Three density

Solid Density. SoHds can be characterized by three densities bulk, skeletal, and particle. Bulk density is a measure of the weight of an assemblage of particles divided by the volume the particles occupy. This measurement includes the voids between the particles and the voids within porous particles. The skeletal, or tme soHd density, is the density of the soHd material if it had zero porosity. Fluid-bed calculations generally use the particle... 

Three density functional theories (DFT), namely LDA, BLYP, and B3LYP, are included in this section. The simplest is the local spin density functional LDA (in the SVWN implementation), which uses the Slater exchange functional [59] and the Vosko, Wilk and Nusair [60] correlation functional. The BLYP functional uses the Becke 1988 exchange... 

The database of the RGNR project consists of 1 datum per 4 km. In this paper three densities of data were used to delineate the different-scale geochemical patterns. 

According to our minimum knowledge assumption, there should not be any correlations in the probability density p. This means that p factorizes into the product of three densities according to ... 

Figure 11.6. Three density maps at differing resolutions a, 1.3 A b, 2.1 A c, 3.0 A. See color insert.

Figure 5. The evolution of the density gradient as a function of time in TLF cell displayed by the positions of the focused zones of three density marker beads suspended in Percoll diluted with water. (Reproduced with permission from reference 14. Copyright 1994 Springer-Verlag.)...

Figure 1. The dependence of the average gas-solid potential energy for methane in a gr hitic slit pore upon position within the pore is shown for three values of the pore filling 0.040 (solid line), 0.059 (short dashes) and 0.077 (long dashes). Also given is the z-dependence of the simulated chemical potential IkT for these three densities in a pore of width 14.8 A From. Ref [22], Sep. ScL and Tech. 27 (1992), 1837-1856.

Figure 13. Part (a) gives the probability of finding a given value of cos P for adsorbed O2 molecules at three values of the layer density (given in units of molecules/A ). The changes in these curves show how these nwlecules reorient from mostly flat on the surface at p = 0.0763 molecules/to nearly all perpendicular at 0.1145 molecules/A. Part (b) shows the distance dependence of the local densities for these molecules at the three densities of part (a) plus a density corresponding to partial bilayer formation The local density p (z)=p(z)x(2.46 A) and the distance z =z/2.46 A. From Ref [54], Can, J. Chera. 66 (1988) 866-874.

Figure 2.2 Selected families (DDj(aj)) of density domains of the water molecule, as calculated with the GAUSSIAN 90 ab initio program [253] and the GSHAPE 90 molecular shape analysis program [254], using a 6-3IG basis set. There are only two topologically different types of families of density domains either a single density domain, or a family of three density domains. The sequence of topologically distinct cases provides a topological description of chemical bonding.

It is worth emphasizing that there are only finitely many (actually only three) density threshold ranges of the water molecule which are distinguishable using the simplest topological criterion of identifying maximum connected components, the density domains DD (a) of the molecular bodies DD(a). 

In the above paragraphs, we have already introduced several approximations in the description of the shift and relaxation rates in transition metals, the most severe being the introduction of the three densities of states Dsp E ),Dt2g(E ), and Deg E ). The advantage is that these values can be supplied by band structure calculations and that the J-like hyperfine field can sometimes be found from experiment. We have no reliable means to calculate the effective Stoner factors ai that appear in Eq. (2), and the disenhancement factors ki in the expression for the relaxation rate, Eq. (4), are also unknown. It is often assumed that k/ can be calculated from some /-independent function of the Stoner parameter k (x), thus k/ = k((X/). A few models exist to derive the relation k((x), all of them for simple metals [62-65]. For want of something better they have sometimes been applied to transition metals as well [66-69]. We have used the Shaw-Warren result [64], which can be fitted to a simple polynomial in rx. There is little fundamental justification for doing so, but it leads to a satisfactory description of, e.g., the data for bulk Pt and Pd. 

Figure 4.5 Simulated densities of Ar at 77.5 K (thick line) and CO2 at 195.5 K (thin line) adsorbed on an fee array of C60 molecules approximated as spherical bodies are plotted as a fimetion of distance from the sohd adsorbent. Adsorption on the outermost layer of C60 spheres produces three density peaks one at 1.4 nm in the deep (and strongly interacting) crevices located in the centers of the squares formed by four C60 molecules one at 1.8 nm in the crevices between pairs of neighboring C60 s and one at 2.1 nm for adsorbed molecules directly over a C60. Peaks at larger distances reflect structure in the adsorbed fluid, with the CO2 density decreasing to zero after second layer formation at 195.5 K because of its reladvely small amount in the simulation box compared to the Ar multilayer densities at 77.5 K.

Three density functions are plotted for t distributions with 5, 30, and 200 df in Figure 6.6. The greater the number of df, the "flatter" the tails. In the figure, the two curves that are closest together are associated with 30 and 200 df. It is interesting to note (and a convenient fact) that the area under the density curve between any two points for the case with 30 df is not appreciably different from the case with 200 df. 

Rates were measured at — 196°C and 1 atm pressure for pellets of three densities. The rate of reaction was also measured for the catalyst in the form of 60-micron (average size) particles. With this small size 1, so that / = 1.0. The rate data and pellet properties are given in Table 11-5. 

Case (C) involves three (electron) densities, one for each labyrinth and one for the surface layer. Because relative intensities are all that are usually measured, only one parameter is needed to allow for any combination of three densities. The scattering function calculated in case (B) is frequently referred to as the structure factor of the surface for a minimal surface characterized by a self-dual skeletal graph, this will correspond to a different space group than that of (A) or (C), or of case (B) with nonzero mean curvature (see the D family, above). 

FIGURE 39 High-load melt index and molecular weight (Mw) values of polymers made at three densities by adjusting the amounts of 1-hexene added to the reactor. The addition of comonomer tended to decrease the polymer MW (raise HLMI). 

Figure 17 shows the time evolution of the potential energy of RDX for temperatures from T = 1200 to 3000 K for the three densities studied. We find that higher densities lead to faster chemistry, and the temporal evolution of the potential energy can be described rather well with a simple exponential function, from which we extract an overall characteristic time of reaction (the solid lines in Fig. 17 shows the exponential functions). 

Figure 18. Characteristic time vs. inverse temperature from ReaxFF MD simulations for three densities 2.11 g/cm 1.68 g/cm, and 0.21 g/cm. We also show the Arrhenius behavior obtained from experimental ignition times [42] in HMX.

Figure 19. Time evolution of products N2 (green), H2O (blue), CO (red), and CO2 (black), and intermediate NO2 (purple), for temperatures 1500 K (left) and 3000 K (right) and three densities 2.11 g/cm (top), 1.68 g/cm (middle), and 0.21 g/cm (bottom).

Figure 26. Characterization of the inherent structures for the model calamitic system GB(3,5,2, 1) ( = 256). (a) Parallel radial distribution function g (/ ) for the inherent structures at all temperatures considered along the isochor at density p = 0.32. Note that the curves for the highest five temperatures are nearly superposed on each other. For others, the amplitude of the peaks gradually increases as the temperature drops, (b) Evolution of the 6-fold bond orientational order parameter 4>6 for the inherent stmctures with temperature at three densities. (Reproduced from Ref. 144.)...

Fig. 9. Three density-of-states curves that fit the hypothetical field-effect data of Fig. 10. The density of states of Spear and LeComber (1976) (Fig. 4) is shown by the solid curve. (From Goodman and Fritzsche ( 980).]...

Fig. 27. Three density-of-states curves with different mobility gaps of1.44,1.54,and 1.64 eV corresponding to the values of a (see Eq. (13)] of 1.2 X 1(F (dotted), 2.0 X 10 (dotted-dashed), and 1.8 X 10 (dashed), as indicated. The original density of states of Spear and LeComber (1976) is shown by the solid curve. Each of these densities of states fits the hypothetical data of Fig. 28. [From Goodman and Fritzsche (1980).]...

The hexagonal phase can be described by a superposition of three density waves with wavevectors... 

Sun, Y.-R, C. E. Bunker, and N. B. Hamilton 1993, Ry scale in vapor phase and in supercritical carbon dioxide. Evidence in support of a three-density-region model for solvation in supercritical fluids . Chem. Phys. Lett. 210, 111. 

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