Born–von Karman boundary conditions are periodic boundary conditions which impose the restriction that a

wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...

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 + n ...

. Named after Max Born and

Theodore von Kármán Theodore von Kármán ( hu, ( szőllőskislaki) Kármán Tódor ; born Tivadar Mihály Kármán; 11 May 18816 May 1963) was a Hungarian-American mathematician, aerospace engineer, and physicist who was active primarily in the fields of aeronaut ...

, this condition is often applied in solid state physics 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, macro ...

. Born and von Karman 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 and the

band gap In solid-state physics, a band gap, also called an energy gap, is an energy range in a solid where no electronic states can exist. In graphs of the electronic band structure of solids, the band gap generally refers to the energy difference ( ...

. 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 In condensed matter physics, Bloch's theorem states that solutions to the Schrödinger equation in a periodic potential take the form of a plane wave modulated by a periodic function. The theorem is named after the physicist Felix Bloch, who d ...

, which is particularly important in understanding the band structure of crystals. However, since any real crystal always has a finite size, the electronic states in the crystal do not satisfy the Born–von Karman boundary condition. Consequently, the conventional theory of electronic states in crystals based on the

has some fundamental difficulties.

References

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