Resonant equations

High-field EPR (HFEPR) spectroscopy greatly improves the resolution of the EPR signals for spectral features such as the g-tensor. Deviations of the g-value from free electron g=2.0023 are due to spin-orbital interactions, which are one of the most important structural characteristics (Kevan and Bowman 1990). Using a higher frequency results in enhanced spectral resolution in accordance with the resonance equation ... 

The resonance equation for a thin film, which is subject to an uniaxial stress, and which exhibits an intrinsic uniaxial perpendicular anisotropy field, may be expressed as (Bushnell and Vittoria 1993)... 

Planar resonators - Equation (5.11) is not only valid for dielectric resonators. Any other type of electromagnetic resonator employing dielectric parts, like metal ceramic coaxial-type resonators (e.g. used as filters in mobile phones) and microstrip or coplanar resonators (used in microwave integrated circuits) have a Q-contribution due to dielectric losses. For the latter type of resonator the dielectric losses are negligible in comparison to metallic losses, unless high temperature superconducting metallization layers are applied. 

This shows that the exponential model (see Section III) can be used to match nonresonant [equation (95)] and resonant [equation (85)] probabilities within the two-state close-coupling approximation. 

The obtained spectra can be explained in the limits of the phenomenological resonance equations for highly anisotropic ferromagnets [14]. Such features are characteristic for imiaxial anisotropy, which rises from the deviation of nanoparticles shape from spherical. The resonance frequency (o of spherical nanoparticles according to the theory of ferromagnetic resonance is determined by the following equation ... 

The high mass sensitivity calculated in the previous example justifies the term microbalance in describing the sensing capabilities of the quartz resonator. Equation 3.9 can be used to calculate fr uency shifts for surface accumulations that behave as ideal mass layers. A real film behaves as an ideal mass layer if it is sufficiently thin and rigid so that it moves synchronously with the oscillating device surface. On a TSM resonator, this condition is realized if the acoustic phase shift across the film is small, i.e., ir. The phase shift is... 

Let us first consider the case where E,n > is dominated by a single resonance. Equation (4.9) now reduces to... 

It should be noted that the frequency shift A/ can be a complex number, and its imaginary part, AT, reflects the half-width of the resonance. Equation 3 shows that the complex frequency shift A/ contains the same information as the mechanical impedance Zl. 

A special case of obvious importance occurs when co = co (i.e., when the ion is in resonance). Equation (10) cannot be solved directly when coi = o)c since it has the value (0/0) under such conditions. However,... 

The calculation of the widths (rates) of tunneling of these vibrational levels has been accomplished recently via semiclassical 11 17 18] as well as quantum 11 19 20] theory. The calculations for He employed extremely accurate PES [1 2 3], since tunneling widths turn out to be very sensitive to the shape of the potential HI. This can be seen in the difference between our results on Be 117] and those of Bauschlicher and Rosi 118]. An important theoretical result from the work of refs 2, 19 and 20 is the fact that the quantum approach of Babb and Du 119] yields reliable results for widths of the order of 10 a.u., something which proved impossible in the case of numerical solution of the complex eigenvalue resonance equation satisfied by the decaying vibrational levels 11 20]. 

Expression (18) has not a direct computational interest. Its usefulness is to provide a general framework to investigate cross sections. It is especially relevant near energy thresholds (see Section 3.2 Giant resonances). Equations (15) and (16) determine the full transition operator (6). The physical quantity of interest is not the transition operator but its matrix elements between the scattering states of the same energy. These matrix elements define the on-shell T matrix, denoted T whose exact expression is... 

Normally, the /xzra-amino substituents additionally weaken the acidity of the carboxyl group " due to resonance (equation 4)... 

The quantity ge(l - o) is denoted g and called simply the g-factor, so that the final resonance equation becomes... 

The sign of the term in brackets is to be retained after the square root is taken [3.6], although it is positive for all stable resonators. Equation (1) presupposes negligible diffraction loss, so it does not apply, for example, to plane-parallel resonators of Fresnel number less than about 20 (The Fresnel number is the mirror radius squared divided by d times the wavelength). 

---