Low-field limit

The assumption of a single electron spin and a single T2 holds usually for S = 1/2 and for S > 1 in certain limits. Let us assume that the instantaneous distortions of the solvation sphere of the ion result in a transient ZFS and that the time-dependence of the transient ZFS can be described by the pseudorotation model, with the magnitude of the transient ZFS equal to At and the correlation time t . The simple picture of electron relaxation for S = 1 is valid if the Redfield condition (Att <5c 1) applies. Under the extreme narrowing conditions ((Os v 1), the longitudinal and transverse electron spin relaxation rates are equal to each other and to the low-field limit rate Tgo, occurring in Eqs. (14) and (15). The low field-limit rate is then given by (27,86) ... 

An analytical theory of the outer-sphere PRE for slowly rotating systems with an arbitrary electron spin quantum number S, appropriate at the limit of low field, has been proposed by Kruk et al. (144). The theory deals with the case of axial as well as rhombic static ZFS. In analogy to the inner sphere case (95), the PRE for the low field limit could be expressed in terms of the electron spin spectral densities s ... 

The field dependence of /2(a), vc)/2 is similar to that of f (a>, xc) (Fig. 3.10), except that the relative height of the two plateaus is 20/7 and at high field the function levels off at a finite value that is one-fifth of the low field limit, because of the field independent term 4rc. 

In the low field limit the ZFS term (the second in the above equation) is dominant, while in the high field limit the Zeeman term (the first in the above equation) is dominant. 

As an alternative method for the determination of the cluster size, we analyzed the low-field magnetic moment data from Fig. 2(c) by means of the modified susceptibility formula [24], derived from equation (1) in the low-field limit g B kB(T — Tc) ... 

Figure 1 The field dependence of the photogeneration efficiency of a dual-layer aggregate photoreceptor in the low-field limit.

Fig. 3.25. The degree of order induced by poling can be determined using the results above to analyse the variation in second harmonic intensity as the poled sample is rotated about an axis normal to the fundamental beam. Generally, the low field limit, given by Equation (3.106), is found to apply and the degree of orientation is low. However, for molecules with large dipole moments th,e divergence from the low field limit shown in Fig. 3.25 can be observed, anil much higher orientation is obtained. The experimentally determined non-linearities are usually the electro-optical (r) and the second harmonic generation (d) coefficients. These are related to the hyperpolarisabilities by ...

If one plotted piT) in an Arrhenius diagram and determined the apparent activation energy A from the tangent at a given temperature T, one would obtain A(F) = (8/9)o cr. For (t=0.1 eV and 7 =29.5 K, A=0.35eV. A is, thus, always a multiple of cr. On the other hand, since any polaron-binding energy Ep would enter the Boltzmann factor for the jump rate as p/2 in the low-field limit, the disorder contribution to A would dominate even if cr and Ep were comparable. 

Although the dispersion (95) and AE cannot be calculated explicitly for arbitrary input intensity N, it is possible to analyze the limit cases [71]. Obviously, both the predictions (90) and (93) will coincide provided that there is almost no information available in the low-field limit N > 0. Not so obvious is the fact that both predictions will coincide also in the high-energy limit N 1 provided the visibility is low V > 0. To see this, let us compare the asymptotic dispersion of the NFM estimator... 

For fluorine nuclei both the correlation time and the type of nuclear-electron interaction will alter with the radical used. To disentangle both these effects it is necessary to study their behaviour over a wide frequency range. This has been done for CgF and CgHjCFj with the radicals TTBP and Galvinoxyl. The most complete sequence of radicals however has been studied only at low field. In the low-field limit, where co5T, 1, the spectral densities and equation (23)... 

Finally, we should enquire as to whether or not it is reasonable to expect a linear response in describing the rotational diffusion of dipoles in the presence of a very strong local field, such as presented by the excess electron. The time window of the optical Kerr gate driven by a picosecond laser pulse depends on the relaxation of the molecules of the Kerr medium from an aligned orientation to an isotropic spatial distribution, once the applied optical field is switched off. For many liquids this relaxation time r is the low field limit, namely, the Debye time. We might anticipate an asynunetry in the temporal response 5 (0 of... 

As discussed by McCauley et al. [87,88] for a triplet state with D > 3E > 0 in the low-field limit, the ODMR transition in the triplet state measured for a sample... 

The induced birefringence is a function of the polarizabilities and the extent of alignment of the resultant dipole moments // of the particles in the solution by the external field E. Under the assumption that interparticle interaction is negligible (dilute solutions) and the energy of interaction U between E and fi is less than the thermal energy kT, the following expressions can be derived [3] for the low-field limit ... 

Low-Field Limit and Onset of High-Field Regime... 

By Equation 2.2, K depends on E at any E. In practice, a quadratic leading term means that the variation of K exceeds the measurement uncertainty and becomes noticeable fairly abruptly above some E/N threshold, as observed in experiment (2.1). At lower E/N, called the low-field limit, K may be deemed independent of E/N. Conventional IMS is usually operated in that regime, as evidenced by linearity of v with respect to E/N varied by changing the drift voltage or gas pressure... 

The low-field limiting case of ion-pair equilibria in water may be derived from Eq. (3.85) ... 

Aspnes, D.E., and J.E. Rowe. 1972. Resonant nonlinear optical susceptibility Electroreflectance in the low-field limit. Phys Rev B 5 4022—4030. 

Low field limit. Minimising Eq. (13.10) with respect to polarization F we find the relation between the tilt and polarization ... 

In the low field limit, expanding (9s + 89) we shall find the soft mode susceptibility of the SmC phase using exactly the same procedure as for crystalline ferroelectrics, see Eq. (13.8) ... 

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