Landau parameters, values

These values allow one to evaluate the penetration depth X = 2400 A and the coherence length I s 26 A, and hence a Ginz-buig-Landau parameter k X/ around 100, an extremely high value approaching that of the high-Tc copper oxides. 

In the latter case, the authors derived Landau equations (3.114) for the am-phtudes of the unstable modes and determined the dependence of the coefficients of these equations on the parameters of the FP problem. Figure 9 depicts the real parts of the coefficients / i and fh as functions of jo (which is proportional to the initial concentration of the initiator) for the typical parameter values [81] (Ei — E2)/ RgqMo) = 19.79 and Ei/ RgqMo) = 58.4. The wavenumber s = 0.55, which is close to the value Sm at which the neutral stabihty curve has a minimum. 

Earlier [14] a ratio connecting the Ginsburg-Landau parameter % of niobium with the value of specific electroresistivity p was obtained ... 

By taking the complex conjugate of this equation, and changing the sign before C2 at the same time, the equation remains invariant. This says that the only relevant parameter is the absolute value of C2. As we see below, sufficiently large c21 causes turbulence. Since C2 = Im /Re, where g is the nonlinear parameter in the original form of the Ginzburg-Landau equation (2.4.10), c2 oo as Re 0 (i.e., as the system approaches the borderline between supercritical and subcritical bifurcations). A number of kinetic models can have parameter values for which Re = 0, so that such systems should in principle exhibit chemical turbulence of the type discussed below. 

As described above, there is some correspondence between the bilinear/ biquadratic P-9 couplings of the phenomenological Landau model and the polar/quadrupolar rotational potentials of the microscopic model. Indeed, the most pronounced deviation from a proportionality between P and 9, the sign inversion of Pg, can be obtained in both models, either by assuming certain values of the Landau parameters [41] or, in the molecular rotation description, by a slight modification of the rotational potential (8.2) [42]. 

In this section we use Landau-de Gennes theory to calculate the thermodynamic properties of liquid-crystal phase transitions in terms of the phenomenological parameters a, B, C, and Tc. In particular, we will derive expressions for Tc, the transition temperature c, the equilibrium order parameter value in the low temperature phase at the transition and As, the transition entropy per molecule. We will also calculate the temperature dependence of (the equilibrium value of S) for T close to Tc, and h, the height of the free-energy barrier between = 0 and = c at T = Tc. 

To calculate the thermodynamic properties listed above, it is sufficient to consider a spatially uniform system in which the order parameter value is spatially invariant. This means that the spatial derivative terms in the Landau free-energy density, which are important for the calculation of fluctuation phenomena as shown in the next section, can be neglected for the present purpose. Therefore, from Eq. [42] we get... 

A detailed quantitative understanding of liquid He at low temperatures requires the application of Landau s Fermi liquid theory, which takes explicit account of the interactions, and parameterises them in the form of a small number of dimensionless constants known as Landau parameters. For most purposes, only three of these parameters are needed (usually written as 7q, F and G ) and almost all of the properties of the interacting Fermi liquid can be calculated in terms of them. Numerical values of the Landau parameters are not predicted by the theory but are to be found by experiment. The crucial test of the theory—a requirement that consistent values of the Landau parameters should be obtained from widely differing kinds of experiment—is convincingly fulfilled. 

Hence, for a fixed total quasi-momentum K, the critical -value or Landau parameter may be caleulated as ... 

This fomi is called a Ginzburg-Landau expansion. The first temi f(m) corresponds to the free energy of a homogeneous (bulk-like) system and detemiines the phase behaviour. For t> 0 the fiinction/exliibits two minima at = 37. This value corresponds to the composition difference of the two coexisting phases. The second contribution specifies the cost of an inhomogeneous order parameter profile. / sets the typical length scale. 

It gives 1 and dir/Lz in the limit of large and small values of the Landau-Zener parameter, respectively. 

The height of the potential barrier is lower than that for nonadiabatic reactions and depends on the interaction between the acceptor and the metal. However, at not too large values of the effective eiectrochemical Landau-Zener parameter the difference in the activation barriers is insignihcant. Taking into account the fact that the effective eiectron transmission coefficient is 1 here, one concludes that the rate of the adiabatic outer-sphere electron transfer reaction is practically independent of the electronic properties of the metal electrode. 

However, coal reactivity as measured by total conversion to liquids and gases becomes less dependent on coal parameters as processing severity increases. The effect of process temperature in the hot-rod reactor was studied using three coals of varying properties. These were Waterberg, Sigma and Landau. At 650°C the conversion yields of these coals were 89, 90 and 88 per cent of the coal (dmmf) respectively. Within experimental error the conversion yields had converged to the same value, whereas at 500°C the conversion yields were 85, 75 and 65 per cent respectively. 

The construction of the phase diagram of a heteropolymer liquid in the framework of the WSL theory is based on the procedure of minimization of the Landau free energy T presented as a truncated functional series in powers of the order parameter with components i a(r) proportional to Apa(r). The coefficients of this series, known as vertex functions, are governed by the chemical structure of heteropolymer molecules. More precisely, the values of these coefficients are entirely specified by the generating functions of the chemical correlators. Hence, before constructing the phase diagram of the specimen of a heteropolymer liquid, one is supposed to preliminarily find these statistical characteristics of the chemical structure of this specimen. Here a pronounced interplay of the statistical physics and statistical chemistry of polymers is explicitly manifested. 

The Landau theory predicts the symmetry conditions necessary for a transition to be thermodynamically of second order. The order parameter must in this case vary continuously from 0 to 1. The presence of odd-order coefficients in the expansion gives rise to two values of the transitional Gibbs energy that satisfy the equilibrium conditions. This is not consistent with a continuous change in r and thus corresponds to first-order phase transitions. For this reason all odd-order coefficients must be zero. Furthermore, the sign of b must change from positive to negative at the transition temperature. It is customary to express the temperature dependence of b as a linear function of temperature ... 

A wide class of analytic second-order phase transitions is characterized by their Landau bifurcational mechanism [38]. According to this mechanism, a system characterized by order parameter r], possesses a single stable equilibrium solution (rje = 0) for a range of the external parameter T (T > Tcr see a schematic illustration in Fig. 2.3.4a). This single solution corresponds to an absolute internal minimum of the system s free energy F as a function of the order parameter (Fig. 2.3.4b, Curve 1). As the external parameter T decreases, at a critical value T = Tcr, the solution with r)e = 0 becomes unstable with two more stable solutions with r e 0 (for T < TCI) bifurcating from it (Fig. 2.3.4a). In the (F, rf) plane this corresponds to the appearance of two new local free energy minima that flank the former one, which now turns into a local maximum (Fig. 2.3.4b, Curve 2). 

Early attempts to develop theories that accounted for the power-law behavior and the actual magnitudes of the various critical exponents include those by van der Waals for the (liquid + gas) transition, and Weiss for the (ferromagnetic + paramagnetic) transition. These and a later effort called the Landau theory have come to be known as mean field theories because they were developed using the average or mean value of the order parameter. These theories invariably led to values of the exponents that differed significantly from the experimentally obtained values. For example, both van der Waals and Weiss obtained a value of 0.50 for (3, while the observed value was closer to 0.35. 

This relaxation time—which, to be specific, we have written in the Landau-Lifshitz representation—has the anticipated behavior the smaller the precession damping constant (the higher the quality factor of the oscillations), the slower does the particle magnetic moment approach its equilibrium position. For ferromagnet or ferrite nanoparticles the typical values of the material parameters are Is Is < 103G, Vm 10 18cm3, and a 0.1. Substituting them in formula (4.28) aty 2 x 108 rad/Oe s and room temperature, one obtains xD 10-9 s. 

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