Magnetic fields theory

Although Beltrami fields have featured prominently in hydrodynamics for over a century, only until relatively recently have they received much attention in experimental and theoretical classical electrodynamics. The reason for the omission of this link in the standard development of electric/magnetic field theory can possibly be traced to a key related deficiency in the structure of vector... 

Such second-order molecular properties as spin-spin coupling depend upon distortion of electron clouds by additional external perturbations that is, in the NMR experiment they depend upon the electronic motion induced by an applied magnetic field. Theories for such second-order molecular properties require a study of the change in the molecular-orbital wavefunctions, which may be found by using a perturbation method to describe the effects occurring when a magnetic field is applied.8-1065-67... 

Grayce and Harris have developed a formal magnetic-field theory... 

The results of measurements and comparisons of the different magnetic field are presented and discussed. Further development will be shown, that this theory can be applied to solve inverse eddy-current problems as well. 

Sensitive parameters are necessary to compare several high resolution magnetic field sensors. Such parameters can be found with methods of signal theory for LTI-systems. The following chapter explains characteristic functions and operations of the signal analysis for linear local invariant systems and their use in non-destructive testing. 

Due to its importance the impulse-pulse response function could be named. .contrast function". A similar function called Green s function is well known from the linear boundary value problems. The signal theory, applied for LLI-systems, gives a strong possibility for the comparison of different magnet field sensor systems and for solutions of inverse 2D- and 3D-eddy-current problems. 

The central equations of electromagnetic theory are elegantly written in the fonn of four coupled equations for the electric and magnetic fields. These are known as Maxwell s equations. In free space, these equations take the fonn ... 

To this point, we have considered only the radiation field. We now turn to the interaction between the matter and the field. According to classical electromagnetic theory, the force on a particle with charge e due to the electric and magnetic fields is... 

As a prelude to discussing mean-field theory, we review the solution for non-interacting magnets by setting J = 0 in the Ising Flamiltonian. The PF... 

Fluctuations in the magnetization are ignored by mean-field theory and there is no correlation between neighbouring sites, so that... 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

The little atomic magnets are of course quantum mechanical, but Weiss s original theory of paramagnetism and ferromagnetism (1907) [7] predated even the Boln atom. He assumed that in addition to the external magnetic field Bq, there was an additional internal molecular field B. proportional to the overall magnetization M of the sample. 

Exponent values derived from experiments on fluids, binary alloys, and certain magnets differ substantially from all those derived from analytic (mean-field) theories. Flowever it is surprising that the experimental values appear to be the same from all these experiments, not only for different fluids and fluid mixtures, but indeed the same for the magnets and alloys as well (see section A2.5.5). 

Luckhurst G R 1985 Molecular field theories of nematics systems composed of uniaxial, biaxial or flexible molecules Nuclear Magnetic Resonance of Liquid Crystals ed J W Emsiey (Dordrecht Reidel)... 

Various data sources (44) on plasma parameters can be used to calculate conditions for plasma excitation and resulting properties for microwave coupling. Interactions ia a d-c magnetic field are more compHcated and offer a rich array of means for microwave power transfer (45). The Hterature offers many data sources for dielectric or magnetic permittivities or permeabiHty of materials (30,31,46). Because these properties vary considerably with frequency and temperature, available experimental data are iasufficient to satisfy all proposed appHcations. In these cases, available theories can be appHed or the dielectric parameters can be determined experimentally (47). 

Fig. 4. (a) Magnetic field dependence of the high- and low-temperature MR, respectively. The solid lines are calculated using a simple two-band model for (a) and the 2D weak localization theory for (b) (after Song et o/.[16]). 

Both spin-lattice and spin-spin relaxation depend on rates of molecular motion, for relaxation results from the interaction of fluctuating magnetic fields set up by nuclei in the spin system and in the lattice. A quantitative theory of this dependence was given by Bloembergen et al., who obtained... 

Vignale, G., and Rasolt, M., 1988, Current- and spin-density-functional theory for inhomogeneous electronic systems in strong magnetic fields , Phys. Rev. B 37 10685. 

This adiabatic principle was one of the corner-stones of the old quantum theory. It allowed one to find the quantum conditions when an adiabatic change was imposed on a system. It was used successfully to account for the Stark and Zeeman effects in the spectrum of atomic hydrogen, resulting from the application of an electric and magnetic field respectively (Schwartzchild [1916] Epstein [1916]). 

---