R-4-dependent interactions

In the sense of RG theory, bare interactions with

long-range interactions [289]. One source for r 4-dependent interactions is the attractive interaction of ions with rotating dipoles in solvent molecules. Hpye and Stell [304] argued that theories for the ion-solvent term can also be profitably employed for modeling ion-ion pair interactions. 

Again the issue of screening is of crucial interest, because one expects the bare r 4-dependent interactions to be screened by the free ions in the same way as the bare Coulombic potential is screened. The problem of screening of the r 4 terms has been treated from various aspects by Fisher and coworkers [170, 171] and by Hpye and Stell [169, 303]. All work indicates that at nonzero ion density these are screened to shorter range by the factor exp(-2IV). 

If the r 4-dependent attractive interactions were unscreened, they would indeed give rise to mean-field criticality. On the other hand, an unscreened repulsive interaction would suppress the development of a critical point, giving rise instead to a charge-density wave instability in the neighborhood of where criticality would take place in the absence of the r-4 cavity term. The latter scenario was developed some time ago by Nabutovskii et al. [305] using a Ginzburg-type analysis. 

For a description of the induced transitions between rotovibrational levels vj) — v f) of H2, radial matrix elements, Eq. 4.17, are needed. Since the H-H interaction potential is well known, the vibrational wave-functions, (pVj(r), which depend on the rotational excitation j, can be computed with the help of digital computers [217], At fixed intermolecu-lar separations R, the AXt(r,R) are known at three vibrational spacings,... 

From the functional form of Eq. 12.5, it is easy to see that as distance between the molecules rjj becomes small, the potential becomes very repulsive due to the dominance of the first term (r 12 dependence). However, the repulsive term drops off very rapidly with distance, and the attractive term dominates at long distances. The interaction potential has a minimum at some intermediate distance, with a characteristic attractive well-depth. The parameter oij represents a net collision diameter, and etj determines the depth (strength) of the interaction. Methods for obtaining these parameters from experiment and other estimation techniques are discussed in Section 12.2.3. Combining rules to estimate the parameters interactions between unlike molecules are given in Section 12.2.4. 

Up to now we have been discussing in this Chapter many-particle effects in bimolecular reactions between non-interacting particles. However, it is well known that point defects in solids interact with each other even if they are not charged with respect to the crystalline lattice, as it was discussed in Section 3.1. It should be reminded here that this elastic interaction arises due to overlap of displacement fields of the two close defects and falls off with a distance r between them as U(r) = — Ar 6 for two symmetric (isotropic) defects in an isotropic crystal or as U(r) = -Afaqjr-3, if the crystal is weakly anisotropic [50, 51] ([0 4] is an angular dependent cubic harmonic with l = 4). In the latter case, due to the presence of the cubic harmonic 0 4 an interaction is attractive in some directions but turns out to be repulsive in other directions. Finally, if one or both defects are anisotropic, the angular dependence of U(f) cannot be presented in an analytic form [52]. The role of the elastic interaction within pairs of the complementary radiation the Frenkel defects in metals (vacancy-interstitial atom) was studied in [53-55] it was shown to have considerable impact on the kinetics of their recombination, A + B -> 0. 

Belorizky E., Fremy M.A., Givord D., Li H.S. (1987) Evidence in rare-earth (R) - transition metal (M) intermetallics for a systematic dependence of R-M exchange interaction on nature of the R atom. J. Appl.Phys. 61(8), 3971-3973. 

Examination of the various contributors to ES reveals that the dipole-dipole R"3 term is particularly sensitive to angular distortions. Whereas the dipole-quadrupole interactions contained in the R 4 term are also sizable, it is important to note that they behave differently depending upon which molecule is rotated. That is, the R 4 term produces a net stabilization if the donor is turned but adds to the destabilization of R-3 if the rotation occurs in the acceptor. Overall, the multipole series, truncated at R 5, provides a reasonable approximation of the full ES distortion energy, particularly at the longer distance. 

K+, Rb+ and Cs+, respectively) and, on the other hand, to the nonlinear distance dependencies of electrostatic interactions. The latter fall off sharply in proportion to R 2 for ion-dipole, R3 for ion-quadrupole and R 4 for ion-induced dipole interactions. 

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