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Worked example: Eigenemittance

A coupled FODO + sextupole + octupole + bi-Gaussian halo lattice that demonstrates all six emittances on the same lattice. Shows when ε_x, ε_y oscillate while the eigenemittances ε_E1, ε_E2, ε_E3 stay invariant.

Source

examples/impactx_features_demo.py and the lattice in examples/impactx_features/coupled_fodo.dat:

python examples/impactx_features_demo.py

Lattice features

  • Coupled FODO — solenoid + skew quadrupole couples (x, x') with (y, y').
  • Thin sextupole and octupole — exercise the Multipole element.
  • Thermal halo distribution — bi-Gaussian with halo_fraction=0.05.

What to plot

Six emittances vs s on the same axes:

import matplotlib
matplotlib.use("Agg")
import matplotlib.pyplot as plt

from linac_gen.io.tracewin_parser import parse_tracewin
from linac_gen.core.config import BeamConfig
from linac_gen.core.simulation import Simulation
from linac_gen.distributions.factory import create_beam

lat, _ = parse_tracewin("examples/impactx_features/coupled_fodo.dat")
beam = create_beam(BeamConfig(n_particles=2000), seed=1)
results = Simulation(lat, beam).run()

fig, ax = plt.subplots()
ax.plot(results.s, results.emit_x, label="ε_x (projected)", lw=0.8)
ax.plot(results.s, results.emit_y, label="ε_y (projected)", lw=0.8)
ax.plot(results.s, results.emit_4d, label="ε_4D (4-D invariant)", lw=1.2)
ax.plot(results.s, results.emit_n1, label="ε₁ (4-D normal-mode)", lw=1.2)
ax.plot(results.s, results.emit_n2, label="ε₂ (4-D normal-mode)", lw=1.2)
ax.plot(results.s, results.emit_e1, "--", label="ε_E1 (6-D Balandin)")
ax.plot(results.s, results.emit_e2, "--", label="ε_E2 (6-D Balandin)")
ax.plot(results.s, results.emit_e3, "--", label="ε_E3 (6-D Balandin)")
ax.legend(); ax.set_xlabel("s [mm]"); ax.set_ylabel("ε [mm·mrad]")

What you should see

  • ε_x, ε_y oscillate strongly through the solenoid (coupling artefact).
  • ε_4D stays nearly flat (4-D invariant — captures the x-y coupling but not dispersion).
  • ε₁, ε₂ stay flat (the normal-mode emittances — these are the textbook "real" transverse emittances).
  • ε_E1, ε_E2, ε_E3 stay flat (full 6-D — invariant under all linear coupling).

For an ideal lattice with no growth, the eigenemittances are constants of motion. The projected ε_x, ε_y are not.

When does this matter?

  • LEBT / solenoid-rich lines — projected ε_x, ε_y wobble badly; eigenemittances are the only meaningful invariant.
  • High-dispersion regions — ε_4D oscillates; need the full 6-D ε_E1/2/3.
  • PIP-II SRF cryomodules with quadrupole-doublet focusing — same story.

Theory

Per V. Balandin, W. Decking, N. Golubeva, "On the calculation of generalized four-dimensional and six-dimensional eigen-emittances" (IPAC 2013 / arXiv:1305.1532), the 6-D eigenemittances are computed from the trace invariants of (J·Σ_6D):

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

These combine into the cubic

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

whose roots are ε_E1², ε_E2², ε_E3².

See Emittances and the source file linac_gen/diagnostics/eigenemittance.py:1.

Cross-references

← Tolerance study · Continue to Validation → TraceWin parity →