The Band Description of Electrons in Narrow Bands

The reader shall be familiar with the concepts of band-theory (which are illustrated in excellent textbooks ) such as the band dispersion E(k) and the Bloch-state solutions of the equation of motion 

If V(R) is small with respect to T in (11) the periodic potential V(R) of the ionic cores is that of a lattice of scattering centres. The electronic excitation, described by is a scattered electron wave of high kinetic energy, traveling through the crystal. Many 

1 For the non-relativistic case (Schrbdinger equation), T = -V. For relativistic case (Dirac equation), T = c a p + 3mc where m is the rest mass of the electron, c is the velocity of light. We have preferred to write the T operator in a general form, covering both cases, given the importance of the relativistic approach in band calculations for actinide solids - see Chap. F 

Narrow bands arise when the overlap of the atomic wave functions is small (as for 5 f s). In this case, the dispersion E(k) is strongly reduced and the bandwidth W becomes very small (zero, in the case of no overlap). The electron charge density, caused by these wave functions, is high in the core region of Fig. 12, and the quasi-particles spend most of their life there, nearly bound to the atom. In case the charge density is all confined within the core region (as for 4f in lanthanides), then the bond description loses its meaning and the atomic description holds. 

Here we shall briefly discuss i. the of (11) ii. the parameters allowing to choose between a band or an atomic description. 

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