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Appendix A — Physics references

Citations for every paper that informs HELIX's physics implementation. Useful when you need to cite HELIX (cite the underlying physics paper directly) or want to understand a particular module's provenance.

Space charge & PIC

  • Qiang, J.; Lidia, S.; Ryne, R.D.; Limborg-Deprey, C. "Three-dimensional quasistatic model for high brightness beam dynamics simulation". Phys. Rev. ST Accel. Beams 9, 044204 (2006); erratum 10, 129901 (2007). Source of HELIX's IGF Green's function. doi:10.1103/PhysRevSTAB.9.044204
  • Hockney, R.W.; Eastwood, J.W. Computer Simulation Using Particles, Adam Hilger (1988). Foundational text on PIC.
  • Wangler, T.P. RF Linear Accelerators, 2nd ed., Wiley-VCH (2008). Standard reference for envelope SC models.
  • Lapostolle, P.M. "Possible emittance increase through filamentation due to space charge in continuous beams". IEEE Trans. Nucl. Sci. NS-18, 1101 (1971).

Continuous-beam envelope (Sacherer ODE)

  • Sacherer, F.J. "RMS envelope equations with space charge". IEEE Trans. Nucl. Sci. NS-18, 1105 (1971).
  • Kapchinskij, I.M.; Vladimirskij, V.V. "Limitations of proton beam current in a strong focusing linear accelerator associated with the beam space charge". Proc. 2nd Int. Conf. on High-Energy Accelerators (1959).

H⁻ stripping

  • Folsom, B.; Eshraqi, M.; Blaskovic-Kraljevic, N.; Gålnander, B. "Stripping mechanisms and remediation for H⁻ beams". Phys. Rev. Accel. Beams 24, 074201 (2021). arXiv:2103.16195. Source of HELIX's magnetic-stripping rate formula.
  • Stinson, G.M. et al. (1969). Original empirical fit for the A₁ / A₂ Lorentz-stripping constants (refit by Scherk 1979, used by Folsom 2021).

Eigenemittances

  • Balandin, V.; Decking, W.; Golubeva, N. "On the calculation of generalized four-dimensional and six-dimensional eigen-emittances". IPAC 2013 / arXiv:1305.1532. Source of the Balandin trace invariants and eigenemittance cubic formula.
  • IMPACT-X code documentation — same eigenemittance convention used in HELIX.

RFQ design

  • Crandall, K.R.; Stokes, R.H.; Wangler, T.P. "RF quadrupole beam dynamics design studies". Linac 1979. Source of the Crandall 2-term potential expansion.
  • Toutatis code documentation — RFQ-design code shipped in Tracewin_code/Toutatis_*.pdf references.

Field maps

  • TraceWin documentation (CEA Saclay) — definitive reference for the FIELD_MAP geom 5-digit encoding and file-naming convention (see the FIELD_MAP and field-map-file sections of the TraceWin manual shipped with the code).

Symplectic integrators

  • Yoshida, H. "Construction of higher order symplectic integrators". Phys. Lett. A 150, 262 (1990).
  • Forest, E. "Geometric integration for particle accelerators". J. Phys. A: Math. Gen. 39, 5321 (2006).

Ring mode

  • Yoshida, H. (1990), above — the compositions of order 2, 4 and 6 of both element models.
  • Forest, E.; Ruth, R.D. "Fourth-order symplectic integration". Physica D 43, 105 (1990). The composition inside a bend's drift.
  • Schachinger, L.; Talman, R. "TEAPOT: a thin-element accelerator program for optics and tracking". Part. Accel. 22, 35 (1987). The thin-lens splitting.
  • Forest, E. Beam Dynamics: A New Attitude and Framework. Harwood Academic (1998). Exact maps: the exact drift and sector bend, the pole-face rotation, the hard-edge fringe.
  • Brown, K.L. "A first- and second-order matrix theory for the design of beam transport systems and charged particle spectrometers". SLAC-75, Rev. 4 (1982). The fringe-field correction of a pole face.
  • Laskar, J. "Frequency analysis for multi-dimensional systems. Global dynamics and diffusion". Physica D 67, 257 (1993). NAFF, the tunes of tracked particles.
  • Qiang, J.; Lidia, S.; Ryne, R.D.; Limborg-Deprey, C. (2006), above — the integrated Green's function, here in two dimensions for ring mode's 2.5-D space charge.
  • Hockney, R.W.; Eastwood, J.W. (1988), above — the convolution on a doubled grid (open boundary).
  • Bassetti, M.; Erskine, G.A. "Closed expression for the electrical field of a two-dimensional Gaussian charge". CERN-ISR-TH/80-06 (1980). The reference field of the space-charge checks.
  • Jones, F.W. "A method for incorporating image forces in multiparticle tracking with space charge". Proc. EPAC 2000, p. 1381. The pipe boundary fitted to a rectangle or an ellipse.
  • Laslett, L.J. "On intensity limitations imposed by transverse space-charge effects in circular particle accelerators". BNL-7534, p. 324 (1963). The incoherent tune shift.
  • Qiang, J. Phys. Rev. Accel. Beams 28, 114602 (2025), doi:10.1103/npx1-nclt. The analytic 2.5-D kick of a Gaussian bunch (eqs. 31–32), a space-charge check.
  • Highland, V.L. "Some practical remarks on multiple scattering". Nucl. Instrum. Methods 129, 497 (1975); Lynch, G.R.; Dahl, O.I. "Approximations to multiple Coulomb scattering". Nucl. Instrum. Methods B 58, 6 (1991). The stripping foil's scattering width θ0 (fitted to the central 98 % of the angles).
  • Gulley, M.S. et al. "Measurement of H⁻, H⁰, and H⁺ yields produced by foil stripping of 800-MeV H⁻ ions". Phys. Rev. A 53, 3201 (1996). Measured stripping yields, close to the two design values the two-step stripping's cross sections are calibrated to.

Boris pusher

  • Boris, J.P. "Relativistic plasma simulation — optimization of a hybrid code". Proc. Conf. Numerical Simulation Plasmas (1970), 3-67.

Halo

  • Wangler, T.P.; Crandall, K.R.; Mills, R.S.; Reiser, M. "Relation between field energy and rms emittance in intense particle beams". IEEE Trans. Nucl. Sci. NS-32, 2196 (1985). Background for kurtosis-style halo measures. Note HELIX's recorded halo parameter is the kurtosis form h = ⟨x⁴⟩/⟨x²⟩² − 1 (Gaussian ⇒ 2, KV ⇒ 1) — see Halo analysis.

Adjacent codes for cross-reference

  • TraceWin — Saclay, closed-source. HELIX inherits its .dat format and matching language.
  • IMPACT-X — Berkeley Lab, open-source C++. Source of HELIX's Misalignment mixin pattern and eigenemittance convention.
  • PyORBIT — open-source. Same H⁻ stripping fit.
  • OPAL — open-source. Same Hockney+IGF PIC algorithm.

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