Spin Quantum Number (ms)

The spin quantum number, denoted m_s, is the fourth and final quantum number needed to completely describe the quantum state of an electron in an atom. Unlike the principal (n), angular momentum (ℓ), and magnetic (m_ℓ) quantum numbers — which describe an electron's energy level, orbital shape, and orbital orientation respectively — the spin quantum number describes an intrinsic property of the electron itself: its spin angular momentum. This property has no classical analogue; electrons do not literally spin like tops, but they behave as though they carry a tiny bar magnet aligned either 'up' or 'down' relative to an external magnetic field. The spin quantum number can take only two possible values: +1/2 (spin-up, often represented by an upward arrow ↑) or −1/2 (spin-down, represented by a downward arrow ↓). This binary nature is a direct consequence of the electron being a fermion — a particle with half-integer spin — governed by the laws of quantum mechanics. The existence of electron spin was first proposed in 1925 by Samuel Goudsmit and George Uhlenbeck to explain the anomalous Zeeman effect, in which spectral lines of atoms split into more components in a magnetic field than classical theory predicted. The spin quantum number is fundamental to the Pauli Exclusion Principle, which states that no two electrons in the same atom can share the same set of all four quantum numbers (n, ℓ, m_ℓ, m_s). Because m_s can only be +1/2 or −1/2, each atomic orbital can hold a maximum of two electrons — one spin-up and one spin-down. This principle explains the entire structure of the periodic table: why electrons fill orbitals in the observed patterns, and why matter has the chemical properties it does. Electron spin also underpins ferromagnetism, electron paramagnetic resonance (EPR) spectroscopy, and MRI technology.

Key takeaways
  • What Is Electron Spin?
  • Allowed Values of m_s
  • The Pauli Exclusion Principle
  • Hund's Rule and Spin
  • Magnetic Properties from Spin

What Is Electron Spin?

Electron spin is an intrinsic form of angular momentum possessed by electrons. It is a purely quantum mechanical phenomenon with no classical counterpart. The spin angular momentum magnitude is given by |S| = ℏ√(s(s+1)), where s = 1/2 for an electron and ℏ is the reduced Planck constant (h/2π ≈ 1.055 × 10⁻³⁴ J·s). Because s is fixed at 1/2 for all electrons, the spin quantum number m_s describes only the component of spin along a reference axis (usually the z-axis defined by an external magnetic field), which equals either +ℏ/2 or −ℏ/2.

Allowed Values of m_s

The spin quantum number m_s has exactly two allowed values: +1/2 and −1/2. These are often called 'spin-up' (↑) and 'spin-down' (↓). Unlike the other quantum numbers, m_s does NOT depend on n, ℓ, or m_ℓ. Every single electron in every orbital of every atom must have m_s = +1/2 or m_s = −1/2. There is no middle ground or other possibility.

The Pauli Exclusion Principle

Wolfgang Pauli (1925) stated that no two electrons in an atom can have the same four quantum numbers (n, ℓ, m_ℓ, m_s). Since m_s has only two values, each orbital (defined by a unique set of n, ℓ, m_ℓ) holds at most two electrons, and they must have opposite spins. For example, the 1s orbital (n=1, ℓ=0, m_ℓ=0) can contain: electron 1 with m_s = +1/2 and electron 2 with m_s = −1/2. A third electron with any m_s would violate the exclusion principle.

Hund's Rule and Spin

When electrons fill orbitals of equal energy (degenerate orbitals, e.g., the three 2p orbitals), Hund's Rule states that electrons occupy separate orbitals with the same spin before pairing up. For nitrogen (N, Z=7), the three 2p electrons each occupy a separate 2p orbital, all with m_s = +1/2. This maximizes total spin and lowers the electron repulsion energy. The resulting half-filled subshell is unusually stable.

Magnetic Properties from Spin

Each spinning electron acts as a tiny magnetic dipole. The magnetic moment associated with electron spin is μ_s = −g_s·μ_B·m_s, where g_s ≈ 2.002 (the electron g-factor) and μ_B = eℏ/(2m_e) = 9.274 × 10⁻²⁴ J·T⁻¹ is the Bohr magneton. Atoms with unpaired electrons (net spin ≠ 0) are paramagnetic — attracted into a magnetic field. Atoms where all electrons are paired are diamagnetic — weakly repelled. For example, O₂ is paramagnetic (2 unpaired electrons), while N₂ is diamagnetic (all paired).

The Key Equation

ms=±12m_s = \pm\frac{1}{2}
  • m_s — Spin quantum number (dimensionless): Describes the z-component of intrinsic spin angular momentum of an electron; always either +1/2 or −1/2.
  • +\frac{1}{2} — Spin-up (dimensionless): Electron spin aligned with the reference magnetic field direction; symbolised by an upward arrow ↑.
  • -\frac{1}{2} — Spin-down (dimensionless): Electron spin opposed to the reference magnetic field direction; symbolised by a downward arrow ↓.

Worked Example

Problem. Write all four quantum numbers for each electron in a ground-state carbon (C, Z=6) atom, and identify whether the atom is paramagnetic or diamagnetic.

C: 1s2 2s2 2p2(2 unpaired electrons, paramagnetic)\text{C: } 1s^2\, 2s^2\, 2p^2 \quad (\text{2 unpaired electrons, paramagnetic})

Answer. Carbon has 2 unpaired electrons (in separate 2p orbitals, both m_s = +1/2) and is paramagnetic.

Figure
Diagram
Orbital Box Diagram for Carbon (C): 1s² 2s² 2p²Illustrating spin quantum numbers m_s = +½ (↑) and m_s = −½ (↓)1s↑↓m_s: +½ −½2s↑↓m_s: +½ −½2p (m_ℓ=−1)↑m_s: +½2p (m_ℓ=0)↑m_s: +½2p (m_ℓ=+1)—emptySpin Direction Visualisatione⁻↑ spin-upm_s = +½e⁻↓ spin-downm_s = −½Pauli RuleSame orbital:must have↑and↓opposite spinsMax 2e⁻/orbitalGreen boxes = unpaired electrons (Hund's Rule) | Blue/red boxes = paired electrons
Origins · 1 minSamuel Goudsmit and how this was discovered
SG
Samuel Goudsmit
1902–1978 · Dutch-American

In 1925, Samuel Goudsmit and George Uhlenbeck jointly proposed the concept of electron spin to explain the anomalous Zeeman effect — the splitting of atomic spectral lines into more components than classical theory predicted when placed in a magnetic field. They suggested that each electron possesses an intrinsic angular momentum of ℏ/2, with only two possible z-projections: +ℏ/2 and −ℏ/2. This idea was later incorporated by Paul Dirac into his relativistic quantum mechanics equation (1928), which derived electron spin naturally from first principles rather than postulating it. Goudsmit later led the Alsos Mission during World War II — a secret Allied scientific intelligence operation that followed troops into Germany to discover how far Nazi Germany had progressed in developing nuclear weapons.

  • Principal Quantum Number (n) — The principal quantum number is the first of the four quantum numbers and determines the energy level and size of an orbital; m_s completes the set.
  • Pauli Exclusion Principle — The Pauli Exclusion Principle directly governs the spin quantum number by requiring that no two electrons in the same atom share all four identical quantum numbers, limiting each orbital to two electrons with opposite spins.
  • Hund's Rule of Maximum Multiplicity — Hund's Rule determines how spin quantum numbers are assigned when electrons fill degenerate orbitals, explaining why electrons spread out with parallel spins before pairing.
  • Electron Configuration — Writing electron configurations requires understanding spin: each orbital holds exactly two electrons with opposite m_s values, which determines how many electrons each subshell can accommodate.
  • Paramagnetism and Diamagnetism — The magnetic properties of atoms and molecules arise directly from electron spin; unpaired electrons (m_s not cancelled) produce net magnetic moments making substances paramagnetic.