Born forces

Coulombic interactions are not the only forces operating in a real ionic lattice. The ions have finite size, and electron-electron and nucleus-nucleus repulsions also arise these are Born forces. Equation 5.12 gives the simplest expression for the increase in repulsive energy upon assembling the lattice from gaseous ions. 

Values of the Bom exponent (Table 5.3) can be evaluated from compressibility data and depend on the electronic configurations of the ions involved effectively, this says that n shows a dependence on the sizes of the ions. 

Using the values given in Table 5.3, determine an appropriate Born exponent for BaO. 

What is the change in the Born exponent in going from BaO to SrO [Ans. -1  

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Chen J, Im W, Brooks CL III (2006) Balancing solvation and intramolecular interactions Toward a consistent generalized Born force field. J Am Chem Soc 128 3728-3736. 

In order to calculate the potential well, Born repulsion should be included in the calculations. Inslead of using an explicit expression for the Born forces, the interaction potential U is cut at a distance of about 4 A. Considering <5 of the order of 24 A, x-4A, >fp-l5-10 20 J and A — 7.5 10 20 J, one obtains that the optimum At is given by the expression... 

Born forces - essentially short-range and repulsive... 

The corresponding energy of attraction between two molecules decreases as the sixth power of the distance between them. However, when the electron clouds of the atoms begin to interpenetrate, strong repulsive forces come into play (Born forces), so that an equilibrium position exists for intermolecular distances of order 0.2-0.4 nm, known as the van der Waals radius. 

Use the Born force-constant model to calculate the phonon frequencies for silicon along the L - r - X directions, where L — (1,1, 1) and X = (1,0,0). Take the ratio of the bond stretching and bond bending force constants to be Kr/Kg = 16. Fit the value of Kr to reproduce the experimental value for the highest optical mode at r, which is 15.53 THz, and use this value to obtain the frequencies of the various modes at X compare these with the values given in Table 6.2. Determine the atomic displacements for the normal modes at F, the lowest acoustic branch at X and L, and the highest optical branch at X and L. 

Energy V varies exponentially with the distance between surfaces. It is a function of the pH and the ionic strength reflected in the potential V o (or i>d) and in k. Energy Fa, in a first approximation, is insensitive to the physicochemical conditions of the medium and varies with the reciprocal of the distance. Therefore, Fa is always predominant at small separation distances, and for short distances (l-2nm) the interaction energy falls drastically and causes F,j to approach minus infinity. In fact, for very short distances, the repulsive forces due to the interpenetration of strongly oriented solvent layers (structural forces) and to the overlap of electron clouds (Born forces) become stronger than the Van der Waals forces [15-17]. As a result, a primary minimum appears on the curve of Ft as a function of distance (Figure 8.6). 

Madeluag constant For an ionic crystal composed of cations and anions of respective change z + and z, the la ttice energy Vq may be derived as the balance between the coulombic attractive and repulsive forces. This approach yields the Born-Lande equation,... 

The discrepancy is not large and the last term is zero for a system without net charge. Thus we see that the use of a shifted Coulomb force is equivalent to a tin-foil reaction field and almost equivalent to a tin-foil Born condition. 

The first requirement is the definition of a low-dimensional space of reaction coordinates that still captures the essential dynamics of the processes we consider. Motions in the perpendicular null space should have irrelevant detail and equilibrate fast, preferably on a time scale that is separated from the time scale of the essential motions. Motions in the two spaces are separated much like is done in the Born-Oppenheimer approximation. The average influence of the fast motions on the essential degrees of freedom must be taken into account this concerns (i) correlations with positions expressed in a potential of mean force, (ii) correlations with velocities expressed in frictional terms, and iit) an uncorrelated remainder that can be modeled by stochastic terms. Of course, this scheme is the general idea behind the well-known Langevin and Brownian dynamics. 

A. rather complex procedure is used to determine the Born radii a values of which. calculated for each atom in the molecule that carries a charge or a partial charge. T Born radius of an afom (more correctly considered to be an effective Born radii corresponds to the radius that would return the electrostatic energy of the system accordi to the Bom equation if all other atoms in the molecule were uncharged (i.e. if the other ato only acted to define the dielectric boundary between the solute and the solvent). In Sti force field implementation, atomic radii from the OPLS force field are assigned to ec... 

Suppose we divide the flow segments into classes according to relaxation times and index the various states by the subscript i. Thus the relaxation time and the component of shear stress borne by the segments in class i are and Fj, respectively. The applied shear force is related to the Fj s through... 

This may also have been a factor in the conquering of the known world by the Roman Legions. Expeditionary and seige forces, even today but much more so then, are at risk for water-borne diseases. The practice of the Roman army to carry with it substantial wine suppHes is seen as sound militarily for health reasons, rather than just from the standpoint of the enjoyment of the soldiers. Recent study indicates that tourist-type diarrheas are less frequently encountered if wine is consumed rather than water, even bottled water (20). 

The basic operations in dust collection by any device are (1) separation of the gas-borne particles from the gas stream by deposition on a collecting surface (2) retention of the deposit on the surface and (3) removal of the deposit from the surface for recovery or disposal. The separation step requires (1) application of a force that produces a differential motion of a particle relative to the gas and (2) a gas retention time sufficient for the particle to migrate to the coUecting surface. The principal mechanisms of aerosol deposition that are apphed in dust collectors are (1) gravitational deposition, (2) flow-line interception, (3) inertial deposition, (4) diffusional deposition, and (5) electrostatic deposition. Thermal deposition is only a minor factor in practical dust-collectiou equipment because the thermophoretic force is small. Table 17-2 lists these six mechanisms and presents the characteristic... 

Despite the attractions of economic forces driving environmental protection, some cautions and failures have been noted. Firstly, the export of hazardous waste to countries where costs for treatment are lower enhances environmental risks during transport and has the potential for transboundary export in the event of pollution. At the same time, the loss of raw material may deprive the home market of an adequate supply of feedstock for the home-based industry. Secondly, there is considerable scepticism that self-regulation of TBT-based antifoulants could be achieved in a timely manner by the shipping industry. This is an instance where the cost benehts to one industry are born by another commercial sector, notably aquaculture. Thus, protection of the marine environment is likely to be aided by economic factors but the role of government, via taxation and standard setting, is not likely to be usurped. Public education and, in turn, pressure, can promote and support corporate environmentalism. 

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