Landau external fields

The most simple way to accomplish this objective is to correct the external field operator post factum, as was repeatedly done in magnetic resonance theory, e.g. in [39]. Unfortunately this method is inapplicable to systems with an unrestricted energy spectrum. Neither can one use the method utilizing the Landau-Teller formula for an equidistant energy spectrum of the harmonic oscillator. In this simplest case one need correct... 

The potential curve for the electrons near the tip surface is shown in Fig. 1.38. The relevant dimensions are much smaller than the radius of the tip end. Therefore, a one-dimensional model is adequate. In the metal, the energy level of the electrons is lower than the vacuum level by the value of the work function c ). From the point of view of classical mechanics, the electrons cannot escape from the metal even with a very high external field, that is, the potential barrier is impenetrable. From the point of view of quantum mechanics, there is always a finite probability that the electrons can penetrate the potential barrier. In the semiclassical (WKB) approximation, the transmission coefficient for a general potential barrier is (Landau and Lifshitz, 1977) ... 

We presented fully self-consistent separable random-phase-approximation (SRPA) method for description of linear dynamics of different finite Fermi-systems. The method is very general, physically transparent, convenient for the analysis and treatment of the results. SRPA drastically simplifies the calculations. It allows to get a high numerical accuracy with a minimal computational effort. The method is especially effective for systems with a number of particles 10 — 10, where quantum-shell effects in the spectra and responses are significant. In such systems, the familiar macroscopic methods are too rough while the full-scale microscopic methods are too expensive. SRPA seems to be here the best compromise between quality of the results and the computational effort. As the most involved methods, SRPA describes the Landau damping, one of the most important characteristics of the collective motion. SRPA results can be obtained in terms of both separate RPA states and the strength function (linear response to external fields). 

An important step in developing the mean-field concept was done by Landau [8, 10]. Without discussing the relation between such fundamental quantities as disorder-order transitions and symmetry lowering, we just want to note here that his theory is based on thermodynamics and the derivation of the temperature dependence of the order parameter via the thermodynamic potential minimization (e.g., the free energy A(r),T)) which is a function of the order parameter. It is assumed that the function A(rj,T) is analytical in the parameter 77 and thus near the phase transition point could be expanded into the series in 77 usually it is a polynomial expansion with temperature-dependent coefficients. Despite the fact that such a thermodynamical approach differs from the original molecular field theory, they are quite similar conceptually. In particular, the r.h.s. of the equation of state for the pressure of gases or liquids and the external field in ferromagnetics, respectively, have the same polynomial form. 

For the magnetic system in zero external field, the Landau expression for the free energy can be written as an expansion in

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For the case considered by Eq. (3.23), explain why the external field makes a contribution Helmholtz free energy per unit volume, AjV. On this basis, give a thermodynamic derivation of Eq. (3.23) (Landau Lifshitz, vol. 8). 

Taking into consideration the invariance of the free energy under rotation, one obtains the following Landau-type expansion correct up to the fourth order in Q in the absence of any external field... 

Due to the effect of external fields, the order can vary in space and gradient terms have to be added to the Landau expansion (8.9). Usually, only the terms up to the quadratic order are considered. There are many symmetry allowed invariants related to gradients of the tensorial order parameter [29]. However, in the vicinity of the phase transition, one is not interested in elastic deformations of the nematic director but rather in spatial variations of the degree of nematic order. Therefore, the pretransitional nematic system is described adequately within the usual one-elastic-constant approximation. 

The supposition about possibility to leave only linear term in Eq. (1.8) is the postulate of Landau theory that gives good agreement with experimental data [10]. Indeed, adding the term r h to thermodynamic potential (1.5), where h is the external field, conjugated to order parameter, one can find that isothermal susceptibility of a system x = r dh equals to... 

There are also other reasons that truncate the order parameter divergence such as spatial inhomogeneities or external fields. For example, to describe a spatial inhomogeneous system, a term quadratic in the gradient of the order parameter G(Vri) must be added to the density of free energy and all the Landau expansion should be integrated over the system volume ... 

In the proper ferroelectrics, the spontaneous polarisation appears as a result of the polarisation catastrophe or, in other words, due to electric dipole-dipole interactions. There are also improper ferroelectrics, in particular, liquid crystalline ones, in which a structural transition into a polar phase occurs due to other interactions and, consequently, appears as a secondary phenomenon. We shall discuss this case later. For simplicity, the square of spontaneous polarisation vector can be taken as a scalar order parameter for the transition from the higher symmetry paraelectric phase to the lower symmetry ferroelectric phase. Therefore, in the absence of an external field, we can expand the free energy density in a series over P (T) and this expansion for ferroelectrics is called Landau-Ginzburg expansion ... 

The total Landau-de Gennes free-energy density for nematic or cholesteric liquid crystals in an external field is given by... 

The third problem is the possible effect of stress or external field on isotropic-nematic phase transition. In equilibrium, this phase transition is usually described by the well-known Landau phenomenology or more specifically (however, less reliably because of large fluctuations) by the Maier-Saupe mean field theory [2] (see also Refs [30,31 ]). The assumption that the transition behavior of nematic elastomers is independent of stress was roughly confirmed while testing the LCE theory [3], where the parameters of anisotropy were assumed to be independent of stress. The possible dependences of scalar/tensor order parameter on stress/extemal field have been considered in molecular Doi theory [9, 11] or phenomenological approach by Ericksen [41]. 

In the absence of an external field, the isotropic phase is characterized by = 0 the minimum of the free energy also corresponds oQ = 0. This means that, in the Landau expansion of the free energy in terms of the order parameter Q, there is no linear term in g that is,... 

Summary. On the basis of phenomenological Ginzburg-Landau approach we investigate the problem of order parameter nucleation in a ferromagnetic superconductor and hybrid superconductor - ferromagnetic (S/F) systems with a domain structure in an applied external magnetic field H. We study the interplay between the superconductivity localized at the domain walls and between the domain walls and show that such interplay determines a peculiar nonlinear temperature dependence of the upper critical field. For hybrid S/F systems we also study the possible oscillatory behavior of the critical temperature TC(H) similar to the Little-Parks effect. 

L.V. Schubnikov and I.E. Nakhutin [24] have corroborated Landau s theory by electroresistivity measurements parallel and perpendicular to the direction of the external magnetic field on reaching the critical field intensity Hc, and called such a domain structure the intermediate state . A.G. Meshkovskyi and Yu.V. Scharvin [25] have demonstrated the existence of the intermediate state during its scanning by drawing a bismuth wire, well known for the remarkable sensitivity of its electroconductivity to magnetic field. 

The simplest form of the Landau free energy functional describing the state of the longitudinal polarization field P(r) in a homogeneous isotropic liquid in the presence of an external electric field E(r) is... 

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