Crystal imaginary terms

Crystals of the compound of empirical formula FiiPtXe are orthorhombic with unit-cell dimensions a = 8-16, h = 16-81. c = 5-73 K, V = 785-4 A . The unit cell volume is consistent with Z = 4, since with 44 fluorine atoms in the unit cell the volume per fluorine atom has its usual value of 18 A. Successful refinement of the structure is proceeding in space group Pmnb (No. 62). Three-dimensional intensity data were collected with Mo-radiation on a G.E. spectrogoniometer equipped with a scintillation counter. For the subsequent structure analysis 565 observed reflexions were used. The platinum and xenon positions were determined from a three-dimensional Patterson map, and the fluorine atom positions from subsequent electron-density maps. Block diagonal least-squares refinement has led to an f -value of 0-15. Further refinements which take account of imaginary terms in the anomalous dispersion corrections are in progress. 

An interesting feature of (4.7) is that the imaginary terms in it are proportional to (cjj - wj). These vanish only if all modes that are coupled to the impurity electron (ki+ 0 4= ki) have identical frequencies (coj = W[), or equivalently if there is one interacting mode. [Actually in an Einstein model of the crystal where all frequencies are equal, one can clearly select one interacting mode]. Otherwise, for a spread of frequencies... 

The idea that crystalline solids have a regular and periodic inner structure is a very old one, predating by a century the discovery of diffraction by crystals. The term lattice is used loosely as equivalent to the crystal structure, but in the strictest sense it is a mathematical and imaginary construct, defined as an ensemble of lattice points which have the same properties as the crystal structure it represents. A lattice point is defined as a random point in the structure all lattice points, however, have exactly the same chemical composition and surroundings. Thus, a lattice is the ensemble of all identical points in the structure. ... 

To determine the structures of drug compounds or protein molecules using X-ray crystallography, it is necessary to have these compounds or molecules available in crystalline form. For example, when crystals of protein are formed, the protein molecules are arranged in orderly fashions like tiny imaginary cubes stacked on top of each other. Each of these building blocks contains a molecule of protein and is termed a unit cell (Fig. 3.3). 

These two systems are examples from non-linear physics, where the equations can be solved in terms of elliptic functions and elliptic integrals. The reader who is not familiar with these functions, which do not arise in the same way as the previously mentioned special functions, is referred to the excellent book by Whittaker and Watson [6]. In that book, the reader will see that there are two flavours of elliptic functions, the Weierstrass and Jacobi representations, three kinds of elliptic integrals, and six kinds of pseudo-periodic functions, the Weierstrass zeta and sigma functions and the four kinds of Jacobi theta functions. Of historical interest for theoretical chemists is the fact that Jacobi s imaginary transformation of the theta functions is the same as the Ewald transformation of crystal physics [7]. 

For tilt boundaries, the value of E can also be calculated if the plane of the boundary is specified in the coordinate systems for both adjoining grains. This method is called the interface-plane scheme (Wolfe and Lutsko, 1989). In a crystal, lattice planes are imaginary sets of planes that intersect the unit cell edges. The tilt and twist boundaries can be defined in terms of the Miller indices for each of the adjoining lattices and the twist angle, , of both plane stacks normal to the boundary plane, as follows ... 

An ideal crystal surface of orientation (hkl) is an imaginary surface formed when all the atoms, ions or molecules on one side of a plane of orientation (hkl) inside the bulk crystal are removed. Such a surface is termed atomically smooth. However, an ideal surface is unstable because the asymmetry of the interatomic forces in the surface region leads to surface relaxation. This usually manifests itself by the movement of the crystal components in a direction normal to the surface plane to enable a reduction... 

There is a close relationship between the adamantane, Ci0Hi6, molecule and the diamond crystal. The Greek word adamant means diamond and diamond has been termed the infinite adamantylogue to adamantane [64], While iceane has D3/, symmetry, adamantane has Td. This high symmetry can be clearly seen when the configuration of adamantane is described by four imaginary cubes packed one inside the other, two of which are shown in Figure 3-23. Similar structures... 

Because the energy state of a Jahn-Teller complex depends on the local lattice distortions, the macroscopic long-distant strain that produces an ultrasonic wave should influence it as well. The cross effect is initiated by the Jahn-Teller complexes (1) the dispersion (i.e., frequency-dependent variation of phase velocity) and (2) attenuation of the wave. In terms of the elastic moduli it sounds as appearance (or account) of the Jahn-Teller contribution to the real and imaginary parts of the elastic moduli. For a small-amplitude wave it is a summand Ac. Obviously, interaction between the Jahn-Teller system and the ultrasonic wave takes place only if the wave, while its propagation in a crystal, produces the lattice distortions corresponding to one of the vibronic modes. 

The small term F has been added to the denominator in step 3. As Eq. 3 shows, the bandwidth can been absorbed into a complex resonance frequency,/r, if one chooses the real part as/r and the imaginary part as F. Since the crystal cannot be excited with a complex frequency, the denominator al-... 

Figure 10.3. The AG/x plots of a few imaginary mixtures to show the contributions of enthalpy (a) and entropy (b) to binary phase diagrams. Top left an ideal mixture (AH = 0) at some temperature (T > 0). Top right a nonideal mixture (AH > 0) also with an entropy term. Bottom right an ideal mixture at nonzero temperature, where A and B have a different crystal structure. ...

The equation is written as the sum of a real and an imaginary part. The real part is the non-optical acitivity term we have dropped a small contribution to it. The imaginary part is the optical activity. The expansion leading to Equation (IT2) is valid if the wavelength of light is large compared to the length of the electronic paths. If the electrons are actually free to move over the entire polymer or crystal, we must use Equation (II-1). 

Thus the number of independent components of the permittivity tensor will depend on the symmetry of the liquid crystal phase. The frequency dependence of the permittivity is described in terms of real and imaginary parts, and these also will be tensor quantities. Apart from complications of anisotropic internal fields, the static or low frequency part of the permittivity tensor can be related to the molecular polarizability and dipole moment averaged over the appropriate orientational distribution functions. 

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