to the sets of crystal lattice planes with Miller indices hkl. The wavelength of the electron-density wave is the spacing of hkl crystal lattice planes, i.e., The amplitude and the relative phase of the electron-... 

In this crystal lattice system, all surfaces with Miller indices, (hkl), satisfying the conditions h x k x 1 and h k l h are chiral [11]. Although such high Miller index surfaces have been studied for decades, it was not until recently that McFadden et al. specifically pointed out and demonstrated that their low synunetry structures render them chiral and, therefore, that they might have enantiospecific interactions with chiral adsorbates [12]. There has been a growing interest in the enantiospecific properties of naturally chiral metal surfaces and in the possibility of using such surfaces for enantioselective chemical processes. 

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. 

A crystallographic plane (hkl) is represented as a light spot of constructive interference when the Bragg conditions (Equation 2.3) are satisfied. Such diffraction spots of various crystallographic planes in a crystal form a three-dimensional array that is the reciprocal lattice of crystal. The reciprocal lattice is particularly useful for understanding a diffraction pattern of crystalline solids. Figure 2.7 shows a plane of a reciprocal lattice in which an individual spot (a lattice point) represents crystallographic planes with Miller indices (hkl). 

This Et has the same signification as the defined below from the Fourier analysis. In both cases, the true mean size is the product of the apparent dimension by the Scherrer constant relative to the variance, K . Tournarie and Wilson have tabulated K for different crystallite shapes as a function of the Miller indices hkl. 

As described in Chapter 1, for a perfect, infinite crystal the reciprocal lattice is made of points, each representing a set of planes with Miller indices hkl). The diffraction condition in reciprocal space is then defined in terms of a geometrical relation diffraction takes place when incident and diffracted beam are such that the scattering vector = (v — Vq) connects the origin with an hkl) point ... 

Because of the perfect periodicity of these profiles, the intensities will be nonzero only at a discrete set of wave vectors. The Miller indices h k 1) of the allowed reflections, as listed in the International T(drles of X-ray Crystallography (1952) for a given space group and Wyckoff position, determine the wave vectors for which intensities must be evaluated. The wave vector that corresponds to the reflection with Miller indices hkl) is h = (2nhla, Ink/a, Inlja), where the lattice parameter a can be taken equal to unity here. 

Table 1. Miller indices hkl) und ratio of Bragg peak positions of cubic (Q) and lamellar (L) lipid structures.

The facets of a well-formed crystal or internal planes through a crystal structure are specified in terms of Miller Indices, h, k and Z, written in round brackets, (hkl). Miller indices, (hkl), represent not just one plane, but the set of all identical parallel planes. 

A beam of radiation will only be diffracted when it impinges upon a set of planes in a crystal, defined by the Miller indices (hkl), if the geometry of the situation fulfils quite specific conditions, defined by Bragg s law ... 

Now we have two different coordinate systems. One, known as the laboratory axes , comprises three orthogonal axes of unit length, the directions of which are uniquely defined with respect to the axes of the diffractometer circles and the direction of the primary beam (although these conventions vary from one diffractometer type to another). A point in these coordinates is described by a vector x. The other system comprises three principal vectors of the reciprocal crystal lattice (see Section 2.1). In this system, a Bragg reflection is expressed by a vector h whose coordinates are the Miller indices hkl (Section 2.2.1). The relation x = Ah between the two systems is defined by the orientation matrix (OM) A,... 

Miller indices (hkl) - A set of indices used to label planes in a crystal lattice. [2]... 

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