Frequency dependence

In Equation 12.11, v is the frequency of incident radiation (cm-1), v o is the frequency corresponding to the energy difference between ground and excited electronic state, the sum is over all excited states, and d0n and d o are the dipole transition moments between the ground and excited state (dno = (n d 0) = / I nd I o dx, the T s are wave functions and d is the dipole moment operator). At low frequency (v - 0) Equation 12.11 reduces to the static field expression 

In Equation 12.13, N is the number density of molecules in the beam of radiation (and is thus inversely proportional to the molar volume, Vm), and o is the permittivity of the vacuum. A useful and widely employed method to evaluate the sum in Equation 12.13 leads via the closure approximation to a one-term equation commonly known as the dispersion relation, 

A and v 2 are fitting parameters amenable to physical interpretation using Equation 12.11. The point of present concern is that the isotope effect on polarizability can now be expressed in terms of isotopic differences in refractive index. It follows from Equation 12.14 that a plot of AR/R = [6n2/((n2—l)(n2 + 2))][An/n] vs. v2 gives an approximately straight line, 

Using Equation 12.12 one obtains (AA/A — Av /v 2) = (Aao/ao). We see that precise refractive index differences measured over a reasonable range of wavelengths allow the recovery of the polarizability isotope effect (i.e. the isotope effect on the electric field induced dipole moment), provided the molar volume and its isotope effect are available. 

In order to obtain a surveyable expression for the faradaic impedance, it is convenient to define the Warburg parameters a0 and oR by 

Note that eqns. (223) also apply to the bimolecular mechanisms mentioned in Sect. 7.2.3 by virtue of the definitions of KR, kR, and kD in eqns. (196) and (197), respectively. 

The complex faradaic impedance is derived from eqn. (224) by substituting s = ico followed by splitting into real and imaginary parts. The result is [53, 175] 

In order to understand the physical background of eqn. (225), let us consider the CE mechanism as an example so according to Table 9 AR = 1, BR = kR = 0. Under d.c.-reversible conditions, i.e. in the absence of control by charge transfer and by coupled chemical reactions on the d.c. process, oQ will be given by 

the term [A0a0 + AR aR ] co-1/2 (1 — i) resembles the Warburg impedance corresponding to diffusional mass transport of A, O and R, with a mobile equilibrium between A and 0, i.e. kQ - °°, whereupon the term in g = kQ /co would vanish. If, however, kQ has a finite value, the faradaic impedance is enlarged by the Gerischer impedance expressed by the term containing g. 

The number of scans (NS) generating this spectrum is 480. RD indicates that the relaxation delay between each scan was 15 seconds. With this delay, there was 

FIGURE 1 Phosphorus NMR spectrum of porcine carotid artery. S/N REF. refers to the resonance of a sealed capillary tube of phe-nylphopshonate (PPA), pH 4.0. The individual naturally occurring peaks are identified by number in the text. From Fisher and Dillon (1988, Fig. 1, p. 123). 

6-10 were easily identified in the first spectra of smooth muscle, because these peaks are commonly present in other tissues. Distinguishing the intracellular and extracellular phosphate peaks requires a well-shimmed sample. There is about a 0.3 pH unit difference between the intracellular pH (7.1) and the extracellular pH (7.4). It has become commonplace to reduce the phosphate content of perfusion solutions to minimize the influence of the extracellular phosphate on the mean position of a P peak, and thereby use it as an intracellular pH marker. 

Peaks 1, 2, and 5 were not as readily identifiable in initial smooth muscle spectra. Although they ap- 

In this case / is inversely proportional to the material would exhibit 

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First, the eddy current density is damped while penetrating into the conductor (penetration effect). Here the frequency dependence of the penetration depth implies that for deep lying cracks low frequencies must be used for obtaining a sufficient current density in the vicinity of the crack. Secondly, due to the induction law the induced current density at the surface jco is diminished when using lower frequencies. Therefore, in total, there is a certain excitation frequency which results in a maximum response field from the crack. 

Figure Al.3.30. Theoretical frequency-dependent conductivity for GaAs and CdTe liquids from ab initio molecular dynamics simulations [42].

In linear, spherical and synnnetric tops the components of a along and perpendicular to the principal axis of synnnetry are often denoted by a and respectively. In such cases, the anisotropy is simply Aa = tty -If the applied field is oscillating at a frequency w, then the dipole polarizability is frequency dependent as well a(co). The zero frequency limit of the dynamic polarizability a(oi) is the static polarizability described above. 

Wormer P E S and Hettema H 1992 Many-body perturbation theory of frequency-dependent... 

We now proceed to the spectrum, or frequency-dependent response [44, 42]. The power, or rate of energy absorption, is given by... 

Poliak E, Grabert H and Hanggi P 1989 Theory of activated rate processes for arbitrary frequency dependent friction solution of the turnover problem J. Chem. Phys. 91 4073... 

Galica G E, Johnson B R, Kinsey J L and Hale M O 1991 Incident frequency dependence and polarization properties of the CH I Raman spectrum J. Phys. Chem. 95 7994-8004... 

Similarly, we can define the corresponding frequency-dependent second-order, and third-order,... 

The sinc fiinction describes the best possible case, with often a much stronger frequency dependence of power output delivered at the probe-head. (It should be noted here that other excitation schemes are possible such as adiabatic passage [9] and stochastic excitation [fO] but these are only infrequently applied.) The excitation/recording of the NMR signal is further complicated as the pulse is then fed into the probe circuit which itself has a frequency response. As a result, a broad line will not only experience non-unifonn irradiation but also the intensity detected per spin at different frequency offsets will depend on this probe response, which depends on the quality factor (0. The quality factor is a measure of the sharpness of the resonance of the probe circuit and one definition is the resonance frequency/haltwidth of the resonance response of the circuit (also = a L/R where L is the inductance and R is the probe resistance). Flence, the width of the frequency response decreases as Q increases so that, typically, for a 2 of 100, the haltwidth of the frequency response at 100 MFIz is about 1 MFIz. Flence, direct FT-piilse observation of broad spectral lines becomes impractical with pulse teclmiques for linewidths greater than 200 kFIz. For a great majority of... 

Neumann M, Steinhauser O and Pawley G S 1984 Consistent calculation of the static and frequency-dependent dielectric constant in computer simulations Mol. Phys. 52 97-113... 

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 simplest example is that of tire shallow P donor in Si. Four of its five valence electrons participate in tire covalent bonding to its four Si nearest neighbours at tire substitutional site. The energy of tire fiftli electron which, at 0 K, is in an energy level just below tire minimum of tire CB, is approximated by rrt /2wCplus tire screened Coulomb attraction to tire ion, e /sr, where is tire dielectric constant or the frequency-dependent dielectric function. The Sclirodinger equation for tliis electron reduces to tliat of tlie hydrogen atom, but m replaces tlie electronic mass and screens the Coulomb attraction. 

The optical properties of metal nanoparticles have traditionally relied on Mie tlieory, a purely classical electromagnetic scattering tlieory for particles witli known dielectrics [172]. For particles whose size is comparable to or larger tlian tire wavelengtli of the incident radiation, tliis calculation is ratlier cumbersome. However, if tire scatterers are smaller tlian -10% of tire wavelengtli, as in nearly all nanocrystals, tire lowest-order tenn of Mie tlieory is sufficient to describe tire absorjDtion and scattering of radiation. In tliis limit, tire absorjDtion is detennined solely by tire frequency-dependent dielectric function of tire metal particles and the dielectric of tire background matrix in which tliey are... 

VER occurs as a result of fluctuating forces exerted by the bath on the system at the system s oscillation frequency O [5]. Fluctuating dynamical forces are characterized by a force-force correlation function. The Fourier transfonn of this force correlation function at Q, denoted n(n), characterizes the quantum mechanical frequency-dependent friction exerted on the system by the bath [5, 8]. 

Equation (C3.5.2 ) is a function of batli coordinates only. The VER rate constant is proportional to tire Fourier transfonn, at tire oscillator frequency Q, of tire batli force-correlation function. This Fourier transfonn is proportional as well to tire frequency-dependent friction q(n) mentioned previously. For example, tire rate constant for VER of tire Emdamental (v = 1) to tire ground (v = 0) state of an oscillator witli frequency D is [54]... 

Chemical reaction dynamics is an attempt to understand chemical reactions at tire level of individual quantum states. Much work has been done on isolated molecules in molecular beams, but it is unlikely tliat tliis infonnation can be used to understand condensed phase chemistry at tire same level [8]. In a batli, tire reacting solute s potential energy surface is altered by botli dynamic and static effects. The static effect is characterized by a potential of mean force. The dynamical effects are characterized by tire force-correlation fimction or tire frequency-dependent friction [8]. 

Figure C3.5.8. Computed frequency-dependent friction (inverseiy proportionai to tire VER iifetime T ) from a ciassicai moiecuiar dynamics simuiation of rigid Hgl moiecuies in etiianoi soiution, from [90]. The Hgl vibrationai...

C3.5.6.5 POLYATOMIC MOLECULES IN LOW-TEMPERATURE CRYSTALS—FREQUENCY DEPENDENCE... 

Much of our knowledge of the frequency dependence of VER rates in polyatomic molecules stems from low-temperature studies of molecular crystals [2] such as pentacene (PTC 221 4) guest molecules in a crystalline naphthalene (N C,., H ) host. In naphthalene, the phonon cut-off frequency is -180 cm [97]. At low temperature,... 

Figure C3.5.10. Frequency-dependent vibronic relaxation data for pentacene (PTC) in naphthalene (N) crystals at 1.5 K. (a) Vibrational echoes are used to measure VER lifetimes (from [99]). The lifetimes are shorter in regime I, longer in regime II, and become shorter again in regime III. (b) Two-colour pump-probe experiments are used to measure vibrational cooling (return to the ground state) from [1021.

The same idea was actually exploited by Neumann in several papers on dielectric properties [52, 69, 70]. Using a tin-foil reaction field the relation between the (frequency-dependent) relative dielectric constant e(tj) and the autocorrelation function of the total dipole moment M t] becomes particularly simple ... 

Neumann, M., Steinhauser, O. On the calculation of the frequency-dependent dielectric constant in computer simulations. Chem. Phys. Lett. 102 (1983) 508-513. 

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