Illustration Allowed k-states of free electrons in a 2d-lattice in reciprocal space

A two-dimensional coordinate system whose axes are the wavenumbers \(k_{\text y}\) and \(k_{\text x}\). The allowed wavenumber states (dots), are states that a free electron can occupy in a 2d crystal (with dimensions \(L_{\text y}\) and \(L_{\text x}\)). These discrete states of electrons result from solving the Schrödinger equation for a free electron gas with periodic boundary conditions.

Each state (represented by the dot) occupies a reciprocal area \( \mathcal{A_1} \) (in 3D: reciprocal volume and in 1D: reciprocal length) given by the distances to the neighboring states, i.e. in y-direction: \(2\pi / L_{\text y}\) and in x-direction \( 2\pi / L_{\text x} \).