Repulsion exponent

From this equation we can calculate the actual crystal radius (for use in normal sodium chloride type crystals) from Ru the univalent crystal radius. A knowledge of the repulsion exponent n is needed, however. This can be derived from the experimental measurement of the compressibility of crystals. Bom11 and Herzfeld,12 give the values in Table III, obtained in this way. 

The forces between ions have been discussed by Lennard-Jones and his collaborators, who have given tables showing the repulsive forces as a function of the repulsion exponent n [Lennard-Jones and Dent, Proc. Roy. Soc., 112A, 230 (1926)]. In conjunction with Wasastjema s radii, these tables have been used in the theoretical treatment of crystals such as caldte, CaCOj, which, however, we consider not to be composed of monatomic ions. Thus, they assume C+1 and O" to be present in caldte [Lennard-Jones and Dent, Proc. Roy. Soc., 113A, 673, 690 (1927)], although the carbonate ion is generally believed by chemists to contain shared-electron bonds. 

For ionic crystals > = 1, and the A are known (Madelung constants). For van der Waals crystals m — 6 (though small terms in and exist) but in view of the difficulties of calculation we obtain B from the observed heat of vapourisation (from A. 3). The repulsion exponent n varies from about 6 for LiF to 12 for Csl, for gases (Lennard-Jones) a value of about 12 seems the best. We assume a constant value of 11 throughout. 

To prevent misunderstanding (94), we emphasize that neither experimental hydration energies nor experimental coordination numbers are necessary for these calculations. Moreover, the coordination numbers obtained are generally not comparable to empirical hydration numbers. The only experimental quantities that enter the calculations are a) cationic radius and charge b) van der Waals radius of water c) dipole and quadrupole moment of water d) polarizabilities e) ionization potentials and f) Born repulsion exponents as well as fundamental constants (see Ref. (92)). 

Variations of the bond angles in the AX E C ) and AX2E (2(3 ) systems versus the ratio of the two charges employing various values for the repulsion exponent in the expression of the potential... 

In Eq. (15) the second term reflects the gain in entropy when a chain breaks so that the two new ends can explore a volume Entropy is increased because the excluded volume repulsion on scales less than is reduced by breaking the chain this effect is accounted for by the additional exponent 9 = y — )/v where 7 > 1 is a standard critical exponent, the value of 7 being larger in 2 dimensions than in 3 dimensions 72 = 43/32 1.34, 73j 1.17. In MFA 7 = 1, = 0, and Eq. (15) simplifies to Eq. (9), where correlations, brought about by mutual avoidance of chains, i.e., excluded volume, are ignored. 

Single chains confined between two parallel purely repulsive walls with = 0 show in the simulations the crossover from three- to two-dimensional behavior more clearly than in the case of adsorption (Sec. Ill), where we saw that the scaling exponents for the diffusion constant and the relaxation time slightly exceeded their theoretical values of 1 and 2.5, respectively. In sufficiently narrow slits, D density profile in the perpendicular direction (z) across the film that the monomers are localized in the mid-plane z = Djl so that a two-dimensional SAW, cf. Eq. (24), is easily established [15] i.e., the scaling of the longitudinal component of the mean gyration radius and also the relaxation times exhibit nicely the 2 /-exponent = 3/4 (Fig. 13). 

The MNDO, AMI and PM3 methods are parameterizations of the NDDO model, where the parameterization is in terms of atomic variables, i.e. referring only to the nature of a single atom. MNDO, AMI and PM3 are derived from the same basic approximations (NDDO), and differ only in the way the core-core repulsion is treated, and how the parameters are assigned. Each method considers only the valence s- and p-functions, which are taken as Slater type orbitals with corresponding exponents, (s and... 

Each of the MNDO, AMI and PM3 methods involves at least 12 parameters per atom orbital exponents, Cj/pi one-electron terms, II /p and j3s/p two-electron terms, Gss, Gsp, Gpp, Gp2, Hsp, parameters used in the core-core repulsion, a and for the AMI and PM3 methods also a, b and c constants, as described below. 

These considerations also explain the occurrence of cases of dimorphism involving the sodium chloride and cesium chloride structures. It would be expected that increase in thermal agitation of the ions would smooth out the repulsive forces, that is, would decrease the value of the exponent n. Hence the cesium chloride structure would be expected to be stable in the low temperature region, and the sodium chloride structure in the high-temperature region. This result may be tested by comparison with the data for the ammonium halides, if we assume the ammonium ion to approximate closely to spherical symmetry. The low-temperature form of all three salts, ammonium chloride, bromide and iodide, has the cesium chloride structure, and the high-temperature form the sodium chloride structure. Cesium chloride and bromide are also dimorphous, changing into another form (presumably with the sociium chloride structure) at temperatures of about 500°. 

The Coulomb repulsion integrals are evaluated using the Mataga-Nishimoto formula The resonance integral is assumed to be of exponential form p=Be , the value of exponent a being taken as 1.7 A... 

Generally, the values of the scaling exponent are smaller for polymers than for molecular liquids, for which 3.2 < y < 8.5. A larger y, or steeper repulsive potential, implies greater influence of jamming on the dynamics. The smaller exponent found for polymers in comparison with small-molecule liquids means that volume effects are weaker for polymers, which is ironic given their central role in the historical development of free-volume models. The reason why y is smaller... 

Recently, Grayce and Schweizer [70] have proposed a liquid-state theory for stars in the melt state, considering only repulsive interactions. They obtained g f-o.64 tjjg exponent of the f-dependence is bracketed by the scaling... 

Figures 6-9 illustrate the use of these finite size scaling relations for the square lattice gas with repulsion between both nearest and next nearest neighbors. In Fig. 6 the raw data of Fig. 5 are replotted in scaled form, as suggested by Eq. (37). Note that neither = TJcc) nor the critical exponents are known in beforehand - the phase transition of the (2x1) phase falls in the universality class of the XY model with uniaxial anisotropy which has nonuniversal exponents depending on R. Clearly, it is desirable to estimate without being biased by the choice of the critical exponents. This is possible...

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