Lorentzian .function

Figure 6.6 Vibrational coherence on a pNB-adsorbed TiO2(110) surface, (a) The raw SH intensity, (b) the modulated component, (c) the Fourier-transformed spectrum, the gray lines show the transformed spectrum. The spectrum simulated with Lorentzian functions is overlaid with broken lines. The pNB-adsorbed surface was irradiated in air with p-polarized pump (8mjcm ) and p-polarized probe (8mjcm ) pulses of a 550-nm wavelength.

Among the various methods, the B3-LYP based DFT procedure appears to provide a very cost-effective, satisfactory and accurate means of determining the vibrational frequencies. As an example. Figures 3.7 and 3.8 display direct comparisons between the ground state experimental and DFT B3-LYP/6-31G calculated Raman spectra for DMABN and its ring deuterated isotopmer DMABN-d4. ° The experimental spectra are normal Raman spectra recorded in solid phase with 532nm excitation. For the calculated spectra, a Lorentzian function with a fixed band width of —10 cm was used to produce the vibrational band and the computed frequencies were scaled by a factor of 0.9614. 

It is also clear from Eq. (2.5.1) that the linewidth of the observed NMR resonance, limited by 1/T2, is significantly broadened at high flow rates. The NMR line not only broadens as the flow rate increases, but its intrinsic shape also changes. Whereas for stopped-flow the line shape is ideally a pure Lorentzian, as the flow rate increases the line shape is best described by a Voigt function, defined as the convolution of Gaussian and Lorentzian functions. Quantitative NMR measurements under flow conditions must take into account these line shape modifications. 

For k states, a relaxation (or noise spectrum) will contain k, exponential (or Lorentzian) components. Thus, the mechanism in Eq. (6.25) above will have two states in the absence of blocker and so give rise to relaxations (or noise spectra) that can be fitted with single exponential (or Lorentzian) functions. Addition of the blocker creates an extra state (the blocked state), giving k = 3. For k = 3, the occupancy of the open state as a function of time will be described by two exponentials ... 

Fourier transforms boxcar function 274 Cauchy function 276 convolution 272-273 Dirac delta function 277-279 Gaussian function 275-276 Lorentzian function 276-277 shah function 277-279 triangle function 275 fraction, rational algebraic 47 foil width at half maximum (FWHM) 55, 303... 

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... 

A. Vijay, J. Chem. Phys., 118, 1007 (2003). A Lorentzian Function Based Spectral Filter for... 

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... 

The diSuse scatter arises because dislocations are defects which rotate the lattice locally in either direction. This gives rise to scatter, from near-core regions, which is not travelling in quite the same direction as the diffraction from the bulk of the crystal. This adds kinematically (i.e. in intensity not amplitude) and gives a broad, shallow peak that mnst be centred on the Bragg peak of the dislocated layer or substrate since all the local rotations are centred on the lattice itself. We can model the diffuse scatter quite well by a Gaussian or a Lorentzian function of the form ... 

Bo is the measurement frequency. Rapid exchange between the different fractions is assumed the bulk, water at the protein surface (s) and interior water molecules, buried in the protein and responsible for dispersion (i). In fact, protons from the protein surface exchanging with water lead to dispersion as well and should fall into this category Bulk and s are relevant to extreme narrowing conditions and cannot be separated unless additional data or estimations are available (for instance, an estimation of fg from some knowledge of the protein surface). As far as quadrupolar nuclei are concerned (i.e., and O), dispersion of Rj is relevant of Eqs. (62) and (63) (this evolves according to a Lorentzian function as in Fig. 9) and yield information about the number of water molecules inside the protein and about the protein dynamics (sensed by the buried water molecules). Two important points must be noted about Eqs. (62) and (63). First, the effective correlation time Tc is composed of the protein rotational correlation time and of the residence time iw at the hydration site so that... 

Fig. 14. Circles typical NMRD data (water longitudinal relaxation rates in a protein aqueous solution adapted from ref. (55)).The curve corresponds to a lorentzian function with tq deduced from the half-height of the experimental data. Note...

The parameters of this expansion, as well as the number N of Lorentzian functions, are determined (from the experimental data) by a non-linear least squares fit along with statistical tests. It can be noticed that this expansion has no physical meaning but is merely a numerical device allowing for smoothing and interpolation of the experimental data. Nevertheless, this procedure proves to be statistically more significant than the Cole-Cole equation and thus to account much better for the representation of experimental data. The two physically meaningful parameters, i.e., C(0) and (Xo), can then be easily deduced from the quantities involved in (71)... 

Indeed, remembering that the Fourier transform of a Lorentzian function is an exponential, we obtain... 

The data were subsequently deconvoluted to the sum of an arctangent and a Lorentzian function as described by Horsley (27) ... 

In fact, in the simulation of the spectrum performed recently[31], the absorption lines were dressed with lorentzian functions which width were fitted to reproduced the experimental spectrum[22], finding that they could be of the order of l-10cm , more or less in agreement with the results obtained here for the A state, but differing significantly with the widths obtained for B state here. 

Once MOD parameters are obtained a spectrum can be simulated with a suitable choice of the band-shape function f. We have used Gaussian and Lorentzian functions for this purpose. In either case, a width parameter must be chosen. This parameter is generally chosen such that the widths of the peaks in the simulated spectrum are similar to those in the observed spectrum. 

Plot the Gaussian and Lorentzian functions (3.89) and (3.86) against v — v0. Use the same axes for both plots and take the peak heights and half-widths to be equal. 

We now replace eqs. (7-25) and (7-26) by a Lorentzian function. As a result of this replacement the discrete sums on the left-hand sides of eqs. (7-24) and (7-25) become continuous, as shown on the respective right-hand sides, and from eq. (7-21)... 

Figure 31 shows the values of HWHM of the Lorentzian function versus the squared momentum transfer Q, at several temperatures. In this figure the data... 

Fig. 30. Quasielastic broadening of the neutron scattering peak observed in the paraelectric phase for the copolymer 60/40. The points represented the experimental results with their statistical error bar. The solid line is the Lorentzian function obtained after the deconvolution process described in the text...

Fig. 31. HWHM of the Lorentzian function versus the momentum transfer Q at several temperatures in the paraelectric phase...

Let us discuss first the case in which only the first term is present. In the Solomon and Bloembeigen equations for / , (i = 1, 2) there is the cos parameter at the denominator of a Lorentzian function. Up to now cos has been taken equal to that of the free electron. However, in the presence of orbital contributions, the Zeeman splitting of the Ms levels changes its value and cos equals xs / o or (g/h)pBBo- When g is anisotropic (see Fig. 1.16), the value of cos is different from that of the free electron and is orientation dependent. The principal consequence is that another parameter (at least) is needed, i.e. the 0 angle between the metal-nucleus vector and the z direction of the g tensor (see Section 1.4). A second consequence is that the cos fluctuations in solution must be taken into account when integrating over all the orientations. Appropriate equations for nuclear relaxation have been derived for both the cases in which rotation is faster [40,41] or slower [42,43] than the electronic relaxation time. In practical cases, the deviations from the Solomon profile are within 10-20% (see for example Fig. 3.14). 

Here A is a constant from the normalization condition j a(r) = 1, rc is the center value of the lifetime distribution, and W is the full-width-at-half maximum for the Lorentzian function. 

Molecular Weight Dependence of Phase Structure. Similar line shape analysis was performed for samples with molecular weight over a very wide range that had been crystallized from the melt. In some samples, an additional crystalline line appears at 34.4 ppm which can be assigned to trans-trans methylene sequences in a monoclinic crystal form. Therefore the spectrum was analyzed in terms of four Lorentzian functions with different peak positions and line widths i.e. for two crystalline and two noncrystalline lines. Reasonable curve fitting was also obtained in these cases. The results are plotted by solid circles on the data of the broad-line H NMR in Fig. 3. The mass fractions of the crystalline, amorphous phases and the crystalline-amorphous interphase are in good accord with those of the broad, narrow, and intermediate components from the broad-line NMR analysis. 

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