Inverse decay length

Figure 7.12 compares the theoretical predictions with the experimental values across the 4d series, assuming one valence s electron per atom and taking x = 12 corresponding to close-packed lattices. The experimental values of the bandwidth are taken from the first principles LDA calculations in Table 7.1. The ratio b2 a is obtained by fitting a bandwidth of 10 eV for Mo with Nd = 5, so that from eqn (7.42) b2/a = eV. The skewed parabolic behaviour of the observed equilibrium nearest-neighbour distance is found to be fitted by values of the inverse decay length that vary linearly across the series as... 

Its Hankel transform has no singularity at p -> 0, and so the expansion of the DCF at p = 0 keeps the analytic form (44). Accordingly, the total correlation function keeps the asymptotics (43). However, the matrices of the expansion coefficients Co, C2, C4,... in (44) have other, modified values. Through Equation (40) this, in general, changes the profile p and hence results in a modified inverse decay length appearing in the asymptotics (42), (43) and (46). 

Gruen and Marcelja considered that the electric and polarization fields are not proportional in the vicinity of a surface and that while the electric field has the ion concentrations as its source, the source of the polarization field is provided by the Bjerrum defects. The coupled equations for the electric and polarization fields were derived through a variational method. Attard et al.14 contested the Gruen—Marcelja model because, to obtain an exponential decay of the repulsion, the nonlocal dielectric function was assumed to have a simple monotonic dependence upon the wavelength (eq 33 in ref 13). This was found to be inconsistent with the exact expression for multipolar models.14 In addition, the characteristic decay length for polarization (denoted in eq 18, ref 13) is inversely proportional to the square of the (unknown) concentration of Bjerrum defects in ice. While at large concentrations of Bjerrum defects the disordered ice becomes similar to water and the traditional Poisson—... 

The Debye-Huckel parameter (/r) describes the decay length of the electrical double-layer, while the inverse of the parameter, k, which is known as the Debye length, indicates the distance away from the surface where the distribution of ions in the solution is affected by the presence of a charged surface ... 

Dielectric constants Liquid fraction at equilibrium Initial liquid fraction in a foam liquid fraction in the dry portion of a foam Polarizability of water Inverse of dimensionless surface viscosity Surface viscosity Reciprocal Debye length Decay length in Eq. (24)... 

A crucial assumption for such a scenario is that the charge-charge correlation function ha(r) decays as exp (—IV), where the inverse charge-charge correlation length T = 1/ does not vanish at the critical point. At low ion densities, where T —> T0, this has meanwhile been proved for DH theory. Then, at the critical point, only the density fluctuations become... 

Figure 5.13 shows a comparison between the echo decay calculated directly from the microstructure as described in the previous section and the measured echo decay using NMR diffusometry at A = 100,200 and 500 msec. It was found that the calculated echo decay fitted reasonably well with the experimental echo decay, especially at intermediate and high ( -values. There is an inverse relation between the ( -values and the length scales in the microstructure, i.e., low ( -values correspond to long distances in the microstructure and vice versa. Another way to look at it is to say that low (j-values correspond to large details in the microstructure, and high ( -values correspond to fine details in the microstructure. Three CLSM micrographs showing different length scales in the structure are inserted in Figure 5.13. They are inserted in order to show which t)q5e of structure affects different...

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