Eigenfrequency

It is considered that the transition from d to d completely removes the fast motion with eigenfrequencies of H0 peculiar to the former, leaving behind only slow decay with relaxation time equal to or greater than V Since the other cofactor in the integrand of Eq. (4.26), M(t), decays much more rapidly (with time tc [Pg.139]

Structure 2.1 Hz eigenfrequency, 14,200 tons moving mass Standard steel beams, serially produced joints ... 

Oscillations of black holes. Non-radial oscillations of black holes can be excited when a mass is captured by the black hole. The so called quasinormal modes have eigenfrequencies and damping times which are characteristic of black holes, and very different of eigenfrequencies and damping times of quasi normal modes of stars having the same mass. Also the eigenmodes being different for a star and a black hole, the associated gw will also exhibit characteristic features. 

The eigenfrequency spectrum of the surface modes of a hollow sphere with gas inside is well known (e.g., see Ref. [109] as well as our Appendix A). If we pretend for a moment that the surface tension coefficient a is curvature independent, the possible values of the eigenfrequency oo are found by solving the following equation ... 

Thus, although Langevin and DPD damping do not alter the eigenfrequencies... 

The exact approach to the problem of dynamic (linear) stability is based on the solution of the equations for small perturbations, and finding eigenvalues and eigenfunctions of these equations. In a conservative system a variational principle may be derived, which determines the exact value of eigenfrequency... 

The width of the gain profile in a CO2 laser is given as 66 MHz (close to the Doppler width of the emission band of the gas). If the eigenfrequency of the laser resonator is tuned to the center of the laser gain profile, what is the maximum length of resonator for which the laser can oscillate in a single mode ... 

The wavelength of a laser line, however, is determined by two factors the fluorescence profile of the corresponding transition in the laser medium and the eigenfrequencies of the laser resonator modes. At normal multimode operation of a laser, where many axial and transverse modes participate in laser oscillation, these eigenfrequencies cover the whole spontaneous line profile nearly uniformly. 

The frequency of a single-mode laser inside the spectral gain profile of its active medium is mainly determined by the eigenfrequency of the active laser cavity mode. Therefore any instability of resonator parameters, such as variation of cavity length, mirror vibrations or thermal drifts of the refractive index will show up as frequency fluctuations and drifts of the laser line. 

The corresponding vibrational eigenmodes and eigenfrequencies at a given point q on the constraint surface are given by the solutions of the eigenvalue equation... 

The nonresonant contributions pertain to electron cloud oscillations that oscillate at the anti-Stokes frequency but do not couple to the nuclear eigenfrequencies. These oscillatory motions follow the driving fields without retardation at all frequencies. The material response can, therefore, be described by a susceptibility that is purely real and does not depend on the frequencies of the driving fields. The resonant contributions, on the other hand, are induced by electron cloud oscillations that are enhanced by the presence of Raman active nuclear modes. The presence of nuclear oscillatory motion introduces retardation effects relative to the driving fields i.e., there is phase shift between the driving fields and the material oscillatory response. 

A one-level system e) that can exchange its population with the bath states [/) represents the case of autoionization or photoionization. However, the above Hamiltonian describes also a qubit, which can undergo transitions between the excited and ground states e) and g), respectively, due to its off-diagonal coupling to the bath. The bath may consist of quantum oscillators (modes) or two-level systems (spins) with different eigenfrequencies. Typical examples are spontaneous emission into photon or phonon continua. In the RWA, which is alleviated in Section 4.4, the present formalism applies to a relaxing qubit, under the substitutions... 

The preceding analysis is just a transformation of one representation of the n-state problem to another representation. To be useful, the new representation must admit simplifying approximations not suggested by the original representation. One such approximation is to replace the frequency variable 03 in M( ) in (7.10) by a typical P space eigenfrequency, say We thereby obtain the frequency-independent effective operator... 

The values of C (r) 2 and IC2WI2 obtained from (7.19) and (7.20) are compared in Figs. 9 and 10. The amplitudes and periods of the temporal evolution predicted by the two approaches to the system dynamics are seen to agree quite well. The differences seen in the amplitudes shown in Fig. 9 are a consequence of the replacement of the exact eigenfrequencies of the Rabi frequency matrix with a typical eigenfrequency from the P subspace. 

Here, v denotes an eigenfunction, co means the eigenfrequency, / is the degree of the spherical harmonic function which describes the pattern of the mode on the stellar surface, and E,(r) is the gravity wave potential which consists of the inverse square of the Brunt-V isfiia frequency, N(r), multiplied by /(7+1) and an /-independent part ... 

DFT calculations were performed for the double proton transfer in bicyclic 2,2 -bis(4,5,6,7-tetrahydro-l,3-diazepine) (Figure 8) <2001CPL591>. Both a concerted and a stepwise mechanism for proton transfer are considered. Though the concerted transition state has two imaginary eigenfrequencies, dynamical calculations have demonstrated that it has to be taken into account in the mechanism of the proton transfer even if it is not a true reaction path. 

Because it contains sin2 ka, the function co = co(k) is periodic. All possible eigenfrequencies are obtained when in Eq. (II.4) sin2ka is varied only in the range 0 < ka < it 12. The magnitude of the wave vector is thus... 

At the edge of BZ 1, according to Eq. (II.10), the eigenfrequency co = /2f/mA for k = n/2a. Insertion of these values in Eqs. (II.3) and (II.2) gives (2 fmB/mA - 2f).UB = 0. For mA i mB it follows that UB = 0 and UA is arbitrary. This means that atoms with the smaller mass mB remain in the equilibrium position and only the heavier A atoms vibrate. For k = ir/2a it follows from k = 2n/ that the wavelength X = 4a. When atoms A and B are charged, no change of electric dipole moment is associated with this vibration, therefore it is also called acoustical. 

The regularity of the array of the atoms in ideal crystals permits the subdivision of the crystal space into equal and equally oriented regions of space, the so-called elementary cells. Each cell contains the same complex of atoms with the same orientation in space. Such an elementary cell may contain s atoms. Each atom has three degrees of freedom, one in each of the directions of the three coordinate axes. Therefore a space lattice has 3 s eigenfrequencies or modes. Of these 3 s modes for k = 0 i.e. in the center of BZ 1) three correspond to the translations into the directions of the coordinate axes. These have the frequency to = 0, which corresponds to the frequencies of the acoustical branches according to Eq. (II.7) for k = 0. The 3 s eigenfrequencies of a crystal with s atoms in the elementary cell correspond to 3 s - 3 optical and 3 acoustical branches. 

This is the background for the Lyddane-Sachs-Teller relation to be treated below. For transverse optical vibrations the origin of an is field is less obvious, but it is also present and its reaction on the eigenfrequency of the TO phonon later gives rise to the polaritons. 

This happens in Acoustic plate-mode (Apm) oscillators, which have a thickness of only a few microns. The eigenfrequency of the Apm oscillators is given by the interdigitated electrode spacing p and by the plate thickness t. 

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