Line Lorentz

L22 (a) Sketch the first derivative of a Lorentz line shape, (b) Some ESR spectrometers record the second derivative (rather than the first derivative) of resonance lines. Sketch the second derivative of a Lorentzian line, indicating the position of Pq. 

Now we shall estimate the frequency xm, at which the absorption A(x) attains its maximum value. In accord with Eq. (53) at small (3, the absorption A(x) is mostly determined by a transverse component of the spectral function, to which the sum of terms with denominators (2/)2(ttw — X)2 — z1 in the integrand of Eq. (51) corresponds. These terms actually present the set of the resonance Lorentz lines, whose center frequencies are given by... 

The effect of the anharmonicity on At follows from estimation (54) of the frequency xm. The right-hand part of this formula is actually the result of gathering of the Lorentz lines in (51), each being determined by a relevant pair h, 1 of arbitrary constants. Evidently, the steeper the dependence of the period on these variables—that is, the greater the (S >-1/0/i) and (0absorption curve, the wider the absorption band. Thus,... 

Taking account of Eq. (343), we express the total complex permittivity in the form of a Lorentz line... 

This is exactly expression for x given in GT, p. 191 or in VIG, p. 195. If we insert here the Lorentz line (355a), we shall have... 

The line forms, described by Eqs. (371) and (373), are illustrated in Figs. 45 and 46. In the first one (Fig. 45) we compare the loss (a) and absorption (b) for the isothermal, Gross, and Lorentz lines see solid, dashed, and dashed-and-dotted curves, respectively. These curves are calculated in a vicinity of the resonance point x = 1. In Figure 45c we show the frequency dependences of a real part of the susceptibility the three curves are extended also to a low-frequency region. The collisions frequency y and the correlation factor g are fixed in Fig. 45 (y = 0.4, g = 2.5). 

In Figs. 45 and 46 we see the frequency the dependences typical for resonance-absorption. The isothermal line (at Ft) is characterized by the loss peak, whose intensity is evidently larger and bandwidth is narrower than those pertinent to two other lines. The Lorentz line (at FB) is characterized by (i) a smallest maximum loss and (ii) an absence of the loss shoulder at lower frequencies. This shoulder appears (at Ft and Fq) due to the denominator in Eqs. (370) and (372), pertinent to the self-consistent collision models. Since the plateau-like loss curve is indeed a feature characteristic for water in the... 

This line has an important advantages over the Lorentz line, since in Eq. (385) (i) the static susceptibility does not depend unlike that in Eq. (355a) on the collision frequency y and (ii) the loss curve is asymmetric. However, in contrast to the formula (382) the integral absorption corresponding to (385) diverges ... 

Boltzmann induced Lorentz line (355a) Van Vleck-Weisskopf... 

The theory of the saturable absorption effect in single-wall CNTs has been developed. The dependence of the CNT surface conductivity on the driving field intensity has been calculated. It has been shown that simple approximation (1) is not valid in the SMT T case. The origin of such behavior is that Eq. (1) is obtained for the case of the two-level system in which the resonant line has the Lorentz form while in the CNT case the resonant line is the superposition of the different Lorentz lines. 

S(co) is called a complex Lorentz line. Its real and imaginary parts, A(co) and D (o), denote the absorption signal and the dispersion signal, respectively (Fig. 2.2.6) ... 

Fig. 2.2.6 The complex Lorentz line, (a) Absorptive real part A(co). (b) Dispersive imaginary part D co).

Narrow band models parameterize the transmission for wavenumber intervals Sv of typically 5 to 20 cm-1. The narrow band model of Elsasser (1942) represents the spectrum by a series of regularly spaced Lorentz lines of the same size and intensity. This model is best applied to linear triatomic molecules such as CO2 and N2O. The model by Goody (1964) is based on the idea that the lines are randomly spaced over a particular wavelength interval, with some exponential distribution of line strength. This model can readily be applied to water vapor and to carbon dioxide. If an exponential distribution of the line intensities... 

The derived system of equations of motion describes simultaneously a periodic variation of the H-bond length and rotation of this bond. We obtain dielectric response both to small translational oscillations of charges and to rigid-dipole reorientations. Each response (for charges and for dipoles) is characterized by two Lorentz lines. 

An important simplification of the VIB state is proposed, which allows consideration of the translational vibration of charges and reorientation of HB rigid dipoles as uncoupled processes. Then the VIB state is determined by two Lorentz lines. 

The b and c vibrations, generated by the harmonic potentials, reveal themselves as the Lorentz lines, respectively, in T- and V-bands. These are... 

In Section VI we study in detail two fast short-lived vibration mechanisms b and c, which concern item 2. The dielectric response to the elastic rotational vibrations of hydrogen-bonded (HB) polar molecules and to translational vibrations of charges, formed on these molecules, is revealed in terms of two interrelated Lorentz lines. A proper force constant corresponds to each line. The effect of these constants on the spectra of the complex susceptibility is considered. The dielectric response of the H-bonded molecules to elastic vibrations is shown to arise in the far IR region. Namely, the translational band (T-band) at the frequency v about 200 cm-1 is caused by vibration of charges, while the neighboring V-band at v about 150 cm-1 arises due to elastic rigid-dipole reorientations. In the case of water these bands overlap, and in the case of ice they are resolved due to longer vibration lifetime. 

In our calculation scheme the quantities x and L(z) are proportional. We shall show that the frequency dependence (83) represents the sum of two Lorentz lines. 

Thus, the spectral function presents a Lorentz line and its intensity is proportional to the average squared relative elongation s defined by (103). 

Thus, b2 +D and b2 — D are both positive. Hence, each partial spectral function can be represented as the sum of two Lorentz lines with the centers at the frequencies... 

Formulas (141), equivalent to the latter and representing the spectral function as a sum of Lorentz lines, gives the same result, as it must. 

Figure 40 (a-d) Frequency dependences of the dimensionless absorption A(x). Solid lines denote absorption due elastic translations of charges and dashed lines due to elastic reorientations. Calculation according to strict theory (a) c = 0.4 (b) c = 0.2 for curves 1 and 2 (c) c = 0.15 for curves 5 and 6 and c — 0.1 for curves 7 and 8 (d) c = 0.05. Approximate calculation (b) for c = 0.2 (curves 3 and 4). Vertical lines refer to the Lorentz line centers estimated as xq = /, xfl = p. (e) Amplitude of angular vibration versus rotary force constant, horizontal line depicts the quantity (158). 

Finally, as seen in Fig. 40b, the smaller maximum of curve 2 is located at the same frequency as the main maximum of curve 1. Analogously, the main maximum of curve 2 coincides with the smaller maximum of curve 1. This property arises due to a pairwise equality of the resonance denominators in the formulas (141a) and (141b). For this reason the dielectric-loss frequency dependence is described by only two Lorentz lines. The difference of the strict theory from the approximate is revealed in that the intensity of each line is determined by vibrations of both types (longitudinal and rotational). As for the simplified representation (148), the intensity of each line is determined by only one vibration type. 

---