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
ε₁, ε₂, ε₃ (canonical)	derived (Results tile)	symplectic eigenvalues of Σ_6D in normalised canonical units	full 6-D, coupled planes

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.

Normalised canonical intrinsic emittances

The recorded Σ is in mixed units (mm, mrad, mm, mrad, deg, MeV) with geometric angles, so its three conjugate pairs carry different phase-space area scales. emit_e1/e2/e3 are therefore invariants only while the longitudinal plane is decoupled from x and y. Through dispersion (x–ΔW) or RF (x–Δφ) coupling, they are not.

canonical_eigenemittances_series(results) first maps Σ to normalised canonical coordinates, all in mm and mrad,

\[ (x,\ eta\gamma x',\ y,\ eta\gamma y',\ c\,\Delta t,\ -10^3\,\Delta E/mc^2), \qquad c\,\Delta t = \Deltaarphi\,\lambda/360 , \]

and then takes the symplectic eigenvalues of Σ·J.

• Sign. The minus sign is required: Δφ > 0 is a late particle and the conjugate of t is −E, so (Δφ, ΔW) is anti-canonical.
• Uncoupled beam. The three modes equal ε_nx, ε_ny and ε_nz exactly.
• Invariance. The modes are invariant under any linear symplectic transport, including dispersion, acceleration and FREQ jumps. HELIX's dipole matrices are symplectic in these coordinates to 1e-16, which is pinned in tests/diagnostics/test_eigenemittance.py.
• Labels. Each mode is labelled x-, y- or z-like by its largest share of symplectic area per plane.
• Coupling ratios. The function also returns

$$ arepsilon_{nx}arepsilon_{ny}arepsilon_{nz}\,/\,(arepsilon_1arepsilon_2arepsilon_3) \ge 1 \quad ext{and}\quad arepsilon_{nx}arepsilon_{ny}\,/\,arepsilon_{4D,n} \ge 1 , $$

where 1 means no coupling. - Mechanical momenta. The momenta are mechanical (x′ = p_x/p_z), so inside solenoid fields the modes are not invariants. - DC beams. Records of a DC beam give NaN for the 6-D quantities.

In the GUI this is the Results tile Intrinsic ε (canonical, norm.). The existing quantities are unchanged.

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.
• linac_gen.diagnostics.eigenemittance.canonical_eigenemittances(sigma, bg, f_MHz, mass) — normalised canonical modes, projected ε_n, ε_4D,n, ε_6D,n, coupling ratios.
• linac_gen.diagnostics.eigenemittance.canonical_eigenemittances_series(results) — the same along a run (vectorised; uses sigma_matrix, ref_beta, ref_gamma, ref_frequency, mass_mev).

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

• Recorder fields
• Worked example: eigenemittance
• linac_gen/diagnostics/eigenemittance.py:1 (source).

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