Isospin

Definition

Isospin (historically "isotopic spin" or "isobaric spin") is an internal quantum number introduced to express the near-symmetry between the proton and the neutron. The strong force that binds nuclei acts almost identically on both, so it is natural to regard them as two states of a single object — the nucleon — that differ only by the orientation of an abstract internal vector called isospin.

The nucleon is assigned isospin I = 1/2. Its third component I₃ (sometimes written T₃) takes two values:

proton  = |I = 1/2, I₃ = +1/2⟩
neutron = |I = 1/2, I₃ = -1/2⟩

These two states form an isospin doublet, exactly analogous to the spin-up and spin-down states of a spin-½ particle. The mathematics is the SU(2) group — the same one that describes ordinary quantum angular momentum — but the "rotations" happen in an abstract internal space, not in physical space.

The SU(2) algebra of isospin

Isospin operators obey the SU(2) Lie algebra. Writing the three generators as I₁, I₂, I₃:

[I_i, I_j] = i ε_ijk I_k

where ε_ijk is the Levi-Civita symbol. This is identical in form to the angular-momentum algebra [J_i, J_j] = iħ ε_ijk J_k, except that isospin is dimensionless (no factor of ħ). The eigenvalues follow the same rules:

I² |I, I₃⟩ = I(I+1) |I, I₃⟩
I₃ |I, I₃⟩ = I₃ |I, I₃⟩       with I₃ = -I, -I+1, ..., +I

A multiplet with total isospin I therefore contains 2I + 1 states — a doublet for I = 1/2, a triplet for I = 1, a quadruplet for I = 3/2, and so on. The nucleon representation is written in terms of Pauli matrices τ (the isospin analogues of the spin matrices σ), with I = τ/2. The raising and lowering operators I₊ and I₋ turn a neutron into a proton and vice versa:

I₊ |neutron⟩ = |proton⟩,   I₋ |proton⟩ = |neutron⟩

Charge, hypercharge, and the Gell-Mann–Nishijima formula

Electric charge is not independent of isospin — it is tied to I₃ through the Gell-Mann–Nishijima relation:

Q = I₃ + Y/2

where Q is charge in units of the elementary charge e = 1.602 × 10⁻¹⁹ C, I₃ is the third isospin component, and Y is the hypercharge, Y = B + S (baryon number plus strangeness, extended to Y = B + S + C + B′ + T when heavy flavours are included). For the nucleon doublet, Y = 1:

proton:  Q = +1/2 + 1/2 = +1
neutron: Q = -1/2 + 1/2 =  0

This formula was central to organizing the "particle zoo" of the 1950s before quarks were understood. Today it emerges naturally from the quark model: the up quark carries I₃ = +1/2, the down quark carries I₃ = -1/2, and their charges +2/3 e and -1/3 e follow directly from Q = I₃ + Y/2 with Y = 1/3 for the light quarks.

Isospin multiplets

Particles that transform into one another under the strong force cluster into multiplets of nearly equal mass. The members differ mainly in electric charge, and their small mass splittings are the fingerprint of isospin breaking.

Multiplet	Isospin I	Members (I₃)	Approx. mass (MeV/c²)
Nucleon	1/2	p (+1/2), n (-1/2)	938.27 / 939.57
Pion	1	π⁺ (+1), π⁰ (0), π⁻ (-1)	139.57 / 135.0 / 139.57
Kaon	1/2	K⁺ (+1/2), K⁰ (-1/2)	493.68 / 497.61
Delta baryon	3/2	Δ⁺⁺, Δ⁺, Δ⁰, Δ⁻	~1232 (all four)
Rho meson	1	ρ⁺, ρ⁰, ρ⁻	~775 (all three)
Sigma baryon	1	Σ⁺, Σ⁰, Σ⁻	1189 / 1193 / 1197
Xi baryon	1/2	Ξ⁰, Ξ⁻	1315 / 1322

The Δ(1232) is the classic I = 3/2 case: four charge states (Δ⁺⁺, Δ⁺, Δ⁰, Δ⁻) with 2I + 1 = 4 members, all at essentially the same mass. The proton–neutron splitting of just 1.293 MeV against a ~938 MeV mass is the ~0.14% breaking that makes isospin such a good approximate symmetry.

Charge independence of the strong force

The physical content of isospin symmetry is charge independence: the strong force between two nucleons depends only on their total isospin state, not on whether each is a proton or a neutron. The proton–proton, proton–neutron, and neutron–neutron strong interactions are the same once you remove Coulomb repulsion. This is seen directly in nuclei: mirror nuclei, which have their proton and neutron numbers swapped (for example ³H with one proton and two neutrons versus ³He with two protons and one neutron), have almost identical nuclear structure and binding energies once the Coulomb energy is subtracted.

Isospin conservation also constrains reaction rates. Because the strong force conserves total isospin, cross-section ratios for related reactions are fixed by Clebsch–Gordan coefficients. For pion–nucleon scattering near the Δ(1232) resonance, where the intermediate state has I = 3/2, isospin predicts:

σ(π⁺p → π⁺p) : σ(π⁻p → π⁰n) : σ(π⁻p → π⁻p) = 9 : 2 : 1

This 9:2:1 ratio is confirmed experimentally to good accuracy — a striking success of isospin symmetry that requires no detailed knowledge of the strong dynamics.

Worked example — combining two nucleons

Two nucleons, each with I = 1/2, combine exactly as two spin-½ particles do. The Clebsch–Gordan decomposition is:

1/2 ⊗ 1/2 = 1 ⊕ 0

So a two-nucleon system is either an isospin triplet (I = 1, symmetric) or an isospin singlet (I = 0, antisymmetric):

I = 1 (triplet):  |1, +1⟩ = pp
                  |1,  0⟩ = (pn + np)/√2
                  |1, -1⟩ = nn

I = 0 (singlet):  |0,  0⟩ = (pn - np)/√2

This has a real physical consequence. The deuteron — the bound state of a proton and a neutron — exists in the I = 0 singlet channel. There is no bound diproton (pp), no bound dineutron (nn), and no bound I = 1 pn state, because the strong force is slightly less attractive in the I = 1 channel and cannot quite form a bound state there. Isospin symmetry demands that if the I = 1 pn combination were bound, so would pp and nn be — and none of them are, consistent with only the I = 0 deuteron existing.

Isospin in the quark era

The modern understanding is that isospin is the approximate symmetry of quantum chromodynamics (QCD) under interchange of the up and down quarks. It would be exact if m_u = m_d and if electromagnetism were switched off. The measured current-quark masses are small and close — roughly m_u ≈ 2.2 MeV/c² and m_d ≈ 4.7 MeV/c² — and both are tiny compared with the QCD scale Λ_QCD ≈ 200 MeV. Because the mass difference and the electromagnetic coupling are small on the hadronic scale, up ↔ down interchange is a near-perfect symmetry of the strong interaction.

In this picture the nucleon doublet is just the doublet of light quarks lifted to the composite level: the proton (uud) and neutron (udd) differ by swapping one up quark for a down quark, exactly the action of the isospin lowering operator. Isospin is the SU(2) subgroup of the larger approximate SU(3) flavour symmetry that also includes the strange quark; SU(3) is more badly broken because the strange quark is heavier (m_s ≈ 95 MeV/c²).

A note on weak isospin

The electroweak theory borrows the same SU(2) mathematics under the name weak isospin, but it is a distinct concept. Weak isospin is an exact (though spontaneously broken) gauge symmetry that groups left-handed fermions into doublets — for instance (ν_e, e⁻)_L and (u, d)_L — and its gauge bosons are the W⁺, W⁻, and Z. Ordinary (strong) isospin, by contrast, is only an approximate global symmetry of the strong force with no gauge bosons of its own. They are mathematically identical SU(2) structures acting in completely different physical arenas; conflating them is a common source of confusion.

Common mistakes

• Thinking isospin is a real spin. Isospin is not angular momentum and carries no units of ħ. Nothing is physically rotating in space; the "rotation" is in an abstract internal charge space. Only the algebra is shared with spin.
• Assuming isospin is exact. It is broken at the ~1% level by the up–down mass difference and by electromagnetism. The proton–neutron mass splitting of 1.293 MeV is a direct measure of this breaking.
• Confusing strong isospin with weak isospin. They use the same SU(2) mathematics but are different symmetries — one an approximate global symmetry of the strong force, the other the gauge symmetry of the electroweak interaction.
• Applying isospin conservation to electromagnetic or weak processes. Only the strong interaction (approximately) conserves isospin. Electromagnetic decays like π⁰ → 2γ and all weak decays violate it.
• Forgetting the 2I + 1 counting. A multiplet of isospin I always has exactly 2I + 1 members. Three pions ⇒ I = 1; four Deltas ⇒ I = 3/2. Miscounting the members means the wrong I.
• Getting the sign convention backwards. By the standard particle-physics convention the proton is I₃ = +1/2 and the neutron I₃ = -1/2 (up quark +1/2, down quark -1/2). Some older nuclear-physics texts use the opposite sign for nucleons; always state the convention.