High field approximation

While all contributions to the spin Hamiltonian so far involve the electron spin and cause first-order energy shifts or splittings in the FPR spectmm, there are also tenns that involve only nuclear spms. Aside from their importance for the calculation of FNDOR spectra, these tenns may influence the FPR spectnim significantly in situations where the high-field approximation breaks down and second-order effects become important. The first of these interactions is the coupling of the nuclear spin to the external magnetic field, called the... 

When the magnetic field is large such that j(Ae + An)2 4a2, then C22 — 1 and C23 0 or 03) aej3n) similarly, C32 0 and C33 1 or 03) (3c n). These high-field approximations to the wavefunctions are shown along with their respective energy levels in Fig. 1. 

Addressing specifically the dipole-dipole coupling interaction, the Hamiltonian in the high-field approximation takes the form... 

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... 

Fig. 7.185. In a self-driving cell, the plot of cell overpotential vs. log cell current density should be a straight line if the charge transfers at both electrodes are both rate controlling and valid under the high-field approximation. An apparent /0 for the cell as a whole can be deduced.

I Vs,ms,1,M/)- In the high field approximation (ge/xsBo -4), which always holds for high field NMR experiments, the contact contribution to the chemical shift is (Appendix III) [1]... 

The expression (12.27) for the corrosion potential can be introduced into Eq. (12.22) and thus, an explicit result for the corrosion current can be obtained. But the resulting equation is quite cumbersome and therefore a simpler equation will be derived by assuming that overpotentials are sufficiently large that the high-field approximation of the Butler-Volmer equation can be used for the electronation- and deelectronation-current densities. Thus, Eqs. (12.22) and (12.23) become... 

What has been presented above is a very elementary account of corrosion under super-ideal conditions. In a few cases, it does give a fairly good agreement with the observed rates of corrosion. Yet, in real systems, corrosion is nearly always too complex a phenomenon for the above simple treatment to be directly applicable. The simple version would be valid if there were no oxide films, if there were a negligible IR drop in the solution, if the corrosion potential dtransfer coefficients of the metal-dissolution and electronation reactions were [cf. Eq. (12.25)]. However, the point of an introductory treatment is not to treat the details and the complex realities, but to present the idealized essence about an electrochemical mechanism that has substantial effects in the everyday world. 

In electrolytic solutions, however, the conditions for the high-field approximation are not often observed. The applied field X is generally relatively small, in which case pX l and the following approximation can be used ... 

When Ip I >100 mV then the Butler-Volmer equation reduces to its high field approximation form ... 

When dealing with a long microscopic time scale (slow motion), in the high-field approximation, the nonsecular terms are usually disregarded. So the Hamiltonian describing the spin system is... 

To obtain the quadrupole Hamiltonian of a spin in a magnetic field the Hamiltonian needs to be transformed from the PAS to the LAB frame, keeping only those terms that commute with L. This is called truncation of a Hamiltonian and is only valid when Hq << Hz (the high field approximation). To perform the transformation it is much more convenient if second-rank irreducible spherical tensors are used. The Cartesian and spherical tensor elements (T) can be related (see Schmidt-Rohr and Spiess 1994 and Eq. 8, in Man 2000), with two of the more common elements being... 

It is convenient to write the Hamiltonian terms in the curly brackets in terms of irreducible tensor operators (ITO), Tm given in Appendix 1, where the Tm a components, with m = -1,0,1, appear in the chemical-shift terms and the components, with m = -2, -1,0,1,2, appear in the dipole-dipole interactions terms. Starting in the rotating frame with operators truncated according to the high-field approximation [14], the transformation of the spin operators maybe written as... 

These studies have shown that, quite often, over relatively wide (e.g., 0.3-0.5 V) ranges of potential, the catalytic rates depend on catalyst-electrode potential in an exponential manner, similar to the high field approximation of the Butler-Volmer equation, that is,... 

For simplicity, the direct Zeeman interaction with the external magnetic field is parametrized out by representing the system in the standard rotating frame. Within the high-field approximation, the Hamiltonian for the external rf irradiation may be expressed as... 

Anisotropy of the Spin-Spin Coupling Tensor. - In NMR experiments performed in anisotropic liquid crystal (LC) phases or in the solid state, the anisotric part of an indirect nuclear spin-spin coupling tensor J appears combined with the direct dipolar coupling D. The NMR spin Hamiltonian appropriate for spin 1/2 nuclei in molecules partially oriented in uniaxial LC solvents can be written in the high field approximation as... 

In the high-field approximation the space-dependent thermodynamic-equilibrium density matrix po(r) is proportional to l fr) (cf. eqn (2.2.58)). A 90° pulse converts this density matrix into p (0+, r) oc U(r) = [l+(r) -I- l (r)] /2, where U is the x-component of the spin operator, 1+ = 1 + il>-, and L = 1 — il. In the subsequent free-precession period the density matrix evolves for a time t under the spin Hamiltonian Hx(r) (cf. eqn (3.1.1)) and under the influence of the applied gradient, which introduces a phase evolution given by kr = —yGrti, where k is the experimental variable,... 

Thus, when making quantitative comparisons between the second-order perturbation and exact approaches for a single crystal containing half-integer quadrupolar nuclei, it is seen that the difference in the expected transition frequencies for the CT at several crystal orientations is minor (< 1%), up to the point at which vq and vq become comparable (about a factor of 1/2 for spin-3/2 and 1/5 for spin-5/2). It is also clear that the CT of a powdered sample is somewhat less affected by the tnmcation. It seems therefore that a common rule-of-thumb regarding the extent to which the second-order approximation (and hence the high-field approximation) is valid (i.e. when vq < 0.1 vq) may in many practical cases be slightly conservative. 

In the high-field approximation with B z, the energy eigenvalues classified by the magnetic spin quantum numbers. Mg andMj, are given by... 

The usual procedure for extracting the exchange current Iq is then to measure q as a function of I and to plot Inl vs q (Tafel plot). Such plots are shown in Figures 3 and 4 for Pt and Ag catalyst electrodes deposited on YSZ and acting as catalyst for C2H4 oxidation. Throughout the rest of this discussion, we omit the subscript "W" from q and simply write q, since the only overpotential of interest is that of the catalyst film. When Iql >100 mV, then the Butler-Volmer equation (16) reduces to its "high field approximation" form, i.e.. 

Note that, in contrast to the low-field approximation, the Tafel equation is sometimes referred to as the high-field approximation since it is only valid for large values of overpotential. 

Special cases of Butler-Volmer equation-high field approximation... 

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