Time-dependent susceptibilities

The time development of these coherences corresponds to a time-dependent susceptibility X (0 of the sample, which affects the polarization characteristics of the probe pulse and appears as quantum beats of the transmitted probe pulse intensity. 

On the other hand, the FC-magnetization has been generally assumed to yield information on the thermal equilibrium behavior of spin glasses because of its reversibility and time-independence (Malozemoff and Imry 1981, Chamberlin et al. 1982, Monod and Bouchiat 1982). Recently, several authors (Lundgren et al. 1982 and 1985, Wenger and Mydosh 1984a, Bouchiat and Mailly 1985, Kinzel and Binder 1984), however, question the validity of this assumption. We shall discuss this point in context with the time-dependent susceptibilities in sec. 6.1. 

The name used for this function a t) is time dependent susceptibility . We obtain ... 

If we eliminate p t) from this equation with the aid of Eq. (5.31) we find the interrelation between the time dependent susceptibility and the time dependent modulus... 

There is little evidence for the operation in reactions of the inducto-meric effect, the time-dependent analogue of the inductive effect. This may be so because the electrons of the delocalized system, and are thus not so susceptible to the demands of the reagent. 

Materials subjected to high temperatures during their service life are susceptible to another form of fracture which can occur at very low stress levels. This is known as creep failure and is a time dependent mode of fracture and can take many hours to become apparent (Fig. 8.88). 

Kimira, T, Yoshino, M., Yamane, T, Yamato, M. andTobita, M. (2004) Uniaxial alignment of the smallest diamagnetic susceptibility axis using time-dependent magnetic fields. Langmuir, 20, 5669-5672. 

Static charge-density susceptibilities have been computed ab initio by Li et al (38). The frequency-dependent susceptibility x(r, r cd) can be calculated within density functional theory, using methods developed by Ando (39 Zang-will and Soven (40 Gross and Kohn (4I and van Gisbergen, Snijders, and Baerends (42). In ab initio work, x(r, r co) can be determined by use of time-dependent perturbation techniques, pseudo-state methods (43-49), quantum Monte Carlo calculations (50-52), or by explicit construction of the linear response function in coupled cluster theory (53). Then the imaginary-frequency susceptibility can be obtained by analytic continuation from the susceptibility at real frequencies, or by a direct replacement co ico, where possible (for example, in pseudo-state expressions). 

New biomarkers will be useful in hepatotoxicity risk assessment if the data quality and validity can be established. The FDA defines a valid biomarker as one that can be measured in an analytical test system with well-established performance characteristics and has an established scientific framework or body of evidence that elucidates the significance of the test results [160]. Although there is no formerly agreed upon path, biomarker validation should include appropriate end-points for study (i.e., toxicology, histopathology, bioanalytical chemistry, etc.) and dose- and time-dependent measurements. An assessment of species, sex and strain susceptibility is also important to evaluate across species differences. More specific considerations for validation of gene and protein expression technologies are reviewed by Corvi et al. and Rifai et al. [144, 147]. 

Thus, a frequency-dependent susceptibility x(to) implies that the polarization P at time t depends on the electric field E at all other times t. This conclusion is consistent with simple physical reasoning. If, for example, a steady electric field is applied to a sample of matter for a sufficient period of time, a steady polarization will be induced in the sample. However, if the electric field were to be suddenly removed, the polarization would not immediately drop to zero but would decay according to characteristic times associated with microscopic processes. In this example it is clear that the polarization is not proportional to the instantaneous field. 

The derivation of Kramers-Kronig relations for the susceptibility was relatively easy, perhaps misleadingly so. With a bit of extra effort, however, we can often derive similar relations for other frequency-dependent quantities that arise in physical problems. Suppose that we have two time-dependent quantities of unspecified origin, which we may call the input X((t) and the output X0(t) the corresponding Fourier transforms are denoted by 9C,(co) and 9Cc(io). If the relation between these transforms is linear,... 

We have shown that a frequency-dependent susceptibility implies temporal dispersion the polarization at time t depends on the electric field at all times previous to t. It is also possible under some circumstances to have spatial dispersion the polarization at point x depends on the values of the electric field at points in some neighborhood of x. This nonlocal relation between P and E... 

To obtain the frequency-dependent susceptibility x(<°), we need the polarization in response to a time-harmonic field E0e ... 

Now a new case is calculated at the best previous point and a new direction of steepest ascent is determined. The process is repeated as many times as seems advisable. The entire procedure is susceptible to automatic treatment on the computer. The only point of uncertainty is the size of the steps to be taken. It may be necessary to revise this from time to time depending upon the progress of the study. As the optimum is approached, the steps should decrease in size. Even if it is decided that for a particular problem and a particular computer completely automatic calculation is impractical, at least some fairly large combination of operations can be programmed for one computer run. 

Although both SH transients in Fig. 5.21 fall to a minimum at about the same time, their form is quite different and qualitative comparisons are useful. The isotropic contribution, /pp(/), decays as a single exponential, in agreement with previous measurements of submonolayer thallium deposition on polycrystalline electrodes [54]. The solid line in Fig. 5.21 a is an exponential fit with r = 10.7 msec. The exponential form suggests that the deposition occurs by an absorption, rather than a nucleation, mechanism [154]. The transient anisotropic response is not as simple. In fact, the initial fall in /ps( ) in Fig. 5.21 b is not a simple decaying exponential. The differing time dependencies for the isotropic and anisotropic responses suggests that f, the bulk anisotropic susceptibility element which is the only common element, is not the main source of the nonlinear response in either case. 

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