Antiresonance states

Due to time-reversal symmetry, every resonance solution has an antiresonance solution, also known as antiresonance states, that occurs at k = —kr — iki. These antiresonance states are incoming states that also diverge at the asymptotes. 

On the other hand, the antiresonance state is generated for zero admittance or infinite impedance ... 

The resonance and antiresonance states are described by the following intuitive model. In a high electromechanical coupling material with k almost equal to 1, the resonance or antiresonance states appear for tan( >L/2v) = 00 or 0 [i.e. (oL 2v = (m — 1/2)7T or mjT (m integer)], respectively. The strain amplitude Xi distribution for each state (calculated using Eq. 28) is illustrated in... 

FIGURE 4.1.9 Strain generation in the resonant or antiresonant state. 

In a typical case, where ksi = 0.3, the antiresonance state varies from the previously mentioned mode and becomes closer to the resonance mode. The low-coupling material exhibits an antiresonance mode where capacitance change due to the size change is compensated completely by the current required to charge up the static capacitance (called damped capacitance). Thus, the antiresonance frequency/a will approach the resonance frequency/r. 

In contrast, the equivalent circuit for the antiresonance state of the same actuator is shown in Figure 4.1.11b, which has high impedance. 

FIGURE 4.1.11 Equivalent circuit of a piezoelectric device for (a) the resonance and (b) the antiresonance states. 

In this section we explicitly demonstrate how the antiresonant line shapes characteristic of configuration interaction between a discrete BO state and BO continuum can be obtained from the Green s function formalism. We restrict attention to the case of one discrete BO state interacting with one BO continuum. We shall assume that the ground state is connected to both the discrete BO state and the BO continuum by nonvanishing dipole matrix elements. 

Similarly, 2 and 2 yield a resonant Ag combination and an antiresonant one. Thus, it is seen that simple qualitative VB considerations not only predict the ground state of 02 to be a triplet, but in addition they yield a correct energy ordering for the remaining low lying excited states. 

The dependence of the angular distribution of the adsorbate - induced photoelectron current on the initial state wave function has been recently calculated by Gadzuk (47—49). He has considered adsorption of atoms on transition metals, assuming the interaction of the adatom with the 7-type orbitals of the substrate to be responsible for the chemisorption. To avoid the difficulties associated with Penn s antiresonance effect he assumed the adsorbate level to be non-degenerate with the metal 7-band. Neglecting the variation of the optical matrix element over the width of the adsorbate level the angular distribution is in this case given by... 

It is assumed that the manifold consists of equally spaced levels with energy separation g-1, and that the interaction V of the discrete state s with the manifold states l is constant, fin and fin are the transition moments to the discrete and manifold states, respectively, it being further assumed that fin is a constant for the manifold. As fin - O, the lineshape index q tends to infinity, that is, to a symmetric Lorentzian form. On the other hand, when the discrete state carries no intensity, that is fiis=O and therefore q=O, a depression, or symmetric antiresonance, centered at the energy of the discrete state is obtained in the absorption band. 

In the sixties Fano and Cooper (79) presented a theoretical explanation for the Beutler bands in the far-UV absorption spectra of rare gasses. Sturge, Guggenheim, and Pryce (80) showed that the Fano theory describing the band profile of a broad ionizing continuum in the vicinity of sharp intraatomic transitions could be applied to the situation of an impurity center for which a broad absorption band is overlapped by a sharp absorption line. This t5q)e of Fano antiresonance in solid state physics has been observed for two cP transition metal ions, viz., by Sturge et al. (80) and Cr by Lempicki et al. (.81). The explanation for the antiresonance observed for Cr " and is based on... 

Recently the observation of Fano antiresonance in the excitation spectra of the luminescence of Eu was reported (82). The two-photon absorption experiments by Downer et al. [37,38], for example, revealed the presence of sharp absorption lines due to transitions from the 87/2 ground state to the Pj, Ij and Dj states within the Af configuration of Eu ". These parity-forbidden transitions are overlapped by the broad 4/ 5d absorption bands of Eu. For this situation the appearance of Fano antiresonance in the vicinity of the sharp absorption lines is to be expected. 

The variable-sign result Eq. (81) produces results that fail to satisfy such time-reversal symmetry, as shown by Andrews et al. [50], The requirement for temporal symmetry remains unequivocal, despite the violation of time-reversal invariance by the system itself (its engagement of molecular interaction with the bath leading to state decay), specifically because of the inclusion of damping. The two conventions agree in ostensibly the most crucial signing, that which relates to potentially resonant denominator terms they differ in antiresonant terms. Nonetheless, in certain processes they can lead to results with experimentally very significant differences. 

Inside the band of two-particle states a channel opens up for polariton decay into two phonons. This process leads to a broadening of the polariton line (see, e.g. (59), (60)), and also a change in the polariton dispersion law. The two most important effects in the latter case are (a) interference of scattering by a polariton and two-particle states (this can lead to drops in intensity of the Fano antiresonance type see (22)), and (b) the presence of singularities in the density of two-particle states (those, in particular, that correspond to quasibiphonons). 

An instructive, albeit trivial, example is the hydrogen exchange reaction, Ha — Hb + He Ha -b Hb — He where the thermal transition state has a collinear geometry, Ha — Hb — He. In the linear structure, the ground state is the resonating combination of R and P, and it constitutes the transition state for the thermal reaction, while the twin-excited state P is the corresponding antiresonant combination that forms the photochemical funnel ... 

The photostimulation of Sn2 systems sueh as X + A — Y (A = alkyl) is a niee example for the utility of the VBSCD. This photoreaetion was analyzed before, using the VBSCD model, for predieting the loeation of eonieal interseetions [30]. Here, the transition state for the thermal reaetion is the eollinear [X-A-Y] strueture, whieh is the P (A ) resonating eombination of the two Lewis struetures, while the twin-exeited state, P (A"), is their antiresonating eombination the symmetry labels refer to Cj symmetry. This latter exeited state is readily written as an A symmetry-adapted eombination of Lewis struetures, Eq. (21) ... 

Fig. 6. Photoemission EDCs taken on twinned crystals of Y, jPr.BajCujOt,, for x=0J22 and 0.50, below (Av = 115eV) and above (Av=124eV) thePr 4d resonance tbieshold. The Pr emtribution to the densHy of states is clearly revealed in the near- p region of the diflerence curves, scaling with the doping level slight antiresonance behavior is seen in the Ba 5p cote levels.

We now tackle the problem of the calculation of W2 and P for the 3-level atom described above, for which we shall use a dressed atom approach. Immediately after the detection of a first fluorescence photon at time t, the system is in the state = g,Ng,N >, i.e. atom in the ground state in presence of Ng blue photons and Ng red photons. Neglecting antiresonant terms, we see that this state is only coupled by the laser-atom interactions to the two other states = e3,Ng-l,Ng> and l4>2> eg,Ng,Ng-l> (the atom absorbs a blue or a red photon and jumps from g> to eg> or leg>). 

Figure 4.1.9. In the resonance state, large strain amplitudes and large capacitance changes (called motional capacitance) are induced, and the current can easily flow into the device. On the other hand, at antiresonance, the strain induced in the device compensates completely, resulting in no capacitance change, and the current cannot flow easily into the sample. Thus, for a high k material the first antiresonance frequency/a should be twice as large as the first resonance frequency/r.

Thus far we have not considered the set of Slater determinants 25 and 26, which are symmetry-equivalent to 25 and 26 by inversion of the and Uy planes. The interactions between the two sets of determinants yield two pairs of resonant-antiresonant combinations that constitute the final low-lying states of dioxygen, as represented in Figure 4. Of course, our effective VB theory was chosen to disregard the bielectronic terms and, therefore, the theory, as... 

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