Lifshitz frequency

Besides cell thickness d of the nematic layer, which has almost no effect in the case ei = es = 0, reveals a strong influence on EC for finite flexocoefficients. This is demonstrated in the lower panel of Fig. 4.5, where for d = 10 /Ltm the conductive branch is totally absent. Then, as with the conductive regime, one can find a transition from oblique to normal dielectric rolls above a Lifshitz frequency In a recent experiment the oblique dielectric rolls at small w have indeed been observed. The threshold characteristics Uc and qc and the obliqueness angle a. could be well reproduced by a theoretical analysis of the nemato-electrohydrodynamic equations including flexopolarization. ... 

Before discussing these experiments we will present new measurements using the nematic Phase 5, which show a similar crossover at ft < 0.1 Hz. Thus for f < ft there are flexodomains as a first instability while for f > ft there are the usual EC roll patterns with conductive symmetry. Above the Lifshitz frequency //, 40 Hz there are normal rolls, which are replaced by... 

The Hamaker constant can be evaluated accurately using tire continuum tlieory, developed by Lifshitz and coworkers [40]. A key property in tliis tlieory is tire frequency dependence of tire dielectric pennittivity, (cij). If tills spectmm were tlie same for particles and solvent, then A = 0. Since tlie refractive index n is also related to f (to), tlie van der Waals forces tend to be very weak when tlie particles and solvent have similar refractive indices. A few examples of values for A for interactions across vacuum and across water, obtained using tlie continuum tlieory, are given in table C2.6.3. 

The transition described by (2.62) is classical and it is characterized by an activation energy equal to the potential at the crossing point. The prefactor is the attempt frequency co/27c times the Landau-Zener transmission coefficient B for nonadiabatic transition [Landau and Lifshitz 1981]... 

With the reader bearing in mind this framework, the Lifshitz theory of van der Waals interactions can readily be understood. According to the Lifshitz theory, van der Waals forces arise from the absorption of photons of frequency tu by a material with a complex dielectric constant... 

Ya.B. applied formal perturbation theory to the interaction of an atom with the electrons of a metal, where the latter are assumed to be free. Meanwhile, Casimir and Polder and Lifshitz neglected the spatial dispersion of the dielectric permittivity of the metal. Therefore, in the region of small distances, frequencies of order ui0 are important at small distances in the sense indicated above, as are arbitrarily small frequencies at large distances. In both limits the dielectric permittivity of the metal is not at all close to one. Meanwhile, the perturbation theory used by Ya.B. corresponds formally to an expansion in powers of e - 1. and is therefore not applicable in this case. Neglecting the spatial dispersion is valid, however, only at distances r > a (a is the Debye radius in the metal) of the atom from the surface. At the opposite extreme, r a, the wave vectors kj 1/r > a vF/u>0 Me of importance (vF is the electron speed at the Fermi boundary). In this region of strong spatial dispersion perturbation theory can be applied, and the (--dependence satisfies Zeldovich s law. 

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. 

Unlike the Lifshitz case, because only the zero-frequency ionic-charge fluctuations count, there is no summation over finite frequencies. There is only the integration over wave-vector magnitudes p to achieve the free energy GLmR(l) [as in Eq. (L3.31)] ... 

P. J. W. Debye, Polar Molecules (Dover, New York, reprint of 1929 edition) presents the fundamental theory with stunning clarity. See also, e.g., H. Frohlich, "Theory of dielectrics Dielectric constant and dielectric loss," in Monographs on the Physics and Chemistry of Materials Series, 2nd ed. (Clarendon, Oxford University Press, Oxford, June 1987). Here I have taken the zero-frequency response and multiplied it by the frequency dependence of the simplest dipolar relaxation. I have also put a> = if and taken the sign to follow the convention for poles consistent with the form of derivation of the general Lifshitz formula. This last detail is of no practical importance because in the summation Jf over frequencies fn only the first, n = 0, term counts. The relaxation time r is such that permanent-dipole response is dead by fi anyway. The permanent-dipole response is derived in many standard texts. 

A fuller theoretical analysis of vdW interactions requires recourse to Lifshitz theory [8[. Lifshitz theory requires a description of the dielectric behavior of materials as a function of frequency, and there are several reviews for the calculation of Hamaker functions using this theory. The method described by Hough and White (H-W) [95], employing the Ninham-Parsegian [96] representation of dielectric data, has proved to be most useful. The nonretarded Hamaker constant (for materials l and 2, separated by material 3) is given by... 

Note first that in this older picture, for both the attractive (van der Waals) forces and for the repulsive double-layer forces, the water separating two surfaces is treated as a continuum (theme (i) again). Extensions of the theory within that restricted assumption are these van der Waals forces were presumed to be due solely to electronic correlations in the ultra-violet frequency range (dispersion forces). The later theory of Lifshitz [3-10] includes all frequencies, microwave, infra-red, ultra and far ultra-violet correlations accessible through dielectric data for the interacting materials. All many-body effects are included, as is the contribution of temperature-dependent forces (cooperative permanent dipole-dipole interactions) which are important or dominant in oil-water and biological systems. Further, the inclusion of so-called retardation effects, shows that different frequency responses lock in at different distances, already a clue to the specificity of interactions. The effects of different geometries of the particles, or multiple layered structures can all be taken care of in the complete theory [3-10]. 

These results leave several basic questions open How to derive a non-Markovian master equation (ME) for arbitrary time-dependent driving and modulation of a thermally relaxing two-level system Would the two-level system (TLS) model hold at all for modulation rates, that are comparable to the TLS transition frequency u)a (between its states e) and g)) which may invalidate the standard rotating-wave approximation (RWA), [to hen-Tannoudji 1992] Would temperature effects, which are known to incur upward g) —> e) transitions, [Lifshitz 1980], further complicate the dynamics and perhaps hinder the suppression of decay How to control decay in an efficient, optimal fashion We address these questions by outlining the derivation of a ME of a TLS that is coupled to an arbitrary bath and is driven by an arbitrary time-dependent field. 

The role of the medium, in which contacting and pull-off are performed, has been mentioned but not considered so far. However, the surroundings obviously influence surface forces, e.g., via effective polarizability effects (essentially multibody interactions e.g., by the presence of a third atom and its influence via instantaneous polarizability effects). These effects can become noticeable in condensed media (liquids) when the pairwise additivity of forces can essentially break down. One solution to this problem is given by the quantum field theory of Lifshitz, which has been simplified by Israelachvili [6]. The interaction is expressed by the (frequency-dependent) dielectric constants and refractive indices of the contacting macroscopic bodies (labeled by 1 and 2) and the medium (labeled by 3). The value of the Hamaker constant Atota 1 is considered as the sum of a term at zero frequency (v =0, dipole-dipole and dipole-induced dipole forces) and London dispersion forces (at positive frequencies, v >0). 

Ajj in condensed phases, called the macroscopic theory. The latter is not limited by the assumption for pairwise additivity of the van der Waals interaction (see also References 34, 254, and 265). The Lifshitz theory treats each phase as a continuous medium characterized by a given uniform dielectric permittivity, which is dependent on the frequency, v, of the propagating electromagnetic waves. Eor the symmetric configuration of two identical phases i interacting across a medium j, the macroscopic theory provides the expression ... 

Values of e, n and ve and Hamaker constants for two identical types of a material in a vacuum, which are calculated from Equation (567) by taking e3 = 1 and 3 = 1, are given in Table 7.1. Unfortunately, the lack of material constants, such as the dielectric constant, as a function of frequency for most of the substances, and also the complexity of the derived formulae have hampered the general use of the Lifshitz model. However, Lifshitz theory made possible the advent of the first theories on the stability of hydrophobic colloids as a balance between London attraction and electrical double-layer repulsion. Later, these theories were further elaborated by Derjaguin and Landau, and independently by Verwey and Overbeek. The general theory of colloidal stability (which is beyond the scope of this book) is based on Lifshitz theory and has become known as the DLVO theory, by combining the initials of these four authors. 

Dzyaloshinski, Lifshitz, and.Pitaevski (5) have quantified van der Waals interactions using the methods of quantum field electrodynamics, characterizing the potentials by summation over dielectric susceptibility functions over all frequencies. This approach generally is referred to as the Lifshitz theory. The van der Waals force at small separations (less than 5 nm), at zero temperature (K), and between two materials (designated by subscripts 1 and 2) immersed in a third medium (designated by subscript 3) can be written... 

N-N zero vibrations M is the nitrogen atomic mass and co is the N2 vibration frequency. To finalize determination of the transition probability between relevant electronic n -> n ) and vibrational (vi V2) states (6-17), the electronic transition probability p"" can be fonnd based on the Landau-Zener formula (Landau Lifshitz, 1981a,b) ... 

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