Third-order magnetic susceptibility

Fig. 55. Tfemperature variation of the third-order magnetic susceptibility for a magnetic field applied along a four-fold axis in cubic TmCu solid lines are calculated taking into account the quadnipolar parameter G, indicated (after laussaud et al. 1980).

Abstract As porphyrins and phthalocyanines possess unique electronic, magnetic and optical properties, supramolecular assembly based on them is subject to intense research targets. Herein, the reviewers focus on the supramolecular architectures of porphyrins, which enable their use as electronic and optical functional materials such as third-order optical susceptibilities, photoenergy conversion systems, and organic field-effect transistors. 

The third-rank diamagnetic and paramagnetic contributions to electric field dependent magnetic susceptibility and nuclear magnetic shielding, to first order in... 

The equation for Qfv is derived in Section V.C together with the scheme of its solution down to the third order in From the solutions obtained, the terms yielding the linear and cubic low-frequency responses to the probing magnetic field H(t) = H cos cot are extracted. In terms of linear and cubic susceptibilities those quantities evaluated numerically are compared in Figures 4.31 and 4.32 primes and double primes there denote, as usual, the in-phase and out-of-phase components of the dynamic susceptibilities. 

In a subsequent paper, Munn [98] showed that the frequency-dependent local-field tensors accounted for the shift of the poles of the linear and nonlinear susceptibilities from the isolated molecular excitation frequencies to the exciton frequencies. The treatment also described the Davydov splitting of the exciton frequencies for situations where there is more than one molecule per unit cell as weU as the band character or wave-vector dependence of these collective excitations. In particular, the direct and cascading contributions to x contained terms with poles at the molecular excitation energies, but they canceled exactly. Combining both terms is therefore a prerequisite to obtaining the correct pole structure of the macroscopic third-order susceptibility. Munn also demonstrated that this local field approach can be combined with the properties of the effective or dressed molecule and can be extended to electric quadrupole and magnetic dipole nonlinear responses [96]. 

However, there are a number of important limitations to this method, which will be discussed in the next subsections. First, the mathematical analysis to obtain the distribution of relaxation times is itself a problem. Secondly, the surface relaxivity parameter is required in order to obtain a pore size distribution. Thirdly, the model assumes that diffusion within the pore is rapid and that interpore coupling can be neglected. Finally, differences in magnetic susceptibility between the solid and fluid phase complicate the interpretation of... 

For TmCd and TmZn a variety of techniques have been applied to determine the important coupling constants and gj. they are listed in table 3. In addition to the temperature dependence of the symmetry elastic constant Cj (T), the parastriction method, the third-order susceptibility and the magnetic field dependence of the structural phase transition temperature Tq B) have been used. The different experimental methods have been described in sect. 2.4.1. It is seen from table 3 that the coupling constants determined with these different methods are in good agreement with each other. 

Details of the synthetic procedures for preparation and characterization of deriva-tized phosphazene trimers and tetramers have been described previously in the open literature [43,44]. Reaction products were characterized using P nuclear magnetic resonance (NMR) spectra and were consistent with previously published results. Third-order susceptibilities were determined from concentrated phosphazene solutions (10% by weight) by degenerate four wave mixing (DFWM) measurements [45]. For acidic conditions, the pH of these solutions was reduced to 1 by the addition of concentrated aqueous HCl. 

It turns out that a third order process is possible which combines Vi and the k-f interaction characterized by (A,. T) = (1,1). Details can be found in Fulde (1975). The net result is that this time the quadrupole susceptibility diverges as TJ T - Tc) in molecular field approximation as the magnetic phase transition is approached from above. This would imply that an elastic constant becomes soft at a second order magnetic phase transition. In practice this requires the presence of the interaction (A, i ) = (1, 1) with reasonable strength. 

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