High-frequency limit

At large a, all the terms on the right-hand side of Equation 5.91, except can be 

it will be shown that the solution to Equation 5.97 yields 

Equation 5.98 means that in the Nyquist plot (-Z,m versus Z ), the high-frequency branch of the spectrum is a straight line with the slope of 45 . This is the general property of all CCL impedance curves (Eikerling and Kornyshev, 1999). 

In the coordinates Zim versus this linear branch has a slope proportional to e. In dimensional form. Equation 5.98 is 

if one of the parameters op or Q/ is known, the slope of the high-frequency line Zim versus yields the other parameter. Physically, in this limit, the electric field changes so fast that the inertial electrochemical processes do not respond to the field changes, and the impedance is determined solely by the double layer charging. Since this charging involves protons. Equation 5.99 contains a.  

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This compensation method now exhibits a -180 degree phase lag at low frequencies, then beginning at one-tenth the error amplifier s Alter pole (/ep) the phase lag increases to its high frequency limit of -270 degrees. 

Hochfrequenzgrenze, /. high-frequency limit, hochfrequenzmassig, a. relating to high frequency. 

It has been suggested [115, 116] to solve the inverse problem using the simplified relation which is actually a high-frequency limit of Eq. (2.73). This relation can be found, if we take into account that... 

On the magnitude plot, the low frequency asymptote is a line with slope -1. The high frequency asymptote is a horizontal line at Kc. The phase angle plot starts at -90° at very low frequencies and approaches 0° in the high frequency limit. On the polar plot, the Gc(jco) locus is a vertical line that approaches from negative infinity at co = 0. At infinity frequency, it is at the Kc point on the real axis. 

By choosing xD < (i.e., comer frequencies l/xD > 1/Xj), the magnitude plot has a notch shape. How sharp it is will depend on the relative values of the comer frequencies. The low frequency asymptote below 1/Xj has a slope of-1. The high frequency asymptote above l/xD has a slope of +1. The phase angle plot starts at -90°, rises to 0° after the frequency l/xIs and finally reaches 90° at the high frequency limit. 

Limitations of the experiment at low frequencies come from the long experimental times, during which the sample structure may change so much that the entire experiment becomes meaningless. At high frequencies, limitations... 

In order to obtain a general model of the creep and recovery functions we need to use a Kelvin model or a Kelvin kernel and retardation spectrum L. However, there are some additional subtleties that need to be accounted for. One of the features of a Maxwell model is that it possesses a high frequency limit to the shear modulus. This means there is an instantaneous response at all strains. The response of a simple Kelvin model is shown in Equation 4.80 ... 

This result is very interesting because whilst we have shown that G(0) has been excluded from the relaxation spectrum H at all finite times (Section 4.4.5), it is intrinsically related to the retardation spectrum L through Jc. Thus the retardation spectrum is a convenient description of the temporal processes of a viscoelastic solid. Conversely it has little to say about the viscous processes in a viscoelastic liquid. In the high frequency limit where co->oo the relationship becomes... 

The fluctuation in elasticity is given by the difference between the elasticity in the high frequency limit and the elasticity of the new configuration ... 

We can consider the friction coefficient to be independent of the molecular weight. At times less than this or at a frequency greater than its reciprocal we expect the elasticity to have a frequency dependence similar to that of a Rouse chain in the high frequency limit. So for example for the storage modulus we get... 

The high frequency limit of this broad band is at 326 cm-1, and there are shoulders at 301 and 271 cm-1. 

The key results from these equations are summarized in Table 11. Naturally, in the 0) 0 limit, results of Table 1 are recovered. Only the high frequency limit is mentioned now. 

In the high frequency limit, the intrinsic viscosity follows from Eq. (306) as... 

With this extension, the complex impedance response of the CCL could be calculated. The model of impedance amplifies diagnostic capabilities— for example, providing the proton conductance of the CCL from the linear branch of impedance spectra (in Cole-Cole representation) in the high-frequency limit. 

Often a non-blocking interface will behave like a resistance (/ ct) and capacitance (Q,) in parallel. This leads to a semicircle in the impedance plane which has a high frequency limit at the origin and a low frequency limit at Z = (Fig. 10.4). At the maximum of the semicircle if the angular frequency is then ctQin>max = fro which dl can be evaluated. 

The 1/to4 high frequency limit for R can be useful in determining optical constants from Kramers-Kronig analysis of reflectance data (see Section 2.7). Reflectances at frequencies higher than the greatest far-ultraviolet frequency for which measurements are made can be calculated from (9.15) and used to complete the Kramers-Kronig integral to infinite frequency. 

These equations are identical with the high-frequency limit (9.13) of the Lorentz model this indicates that at high frequencies all nonconductors behave like metals. The interband transitions that give rise to structure in optical properties at lower frequencies become mere perturbations on the free-electron type of behavior of the electrons under the action of an electromagnetic field of sufficiently high frequency. 

A high-frequency limit for the applied potential is encountered above several kilohertz where the impedance of the conductance cell again begins to deviate from the resistance R. Since the solution medium itself is a dielectric situated between two parallel charged surfaces, it can assume the characteristics of a capacitor placed in parallel across the solution resistance as shown in Figure 8.9a. The magnitude of this capacitance is given by... 

The above description is obviously an extension of the low frequency description, but it can also be connected to the high frequency limit. Consider the case in which a) 1 IT, i.e., there are many cycles of the field, and ES — ER rf, the static tuning field is near resonance compared to the rf field amplitude. For a large transition probability the transition amplitudes of successive rf cycles must add constructively. This only happens when the phase difference over an rf cycle satisfies = 2jzN with N an integer. Using the energies of Eq. (15.35),... 

We now consider the high-frequency limit e/Lw — 0 of the scaled field v, where Lw is a characteristic pore size. As argued by [6], the fluid motion... 

Eq. (4) gives k (co) for a relaxation process, with a relaxation time and with zero-frequency and high-frequency limits for k (w) of Kg and k, respectively. Two complicating features render the Debye equation, Eq. (4), approximate, and require emendation (a) a given relaxation process may be associated with a... 

Here, Eq is the permittivity of free space. For a simple Debye-type relaxation process, Eq. (4), and owing to the incorrect representation of the high-frequency limit inherent in any expression for K (to) consistent with an exponential decay function for the electric moment, one obtains for the high-frequency limit of a(to) from Eqs. (4) and (6) (30) ... 

The high frequency limit of for this second process is therefore n. The result of the fit is shown in Table III where the mean values of the various parameters and their associated 95% confidence intervals are given. Considering the small amplitude of the second dispersion both in absolute t rms and in relation to the main dispersion the parameters 6m, n and Y are quite well defined, and therefore it may be concluded that the double Debye representation is an acceptable description of the dielectric behaviour of water up to around 2THz. Other alternative interpretations are clearly possible but no attempt has been made here to follow these up at this stage. What is clear is that a small subsidiary dispersion region in the far infrared is necessary to account for all the presently available permittivity data, and that such a dispersion is centred around 650GHz and has an amplitude of about 2.4 in comparison with that of the principal dispersion which is approximately 75. 

The polar lattice modes split into TO- (cuto,%j) and LO-modes (u>lo,ij), with broadening parameters 7to,ij and 7lo,ij, respectively [73]. The parameters oo,i denote the high-frequency limits in this model approach, which are related to the static dielectric constants to, by the Lydanne-Sachs-Teller relation [110] (Sect. 3.3)... 

Figure 3.6. Complex plane graph of the total impedance at low frequencies 3.3.2.2 High-Frequency Limit...

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