pyffag

DA-based FFAG accelerator tracking using differential algebra.

Built on daceypy for arbitrary-order transfer map computation through FFAG sector magnets via integration of the exact midplane Hamiltonian.

Installation

pip install pyffag

Quick start

import numpy as np
from daceypy import DA
from pyffag import sector_map, compose_sequence, compose_n, tune, twiss
from pyffag.constants import kinetic_to_brho, M_PROTON

# 250 MeV proton FFAG ring: 8 FD doublet cells
DA.init(5, 2)  # DA order 5, 2 variables (x, px)
Brho = kinetic_to_brho(250.0, M_PROTON)

# F magnet: horizontally focusing sector, 25 degrees
F = sector_map([0.9, 2.0], Brho, angle=np.radians(25.0))

# D magnet: horizontally defocusing sector, 20 degrees
D = sector_map([0.9, -3.0], Brho, angle=np.radians(20.0))

cell = compose_sequence([F, D])
ring = compose_n(cell, 8)

print(f"Cell tune: {twiss(cell)['tune']:.4f}")
print(f"Ring tune: {tune(ring):.4f}")

Features

• Sector tracking — exact curvilinear Hamiltonian integration through sector magnets with polynomial midplane field By(x) = Σ B_k x^k. Midplane 2-DOF (sector_map) and 4-DOF (sector_map_4d, Maxwell-consistent off-midplane expansion).
• Element maps — exact drift, thin quadrupole / sextupole / octupole, edge kicks.
• Ring operations — map composition, N-fold composition, closed-orbit finding (Newton with DA Jacobian; see below).
• Optics — transfer matrix, tune, Twiss (β, α, γ), stability check, symplecticity error.
• Shared lattice format — JSON schema that defines a sequence of elements and drives pyffag.build_map(). Portable across pyffag versions and, when paired with external cross-validation harnesses, across codes.

Lattice format (JSON)

{
  "name": "fd_doublet",
  "brho": 2.0,
  "elements": [
    {"type": "sector", "B": [1.0, 2.0], "angle_deg": 15.0},
    {"type": "drift",  "length": 0.3},
    {"type": "sector", "B": [1.0, -1.0], "angle_deg": 15.0},
    {"type": "thin_sext", "kL": 0.05}
  ]
}

import pyffag
from daceypy import DA

lattice = pyffag.load("fd_doublet.json")
DA.init(5, 2)
ring = pyffag.build_map(lattice)
print(pyffag.tune(ring), pyffag.twiss(ring))

Element types: sector, drift, drift_paraxial, thin_quad, thin_sext, thin_oct, edge_kick. See pyffag.lattice for the full schema.

Closed-orbit finding

from daceypy import DA
from pyffag import sector_map, find_closed_orbit

DA.init(5, 2)
x_co, px_co, converged = find_closed_orbit(
    lambda: sector_map([1.0, 2.0], 2.0, angle=0.4363),
    x_guess=0.0, px_guess=0.0,
)

The solver expands the DA map internally about the current guess and uses the DA Jacobian for a Newton step. Converges in one iteration for linear maps; quadratic convergence for nonlinear.

Physics

sector_map() integrates the midplane equations of motion in Frenet-Serret (curvilinear) coordinates with arc length as the independent variable. Sector magnets have radial edge faces (no edge focusing); non-radial edges are modelled explicitly via edge_kick.

Integration with danf

For nonlinear normal-form analysis (detuning, chromaticity, complex-basis NF) use danf:

from danf import NormalForm, chromaticity_1d

nf = NormalForm(ring)
nf.compute()
print(nf.tunes, nf.detuning)

License

MIT