Cubic field

Fig. 20. Magnetic moment as function of kF/A for weak-field cubic terms (de ) under the combined action of spin-orbit coupling and an axial ligand field. Values of d/A are indicated at the end of each curve. k=1.0 [5,35],...

M = DlS -iS S+iy] + E S -S )+g)iBS-H as a consequence of axially symmetric and rhombic contributions to the crystal field cubic ligand field parameter electronic charge... 

Finally, by volumetric imaging Three-dimensional information was obtained by stacking reflection tomograms from multiple planes. Using this stacking technique, cubic voxels were obtained and could be numerically dissected in any plane. Although there are several attractive features related to this technique, there are also several questions which need to be addressed before it can be used for industrial applications. For example, the applied sound field must be further characterized. 

Any cavity contains an infinite number of electromagnetic modes. For radiation confined to a perfectly conducting cubical cavity of volume V= L, the modes are given by the electric field components of the fomi ... 

The most widely used type of trap for the study of ion-molecule reactivity is the ion-cyclotron-resonance (ICR) [99] mass spectrometer and its successor, the Fourier-transfomi mass spectrometer (FTMS) [100, 101]. Figure A3.5.8 shows the cubic trapping cell used in many FTMS instmments [101]. Ions are created in or injected into a cubic cell in a vacuum of 10 Pa or lower. A magnetic field, B, confines the motion in the x-y... 

The higher-order bulk contribution to the nonlmear response arises, as just mentioned, from a spatially nonlocal response in which the induced nonlinear polarization does not depend solely on the value of the fiindamental electric field at the same point. To leading order, we may represent these non-local tenns as bemg proportional to a nonlinear response incorporating a first spatial derivative of the fiindamental electric field. Such tenns conespond in the microscopic theory to the inclusion of electric-quadnipole and magnetic-dipole contributions. The fonn of these bulk contributions may be derived on the basis of synnnetry considerations. As an example of a frequently encountered situation, we indicate here the non-local polarization for SFIG in a cubic material excited by a plane wave (co) ... 

As an illustration, we consider the case of SFIG from the (111) surface of a cubic material (3m. syimnetry). More general treatments of rotational anisotropy in centrosymmetric crystals may be found in the literature [62. 63 and M]- For the case at hand, we may detennine the anisotropy of the radiated SFl field from equation Bl.5.32 in conjunction with the fonn of -)from table Bl.5.1. We fmd, for example, for the p-in/p-out and s-... 

Excitable media are some of tire most commonly observed reaction-diffusion systems in nature. An excitable system possesses a stable fixed point which responds to perturbations in a characteristic way small perturbations return quickly to tire fixed point, while larger perturbations tliat exceed a certain tlireshold value make a long excursion in concentration phase space before tire system returns to tire stable state. In many physical systems tliis behaviour is captured by tire dynamics of two concentration fields, a fast activator variable u witli cubic nullcline and a slow inhibitor variable u witli linear nullcline [31]. The FitzHugh-Nagumo equation [34], derived as a simple model for nerve impulse propagation but which can also apply to a chemical reaction scheme [35], is one of tire best known equations witli such activator-inlribitor kinetics ... 

Force fields like MM3, MM4, CFF, or MMFF therefore use cubic and/or quartic or even higher contributions, up to the sixth power. Special attention has to be paid in the case of reference angles approaching 180°, e.g., for molecules with linear firag-ments such as acetylene compounds. In this circumstance, replacing Eq. (23) by a... 

Hach molecular mechanics method has its own functional form MM+. AMBER, OPL.S, and BIO+. The functional form describes the an alytic form of each of th e term s in th e poteri tial. For exam pie, MM+h as both a quadratic and a cubic stretch term in th e poten tial whereas AMBER, OPES, and BIO+ have only c nadratic stretch term s, I h e functional form is referred to here as the force field. For exam pie, th e fun ction al form of a qu adratic stretch with force constant K, and equilibrium distance i q is ... 

The default parameters for bond stretching are an ec iiilibriiim bond length an d a stretch in g force eon starit. fb e fun etion al form isjiist that of the. M.M+ force field including a correction for cubic stretches. The default force constant depends only on the bond... 

In the case of ethylene, because of 2-fold symmetry, odd terms drop out of the series, V3, V5,... = 0. In the case of ethane, because of 3-fold symmeti-y, even temis drop out, V2, V4,... = 0. Terms higher than three, even though permitted by symmetry, are usually quite small and force fields can often be limited to three torsional terms. Like cubic and quaitic terms modifying the basic quadratic approximation for stretching and bending, terms in the Fourier expansion of Ftors (to) beyond n = 3 have limited use in special cases, for example, in problems involving octahedrally bound complexes. In most cases we are left with the simple expression... 

FIGURE 6.2 Hannonic, cubic, and Morse potential curves used to describe the energy due to bond stretching in molecular mechanics force fields. 

The direction of the alignment of magnetic moments within a magnetic domain is related to the axes of the crystal lattice by crystalline electric fields and spin-orbit interaction of transition-metal t5 -ions (24). The dependency is given by the magnetocrystalline anisotropy energy expression for a cubic lattice (33) ... 

With the Monte Carlo method, the sample is taken to be a cubic lattice consisting of 70 x 70 x 70 sites with intersite distance of 0.6 nm. By applying a periodic boundary condition, an effective sample size up to 8000 sites (equivalent to 4.8-p.m long) can be generated in the field direction (37,39). Carrier transport is simulated by a random walk in the test system under the action of a bias field. The simulation results successfully explain many of the experimental findings, notably the field and temperature dependence of hole mobilities (37,39). 

Barium titanate [12047-27-7] has five crystaUine modifications. Of these, the tetragonal form is the most important. The stmcture is based on corner-linked oxygen octahedra, within which are located the Ti" " ions. These can be moved from their central positions either spontaneously or in an apphed electric field. Each TiO octahedron may then be regarded as an electric dipole. If dipoles within a local region, ie, a domain, are oriented parallel to one another and the orientation of all the dipoles within a domain can be changed by the appHcation of an electric field, the material is said to be ferroelectric. At ca 130°C, the Curie temperature, the barium titanate stmcture changes to cubic. The dipoles now behave independentiy, and the material is paraelectric (see Ferroelectrics). 

The initial configuration is set up by building the field 0(r) for a unit cell first on a small cubic lattice, A = 3 or 5, analogously to a two-component, AB, molecular crystal. The value of the field 0(r) = at the point r = (f, 7, k)h on the lattice is set to 1 if, in the molecular crystal, an atom A is in this place if there is an atom B, 0, is set to —1 if there is an empty place, j is set to 0. Fig. 2 shows the initial configuration used to build the field 0(r) for the simple cubic-phase unit cell. Filled black circles represent atoms of type A and hollow circles represent atoms of type B. In this case all sites are occupied by atoms A or B. 

The symmetry of the structure we are looking for is imposed on the field 0(r) by building up the field inside a unit cubic cell of a smaller polyhedron, replicating it by reflections, translations, and rotations. Such a procedure not only guarantees that the field has the required symmetry but also enables substantial reduction of independent variables 0/ the function F (f)ij k )- For example, structures having the symmetry of the simple cubic phase are built of quadrirectangular tetrahedron replicated by reflection. The faces of the tetrahedron lie in the planes of mirror symmetry. The volume of the tetrahedron is 1 /48 of the unit cell volume. 

In a cubic field three spin-allowed transitions are expected because of the splitting of the free-ion, ground term and the presence of the term. In an octahedral field the splitting is the same as for the octahedral d ion and the same energy level diagram (p. 1029) can be used to interpret the spectra as was used for octahedral Cr Spectra of octahedral Ni usually do consist of three bands which are accordingly assigned as ... 

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