Bloch term

The NMR phenomenon can be quantitatively described in classical terms. This was first done by Bloch.The approach is helpful in developing an understanding of nuclear relaxation processes. 

The form of the functions may be closely similar to that of the molecular orbitals used in the simple theory of metals. If there are M interatomic positions in the crystal which might be occupied by any one of the N electron-pair bonds, then the M functions linear aggregates that approximate the solutions of the wave equation with inclusion of the interaction terms representing resonance. This combination can be effected with use of Bloch factors ... 

The resonating-valence-bond theory of the electronic structure of metals is based upon the idea that pairs of electrons, occupying bond positions between adjacent pairs of atoms, are able to carry out unsynchronized or partially unsynchronized resonance through the crystal.4 In the course of the development of the theory a wave function was formulated describing the crystal in terms of two-electron functions in the various bond positions, with use of Bloch factors corresponding to different values of the electron-pair momentum.5 The part of the wave function corresponding to the electron pair was given as... 

Whenever the system is no longer constituted by single non-interacting spins, the simple Bloch Equation (2) must be completed by additional coupling terms. Let us consider the dipolar interaction between two spins... 

Suppose we have a system in which a spin can exist in either of two different sites, A or B, and that these are distinguished by different resonant frequencies, coa and coB, and/or by different relaxation times, T2a and T2b If there is no exchange between sites, site A spins and site B spins can be described separately and independently by sets of Bloch equations. When exchange takes place, however, additional rate terms - completely analogous to terms in chemical rate equations - must be added to the Bloch equations. 

As with any other homonuclear decoupling, the transverse magnetization of the 13C , for example in the evolution time, is perturbed by the decoupling pulse, resulting in an additional precession virtually around the z axis. Since the adiabatic decoupling is applied during the entire evolution time of the 13C , a non-linear frequency shift rather than a phase shift appears in the spectrum. This is termed the Bloch-Siegert (frequency) shift in the NMR literature in honour of their discovery of the phenomenon. 

While the Bloch correction represents a series expansion in powers of k, the leading term in the Barkas-Andersen correction was found [22] to be oc B. 

The theory of nuclear spin relaxation (see monographs by Slichter [4], Abragam [5] and McConnell [6] for comprehensive presentations) is usually formulated in terms of the evolution of the density operator, cr, for the spin system under consideration from some kind of a non-equilibrium state, created normally by one or more radio-frequency pulses, to thermal equilibrium, described by Using the Bloch-Wangsness-Redfield (BWR) theory, usually appropriate for the liquid state, we can write [7, 8] ... 

The Bloch functions near the K points have a long decay length and contribute to the second term of Eq. (5.10). Following Eq. (5.12), in general, a surface Bloch function at that point has the form ... 

In addition to the term with n = 0, the term with n=-l have the same decay length, and thus have the same magnitude. Also, the Bloch function that generates the symmetric charge density must also be s metric. The lowest-order symmetric Fourier sum of the Bloch function near K is ... 

The function 4>< ( x) has a value I at each lattice point in real space, and 0 at the center of four neighboring lattice points. Similar to the one-dimensional case, the ao(z) term in Eq. (5.29) comes mainly from the Bloch functions near... 

The problem of the electron charge-density distribution of a surface with hexagonal symmetry has been treated by Liebsch, Harris, and Weinert (1984). Similar to previous cases, the oo(z) term in Eq. (5.41) comes mainly from the Bloch functions near E, whose lowest Fourier component is ... 

The local modification of sample wavefunctions due to the proximity of the tip, and consequently the involvement of the Bloch functions outside the energy window Er eV in the tunneling process, has an effect on the limit of the energy resolution of scanning tunneling spectroscopy. This effect is discussed in detail by Ivanchenko and Riseborough (1991). First, if the tunneling current is determined by the bare wavefunctions of the sample and the tip, the process is linear, and there is no effect of quantum uncertainty. The effect of quantum uncertainty is due to the modification or distortion of the sample wavefunction due to the existence of the tip. Here, we present a simple treatment of this problem in terms of the MBA. 

Therefore, considering Bloch states, a positive exchange term coupling opposite spins is added to the Hamiltonian (11) ... 

The definition of the Stokes vector in terms of bilinear products of field amplitudes is analogous to the definition of the two-state Bloch vector r = [u, v, iv] from products of probability amplitudes. The conventional numbering of the two vectors differs the connection is... 

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