Analytical dipolar calculation

Feg). Subsequently, thermodynamic properties of spins weakly coupled by the dipolar interaction are calculated. Dipolar interaction is, due to its long range and reduced symmetry, difficult to treat analytically most previous work on dipolar interaction is therefore numerical [10-13]. Here thermodynamic perturbation theory will be used to treat weak dipolar interaction analytically. Finally, the dynamical properties of magnetic nanoparticles are reviewed with focus on how relaxation time and superparamegnetic blocking are affected by weak dipolar interaction. For notational simplicity, it will be assumed throughout this section that the parameters characterizing different nanoparticles are identical (e.g., volume and anisotropy). 

With regard to real electrolytes, mixtures of charged hard spheres with dipolar hard spheres may be more appropriate. Again, the MSA provides an established formalism for treating such a system. The MSA has been solved analytically for mixtures of charged and dipolar hard spheres of equal [174, 175] and of different size [233,234]. Analytical means here that the system of integral equations is transformed to a system of nonlinear equations, which makes applications in phase equilibrium calculations fairly complex [235]. 

A general approach (VIG, GT) to a linear-response analytical theory, which is used in our work, is viewed briefly in Section V.B. In Section V.C we consider the main features of the hat-curved model and present the formulae for its dipolar autocorrelator—that is, for the spectral function (SF) L(z). (Until Section V.E we avoid details of the derivation of this spectral function L). Being combined with the formulas, given in Section V.B, this correlator enables us to calculate the wideband spectra in liquids of interest. In Section V.D our theory is applied to polar fluids and the results obtained will be summarized and discussed. 

Nitzan and Brus developed an analytical formula for the molecular absorption cross section given the model defined above [14]. Figure 9.2 is taken fi"om Ref. [13] and shows the calculated absorption cross section based on the model associated with the photodissociation of I2. (The I2 formed through the absorption process is very short lived.) Photodissociation predicted to be enhanced as the molecule is placed near a silver metal nanoparticle of radius a - 50 nm near the electronic transition resonance position of cat) 22,200 cm . If e eiai(co) is the dielectric fiinction for the metal, a small metal nanoparticle plasmon in air will have its dipolar surface plasmon resonance at frequency <24 such that [1]... 

The analytic solution of the SSOZ-MSA equation for polar hard dumbbells came before any serious consideration was given to calculating the dielectric constants of such systems by computer simulation. At the time, there was considerable controversy about the simulation methods used to calculate the dielectric constant, and for the model systems then in vogue (dipolar hard spheres and the Stockmayer fluid) there was also debate about the correct value of the dielectric constant. Today, this problem is becoming better understood in particular, the quality of the simulation work has improved greatly, and this has allowed meaningful conclusions to be drawn about the relative merits of simulation methods. 

The cross sections and trajectories for the Langevin case are completely determined by analytical techniques. On the other hand, no general analytical treatments are available for collisions between ions and dipolar molecules. We have investigated the use of numerical methods to obtain information on these collisions and most of the results presented here are obtained from computer calculations. First, however, we shall review some analytical considerations which yield useful approximate results. 

The Langevin trajectories for ion-molecule collisions (described in the Introduction) have a very simple property they either lead to a capture collision or they do not. It is a simple all-or-nothing situation. The treatment leading to the analytical maximum cross section for a dipolar molecule assumed that these collisions have the same property it is also for an all-or-nothing situation. The calculated ion-dipolar molecule collisions do not have this property. There is no all-or-nothing answer to the question of capture in an ion-permanent-dipole collision for a fixed initial impact parameter. For... 

When only inhomogeneous interactions are involved and the dynamic effects are neglected, the Hamiltonian becomes exactly solvable and frequently in analytical forms. For example, the lineshapes of the chemical shift, heteronuclear dipolar and quadrupolar interactions can be calculated exactly and analytically, either for static or for rotating samples. ... 

In many cases, the shape of the solute molecule may be very different from the sphere and therefore, it is necessary to develop the methods of calculation of the electrostatic solvation energy for more complex cavities. In the case of the ellipsoidal cavity with main semiaxes a, b, and c, the analytical formulas are still available for the calculation of the eharge and dipolar terms of the electrostatic interaction with the reaction field. The charge term is simply... 

The general problem of calculating the lineshape for dipolar-coupled spins is not capable of analytic solution except for very simple cases, such as isolated pairs. This is because it is a many-body problem (1,2). Useful quantities, in such circumstances, are the moments of the lineshape. In particular the second moment, M2, which is the mean-squared linewidth, is often used. This, we can see from the above discussion, is a measure of the strength of the dipolar local field, and its utility lies in the fact that its calculation does not require the solution of the many-body problem (1,2). For a single crystal eq. 3 holds. 

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