Complex Resonance Frequencies

We use complex resonance frequencies, where the real part, /r, is the series resonance frequency and the imaginary part, F, is half the bandwidth at half maximum of the resonance (half-band-half-width, HBH width, also termed bandwidth for short). In the following, we comment on why—and under what conditions—the imaginary part of the resonance frequency is equal to the half-band-half-width [37]. 

Consider a forced resonator obeying the force-balance equation d x t) 

The small term F has been added to the denominator in step 3. As Eq. 3 shows, the bandwidth can been absorbed into a complex resonance frequency,/r, if one chooses the real part as/r and the imaginary part as F. Since the crystal cannot be excited with a complex frequency, the denominator al- 

It is instructive to go through a similar set of equations in the time domain. Assume that the excitation of the resonance is carried out with a radiofre-quency pulse (rather than a continuous sine wave). After the excitation has been turned off, the resonator rings down according to a decaying complex exponential  

the imaginary part of cwr is one half of the decay constant, y, provided that the resonance is sharp. Sharp resonances are always found for the QCM. A quick estimate shows that the error caused by neglecting y /4 in comparison to col negligible in all cases of practical interest. The complex resonance frequency, /r, also describes the ring-down of a freely oscillating resonator. Since the decay time, r, is equal to F can be determined 

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In order to analyze the influence of the different loading mechanisms on the QCM response one has to model a dependence of the mechanical impedance Zl or the complex resonance frequency shift on the chemical and physical properties of the contacting mediiun. Various models for the mechanical contact between the oscillating quartz crystal and the outer medium are discussed below. The QCM is now so widely and extensively used that, in the framework of this chapter, it is not possible to review all the available literature. Hence we limited ourselves here to a review of the experimental data and theoretical ideas concerning the studies of structure and interaction at solid-hquid interface. Furthermore, we did not present here studies on... 

Above, we discussed the situation where the adsorbed layer is rigidly attached to the oscillating crystal surface, and there is finite slippage at the adsorbate-liquid interface. An alternative model, based on the assumption that slippage occurs at the crystal-adsorbed interface and non-sUp boundary conditions apply to the adsorbate-liquid interface, can also be considered. Eor a small slip length, bs <3C 3, this model leads to the same results for the shift of the complex resonance frequency as the model discussed above and measurements employing the QCM cannot distinguish between them. How-... 

The parameter A/ = A/ + iAF is a complex resonance frequency, r is the half-band-half-width (cf. Sect. 2 in Chap. 2 in this volume). The drag coefficient may describe interfacial drag, but also the withdrawal of energy from the crystal via radiation of sound. Equation 5 can be inverted, leading to ex-pUcit formulas for ks(co) and s([Pg.157]

In order to analyze the influence of the diflferent loading mechanisms on the QCM response, one has to model a dependence of the mechanical impedance Zout or the complex resonance frequency shift on the chemical and physical properties of the contacting medium. Various models for the mechanical contact between the oscillating quartz crystal and the outer medium are discussed below. 

W. Zheng, Computation of complex resonance frequencies of isolated composite objects, IEEE Trans. Microwave Theory Tech. 37, 953 (1989)... 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

Substituent effects hae been observed in a series of substituted malonato complexes (167). The 9Be resonance frequencies move to higher field as the basicity of the ligand increases, as can be seen in Table XI. Quadrupolar broadening is considerable in complexes of the hydrolyzed trimer, so much so that unless a species of this sort is present in high concentration relative to the others, its signal may be buried in the baseline noise. The tetrahedron is significantly distorted from Td symmetry in these compounds. 

Another type of DOUBLE ENDOR, called special TRIPLE , has been introduced by Dinse et al.90 to study proton hf interactions of free radicals in solution. In a special TRIPLE experiment two rf fields with frequencies vp + Av and vp — Av are swept simultaneously. For systems with Tln < T,i this leads to a considerable signal-to-noise improvement and to TRIPLE line intensities which are directly proportional to the number of nuclei with the same hf coupling constant. It should be remembered, however, that in transition metal complexes in the solid state the resonance frequencies are not, in general, symmetrically placed about the free proton frequency vp and that the condition Tln < Tj,i is not always fulfilled. 

Boriskina, S.V., Benson, T.M., SeweU, P., and Nosich, A.I., 2002, Effect of a layered environment on the complex natural frequencies of 2D WG-mode dielectric-ring resonators, J. Lightwave Technol. 20 1563-1572. 

Here r is the internuclear distance, C is a combination of physical constants, and T1M is the longitudinal relaxation time. The complex function/(tc) depends upon the correlation time Tc, the resonance frequency of the nucleus being observed, and the frequency of precession of the electron spins at the paramagnetic centers. The value of Tc can be estimated (Chapter 3) and, in turn, the distance r according to Eq. 12-30. 

Complex eigen-frequencies of overstable convective modes coupled with envelope g modes as a function of the ratio q (thick curves). heal frequencies of high order envelope g modes are also given (thin curves). Resonance couplings find themselves in the wavy features in the imaginary part of the frequency. 

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