Electron density profile

Figure Bl.9.12. The schematic diagram of the relationships between the one-dimensional electron density profile, p(r), correlation fiinction y (r) and interface distribution fiinction gj(r).

The important information about the properties of smectic layers can be obtained from the relative intensities of the (OOn) Bragg peaks. The electron density profile along the layer normal is described by a spatial distribution function p(z). The function p(z) may be represented as a convolution of the molecular form factor F(z) and the molecular centre of mass distribution f(z) across the layers [43]. The function F(z) may be calculated on the basis of a certain model for layer organization [37, 48]. The distribution function f(z) is usually expanded into a Fourier series f(z) = cos(nqoz), where the coefficients = (cos(nqoz)) are the de Gennes-McMillan translational order parameters of the smectic A phase. According to the convolution theorem, the intensities of the (OOn) reflections from the smectic layers are simply proportional to the square of the translational order parameters t ... 

In recent years, high-resolution x-ray diffraction has become a powerful method for studying layered strnctnres, films, interfaces, and surfaces. X-ray reflectivity involves the measurement of the angnlar dependence of the intensity of the x-ray beam reflected by planar interfaces. If there are multiple interfaces, interference between the reflected x-rays at the interfaces prodnces a series of minima and maxima, which allow determination of the thickness of the film. More detailed information about the film can be obtained by fitting the reflectivity curve to a model of the electron density profile. Usually, x-ray reflectivity scans are performed with a synchrotron light source. As with ellipsometry, x-ray reflectivity provides good vertical resolution [14,20] but poor lateral resolution, which is limited by the size of the probing beam, usually several tens of micrometers. 

Figure 6.4 shows a third and commonly used way of representing electron density profiles the two-dimensional contour map. This map for the SCI2 molecule corresponds to the relief map in Figure 6.2. Although this map is able to show very detailed information, we are restricted to a particular choice of plane, or to a selection of planes. To obtain an approximately equally dense distribution of contour lines, contour values used in this book increase in the nearly geometrical sequence, I0 3, 2 X 10 3, 4 X 10 3, 8 X 10-3. 2 X 10-3.

Figure 7.1 shows a hypothetical monotonically decreasing function mimicking a onedimensional electron density profile for a period 2 element. The value of the function f(x)... 

This contribution involves the positive-ion and electron density profiles of the metal, and the former is often assumed not to change with charging of the interface. In 1983 and 1984, several workers30-32,79 showed how certain features of the interfacial capacity curves should depend on the metal. 

The electron density profile was assumed to have the exponential form... 

The dependence of dx on qM is central in a model, proposed by Price and Halley,93 for the metal surface in the double layer which is related to that discussed above. The positively charged ion background profile p+(z) is assumed uniform, with a value equal to the bulk density pb, from z = -oo to z = 0, with the electronic density profile n(z) more diffuse. In contrast to the previous model30 which emphasizes penetration by the conduction electrons of the region of solvent, this model93 supposes that the density profile n(z) is zero for z > dx, where z > dx defines the region of the electrolyte. Then the potential at dx is given by... 

Modern theories of electronic structure at a metal surface, which have proved their accuracy for bare metal surfaces, have now been applied to the calculation of electron density profiles in the presence of adsorbed species or other external sources of potential. The spillover of the negative (electronic) charge density from the positive (ionic) background and the overlap of the former with the electrolyte are the crucial effects. Self-consistent calculations, in which the electronic kinetic energy is correctly taken into account, may have to replace the simpler density-functional treatments which have been used most often. The situation for liquid metals, for which the density profile for the positive (ionic) charge density is required, is not as satisfactory as for solid metals, for which the crystal structure is known. 

A comparison of the calculated relative peak heights using the very simplified electron density profiles of Figs. 23b and 23d with the SAXS pattern of the sample, Fig. 24, unequivocally indicates the presence of the non-centrosysmmetric lamellae structure depicted in Fig. 23a. In agreement with SCMF calculations [90], the centrosymmetric two-way arrangement (Fig. 23c) of A-B-C-A-C-B pattern was not observed. 

This energy functional attains its minimum for the true electronic density profile. This offers an attractive scheme of performing calculations, the density functional formalism. Instead of solving the Schrodinger equation for each electron, one can use the electronic density n(r) as the basic variable, and exploit the minimal properties of Eq. (17.8). Further, one can obtain approximate solutions for n(r) by choosing a suitable family of trial functions, and minimizing E[n(r)] within this family we will explore this variational method in the following. 

Within this local-density approximation one can obtain exact numerical solutions for the electronic density profile [5], but they require a major computational effort. Therefore the variational method is an attractive alternative. For this purpose one needs a local approximation for the kinetic energy. For a one-dimensional model the first two terms of a gradient expansion are ... 

Electron density profile of the quartz-PTA interface. (From Park, C., Fenter, P, Sturchio, N., and Regalbuto, J.R., Phys. Rev. Lett. 94, 2005, 076104.) A schematic of PTA adsorbed with one or two hydration sheaths. 

MD simulations of model membrane systems have provided a unique view of lipid interactions at a molecular level of resolution [21], Due to the inherent fluidity and heterogeneity in lipid membranes, computer simulation is an attractive tool. MD simulations allow us to obtain structural, dynamic, and energetic information about model lipid membranes. Comparing calculated structural properties from our simulations to experimental values, such as areas and volumes per lipid, and electron density profiles, allows validation of our models. With molecular resolution, we are able to probe lipid-lipid interactions at a level difficult to achieve experimentally. 

Esbjerg, N., and Nprskov, J. K. (1980). Dependence of the He-scattering potential at surfaces on the surface-electron-density profile. Phys. Rev. Lett. 45, 807-810. 

Figure 2.8. Electron density contours for atomic chemisorption on jellium with electron density that corresponds to A1 metal. Upper row Contours of constant electron density in the plane normal to the surface. Center row Difference in charge density between isolated adatom and metal surface, full line gain and dashed line loss of charge density. Bottom row Bare metal electron density profile. Reproduced from [30].

Fig. 5.17 Tolal electron density profiles of simple molecules along the intemuclcar axis. [From Ransil, B. J. Sinai, J. J. J. Chem. Phys. 1967, 46, 4050. Reproduced with permission.]...

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