Periodic structures, external fields

One-electron atoms subjected to a time-dependent external field provide physically realistic examples of scattering systems with chaotic classical dynamics. Recent work on atoms subjected to a sinusoidal external field or to a periodic sequence of instantaneous kicks is reviewed with the aim of exposing similarities and differences to frequently studied abstract model systems. Particular attention is paid to the fractal structure of the set of trapped unstable trajectories and to the long time behavior of survival probabilities which determine the ionization rates of the atoms. Corresponding results for unperturbed two-electron atoms are discussed. 

The studies of Wiesenfeld [28] and Lai et al. [43] on the classical dynamics of a one-electron atom in a sinusoidal external field provide a physically realistic example in which the presence of KAM tori surrounding stable periodic orbits leads to deviations from the generic behaviour characteristic of a hyperbolic scattering system as discussed in Sect. 2. Although this system (10) seems simple, further studies illuminating the mathematical structures behind the scattering process, e.g. calculation of the Liapunov exponents of the unstable trapped orbits and the fractal dimension of the trapped set, have yet to be performed. 

The next slightly more complicated situation concerns a fluid confined to a nanoscopic slit-pore by structured rather than unstructured solid surfaces. For the time being, we shall restrict the discussion to cases in which the symmetry of the external field (represented by the substrates) i)reserves translational invarianee of fluid properties in one spatial dimension. An example of such a situation is depicted in Fig. 5.7 (see Section 5.4.1) showing substrates endowed with a chemical structm e that is periodic in one direction (x) but quasi-infinite (i.e., macroscopically large) in the other one (y). 

Another important application of all-orders in aZ atomic QED is the theory of the multicharged ions. Nowadays all elements of the Periodic Table up to Uranium (Z=92) can be observed in the laboratory as H-like, He-like etc ions. The recent achievements of the QED theory of the highly charged ions (HCI) are summarized in [11], [12]. In principle, the QED theory of atoms includes the evaluation of the QED corrections to the energy levels and corrections to the hyperfine structure intervals, as well as the QED corrections to the transition probabilities and cross-sections of the different atomic processes photon and electron scattering, photoionization, electron capture etc. QED corrections can be evaluated also to the different atomic properties in the external fields bound electron -factors and polarizabilities. In this review we will concentrate mainly on the corrections to the energy levels which are usually called the Lamb Shift (here the Lamb Shift should be understood in a more broad sense than the 2s, 2p level shift in a hydrogen). 

As a result of the compression plasma flow action on the sample, highly oriented periodic structures are formed on the silicon surface (Fig. 1, a-c). The structure fragments measure 100-800 nm in diameter and 50-100 pm in length. Application of steady external magnetic field (5=0.1 T) causes the surface structures diameter to decrease and their surface density to enhance. 

One of the results of the first order perturbation theory calculations is the equality C]=C2. This corresponds to the formation of the vortex chain in the plane symmetry XOY of the SMN. This one-dimensional vortex lattice is not stable with respect to external forces as a usual two-dimensional lattice. The bias current creates Lorentz force, which acts on the vortex lattice. Moreover, vortices are situated in the field of the piiming force created by the periodical structure of the SMN. 

Otto [26] and later Kudin and Scusseria [27] realized that the major problem for directly including the field in an electronic-structure calculation is related to the fact that the field destroys the periodicity. On the other hand, as mentioned above, both mathematical arguments and actual calculations have found that the charge distribution inside an extended system remains periodie also in the presence of an external field. Therefore, Otto sought a separation of the form... 

When an external electric field E higher than a threshold field Ec is applied in the jc direction, the liquid crystal will reorient as shown in Figure 11.35(b). Because of the periodical structure, we only have to consider the liquid crystal director configuration in the region (0 < x < L, 0 y < L). The liquid crystal director rt varies in space and is described by... 

Both cholesteric and smectic mesophases are layered. In the former case, the periodicity arises from a natural twist to the director field, and in the latter, from a center-of-mass correlation in one dimension. There are many types of smectic phases distinguished by their symmetry and order. The set of field-induced phenomena is quite different for these two materials, owing primarily to the very different layer compressibility. That is, the cholesteric pitch can be unwound by an external field, whereas the smectic layering is typically too strong to be altered significantly. However, because of the common layered structure, there are also strong similarities. 

Of special interest is the case where the electrohydrodynamic instability is caused by a spatially periodic external field. Then there begins an interplay of two periods, the period of the vortex structure w d and the field period A which may be varied in experiment [61]. In such a situation the resulting period of the domain structure may be either commensurate or incommensurate with the field period. So, using electrohydrodynamic patterns with a controlled period, e.g., by a change in the layer thickness, it is possible to model the appearance of different phases in the physical systems, investigate the motion of solitary waves, etc. [62]. 

Cholesteric liquid crystals (CLCs) show very distinctly that molecular structure and external fields have a profound effect on cooperative behavior and phase structure (see also Chapters 2 and 3). CLCs possess a supermolecular periodic helical structure due to the chirality of molecules. The spatial periodicity (helical pitch) of cholesterics can be of the same order of magnitude as the wavelength of visible light. If so, a visible Bragg reflection occurs. On the other hand, the helix pitch is very sensitive to the influence of external conditions. A combination of these properties leads to the unique optical properties of cholesterics which are of both scientific and practical interest. 

It should be noted that a wide range of external field behavior of the specific conductivity a(E) for nanotubes with hydrogen adatoms has the same qualitative nonlinear dependence as for the ideal case of nanoparticles, which was discussed in detail in Ref [11], In general, the dependence of conductivity on the electric field has a characteristic for semiconductors form tends to saturate and decreases monotonically with increasing intensity. This phenomenon is associated with an increase in electrons fill all possible states of the conduction band. Behavior of electrical conductivity under the influence of an external electric field is typical for semiconductor structures with periodic and limited dispersion law [17]. 

---