Born–von Karman boundary conditions are

periodic boundary conditions Periodic boundary conditions (PBCs) are a set of boundary conditions which are often chosen for approximating a large (infinite) system by using a small part called a ''unit cell''. PBCs are often used in computer simulations and mathematical mod ...

which impose the restriction that a

wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...

must be periodic on a certain

Bravais lattice In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by : \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 ...

. Named after

Max Born Max Born (; 11 December 1882 – 5 January 1970) was a German-British theoretical physicist who was instrumental in the development of quantum mechanics. He also made contributions to solid-state physics and optics, and supervised the work of a ...

and Theodore von Kármán, this condition is often applied in

solid state physics Solid-state physics is the study of rigid matter, or solids, through methods such as solid-state chemistry, quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state p ...

to model an ideal

crystal A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macros ...

. Born and von Kármán published a series of articles in 1912 and 1913 that presented one of the first theories of specific heat of solids based on the crystalline hypothesis and included these boundary conditions. The condition can be stated as :

\psi(\mathbf+N_i \mathbf_i)=\psi(\mathbf), \,

where ''i'' runs over the dimensions of the

, the a_''i'' are the primitive vectors of the lattice, and the ''N_i'' are integers (assuming the lattice has ''N'' cells where ''N=N₁N₂N₃''). This definition can be used to show that :

\psi(\mathbf+\mathbf)=\psi(\mathbf)

for any lattice translation vector T such that: :

\mathbf = \sum_i N_i \mathbf_i.

Note, however, the Born–von Karman boundary conditions are useful when ''N_i'' are large (infinite). The Born–von Karman boundary condition is important in solid state physics for analyzing many features of crystals, such as

diffraction Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...

and the

band gap In solid-state physics and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to t ...

. Modeling the

potential Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...

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

* * * Condensed matter physics Boundary conditions Max Born {{CMP-stub