Width of the resonance curve

Note 3 For the Voigt-Kelvin behaviours specified in notes 1 and 2, the ratio of the width of the resonance curve ( Avr ) to the resonance frequency (vr) is equal to the loss tangent (tan S). 

Voigt-Kelvin element Voigt-Kelvin model Voigt element Voigt model volume compression vorticity tensor width of the resonance curve Young s modulus zero-shear viscosity... 

Ap = width of the resonance curve Po = density of polymer within the solid dg = thickness of surface layer... 

As illustrated in Figure 44.42, a resonance peak represents a large amount of energy. This energy is the result of both the amplitude of the peak and the broad area under the peak. This combination of high peak amplitude and broad-based energy content is typical of most resonance problems. The damping system associated with a resonance frequency is indicated by the sharpness or width of the response curve, ci) , when measured at the half-power point. i MAX is the maximum resonance and Rmax/V is the half-power point for a typical resonance-response curve. 

In a basic Mossbauer experiment, the reduction in transmission (9) (Figure 2) or the increase in scattered intensity of radiation (2) (Figure 3) is observed as a function of the relative velocity between a source and an absorber. The full width at half maximum of the resonance curve r is related to the mean life of the radiating state by the uncertainty relation r 2h/r. The depth of the curve, c, is related to /, the magnitude of the recoilless fraction of gamma rays emitted, and hence to the crystalline properties of the solid. Finally, the displacement of the curve from zero relative velocity indicates the energy difference between emitted and absorbed radiation and is proportional to the s-electron... 

Figure 5 shows that there is no way to fit the experimental data assuming that only one type of roughness is presented on the surface. We are thus forced to conclude that, in these experiments the surface has a multiscale roughness, shown schematically in Fig. 6. The structure of this rough surface is a combination of a slight and a strong roughness shown in Fig. 3a,b. When this is taken into account, it is possible to use Eqs. 33, 34,43, and 44 to calculate the shift in resonance frequency and shift in the width of the resonance, and fit the experiments to the calculated curves with properly chosen values of the parameters of strong roughness. The result of such a fit is shown in Fig. 4, curves 2 and 3. For details of the fitting procedure, the limitations associated with the use of a simplified model, and the comparison with STM data see [27].

If we take the same damped mass and spring system and, instead of driving the system with a motor, simply displace the mass and let it go, we find that it will oscillate with its natural resonance frequency and the amplitude of the oscillation will decay in time. The decay will be exponential with the amplitude proportional to exp(-t/T2p where the viscosity of the liquid will determine T. It is interesting to note that if you pluck the system, its oscillation and decay contains all the information of the system s behavior just as the frequency dependence of the amplitude in the driven simple harmonic oscillator experiment does. In the one case, the full width at half height of the resonance curve is 2/T, while in the other, the decay time constant is T. ... 

Figure 2 (A) Refractive index, n, and (B) dielectric constant, E = (n-ik), related to absorption coefficient, k, around a resonance frequency, co for a Lorentzian oscillator. Half-width of the absorption curve Is V).

Fig. 31. Calculated potential energy curve for the C-C stretching coordinate for the neutral C2H2 molecule and its negative ion. The calculated width of the resonance state is also given.

Curves 2-A in Fig. 19b were calculated for different values of the film thickness, L, and a constant value of the roughness correlation length, according to Eq. (49). Lines 1 and 5 are the same as in Fig. 19a. The width of the resonance is seen to increase with increasing film thickness. 

A plot of the variation of the half peak width of the resonances with frequency is shown for water. The technique is sensitive to smal1 change s in the absorption as shown by the acoustic loss curves for a solution of 0,005 molar 4 methyl piperidine in water. 

The sharpness of the frequency response of a resonant system is conunonly described by a factor of merit, called the quality factor, Q=v/Av. It may be obtained from a measurement of the frill width at half maxuuum Av, of the resonator frequency response curve obtained from a frequency sweep covering the resonance. The sensitivity of a system (proportional to the inverse of tlie minimum detectable number of paramagnetic centres in an EPR cavity) critically depends on the quality factor... 

The generalization to the case of a thermally averaged parent state describes an interesting modulation curve that reflects in position and width the rotational eigenvalue spectrum of the resonant intermediate [31]. This structure has been observed in studies of HI ionization in Ref. 33. A schematic cartoon depicting the excitation scheme and the form of the channel phase for the case of a thermally averaged initial state is shown in Fig. 5g. 

The curves 1 in Figs. 4.6a and b show the functions Fr and FA calculated by formulae (4.3.35) and (4.3.38) for the case of normal molecular orientations (e Oz) and plotted versus the argument AQ/( +AQ). The dimensionless argument and functions of this kind normalized with respect to the sum of the resonance and the band widths were introduced so as to depict their behavior in both limiting cases, ACl rj and Af2 77. The deviation of the solid lines from the dotted ones indicates to which degree the one-parameter approximation defined by Eq. (4.3.38) differs from the realistic dispersion law. As seen, this approximation shows excellent adequacy, but for the region AQ r/, where the asymptotic behavior of the approximation (4.3.38) and Eq. (4.3.35) are as follows ... 

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