Free electrons: from colliding balls to dancing waves

Representation of electrons in the Drude model — In the **Drude model**, electrons are like tiny balls bumping into the metal's atoms… but experimental measurements do not fit this interpretation.
Image source: astronoo.com (new window) - AI-generated image, public domain.

Scientific abstract

The article traces the evolution of the understanding of free electrons, the weakly bound valence electrons in metals. Drude's model (1900) described them as "classical balls" forming a gas striking the atoms (explains Ohm's law, but fails on specific heat). Sommerfeld's model (1927) applies quantum mechanics and Pauli's exclusion principle : only electrons close to the Fermi level (~1% at room temperature) take part in thermal phenomena, correcting the specific heat problem. Bloch's model (1928) introduces the periodic potential of the crystal lattice, giving birth to band theory, which explains the difference between conductors, semiconductors and insulators. The free electron is described there as a delocalized quantum entity, governed by the periodic structure of the crystal.

What are free electrons, and how has our understanding of them evolved from the classical ball to the quantum wave?

Free electrons are the valence electrons (the outer shell of atoms) which, in metals, are weakly bound to the nucleus and can move from one atom to another. Our understanding of them has gone through three major stages. The Drude model (1900) described them as "classical balls" moving as a gas and striking the atoms (explains Ohm's law, but fails on specific heat). The Sommerfeld model (1927) applied quantum mechanics and Pauli's exclusion principle : only electrons close to the Fermi level (~1% at room temperature) take part in thermal phenomena, correcting the specific heat problem. The Bloch model (1928) introduced the periodic potential of the crystal lattice, giving birth to band theory, which explains the difference between conductors, semiconductors and insulators.

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Why are free electrons free?

Electron shells

The electrons of an atom are distributed across successive electron shells, each of which can hold a maximum number of electrons (2, 8, 18, 32, 50, 72 electrons depending on the shell). The valence shell is the outer shell, the farthest from the nucleus. The electrons located there are less strongly bound to the nucleus by electrostatic attraction than those of the inner shells. Consequently, only a small amount of energy is needed to detach them from the atom or make them move from one atom to another.

The example of copper (Cu)

Copper has 29 protons and therefore 29 electrons, distributed as follows : 2 electrons on the first shell, 8 on the second, 18 on the third, i.e. 28 electrons strongly bound to the nucleus. The 29th electron sits alone on the fourth shell, much farther from the nucleus and therefore weakly bound. It is the electrons located on the outer shell (the valence shell) that become free electrons, and are therefore easily mobile.

Free electrons are not electrons in directed motion

In a metal, valence electrons are already free at room temperature. They move randomly and disorderly throughout the metal structure (Brownian motion). Their thermal velocity is very high (~10⁶ m/s) but has no preferred direction, so that the average net drift velocity is zero. There is therefore no electric current, because the movements cancel each other out statistically.

N.B.:
The potential difference (voltage) does not free the electrons (they are already free); it imposes a preferential direction on them.

Historical evolution of the understanding of free electrons

The table below summarizes the three major stages that marked the evolution of the understanding of free electrons in metals, from Drude's classical model to Bloch's quantum band theory.

Three major stages in the theory of free electrons in metals
Year	Scientist	Proposed model	Contributions and limitations
1900	Paul Drude	Classical free-electron model Valence electrons move freely like a gas of classical particles in the metal	Contributions: Explains Ohm's law and electrical conductivity Limitations: Incorrectly predicts the specific heat of metals and does not explain the variation of conductivity with temperature
1927	Arnold Sommerfeld	Quantum free-electron model Applies Fermi-Dirac quantum statistics to the free-electron gas	Contributions: Corrects the specific heat problem, explains why only electrons close to the Fermi level take part in thermal properties Limitations: Still ignores the periodic structure of the crystal lattice
1928	Felix Bloch	Band theory Electrons evolve in the periodic potential created by the ions of the crystal lattice, which represents the average effect of the electrostatic force exerted by the ions on the electrons	Contributions: Explains the difference between conductors, semiconductors and insulators thanks to allowed and forbidden energy bands Note: The current model used in solid-state physics

The Drude model (1900): Electrons as billiard balls

The mental image of billiard balls

Paul Karl Ludwig Drude (1863-1906) imagined free electrons as tiny solid balls moving through the metal like gas molecules. These "electronic balls" bounce off the atoms of the metallic lattice, a bit like billiard balls on a table strewn with fixed obstacles. A cloud of little balls flying in all directions inside the metal, colliding with the atoms and with each other, exactly like the molecules of a gas confined in a bottle.

What the Drude model explains

This model explains Ohm's law and basic electrical conductivity. When an electric field is applied, the balls drift slowly in a preferred direction while continuing their disorderly motion.

The limitations of the Drude model

The problem: if electrons really behaved like classical balls, metals should absorb far more heat than they actually do. The model also predicts poorly how electrical resistance varies with temperature.

The Sommerfeld model (1927): Electrons become quantum wave-particles

Pauli's exclusion principle

Arnold Johannes Wilhelm Sommerfeld (1868-1951) abandoned the image of classical balls and applied quantum mechanics to the electron gas. Electrons are no longer mere balls: they obey Pauli's exclusion principle, which states that two electrons cannot occupy exactly the same quantum state.

The mental image of the Fermi gas

Imagine a giant human pyramid where each person represents an electron and occupies a unique position (Pauli's principle). The lower levels are completely filled and frozen : these people cannot move without destabilizing the whole structure. Only the people on the upper levels, near the top (the Fermi energy), have the freedom of movement needed to change position. At room temperature, about 99% of electrons are "locked" at the lower levels, and only the last 1% at the top can actually take part in thermal and electrical phenomena.

What the Sommerfeld model explains

This model explains why the specific heat of metals is so low even though they contain billions of free electrons. Only a tiny fraction of the electrons (about 1% at room temperature) can absorb thermal energy. The model also explains thermal and electrical conductivity better.

The limitations of the Sommerfeld model

This model still treats the metal as an empty space where electrons move freely, completely ignoring the ordered and periodic presence of the atoms of the crystal lattice.

The Bloch model (1928): Electrons in a crystal lattice

The periodic potential of the crystal

Felix Bloch (1905-1983) revolutionized the classical view by considering that electrons do not move in an empty space, but in the periodic potential created by the atoms of the crystal lattice. Each electron therefore "swims" in an organized electrostatic field, which profoundly changes its motion and energy properties.

The mental image of waves

Imagine a rolling sea in which fish (the electrons) swim. The regular waves represent the periodic potential of the atoms. The fish cannot move just any way they like: they must follow the undulating structures of the sea, some zones being forbidden, others favorable to circulation. This image illustrates the formation of the allowed and forbidden energy bands in the metal.

What the Bloch model explains

This model explains electrical conduction and the electronic structure of solids, including the existence of conduction bands and forbidden bands. It accounts for many phenomena intertwined with quantum mechanics, such as the properties of semiconductors and the insulation of materials.

The limitations of the Bloch model

The model is complex to treat analytically, and often requires approximations or numerical calculations. It remains theoretical and abstract for directly visualizing the individual behavior of electrons.

The end of the little-ball picture

The delocalized electron

The free electron turns out to be a delocalized quantum entity, like a musical note that fills the entire concert hall, present everywhere simultaneously.

The periodic order of the lattice

Its behavior is entirely governed by the periodic order of the crystal lattice, where the atoms stack identically in the three dimensions of space.

References

FAQ: Everything about free electrons

Why are valence electrons "free" in metals ?

Electrons are distributed in electron shells around the nucleus. The outermost shell (the valence shell) is the farthest from the nucleus, so the electrons located there are weakly bound by electrostatic attraction. For example, copper (29 electrons) has 28 electrons strongly bound on the first three shells, and a single valence electron on the fourth shell. A small amount of energy is enough to detach it from the atom or to make it move from one atom to another. At room temperature, these electrons are already free and move randomly (Brownian motion) throughout the metal.

What is the difference between the Drude, Sommerfeld and Bloch models ?

The Drude model (1900) treats electrons as classical balls in gaseous motion, striking the atoms (explains Ohm's law, but fails on specific heat). The Sommerfeld model (1927) applies quantum mechanics and Pauli's exclusion principle : electrons are wave-particles and only those close to the Fermi level (about 1% at room temperature) can absorb thermal energy, correcting the specific heat problem. The Bloch model (1928) introduces the periodic potential of the crystal lattice : electrons evolve in an organized electrostatic field, which creates allowed and forbidden energy bands, explaining the difference between conductors, semiconductors and insulators.

Why don't free electrons produce an electric current all the time ?

At room temperature, free electrons move randomly and disorderly in the metal (Brownian motion). Their thermal velocity is very high (~10⁶ m/s), but with no preferred direction. The average net drift velocity is therefore zero : the movements cancel each other out statistically. There is therefore no electric current. A potential difference (voltage) does not free the electrons (they are already free); it collectively imposes a preferential direction on them, thereby creating a net electric current.

What is the Fermi level and what is its role ?

The Fermi level is the highest energy level occupied by electrons at a temperature of 0 K. In a metal, it lies inside the conduction band. At room temperature, only electrons whose energy is close to this level can be thermally excited and take part in conduction phenomena. The Fermi level plays a central role in understanding the electrical and thermal properties of solids, because it determines which electronic states are available for conduction.

How does band theory explain semiconductors ?

In a semiconductor, the valence band is completely filled and the conduction band is empty, but the energy gap between the two is relatively small (about 1 eV). A small amount of thermal energy or a moderate electric voltage can move electrons from the valence band to the conduction band, thereby creating an electric current. This intermediate behavior between conductors (where the bands overlap) and insulators (where the gap is very large) is the basis of modern electronics.

Can free electrons really be considered "free" ?

The term "free electron" is a historical simplification. In reality, even in metals, electrons are subject to the periodic potential of the crystal lattice and interact with each other. The Bloch model shows that they are not totally free but behave like delocalized waves whose energy is organized into bands. They are "free" in the sense that they can travel long distances without being bound to a particular atom, but their behavior is profoundly influenced by the crystalline environment.