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Space-charge models

HELIX has four distinct space-charge models, each appropriate for a different physical regime. Picking the right one is critical: mismatch between physics and model is the most common source of "why does my simulation disagree with TraceWin?" complaints.

TL;DR — pick by regime

Beam regime Tracker SC model When
Bunched, RMS only EnvelopeSolver uniform-ellipsoid Σ kick matching, parameter sweeps
Bunched, multi-particle Tracker (with SpaceChargeConfig) 3-D PIC (Hockney FFT) production tracking
Continuous (DC), RMS Sacherer ODE KV-uniform analytic pre-RFQ LEBT envelope
Continuous (DC), multi-particle Tracker (with SpaceChargeConfig, continuous=True) 2-D analytic kick pre-RFQ LEBT MP

Model 1: Uniform triaxial ellipsoid (envelope)

Used by EnvelopeSolver for bunched beams. Treats the beam as a uniform-density ellipsoid with rms-equivalent semi-axes (√5·σ_x, √5·σ_y, √5·γ·σ_z) and applies an analytic depolarisation-factor kick using the Carlson R_D elliptic integral (Lapostolle / Wangler formulation).

  • Pros: closed-form, exact for matched RMS Gaussians.
  • Cons: doesn't capture non-Gaussian effects, halo, or filamentation.

Model 2: 3-D PIC (Hockney FFT)

Used by the multi-particle tracker (Tracker + SpaceChargeConfig). Solves Poisson's equation on a 3-D Cartesian grid via doubled-grid FFT convolution with one of two Green's functions.

The full 8-step PIC cycle:

  1. Lorentz boost on z to enter the beam rest frame (γ on z only).
  2. Grid setup on the boosted coordinates.
  3. Charge deposition on the grid (CIC or TSC kernel).
  4. Doubled-grid FFT convolution with Green's function.
  5. Field gather back to particles (CIC or TSC).
  6. Transverse kick applied directly to the lab-frame x′/y′ — the boost back is folded into the β²γ² factor of the kick formula (one γ from the E+v×B force cancellation, one βγ from the Δp⊥ → x′ conversion).
  7. Longitudinal kick applied directly to the lab-frame ΔW (E_z is invariant under the z-boost).
  8. The reference particle is never modified — SC kicks touch only the deviation coordinates of the macroparticles.

For details see PIC solver and Kernels.

  • Pros: captures non-axisymmetric, non-Gaussian, halo, filamentation effects.
  • Cons: ~10× slower than envelope; statistical noise floor ≈ 1/√N on σ.

Model 3: Sacherer / KV ODE (continuous beam)

For pre-RFQ continuous beams, linac_gen.tracking.sacherer solves the continuous-beam envelope ODE:

\[ \sigma_x'' + \kappa_x(s)\,\sigma_x - \frac{\varepsilon_x^2}{\sigma_x^3} - \frac{2K}{\sigma_x + \sigma_y} = 0 \]

(and the analogous equation in y) with generalised perveance K = qI / (2π ε₀ m c³ (βγ)³).

  • Pros: textbook DC-beam benchmark, fast.
  • Cons: no acceleration (assumes constant β, γ), only handles drifts + hard-edge quads + 1-D/3-D solenoid maps.

Model 4: 2-D analytic DC kick (continuous beam, MP)

For continuous-beam multi-particle tracking, linac_gen.pic.pic_solver.kick_continuous_2d applies the closed- form transverse kick for a uniform-density elliptical-cylinder beam. Activated automatically when Beam.continuous == True.

Three DC-kernel choices via SpaceChargeConfig.dc_kernel:

dc_kernel Model
"uniform" analytic linear uniform-elliptical kick (matches rigid-Σ envelope)
"gaussian" Bassetti-Erskine field of a 2-D Gaussian (rigid σ; per-particle non-linear)
"pic2d" 2-D Hockney FFT PIC over the actual particle distribution (most accurate)

For details see DC mode.

Coherent synchrotron radiation (CSR)

CSR is a separate collective effect — not one of the four space-charge models above. When a short bunch travels around a bend, radiation emitted toward the tail catches up with the head along the shorter chord and acts back on the bunch, driving energy spread and — through the bend dispersion — transverse emittance growth.

HELIX models CSR with a 1-D steady-state wake (Saldin–Schneidmiller–Yurkov 1997; Derbenev 1995):

\[ \frac{dW}{ds}(s) = -A \int_{-\infty}^{s} \lambda'(s')\,(s - s')^{-1/3}\,ds' \]

where λ(s) is the bunch line-charge density and the prefactor scales as A ∝ R^{-2/3} with R the bend radius. The kick is applied per sub-step inside every Dipole.

  • Multi-particle only. CSR needs the actual bunch line-density profile, so it acts only in Tracker runs — the envelope solver ignores it.
  • Steady-state only. Transient entrance/exit (drift→bend, bend→drift) fields are not modelled; the model is valid when the bend is long compared with the CSR formation length.
  • Enable it with SpaceChargeConfig(csr_enabled=True), or the "CSR in bends" checkbox on the Numerics tab.
SpaceChargeConfig field Default Meaning
csr_enabled False apply the CSR kick in bends
csr_bins 200 longitudinal line-density bin count
csr_model "1d_steady" model selector (only value for now)

Decision summary

Is the beam bunched?
├── Yes (post-RFQ)
│   ├── RMS only? → EnvelopeSolver (Model 1)
│   └── Multi-particle? → Tracker + SpaceChargeConfig (Model 2)
└── No (pre-RFQ, DC)
    ├── RMS only? → Sacherer ODE (Model 3)
    └── Multi-particle? → Tracker + SpaceChargeConfig with continuous=True (Model 4)

Cross-references

← Beam → .dst loading · Continue to PIC solver →