Rigid sphere atomic model

Here Hd is the number of atoms in a unit cell, the volume of which is V, and is the shortest interatomic distance in the arrangement. The definition contains a division by /2 so that the parameter D becomes unity for close-packing structures. Kepler s conjecture ensures that the parameter D is always less than or equal to unity. The fraction of space occupied (fi in the rigid-sphere model, which is often used in the discussion of metallic structures, is proportional to the parameter D and the relation is as follows. 

The natural way to seek improvement on the plain FeS2—m, hep X2 model would be to take into account the T atoms in the form of rigid spheres whose size exceeds those of the original octahedral cavities. (The clearly less realistic FeS2—m, hep X model need not be considered separatly in this connection, but it should be noted that all conclusions for this model would be virtually identical with those drawn for FeS2—m, hep X2.) However, if the only modifying influence of T was due to its attributed spherical shape, this would produce a uniform overall expansion of all cell dimensions without appreciable alterations in the axial ratios. The thus modified atomic arrangement would fit the experimental... 

This model is frequently used considering mono-atomic uncharged molecules. However, this model gives a very crude representation of the actual physics (e.g., repulsive forces and volume of sphere), since molecules in fact are complicated electronic structures, and can by no means resemble rigid spheres. 

Alternative estimates of the transport coefficients can be obtained from the rigorous Chapman-Enskog expansion method of mono-atomic gases at low densities (e.g., [24] [25] [12] [61] (p 202) [28]). The transport coefficients deduced from the Chapman-Enskog kinetic theory with the rigid elastic spheres interaction model yield (e.g., [39] sect 8.2 [61], p 202) ... 

We know from quantum mechanics that atoms and ions do not have precisely defined radii. However, the concept of an ion as a hard sphere with a fixed radius is very useful when predicting crystal structures. Experimental evidence shows that such a model has some justification the model often works. Nevertheless, always bear in mind that atoms and ions are not rigid spheres and their size will be affected by their local environment. 

In this case we do not rely on facts observed directly but begin by designing on the basis of reasonable assumptions a model of the perfect gas on the proper choice of this model depends the success of the whole undertaking. Wc assume that both the gases enclosed in volumes V and V2 consist respectively of n and ri2 spherical atoms or molecules, of very small total volume as compared with Vi and V2 (e.g. roVir) which exert no forces on one another and behave on collision as elastic rigid spheres following the laws of classic mechanics. In each of the partial volumes these particles should execute entirely random motions at velocities w characterized by MaxwelPs law of distribution, the average value w of which can be calculated simply from the Boyle-Charles equation... 

While adhesion between spheres was not considered in the Hertzian contact model, understanding of interactive forces between solids, and particularly colloids, showed a big progress in the early twentieth century. By integrating attractive interactions that follow the power law, Bradley found the total force between two rigid spheres separated at an equilibrium distance Zq is given by Ppuii-off = - JirwR, where w is the work of adhesion [11]. While Bradley only took Zq to be the atomic equilibrium separation, that is, a constant value, one can obtain... 

Models, derived from STM observations or from the fitting of diffraction (such as LEED and XRD), provide information about static atomic structures of the surface. These structural models succeed in describing specific situations in terms of the static positions of the adsorbates that are often assumed rigid spheres. The general characteristics of electronic structures and atomic arrangement on a variety of chemisorbed metal surfaces have been now fairly determined [20, 21] with many landmark reviews on the progress in this field [15, 17, 20, 22-32]. 

Fig. 55. Debye temperature, d, and density of states at the Fermi level, N(Ep), for Y(Ni xCox)2B2C and Y(Ni xCux )2B2C as a function of the Co/Cu substitution level x. Symbols results derived from a relativistic band calculations in the atomic sphere approximation. Curves (in lower panel) rigid band model. After Ravindran...

A1P04-31 (structure type ATO) has unidimensional channels with nominal diameter 5.4 A. To model Xe/AlP04-31 atomistically, we assume that A1P04-31 is rigid and defect free with the experimentally determined crystal structure [7]. Xe atoms are represented as spheres, and Xe-Xe and Xe-0 interactions are taken to be Lennard-Jones potentials using previously derived parameters [5,8]. 

Assuming a fixed band structure (the rigid band model), a decrease in the density of states is predicted for an increase in the electron/atom ratio for a Fermi surface that contacts the zone boundary. It will be recalled that electrons are diffracted at a zone boundary into the next zone. This means that A vectors cannot terminate on a zone boundary because the associated energy value is forbidden, that is, the first BZ is a polyhedron whose faces satisfy the Laue condition for diffraction in reciprocal space. Actually, when a k vector terminates very near a BZ boundary the Fermi surface topology is perturbed by NFE effects. For k values just below a face on a zone boundary, the electron energy is lowered so that the Fermi sphere necks outwards towards the face. This happens in monovalent FCC copper, where the Fermi surface necks towards the L-point on the first BZ boundary (Fig. 4.3f ). For k values just above the zone boundary, the electron energy is increased and the Fermi surface necks down towards the face. 

In addition to the needs following from the stoichiometry the VEP also takes those requirements of structural geometry into consideration which coincide with the principles of symmetry and connection. The space principle is also discussed here, however, with the difference that we do not start from a model of rigid mutually contacting spheres but also plastical or soft contacts between atoms and ions are admitted. Let us try and delimit the physical meaning of the VEP by the following consideration ... 

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