The quantum free electron theory, also known as the free electron model or the free electron gas model, is a simplified theoretical model used to describe the behavior of electrons in a solid material. It is an approximation that helps explain certain electronic properties of metals.
According to the quantum free electron theory, electrons in a metal are considered to be free and independent particles that move within a rigid crystal lattice. This model assumes that the crystal lattice structure has a regular arrangement of positive ions, and the negatively charged electrons move in the interstitial spaces between these ions.
Key features of the quantum free electron theory include:
Independent electrons: Each electron is treated as an independent entity and does not interact with other electrons or the lattice structure, except through collisions.
No potential barrier: The model assumes that the potential energy barrier for the electrons moving within the crystal lattice is negligible. In other words, the electrons are free to move throughout the material without significant resistance.
Constant potential: The model assumes a constant positive potential due to the stationary positive ions in the lattice. This potential is responsible for confining the electrons within the crystal.
Energy quantization: The energy levels of the electrons are quantized, meaning they can only occupy specific energy states within the crystal. These energy states form a continuous band known as the energy band or energy spectrum.
Fermi-Dirac statistics: The model utilizes Fermi-Dirac statistics, which describe the behavior of particles with half-integer spin, such as electrons. This statistical distribution determines the probability of an electron occupying a particular energy state at a given temperature.
The quantum free electron theory helps explain several electronic properties of metals, including electrical conductivity, thermal conductivity, and electronic specific heat. However, it is a highly simplified model and does not account for many important factors, such as electron-electron interactions, electron-lattice interactions, and the presence of energy bands and band gaps in real materials. More advanced models, such as the band theory and density functional theory, provide a more accurate description of electron behavior in solids.