Short-range force assumption

In order to make use of the flux expressions in Sects. 6, 7, and 8, it is necessary to have the singlet distribution function and - unless the short-range force assumption is used - the doublet distribution function as well. Virtually nothing is known about the doublet distribution function. If we knew how to make a reasonable guess of this function (possibly obtainable from molecular or Brownian dynamics), then we could estimate the contributions to the fluxes in Table 1 that involve the molecule-molecule interactions. 

In other words, the electrons confined to their ground state are only slightly disturbed by the incident photons Hence the NLO effects should be classified into the second category where short-range forces play a decisive role We therefore make the assumption that, in the NLO effects, the electron motion may be regarded as confined to small regions In other words, any NLO susceptibility (or second-order susceptibility) in crystals is a localized effect arising from the action of incident photons on the electrons in certain orbitals of atomic clusters ... 

These rules are simple and immediately intuitive, once the electrical characterization of a molecule in terms of its point-like multipoles is accepted. The underlying physical assumption is that the electrostatic interaction is the dominant attractive component of the intermolecular potential determining the angular shape of the dimer, while short-range forces are assumed to provide a repulsive uniform background balancing attraction at the VdW minimum. Monomer size enters the model through rule 7, which corrects for deviation from uniform repulsion when steric interactions occur below the sum of the respective VdW radii.10... 

In fig. 60 for the case of strong substrate attraction all layering transition lines accumulate at the point T = 0, H = 0 (i.e., ix = ixC0ltx). However, this is only true due to the specific assumption of a short range force arising from the surface, which in the Ising lattice gas framework leads to a surface... 

The numerator in this equation takes account of the contribution to non-ideality arising from the long range electrostatic forces obeying Coulomb s inverse square law, while the denominator takes account, in a rather crude manner, of the non-ideality introduced by short range forces when the ions come close together. This corresponds to the assumption that the ions are hard spheres, non-polarisable, non-deformable and spherically symmetrical, and manifests itself as the distance of closest approach, a. There are, of course, other short range... 

The Maier-Saupe theory of nematic liquid crystals is founded on a mean field treatment of long-range contributions to the intermolecular potential and ignores the short-range forces [88, 89]. With the assumption of a cylindrically symmetrical distribution function for the description of orientation of the molecules and a nonpolar preferred axis of orientation, an appropriate order parameter for a system of cylindrically symmetrical molecules is... 

The basic assumption of the DLVO theory used for interpretation of the particle aggregation and adsorption phenomena [17,18] is that the net interaction energy can be obtained as a superposition of the electrostatic and dispersion contributions discussed earher. AH additional short-range forces are neglected—in particular, the Bom repulsion, the forces arising due to surface deformations, chemical interactions, and so forth. 

We have assumed so far, implicitly, that the interactions are strictly local between neighboring atoms and that long-ranged forces are unimportant. Of course the atom-atom interaction is based on quantum mechanics and is mediated by the electron as a Fermi particle. Therefore the assumption of short-range interaction is in principle a simplification. For many relevant questions on crystal growth it turns out to be a good and reasonable approximation but nevertheless it is not always permissible. For example, the surface of a crystal shows a superstructure which cannot be explained with our simple lattice models. 

In addition, several new forms of atomism or corpuscularism were also introduced, the most famous of which were Descartes plenum theoiy and Newton s dynamic atomism, both of which rejected one or more of the basic assumptions of Epicurean atomism. Thus Descartes rejected both a lower hmit to particle divisibility and the existence of an interparticle vacumn or void, as well as insisting on a strong dichotomy between matter and soul, whereas Newton replaced mechanical entanglement with short-range interparticle forces of attraction and repulsion. 

Finally, we must consider the contribution of the electrostatic work required to transfer one electron into free space. After overcoming the short range chemical forces, the electron must be moved a certain distance against the electric field in the surface. Under the assumption that the lines of force of the electric field are located between the ion defects in the boundary layer and the surface charges represented by the chemisorbed gas atom, we obtain the expression afi for this electrostatic work term. is the boundary field strength represented in Equation (11), and a is the distance between the surface of the oxide and the centers of charge of the chemisorbed atoms in the a-phase. 

The traditional method of dealing with irreversible processes is, of course, the use of the Boltzmann integro-differential equation and its various extensions. But this method leads to two serious difficulties. The first is that Boltzmann s equation is neither provable nor even meaningful except in the context of molecular encounters, i.e., under the assumption that the intermolecular forces are of such short range in comparison with molecular distances that a molecule spends only a negligible fraction of its time within the influence of others. This drastically restricts the field of applicability, confining the treatment to gases close to the ideal state. But even then the equation can only be established on the basis of an essential assumption of molecular probabilistic independence ( micromolecular chaos ).3... 

In the first instance, and as a first approximation valid for very dilute solutions, one may ignore all types of ion-ion interactions except those deriving from simple Coulombic" forces. Thus, short-range interactions (e.g dispersion interactions) are excluded. This is a fundamental assumption of the Debye-Hiickel theory. Then the potential of average force U simply becomes the Coulombic potential energy of an ion of charge z, q in the volume element dV, i.e., the charge on the ion times the electrostatic potential in the volume element dV. That is,... 

As a result of assumption (ii) above, it will be shown that the value of can be expressed as the sum over the contributions of all the atoms present in the molecule, with these contributions being induced by the potential field of the surface. It should be noted here that although the interactions can be divided into two major classes, via pair or atom-atom interactions, such as dispersion and short-range repulsion forces, and an essentially non-pair interaction, that due to polarization, they can both be treated in a similar fashion within the framework of the proposed method. 

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