Core density

This treatment may be extended to spheres core-shell structure. If the core density is p 0 to fp the shell density is p2 in the range o density of the surrounding medium is Pq, th of the structure factor becomes... 

The MaxEnt valence density for L-alanine has been calculated targeting the model structure factor phases as well as the amplitudes (the space group of the structure is acentric, Phlih). The core density has been kept fixed to a superposition of atomic core densities for those runs which used a NUP distribution m(x), the latter was computed from a superposition of atomic valence-shell monopoles. Both core and valence monopole functions are those of Clementi [47], localised by Stewart [48] a discussion of the core/valence partitioning of the density, and details about this kind of calculation, may be found elsewhere [49], The dynamic range of the L-alanine model... 

At Leica Biosystems Newcastle Ltd., all control cell lines undergo strict quality control evaluation using haematoxylin eosin (H E) and Oracle HER2 Bond IHC System (Leica Microsystems Newcastle, UK) stained sections. This allows for evaluation of the three main cell line characteristics cellular morphology, IHC profile, and core density (see Table 6.1)... 

TABLE 6.1 IHC Profile, Cellular Morphology, and Core Density Quality Control Procedures employed at Leica Microsystems... 

Core Density. At Leica Biosystems Newcastle Ltd., the density of viable cell numbers within each core is strictly regulated, yielding consistent and reliable material for EQA assessment. 

At 2000 K there is sufficient energy to make the H2 molecules dissociate, breaking the chemical bond the core density is of order 1026 m-3 and the total diameter of the star is of order 200 AU or about the size of the entire solar system. The temperature rise increases the molecular dissociation, promoting electrons within the hydrogen atoms until ionisation occurs. Finally, at 106 K the bare protons are colliding with sufficient energy to induce nuclear fusion processes and the protostar develops a solar wind. The solar wind constitutes outbursts of material that shake off the dust jacket and the star begins to shine. 

Thermoplastic structural foams with bulk densities not less than 50% of the solid resin densities are considered. Cellular morphology, uniform-density cell behaviour, the I-beam concept in designing, core-density profile and the role of the skin, mechanical properties, and ductile-brittle transitions are discussed. 63 refs. 

A number of different atom-centered multipole models are available. We distinguish between valence-density models, in which the density functions represent all electrons in the valence shell, and deformation-density models, in which the aspherical functions describe the deviation from the IAM atomic density. In the former, the aspherical density is added to the unperturbed core density, as in the K-formalism, while in the latter, the aspherical density is superimposed on the isolated atom density, but the expansion and contraction of the valence density is not treated explicitly. 

An Earth example not previously discussed deals with the roles of temperature and pressure on the density of ice cores (Marion and Jakubowski 2004). Gow (1971) has shown that the density of deep ice cores under pressure relaxes elastically as soon as the cores are extracted. In Fig. 5.9, we used our model parameters to calculate how the density of an ice core from Antarctica (Gow et al. 1968 Gow 1971) would vary with core temperature at 1 atm, which is what is measured at the surface with corrections for temperature, to the same core under both temperature and pressure constraints. At 1 atm pressure, the core density changes linearly with temperature (Fig. 5.9), in agreement with our model (Fig. 3.2) and the Gow (1971) results (see his table 1). In contrast, the density of the ice core subjected to both temperature... 

As HJ point out, in practice the above scheme is complicated by the slight overlap of the core densities on different atoms. In their work, the ground-state energy EA> its components EA and and core and valence densities Py(r) and Pc(r), respectively, are determined self-consistently for each constituent atom. The core density is then renormalized to a suitable radius Rc by the addition of a term... 

This core renormalization is such that the core density is changed predominantly in the region near Rc. The renormalized core then has the correct total charge and the core density vanishes for r>Rc. 

The binding energy curve E °(r) in equation (183) was then found by HJ by evaluating Ev(r) using the frozen core density determined above. The energy curve depends on Rc and can be calculated only for r>2Rc. The usefulness of the procedure lies in the large cancellation of the effects of core renormalization in the molecule and atoms, so that c (r) is much less dependent on Rc than its two components separately. In some cases, such as Cua, HJ find that this error cancellation is almost complete, and the cancellation is substantial even in the heavier alkali dimers, which have very extended cores. 

To assess the effect of chemical bonding on the electron density it is assumed that the effect on core density is negligible, and that the total distortion will be due to valence-charge migration. The molecular charge density may hence be written as... 

A disadvantage of the (LF) theory is the prediction of AV from the close-packed densities. Most of the hard core densities, q, predicted by the (LF) theory are about 10% smaller than their known crystalline densities, which is most probably due to the packing factor of the lattice. There have been few applications of this theory to a real mixture, but from the work done by Sanchez it seems that the introduction of an entropy correction factor into the model is inevitable if it is going to be appUed to a system with specific interactions. 

We turn next to the kinetic energy of the electrons. In Eq. (15-5) was given the kinetic energy for the uniform valencc-clcctron gas, and we might at first think that this could be directly added to the kinetic energy of the electrons in the cores. This would not be consistent with the way we calculated kinetic energy for the overlap interaction in Chapter 7, however, and would not be correct. In Appendix C we computed the kinetic energy locally in terms of the five-thirds power of the total electron density at each point. Let us write the valence density N and the core density N r)-, notice that [N + Thus, even in the free-... 

In the region between cores, N r) is zero and the contribution is equivalent to Eq. (15-5). Within the cores, the core density may be very much larger than the valence electron density and we may expand Eq. (15-10) in fV/Nc, keeping only the first two terms ... 

As is seen, the difference in the eigenvalues calculated with both approaches does not exceed 0.1 eT, as a rule. We did not find any essential dependence on the form of the core density added to the valence pseudodensity when calculating the exchange (i. e., as a superposition of the numerical atomic densities or simulated by a spherical Bessel function pc = Asin Br)fr [58]). The orbital energy of the 2cr LUMO calculated in both approaches differs by 1 eV, and the difference depends on the shape of the innermost section of the pseudoorbitals. Such a behavior is rather common [59] - [61]. 

Bertka and Fei (1997) experimentally determined mantle mineral stabilities using the Wanke and Dreibus (1988) model composition. The mineral stability fields and resulting mantle density profile, as well as core densities and positions of the core-mantle boundary for a range of model core compositions, are illustrated in Figure 9. The moment of inertia calculated from these experimental data (0.354) is consistent with the Mars Pathfinder measurement (Bertka and Fei, 1998). However, this model requires an unrealistically thick crust. 

Anderson O. E. and Isaak D. G. (2002) Another look at the core density dehcit of Earth s outer core. Phys. Earth Planet. Int. 131, 10-27. 

Presently EOS data are limited to crystalline phases at relative low pressures and/or low temperatures (Eigure 7). These data demonstrate that all the proposed light elements are capable of reducing the density of iron as expected. The efiftciency of density reduction (or the amount of light element needed to account for the core density deficits) depends on the stmcture and EOS of the alloys or compounds. Assuming that the... 

---