Zero-frequency conductivity

Fig. 8. Temperature dependence of the zero-frequency conductivity of the paired holon superconductor. The curve is indistinguishable for the three coupling strengths X 0.1, X= 0.5 and X= 1.2.

Ionic conduction has been shown to occur in MHFg (M = K, Rb, or Cs) in both the a (tetragonal) and the / (cubic) phases. Plots of the zero-frequency conductivities versus T show discontinuities at the a-/S transition temperatures moreover, the gradients for the phases are all essentially the same (A 20 kcal), whereas there are considerable differences between the... 

For mullite Arrhenius plots of the extracted zero frequency conductivity data Odc T Odo and Vc shown in Fig. 7. For Vc open circles denote values directly optained from modulus M peak... 

The (zero frequency) conductivity is defined as the steady-state current which flows in unit field. The measured quantity is usually the conductance (G), which is the ratio of steady-state current to... 

The type and magnitude of frequency dependence upon measured resistance depends upon the design of the conductance cell. Generally, measured resistance decreases with increasing frequency, although the opposite effect is observed in some cases with Erlenmeyer-type cells 21>2S>. Mysels et al. 2S> analyze this effect and extrapolate to zero frequency on a plot of resistance vs. f 2. 

Fig. 21. Real part of the conductivity of YbFe4St>i2- The symbols on die left axis represent dc values at different temperatures. Below T (fv 50 K), a narrow peak at zero frequency and a gap-like feature at 18 meV gradually develop. Inset Renormalized band structure calculated from die Anderson lattice Hamiltonian. % and f denote bands of free carriers and localized electrons, respectively. At low temperatures a direct gap A opens. The Fermi level, Ep is near die top of die lower band,, resulting in hole-like character and enhanced effective mass of die quasiparticles (Dordevic et al., 2001).

Eq. (4), frequency-dependent, such that the limit for a(w) in Eq. (8) becomes physically acceptable. Under conditions appropriate to the correct limit, the normalized real and imaginary parts of the complex permittivity and the normalized dielectric conductivity take on the form depicted in Fig. (1). Here, is the relaxation time in the limit of zero frequency (diabatic limit). Irrespective of the details of the model employed, both a(w) and cs(u>) must tend toward zero as 11 + , in contrast to Eq. (8), for any relaxation process. In the case of a resonant process, not expected below the extreme far-infrared region, a(u>) is given by an expression consistent with a resonant dispersion for k (w) in Eq. (6), not the relaxation dispersion for K (m) implicit in Eq. 

Both formulations stumble when the materials are real conductors such as salt solutions or metals. In these cases important fluctuations can occur in the limit of low frequency where we must think of long-lasting, far-reaching electric currents. Unlike brief dipolar fluctuations that can be considered to occur local to a point in a material, walls or discontinuities in conductivity at material interfaces interrupt the electrical currents set up by these longer-lasting "zero-frequency" fields. It is not enough to know finite bulk material conductivities in order to compute forces. Nevertheless, it is possible to extend the Lifshitz theory to include events such as the fluctuations of ions in salt solutions or of electrons in metals. 

The choice among these equivalent forms is a matter of convenience. When the s s are introduced at zero frequency, exclusive of ionic-conductance terms, the free energy... 

The third term is a damping term, which allows for the possibility that a wave is absorbed by the medium this is called the evanescent wave and the quantity a is also called the optical conductivity (at zero frequency it becomes the electrical conductivity). The evanescent wave is exploited in near-field scanning optical microscopy. If waves propagate along x, so that <)/ <)y = 0, d/dz = Q, then EX = HX = 0. Next, assume Ey(x,t)=f(x)exp(icot) = 0 and Ez(x,t)=g(x)exp(icot) = 0 that is, assume plane-polarized light with the E vector in the xy plane Then the differential equation to be solved is more simply... 

Therefore, in the frequency range (o> 30 to 40 cm-1) the collective mode contribution to the conductivity becomes negligible compared to the single-particle contribution (see Fig. 30) and the single-particle contribution in the oscillator strength is shifted down to the narrow zero-frequency mode. 

However, in real materials the interaction between a SDW (or the associated CDW) and impurities or crystal defects provides a finite pinning energy, and a threshold electric field ET must be reached before the condensate can contribute to the conduction at zero frequency. It is given by the balance between the energy provided by the electric field when the condensate is moved by the length 2tt//cf and the pinning energy ... 

Conductivity at non-zero frequency is not constrained by the requirement that carriers must have a conducting path completely through the material. Hopping back and forth between two localized states contributes to the ac conductivity ), but makes no contribution to the dc conductivity. Consequently ct((o) is larger than a(0) and is often dominated by hopping between pairs of states. The conductivity due to hopping near E in a uniform density of states is (Austin and Mott 1969). 

Fig.20 Proton conductivity of P4VP(MSA)i.o(PDP)i.o-fc-PS recorded during heating at 5 °C/min based on AC impedance measurements extrapolated to zero frequency. The cartoons show the dimensionality transitions occurring upon heating from one-dimensional slabs to two-dimensional lamellae to one-dimensional cylinders. The order-disorder transitions at ca. 100 °C and the order-order transition at ca. 150 °C observed by SAXS (Fig. 17) are distinctly reflected in the conductivity [46]...

Other zero-frequency transport coefficients (thermal conductivity, viscosity, etc.) may also be expressed as areas under time-correlation functions by use of the methods described in Chapter 11. 

In heterogeneous systems, an interfacial polarisation is Created due to the space charges. This polarisation corresponds to the electron motion inside conductive charges, dispersed in an insulated matrice (Maxwell-Wagner Model). In fact, this phenomenon will appear as soon as two materials I and 2 are mixed so that c7]/ei C2le.2 with a conductivity and e dielectric constant at zero frequency [ 123]. 

This model predicts that the conductivity is proportional to 0) (tT(m)ao) ) from low frequencies (zero conductivity at d.c.) until the relaxation frequency and that beyond this frequency conductivity is constant. 

Another, more realistic, model takes into account the geometry of the inclusions [125] and a low conductivity for the matrix so that a non-zero static conductivity can be modelled. In this model, the conductivity is first constant, then increases as an law and finally stabilised beyond the relaxation frequency. 

The Brillouin linewidth Td) depends on the dynamic shear and volume viscosities r s(w) and t v(w). If the hypersonic shear viscosity r s(Aa)(i)) is equal to the zero frequency shear viscosity and the small term caused by thermal conductivity is neglected, then measurements of Td) can be used to obtain the volume viscosity. Champion and Jackson (8) noticed that the volume viscosities determined in the above manner for the n-alkanes were essentially independent of temperature. The values of r(i) measured in the authors laboratory for n-hexadecane are plotted... 

At zero frequency the final term dominates (9.28) so that, essentially, the ion state current must equal the defect current and the two mechanisms are effectively in series rather than in parallel as in (9.30) so that the static conductivity is given by... 

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