Bloch-type equations

Yafet (1963) calculated the relaxation of the conduction electrons to the lattice <5 l due to spin-orbit scattering. Set can be separated into (intrinsic) and S eB-X. Hereby x denotes the concentration of the extrinsic scattereis. Has awa (1959) analyzed the situation which is shown in fig. 1 by Bloch-type equations. In this scenario the paramagnetic ions are... 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

At low temperatures, the small-polaron moves by Bloch-type band motion, while at elevated temperatures it moves by thermally activated hopping mechanism. Holstein (1959), Friedman and Holstein (1963), Friedman (1964) performed the theoretical calculations of small-polaron motion and showed that the temperature dependencies of the small-polaron mobility in the two regimes are different. In the high-temperature hopping regime, the electrical conductivity is thermally activated and it increases with increasing temperature. As shown by Naik and Tien (1978), its temperature dependence is characterized by the following equation... 

Let us summarise in the effective Hamiltonian language [20] the general fea-tmes of a CP /SO method, in its simple Bloch-type version. In a first step, the scalar relativistic secular equations for states under interest are solved, and extensive Cl calculations define a determinant target space dim providing accurate energies and the corresponding multiconfigurational states of interest Om) m E In a second step, a determinant inter-... 

Faced with a broad range of prospective spin-lattice relaxation times, the investigator needs two types of spectrometers, a situation that is further complicated if multi-frequency measmements are required. Furthermore, the phenomenological descriptions of measurements made by cw and transient spectrometers differ, as they correspond to separate solutions to Bloch s equations. This chapter describes refinements of both instrumental and theoretical/computational techniques that facilitate the measurement of spin-lattice relaxation times. 

These are Green s functions diffusing in what is interpreted as an imaginary time [3, according to the Bloch equation, dp/8(3 = JYp (a diffusion-type partial differential equation). These Green s functions satisfy the equation... 

The first approach, taking the advantage of the BCH formula, was initiated hy Jeziorski and Monkhorst [23] and, so far, it has been intensively developed within Paldus s group [5,51-55] who formulated an orthogonally spin-adapted Hilbert space MR CC method for a special case of a two-dimensional model space spanned by closed-shell-type reference configurations. The unknown cluster amplitudes are obtained by the solution of the Bloch equation [45-49]... 

For the cluster expansion of the type (a2), we may thus conceive of two approaches. One is to invoke a single root strategy, and use either anonymous parentage or preferred parentage approximation. This approach by-passes the need for H, but by its very nature cannot generate a potentially exact formalism. The other is to use the multi—root strategy through the Bloch equation, and thereby produce a formally exact theory. We shall review these two types of schemes in Secs. 6.2 and 6.3 respectively. 

This behaviour is reminiscent of dynamical averaging seen in NMR spectroscopy, which is treated using a Bloch-equation analysis. Turner and coworkers were the first to use this type of analysis of dynamic infrared spectra to calculate rate constants and self-isomerisation barrier heights in the case of a turnstile -type exchange of CO ligands observed in trigonal bipyramidal [(q -diene)Fe(CO)3] complexes. In a similar fashion, rate constants for electron transfer in the ruthenium cluster dimers were calculated from the lineshapes of the (—1) states. As expected from the electrochemical data, the rate constant was fastest for the most electronically coupled dimer (1) and slowest for the... 

The Bloch Simulator is used primarily to calculate and visualize the basie effects of different types of pulses and small pulse sequence fragments using a purely elassical macroscopic magnetization approach. To simplify the calculations the simulator ignores the relaxation processes that formed part of the original Bloch equations. There are three main applications for the Bloch simulator ... 

For these and other reasons, much attention was given to the so-called state-selective or state-specific (SS) MR CC approaches. These are basically of two types (i) essentially SR CCSD methods that employ MR CC Ansatz to select a subset of important higher-than-pair clusters that are then incorporated either in a standard way [163,164], or implicitly [109-117], or via the so-called externally corrected (ec) approaches of either the amplimde [214-219] or energy [220,221] type, and (ii) those actually exploiting Bloch equations, but focusing on one state at a time [222]. The energy-correcting ec CC approaches [220,221] are in fact very closely related to the renormalized CCSD(T) method of Kowalski and Piecuch mentioned earlier [146,147]. 

Relaxation experiments were among the earliest applications of time-domain high-resolution NMR spectroscopy, invented more than 30 years ago by Ernst and Anderson [23]. The progress of the experimental methodology has been enormous and only some basic ideas of the experiment design will be presented here. This section is divided into three subsections. The first one deals with Bloch equation-type experiments, measuring and Tj when such quantities can be defined, i.e. when the relaxation is monoexponential. As a slightly oversimplified rule of thumb, we can say that this happens in the case of isolated spins. The two subsections to follow cover multiple-spin effects. 

In this context, it is worthwhile to recall the quantum jump approach developed in the quantum optics community. In this approach, an emission of a photon corresponds to a quantum jump from the excited to the ground state. For a molecule with two levels, this means that right after each emission event, = 0 (i.e., the system is in the ground state). Within the classical approach this type of wave function collapse never occurs. Instead, the emission event is described with the probability of emission per unit time being Fp (t), where Pee(0 is described by the stochastic Bloch equation. At least in principle, the quantum jump approach, also known as the Monte Carlo wave function approach [98-103], can be adapted to calculate the photon statistics of a SM in the presence of spectral diffusion. 

The solution of this two-equation system with two unknown coefficients is a fully constrained algebraic problem, which leads to a symmetric and an antisymmetric linear combination of Bloch functions, as a consequence of the equivalence of carbon atoms of types A and B. These solutions correspond to 7i-bonding and 7i-antibonding COs having the following form ... 

Quantitative MRI is possible by calculating the real T1 and T2 figures from the T1 and T2 weighted acquisitions, using the Bloch equation of MRI physics. Multi-modal MRI scans can be exploited for tissue classification when different MRI techniques are applied to the same volume, each voxel is measured with a different physical property, and a feature space can be constructed with the physical units along the dimensional axes e.g. in the characterization of tissue types in atherosclerotic lesions with Tl, T2 and proton density weighted acquisitions, fat pixels tend to cluster, as do blood voxels, muscle voxels, calcified voxels, etc., see Figure 9.8. 

In this part, we show how ultrashort pulses on resonant intersubband (IS) transitions propagate nonlinearly in multiple symmetric double quantum wells (QWs). A -type modulation-doped multiple QWs sample consisting of N are equally spaced electronically uncoupled symmetric double semiconductor GaAs/AlGaAs QWs with separation d, as shown in Fig. 13. There are only two lower energy subbands contribute to the system dynamics, W = 0 for the lowest subband with even parity and = 1 for the excited subband with odd parity. The Fermi level is below the n = subband minimum, so the excited subband is initially empty. This is succeeded by a proper choice of the electron sheet density. The nonlinear propagation of ultrashort pulses on resonant IS transitions in multiple semiconductor quantum wells is described by the fiill Maxwell-Bloch equations ... 

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