Angular dependence

There is, of course, a mass of rather direct evidence on orientation at the liquid-vapor interface, much of which is at least implicit in this chapter and in Chapter IV. The methods of statistical mechanics are applicable to the calculation of surface orientation of assymmetric molecules, usually by introducing an angular dependence to the inter-molecular potential function (see Refs. 67, 68, 77 as examples). Widom has applied a mean-held approximation to a lattice model to predict the tendency of AB molecules to adsorb and orient perpendicular to the interface between phases of AA and BB [78]. In the case of water, a molecular dynamics calculation concluded that the surface dipole density corresponded to a tendency for surface-OH groups to point toward the vapor phase [79]. 

Finally, the assumed spherical synnnetry of the interactions implies that the volume element r 2 is dri2- For angularly-dependent potentials, the second virial coefficient... 

For molecules having dimensions comparable with the wavelength, phase differences will occur between waves scattered from different regions of the molecule. These phase differences result in an angular dependence of the scattered intensity. The reduction may be expressed in temis of a particle interference factor P(2Q) such that... 

This angular dependence is different from the first-order perturbations so that the conventional teclmique of removing linebroadening in solids, MAS (see below), caimot completely remove this interaction at the same time as removing the first-order broadening. Flence, the resolution of MAS spectra from quadnipolar nuclei is usually worse than for spin-2 nuclei and often characteristic lineshapes are observed. If this is the case, it is... 

The interpretation of MAS experiments on nuclei with spin / > Fin non-cubic enviromnents is more complex than for / = Fiuiclei since the effect of the quadnipolar interaction is to spread the i <-> (i - 1) transition over a frequency range (2m. - 1)Vq. This usually means that for non-integer nuclei only the - transition is observed since, to first order in tire quadnipolar interaction, it is unaffected. Flowever, usually second-order effects are important and the angular dependence of the - ytransition has both P2(cos 0) andP Ccos 9) terms, only the first of which is cancelled by MAS. As a result, the line is narrowed by only a factor of 3.6, and it is necessary to spin faster than the residual linewidth Avq where... 

Dharmasena G, Copeland K, Young J H, Lasell R A, Phillips T R, Parker G A and Keil M 1997 Angular dependence for v /-resolved states in F + H2 -> HF(v /) + H reactive scattering using a new atomic beam source J. Rhys. Chem. A101 6429—40... 

To incorporate the angular dependence of a basis function into Gaussian orbitals, either spherical haimonics or integer powers of the Cartesian coordinates have to be included. We shall discuss the latter case, in which a primitive basis function takes the form... 

Describe the angular dependence of the vertically and horizontally polarized light scattered by a molecule and their resultant by considering the intensity as a vector anchored at the origin whose length in various directions is given by the trigonometric terms in Eqs. (10.25), (10.26), and (10.30),... 

Thus Rg is a constant in any particular experiment where Rayleigh scattering is obtained, since the entire angular dependence of the light intensity is correctly contained in the 1 + cos 6 term. 

The angular dependence of the fluorescence yield in the ne borhood of the critical angle should be considered in detail to establish the chemical nature of surface impurities, as well as for quantitation in terms of their concentrations (Figure 1). 

Experimental curves for the angular dependence of the fluorescence intensity from plated or sputtered submonatomic Ni layers (open triangles), layers produced by the evaporation of a Ni salt solution (open circles), and the silicon substrate (filled circles). 

With VPD preconcentration, the angular dependence of the impurity fluorescence yield foUows the curve for residue impurities, as shown in Figure 1, in contrast to the plated-impurity case using direct TXRF. 

E. L. Church, H. A. Jenkinson, and J. M. Zavada. Relation Berween the Angular Dependence of Scattering and Microtopographic Features. 

One of the most fascinating applications of channeling RBS is the study of lattice locations of impurity atoms. By measuring the angular dependence of the back-scattering yield of the impurity and host atoms around three independent channeling axes it is possible to calculate the position of the impurity. Details can be found elsewhere [3.122]. 

Since then, TXRE has become the standard tool for surface and subsurface microanalysis [4.7-4.11]. In 1983 Becker reported the angular dependence of X-ray fluorescence intensities in the range of total reflection [4.12]. Recent demands have set the pace of further development in the field of TXRE - improved detection limits [4.13] in combination with subtle surface preparation techniques [4.14, 4.15], analyte concentrations extended even to ultratraces (pg) of light elements, e. g. A1 [4.16], spe-dation of different chemical states [4.17], and novel optical arrangements [4.18] and X-ray sources [4.19, 4.20]. 

Fig. 4.15. Angular dependence of the fluorescence radiation emitted from a Co-layered Si substrate. The Co-Ka intensity is plotted semi-logarithmically for layers of different thickness (mm). The maxima forthe ultra-thin Co-layers are located at the critical angle of Si (dashed vertical line). They are shifted to the critical angle of Co (dotted vertical line) ifthe layer is more than 10 mm thick ([4.21], after Ref [4.41]).

Although CNTs showed similar EELS pattern in plasmon-loss and core-loss regions to graphite, SWCNT and fine MWCNT with a diameter less than 5 nm had different features. Furthermore, it has been found out that the angular-dependent EELS along the direction normal to the longitudinal axis of CNT shows stronger contribution from Jt electrons than [Pg.38]

Let us consider a one-component system of hard spheres of unit diameter (7=1 with angular-dependent associative potential. The nonassociative potential is thus given by Eq. (35), whereas the associative forces are described by Eq. (61) with d = a = I, a = 1.05, and 6 = 21°. 

We report here some results for a simple model of a one-component fluid interacting via a slightly modified Lennard-Jones potential, with angular-dependent associative forces. The model is considered in contact with the adsorbing surface. The principal aim of the simulation is to investigate the... 

To find the equilibrium form of a crystal, the following Wullf construction [20] can be used, which will be explained here, for simplicity, in two dimensions. Set the centre of the crystal at the origin of a polar coordinate system r,6. The radius r is assumed proportional to the surface tension 7( ), where 6 defines the angle between the coordinate system of the crystal lattice and the normal direction of a point at the surface. The anisotropy here is given through the angular dependence. A cubic crystal, for example, shows in a two-dimensional cut a clover-leaf shape for 7( ). Now draw everywhere on this graph the normals to the radius vector r = The... 

Figure A1.2 Models schematically illustrating the angular dependence functions A ,(6, 0). There is no unique way of representing the angular dependence functions of all seven f orbitals. An alternative to the set shown is one f 3, three f i, fy i, and three fx(i2 3j,2), fj.(j,2 3x2), and fj(x2 3x2).

Orbital Angular dependence function Orbital Angular dependence function... 

It is now generally realized that the angular dependence of vicinal coupling constants is considerably mare complex than was at first appreciated (24, 38, 39) and that care must be exercised in utilizing... 

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