Diffraction by lattice

Fig. 4.3 Schematic illustration of the formation of the layer lines on an oscillation photograph. (a) Diffraction by lattice in one dimension, (b) Experimental arrangement.

The notion of a reciprocal lattice cirose from E vald who used a sphere to represent how the x-rays interact with any given lattice plane in three dimensioned space. He employed what is now called the Ewald Sphere to show how reciprocal space could be utilized to represent diffractions of x-rays by lattice planes. E vald originally rewrote the Bragg equation as ... 

Using this equation, E)wald applied it to the case of the diffraction sphere which we show in the following diagram as 2.1.10. on the next page. Study this diagram carefully. In this case, the x-ray beam enters the sphere enters from the left and encounters a lattice plane, L. It is then diffracted by the angle 20 to the point on the sphere, P, where it... 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the sphere of reflection, is reciprocal to the wavelength of X-ray radiation—that is, IX. The reciprocal lattice rotates exactly as the crystal. The direction of the beam diffracted from the crystal is parallel to MP in Figure 3.7 and corresponds to the orientation of the reciprocal lattice. The reciprocal space vector S(h k I) = OP(M/) is perpendicular to the reflecting plane hkl, as defined for the vector S. This leads to the fulfillment of Bragg s law as S(hkI) = 2(sin ())/X = 1 Id. 

The term exp(-2k2c ) in (6-9) accounts for the disorder of the solid. Static disorder arises if atoms of the same coordination shell have slightly different distances to the central atom. Amorphous solids, for instance, possess large static disorder. Dynamic disorder, on the other hand, is caused by lattice vibrations of the atoms, as explained in Appendix 1. Dynamic disorder becomes much less important at lower temperatures, and it is therefore an important advantage to measure spectra at cryogenic temperatures, especially if a sample consists of highly dispersed particles. The same argument holds in X-ray and electron diffraction, as well as in Mossbauer spectroscopy. 

Crystals are made of atoms, ions or groups of atoms which repeat along the three dimensions to form a 3D lattice. Since the repeat distances in these crystal lattices are about 0.2 nm, diffraction by visible light is impossible. 

As mentioned above, the formalism of the reciprocal lattice is convenient for constructing the directions of diffraction by a crystal. In Figure 3.4 the Ewald sphere was introduced. The radius of the Ewald sphere, also called the... 

Table 5.1. Adsorption properties of metal monolayers on metal substrates. The clean substrate properties are also given for comparison. Substrates are ordered by lattice type (fee, bcc, hep, cubic, diamond and rhombic). The structures, nearest neighbor distances and heats of vaporization refer to the bulk material of the substrate or the adsorbate. VD, ID and S stand for vapor deposition, ion beam deposition and surface segregation, respectively. TD, WF and TED stand for thermal desorption, work function measurements and transmission electron diffraction, respectively...

The temperature variation of the lattice parameters, as measured by x-ray diffraction by Merlo et al. (1998) and Jarosz et al. (2000), shows no significant anisotropic spontaneous... 

The contrast observed on the micrographs results essentially from the variations in intensity of the electron beam diffracted by the 002 interferences as a function of the direction of the C-axis, both in bright field where the diffracted rays are stopped by the contrast diaphragm and in dark field where the image is formed by these rays alone. This result has been demonstrated theoretically, at least, in the case of elastic diffusion. It is found that the energy scattered in a given direction by a pregraphitic structure depends on the orientation of the lattice in relation to the incident beam (17). 

The classical theory for electronic conduction in solids was developed by Drude in 1900. This theory has since been reinterpreted to explain why all contributions to the conductivity are made by electrons which can be excited into unoccupied states (Pauli principle) and why electrons moving through a perfectly periodic lattice are not scattered (wave-particle duality in quantum mechanics). Because of the wavelike character of an electron in quantum mechanics, the electron is subject to diffraction by the periodic array, yielding diffraction maxima in certain crystalline directions and diffraction minima in other directions. Although the periodic lattice does not scattei the elections, it nevertheless modifies the mobility of the electrons. The cyclotron resonance technique is used in making detailed investigations in this field. 

This relationship between (1) diffraction by a single object and (2) diffraction by many identical objects in a lattice holds true for complex objects also. Figure 2.9 depicts diffraction by six spheres that form a planar hexagon, like the six carbons in benzene. 

Notice the starlike six-fold symmetry of the diffraction pattern. Again, just accept this pattern as the diffraction signature of a hexagon of spheres. (Now you know enough to recognize two simple objects by their diffraction patterns.) Figure 2.10 depicts diffraction by these hexagonal objects in a lattice of the same dimensions as that in Fig. 2.8. 

As mentioned earlier, the phase of a wave is implicit in the exponential formulation of a structure factor and depends only upon the atomic coordinates (Xj.,) Z-) of the atom. In fact, the phase for diffraction by one atom is 2tt(Hx- + ky. + Izj), the exponent of e (ignoring the imaginary i) in the structure factor. For its contribution to the 220 reflection, an atom at (0, /2, 0) has phase 2tt(/zx. + ky. + Izj) or 2tt(2[0] + 2[V2l + 0[0]) = 2tt, which is the same as a phase of zero. This atom lies on the (220) plane, and all atoms lying on (220) planes contribute to the 220 reflection with phase of zero. [Try the above calculation for another atom at (V2, 0, 0), which is also on a (220) plane ] This is in keeping with Bragg s law, which says that all atoms on a set of equivalent, parallel lattice planes diffract in phase with each other. 

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