Complex frequency dependence

Treating the free electrons in a metal as a collection of zero-frequency oscillators gives rise51 to a complex frequency-dependent dielectric constant of 1 - a>2/(co2 - ia>/r), with (op = (47me2/m)l/2 the plasma frequency and r a collision time. For metals like Ag and Au, and with frequencies (o corresponding to visible or ultraviolet light, this simplifies to give a real part... 

The combined effects of the dipolar and exchange interactions produce a complex frequency-dependent EPR spectrum, which can however be analysed by performing numerical simulations of spectra recorded at different microwave frequencies. When centre A is a polynuclear centre, the value of its total spin Sa = S, is determined by the strong exchange coupling between the local spins S, of the various metal sites. In this case, the interactions between A and B consist of the summation of the spin-spin interactions between Sb and all the local spins S, (Scheme II). The quantitative analysis of these interactions can therefore yield the relative arrangement of centres A and B as well as information about the coupling within centre A. 

Equation (2) is, strictly speaking, not suitable for optical fields, which are rapidly varying in time. The damping of the oscillating dipole, and the resultant phase shift, is then conveniently expressed by treating the hyperpolarizabilities as complex, frequency-dependent quantities. For the cubic hyperpolarizability, the relation between the Fourier components of the electric field and the Fourier amplitude of the oscillation of the electric dipole gives... 

Equation (187) defines a complex frequency-dependent function T (co), in which terms is the velocity spectral density Cvv(oo) = y.i v(t)v)eu" dt... 

The response of a pure, homogeneous medium to the applied fields may be characterized quite generally by a complex frequency-dependent dielectric function e(o>), which can be written in terms of its real and imaginary parts as... 

A measurement of the optical conductivity of PAni samples over the range 2 meV to leV is illustrated in Fig. 9.38. The reflection spectra were obtained using a Fourier transform interferometer and a synchrotron radiation source. The Kramers-Kroiiig transformation, with appropriate extrapolation of the data to zero and high energy, was used to extract the complex frequency-dependent conductivity and its real part, the optical conductivity. The optical conductivity is related to the imaginary component of the electric susceptibility, since the power dissipated by an optical field of amplitude E is given by ... 

It should be noted that polycarbonate has a strong beta transition near -100 C. The polymers least mobile state is, therefore, below the beta transition. Further experimentation, not included here, indicates that below the beta transition the magnitude of E" and tan 8 are far less frequency dependent than above the transition. The poor fit seen in the "glassy" region of Figure 6 appears to be due to the presence of the beta transition. The difference in activation energies for the a and P transitions result in tan 8 having a more complex frequency dependence in the... 

If the nanocrystal permittivity is greater than the host permittivity, snc > Shost, the screening factor S describes depression of the decay due to screening of the radiation field inside the nanocrystal. Frohlich was the first to note that for a metal nanocrystal with the complex frequency-dependent dielectric function... 

Let us, at this point, return to the example described above (Fig. 8A), for which an exponentially decaying response function was obtained. This response function is here written as (r) = (Ax/T)e /, where t is the relaxation time and A/ is a constant that may be interpreted as the maximal change in /, hence the notation. As equation (20) shows, the corresponding complex frequency-dependent susceptibility is... 

When a field E is applied across a dielectric (a simple parallel plate condenser, for example), the resulting displacement current, D is related to E as D = eE. In glasses, which are dielectrically isotropic, the permittivity, 8 behaves as a scalar quantity and is equal to D E. While E is an experimentally controlled alternating field of arbitrary frequency, D and s are the material dependent responses and Z), which represents the polarization current is not always in phase with E. s is, therefore, a complex frequency dependent quantity. The complex permittivity, e, is defined as... 

Furthermore, in a non-magnetic (where the permeability 4= 1) medium, n is related to the complex frequency-dependent dielectric constant (cop), and the dielectric susceptibility Xg, by... 

D and s are the material dependent responses and D, which represents the polarization current is not always in phase with E. s is, therefore, a complex frequency dependent quantity. The complex permittivity, e, is defined as... 

Second, the positions and llneshapes of resonances arising from potentially mobile parts of the peptide (e,g, side chains) have revealed dynamical aspects of the solid-state structures of peptides. The analysis of molecular motions is simplified In the solid state by the absence of overall molecular tumbling, which modulates spin interactions and leads to complex frequency -dependent spectral responses. In particular, signals arising from aromatic ring side chains are well separated from other resonances, and may be interpreted in terms of reorientation models of these side chains. Such ring dynamics are of great importance in protein structures, and studies with model peptides can help elucidate fundamental aspects of these processes. 

Figure 8.3. The real part of the complex frequency-dependent dielectric function [e (co)] of aqueous myoglobin solution for different concentrations. Concentrations are (from top to bottom) 161, 99, and 77 mg/mL at 293.15 K. The symbols denote experimental results while the solid line is a fit to the theory of dynamics exchange model developed by Nandi and Bagchi. Adapted with permission from J. Phys. Chem. A, 102 (1998), 8217-8221. Copyright (1998) American Chemical Society.

The so-called Hanai equation (8,9) gives the complex (frequency dependent) permittivity of an emulsion as... 

The physical quantities describing passive electrical behavior of material in time and frequency domain are clearly distinguished. The complex, frequency-dependent impedance (impedance spectrum) exists only in the frequency domain, whereas the impulse answer is the respective property in the time domain. A single relaxation process yields a dispersion region (e.g., 3-dispersion) with a characteristic frequency (e.g., wb) in the frequency domain that corresponds to a relaxation strength and relaxation time (time constant) in the time domain. 

Where u is the Fourier transformed displacement vector, p is the density of material, V is the three dimensional differential operator and O) is the angular frequency. The complex frequency dependent functions X and p are related to the relaxation functions of the material and //(f). 

Both the phase shift and the amplitude of the dynamic component are used for the calculation of a complex (frequency-dependent) heat capacity. These quantities can be interpreted in the context of the relaxation theory or irreversible thermodynamics. 

Kramers-Kronig Relations. The Kramers-Kronig (KK) relations are derived from the basic causality condition that the output strain cannot precede the input stress in any physical material (13-15). These relations apply to the complex, frequency-dependent elastic moduli of any material, and relate the real and imaginary components of the modulus. For example, for the complex shear modulus, G (co) = G (co) + iG" co), the Kramers-Kronig relations are... 

The temperature-modulated mode of operation has been well known for many decades in calorimetry [33], but became well established only during the 1990s, when commercial DSC was modified this way [34], The idea is to examine the behavior of the sample for periodic rather than for isothermal or constant-heating-rate temperature changes. In this way it is possible to obtain information on time-dependent processes within the sample that result in a time-dependent generalized (excess) heat capacity function or, equivalently, in a complex frequency-dependent quantity. Similar complex quantities (electric susceptibility, Young s modulus) are known from other dynamic (dielectric or mechanical) measurement methods. They are widely u.sed to investigate, say, relaxation processes of the material. 

This is the reason why frequency-dependent measurements are of increasing interest, particularly in polymer science. A drawback of temperature-modulated calorimetry is the rather unclear complex frequency-dependent function and the theory behind it. 

The normally used dynamic specific heat c(tu) as given by the heat capacity per unit mass is in general a complex, frequency-dependent quantity of the... 

In this section we shall consider how the results obtained above for reflection from a plane surface are modified in the case of rough surfaces. We shall assume that the amplitude of surface roughness, (ry), can be treated as small and thus the solutions of Maxwell s equations can be expanded as a Taylor series in it (Maradudin and Mills 1975). We suppose for simplicity that above the surface z = (ry) is vacuum, while below it is the isotropic medium with a complex frequency-dependent dielectric function e = e co). The total dielectric function can then be written as... 

The complex, frequency dependent, dipole polarizability, a(o)), is a useful quantity to study since it is simply related to the total photoabsorption cross section by... 

Equations (8) and (9) constitute the TDLBA which requires their simultaneous solution. The resulting 6n(x o)) is then combined with (5) and (6) to yield the total photoabsorption. Alternatively, it can be shown that the Golden Rule expression, (4), remains valid if uext(x) is replaced by the complex, frequency dependent effective field... 

Figure 11.10 Complex, frequency dependent conductivities for PbS measured 10 ps after photoexcitation for two different 266 nm excitation fluences (black and grey dots). The data is described well by the Drude expression (solid lines) yielding the plasma frequency (directly related to density) and the carrier scattering time.

The optical properties of all linear optical materials can be described by the complex, frequency-dependent permittivity or dielectric constant ... 

---