The Bloch theory

Electrons whose k values satisfy equation (5.13) cannot travel through the crystal since they suffer reflexion in the atomic planes. Physically this means that such electrons have no existence in the crystal and that the corresponding energy values are missing from the energy distribution. 

A convenient way to represent this state of affairs is to plot a diagram in k space on which are shown the forbidden values of k. Such a diagram, corresponding to our one-dimensional model, is shown in fig. 5.10J. O is the origin, and the points kl9 2, represent the values of k given by equation (5.13). It must be remembered that k is a vector, and such a diagram must therefore be interpreted in the sense that the 

More refined arguments show that the energies of electrons whose 

In the three dimensions of a real crystal similar arguments again apply and we find that the Brillouin zones are now defined by a series of concentric polyhedra in k space. 

Suppose, however, that the E/k curves take the form shown in fig. 5-i2 . It is still true that in any given direction the curve is as shown in fig. 5.11, but we now note that for certain directions in the first zone (e.g. kc) the energy is greater than for other directions in the second zone (e.g. ka). There is therefore now no gap of forbidden energy values and the two zones overlap (fig. 5.12d). This distinction is of profound importance, for it enables us to discuss what happens as the energy states in a metal are successively filled with electrons. 

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While this has been known for a long time, recent developments in heavy-ion stopping [56] stimulated a reevaluation of the Bloch theory which resulted in a major revision of relativistic stopping theory, giving rise to substantial changes in quantitative predictions, in particular for the heaviest ions [9]. 

The Bloch theory of NMR assumes that the recovery of the -f z-magnetisa-tion, Mj, follows exponential behaviour, described by ... 

Fig. 5.09. Form of potential well in a one-dimensional metal on the basis of the Bloch theory.

Fig. 5.11. (a) Variation of electron energy with wave number for a one-dimensional metal on the basis of the Bloch theory. The first two forbidden k values are shown and in (forbidden energy are indicated. The only permissible energy levels are those within the shaded regions. 

This equation is a generalization of an equation previously found by Bloch and generalized by Lindgren. Solving the operator equation (22) appears as the main task in the Bloch theory of effective Hamiltonians since H can immediately be deduced from Cl by means of (20). It will also be shown below that the wave operator 12 plays a fundamental role in the whole theory of effective Hamiltonians. 

H is the Bloch operator associated with the main model space. In close analogy with the Bloch theory, an equation for R can be chosen in the form... 

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 ... 

It was pointed out in my 1949 paper (5) that resonance of electron-pair bonds among the bond positions gives energy bands similar to those obtained in the usual band theory by formation of Bloch functions of the atomic orbitals. There is no incompatibility between the two descriptions, which may be described as complementary. It is accordingly to be expected that the 0.72 metallic orbital per atom would make itself clearly visible in the band-theory calculations for the metals from Co to Ge, Rh to Sn, and Pt to Pb for example, the decrease in the number of bonding electrons from 4 for gray tin to 2.56 for white tin should result from these calculations. So far as I know, however, no such interpretation of the band-theory calculations has been reported. 

Bloch, F., 1929, The Electron Theory of Ferromagnetism and Electrical Conductivity , Z. Physik, 57, 545. 

Since Aa> is proportional to the magnetic field, higher fields allow for exploiting faster exchanges which consequently leads to a more important CEST effect. More detailed explanation of the CEST theory (52,159) and numerical solution of the Bloch equations describing the CEST effect (160) can be found in the literature. 

To verify the theory of PIP, a computer program using C language was developed. It can be used to directly calculate the excitation profiles by PIPs or any other RF pulses. The calculation is based on the Bloch vector model for a non-interacting spin-1/2 system, where the spin-lattice relaxation during the pulse is neglected. The basic idea of the program is discussed as follows. 

Finally - and equally important - Jens contribution to the formal treatment of GOS based on the polarization propagator method and Bethe sum rules has been shown to provide a correct quantum description of the excitation spectra and momentum transfer in the study of the stopping cross section within the Bethe-Bloch theory. Of particular interest is the correct description of the mean excitation energy within the polarization propagator for atomic and molecular compounds. This motivated the study of the GOS in the RPA approximation and in the presence of a static electromagnetic field to ensure the validity of the sum rules. 

This chapter is organized in 6 sections. Section 2 describes the geometry of (CBED). Section 3 covers the theory of electron diffraction and the principles for simulation using the Bloch wave method. Section 4 introduces the experimental aspect of quantitative CBED including diffraction intensity recording and quantification and the refinement technique for extracting crystal stmctural information. Application examples and conclusions are given in section 5 and 6. 

Electron Dynamic Theory - The Bloch wave method... 

In Chapter 3 we went as far as we could in the interpretation of rocking curves of epitaxial layers directly from the features in the curves themselves. At the end of the chapter we noted the limitations of this straightforward, and largely geometrical, analysis. When interlayer interference effects dominate, as in very thin layers, closely matched layers or superlattices, the simple theory is quite inadequate. We must use a method theory based on the dynamical X-ray scattering theory, which was outlined in the previous chapter. In principle that formrrlation contains all that we need, since we now have the concepts and formtrlae for Bloch wave amplitude and propagatiorr, the matching at interfaces and the interference effects. 

A more general formulation of relaxation theory, suitable for systems with scalar spin-spin couplings (J-couplings) or for systems with spin quantum numbers higher than 1/2, is known as the Wangsness, Bloch and Redfield (WBR) theory or the Redfield theory 17). In analogy with the Solomon-Bloembergen formulation, the Redfield theory is also based on the second-order perturbation approach, which in certain situations (not uncommon in... 

Various theoretical formalisms have been used to describe chemical exchange lineshapes. The earliest descriptions involved an extension of the Bloch equations to include the effects of exchange [1, 2, 12]. The Bloch equations formalism can be modified to include multi-site cases, and the effects of first-order scalar coupling [3, 13, 24]. As chemical exchange is merely a special case of general relaxation theories, it may be compre-... 

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] ... 

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