Electron spin and the Pauli principle

The Bohr model solves the hydrogen atom and explains its spectral lines, but leaves open an enormous question: why does the periodic table have that shape? Why does helium behave as a noble gas and not as a metal? Why does oxygen, the eighth element, chemically resemble sulfur, which has sixteen electrons? The answer requires two new ingredients, both absent from the Bohr model: spin and the Pauli principle.

Spin, an “intrinsic” angular momentum

In 1922 Otto Stern and Walther Gerlach sent a beam of silver atoms through a strongly non-uniform magnetic field. The classical expectation was clear: the magnetic moments of the atoms (linked to the angular momentum of the electrons) should orient themselves continuously over a whole range of angles, so the beam should spread out on the screen into an elongated smear. Instead something surprising happened: the beam split into exactly two distinct beams, one deflected upwards and one downwards — never an intermediate one. It was as if every atom, along the direction of the field, could only take two values of magnetic moment.

Stern-Gerlach experiment (1922). The beam of silver atoms, passing through a non-uniform magnetic field, splits into only two components rather than spreading continuously: the component of angular momentum along the field axis is quantised.

The interpretation, due to Samuel Goudsmit and George Uhlenbeck (1925), is that the electron carries an intrinsic angular momentum — spin — independent of its orbital motion around the nucleus. The component of this angular momentum along an axis can take only two quantised values:

$\ev{s_z = \pm\frac{\hbar}{2}}$

often called spin up ($\uparrow$) and spin down ($\downarrow$). Beware of intuition here: spin is not a little sphere spinning on itself. It is a genuinely quantum property with no classical analogue. It does, however, produce a small magnetic moment, and it is precisely this moment that is deflected in the non-uniform Stern-Gerlach field.

Fermions and bosons

According to the spin-statistics theorem (Pauli, 1940), all the particles in the universe fall into two great families depending on the value of their spin:

• Fermions — half-integer spin ($\hbar/2,\ 3\hbar/2,\dots$). Examples: electron, proton, neutron, quark, neutrino. They are the building blocks that make up matter.
• Bosons — integer spin ($0,\ \hbar,\ 2\hbar,\dots$). Examples: photon, gluon, Higgs boson. They mediate the fundamental interactions.

The distinction is not a taxonomic detail: it determines how particles can share quantum states, and it is the root of the principle that follows.

The Pauli exclusion principle

Principle — Pauli exclusion (1925)

Two identical fermions cannot simultaneously occupy the same quantum state. For an electron in an atom, the state is specified by four quantum numbers $(n,\ell,m,s_z)$: there cannot exist two electrons with all four numbers equal.

Key formula

Electron spin: $s_z = \pm\hbar/2$. Pauli: no two identical fermions in the same state of an atom. Bosons (e.g. photons): no exclusion, they can pile up as many as desired in the same state.

The periodic table as a consequence

Combining the Pauli principle with the atomic model builds up the electron shells. Each level $n$ of the Bohr model admits at most $2n^2$ electrons — two for each orbital, one with $s_z=+\hbar/2$ and one with $s_z=-\hbar/2$:

• $n=1$ (shell K): 2 electrons. Filled at helium (He).
• $n=2$ (shell L): 8 electrons. Filled at neon (Ne).
• $n=3$ (shell M): 18 electrons in principle, but in practice filled in stages because of the structure of the $s,p,d$ subshells. (Partially) completed at argon (Ar).

When a shell is full, the atom is particularly stable and unreactive: these are the noble gases. When a shell has just one electron too many (alkali metals: Li, Na, K) or just one electron too few (halogens: F, Cl, Br), chemistry becomes very active. Mendeleev had already observed the periodicity in 1869, but its explanation only arrived in 1925 with Pauli (Simonyi 2012).

Curiosity — Without Pauli none of this

If electrons were not fermions, all the electrons of an atom would fall into the ground state $n=1$. All atoms would be indistinguishable in their chemical behaviour: no chemistry, no bonds, no molecules, no life. The Pauli principle is what prevents matter from “collapsing in on itself”: the degeneracy pressure that comes with it is what holds up white dwarfs and stops neutron stars from turning into black holes.

In summary: spin is an intrinsic quantum angular momentum, with $s_z=\pm\hbar/2$ for electrons. Fermions (half-integer spin, subject to Pauli exclusion) make up matter; bosons (integer spin) mediate interactions. The Pauli principle is the key that explains the shape of the periodic table.

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Topics: Quantum physics Concepts: Spin and the Pauli principle · Angular momentum · Bohr model

Related exercises: Problem — Speculative physics: a universe without Pauli · Hydrogen levels, Balmer and Lyman · Derive the radius of the Bohr orbit