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Twiss & emittance conventions

Every beam in HELIX is described by 9 numbers per plane: ε (emittance) + (α, β) (Twiss). This page nails down the conventions — units, sign, normalisation — so you never have to wonder if your input values mean the same thing HELIX thinks they do.

TL;DR

Transverse (x, y) Longitudinal (z)
Geometric emittance ε mm·mrad deg·MeV (native)
Normalised emittance ε_n = β·γ·ε mm·mrad mm·mrad (after deg·MeV → mm·mrad conv.)
Twiss α dimensionless dimensionless
Twiss β mm/mrad = m deg/MeV (native)
Twiss γ mrad/mm = 1/m MeV/deg

BeamConfig.emit_nx and emit_ny are normalised. HELIX divides by β·γ before passing to the distribution generator (geometric ε is what the matched-Twiss ellipse uses).

Tutorial

What are α, β, ε?

The Courant-Snyder parameters describe a phase-space ellipse:

\[ \gamma\, x^2 + 2\alpha\, x\,x' + \beta\, x'^2 = \varepsilon \]

with γ = (1+α²)/β as a dependent quantity. Geometrically:

  • β is the "size" envelope — RMS beam half-width is √(β·ε).
  • α = -dβ/(2ds) characterises whether the beam is converging (α > 0) or diverging (α < 0).
  • ε is the area/π of the ellipse — invariant under linear symplectic transport (in the absence of coupling).

The ellipse rotates and shears as the beam propagates; the Courant-Snyder invariant (the area) stays the same — that's why ε is the natural beam-quality metric.

Geometric vs normalised emittance

For relativistic beams (γ ≫ 1), the geometric ε shrinks during acceleration (adiabatic damping). To compare beams at different energies, use the normalised emittance:

\[ \varepsilon_n = \beta\,\gamma\,\varepsilon \]

ε_n is invariant under acceleration — same beam quality before and after a cavity. For a 3 MeV proton (β = 0.0796, γ = 1.003) the factor βγ ≈ 0.0799; for a 1 GeV proton (β = 0.875, γ = 2.067) it's ≈ 1.81. So ε_n / ε ≈ 23× difference between the two energies.

Longitudinal emittance

HELIX records longitudinal emittance natively in deg·MeV (matches TraceWin) and converts to mm·mrad for some plots.

\[ \varepsilon_z\,[\text{mm·mrad}] = \frac{\varepsilon_z\,[\text{deg·MeV}] \cdot 1000}{k_\phi} \]

where k_φ = 360° / (β·λ_RF) is the phase-to-length conversion.

For β·λ_RF = β · c / f, with λ_RF in mm.

The normalised longitudinal emittance is then ε_nz = β·γ·ε_z(mm·mrad).

Where it bites

BeamConfig.emit_nx is normalised, not geometric

The most common confusion in HELIX usage. If you write emit_nx=0.21 thinking it's the geometric value, you'll over-predict σ_x by a factor of 1/βγ ≈ 11× at LEBT entrance. The factory always divides by ref.bg before using it.

BeamConfig.emit_z is in deg·MeV

Not normalised, not in mm·mrad. Native TraceWin units. HELIX converts internally where needed.

Sample values for PIP-II

Stage Energy (MeV) β·γ ε_nx (mm·mrad) β_x (m) α_x
LEBT entrance (after source) 0.030 0.0080 0.20 (varies) (varies)
MEBT entrance 2.10 0.0669 0.21 0.32 1.23
MEBT exit 2.45 0.0723 0.21 (matched) (matched)
HWR entrance 2.45 0.0723 0.21 (matched) (matched)
End of LB650 200 0.686 0.25 (varies) (varies)
End of HB650 800 1.558 0.30 (varies) (varies)

The slight ε_n growth (0.21 → 0.30 mm·mrad over 256m) is emittance-growth from chromaticity, RF gymnastics, and PIC noise — within engineering tolerance for the design.

Cross-references

← Distributions · Continue to BeamConfig reference →