Angular interactions

The Finnis-Sinclair analytic functional form was introduced at about the same time as two other similar forms, the embedded-atom method > and the glue model." ° However, the derivation of the Finnis-Sinclair form from the second-moment approximation is very different from the interpretation of the other empirical forms, which are based on effective medium theory as discussed later. This difference in interpretation is often ignored, and all three methods tend to be put into a single class of potential energy function. In practice, the main difference between the methods lies in the systems to which they have been traditionally applied. In developing the embedded-atom method, for example, Baskes, Daw, and Foiles emphasized close-packed lattices rather than body-centered-cubic lattices. Given that angular interactions are usually ig-... 

There has been considerable effort since the introduction of the Finnis-Sinclair potential to develop expressions that include angular interactions and higher moments of the local density of states. " Carlson and coworkers, for example, have introduced a matrix form for the moments of the local density of states from which explicit environment-dependent angular interactions can be obtained. The role of the fourth moment, in particular, has been stressed for half-filled bands because it describes the tendency to introduce an energy gap. These investigations have led to improved models that describe local bonding in both covalent and body-centered-cubic materials. 

The second innovation by Tersoff was to introduce a functional form different from that given above for the bond order his form incorporates angular interactions while still maintaining coordination as the dominant featuredetermining structure. With this functional form, Tersoff was able not only to model stabilization of the diamond lattice against shear, he also was able to ob-... 

As one can see both contribution to the force depend on the relative displacements of two atoms so that an uniform translation does not produce any force on the crystal. Similarly a rotation of the crystal does not introduce any force in the crystal. For this reason the dynamical matrix which can be obtained from Fqs. (5 S) and (3 9) is invariant under rotation and translation of the crystal. The general expression of the dynamical matrix including both central and angular interaction is very cumberstome. For this reason we specialize the results to FCC crystals for which experimental results are available at present. We will consider interactions extending up to second nearest neighbors for the... 

The parameters involving central and angular interactions up to second neighbors are ... 

The term symbols used to describe these levels to the first order are Russell Saunders type (i. e., states described by total spin S and total angular momentum L of the / electrons). Splittings between these levels, a result of electrostatic repulsion are large, in the order of 1000-50000 cm-. In the rare earths spin orbital interaction is also important, and the / (spin and angular interaction) values for a given LS term are split in the order of 1000 cm". ... 

Here V, V2, and represent the single-body interaction with external fields, two-body separation interactions, and three-body angular interactions respectively. Classical examples of such potentials include the simple two-body Lennard-Jones potential [7], three-body Stillinger-Weber potential for silicon [8], and multibody environment-dependent interatomic potential (EDIP) for silicon [9] and carbon [10], the forces being simply the negative gradient of the potential energy function with respect to position. 

The problem, in essence, is to provide for an angular interaction by an accurate quantum mechanical description of the transition matrix for absorbance and reflectance. In other words, the field of the electromagnetic wave which penetrates the bulk metal will determine the photocurrent, provided that a correction is made for the reflected light. It has been shown that, approximately," ... 

For interactions between two quadmpolar molecules which have and 0g of the same sign, at a fixed separation r, the angular factor in equation (Al.5.131 leads to a planar, T-shaped stmcture, 0 = 0, 0g = nil, (ji = 0, being preferred. This geometry is often seen for nearly spherical quadmpolar molecules. There are other planar (ij) = 0) configurations with 0 = jr/2-6g that are also attractive. A planar, slipped parallel stmcture,... 

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... 

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... 

Wlien the atom-atom or atom-molecule interaction is spherically symmetric in the chaimel vector R, i.e. V(r, R) = V(/-,R), then the orbital / and rotational j angular momenta are each conserved tln-oughout the collision so that an i-partial wave decomposition of the translational wavefiinctions for each value of j is possible. The translational wave is decomposed according to... 

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