Anisotropic reflectance functions

In order to calculate the reflectance function for waves from a fluid incident on an anisotropic surface, the Christoffel equation must be solved with the... 

In isotropic materials, and along symmetry directions in anisotropic materials, all the traction components in the third equation vanish, as does the component of A associated with the SH mode. There are then three equations for three unknowns. For the general anisotropic case the four equations can be solved to give the amplitude A4 of the reflected wave. The amplitude is a complex quantity, because there may be a change in phase upon reflection. For an incident wave at an angle d to the normal and 0 to some direction lying in the surface (usually the lowest index direction available) the reflectance function may be written... 

Complex mean reflectance functions for aluminium, nickel, and copper are presented in Fig. 11.10. These represent a series of increasingly anisotropic... 

History. Wilke [129] considers the case that different orders of a reflection are observed and that the orientation distribution can be analytically described by a Gaussian on the orientation sphere. He shows how the apparent increase of the integral breadth with the order of the reflection can be used to separate misorientation effects from size effects. Ruland [30-34] generalizes this concept. He considers various analytical orientation distribution functions [9,84,124] and deduces that the method can be used if only a single reflection is sufficiently extended in radial direction, as is frequently the case with the streak-shaped reflections of the anisotropic... 

The definition of y is arbitrary and varies between researchers although it is usually taken as the angle between an axis in the laboratory frame, such as parallel or perpendicular to the plane of incidence, and a mirror plane or crystal axis direction in the surface. (This angle is labelled in Fig. 4.2 b as .) The function f(y/) in Eq. (3.10) then reflects the 2 mm, 3 m, and 4 mm symmetry of the (110), (111), and (100) surfaces, respectively. The constants A and B represent the isotropic and anisotropic contributions to the SH intensity which depend on the crystal, the experimental geometry, the frequencies used and the fundamental and SH polarizations. The data is then fit to this functional form and relative magnitudes for the phenomenological constants, A and B, determined. 

Figure 14.2 Scheme of a two-step decay of the correlation function S of a unit undergoing anisotropic motion before finally full isotropisation is achieved. The dashed line indicates the case of a one-step decay by isotropic motion. For the study of viscoelastic polymers the intermediate plateau that reflects residual couplings is the... 

Finally, n was determined by spectroscopic ellipsometry. The main drawback with this technique when applied to anisotropic samples is that the measured ellipsometric functions tanlF and cos A are related both to the incidence angle and the anisotropic reflectance coefficient for polarizations parallel and perpendicular to the incidence plane. The parameters thus have to be deconvolved from a set of measurements performed with different orientations of the sample [see (2.15) and (2.16)]. The complex refractive index determined by ellipsometry is reliable only in the spectral region where the sample can be considered as a bulk material. In fact, below the absorption... 

As the resolution of the Bragg reflection data is improved, it becomes possible to obtain information on the more minute details of electron density in a molecule. At high enough resolution information can be obtained on the redistribution of electron density (deformation density) around atoms when they combine to form a molecule. Electrons in molecules ma -form bonds or exist as lone pairs, thereby distorting the electron density around each atom and requiring a more complicated function to describe this overall electron density than normally used, in which it is treated as if it were spherically symmetrical (deformed to an ellipsoid in order to account for anisotropic displacements). This assumption is inherent in the use of spherically-symmetrical scattering factors although the elec-... 

Graphite is thermodynamically stable under usual conditions, but its structure is typically anisotropic as it is reflected, for instance, by the Young s modulus that is 10.3 X 10 MPa along the basal plane and 0.3 x 10 MPa in the direction perpendicular to the basal plane. A similar effect occurs with the capacitance of the HOPG/aqueous electrolyte interface, the potential of zero charge, and the work function values (see Chapter 21). 

The g-factors of electrons and holes reflect the nature of the conduction and valence bands in much the same way as the effective masses. Thus in AgF, AgCl, and AgBr, free electrons and shallowly trapped electrons whose wavefunctions are made up largely of conduction band functions are expected to be isotropic in nature. A free electron at the bottom of the band will have a single effective mass and g-factor. In contrast, free holes near the L-point and shallowly trapped holes whose wavefunctions are largely valence band functions are expected to show anisotropic behavior. A free hole will have parallel and perpendicular g-factors. The available data on electron and hole masses were given in Table 1 and the data on g-factors are given in Table 9. Thermalized electrons and holes in both modifications of Agl will be at the zone center. The anisotropic nature of the wurtzite crystal structure will be reflected in the effective masses and g-factors. 

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