Localization length

The consequence of the presence of scroll helicity in a tubule is expected to be that any increase (decrease) of the intralayer C—C distance G will increase (decrease) the local length of the spiral, but not necessarily the mean interlayer distance, since the scroll can easily adapt its radius of curvature to minimize, if necessary, any energetic strain due to a stress in the local bond lengths. 

Before describing the transition prediction methods, it is instructive to describe how a diabatic map is used. One chooses a desired mass flux and sets the heat flux to be dissipated (assumed uniform along and around the circular channel) up to the desired local length from the inlet to find the corresponding local vapor quality (from an energy balance) and thus the location of this process condition on the map. The... 

Fig. 7.1. Tilted-band picture of WSL energy spectrum showing Wannier wave functions and localization length L = 4/3/T. Reprinted from Hacker and Obermair (1970) with permission from Springer.

In the discussion on the dynamics in the bead-spring model, we have observed that the position of the amorphous halo marks the relevant local length scale in the melt structure, and it is also central to the MCT treatment of the dynamics. The structural relaxation time in the super-cooled melt is best defined as the time it takes density correlations of this wave number (i.e., the coherent intermediate scattering function) to decay. In simulations one typically uses the time it takes S(q, t) to decay to a value of 0.3 (or 0.1 for larger (/-values). The temperature dependence of this relaxation time scale, which is shown in Figure 20, provides us with a first assessment of the glass transition... 

We now give evidence to show that, when EF lies AE above Ec and AE is small we can define a length , called the coherence length, which is the localization length for an energy AE below Ec. We show again that at T=0, i.e. when Ls is infinite,... 

Fig. 1.26 Wave functions for energies just above Ec according to Mott (1983). ( is the localization length for a function at the same energy below e.

Since oc in (81) is the reciprocal of the localization length , measurements of T0 near to the (Anderson) metal-insulator transition can determine how varies with nc—n. Thus Castner and co-workers (Shafarman and Castner 1986, Shafarman et al 1986) have made measurements in Si P and found, as expected, that for systems just below the Anderson transition varies as (nc— n)-v, with v l. 

The analysis of the classical dynamics shows a transition to chaotic motion leading to diffusion and ionization [6]. In the quantum case, interference effects lead to localization and the quantum distribution reaches a steady state that is exponentially localized (in the number of photons) around the initially excited state. As a consequence, ionization will take place only when the localization length is large enough to exceed the number of photons necessary to reach the continuum. 

Integration of the flow equations gives for small initial disorder and K -C K an effective correlation or localization length at which u becomes of the order unity. This can be extracted from (23b), neglecting the flow of K. 

Figure 27 Geometrical structures of the 0.4 nm Ge wires in the [110] (top), [111] and [100] (bottom) directions shown from the side (left) and from the top (right). Large spheres represent Ge atoms small spheres are H atoms used to saturate the dangling bonds. The grey isosurface gives the probability distribution h//Cxc(r>, r/,) 2 of finding the electron when the hole is fixed in a given position (the e-h localization length L is reported for each wire). The hole positions lie on the dashed line in the left panel and are represented by the crosses in the right panel.

Fig. 6 Comparison of the localization length vs. exciton energy for a gaussian-noise model for dAdT.

As noted in Section 7.2, the RB44Si2 samples that can be grown are not single crystals but are polycrystalline. Therefore, we note that there is sample/measurement (configuration of electrodes and preferred orientation) dependence and this is reflected in differences in the absolute values of the resistivity for the different lanthanide phases. However, the To of crystals are similar and should be considered to be close to the intrinsic values. That is, To is a microscopic parameter reflective of the localization length which is dependent on intrinsic disorder throughout the compound, while po in Eq. (3) will cumulatively reflect contributions from the grain boundaries. The values of po and Tq are listed in Table 5. 

Figure 14. Localization of spin-wave modes with k vectors along the wire axis. All modes are localized, but the localization length is smallest for low-lying modes.

Predictions for physical properties near the transition were first made by Mott. He argued for a nonzero minimum conductivity of the metal at the transition point and for a divergence of localization length in the nonmetal with a consequent divergence in the dielectric constant. 

The corresponding local length scale is given by A — ux)tn while the diffusivity may be written as Aff = (Mx)2tD. When axial molecular diffusion is neglected (or equivalently, in the limit of Per —> oo) Eq. (46) simplifies to... 

---