The Fermi energy in metals is much higher than the thermal energy at typical temperatures. Why does this make the mean speed of conduction electrons nearly independent of temperature?

Thermal energy is a form of kinetic energy present in the particles that make up all forms of matter. At an atomic level, it's the energy possessed due to the vibration and movement of particles. This form of energy is directly related to temperature, an observable measure of the average kinetic energy among particles in a substance.

In the metal, when we provide heat, the atoms in the solid lattice vibrate more rapidly, because an increase in temperature boosts the kinetic energy of these particles. As a consequence, even electrons, sometimes called conduction electrons when talking about metals, absorb some of this thermal energy, increasing their kinetic energy.

Normally, you might expect that if electrons gain energy, they would move faster, and hence the mean speed of all the electrons would change with temperature. However, in the context of Fermi energy, thermal energy at everyday temperatures is relatively small, and hence most conduction electrons do not gain enough energy to significantly alter their average speed, maintaining the mean speed of conduction electrons nearly independent of temperature.

This understanding underscores the relative insensitivity of metallic conduction properties to temperature changes, which is a crucial concept for various applications in electronics and materials engineering.

Conduction electrons are the electrons available for electrical conduction in a metal. These are often described using a 'sea of electrons' model, where the conduction electrons are not bound to individual atoms but move freely throughout the metal, contributing to its ability to conduct electricity.

Metals have a structure of closely packed atoms with their outer electrons squandering their individual loyalties to atoms and becoming delocalized. This delocalization is necessary for the high conductivity properties of metals. Conduction electrons can be thought of as the 'fluid' that carries electric charge through the solid structure of positively charged ion cores.

Even though thermal energy gives these electrons additional kinetic energy, their overall behavior in metal is largely governed by the quantum mechanical properties of the material, specifically the distribution of energy states available for them to occupy, which is described by Fermi-Dirac statistics. Hence, the structure and properties of energy states in solids are crucial in dictating the response of conduction electrons to thermal energy inputs.

In solid-state physics, the energy states available to electrons are quantized and are dictated by the unique quantum mechanical nature of solids. The arrangement of these energy levels, particularly in metals, is what leads to the concept of bands, including the conduction and valence bands.

The Fermi energy is the highest energy level that electrons occupy at absolute zero temperature. Above this energy, states are empty, and below it, they are filled. The distribution of electrons in these states at finite temperatures follows what is known as the Fermi-Dirac distribution.

When a metal is heated, only electrons near the Fermi surface—those at the highest occupied energy levels—have a significant probability of being excited to higher energy levels. At normal temperatures, these thermally excited electrons are very few compared to the total number of conduction electrons, due to the sheer number of available states near the Fermi level. As a result, changes in their average speed, due to increased thermal energy, are minimal. Consequently, the overall electrical conductivity remains nearly unchanged with temperature variation. This insight is vital for predicting the behavior of metals in electrical circuits and understanding their thermal properties.