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advanced30 min studyLesson 574 in Physics

Isospin and charge symmetry

Organize proton-neutron states by approximate internal symmetry without identifying it with spatial spin or exact mass degeneracy.

Before you begin
On this page

The question for today

What you’ll understand

  • State a consistent neutron-proton isospin convention.
  • Couple two nucleon isospins and enforce fermionic exchange symmetry.
  • Distinguish charge symmetry, charge independence, and physical symmetry breaking.

Protons and neutrons have different electric charges, yet much of their strong interaction can be organized as if they were two states of one nucleon. Isospin formalizes this approximate internal symmetry using the same angular-momentum algebra as spin. The resemblance concerns mathematics, not rotation in ordinary space. Its value is predictive organization across related nuclei, provided electromagnetic effects, mass differences, and residual charge-dependent strong terms are handled rather than silently removed from the evidence.

Declare the projection convention

Choose neutron isospin projection plus one half and proton projection minus one half. The many-nucleon projection is then half the neutron excess. Total isospin T is at least the magnitude of this projection, but a measured neutron and proton count does not uniquely fix T. Different states of the same nucleus can have different total isospin, just as a fixed spin projection need not determine the total angular momentum.

In the ideal charge-independent strong Hamiltonian, all generators of isospin commute with the dynamics. States in one multiplet then share the same strong-interaction energy after specifying the same other quantum numbers and reference conventions. The multiplet has two T plus one projections, often spread over several isobars. Its members are analog states, which need not all be ground states. Physical energies are split by effects absent from this ideal symmetry statement.

tz(n)=12,tz(p)=−12,Tz=N−Z2,T≥∣Tz∣,[Hstrong,T]=0 in the symmetry limitt_z(n)=\tfrac12,\quad t_z(p)=-\tfrac12,\qquad T_z=\frac{N-Z}{2},\quad T\ge|T_z|,\quad [H_{\rm strong},\mathbf T]=0\ \text{in the symmetry limit}
Isospin quantum numbers are dimensionless. This neutron-positive convention differs by a sign from some particle-physics conventions and must be retained throughout a calculation.↔ Scroll wide equations horizontally.

Couple states and respect exchange

Two nucleon isospins combine into a symmetric triplet and an antisymmetric singlet. The mixed neutron-proton products therefore form both a symmetric and an antisymmetric combination at zero projection. They have identical particle counts but different total isospin. In contrast, two neutrons or two protons have projection magnitude one and belong to the triplet. Their comparison with neutron-proton scattering must match the isospin channel, spin, and orbital state.

Treating proton and neutron as internal states of an identical fermionic nucleon requires the complete space-spin-isospin wave function to be antisymmetric on exchange. The resulting rule is that orbital angular momentum plus total spin plus total isospin is odd. For an S wave, a spin triplet pairs with isospin zero, while a spin singlet pairs with isospin one. This explains why the deuteron channel cannot be compared indiscriminately with two-proton scattering in a different spin-isospin sector.

∣1,0⟩=∣np⟩+∣pn⟩2,∣0,0⟩=∣np⟩−∣pn⟩2,(−1)L+S+T=−1|1,0\rangle=\frac{|np\rangle+|pn\rangle}{\sqrt2},\qquad |0,0\rangle=\frac{|np\rangle-|pn\rangle}{\sqrt2},\qquad (-1)^{L+S+T}=-1
The ket labels here are total isospin and its projection. The exchange rule uses orbital angular momentum L and pair spin S as integer quantum numbers in the isospin-symmetric nucleon description.↔ Scroll wide equations horizontally.

Separate two strengths of symmetry

Charge symmetry exchanges protons with neutrons, corresponding to a particular isospin rotation. Charge independence requires invariance under arbitrary isospin rotations and is a stronger condition. Approximate equality of corresponding proton-proton and neutron-neutron strong interactions tests the former; matching the neutron-proton member in the same isospin-one channel additionally tests the latter. Electromagnetic contributions must be treated consistently before attributing a difference to the strong interaction.

Analog-state mass energies are often summarized by a quadratic expression in isospin projection. Its linear and quadratic coefficients parameterize symmetry-breaking patterns within a specified multiplet; the form is an approximation, not an identity for arbitrary neighboring ground states. Coulomb effects and charge-dependent strong terms can also mix states of different T with the same conserved quantum numbers. Comparisons need measured uncertainties and a definition of atomic or nuclear masses, since electron and neutron-proton mass terms affect fitted coefficients.

E(Tz)=a+bTz+cTz2E(T_z)=a+bT_z+cT_z^2
This is the quadratic isobaric-multiplet mass-energy parameterization for matched analog states. Coefficients have energy units and include the stated reference and symmetry-breaking conventions.↔ Scroll wide equations horizontally.

Work it through

Fit a fictional analog triplet

Matched toy T=1T=1 analog states have energies 100, 101, and 104 MeV at Tz=−1,0,+1T_z=-1,0,+1. Find a, b, c in the quadratic expression and interpret whether the triplet is degenerate.

  1. At zero projection the linear and quadratic terms vanish, so a=101 MeVa=101\,\mathrm{MeV}. These are supplied matched-state energies, not a claim about ground states of three real nuclei.

  2. Subtracting the two endpoint equations gives 2b=104−100=4 MeV2b=104-100=4\,\mathrm{MeV}, hence b=2 MeVb=2\,\mathrm{MeV}. Averaging them gives a+c=(104+100)/2=102 MeVa+c=(104+100)/2=102\,\mathrm{MeV}, so c=1 MeVc=1\,\mathrm{MeV}.

  3. Substitution returns 101−2+1=100101-2+1=100 and 101+2+1=104 MeV101+2+1=104\,\mathrm{MeV}. The unequal energies display symmetry breaking in the modeled mass energies, while the shared T label organizes their analog relation.

The result

The fitted coefficients are a=101 MeV, b=2 MeV, and c=1 MeV; an analog multiplet need not be physically degenerate.

The living notebook

Explore the model

Interactive lab

Interactive lab · quantum

Symmetric and antisymmetric internal combinations

Interpret the two basis labels abstractly as np and pn at fixed total charge. Reset and set 90∘,0∘90^\circ,0^\circ for the symmetric combination, then compare a fresh Reset run at 90∘,180∘90^\circ,180^\circ for the antisymmetric combination. Predict equal basis probabilities but opposite Bloch x. The instrument does not implement isospin rotations of nuclei or their spatial-spin antisymmetrization.

Symmetric and antisymmetric internal combinationsA normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Current readings: Probability |0⟩: 71.1%; Probability |1⟩: 28.9%; Bloch (x, y, z): 0.78, 0.45, 0.42. All parameters are adjustable with the labeled controls below.xyz|0⟩|1⟩Basis probabilitiesP(0)71.1%P(1)28.9%
Bloch vector / P(0)P(1)Phase latitude
Probability |0⟩
71.1%
Probability |1⟩
28.9%
Bloch (x, y, z)
0.78, 0.45, 0.42
t = 0 s

A normalized two-level pure state, not an interacting field simulation. Playback adds one radian per second to the relative phase; basis probabilities stay fixed. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forEqual neutron-proton counts do not specify the relative phase or the total-isospin coupling.

Pause & reason

Check your understanding

Which statement correctly distinguishes the two approximate nuclear symmetries?

Your turn

Try it yourself

A nucleus has N=8,Z=6N=8,Z=6. Find TzT_z and the minimum allowed T. For a separate fictional T=1T=1 analog triplet use a=50,b=−0.3,c=0.1 MeVa=50,b=-0.3,c=0.1\,\mathrm{MeV} to calculate its three energies.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

Apply the declared neutron-positive projection convention, then substitute each of minus one, zero, and plus one into the fitted expression.

Compare with the solution

The projection is (8−6)/2=1(8-6)/2=1, so T must be at least one; particle counts alone do not prove it equals one. The separate triplet energies are 50+0.3+0.1=50.4 MeV50+0.3+0.1=50.4\,\mathrm{MeV} at minus one, 50 MeV at zero, and 50−0.3+0.1=49.8 MeV50-0.3+0.1=49.8\,\mathrm{MeV} at plus one. Their endpoint difference is minus 0.6 MeV, consistent with twice the supplied linear coefficient.

Keep the idea

Take it with you

  • Isospin is an approximate internal symmetry with a declared projection convention.
  • The full nucleon wave function must satisfy exchange antisymmetry.
  • Analog-state splittings require symmetry-breaking terms and consistent mass references.

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