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

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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 $$2\pi / L_{\text x}$$.