Free characteristic time constant

The principle of Fourier transform (FT) NMR spectroscopy is the observation of the so-called free induction decay (FID) after the application of radio frequency (rf) pulses to the resonating nuclei. The carrier frequency of the rf-pulses is the Larmor frequency. In many cases, the FID is observed after single-pulse (SP) excitation, e.g., after application of a so-called 7r/2-pulse which rotates the magnetization by 90° from the direction of the external magnetic field (z-direction) into the x,y-plane. The characteristic time constant for the free induction decay is the transverse relaxation time, T2, which is given by T2=(2/M2) =0.53 (A Vi/2)" for Gaussian lines. Fourier transformation of the FID yields the common absorption spectrum. 

From the discussion so far, it might appear that stratification is advantageous only if the free energy changes as a function of . This is, however, not so. Stratification improves efficiency even if the free energy is constant and the motion along is strictly diffusive. If the full range of the order parameter is divided into L windows of equal size, the computer time needed to acquire the desired statistics in each window, tw, is proportional to the characteristic time of diffusion within a window... 

For some reactions the rate constant kj can be very large, leading potentially to very rapid transients in the species concentrations (e.g., [A]). Of course, other species may be governed by reactions that have relatively slow rates. Chemical kinetics, especially for systems like combustion, is characterized by enormous disparities in the characteristic time scales for the response of different species. In a flame, for example, the characteristic time scales for free-radical species (e.g., H atoms) are extremely short, while the characteristic time scales for other species (e.g., NO) are quite long. It is this huge time-scale disparity that leads to a numerical (computational) property called stiffness. 

Figure 15.1 shows families of solutions to the model problem for different values of X and different initial conditions. The family of solutions can be thought of as a manifold of solutions, all of which, regardless of the initial condition, tend toward the slowly varying y = t2 + 1 solution. In chemical kinetics, the behavior illustrated in Fig. 15.1 is exhibited by certain species, like the free radicals. After initial very rapid transients such as a combustion ignition, the free-radical concentrations often vary slowly, with their behavior controlled by steady-state or partial-equilibrium conditions. The faster the characteristic scales, the more rapidly the fast-time-constant species come into equilibrium with the major species (i.e., approach a slowly varying solution). 

Also discussed are precipitation specific experimental techniques, such as supersaturation measurements, constant composition (CC) method, instantaneous mixing devices, maximum (critical) growth rate experiments, and sizing. Due to the intrinsic difficulties with the direct supersaturation measurements and the microsecond characteristic time scale of precipitation reaction and nucleation, the CC method is used to study the precipitation kinetics. For the same reasons, the critical growth experiments are used to delineate the domain of the reactant feed rate that assures a renucleation-free process and a unimodal CSD. 

This model with a free variable conductance in parallel with a CPE or CPEp is often found in the literature and analyzed as if being in agreement with the Cole model which it is not. Equation 9.35 is a non—Cole-compatible model that can be used if both conductance and characteristic frequency or time constant are found to vary during an experiment. 

Modern NMR spectrometry uses the pulsed Fourier method, in which a carefully shaped pulse of radio-frequency energy, tuned to the characteristic NMR frequency called the Larmor frequency (co), is pumped into the sample. The sample then responds by sending out a very much weaker signal called a free-induction decay (FID). This FID signal appears at the same Larmor resonance frequency but with an amplitude that decays approximately as a decreasing exponential. Depending on a number of experimental parameters, the FID time constant may range from milliseconds to seconds. 

Cao et al. [21] analyzed the PSD for some AISI 304 (MnS-containing) and 321 (Ti-bearing, then MnS-free) steels. No reference is made to the difference in inclusions between the two steels. For 321 steel, the elementary prepitting event is found to consist of a linear increase in anodic current, up to some pA in the tested conditions, followed by an exponential decrease. The frequency dependence of the PSD depends on the time characteristics of these two processes (growth rate of the micropit and repassivation time constant), which are potential dependent. Following the values of these time characteristics, and then the electrode potential, the PSD varies asat the high-frequency limit, with = 2 to 4. A white noise (no frequency dependence) is found at very low frequency (some 0.1 Hz). From this work, it is also inferred that the solution chloride content affects the nucleation frequency but not the growth or the repassivation kinetics of the micropit. 

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