Frequency-dependent electric-field

Although the Kramers-Kronig relations do not follow directly from physical reasoning, they are not devoid of physical content underlying their derivation are the assumptions of linearity and causality and restrictions on the asymptotic behavior of x> As we shall see in Chapter 9, the required asymptotic behavior of x is a physical consequence of the interaction of a frequency-dependent electric field with matter. 

General theory of the response to frequency-dependent electric fields... 

In the present study we have used a relatively new technique to study the morphology of these systems. Application of an alternating electric field to a multiphase morphology results in a frequency-dependent electric-field distribution due to the different permittivities and conductivities of dissimilar phases. Measured permittivity increases with decreasing frequency v = 0)/2ir, approaching a low frequency limit eQ as the field distribution transits from permittivity dominated to conductivity dominated. 

However, the equivalence of the response functions to the property derivatives is in approximate methods not always strict, as, for example, CC response functions as defined in Section 2 do not involve contributions due to orbital relaxation while property derivatives usually do. The incorporation of orbital relaxation effects in the property derivatives is mandatory when perturbation-dependent basis functions such as GIAOs/LAOs are used. Applying the above reformulation to the expressions for a(-u), w) and obtained from the CC response functions takes only relaxation with respect to the (static) external magnetic field into account [70, 71]. The frequency-dependent electric fields are treated in an unrelaxed manner, which avoids spurious poles due to orbital relaxation (see Section 2.2). 

For a long time the finite oligomer approach was the only method available for determining linear and nonlinear polarizabilities of infinite stereoregular polymers. Recently, however, the problem of carrying out electronic band structure (or crystal orbital) calculations in the presence of static or frequency-dependent electric fields has been solved [115, 116]. A related discretized Berry phase treatment of static electric field polarization has also been developed for 3D solid state systems... 

A popular alternative to variational approaches in the study of excited state properties is time-dependent DFT (TD-DFT) in its adiabatic formulation [122-128]. In this scheme, one considers the first-order change in the ground state density (co. r) due to a perturbation from the 7-component Ey(co) of a frequency-dependent electric field and the associated frequency-dependent polarizability ocyP(co). The excitation energies ojq are subsequently found as poles or resonances for ocy (co). 

In this section we consider molecular properties which characterize the interactions with static and/or frequency-dependent electric fields. The electric properties of a molecule determine the electric properties of the bulk sample, such as the relative permittivity (dielectric constant) and the refractive index. In addition, the electric properties can be used to describe intermolecular forces. 

The consequences of the fact that an electromagnetic wave has both a frequency-dependent electric field component and a frequency-dependent magnetic field component have so far not been discussed. The two components are perpendicular to each other, as well as perpendicular to the direction of the propagation of the light beam. Moreover, we have not discussed the implications of the possible polarization of the light beam. Light experienced in everyday life... 

Depending on field strengths, it can be arranged that only ions of one selected mass can pass (filter) through the rod assembly while all others are deflected to strike the rods. By changing the strengths and frequencies of electric fields, different masses can be filtered through the system to produce a mass spectrum. 

We only consider static response properties in this chapter, which arise from fixed external field. Their dynamic counterparts describe the response to an oscillating electric field of electromagnetic radiation and are of great importance in the context of non-linear optics. As an entry point to the treatment of frequency-dependent electric response properties in the domain of time-dependent DFT we recommend the studies by van Gisbergen, Snijders, and Baerends, 1998a and 1998b. 

If the electric and magnetic dipole moments in the presence of frequency-dependent electric and static magnetic fields are expanded in a series, the leading terms give the following expression for (9)... 

When a low frequency AC electric field is imposed, the particle oscillates around its mean position and platy particles may become optimally aligned with the field. At high frequencies, neither particle shift nor alignment takes place. However, translational movement of dispersed particles can be attained in an asymmetric AC field (without a DC component). The observed drift is attributed to the velocity-dependent viscous drag force in relation to double layer polarization as sketched in Figure 2 for reference, bacteria swim at 0.02-1 mm/s. For more details see Palomino [2], The field frequency co must be low enough such that ionic concentrations and hydrodynamic fields may adjust to... 

The quantity on the left is the Fourier component of the dipole moment induced by the optical field Max(w). These equations can be generalized to mixed frequency-dependent electric dipole, electric quadrupole, magnetic dipole properties, and similar equations can be written for the Fourier components of the permanent electric quadrupole, aj8(magnetic dipole, ma(co). For static Maxwell fields similar expansions yield effective (starred) properties, defined as derivatives of the electrostatic free energies. 

For (2116) faces, the frequencies and electric fields critically depend on the relaxation because the Cr3+ ions located in exposed positions undergo a remarkable inward relaxation. 

If a sample is subjected to the sum of two electrical fields, a direct current (DC) (i.e., frequency-independent component EDC) and a frequency-dependent "optical field" E(co),... 

Fundamental Formulae. We have seen that a strong electric field makes an isotropic body optically anisotropic. By this we mean that its frequency-dependent electric permittivity and magnetic permeability take the form... 

In a subsequent paper, Munn [98] showed that the frequency-dependent local-field tensors accounted for the shift of the poles of the linear and nonlinear susceptibilities from the isolated molecular excitation frequencies to the exciton frequencies. The treatment also described the Davydov splitting of the exciton frequencies for situations where there is more than one molecule per unit cell as weU as the band character or wave-vector dependence of these collective excitations. In particular, the direct and cascading contributions to x contained terms with poles at the molecular excitation energies, but they canceled exactly. Combining both terms is therefore a prerequisite to obtaining the correct pole structure of the macroscopic third-order susceptibility. Munn also demonstrated that this local field approach can be combined with the properties of the effective or dressed molecule and can be extended to electric quadrupole and magnetic dipole nonlinear responses [96]. 

Transient terahertz spectroscopy Time-resolved terahertz (THz) spectroscopy (TRTS) has been used to measure the transient photoconductivity of injected electrons in dye-sensitised titanium oxide with subpicosecond time resolution (Beard et al, 2002 Turner et al, 2002). Terahertz probes cover the far-infrared (10-600 cm or 0.3-20 THz) region of the spectrum and measure frequency-dependent photoconductivity. The sample is excited by an ultrafast optical pulse to initiate electron injection and subsequently probed with a THz pulse. In many THz detection schemes, the time-dependent electric field 6 f) of the THz probe pulse is measured by free-space electro-optic sampling (Beard et al, 2002). Both the amplitude and the phase of the electric field can be determined, from which the complex conductivity of the injected electrons can be obtained. Fitting the complex conductivity allows the determination of carrier concentration and mobility. The time evolution of these quantities can be determined by varying the delay time between the optical pump and THz probe pulses. The advantage of this technique is that it provides detailed information on the dynamics of the injected electrons in the semiconductor and complements the time-resolved fluorescence and transient absorption techniques, which often focus on the dynamics of the adsorbates. A similar technique, time-resolved microwave conductivity, has been used to study injection kinetics in dye-sensitised nanocrystalline thin films (Fessenden and Kamat, 1995). However, its time resolution is limited to longer than 1 ns. 

Furthermore, in treating the electrical conductivity we have thus far considered only single-particle excitations and, in particular at T< T, only thermal excitations of the charge carriers across the Peierls band gap 2A. As we shall see later, the charge-density wave itself can also be transported. This charge-density-wave transport is strongly frequency and electric-field dependent (see Sect. 9.6.6). 

Xu, H., Zhang, S.X., Anlage, S.M., Hu, L.B., and Gruner, G. (2008) Frequency-and electric-field-dependent conductivity of single-walled carbon nanotube networks of varying density. Phys. Rev. 

Sohn and co-workers reported a novel device which can quantify the DNA content within the nucleus of single eukaryotic cells [3]. Since DNA molecules are highly charged in intracellular environment, they will he polarized in an applied low-frequency AC electric field. This polarization response can he measured as a change in total capacitance ACj, across a pair of microelectrodes as individual cells suspended in buffer solution flow one by one through a microchannel (as shown in Fig. 1 and Fig. 2). They found that there is a linear relationship between the capacitance and the DNA content of a cell. And they further showed that this relationship is not species-dependent. This innovative technique is termed as capacitance cytometry, which helps to reveal changes in cellular internal properties and determine the phase of individual cell in cell-cycle. 

On the contrary, when the time-dependent electric field varies on a time scale faster than the relaxation time of one or more molecular degrees of freedom there is not time to reach at any moment a time-dependent polarization which is in equilibrium with the electric field. In this regime, which is called non-equilibrium polarization, the actual value of polarization will also depend values of the electric field at previous time, and the relation between the polarization of a dielectric medium and the time-dependent polarizing field is phenomenologically described in terms of the whole specuiim of the dielectric permittivity as a function of the frequency co of the oscillating electric field. 

In electrical applications, polymers are mostly used as isolators. Since it is then important to be informed about possible electric losses, one needs to know their dielectric properties in dependence on frequency and temperature. As we shall see, description of the response of dielectric materials to applied time dependent electric fields is formally equivalent to the treatment of time dependent mechanical responses. Therefore, we shall discuss both together in one chapter. 

In order to derive a quantum mechanical expression for the frequency-dependent polarizability we can make use of time-dependent response theory as described in Section 3.11. We need therefore to evaluate the time-dependent expectation value of the electric dipole operator (4 o(i (f)) Pa o( (t))) in the presence of a time-dependent electric field, Eq. (7.11). Employing the length gauge, Eqs. (2.122) - (2.124), which implies that the time-dependent electric field enters the Hamiltonian via the scalar potential in Eq. (2.105), the perturbation Hamilton operator for the periodic and spatially uniform electric field of the electromagnetic wave is given as... 

The linear response function describing the first-order induced electric dipole moment due to an oscillating and spatially uniform electric field is related to the frequency-dependent electric... 

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