Empty lattice approximation: Difference between revisions

Revision as of 23:43, 5 September 2018

Revision as of 21:32, 14 November 2018

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In theory the lattice is infinitely large, so a weak periodic scattering potential will eventually be strong enough to reflect the wave. The scattering process results in the well known [[Bragg's law|Bragg reflections]] of electrons by the periodic potential of the [[crystal structure]]. This is the origin of the periodicity of the dispersion relation and the division of [[Reciprocal lattice|k-space]] in Brillouin zones. The periodic energy dispersion relation is expressed

as:

:<math>E_n(\boldmathbf{k}) = \frac{\hbar^2 (\boldmathbf{k} + \boldmathbf{G}_n)^2}{2m}</math>

The <math>\boldmathbf{G}_n</math> are the [[reciprocal lattice]] vectors to which the bands{{clarify|date=November 2014}} <math>E_n(\boldmathbf{k})</math> belong.

The figure on the right shows the dispersion relation for three periods in reciprocal space of a one-dimensional lattice with lattice cells of length ''a''.

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==The energy bands and the density of states==

In a one-dimensional lattice the number of reciprocal lattice vectors <math>\boldmathbf{G}_n</math> that determine the bands in an energy interval is limited to two when the energy rises. In two and three dimensional lattices the number of reciprocal lattice vectors that determine the free electron bands <math>E_n(\boldmathbf{k})</math> increases more rapidly when the length of the wave vector increases and the energy rises. This is because the number of reciprocal lattice vectors <math>\boldmathbf{G}_n</math> that lie in an interval <math>[\boldmathbf{k},\boldmathbf{k} + d\boldmathbf{k}]</math> increases. The [[density of states]] in an energy interval <math>[E,E + dE]</math> depends on the number of states in an interval <math>[\boldmathbf{k},\boldmathbf{k} + d\boldmathbf{k}]</math> in reciprocal space and the slope of the dispersion relation <math>E_n(\boldmathbf{k})</math>.

[[Image:Free-electron DOS.svg|thumb|300px|right|Figure 3: Free-electron DOS in 3-dimensional k-space]]

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==Second, third and higher Brillouin zones==

[[Image:Brillouin Zone (1st, FCC).svg|thumb|300px|right|FCC Brillouin zone]]

"Free electrons" that move through the lattice of a solid with wave vectors <math>\boldmathbf{k}</math> far outside the first Brillouin zone are still reflected back into the first Brillouin zone. See the [[#External links|external links]] section for sites with examples and figures.

{{clear}}

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:<math>V(r) = \frac{Z e}{r} e^{-q r}</math>

where ''Z'' is the [[atomic number]], ''e'' is the elementary unit charge, ''r'' is the distance to the nucleus of the embedded ion and ''q'' is a screening parameter that determines the range of the potential. The [[Fourier transform]], <math>U_{\boldmathbf{G}}</math>, of the lattice potential, <math>V(\boldmathbf{r})</math>, is expressed as

:<math>U_{\boldmathbf{G}} = \frac{4 \pi Z e}{q^2 + \boldmathbf{G}^2}</math>

When the values of the off-diagonal elements <math>U_{\boldmathbf{G}}</math> between the reciprocal lattice vectors in the Hamiltonian almost go to zero. As a result, the magnitude of the band gap <math>2|U_{\boldmathbf{G}}|</math> collapses and the empty lattice approximation is obtained.

==The electron bands of common metal crystals==