Dominant terms

The essence of exchange interaction is the formation of a weak bond between magnetic moments which may be spread over quite distant centres. Several types of exchange interaction are distinguished (Table 10.1) however, a sharp border between them does not exist [1]. 

The spin Hamiltonian that describes the exchange interaction in a dinuclear system is 

The Cartesian spin-spin interaction tensor D, however, can be decomposed into its irreducible components and the spin Hamiltonian rewritten as follows 

direct exchange coupling of localized magnetic moments in insulators through space 

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There are two enthalpy corrections for strongly associating vapors. The dominant term is due to the combined enthalpies of reaction of the stoichiometric species, Ah, to form the true... 

Clearly, the pseudoscalar term vanishes at these points so the ci character at the roots is maintained, no matter whether there are or are not A-i terms. Also, the vanishing of Ai terms will not lead to new ci s.) On the other hand, by circling over a large radius path q oo,so that all ci s are enclosed, the dominant term in Eq. (84) is the last one and the acquired Berry phase is —4(2tc)/2 = —4ti. 

Use nonbonded (NB) tmncation methods to reduce size of NB pairHst it is a dominating term in the calculation. It is important to remember the pairHst i A/, consider tmncation of NBs at 100—120 nm (10—12E), and to experiment with electrostatic cutoffs independentiy of van der Waals. 

The character of AV will determine the type of stationary value at x = Xi. Specifically, the dominant term in the Taylor series for AV must be examined in order to determine whether AV is always positive (a relative minimum), always negative (a relative maximum), sometimes negative and sometimes positive (an inflection point), or always zero (a neutral point). For AV to be positive, the leading term in the Taylor series, Equation (B.4), which is by inspection the largest term because h is a very small number, must be positive, i.e.. 

The Bom-Haber cycle is also useful in examining the possibility of forming alkali-metal halides of stoichiometry MX2- The dominant term will clearly be the very large second-stage... 

Equation (12.14) applies to all the elements that constitute a cell, but it is normally applied in the form of a carbon balance. The dominant terms in the balance are carbon in the cells, measured through Yxjs, and carbon in the primary metabolites—e.g., ethanol and CO2—measured through the Yp/s terms for these metabolites. 

In order to seek the most soft nuclear deformation in an excited state, the approximation is again made of replacing the sum over excited states in Eq. (17) by a dominant term corresponding to the next higher excited state. Now, the transition density p between the nth excited state corresponding to the orbital jump and the mth... 

In fluid dynamics the behavior in this system is described by the full set of hydrodynamic equations. This behavior can be characterized by the Reynolds number. Re, which is the ratio of characteristic flow scales to viscosity scales. We recall that the Reynolds number is a measure of the dominating terms in the Navier-Stokes equation and, if the Reynolds number is small, linear terms will dominate if it is large, nonlinear terms will dominate. In this system, the nonlinear term, (u V)u, serves to convert linear momentum into angular momentum. This phenomena is evidenced by the appearance of two counter-rotating vortices or eddies immediately behind the obstacle. Experiments and numerical integration of the Navier-Stokes equations predict the formation of these vortices at the length scale of the obstacle. Further, they predict that the distance between the vortex center and the obstacle is proportional to the Reynolds number. All these have been observed in our 2-dimensional flow system obstructed by a thermal plate at microscopic scales. ... 

Notice that the usually dominant term, the one containing is not approximated. 

A sequence of approximations, using properties of the confluent hypergeometric function, integration by steepest descents, and judicious discard of all but the dominant terms, gives one the asymptotic form... 

The first term is characterized by a scalar, 7, and it is the dominant term. Be aware of a convention disagreement in the definition of this term instead of -27, some authors write -7, or 7, or 27, and a mistake in sign definition will turn the whole scheme of spin levels upside down (see below). The second and third term are induced by anisotropic spin-orbit coupling, and their weight is predicted to be of order Ag/ge and (Ag/ge)2, respectively (Moriya 1960), when Ag is the (anisotropic) deviation from the free electron -value. The D in the second term has nothing to do with the familiar axial zero-field splitting parameter D, but it is a vector parameter, and the x means take the cross product (or vector product) an alternative way of writing is the determinant form... 

The firsttwo terms on the right-hand side of this expression are responsible for spatial transport of scalar dissipation. In high-Reynolds-number turbulent flows, the scalar-dissipation flux (iijC ) is the dominant term. The other terms on the right-hand side are similar to the corresponding terms in the dissipation transport equation ((2.125), p. 52), and are defined as follows. 

For surfaces for which S is small, this new expression for the E-function is in fact similar to the old one, since then the dominant term under the square root is (x/ )2-... 

The Debye-Hiickel term, which is the dominant term in the expression for the activity coefficients in dilute solution, accounts for electrostatic, nonspecific long-range interactions. At higher concentrations, short-range, nonelectrostatic interactions have to be taken into account. This is usually done by adding ionic strength dependent terms to the Debye-Hiickel expression. This method was first outlined by Bronsted [5,6], and elaborated... 

For large values of r, the dominant terms in the density expansions (see Eq. (72)) are those with the smallest exponents. Moreover, because the exponential term is the dominant one, we can approximate F(/(r)) in Eq. (71) by ... 

The probability of a background signal overlapping with a target signal once [e.g., P(a = 1)] will normally be the dominant term in Eq. (11.3), but the higher... 

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