Top Qs
Timeline
Chat
Perspective
Born–von Karman boundary condition
Periodic boundary condition in solid-state physics From Wikipedia, the free encyclopedia
Remove ads
The Born–von Karman boundary condition requires the wave function to be periodic on a certain Bravais lattice. Named after Max Born and Theodore von Kármán, this periodic boundary condition is often applied in solid-state physics to model an ideal crystal. Born and von Kármán published a series of articles in 1912 and 1913 that presented this model of the specific heat of solids based on the crystalline hypothesis and included this boundary condition.[1][2] Historically, the Born-von Karman boundary condition is, like the Debye model, an improvement upon the Einstein model of solids, the first quantum theory of specific heats.[3]
The condition can be stated as
where i runs over the dimensions of the Bravais lattice, the ai are the primitive vectors of the lattice, and the Ni are integers (assuming the lattice has N cells where N=N1N2N3). This definition can be used to show that
for any lattice translation vector T such that:
Note, however, the Born–von Karman boundary conditions are useful when Ni are large (infinite).
The Born–von Karman boundary condition is important in solid-state physics for analyzing many features of crystals, such as diffraction and the band gap. Modeling the potential of a crystal as a periodic function with the Born–von Karman boundary condition and plugging in Schrödinger's equation results in a proof of Bloch's theorem, which is particularly important in understanding the band structure of crystals.
Remove ads
References
Wikiwand - on
Seamless Wikipedia browsing. On steroids.
Remove ads