Imaginary frequency, definition

Imaginary frequencies are listed in the output of a frequency calculation as negative numbers. By definition, a structure which has n imaginary frequencies is an n order saddle point. Thus, ordinary transition structures are usually characterized by one imaginary frequency since they are first-order saddle points. 

This degree of freedom is the reaction coordinate (note that this definition coincides with the definition in Chapter 3). In Appendix E, we show that a multidimensional system close to a stationary point can be described as a set of uncoupled harmonic oscillators, expressed in terms of so-called normal-mode coordinates. The oscillator associated with the reaction coordinate has an imaginary frequency, which implies that the motion in the reaction coordinate is unbound. 

For each EA spectrum, the transmission T was measured with the mechanical chopper in place and the electric field off. The differential transmission AT was subsequently measured without the chopper, with the electric field on, and with the lock-in amplifier set to detect signals at twice the electric-field modulation frequency. The 2/ dependency of the EA signal is due to the quadratic nature of EA in materials with definite parity. AT was then normalized to AT/T, which was free of the spectral response function. To a good approximation [18], the EA signal is related to the imaginary part of the optical third-order susceptibility ... 

To make the phase angle plot, we simply use the definition of ZGp(joo). As for the polar (Nyquist) plot, we do a frequency parametric calculation of Gp(jco) and ZGp(joo), or we can simply plot the real part versus the imaginary part of Gptjco).1 To check that a computer program is working properly, we only need to use the high and low frequency asymptotes—the same if we had to do the sketch by hand as in the old days. In the limit of low frequencies,... 

From our definition of a pure capacitor (i.e. one having no resistive component), we can say that the real impedance Z is zero. We see straightaway from equation (8.8) that the impedance is a function of frequency. The impedance of a capacitor is infinite when a DC voltage is applied (just put ru = 0 into equation (8.8)), while the imaginary impedance Z" decreases as the frequency co is increased. 

The most obvious problem of non-linearity is the definition of a modulus. For a linear viscoelastic material we need to define not only a real and an imaginary modulus but also a spectrum of relaxation times if we are fully to describe the material - although it is more usual to quote either an isochronous modulus or a modulus at a fixed frequency. We must, for a full description of a non-linear material give the moduli (and relaxation times) as a function of strain as well this will not usually be practicable so we satisfy ourselves by quoting the modulus at a given strain. The question then arises as to whether this... 

FIGURE 5.69 Schematic presentation of the electric conductivity, A%, the real part, Ae, and the imaginary part, Ae", of the dielectric permittivity increments of dispersion as functions of the frequency of the electric field, CO. For definitions of A%, Ae, and Ae", see Equation 5.390. 

Fig. 7.12 Frequency dependence of the real solid line) and imaginary dash curve) parts of the dielectric permittivity of an isotropic liquid (a) and the definition of the phase angle < ) (b)...

Impedance is by definition a complex quantity and is only real when 0=0 and thus Z(m) = Z(a>), that is, for purely resistive behavior. In this case the impedance is completely frequency-independent. When Z is found to be a variable function of frequency, the Kronig-Kramers (Hilbert integral transform) relations (Macdonald and Brachman [1956]), which holistically connect real and imaginary parts with each other, ensure that Z" (and 9) cannot be zero over all frequencies but must vary with frequency as well. Thus it is only when Z(linear resistance, that Z(m) is purely real. 

For the definition of a distance measure for two impedance spectra Z, and Z, the complex nature of an impedance point Z (o) has to be taken into account. The distance between the two complex impedance points at a certain frequency the inner distance 8 ( .), is determined first. The complex impedance points are two-dimensional vectors containing real and imaginary part of the impedance, see the second column of Table 1 for examples of vector distances. This inner distance evaluated at all frequencies is then a real-valued function over the frequency domain on which then an outer norm is applied. This outer norm includes an integral over the inner distance function 5 (0). 

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