Empty lattice approximation: Difference between revisions

Revision as of 12:47, 25 December 2010

Revision as of 12:53, 25 December 2010

Line 1:

The '''Empty Lattice Approximation''' is a theoretical [[electronic band structure]] model in which the periodic potential of the crystal lattice is not defined more precisely than '''"periodic"''' and it is assumed that the potential is '''weak'''. This model mainly serves to illustrate a number of concepts which are fundamental to all electronic band structure phenomena.

==Introduction==

[[Image:Empty-Lattice-Approximation-FCC-bands.svg|thumb|300px|left|Free electron bands in a FCC crystal structure]]

[[Image:Empty-Lattice-Approximation-BCC-bands.svg|thumb|300px|right|Free electron bands in a BCC crystal structure]]

[[Image:Empty-Lattice-Approximation-HCP-bands.svg|thumb|500px|center|Free electron bands in a HCP crystal structure]]

ItThe potential must be weak because otherwise the electron wouldn't be free, but it is just strong enough to '''scatter''' the electrons. How strong must a potential be to be able to [[Scattering theory|scatter]] an electron? The answer is that it depends on the topology of the system how large topologically defined parameters, like [[Scattering cross-section|scattering]] [[Cross section (physics)|cross sections]], depend on the magnitude of the potential and the size of the [[potential well]]. One thing is clear for currently known 1, 2 and 3-dimensional spaces: '''potential wells do always scatter waves''' no matter how small their potentials are or what their sign is and how limited their sizes are. For a [[Particle in a one-dimensional lattice (periodic potential)|particle in a one-dimensional lattice]], like the Kronig-Penney model, it is easy to substitute the values for the potential and the size of the potential well.<ref name=Kittel>

{{cite book |author=C. Kittel |title=Introduction to Solid State Physics |year= 1953-1976 |publisher=Wiley & Sons |isbn=0-471-49024-5 }}

</ref>

Line 11:

:<math>E_n(\bold{k}) = \frac{\hbar^2 (\bold{k} + \bold{G_n})^2}{2m}</math>

and consists of a increasing number of free electron bands <math>E_n(\bold{k})</math> when the energy rises. <math>\bold{G}_n</math> is the [[reciprocal lattice]] vector to which the band <math>E_n(\bold{k})</math> belongs. Electrons with larger wave vectors outside the first [[Brillouin zone]] are mapped back into the first Brillouin zone by a so called [[Umklapp scattering|Umklapp process]].

[[Image:Empty-Lattice-Approximation-HCP-bands.svg|thumb|500px|center|Free electron bands in a HCP crystal structure]]

;Nearly free electron model