Lorentzian transition probability

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Normally one might expect that if the transition probability vanishes on resonance it also vanishes off resonance. However, such is not the case. When the transition probability is calculated off resonance, by numerically solving Eqs. (14.16) using a Taylor expansion method, it is nonzero for both v E and v 1E.14,16 In Fig. 14.6 we show the transition probabilities obtained using two different approximations for v E, and vlE for the 17s (0,0) collisional resonance.16 To allow direct comparison to the analytic form of Eq. (14.21) we show the transition probabilities calculated with EAA = VBB = 0. For these calculations the parameters ju2l = pLz, = 156.4 ea0, b = 104ao, and v = 1.6 x 10-4 au have been used. The resulting transition probability curves are shown by the broken lines of Fig. 14.6. As shown by Fig. 14.6 these curves are symmetric about the resonance position. The vlE curve of Fig. 14.6(b) has an approximately Lorentzian form, but the v E curve of Fig. 14.6(a), while it vanishes on resonance as predicted by Eq. (14.24), has an unusual double peaked structure. 

The observed cross sections for the 18s (0,0) collisional resonance with v E and v 1 E are shown in Fig. 14.12. The approximately Lorentzian shape for v 1 E and the double peaked shape for v E are quite evident. Given the existence of two experimental effects, field inhomogeneties and collision velocities not parallel to the field, both of which obscure the predicted zero in the v E cross section, the observation of a clear dip in the center of the observed v E cross section supports the theoretical description of intracollisional interference given earlier. It is also interesting to note that the observed v E cross section of Fig. 14.12(a) is clearly asymmetric, in agreement with the transition probability calculated with the permanent electric dipole moments taken into account, as shown by Fig. 14.6. 

Fig. 15.11 The K 29s + 27d resonance in the presence of a low frequency rf field. In zero rf field (a), the FWHM is 1.6 MHz. In (b)-(d), a 1.0 MHz field of strength 0.05 V/cm, O.lV/cm, and 0.2 V/cm respectively is present. The solid line in (b) is a numerical integration of the transition probability, and the bold line is the convolution of a Lorentzian lineshape with a sinusoidal shift from resonance. In (e), the rf frequency is 0.5 MHz and its strength is 0.2 V/cm. For these low frequencies, the features are no long frequency dependent but rather are field strength dependent (from ref. 18).

Numerical studies allow us to explore aspects of these models for a number df molecular continua and pulse configurations. Consider first the effect of the puUe intensity on transition probabilities to a slowly varying continuum by considering continuum composed of single broad Lorentzian [Eq. (10.21)] of wid r, = 2000 cm-1, excited by a 120 cm-1 wide pulse (i.e., a pulse of 80fs durp tion). The central frequency of the pulse is tuned to the center of the continuutfi) (A, = 0) and the pulse peaks at t = 0. 

The laser-induced X-ray count rate data, normalized to laser power and beam current, were fitted with a Lorentzian using a least-squares technique to obtain the resonance centroid in the laboratory frame. The average resonance width, corrected to the rest frame of the ions, was 8.5 0.4 cm-1, compared to the natural width of 8.0 cm-1. This is consistent with some saturation in the transition probability and also in the detection sensitivity of the proportional counter at increased count rate. The wavenumber of the resonance centroid, in the rest frame of the moving ion, is obtained using the relativistic Doppler formula,... 

Still assuming that a Lorentzian distribution of vibrational energies and the dipole approximation are employed. In this expression is the IR transition momenL Mu is the Raman transition probability, is the resonant mode frequency and is the natural line width of the transition. Since sum-frequency active modes must be both IR- and Raman-active, any vibrational mode that has an inversion centre cannot be sum-frequency-active. This result coupled with the coherent nature of sum-frequency generation precludes any sum-frequency response from bulk isotropic media. 

The main difference between (2S-D1ABAT1C) and (IS-GP) results is the appearance of broad Fano profiles on the (2S-D1ABAT1C) transition probabilities, which suggests that the upper adiabatic PES can support resonances which do not exist in the single ground adiabatic surface calculation. This can be investigated further with the lifetime matrix formalism described in Sect. 3.4. Smith lifetime matrices for the (2S-DIABATIC) case differ from the (IS-GP) ones only by the appearance of Lorentzian-shape eigenvalues near and above 4 eV. 

Equation (53) shows that the dependence of the transition probability on the reaction free energy or on A/ is of the form of the sum of Lorentzian maxima weighted by the factors It should be noted... 

Note The real collision-induced line profile depends on the interaction potential between A and B. In most cases it is no longer Lorentzian, but has an asymmetric profile because the transition probability depends on the intemu-clear distance and because the energy difference A (/ ) = Ei R) — Ek(R) is generally not a uniformly rising or falling function but may have extrema. 

The transition probability shows a series of Lorentzian resonance peaks. Those appear when the energy of the incoming electron in the x direction, E — is close to the real part of the resonance energy, e. Due to the symmetry of the problem, tunneling is allowed only through even resonance states. 

Fermi level. In a coarse approximation the p-level distribution can be assumed to correspond to that of a Fermi gas of free electrons, or even simpler, to that of a rectangular shaped distribution of equally spaced states (Richtmyer et al. 1934). Assuming equal transition probabilities and a Lorentzian lifetime broadening, the K absorption coefficient follows an arctan curve as a function of energy (cf. fig. 7, Ho) ... 

The line profile of the transition probability is Lorentzian (Sect.3.1) with a halfwidth, depending on 7 = 7a 7b and on the strength of the interaction. 

Equation 3.75 in conjunction with Equation 3.77 provides an explicit expression for the nonradiative transition probability, where the width y is assumed to be independent of the particular vibronic levels lwn). This expression is general, being applicable for both the statistical limit and for the small molecule cases. The Lorentzian function in (3.77) exhibits a sharp peak around Q ip cOy, = —Ths height of this peak is 2/y, while its width is given by y. The sharp peak of the Lorentzian function around Q = C0y —njiCo j) very strongly favors the transitions toward those final resonance levels riy of the I electronic state, the so-called close coupled levels, for which the quantity Q =p co, + deviate from zero by an... 

To consider gas molecules as isolated from interactions with their neighbors is often a useless approximation. When the gas has finite pressure, the molecules do in fact collide. When natural and collision broadening effects are combined, the line shape that results is also a lorentzian, but with a modified half-width at half maximum (HWHM). Twice the reciprocal of the mean time between collisions must be added to the sum of the natural lifetime reciprocals to obtain the new half-width. We may summarize by writing the probability per unit frequency of a transition at a frequency v for the combined natural and collision broadening of spectral lines for a gas under pressure ... 

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