Dispersion relation three-dimensional crystals

As a case study in the calculation of the relation between co and q (also known as phonon dispersion relations) for three-dimensional crystals, we consider the analysis of normal modes of vibration in fee Al. As will be evident repeatedly as coming chapters unfold, one of our principal themes will be to examine particular problems from the perspective of several different total energy schemes simultaneously. In the present context, our plan is to consider the dispersion relations in Al as computed using both empirical pair functional calculations as well as first-principles calculations. 

In Chap.3, the theory is generalized to three-dimensional crystals with a basis. Chapter 3 also contains a section dealing with the connection of lattice dynamics and the theory of elasticity. The force constants, dynamical matrix and dispersion relation are illustrated with the help of monoatomic crystals with fee structure. 

We recognize that there are applications in two- and three-dimensional waveguides (12,13) which do not have the same criteria of phase-matching as in simple crystals or that one may just as well be interested in screening these materials for the related electrooptic performance by the simple SHG powder method. (It has been shown for several organic materials that although the electro-optic and SHG x tensors are in principle unequal, due to dispersion and due to the possible contribution of atomic and molecular distortions... 

Anomalous scattering can also be used directly if the protein is small and a suitable anomalous scatterer can be used. The three-dimensional structure of the small protein, crambin, was determined by W ayne A. Hendrickson and Martha Teeter by the use of anomalous dispersion measurements. This protein contains 45 amino acid residues and diffracts to 0.88 A resolution. It crystallizes with 72 water and four ethanol molecules per protein molecule. Since there is a sulfur atom in the protein molecule, the use of its anomalous scattering was made. The nearest absorption edge of sulfur lies at 5.02 A, but for Cu Ka radiation, wavelength 1.5418 A, values of A/ and A/" for sulfur are 0.3 and 0.557, respectively. Friedel-related pairs of reflections were measured to 1.5 A resolution, and sulfur atom positions were computed from difference Patterson maps. The structure is now fully refined and a portion of an a helix was shown in Figure 12.27 (Chapter 12). 

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