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Emittances

Emittance is HELIX's primary beam-quality metric. But "emittance" covers six different things in HELIX, each useful in a different regime. This page disambiguates and explains when to use which.

Six emittances at a glance

Symbol Field Definition Use case
ε_x emit_x √(⟨x²⟩⟨x'²⟩ − ⟨xx'⟩²) basic transverse RMS
ε_n,x emit_nx β·γ·ε_x invariant under acceleration
ε_z emit_z longitudinal RMS in deg·MeV native long. units
ε_z(mmmrad) emit_z_mmmrad ε_z converted to mm·mrad unified plotting
ε_4D emit_4d √det(Σ_4D) invariant under x-y coupling
ε_1, ε_2 emit_n1, emit_n2 symplectic eigenvalues of J·Σ_4D 4-D normal-mode
ε_E1, ε_E2, ε_E3 emit_e1/e2/e3 Balandin trace invariants of Σ_6D full 6-D

When projected ε_x is enough

For an uncoupled lattice (no solenoids, no skew quads, no dispersion- energy coupling), the projected ε_x and ε_y are constant under linear symplectic transport. This is the textbook case and ε_x is all you need.

When projected ε_x lies — coupling

The moment a solenoid couples (x, x') with (y, y'), the projected ε_x and ε_y oscillate visibly through the lattice — even for an ideal beam with no real growth. Symptom: σ_x and σ_y wobble; ε_x, ε_y wobble; you'd swear the beam is heating up but actually nothing is wrong.

The 4-D invariant captures this:

\[ \varepsilon_{4D} = \sqrt{\det \Sigma_{4D}} \]

It stays constant through any linear x-y coupling. And the 4-D normal-mode eigenemittances (ε₁, ε₂) are the genuine "transverse emittances" — what would σ_x and σ_y be if you diagonalised away the coupling.

For solenoid-rich LEBT analyses, always plot ε₁ and ε₂ instead of ε_x and ε_y.

When the 4-D isn't enough — 6-D coupling

Dispersion couples (x, x') with (Δφ, ΔW). In high-dispersion regions, even ε_4D oscillates while the full 6-D emittance is invariant.

The Balandin trace invariants (I₂, I₄, I₆) of Σ_6D give three constants:

\[ I_2 = -\tfrac{1}{2}\,\mathrm{tr}\!\left((J\Sigma)^2\right),\quad I_4 = +\tfrac{1}{2}\,\mathrm{tr}\!\left((J\Sigma)^4\right),\quad I_6 = -\tfrac{1}{2}\,\mathrm{tr}\!\left((J\Sigma)^6\right) \]

These combine into a cubic equation whose roots are the squared eigenemittances ε_E1², ε_E2², ε_E3²:

\[ x^3 - I_2\,x^2 + \frac{I_2^2 - I_4}{2}\,x - \left(\frac{I_2^3}{6} - \frac{I_2 I_4}{2} + \frac{I_6}{3}\right) = 0 \]

(See linac_gen/diagnostics/eigenemittance.py:1 for the full derivation.) HELIX records ε_E1 ≥ ε_E2 ≥ ε_E3 as emit_e1, emit_e2, emit_e3.

In a fully uncoupled lattice, ε_E1 = ε_x, ε_E2 = ε_y, ε_E3 = ε_z. With coupling, the eigenemittances stay invariant while the projected ε_x/y/z visibly wobble.

Reference plots

For an example of all six emittances on the same plot — illustrating exactly when each is meaningful — see Worked example: Eigenemittance demo.

Implementation

  • linac_gen.diagnostics.eigenemittance.kinetic_invariants(sigma_6d) — computes (I₂, I₄, I₆).
  • linac_gen.diagnostics.eigenemittance.eigenemittances(sigma_6d) — returns sorted (ε_E1, ε_E2, ε_E3).
  • linac_gen.diagnostics.eigenemittance.eigenemittances_4d(sigma_4d) — 4-D version.
  • linac_gen.diagnostics.eigenemittance.eigenemittances_series(...) — per-step over the full lattice.

Reference

  • V. Balandin, W. Decking, N. Golubeva, "On the calculation of generalized four-dimensional and six-dimensional eigen-emittances", IPAC 2013 / arXiv:1305.1532.
  • IMPACT-X documentation — same eigenemittance convention.

Cross-references

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