Free-atom model

In the MEM/Rietveld analysis, each of the observed structure factors of intrinsically overlapped reflections (for instance, 333 and 511 in a cubic system) can be deduced by the structure model based on a free atom model in the Rietveld refinement. In such a case, the obtained MEM charge density will be partially affected by the free atom model used. In order to reduce such a bias, the observed structure factors should be refined based on the deduced structure factors from the obtained MEM charge density. The detail of the process is described in the review article [9,22-24]. In addition, the phased values of structure factors based on the structure model used in Rietveld analysis are used in the MEM analysis. Thus, the phase refinement is also done for the noncentrosymmetric case as P2, of Sc C82 crystal by the iteration of MEM analysis. The detail of the process is also described elsewhere [25]. All of the charge densities shown in this article are obtained through these procedures. 

Provided the free atom model is valid, the ratio of number of photons in the first band to the number of photons in the second band is directly related to an effective atomic number, Z, whose value can be determined using either a curve linking the theoretical points of Fig. 10 or its linear approximation. Because this procedure measures photons originating preferentially in the so-called tip region of the source spectrum, it is denoted here by the acronym HETRA - High Energy Tip Region Analysis. 

Crystal-field theory (CFT) was constructed as the first theoretical model to account for these spectral differences. Its central idea is simple in the extreme. In free atoms and ions, all electrons, but for our interests particularly the outer or non-core electrons, are subject to three main energetic constraints a) they possess kinetic energy, b) they are attracted to the nucleus and c) they repel one another. (We shall put that a little more exactly, and symbolically, later). Within the environment of other ions, as for example within the lattice of a crystal, those electrons are expected to be subject also to one further constraint. Namely, they will be affected by the non-spherical electric field established by the surrounding ions. That electric field was called the crystalline field , but we now simply call it the crystal field . Since we are almost exclusively concerned with the spectral and other properties of positively charged transition-metal ions surrounded by anions of the lattice, the effect of the crystal field is to repel the electrons. 

It is well known that the energy profiles of Compton scattered X-rays in solids provide a lot of important information about the electronic structures [1], The application of the Compton scattering method to high pressure has attracted a lot of attention since the extremely intense X-rays was obtained from a synchrotron radiation (SR) source. Lithium with three electrons per atom (one conduction electron and two core electrons) is the most elementary metal available for both theoretical and experimental studies. Until now there have been a lot of works not only at ambient pressure but also at high pressure because its electronic state is approximated by free electron model (FEM) [2, 3]. In the present work we report the result of the measurement of the Compton profile of Li at high pressure and pressure dependence of the Fermi momentum by using SR. 

Kaminski, G. Duffy, E. M. Matsui, T. Jorgensen, W. L., Free-energies of hydration and pure liquid properties of hydrocarbons from the OPLS all-atom model, J. Phys. Chem. 1994, 98, 13077-13082. 

In the free electron model, the electrons are presumed to be loosely bound to the atoms, making them free to move throughout the metal. The development of this model requires the use of quantum statistics that apply to particles (such as electrons) that have half integral spin. These particles, known as fermions, obey the Pauli exclusion principle. In a metal, the electrons are treated as if they were particles in a three-dimensional box represented by the surfaces of the metal. For such a system when considering a cubic box, the energy of a particle is given by... 

These three structures are the predominant structures of metals, the exceptions being found mainly in such heavy metals as plutonium. Table 6.1 shows the structure in a sequence of the Periodic Groups, and gives a value of the distance of closest approach of two atoms in the metal. This latter may be viewed as representing the atomic size if the atoms are treated as hard spheres. Alternatively it may be treated as an inter-nuclear distance which is determined by the electronic structure of the metal atoms. In the free-electron model of metals, the structure is described as an ordered array of metallic ions immersed in a continuum of free or unbound electrons. A comparison of the ionic radius with the inter-nuclear distance shows that some metals, such as the alkali metals are empty i.e. the ions are small compared with the hard sphere model, while some such as copper are full with the ionic radius being close to the inter-nuclear distance in the metal. A consideration of ionic radii will be made later in the ionic structures of oxides. 

All models of this type have become known colloquially by the misnomer free-particle model. Diverse objects with formal resemblance to chemical systems are included here, such as an electron in an impenetrable sphere to model activated atoms particle on a line segment to model delocalized systems particle interacting with finite barriers to simulate tunnel effects particle interacting with periodic potentials to simulate electrons in solids, and combinations of these. 

The spectra of linear polyenes are modelled well as one-dimensional free-electron systems. The cyanine dyes are a classical example. They constitute a class of long chain conjugated systems with an even number n of 7r-electrons distributed over an odd number N = n — 1 of chain atoms. The cyanine absorption of longest wavelength corresponds to promotion of an electron from the highest occupied energy level, En/2 to the lowest unoccupied level, such that in terms of a free-electron model... 

The fact that evaporated potassium arrives at the surface as a neutral atom, whereas in real life it is applied as KOH, is not a real drawback, because atomically dispersed potassium is almost a K+ ion. The reason is that alkali metals have a low ionization potential (see Table A.3). Consequently, they tend to charge positively on many metal surfaces, as explained in the Appendix. A density-of-state calculation of a potassium atom adsorbed on the model metal jellium (see Appendix) reveals that the 4s orbital of adsorbed K, occupied with one electron in the free atom, falls largely above the Fermi level of the metal, such that it is about 80% empty. Thus adsorbed potassium is present as K, with 8close to one [35]. Calculations with a more realistic substrate such as nickel show a similar result. The K 4s orbital shifts largely above the Fermi level of the substrate and potassium becomes positive [36], Table 9.2 shows the charge of K on several metals. 

Single crystals of free lipid A or LPS are as yet not available. Therefore, the most promising approach to obtain molecular models is to perform theoretical calculations. After the chemical structures of enterobacterial lipid A had been elucidated, this methodology was successfully applied with heptaacyl S. minnesota lipid A (220) and hexaacyl E. coli Re LPS (221). As an example, Fig. 13 shows the atomic model of the E. coli lipid A molecule, as calculated by Kastowsky et al. (221) using energy-minimization techniques. 

The free-electron model is a simplified representation of metallic bonding. While it is helpful for visualizing metals at the atomic level, this model cannot sufficiently explain the properties of all metals. Quantum mechanics offers a more comprehensive model for metallic bonding. Go to the web site above, and click on Web Links. This will launch you into the world of molecular orbitals and band theory. Use a graphic organizer or write a brief report that compares the free-electron and band-theory models of metallic bonding. 

The first simplification in the TDAN model is to consider only a few electronic orbitals on the scattered atom. For many applications, it is sufficient to consider one only, that from which, or into which, an electron is transferred. Let the ket 10 > denote the spatial part of the orbital. When far from the surface, suppose its energy is So> let Uq be the Coulomb repulsion integral associated with the energy change when it is occupied by two electrons of opposite spin. In terms of creation and annihilation operators and Co for 0>, with ff( = aorfi)a spin index, that part ofJt which refers to the free atom is... 

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