Physics:Born–von Karman boundary condition
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.
References
- ↑ von Kármán, Theodore; Born, Max (1912-04-15). "Über Schwingungen in Raumgittern" (in de). Physikalische Zeitschrift 13 (8): 297-309. https://hdl.handle.net/2027/mdp.39015023176806?urlappend=%3Bseq=365%3Bownerid=13510798901083269-399.
- ↑ von Karman, Theodore; Born, Max (1913-01-01). "Zur Theorie der spezifischen Wärme" (in de). Physikalische Zeitschrift 14 (1): 15-19. https://hdl.handle.net/2027/mdp.39015021268936?urlappend=%3Bseq=49%3Bownerid=13510798901066996-55.
- ↑ Jammer, Max (1966). The Conceptual Development of Quantum Mechanics. McGraw-Hill. pp. 58-9.
- Ashcroft, Neil W.; Mermin, N. David (1976). Solid state physics. New York: Holt, Rinehart and Winston. pp. 135. ISBN 978-0-03-083993-1. https://archive.org/details/solidstatephysic00ashc/page/135.
- Leighton, Robert B. (1948). "The Vibrational Spectrum and Specific Heat of a Face-Centered Cubic Crystal". Reviews of Modern Physics 20 (1): 165–174. doi:10.1103/RevModPhys.20.165. Bibcode: 1948RvMP...20..165L. https://authors.library.caltech.edu/47755/1/LEIrmp48.pdf.
- Ren, Shang Yuan (2017). Electronic States in Crystals of Finite Size: Quantum Confinement of Bloch Waves. (2 ed.). Singapore: Springer.
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