Solenoid¶

A solenoid focuses the beam in both transverse planes simultaneously via the on-axis B_z field plus the rotational coupling at entrance and exit. The standard focusing element of a low-energy LEBT (where quadrupoles are too weak) and a common companion to RF cavities in SRF cryomodules.

TL;DR (TraceWin users)¶

	TraceWin	HELIX
Keyword	`SOLENOID L B aperture`	`Solenoid(name, length, field, ...)`
Length L	mm	mm
Field B	T (on-axis B_z)	T

Conventions:

Always coupled: x and y are linked through Larmor rotation — σ_x and σ_y individually oscillate even if the beam is round. Use eigenemittances for the invariants.
field_rel adds a fractional field error (effective_B = B·(1 + field_rel)).

Tutorial (newcomers)¶

A solenoid produces an axial magnetic field B_z along the beam axis. The transverse 6-D dynamics combine three effects:

Entrance kick — radial fringe field gives x' a kick proportional to y, and vice versa.
Body — particles spiral around the axis at the Larmor frequency ω_L = q·B / (2m) (in the rest frame). Effective focusing strength k_s = q·B / (2·Bρ).
Exit kick — opposite of entrance kick.

The net effect is focusing in both planes — the axisymmetric counterpart to a quadrupole pair. Unlike a quadrupole, the four transverse coordinates (x, x', y, y') are fully coupled: no purely-x or purely-y subspace is invariant.

Full 6×6 transfer matrix¶

Define the Larmor wavenumber

\[ k_s \;=\; \frac{q\,B_z}{2\,B\rho} \quad \text{(rad/m)} \]

and let C = cos(k_s L), S = sin(k_s L). The full 6×6 in coordinates (x, x', y, y', Δφ, ΔW) is:

\[ M_{\text{sol}} = \begin{pmatrix} C^2 & SC/k_s & SC & S^2/k_s & 0 & 0 \\ -k_s SC & C^2 & -k_s S^2 & SC & 0 & 0 \\ -SC & -S^2/k_s & C^2 & SC/k_s & 0 & 0 \\ k_s S^2 & -SC & -k_s SC & C^2 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & M_{56} \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix} \]

with the same drift-like longitudinal phase slip as every other thick element:

\[ M_{56} = -\,\frac{360 \, L \, [\text{mm}]}{\beta^{3} \, \gamma^{3} \, m \, \lambda_{\text{RF}}} \quad [\text{deg}/\text{MeV}] \]

Block	Effect
(x, x', y, y') 4×4	fully coupled focusing — Larmor rotation + radial focus
(x, y), (x, y'), (x', y), (x', y')	non-zero (this is what couples the planes)
(Δφ, ΔW) 2×2	drift-like phase slip identical to any thick element
Cross to longitudinal	zero (no acceleration; on-axis B_z does no work)

The C² diagonal terms are why σ_x and σ_y individually appear to "breathe" through the solenoid even for a round, matched beam: the projected emittances ε_x and ε_y are no longer invariants of the motion — only the 4-D ε_⊥ and the symplectic eigenemittances ε_1, ε_2 are.

Key consequence: ε_x and ε_y "wobble"¶

Because the (x, x') and (y, y') subspaces are coupled, the projected emittances ε_x and ε_y oscillate through the solenoid even for an ideal beam — the true invariants are the 4-D ε_⊥ = √det(Σ_4D) and the symplectic normal-mode emittances ε_1, ε_2. See Emittances.

Example: PXIE-style LEBT solenoid¶

from linac_gen.elements.solenoid import Solenoid
from linac_gen.elements.drift import Drift
from linac_gen.core.lattice import Lattice

lat = Lattice()
lat.add(Drift(name="D0", length=200.0))
lat.add(Solenoid(name="SOL1", length=300.0, field=0.20))   # 0.20 T
lat.add(Drift(name="D1", length=400.0))
lat.add(Solenoid(name="SOL2", length=300.0, field=0.20))
lat.add(Drift(name="D2", length=200.0))

print(f"Lattice has {len(lat.elements)} elements, "
      f"{lat.total_length:.0f} mm long.")

For a complete worked LEBT example see DC LEBT walkthrough.

API reference (developers)¶

linac_gen.elements.solenoid.Solenoid(
    name: str,
    length: float,                # mm
    field: float,                 # T (on-axis B_z)
    aperture: float = 0.0,        # mm
    dx: float = 0.0, dy: float = 0.0, dz: float = 0.0,
    tilt_deg: float = 0.0,
    pitch_deg: float = 0.0, yaw_deg: float = 0.0,
    field_rel: float = 0.0,
    n_steps: int = 5,
)

Parameter	Default	Units	Notes
`name`	(required)	—	identifier
`length`	(required)	mm	physical length
`field`	(required)	T	on-axis B_z (design value)
`aperture`	0.0	mm	round-aperture radius
`dx..yaw_deg`	0.0	mm/deg	misalignment
`field_rel`	0.0	—	fractional field error (FieldError mixin)
`n_steps`	5	—	tracker substeps

Properties¶

effective_field — field * (1 + field_rel).

Source¶

linac_gen/elements/solenoid.py:1 (~120 lines).