Lorentzian dispersion function

The Fourier transform of a decaying sine function sini2r exp(-f/T2) is a dispersion mode Lorentzian centred at frequency Q. 

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... 

Song et al.172 theoretically calculated the impedance spectra based upon Eq. (34) with such distribution functions of PSD as normal, lognormal, Lorcntzian, log Lorentzian distributions. They concluded that the impedance spectra simulated based upon the transmission line model (TLM) with different PSD functions share a common point that the wider PSD leads to the more frequency dispersion in the impedance spectra. 

For a Voigt function that is almost Lorentzian, the extent of Gaussian broadening can be visualized by plotting the dispersion of the lineshape, D f) against the absorption, A(f).76,77 For a pure Lorentzian lineshape, a circle is obtained. Hence, the extent of the departure from this circular shape indicates the extent of the Gaussian broadening. 

These functions describe an absorption with a Lorentzian line shape for which n and k vary, as shown in Fig. 3.19. This is identical to the dispersion illustrated earlier in Fig. 2.4. Quite clearly this model will also apply to molecular vibrations that produce an oscillating dipole. Hence the dispersion associated with vibrations that give rise to infra-red absorption will be of the same form, see Fig. 2.4. 

If the incident beam corresponds to several radiations with different mean wavelengths, the beam s wavelength dispersion is then described as the sum of all the Lorentzian functions associated with each radiation. 

The function S(co) can be subdivided into a real and imaginary part. The real part R(Aco) corresponds to a signal with a absorptive Lorentzian lineshape which is normally displayed while the imaginary part I(Aco) corresponds to a dispersive Lorentzian lineshape. To phase a spectrum both real and imaginary parts are required. 

Finally, when studying dilute solutions of high-molecular-weight flexible coil macromolecules, an additional contribution to the spectral dispersion of the scattered light can arise from the dynamic behavior of the low frequency, long wavelength internal vibrational motions of the chains (70). Additional Lorentzian spectral components (or exponential components of the correlation function) arise through this mechanism characterized by halfwidth (time constant) ... 

Sensitivity analysis is also a tool that can help to refine potential energy functions for (bio)molecular simulations. Sensitivity analysis can help one decide whether a specific feature needs to be included in a potential function for describing a specified set of properties of a given class of molecules. For example, because point charge models are commonly used in bio(molecular) modeling, it is useful to inquire whether a dispersed charge representation would improve the description of intra- and intermolecular electrostatic interactions. One study of this type was carried out by Zhu and Wong," ° who included in the force field a squared Lorentzian function f r - f/ ) of the form... 

In this model, the symbols have the following meaning. ( )R(a3) ao(l)HD(a3) is the optical rotation produced by the vapour and depends on the number of absorption lengths ao and on the lineshape function D(o)) which takes the form of a Doppler-broadened dispersion curve for magnetic and electric field induced rotation ( )r will depend on the strength of the field and D(oa) on the direction and type of field (see table 2). The transmitted intensity 1 - Ij exp[-aoG(o))] where the lineshape function G(o)) for a single spectral component can usually be accurately described by a Doppler-broadened Lorentzian curve. Finally the terms B and C in equation (53) represent respectively the finite extinction ratio of the polarisers and a laser independent... 

As usual there is a reciprocal relationship between the time domain and the frequency domain. The more rapid is the damping (decay) of the signal (i.e., larger a and shorter lifetime t = 1/a), the wider the Lorentzian and dispersion lineshape functions become in the spectral domain (Figure 4). 

The transformation to an axial symmetry of a paramagnetic centre means that the dispersion (j,p tends to infinity and all

line shape function can be Lorentzian or Gaussian. 

For T = 0 the cosine Fourier-transform represents the normal Lorentzian profile and the sine-transform the dispersion profile. For T > 0 the functions (13.38) show an oscillatory structure (see Fig.13.16) where the central peak of the real part can still be approximately described for u)q-u)1 < (1/T)[Pg.634]

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