freeelectronmodell

The free electron model, also known as the free electron gas model, is a simple quantum mechanical description of conduction electrons in metals. It treats valence electrons as a degenerate Fermi gas of non-interacting particles moving in a uniform positive background charge, neglecting the lattice potential and electron–electron interactions. It provides a starting point for understanding many metallic properties.

In this model the electrons are confined in a box with periodic boundary conditions. Energies are Ek = ħ^2 k^2 / 2m, with k-space quantized by the box size. At zero temperature, electrons fill states up to the Fermi energy EF. In three dimensions, the density of states g(E) ∝ sqrt(E) and EF ∝ n^(2/3), where n is the electron density. The Fermi surface is a sphere in k-space. The model predicts a linear electronic contribution to the low-temperature heat capacity and offers a qualitative explanation for metallic conductivity.

Limitations include the neglect of the lattice periodic potential, which in real metals creates band structure with allowed and forbidden energy regions. Electron–electron interactions are ignored, limiting accuracy for many properties and materials, especially transition metals and insulators. Extensions such as the nearly free electron model and Bloch's theorem incorporate lattice periodicity and yield more accurate band structures. The jellium model treats the positive ion cores as a uniform background, further refining the framework.

Despite its simplifications, the free electron model remains foundational for understanding simple metals, helping to estimate Fermi energies, densities of states, and basic electronic behavior, and it underpins more advanced band-structure theories.