Bloch

A1.3.4 ELECTRONIC STATES IN PERIODIC POTENTIALS BLOCH S THEOREM... 

The periodic nature of crystalline matter can be utilized to construct wavefunctions which reflect the translational synnnetry. Wavefiinctions so constructed are called Bloch functions [1]. These fiinctions greatly simplify the electronic structure problem and are applicable to any periodic system. 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In tln-ee dimensions, four quantum numbers are required to characterize an eigenstate. In spherically syimnetric atoms, the numbers correspond to n, /, m., s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent... 

By taking the [Pg.101]

One of the first models to describe electronic states in a periodic potential was the Kronig-Penney model [1]. This model is commonly used to illustrate the fundamental features of Bloch s theorem and solutions of the Schrodinger... 

Equation (A1.6.64) describes the relaxation to equilibrium of a two-level system in tenns of a vector equation. It is the analogue of tire Bloch equation, originally developed for magnetic resonance, in the optical regime and hence is called the optical Bloch equation. 

Figure Al.6.32. (a) Initial and (b) final population distributions corresponding to cooling, (c) Geometrical interpretation of cooling. The density matrix is represented as a point on generalized Bloch sphere of radius R...

Comparing this with the Bloch equation establishes a correspondence between t and /p/j. Putting t = ip/j, one finds... 

To evaluate the density matrix at high temperature, we return to the Bloch equation, which for a free particle (V(x) = 0) reads... 

The solution to this is a Gaussian function, which spreads out in time. Hence the solution to the Bloch equation for a free particle is also a Gaussian ... 

Ulness D J and Albrecht A C 1996 Four-wave mixing in a Bloch two-level system with incoherent laser light having a Lorentzian spectral density analytic solution and a diagrammatic approach Rhys. Rev. A 53 1081-95... 

The concept of relaxation time was introduced to the vocabulary of NMR in 1946 by Bloch in his famous equations of motion for nuclear magnetization vector M [1] ... 

The spin-spin relaxation time, T, defined in the Bloch equations, is simply related to the width Av 2 Lorentzian line at the half-height T. Thus, it is in principle possible to detennine by measuring... 

Szymanski S, Gryff-Keller A M and Binsch G A 1986 Liouville space formulation of Wangsness-Bloch-Redfield theory of nuclear spin irelaxation suitable for machine computation. I. Fundamental aspects J. Magn. Reson. 68 399-432... 

The classical description of magnetic resonance suffices for understanding the most important concepts of magnetic resonance imaging. The description is based upon the Bloch equation, which, in the absence of relaxation, may be written as... 

The Bloch equation is simplified, and the experiment more readily understood, by transfonnation into a frame of reference rotating at the frequency ciDq=X Bq about die z-axis whereupon ... 

For example, if the molecular structure of one or both members of the RP is unknown, the hyperfine coupling constants and -factors can be measured from the spectrum and used to characterize them, in a fashion similar to steady-state EPR. Sometimes there is a marked difference in spin relaxation times between two radicals, and this can be measured by collecting the time dependence of the CIDEP signal and fitting it to a kinetic model using modified Bloch equations [64]. 

With this definition, the Bloch equations can be written as in equation (B2.4.4)). 

In chemical exchange, tire two exchanging sites, A and B, will have different Lannor frequencies, and cOg. Assuming equal populations in the two sites, and the rate of exchange to be k, the two coupled Bloch equations for the two sites are given by equation (B2.4.5)). 

This Liouville-space equation of motion is exactly the time-domain Bloch equations approach used in equation (B2.4.13). The magnetizations are arrayed in a vector, and anything that happens to them is represented by a matrix. In frequency units (1i/2ti = 1), the fomial solution to equation (B2.4.26) is given by equation (B2.4.27) (compare equation (B2.4.14H. 

The Bloch equation approach (equation (B2.4.6)) calculates the spectrum directly, as the portion of the spectrum that is linear in a observing field. Binsch generalized this for a frilly coupled system, using an exact density-matrix approach in Liouville space. His expression for the spectrum is given by equation (B2.4.42). Note that this is fomially the Fourier transfomi of equation (B2.4.32). so the time domain and frequency domain are coimected as usual. 

Reeves L W and Shaw K N 1970 Nuclear magnetic resonance studies of multi-site chemical exchange. I. Matrix formulation of the Bloch equations Can. J. Chem. 48 3641-53... 

We wish to construct linear combinations of the atomic orbitals such that the overall wavefunction meets the Bloch requirement. Suppose the s orbitals in our lattice are labelled X , where the wth orbital is located at position x = na. An acceptable linear combination of these orbitals that satisfies the Bloch requirements is ... 

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