Correlation time, definition

Fig. 23. Experimental and calculated methyl-deuteron spectra of polycarbonate for different temperatures and different evolution times Tr For the definition of cf. Fig. 13. The width of the distribution of correlation times is 2.7 decades...

The physical meaning of and f L.., is obvious they govern the relaxation of rotational energy and angular momentum, respectively. The former is also an operator of the spectral exchange between the components of the isotropic Raman Q-branch. So, equality (7.94a) holds, as the probability conservation law. In contrast, the second one, Eq. (7.94b), is wrong, because, after substitution into the definition of the angular momentum correlation time... 

The relaxation rates calculated from Eq. (15) are smaller than the measured ones at low field, while they are larger at high field. OST is thus obviously unable to match the experimental results. However, water protons actually diffuse around ferrihydrite and akaganeite particles and there is no reason to believe that the contribution to the rate from this diffusion would not be quadratic with the external field. This contribution is not observed, probably because the coefficient of the quadratic dependence with the field is smaller than predicted. This could be explained by an erroneous definition of the correlation length in OST, this length is the particle radius, whilst the right definition should be the mean distance between random defects of the crystal. This correlation time would then be significantly reduced, hence the contribution to the relaxation rate. 

The amount of decrease of the resonance width may be simply estimated in the following way 50). Let the motion of the spins be characterized by a time tc, that is t is the average time a spin stays in a definite environment or the correlation time for the motion. This environment will cause a difference 5w in the precessional frequency of the spin which may be positive or negative from some average value to. During the time Tc the spin acquires a phase angle 60 = TcSu in addition to that acquired by the uniform precession at to. If we consider the motion to be a random walk process (51), after n such intervals during a time t the mean square phase acquired will be... 

Bloembergen et al. (S) have presented a relationship between the correlation time for molecular rotation in liquids and the relaxation times assuming that relaxation takes place via mechanism (i) of Section II,A,3. Although the theory can be at best semiquantitative when applied to the protons of water molecules adsorbed on silica gel, values of the nuclear correlation time have been calculated 18) from the T data. These correlation times as a function of x/m show a definite change of slope near a monolayer coverage. This result, if corroborated by data on other solids, may provide a rather unique method for the determination of monolayer coverage. 

Indeed, 13C spin-lattice relaxation times can also reflect conformational changes of a protein, i.e. helix to random coil transitions. This was demonstrated with models of polyamino acids [178-180], in which definite conformations can be generated, e.g. by addition of chemicals or by changes in temperature. Thus effective molecular correlation times tc determined from spin-lattice relaxation times and the NOE factors were 24-32 ns/rad for the a carbons of poly-(/f-benzyl L-glutamate) in the more rigid helical form and about 0.8 ms/rad for the more flexible random coil form [180],... 

An important goal of studies with intermediate- and short-term biomarkers is validation of the markers by demonstrating their correlation with definitive endpoints. It is reasonable to expect that these biomarkers will enable differences in human susceptibilities to exogenous chemicals to be detected, or suspected, at an early stage of exposure, and the liability of susceptible persons to be predicted long before the definitive endpoint is reached. It follows that studies using such biomarkers could be accomplished in a much shorter time and at a much lower cost than those aimed at determining definitive end-points. 

Backbone fluctuations with correlation times in the nanosecond regime are revealed in variations of the R1 residue scaled mobility, Ms, along the sequence (see Section III,A,1 for definition). Figure 15A (see color insert) shows a plot of Ms versus sequence for C1-C3, H8, and adjacent sequences in the TM helices. Figure 15B shows the cytoplasmic surface of rhodopsin color-coded according to Ms values. In this figure, C3 was modeled from the SDSL data, as in Fig. 9. 

In many cases the auto-correlation function is an exponential function with a time constant Tc, which is called the correlation time of the process. Following the definition of the correlation time for an exponential correlation function (3.2.10) the correlation time for a nonexponential correlation function is defined as... 

Ito [51] obtains different expressions for the Kramers-Moyal coefficients in which the spurious drift term is absent. However use of Ito coefficients involves new rules for calculus and so Stratonovich s method will be used here since it is also in agreement with the original method of Brown [8] and is the correct definition to use in the case of a physical noise which always has a finite correlation time [58] (see B.2). 

Two different definitions are used for downcomer residence time (371). The "apparent residence time is the ratio of the downcomer volume to the clear liquid flow in the downcomer. The downcomer volume is based on the tray spacing times the average downcomer cross section. The true residence time is the ratio of froth volume in the downcomer to the frothy liquid flow in the downcomer. The true residence time can alternatively be expressed as the ratio of the clear liquid volume in the downcomer to the clear liquid flow. Different literature sources use different definitions the definition adopted here is that of the apparent downcomer residence time. The author found this definition easier to apply, and to give a better correlation with the guidelines below [which were based on apparent residence times (49,86)]. Further, the author found that applying the true residence time definition as outlined by some early sources (371) can lead to oversized downcomers. 

A = and M.22 z) is the matrix element (2 M(z) 2). All the other matrix elements ot the memory operator are zero [63]. The appearance ot the tactor i corresponds to the Hermitian definition ot the Liouvillian which is not universal. Equation (98) is an exact z-dependent expression. This z-dependence can be eliminated by approximating M22(z) by a positive constant r (Markovian approximation). This approximation is justified since the microscopic correlation time is much shorter than the regression time ot the fluctuation. Then Eq. (98) transtorms into the z-independent effective Liouvillian... 

Water residence times on the protein surface are not directly measurable experimentally, bnt can be defined as the relaxation time of time correlation functions of the popnlation of the hydration shell [5], Due to differences in the definition from one investigation to another, the values reported in the literature exhibit considerable variability. Nonetheless, heterogeneity in water dynamics near the protein surface is clearly manifested in distributions of water residence times. The distribution we have constrncted for the N state of HocLA in solution from the residence time of water next to each residue is plotted up to lOOps in Figure 16.1d. The distribution is very broad, ranging from 2.6 to 241 ps, but highly skewed toward shorter residence times. The mean residence time of 23 ps is about 2.5 times longer than the rotational correlation time for hydration water. 

T2 transvers relaxation time) The value Tz/Tj- = 5 has been assumed, where T(- is the correlation time (reciprocal spectral width) of light sources. The coherence parameter P represents the extent of random phase distribution, and the dispersion parameter W represents the degree of regular phase-modulation due to material dispersion (see text for definition). The cross relaxation effect has been neglected. 

Bond orientational correlation functions and the associated correlation times for polyethylene have been evaluated by Bahar et al. [10]. The former are determined from the definitions given by Eqs. (8) and (9) where nii in Eq. (9) is replaced by the bond vector Ij. Cross-correlation times for orientations of bonds i and k are obtained as the area under the corresponding OACF decay curves according to Eq. (3). The analysis has also been extended to the domain of normal modes by adopting the transformation... 

The conventional, and very convenient, index to describe the random motion associated with thermal processes is the correlation time, r. This index measures the time scale over which noticeable motion occurs. In the limit of fast motion, i.e., short correlation times, such as occur in normal motionally averaged liquids, the well known theory of Bloembergen, Purcell and Pound (BPP) allows calculation of the correlation time when a minimum is observed in a plot of relaxation time (inverse) temperature. However, the motions relevant to the region of a glass-to-rubber transition are definitely not of the fast or motionally averaged variety, so that BPP-type theories are not applicable. Recently, Lee and Tang developed an analytical theory for the slow orientational dynamic behavior of anisotropic ESR hyperfine and fine-structure centers. The theory holds for slow correlation times and is therefore applicable to the onset of polymer chain motions. Lee s theory was generalized to enable calculation of slow motion orientational correlation times from resolved NMR quadrupole spectra, as reported by Lee and Shet and it has now been expressed in terms of resolved NMR chemical shift anisotropy. It is this latter formulation of Lee s theory that shall be used to analyze our experimental results in what follows. The results of the theory are summarized below for the case of axially symmetric chemical shift anisotropy. 

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