Lattice beams notes

As noted in the introduction, vibrations in molecules can be excited by interaction with waves and with particles. In electron energy loss spectroscopy (EELS, sometimes HREELS for high resolution EELS) a beam of monochromatic, low energy electrons falls on the surface, where it excites lattice vibrations of the substrate, molecular vibrations of adsorbed species and even electronic transitions. An energy spectrum of the scattered electrons reveals how much energy the electrons have lost to vibrations, according to the formula ... 

Note that parallel lines b and c have the same set of indices. Similarly, in three-dimensional lattices, planes with the same sets of indices are parallel a beam of x-rays striking one plane at an angle permitting diffraction must strike all planes having the same set of indices at the same angle, also permitting diffraction. 

FIGURE 7.1. The relative orientations of the reciprocal lattice of a crystal (expressed as a and b ), and its indexed X-ray diffraction pattern (expressed as h and k). In the diffraction pattern the intensities of the diffracted beams (/) (the blackness of spots on X-ray film, for example) and the directions of travel (sin 6) (positions of spots on the X-ray film) are measured. Note the relationship of a to h, and b to k. From the positions of spots on the photographic film it is possible to deduce the dimensions of the reciprocal lattice, hence of the crystal lattice, hence the indices hkl of each Bragg reflection. 

FIGURE 4.20 Constructive interference of x-rays scattered by atoms in lattice planes. Three beams of x-rays, scattered by atoms in three successive layers of a crystal are shown. Note that the phases of the waves are the same along the line CH, indicating constructive interference at this scattering angle 26. 

FIGURE 1.14 Seen here is the hk0 zone diffraction pattern from a crystal of M4 dogfish lactate dehydrogenase obtained using a precession camera. It is based on a tetragonal crystal system and, therefore, exhibits a fourfold axis of symmetry. The hole at center represents the point where the primary X-ray beam would strike the film (but is blocked by a circular beamstop). Note the very predictable positions of the diffraction intensities. All the intensities, or reflections, fall at regular intervals on an orthogonal net, or lattice. This lattice in diffraction space is called the reciprocal lattice. 

It is important to note the features of diffraction pattern of a single crystal in TEM the diffraction pattern represents a reciprocal lattice plane with reciprocal lattice points which are shown as the diffraction spots, and the reciprocal lattice plane contains the diffraction of lattice planes belonging to one crystal zone, of which the axis is parallel to the transmitted beam. These features are well illustrated in Figure 2.9 in which the zone axis is [001] of a cubic crystal and the reciprocal lattice plane consists of diffraction spots from lattice planes (hkO). 

The most important consequence the symmetry elements present in a crystal is that some (hkl) planes have F(hkl) = zero, and so will never give rise to a diffracted beam, irrespective of the atoms present. Such missing diffracted beams are called systematic absences. This can most easily be understood in terms of the vector representation described above. Suppose that a crystal is derived from a body centred lattice. In the simplest case, the motif is one atom per lattice point, and the unit cell contains atoms at 000 and Zz V2 Zz, (Figure 6.13a). [Note the cell can have any symmetry]. The structure factor for each (hkl) set is given by ... 

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