Bloch dispersion relation

In disordered metals, the electrons couple with the static structure at any. K-value and are heavily damped close to pseudo Brillouin-zone boundaries. A definite dispersion relation does not exist (Fig. 5.3bl) and Bloch s theorem is no longer valid. Electronic states cannot be described as eigenstates of the system and, strictly speaking, cannot be translated to the reduced-zone scheme. 

The effect of inter-site hopping is then introduced into the system. The manifold of basis states are limited to those in which the local correlations have been diagonalized. The wave functions for the composite particles then obey Bloch s theorem, which results in the formation of a dispersion relation consisting of two bands for the quasi-bosons the first band describes spinless quasi-boson excitations, the second band describes the magnetic quasi-bosons. Although these composite particles are bosons in that they commute on different sites, they nevertheless have local occupation numbers which are Fermi-Dirac like. 

The dispersion relation (Eq. (5.34)) yields two bands, which for an infinite ( bulk ) crystal are displayed by the thick lines in the left part of Eigure 5.12. As can be seen, an energy gap of width 2Vg opens up at the Brillouin zone boundary k = f Note that for an infinite crystal the wave vector k has to be real, since otherwise the Bloch wave function F(z) = e" Ut(z) would diverge exponentially for either z -r- +00 or z —00. 

In an NMR experiment the resonance signal contains components in-phase and out-of-phase with the incident radio frequency radiation so that a complex susceptibility x can usefully be defined, such that x = x x"- The dispersion component x is in-phase, and the absorption component x is out-of-phase. By introducing phenomenologically the exponential decay constants Ti and T2 for the nuclear magnetization parallel and perpendicular to the applied field Ho, Bloch et al. (1946) derived expressions for the magnetization as a function of frequency. These may be related to expressions for x and x", the results being ... 

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