High-field equation

This is commonly known as the high field equation. It is of similar form to the Tafel equation for activation controlled electrochemical reactions with... 

Lohrengel [12] proposed an extension of the high-field model where the concentration of defects varies with the electric field strength. When the concentration reaches a constant value, the high-field equations may be valid. D Alkaine et al. [13] developed a model where migration is the main type of transport, applied for both stationary [13,14] and potentiodynamic [14,15] conditions of film growth. In this case, it was assumed that the movement of ions inside the film presents characteristics similar to their movement in solution and that the potential gradient across the film obeys a linear relationship. 

Figure 6 shows the field dependence of hole mobiUty for TAPC-doped bisphenol A polycarbonate at various temperatures (37). The mobilities decrease with increasing field at low fields. At high fields, a log oc relationship is observed. The experimental results can be reproduced by Monte Carlo simulation, shown by soHd lines in Figure 6. The model predicts that the high field mobiUty follows the following equation (37) where d = a/kT (p is the width of the Gaussian distribution density of states), Z is a parameter that characterizes the degree of positional disorder, E is the electric field, is a prefactor mobihty, and Cis an empirical constant given as 2.9 X lO " (cm/V). ...

In order to use Eqs. (2.6.1) and (2.6.2) to estimate the sensitivity gain obtained by remote detection, knowledge of the relative sensitivity of the detector and the encoding circuit, A, is required. Here we discuss the sensitivity of an rf coil detector as an example, because all the experiments presented in this text use inductive detection at high field. The signal-to-noise ratio of inductive NMR detection can be approximated by the following simplified equation [12] ... 

High-field EPR (HFEPR) spectroscopy greatly improves the resolution of the EPR signals for spectral features such as the g-tensor. Deviations of the g-value from free electron g=2.0023 are due to spin-orbital interactions, which are one of the most important structural characteristics (Kevan and Bowman 1990). Using a higher frequency results in enhanced spectral resolution in accordance with the resonance equation ... 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

Electronic relaxation is a crucial and difficult issue in the analysis of proton relaxivity data. The difficulty resides, on the one hand, in the lack of a theory valid in all real conditions, and, on the other hand, by the technical problems of independent and direct determination of electronic relaxation parameters. At low fields (below 0.1 T), electronic relaxation is fast and dominates the correlation time tc in Eq. (3), however, at high fields its contribution vanishes. The basic theory of electron spin relaxation of Gdm complexes, proposed by Hudson and Lewis, uses a transient ZFS as the main relaxation mechanism (100). For complexes of cubic symmetry Bloembergen and Morgan developed an approximate theory, which led to the equations generally... 

Longitudinal and transverse nuclear relaxation profiles differ in the high field part. In fact, the equation for the transverse nuclear relaxation rates contains a non-dispersive term, depending only on Xd. Therefore the transverse relaxation does not go to zero at high fields, as longitudinal relaxation does, but increases because Tie increases (until it increases to the point where it becomes longer than x or Xm)-... 

Fig. 29. Water NMRD longitudinal profiles for Fe(H20)g" solutions at 278 ( ), 288 ( ), 298 (A), and 308 ( ) K. High field transverse relaxivity data at 308 K ( ) are also shown. The lines represent the best fit curves using the Solomon-Bloembergen-Morgan equations (Eqs. (1-6)) (36).

The NMRD profiles of water solution of Ti(H20)g" have been shown in Section I.C.7 and have been already discussed. We only add here that the best fit procedures provide a constant of contact interaction of 4.5 MHz (61), and a distance of the twelve water protons from the metal ion of 2.62 A. If a 10% outer-sphere contribution is subtracted from the data, the distance increases to 2.67 A, which is a reasonably good value. The increase at high fields in the i 2 values cannot in this case be ascribed to the non-dispersive term present in the contact relaxation equation, as in other cases, because longitudinal measurements do not indicate field dependence in the electron relaxation time. Therefore they were related to chemical exchange contributions (see Eq. (3) of Chapter 2) and indicate values for tm equal to 4.2 X 10 s and 1.2 X 10 s at 293 and 308 K, respectively. 

In natural melts, the presence of high field strength ions such as Fe , TF, and P, which, like silicon, preferentially assume a tetrahedral coordination with oxygen, complicates the structure, and the constitution of the anion matrix may not be deduced on the basis of the equations in section 6.1.2. Structural parameters valid for compositionally complex melts were proposed by Mysen et al. (1980) and Virgo et al. (1980) on the basis of the results of Raman spectroscopy. These parameters are NBO/Si and NBO/T (NBO = Non-Bridging Oxygen T groups all tetrahedrally coordinated cations—i.e., Fe ", TF, P ", ... 

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