Isotropic pair interactions

In the experiment discussed above, no directional dependence of the pair interaction is attempted. Pair interactions are simply assumed to be isotropic on the W (110) surface. The pair interaction, in general, should depend both on the direction of the adatom-adatom pair bond and on the bond length. Thus pair energies should therefore be measured for each possible pair bond. A preliminary study in this direction has been reported by the same authors for Si-Si interaction on the W (110) surface.94 Si-Si interaction is of particular interest since (1) Si atoms interact with one another in solid state by forming covalent bonds rather than metallic bonds it would be interesting to see how the interaction of Si adatom pairs on a metal surface is different from that of metal adatom pairs (2) semiconductor-metal interfaces are technologically important... 

The molecular DFT approach describes the attractive interaction separating the isotropic part as the orientation average from the interaction between pairs of molecules. Regarding the anisotropic part of an attractive pair interaction as an excess to the isotropic part, it contributes the essentially smaller share to the energy of the system in comparison with the isotropic part. [7] shows that the anisotropic parts of the dispersive and quadrupolar interaction only very slightly affect the calculated isotherms. 

This result proves to be valid for all MV — d" pairs with n < 4, provided that the spin core is defined as an ion without extra electron. It is also valid for n > 4 but in such case the spin core must be defined as an ion without extra hole. The ferromagnetic effect of the double exchange is illustrated by Fig. 7. Along with the double exchange, the isotropic exchange interaction plays an important role in MV clusters. This interaction is described by the HDVV spin Hamiltonian ... 

The method used here for considering conformal solution models for fluids with molecular anisotropies is based on the method used by Smith (4) for treating isotropic one-fluid conformal solution methods as a class of perturbation methods. The objective of the method is to closely approximate the properties of a mixture by calculating the properties of a hypothetical pure reference fluid. The characterization parameters (in this case, intermolecular potential parameters) of the reference fluid are chosen to be functions of composition (i.e., mole fractions) and the characterization parameters for the various possible molecular pair interactions (like-like and unlike-unlike). In principle, all molecular anisotropies (dipole-dipole, quadrupole-quadrupole, dipole-quadrupole, and higher multipole interactions, as well as overlap and dispersion interactions ) can be included in the method. Here, the various molecular anisotropies are lumped into a single term, so that the intermolecular potential energy uy(ri2, on, a>2) between Molecules 1 and 2 of Species i and / can be written in the form... 

Restricting ourselves only to pair interactions, the spin Hamiltonian appropriate for the isotropic exchange interaction in polynuclear systems becomes... 

As far as isotropic exchange is concerned the interaction matrix consists of all combinations of the pair-interaction matrices... 

The six terms of the isotropic exchange corresponding to the pair-interactions become simplified for some special cases ... 

The isotropic approximation to elastic motion makes the treatment remarkably simple by converting the evaluation of the solute distribution function p(r) to a trivial multiplication of independent terms describing the pair interaction between the solute molecule and the polymer atoms localized at their average positions . A set of mean positions describes the structure of the... 

Let us examine the pair interactions in detail. The polarizability of an atom pair differs from the sum of the atomic polarizabilities because of the anisotropy of a collisional pair. Accordingly, the diatomic polarizability is defined as the polarizability of the interacting pair minus the sum of the polarizabilities of the two isolated atoms. The diatomic polarizability is a tensor. The invariants of the tensor, namely, the isotropic (trace) and anisotropic polzirizabilities, a R) and /3(/ ), respectively, are defined by... 

For the pair, and the respective isotropic (contact) interactions of 26.5 and 18.8 MHz. The exact cancellation condition for the two nitrogen nuclei would therefore occur at v= Via, or 13 and 9.4 MHz, respectively, and the field required to promote the nuclear Zeeman energy to 13 MHz is 4.25T, which is a 122-GHz experiment at g=2.05. The corresponding field and spectrometer operating frequency that would cross levels of the more weakly coupled nitrogen is 3.07T (88 GHz). The predicted splitting of the Am/ =2 transition of the m/ spin manifold at 4.2r is approximately 52 MHz, and the three ground states are therefore within the excitation bandwidth of a 15-ns microwave pulse, which is routinely achieved. 

Hereby 4o is the superconducting order parameter at T = 0 in the absence of any impurities. Furthermore an isotropic exchange interaction has been assumed. The dependence of Tc as function of n or a is shown in fig. 17.13. acrit denotes the critical value of the pair-breaking parameter a beyond which superconductivity can no longer exist. It is a it = i The functional dependence of (nO is described by... 

If instead of the isotropic exchange interaction (A, S) = (0,1) we consider the aspherical Coulomb interaction (A, 2 ) = (2,0) we obtain fig. 17.15 instead. The negative sign for y indicates pair-enhancement instead of pair-breaking. Since this is a dynamical effect it vanishes for 5 = 0. The maximum effect occurs for 8 = lOTc. It resembles the one of additional Einstein oscillators put into the matrix. The quantity t, in the definition of y has to be replaced by to which is defined in a similar way as Ts (see eq. (17.66)) but now with Hac instead of He. ... 

Table 1 summarizes the classes of phase behavior found for these polar/nonpolar systems, using an argon-krypton reference system, and compares it with the behavior for simple nonpolar Lennard-Jones systems. An important difference between the two types of systems is that the Lennard-Jones mixtures do not form azeotropes, and appear to exhibit class II behavior only when the components have very different vapor pressures and critical temperatures (T j /Ta > 2). In practice, the liquid ranges of the two components would not overlap in such cases, so that liquid-liquid immiscibility (and hence class II behavior) would not be observed in Lennard-Jones mixtures (the only exception to this statement seems to be when the unlike pair Interaction is improbably weak). Thus, the use of theories based on the Lennard-Jones or other Isotropic potential models cannot be expected to give good results for systems of class II, and will probably give poor results for most systems of classes III, IV and V also. 

The first stage (O Eq. 29.4) is thought to be magnitosensitive (Afanasyeva et al. 2006). Electron transfer from NADH to native HRP produces the quartet (Q) radical pair of Per and NADH, which is imreactive toward recombination. The quartet - doublet (D) transition produces a radical pair that can recombine and quench the enzymatic process. It is claimed (Afanasyeva et al. 2006) that the Q-D transition is governed by isotropic hyperfine interaction and radical g-factors in terms of radical pair theory (Kumar et al. 2005). This Q-D transition is presented in the left part of the following scheme ... 

Andrienko, D., M. Tasinkevych, and S. Dietrich. 2005. Effective pair interactions between colloidal particles at a nematic-isotropic interface. Europhys. Lett. 70 95-101. 

Closs G L and Trifunac A D 1970 Theory of chemically Induced nuclear spin polarization. III. Effect of Isotropic g shifts In the components of radical pairs with one hyperfine Interaction J. Am. Chem. Soc. 92 2183-4... 

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