Lorentzian dependence

Broadening effects, other than from g- or hyperfine-anisotropies, generally lead to symmetrical absorption curves that are Gaussian or Lorentzian depending upon the broadening mechanism. 

Figure 4. Dispersive-Lorentzian dependence of NMOR on the longitudinal magnetic-field.

Deep centres are often present in photoconductors and they can trap the photo-generated carriers. The statistical trapping (recombination or capture) and subsequent release (generation or emission) of these carriers leads to an extra source of noise called generation-recombination (g-r) noise. The presence of this noise depends on the purity of the material used as a photoconductor, but in some cases, it is inherent to the deliberate technological process as recombination centres can be added to reduce the time constant of the detector for specific applications. The time constant r of a single trap is related to its capture and emission time constants rc and re by r 1 = r 1 + t"1, and when the g-r noise arises from a trap with a definite value of r, the observed noise spectrum has a Lorentzian dependence on the modulation frequency /, peaking at /o = l/2nr. 

The role of vibrational relaxation and solvation dynamics can be probed most effectively by fluorescence experiments, which are both time- and frequency-resolved,66-68 as indicated at the end of Sec. V. We have recently developed a theory for fluorescence of polar molecules in polar solvents.68 The solvaion dynamics is related to the solvent dielectric function e(co) by introducing a solvation coordinate. When (ai) has a Lorentzian dependence on frequency (the Debye model), the broadening is described by the stochastic model [Eqs. (113)], where the parameters A and A may be related to molecular... 

Thus J(co) has a Lorentzian dependence upon the value of Tr, just as an exponentially decaying free induction decay gives rise to a Lorentzian line-shape, i.e. /(x) = 1/(1 -h x ). 

Calculations of (pickup) and 5), (resonator) were made for a spin packet for the same values of 7), T2, Bi, /w and Mq as those for set (a) listed above, off-resonance by Acd=20 MHz. The resulting calculated signals are included in Figures 5 and 6, respectively. The required initial value of magnetization. Mo, off resonanee is expected to be smaller than that at resonanee, as it depends on deviation Aco and the EPR lineshape (e.g. Gaussian, Lorentzian) depending on the properties of the sample. But in order to demonstrate the dependence of the signals on T, T2 and facilitate easy comparison with the situation when the spin packet is at resonance. 

The steady-state rate of population of state 2 thus has a Lorentzian dependence on the energy gap 12. As we discussed in Qiap. 2, the Lorentzian function can be equated to the homogeneous distribution of 12 when the mean value of 12 is zero and state 2 has a lifetime of 2/2. Note that, according to Eq. (10.29b), T2II — when pure dephasing is negUgible. If we identily the time ccaistant T2 in Eq. (10.35) with 2T in Eq. (2.71), and identify the energy difference 12 with ( — ), then the factor in the second set of parentheses in Eq. (10.35) must be lulh times the distribution function Re[G( )] in Eq. (2.71). 

Fig. 3.6.1 Collision, Voigt, and Doppler line shapes. All three are shown with the same maximum amplitude, and with the same width at half maximum amplitude. The Voigt line shape is one of a continuum of profiles between Gaussian and Lorentzian, depending on the degree of collision broadening.

Specification of. S SkCG, CO) requires models for the diffusive motions. Neutron scattering experiments on lipid bilayers and other disordered, condensed phase systems are often interpreted in terms of diffusive motions that give rise to an elastic line with a Q-dependent amplitude and a series of Lorentzian quasielastic lines with Q-dependent amplitudes and widths, i.e.. 

The widths of the broad Lorentzians representing fast motions in the plane and perpendicular to the plane of the bilayer are compared in Ligures 12a and 12b, respectively. Only data at the lower hydration (23%) are available for comparison, and these agree well with the MD results, which show a slow, monotonic increase with Q. Although we expect the fast process to be at most only weakly dependent on hydration, it is not clear to what extent the comparison validates the simulation. 

Fig. 2.7 Dependence of the experimental line width Cexp on the effective absorber thickness t for Lorentzian lines and inhomogenously broadened lines with quasi-Gaussian shape (from [9])...

Although Lorentzian line shapes should be strictly expected only for Mossbauer spectra of thin absorbers with effective thickness t small compared to unity, Margulies and Ehrman have shown [9] that the approximation holds reasonably well for moderately thick absorbers also, albeit the line widths are increased, depending on the value of t (Fig. 2.7). The line broadening is approximately... 

In the case of resonance absorption of synchrotron radiation by an Fe nucleus in a polycrystalline sample, the frequency dependence of the electric field of the forward scattered radiation, R(oj), takes a Lorentzian lineshape. In order to gain information about the time dependence of the transmitted radiation, the expression for R(oj) has to be Fourier-transformed into R(t) [6]. 

Figure 3.17 presents ps-TR spectra of the olehnic C=C Raman band region (a) and the low wavenumber anti-Stokes and Stokes region (b) of Si-rra i-stilbene in chloroform solution obtained at selected time delays upto 100 ps. Inspection of Figure 3.17 (a) shows that the Raman bandwidths narrow and the band positions up-shift for the olehnic C=C stretch Raman band over the hrst 20-30 ps. Similarly, the ratios of the Raman intensity in the anti-Stokes and Stokes Raman bands in the low frequency region also vary noticeably in the hrst 20-30 ps. In order to better understand the time-dependent changes in the Raman band positions and anti-Stokes/Stokes intensity ratios, a least squares htting of Lorentzian band shapes to the spectral bands of interest was performed to determine the Raman band positions for the olehnic... 

There is a second relaxation process, called spin-spin (or transverse) relaxation, at a rate controlled by the spin-spin relaxation time T2. It governs the evolution of the xy magnetisation toward its equilibrium value, which is zero. In the fluid state with fast motion and extreme narrowing 7) and T2 are equal in the solid state with slow motion and full line broadening T2 becomes much shorter than 7). The so-called 180° pulse which inverts the spin population present immediately prior to the pulse is important for the accurate determination of T and the true T2 value. The spin-spin relaxation time calculated from the experimental line widths is called T2 the ideal NMR line shape is Lorentzian and its FWHH is controlled by T2. Unlike chemical shifts and spin-spin coupling constants, relaxation times are not directly related to molecular structure, but depend on molecular mobility. 

To uniquely associate the unusual behavior of the collision observables with the existence of a reactive resonance, it is necessary to theoretically characterize the quantum state that gives rise to the Lorentzian profile in the partial cross-sections. Using the method of spectral quantization (SQ), it is possible to extract a Seigert state wavefunction from time-dependent quantum wavepackets using the Fourier relation Eq. (21). The state obtained in this way for J = 0 is shown in Fig. 7 this state is localized in the collinear F — H — D arrangement with 3-quanta of excitation in the asymmetric stretch mode, and 0-quanta of excitation in the bend and symmetric stretch modes. If the state pictured in Fig. 7 is used as an initial (prepared) state in a wavepacket calculation, one observes pure... 

The explicit form of the function f (Hr — H, AH) depends on the shape of the individual derivative curves. If the absorption curve can be described by a Lorentzian function, then... 

In the practice of solid-state bioEPR, a Lorentzian line shape will be observed at relatively high temperatures and its width as a function of temperature can be used to deduce relaxation rates, while a Gaussian line will be observed at relatively low temperatures and its linewidth contains information on the distributed nature of the system. What exactly is high and low temperature, of course, depends on the system for the example of low-spin cytochrome a in Figure 4.2, a Lorentzian line will be observed at T = 80°C, and a Gaussian line will be found at T 20°C, while at T 50°C a mixture (a convolution) of the two distributions will be detected. 

Fig. A2.2. Temperature dependences measured for the haliwidths Av n of the IR absorption bands for valence vibrations of OH(D) groups on Si02 surface (filled markers) and recalculated for the halfwidths w of three components of Lorentzian lines (empty markers) for OH (1) and (OD) groups of high concentration (2), and for (OD) groups of low concentration (3).2"2...

We have investigated the influence of diquark condensation on the thermodynamics of quark matter under the conditions of /5-equilibrium and charge neutrality relevant for the discussion of compact stars. The EoS has been derived for a nonlocal chiral quark model in the mean field approximation, and the influence of different form-factors of the nonlocal, separable interaction (Gaussian, Lorentzian, NJL) has been studied. The model parameters are chosen such that the same set of hadronic vacuum observable is described. We have shown that the critical temperatures and chemical potentials for the onset of the chiral and the superconducting phase transition are the lower the smoother the momentum dependence of the interaction form-factor is. 

It has been shown for a hybrid star model which uses the quark matter EoS presented in this work that the possibility to obtain a stable star configuration with 2SC quark matter core depends on the form-factor of the quark interaction [34], The Gaussian and Lorentzian form-factor models do allow a quark matter core, whereas the NJL form-factor model does not. 

Long-range order theory, 35 4-5 Looper s walk capping, 32 438-445 Lorentzian energy averaging, 34 217 Lorentzian function, energy-dependent, 34 243 Losod, 33 215, 224, 258 Low-coordinated transition-metal ions, 34 133 Low energy... 

Table I lists the comparative parameters for the various indochinite spectra. Two methods were used in preparing these samples. The first two samples listed were prepared by grinding the indochinite specimen and binding the powder with water glass. The other samples were sliced with a diamond saw. The two spectral lines are given with their position, width, height, and area. The quadrupole splitting and isomer shift are listed in the columns labeled QS and IS. (The isomer shift is really a combination of isomer shift and temperature-dependent shift, and the values are relative to iron in palladium.) The raw data points were fitted with a two-peak Lorentzian using an IBM 7094 least-squares fit.

Fig. 3. The transmittance function and its variation with length for the tight binding model. For the two-site case, the exact result demonstrates the transmittance as a Lorentzian. However, for longer chains the transmittance (as in Fig. 2) varies weakly within the band, and drops quite sharply outside the band - this latter dependence dominates the overall transport in this region.

This is a Gauss profile with a lorentzian hole, the width of which is determined by the homogeneous linewidth parameters 7ab> T with 7ab " rb depending on the lifetimes 7a, Tb of... 

The relaxation rates in Eqs. (12) and (13) depend now on the magnetic field in a more complicated way. Not only are the Larmor frequencies in the denominators of the Lorentzians proportional to the field, the electron spin relaxation rates are, in principle, also field-dependent. 

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