Orientational dependence

In the very same way as the Born-Oppenheimer approximation allows the definition of a potential energy surface for a Van der Waals molecule, it enables, too, the concept of an interaction tensor field. This is a field dependent on the relative coordinates of the monomers and transforming as a tensor under rotation of the complex as a whole. (The potential energy surface is an example of a rank zero interaction tensor field). In the case of tensor fields it is also convenient to base the theory on irreducible tensors and to use an expansion in terms of a complete set of functions of the five angular coordinates describing a Van der Waals dimer. 

The generalization of the scalar-valued angular functions (1 b) to arbitrary rank J is  

It is possible to apply the multipole expansion and perturbation theory in order to derive long range expressions for T (R), thus relating this quantity to monomer properties. A simple example of such a procedure can be found in the appendix of ref. , where the induction contribution to the dipole moment (J = 1) of an arbitrary Van der Waals dimer has been evaluated. 

Because not much is known experimentally about general interaction tensors, and especially not about their long range behaviour, we will not pursue this line of approach, but rather give a brief review of the existing work which has concentrated on two different tensors the pair dipole (order 1 tensor) and the pair polarizability (order 2 plus order 0 tensor). 

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There has been considerable elaboration of the simple Girifalco and Good relationship, Eq. XII-22. As noted in Sections IV-2A and X-6B, the surface ftee energies that appear under the square root sign may be supposed to be expressible as a sum of dispersion, polar, and so on, components. This type of approach has been developed by Dann [70] and Kaelble [71] as well as by Schonhom and co-workers (see Ref. 72). Good (see Ref. 73) has preferred to introduce polar interactions into a detailed analysis of the meaning of in Eq. IV-7. While there is no doubt that polar interactions are important, these are orientation dependent and hence structure sensitive. 

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. 

Csepregi L, Kennedy E F, Gallagher T J, Mayer J W and Sigmon T W 1978 Substrate orientation dependence of the epitaxial regrowth rate from Si-implanted amorphous Si J. Appi. Phys. 49 3906... 

Feibelman P J 1991 Orientation dependence of the hydrogen molecule s interaction with Rh(OOI) Phys. Rev. Lett. 67 461... 

Ochiai and Okamoto showed that nitration of quinoline i-oxide in sulphuric acid at o °C gave 5- and 8-nitroquinoline i-oxides with a trace of the 4-isomer, but that at 60-100 °C 4-nitration became overwhelmingly dominant. The orientation depends not only upon temperature but also upon acidity, and kinetic studies (table 8.4 table 10.3) show that two processes are occurring the nitration of the free base (vil, R = O at C(4), favoured by low acidities and high temperatures, and the nitration of the cation (vil, R = OH), favoured by high acidities and low temperatures. ... 

Flow processes iaside the spinneret are governed by shear viscosity and shear rate. PET is a non-Newtonian elastic fluid. Spinning filament tension and molecular orientation depend on polymer temperature and viscosity, spinneret capillary diameter and length, spin speed, rate of filament cooling, inertia, and air drag (69,70). These variables combine to attenuate the fiber and orient and sometimes crystallize the molecular chains (71). 

In the section dealing with electrophilic attack at carbon some results on indazole homocyclic reactivity were presented nitration at position 5 (Section 4.04.2.1.4(ii)), sulfon-ation at position 7 (Section 4.04.2.1.4(iii)) and bromination at positions 5 and 7 (Section 4.04.2.1.4(v)). The orientation depends on the nature (cationic, neutral or anionic) of the indazole. Protonation, for instance, deactivates the heterocycle and directs the attack towards the fused benzene ring. A careful study of the nitration of indazoles at positions 2, 3, 5 or 7 has been published by Habraken (7UOC3084) who described the synthesis of several dinitroindazoles (5,7 5,6 3,5 3,6 3,4 3,7). The kinetics of the nitration of indazole to form the 5-nitro derivative have been determined (72JCS(P2)632). The rate profile at acidities below 90% sulfuric acid shows that the reaction involves the conjugate acid of indazole. 

Wecansay then that reactionshavestrongtemperature and orientational dependence, and thatcollisionsalonearenotenoughforreactiontotakeplace. 

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. ... 

K. M. Beatty, K. A. Jackson. Orientation dependence of the distribution coefficient obtained from a spin-1 Ising model. J Cryst Growth 774 28,... 

Cottrell, A.H., 1996, Point defects in Al-Ni-Cu alloys based on the NiAl phase, Intermetallics, 4 1 Leapman, R.D., and Silcox, J.,1979, Orientation dependence of core edges in electron energy loss spectra from anysotropic materials, Phys. Rev. Lett., 42 1361. 

In this case the interaction between neighbouring molecules is not only stronger but also orientation dependent. The mean statistical energy of dipole-dipole interactions 0dd also decreases with r 6, but depends on i2 (p dipole moment) according to 3>... 

OIDEP usually results from Tq-S mixing in radical pairs, although T i-S mixing has also been considered (Atkins et al., 1971, 1973). The time development of electron-spin state populations is a function of the electron Zeeman interaction, the electron-nuclear hyperfine interaction, the electron-electron exchange interaction, together with spin-rotational and orientation dependent terms (Pedersen and Freed, 1972). Electron spin lattice relaxation Ti = 10 to 10 sec) is normally slower than the polarizing process. 

In contrast, the second term in (4.6) comprises the full orientation dependence of the nuclear charge distribution in 2nd power. Interestingly, the expression has the appearance of an irreducible (3 x 3) second-rank tensor. Such tensors are particularly convenient for rotational transformations (as will be used later when nuclear spin operators are considered). The term here is called the nuclear quadrupole moment Q. Because of its inherent symmetry and the specific cylindrical charge distribution of nuclei, the quadrupole moment can be represented by a single scalar, Q (vide infra). 

An instructive description of the first-order perturbation treatment of the quadrupole interaction in Ni has been given by Travis and Spijkerman [3]. These authors also show in graphical form the quadrupole-spectrum line positions and the quadrupole-spectrum as a function of the asymmetry parameter r/ they give eigenvector coefficients and show the orientation dependence of the quadrupole-spectrum line intensities for a single crystal of a Ni compound. The reader is also referred to the article by Dunlap [15] about electric quadrupole interaction, in general. 

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