Crystallographic plane interplanar spacing

A simple correlation was noticed by Donnay and Harker, (10) between the interplanar spacing of a crystallographic plane, d, and its area on an average crystal. A similar correlation holds between d and the frequency with which the plane (hkl) appears in an ensemble of crystals. Since an area of a plane is roughly proportional to the inverse of its (linear) growth velocity R, the Donnay-Harker law is equivalent to stating that R 1/dj. 

According to the Braggs, diffraction from a crystalline sample can be explained and visualized by using a simple notion of mirror reflection of the incident x-ray beam from a series of crystallographic planes. As established earlier (see Chapter 1, section 1.14.1), all planes with identical triplets of Miller indices are parallel to one another and they are equally spaced. Thus, each plane in a set hkJ) may be considered as a separate scattering object. The set is periodic in the direction perpendicular to the planes and the repeat distance in this direction is equal to the interplanar distance dhki- Diffraction from a set of equally spaced objects is only possible at specific angle(s) as... 

The important characteristics of the reciprocal lattice that should be noted are (1) that the vector r kl is normal to the crystallographic plane whose Miller indices are (hkl), and (2) that the length r%kl of the vector is equal to the reciprocal of the interplanar spacing dh]d. 

When a crystal is submitted to a monochromatic x ray beam with wavelength A, a given crystallographic family plane hkl, with an interplanar spacing dhki, gives rise to a cooperative diffusion, i.e. to diffraction, if the normal to these hkl planes makes an angle (7t/2) — 6 with the beam this is the well known Bragg s law,... 

The introduction of reciprocal space allows for both the characterization by unitary projection (planes are represented by dots) and completes projection (both information of orientation and interplanar distance are present) for the crystallographic planes of direct space as well as the possibility of their quantum representation by the characterization of Brillouin Zones of wave vectors associated to the crystal s eigen-states (Pettifor, 1995). 

Under these conditions, the eigen-functions, not being modified by the existence of the potential perturbation of first order, there remains the energetic correction (7.8) to be evaluate for the energy of free electrons (3.58) at the frontier of first Brillouin zone, i.e., in the dot of reciprocal space corresponding to entire family of crystallographic planes of direct space, with interplanar a-distance, thus evaluating the entire periodical potentials influence of type (3.75) upon the quasi-free electrons in crystal. 

In a simplified representation, the principles of an electron diffraction device are depicted in Figure 7.1. The main part of the instrument is the vacuum column, in which all the elements of the device are contained. The heating filament 1 (cathode) emits electrons by thermoemission, which then speed in an electrostatic field with the potential difference U. Passing through the diaphragm, monochromatic electrons (i.e., electrons with constant wavelength, refer to formula (7.1.2)) fall onto the polycrystaUine film sample 3. The polycrystalline sample contains an enormous number of small microcrystals, absolutely chaotically oriented in space. From the whole set of microcrystals, some are oriented with their crystallographic planes with interplanar distance with respect to... 

The expression (3.51) can be further used for rewriting the interplanar distance which in the direct space is measured on the normal direction to the crystallographic hkl) plane, for example, through the identities based... 

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