Splitting, second-order

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... 

A different long-time-step method was previously proposed by Garci a-Archilla, Sanz-Serna, and Skeel [8]. Their mollified impulse method, which is based on the concept of operator splitting and also reduces to the Verlet scheme for A = 0 and admits second-order error estimates independently of the frequencies of A, reads as follows when applied to (1) ... 

In order to determine the phonon dispersion of CuZn and FeaNi we made use of an expanded tight binding theory from Varma and Weber . In the framework of a second order perturbation theory the dynamical matrix splits in two parts. The short range part can be treated by a force constant model, while the T>2 arising from second order perturbation theory is given by... 

The contributions of the second order terms in for the splitting in ESR is usually neglected since they are very small, and in feet they correspond to the NMR lines detected in some ESR experiments (5). However, the analysis of the second order expressions is important since it allows for the calculation of the indirect nuclear spin-spin couplings in NMR spectroscoi. These spin-spin couplings are usually calcdated via a closed shell polarization propagator (138-140), so that, the approach described here would allow for the same calculations to be performed within the electron Hopagator theory for open shell systems. 

Some situations, however result in the form of second-order diffferential equations, which often give rise to problems of the split boundary type. In order to solve this type of problem, an iterative method of solution is required, in which an unknown condition at the starting point is guessed, the differential equation integrated twice and the resulting solution compared with a known boundary condition, obtained at the end point of the calculation. Any error between the known value and the calculated value can then be used to revise the initial starting guess for the next iteration. This procedure is then repeated until... 

Cahbration spectra must be measured at defined temperamres (ambient temperature for a-iron) because of the influence of second-order Doppler shift (see Sect. 4.2.1) for the standard absorber. After folding, the experimental spectrum should be simulated with Lorentzian lines to obtain the exact line positions in units of channel numbers which for calibration can be related to the hteramre values of the hyperfine splitting. As shown in Fig. 3.4, the velocity increment per channel, Ostep, is then obtained from the equation Ustep = D,(mm s )/D,(channel numbers). Different... 

For vanadocene second-order perturbation calculations of the spin-orbit contribution to the zero field splitting, DLS, showed the II ( 2) level to lie below A ( - ), but the estimated value, 1.4 to 1.5 cm-1, constitutes only about half of the observed result. How-... 

Here, A is the nearly isotropic nuclear coupling constant, I is the nuclear spin (Iun = I), and m is the particular nuclear spin state. It may be observed that the zero field splitting term D has a second-order effect which must be considered at magnetic fields near 3,000 G (X-band). In addition to this complication nuclear transitions for which Am = 1 and 2 must also be considered. The analysis by Barry and Lay (171) of the Mn2+ spectrum in a CsX zeolite is shown in Fig. 35. From such spectra these authors have proposed that manganese is found in five different sites, depending upon the type of zeolite, the primary cation, and the extent of dehydration. 

FIGURE 5.1 Isotropic hyperfine pattern for 51VIV in S-band. The spectrum is from V0S04 in aqueous solution. Use of the low frequency enhances the second-order effect of unequal splitting between the eight hyperfine lines. 

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