Orientational dependence, anisotropic

Since the etch rate in isotropic etching is the same in all directions, its undercut rate is 1. For orientation-dependent anisotropic wet etching, the undercut rate is basically < 1. If the undercut rate can be controlled in the etching process, the lateral contours along the z direction will be changed, and a special 3D structure can be created with a 2D mask pattern. 

Orientation-dependent, anisotropic shrinkage occurs when fibrous reinforcement materials (in the form of short glass fibers) are used. The shrinkage for shell-shaped, solid molded parts can be determined using the finite element method (FEM) in the design phase [4, 5]. 

A fonn of anisotropic etching that is of some importance is that of orientation-dependent etching, where one particular crystal face is etched at a faster rate than another crystal face. A connnonly used orientation-dependent wet etch for silicon surfaces is a mixture of KOH in water and isopropanol. At approximately 350 K, this etchant has an etch rate of 0.6 pm min for the Si(lOO) plane, 0.1 pm min for the Si(l 10) plane and 0.006 pm miiG for the Si(l 11) plane [24]. These different etch rates can be exploited to yield anisotropically etched surfaces. 

Crystals have spatially preferred directions relative to their internal lattice structure with consequences for orientation-dependent physico-chemical properties i.e., they are anisotropic. This anisotropy is the reason for the typical formation of flat facetted faces. For the configuration of the facets the so-called Wullf theorem [20] was formulated as in a crystal in equihbrium the distances of the facets from the centre of the crystal are proportional to their surface free energies. ... 

The diffraction lines due to the crystalline phases in the samples are modeled using the unit cell symmetry and size, in order to determine the Bragg peak positions 0q. Peak intensities (peak areas) are calculated according to the structure factors Fo (which depend on the unit cell composition, the atomic positions and the thermal factors). Peak shapes are described by some profile functions 0(2fi—2fio) (usually pseudo-Voigt and Pearson VII). Effects due to instrumental aberrations, uniform strain and preferred orientations and anisotropic broadening can be taken into account. 

Anisotropic materials have different properties in different directions (1-7). 1-Aamples include fibers, wood, oriented amorphous polymers, injection-molded specimens, fiber-filled composites, single crystals, and crystalline polymers in which the crystalline phase is not randomly oriented. Thus anisotropic materials are really much more common than isotropic ones. But if the anisotropy is small, it is often neglected with possible serious consequences. Anisoiropic materials have far more than two independent clastic moduli— generally, a minimum of five or six. The exact number of independent moduli depends on the symmetry in the system (1-7). Anisotropic materials will also have different contractions in different directions and hence a set of Poisson s ratios rather than one. 

The second term in (29) is the anisotropic, orientation-dependent frequency shift due to the first-order quadrupolar interaction... 

The anisotropy of the overall tumbling will result in the dependence of spin-relaxation properties of a given 15N nucleus on the orientation of the NH-bond in the molecule. This orientational dependence is caused by differences in the apparent tumbling rates sensed by various internuclear vectors in an anisotropically tumbling molecule. Assume we have a molecule with the principal components of the overall rotational diffusion tensor Dx, Dy, and l)z (x, y, and z denote the principal axes of the diffusion tensor), and let Dx< Dy< Dz. 

While many chemical investigations of solids seek to emulate solution investigations by averaging all orientation-dependent properties, this approach may be shortsighted in that it necessarily reduces the information conveyed by the full anisotropic tensor quantities. 2D MAS experimental techniques originally developed by Bax, Szeverenyi, and Maciel (40) and refined by Grant and co-workers (41) can provide isotropic chemical shifts... 

Weighted Mean Curvature of an Interface. The weighted mean curvature, k7, has exactly the same geometrical properties as the mean curvature except that it is weighted by the possibly orientation-dependent magnitude of the interfacial tension. It is particularly useful for addressing capillarity problems when the interfacial energy is anisotropic, that is, dependent upon the interface orientation (Section C.3). 

We should note that if g = ge, the contact shift is isotropic (independent of orientation). If g is different from ge and anisotropic (see Section 1.4), then the contact shift is also anisotropic. The anisotropy of the shift is due to the fact that (1) the energy spreading of the Zeeman levels is different for each orientation (see Fig. 1.16), and therefore the value of (Sz) will be orientation dependent and (2) the values of (5, A/s Sz S, Ms) of Eq. (1.31) are orientation dependent as the result of efficient spin-orbit coupling. On the contrary, the contact coupling constant A is a constant whose value does not depend on the molecular orientation. 

Pseudocontact shift is expected every time there are energy levels close to the ground state. This causes orbital contributions to the ground state, and such contributions are orientation dependent. Therefore, the magnetic susceptibility tensor is anisotropic. Anisotropy of the magnetic susceptibility tensors arises also from sizable ZFS of the S manifold (Section 1.4). 

Let us discuss first the case in which only the first term is present. In the Solomon and Bloembeigen equations for / , (i = 1, 2) there is the cos parameter at the denominator of a Lorentzian function. Up to now cos has been taken equal to that of the free electron. However, in the presence of orbital contributions, the Zeeman splitting of the Ms levels changes its value and cos equals xs / o or (g/h)pBBo- When g is anisotropic (see Fig. 1.16), the value of cos is different from that of the free electron and is orientation dependent. The principal consequence is that another parameter (at least) is needed, i.e. the 0 angle between the metal-nucleus vector and the z direction of the g tensor (see Section 1.4). A second consequence is that the cos fluctuations in solution must be taken into account when integrating over all the orientations. Appropriate equations for nuclear relaxation have been derived for both the cases in which rotation is faster [40,41] or slower [42,43] than the electronic relaxation time. In practical cases, the deviations from the Solomon profile are within 10-20% (see for example Fig. 3.14). 

The surface energy of a crystal is anisotropic. For planes with low indices, it is possible to calculate by trigonometry the number of near-neighbor bonds missing per area and follow the procedure above to estimate the relative surface energies of different low-index planes. The orientation dependence of the free surface energy of a two-dimensional square lattice depends on the orientation of the surface and can be found as follows. 

In almost all cases the admixture of excited states is anisotropic that is, the observed g value varies according to the orientation of the paramagnetic species in relation to the applied magnetic field (orientation-dependent). The g-factor anisotropy is characterized by three principal g values, namely, gxx, gyy, and g--. When these three values are different, the symmetry is defined as rhombic and in the case of axial symmetry, gxx = gyy gzz. In the orientation-independent (isotropic) situation the g factor is represented by a single value. This is also true if the species paramagnetic is in a solution of low viscosity (water) where the molecular tumbling causes all the g factor anisotropy to be averaged out (Knowles et al., 1976 Campbell and Dwek, 1984). 

The coefficients a and b (see Table 3.2) take into account the restrictions in spin dimensionality. For a = b= 1, the Heisenberg model with isotropic exchange interaction and isotropic susceptibility results. The combination of a = 1 and b = 0 yields the strictly anisotropic Ising model, in which the orientation of the spins is restricted to the z-axis. Consequently, the susceptibility is strongly orientation dependent and one needs to differentiate between x" in the direction of the z-axis ( easy axis ) and x perpendicular to z. The molar susceptibilities are then related as... 

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