Cell volumes, equations

In this equation is the 6x6 matrix of second derivatives (elements Ide j), emd are the corresponding 3N x 6 and 6 x,3N mixed coordinate/strain matrices, is the 3iV X 3N second-derivative coordinate matrix and V is the unit cell volume. It is the second term in Equation (5.54) that accounts for internal atomic relaxations as the cell distorts. 

Several methods are also available for determination of the isothermal compressibility of materials. High pressures and temperatures can for example be obtained through the use of diamond anvil cells in combination with X-ray diffraction techniques [10]. kt is obtained by fitting the unit cell volumes measured as a function of pressure to an equation of state. Very high pressures in excess of 100 GPa can be obtained, but the disadvantage is that the compressed sample volume is small and that both temperature and pressure gradients may be present across the sample. 

Equation 5.15 is based on 13 values, has a correlation coefficient R2 = 0.99, and does not include the cell volume of the pure compound Ca2Si04 (see figure 5.6). It can be opportunely modified to give molar volumes according to... 

Equation (67) clearly shows that when d is decreased either the detector cell volume has to be decreased proportionally or the column diameter should be increased in order to avoid significant peak distortion. With typical values such as 6 = 0.1 [Pg.26]

Substitution of this value for into equation (21.9) gives the cell volume Vc for subsequent use in the working equation (21.1). 

Equation (1) expresses the crystal polarization (P, C/iiF). as a function of the dipole moment (p, Cm) and the unit cell volume (V, iif). In PVDF, it suffices to express Eq. (1) in scalar form, where it is miderstood that P and p represent the components of the polarization and dipole moment vectors parallel to the ( -crystal axis. This arrangement of dipoles produces a significant local electric field in the... 

Equation (4) expresses G as a function of temperature and state of applied stress (pressure) (o. Pa), (/(a) is given by the force field for the set of lattice constants a, Vt is the unit cell volume at temperature T, and Oj and are the components of the stress and strain tensors, respectively (in Voigt notation). The equilibrium crystal structure at a specified temperature and stress is determined by minimizing G(r, a) with respect to die lattice parameters, atomic positions, and shell positions, and yields simultaneously the crystal structure and polarization of minimum free energy. 

Equation (5.15) describes one structure factor in terms of diffractive contributions from all atoms in the unit cell. Equation (5.16) describes one structure factor in terms of diffractive contributions from all volume elements of electron density in the unit cell. These equations suggest that we can calculate all of the structure factors either from an atomic model of the protein or from an electron density function. In short, if we know the structure, we can calculate the diffraction pattern, including the phases of all reflections. This computation, of course, appears to go in just the opposite direction that the crystallographer desires. It turns out, however, that computing structure factors from a model of the unit cell (back-transforming the model) is an essential part of crystallography, for several reasons. 

During synthesis of a polymer, particularly of polyurethane, gaseous products can appear. Therefore, a complete model of the process must take into account (at least in some cases) the possibility of local evaporation and condensation of a solvent or other low-molecular-weight products. Such a complex model is discussed for chemical processing of polyurethane that results in formation of integral foams in a stationary mold.50 In essence, the model is an analysis of the effects of temperature in a closed cell containing a solvent and a monomer. An increase in temperature leads to an increase in pressure which influences the boiling temperature of the solvent and results in an increase in cell volume. The kinetics of polymerization is described by a simple second-order equation. The... 

It is important, first, to realize that efficiency is not a function solely of the column. Bad extracolumn parameters, such as detector cell volume or tubing diameters, can make the best column in the world look terrible. Second, efficiency measurements are very poor ways of comparing or purchasing columns unless all other parameters are constant. Many columns are bought and sold because they have a higher plate count than someone else s column. The efficiency calculations could have been made with different equations, on different compounds, on different machines, at different flow rates, all of which will have a profound effect on efficiency. The only valid use of plate counts that I have found is in column comparisons where all other variables are equal, or in following column aging over a period of days or months. 

Lattice coordination numbers (2) and the cell volumes (vjj) for both the pure components and mixture lattices are assumed to have the same value. The partition function for this ensemble can be formulated following Equation 2. It is assumed now that the partition function, far from the binary critical point can be approximated by its largest term. Since molecule segments and holes can distribute themselves non-randomly, the partition function must incorporate terms to account for this effect. The nonrandomness correction rjj... 

Once the lattice parameters are measured at various pressures, the pressure dependence of the unit cell volume is fitted with an equation of state (EOS). The simplest, and most used, is the Mumaghan [9] EOS based on the assumption that the bulk modulus has a linear dependence with the pressure ... 

Fig. 4 Effect of temperature on the cellular DMSP C-ratio (mohmol) in Emiliania huxleyi (closed symbols recalculated from van Rijssel and Gieskes (2002). For comparison, data from experiments with Phaeocystis antarctica (4°C Stefels and van Leeuwe 1998) and P. globosa (10°C Stefels and van Boekel 1993) are included. Except for P. antarctica, carbon data are calculated from cell volume data (see Table 1). Equation of the power fit to the E. huxleyi data is DMSP C = 0.009 + 0.17 T1 6...

---