Quadrupole

This module describes the PASS quadrupole element Quadrupole, used to simulate the motion of charged particles in a quadrupole magnet. The quadrupole is the most fundamental focusing element in accelerators, providing linear focusing force through a gradient magnetic field.

The PASS quadrupole supports both thick element (length > 0) and thin lens (length = 0) modes. The thick element provides two tracking models: drift-kick-drift-exact (DKD-exact) uses the exact drift-kick-drift symplectic integration scheme, supporting both uniform (2nd-order) and yoshida4 (4th-order) symplectic integrators; mat-kick-mat (MKM, default) uses the exact linear transport matrix scheme, which is exact for a purely linear field with a single slice.

Code Location

• Source file: PASS/commands/element/quadrupole.py

• Class name: Quadrupole (inherits from Command)

• Registration name: quadrupole

• Key features:

  • Supports thin lens mode (length = 0, applies only a quadrupole kick)

  • Supports thick lens mode (length > 0, DKD-exact symplectic integration or MKM exact linear matrix)

  • Supports uniform (2nd-order leapfrog) and yoshida4 (4th-order Yoshida composition) integrators

  • Supports mat-kick-mat (MKM) model (exact linear transport matrix, including chromaticity)

  • MKM model handles k1+k1s combinations via rotational diagonalization

  • Supports normal quadrupole (k1l) and skew quadrupole (k1sl) and their combinations

  • Zero field (k1l = k1sl = 0) automatically degenerates to a pure drift

  • Chromaticity effects naturally introduced through exact drift

  • Supports aperture check

Coordinate Convention

PASS uses normalized curvilinear coordinates consistent with Xsuite. The six-dimensional phase-space variables are \((x, p_x, y, p_y, z, \delta)\):

Variable	Symbol	Definition
x	\(x\)	Horizontal offset (relative to the reference orbit)
px	\(p_x\)	Normalized horizontal momentum, \(p_x = P_x / P_0\)
y	\(y\)	Vertical offset
py	\(p_y\)	Normalized vertical momentum, \(p_y = P_y / P_0\)
z	\(\zeta\)	Longitudinal coordinate, \(\zeta = s - \beta_0 c t\)
dp	\(\delta\)	Relative momentum deviation, \(\delta = P / P_0 - 1\)

where \(P_0\) is the reference particle momentum, \(\beta_0 = v_0 / c\) is the reference particle normalized velocity, \(s\) is the arc length along the reference orbit, and \(t\) is time.

The longitudinal momentum component is defined as:

\[p_z = \sqrt{(1+\delta)^2 - p_x^2 - p_y^2}\]

Charge-to-mass ratio factor:

\[\chi = \frac{q}{q_0} \cdot \frac{m_0}{m}\]

For a beam of identical particle species, \(\chi = 1\).

Quadrupole Field and Normalized Strength

The magnetic field of a quadrupole magnet is linearly distributed in the transverse plane. In complex notation:

\[B_y + i B_x = (G + i G_s)(x + i y)\]

where \(G\) is the normal quadrupole gradient and \(G_s\) is the skew quadrupole gradient. Expanding:

\[B_y = G \cdot x - G_s \cdot y\]

\[B_x = G \cdot y + G_s \cdot x\]

The normalized quadrupole strength is defined as:

\[K_1 = \frac{q_0 G}{P_0}\]

\[K_{1s} = \frac{q_0 G_s}{P_0}\]

The integrated strength is:

\[K_{1L} = K_1 \cdot L, \qquad K_{1sL} = K_{1s} \cdot L\]

where \(L\) is the magnet length. In PASS, the user directly specifies \(K_{1L}\) (k1l) and \(K_{1sL}\) (k1sl); for thick lenses, \(K_1 = K_{1L} / L\) and \(K_{1s} = K_{1sL} / L\) are solved internally.

Overall Tracking Flow

Depending on the magnet length, the quadrupole has two tracking modes:

Thin lens mode (\(L = 0\))

====== Thin lens (length = 0) ======

Single quadrupole kick Kick(K1L, K1sL)
[Position unchanged, momentum jump only]

Thick lens mode (\(L > 0\))

DKD-exact model:

====== Thick lens DKD-exact (length > 0) ======

Slice 1 → Slice 2 → ... → Slice N
(Each slice: Drift(ds/2) → Kick(ds) → Drift(ds/2))

where ds = L / N

If K1L = 0 and K1sL = 0: degenerates to a single exact drift Drift(L)

MKM model:

====== Thick lens mat-kick-mat (length > 0) ======

Slice 1 → Slice 2 → ... → Slice N
(Each slice: M(ds), exact linear transport matrix)

For pure k1 + k1s (no higher-order multipoles): M(L) = M(ds)^N
Therefore num_slice = 1 is sufficient; multiple slices do not change the result.

The complete map is:

Thin lens:

\[\mathcal{M}_{\text{thin}} = \text{Kick}(K_{1L}, K_{1sL})\]

Thick lens (N slices):

\[\mathcal{M}_{\text{thick}} = \left[\mathcal{M}_{\text{DKD}}(\Delta s)\right]^N\]

where the DKD map for each slice is:

\[\mathcal{M}_{\text{DKD}}(\Delta s) = D\!\left(\frac{\Delta s}{2}\right) \circ K(\Delta s) \circ D\!\left(\frac{\Delta s}{2}\right)\]

Note

• Thin lens mode does not change the particle position coordinates \((x, y, z)\), only applies momentum kicks

• Chromaticity effects in thick lens mode are naturally introduced through the \(p_z\) expression in exact drift (see chromaticity section)

• When \(K_{1L} = 0\) and \(K_{1sL} = 0\), the thick lens degenerates to a pure drift, avoiding meaningless empty kick loops

Physical Derivation

Hamiltonian

In the Cartesian coordinate system (quadrupole has no curvature, \(h = 0\)), the quadrupole Hamiltonian is:

\[H_{\text{quad}} = \frac{p_\tau}{\beta_0} - \sqrt{(1+\delta)^2 - p_x^2 - p_y^2} + \frac{\chi}{2}\left(K_1 x^2 - K_1 y^2 + 2 K_{1s} x y\right)\]

Splitting it into the propagation part (exact drift \(H_D\)) and the kick part (\(H_K\)):

\[H_D = \frac{p_\tau}{\beta_0} - \sqrt{(1+\delta)^2 - p_x^2 - p_y^2}\]

\[H_K = \frac{\chi}{2}\left(K_1 x^2 - K_1 y^2 + 2 K_{1s} x y\right)\]

where \(H_D\) is the exact drift Hamiltonian (preserving the \(p_z\) square root without small-momentum expansion), and \(H_K\) is the quadrupole kick. This is the standard split-operator method: the Hamiltonian is split into analytically solvable parts, maps are applied separately, and then combined into a symplectic integrator.

Exact Drift Map D

The Hamilton’s equations of the propagation part give the exact drift:

\[p_z = \sqrt{(1+\delta)^2 - p_x^2 - p_y^2}\]

\[x \leftarrow x + \frac{p_x}{p_z} \cdot L_D\]

\[y \leftarrow y + \frac{p_y}{p_z} \cdot L_D\]

\[\zeta \leftarrow \zeta + L_D \cdot \left(1 - \frac{\beta_0}{\beta} \cdot \frac{1+\delta}{p_z}\right)\]

where \(L_D\) is the drift length, and \(\beta\) is the particle’s actual normalized velocity:

\[\beta = \frac{(1+\delta) \, \beta_0 \gamma_0}{\sqrt{1 + \left[(1+\delta) \, \beta_0 \gamma_0\right]^2}}\]

Note

The meaning of “exact”: the drift part preserves the exact square root \(p_z = \sqrt{(1+\delta)^2 - p_x^2 - p_y^2}\) without small-momentum expansion \(p_x \ll 1\). The approximation lies only in separating the propagation part from the kick part (split-operator method). This formula is identical to the exact drift in the Drift element and the SBend element.

Quadrupole Kick Map K

The kick part is a thin lens map (position unchanged, momentum jump only). From Hamilton’s equations \(\dot{p}_x = -\partial H / \partial x\), \(\dot{p}_y = -\partial H / \partial y\):

\[\Delta p_x = -\chi K_1 L_K \cdot x + \chi K_{1s} L_K \cdot y\]

\[\Delta p_y = +\chi K_1 L_K \cdot y + \chi K_{1s} L_K \cdot x\]

where \(L_K\) is the kick effective length.

Physical meaning of each term:

Term	Source	Physical Meaning
\(-\chi K_1 L_K \cdot x\)	\(\frac{\chi K_1 x^2}{2}\)	Horizontal normal quadrupole focusing (focusing when \(K_1 > 0\), defocusing when \(K_1 < 0\))
\(+\chi K_1 L_K \cdot y\)	\(-\frac{\chi K_1 y^2}{2}\)	Vertical normal quadrupole defocusing (opposite to horizontal)
\(+\chi K_{1s} L_K \cdot y\)	\(\chi K_{1s} x y\)	Skew quadrupole horizontal coupling kick
\(+\chi K_{1s} L_K \cdot x\)	\(\chi K_{1s} x y\)	Skew quadrupole vertical coupling kick

For thin lens mode, \(L_K = 1\), using the integrated strengths \(K_{1L}\) and \(K_{1sL}\) directly. For DKD mode, \(L_K = \Delta s\), using \(K_1 \Delta s\) and \(K_{1s} \Delta s\).

Note

A normal quadrupole (\(K_1 > 0\)) focuses in the horizontal direction and defocuses in the vertical direction. This is a direct result of the quadrupole field \(B_y = G \cdot x\): particles deviating from the axis experience a force proportional to their offset, with a restoring force (focusing) in the horizontal direction and a repulsive force (defocusing) in the vertical direction. To achieve focusing in both directions simultaneously, focusing quadrupoles (F) and defocusing quadrupoles (D) must be alternately arranged, i.e., the FODO structure.

A skew quadrupole (\(K_{1s} \neq 0\)) rotates the focusing action by \(\pi / 4\), producing \(x\)-\(y\) coupling. In practice, it is often used for coupling correction or simulating installation rotation errors.

Uniform Integrator (2nd-order symplectic)

Each slice uses the drift-kick-drift (DKD) structure, i.e., 2nd-order leapfrog:

\[S_2(\Delta s) = D\!\left(\frac{\Delta s}{2}\right) \circ K(\Delta s) \circ D\!\left(\frac{\Delta s}{2}\right)\]

The per-slice error is \(O(\Delta s^3)\), and the global error is \(O(\Delta s^2)\). A 2nd-order symplectic integrator where every step is a canonical transformation.

yoshida4 Integrator (4th-order symplectic)

A 4th-order symplectic map is constructed by composing three 2nd-order DKD steps [Yoshida 1990]:

\[S_4(\Delta s) = S_2(z_1 \Delta s) \circ S_2(z_0 \Delta s) \circ S_2(z_1 \Delta s)\]

where the Yoshida coefficients are:

\[z_1 = \frac{1}{2 - 2^{1/3}} \approx 1.3512\]

\[z_0 = 1 - 2 z_1 \approx -1.7024\]

Note

\(z_0 < 0\) means the middle step is a backward tracking (the drift and kick “lengths” are negative). This is a mathematical requirement of the Yoshida composition method and is fully self-consistent in the symplectic map group. The per-slice error is \(O(\Delta s^5)\), and the global error is \(O(\Delta s^4)\).

Integrator Selection Recommendations:

Scenario	Recommended Integrator	Reason
Fast simulation	uniform	2 drifts + 1 kick per slice, low computational cost
High-precision simulation	yoshida4	4th-order accuracy, but 6 drifts + 3 kicks per slice
With space charge	uniform + more slices	PIC solve cost far exceeds drift; 4th-order Yoshida requires 3 PIC solves

Chromaticity Effects

Chromaticity describes the dependence of the particle tune on the momentum deviation \(\delta\). The PASS DKD-exact model naturally introduces chromaticity through the exact drift’s \(p_z\) expression, without any additional treatment.

Physical Mechanism

In DKD integration, the drift uses the exact expression \(p_z = \sqrt{(1+\delta)^2 - p_x^2 - p_y^2}\), while the kick in \((x, p_x)\) space is \(\Delta p_x = -\chi K_1 \Delta s \cdot x\) (not divided by \(1+\delta\)).

Transforming to \((x, x')\) space (where \(x' = p_x / (1+\delta)\)), the equivalent focusing strength automatically becomes:

\[K_{1,\text{eff}} = \frac{K_1}{1+\delta}\]

This is the physical origin of the natural chromaticity \(Q'_x = -\frac{1}{4\pi}\oint \beta_x K_1 \, ds\). No explicit division is needed in the code—the exact drift’s \(p_z\) expression automatically accomplishes this.

Thin Lens vs. Thick Lens Chromaticity Comparison

Effect	Thin Lens	Thick Lens DKD-exact
Natural chromaticity (normal quadrupole)	Not introduced	Introduced (\(K_{1,\text{eff}} = K_1/(1+\delta)\))
Coupling chromaticity (skew quadrupole)	Not introduced	Introduced (coupling transport contains \(\delta\) dependence)
Higher-order nonlinear dispersion	Not introduced	Introduced (\(p_z\) preserves full square root)
Path-length effects (\(R_{56}\), etc.)	Not introduced	Introduced

Physically, a thin lens has zero length with no drift space, and the kick \(\Delta p_x = -K_{1L}\,x\) does not contain \(\delta\). In a thick lens, particles have a drift path inside the magnet, and particles with different momenta travel different paths and experience different effective focusing—this is the source of chromaticity. The same applies to skew quadrupoles: the \(p_z\) dependence in drift makes the coupling transport also contain \(\delta\) dependence, introducing coupling chromaticity.

Note

• Thin lens mode (length = 0) has no path-length effects, so the thin lens quadrupole itself does not introduce natural chromaticity—whether normal or skew quadrupole

• Thick lens DKD-exact mode fully includes natural chromaticity effects, including higher-order nonlinear dispersion terms

• Unlike the mat-kick-mat model (which explicitly divides by \(1+\delta\)), PASS’s DKD-exact introduces chromaticity implicitly through exact \(p_z\), and also includes the higher-order nonlinear effects of \(p_z\)

• In PASS’s Twiss linear transport framework, natural chromaticity is introduced through DQx / DQy parameters (\(\delta\) terms in the phase advance), not through the element itself. If a thin lens quadrupole is additionally inserted in Twiss transport, it will not double-count chromaticity with DQx—because the thin lens itself does not introduce chromaticity. However, if the inserted quadrupole strength is large enough to significantly change the lattice tune and \(\beta\) functions, the original Twiss parameters (including DQx) are no longer accurate and need to be recomputed

Tracking Model Comparison

The PASS quadrupole supports two thick lens body models, suitable for different precision and speed requirements.

Model Overview

drift-kick-drift-exact (DKD-exact): Splits the Hamiltonian into exact drift and thin lens kick, combined via symplectic splitting. Preserves the full \(p_z\) nonlinear kinematics, but linear focusing and chromaticity have symplectic splitting errors (controllable by increasing the number of slices or using higher-order integrators).

mat-kick-mat (MKM): Solves the analytical exact solution of the linearized equation of motion \(u'' + K_{\text{eff}} \chi/(1+\delta) \cdot u = 0\), constructing the transport matrix using trigonometric functions (focusing plane) and hyperbolic functions (defocusing plane). Linear focusing, linear chromaticity, and \(R_{56}\) are all exact solutions, but higher-order nonlinear terms of \(p_z\) are not included.

Comparison Table

Feature	drift-kick-drift-exact	mat-kick-mat
Linear focusing	2nd-order symplectic splitting approximation	Exact (analytical matrix)
Linear chromaticity	Approximate (\(O(1/N^2)\) splitting error)	Exact (\(K_1/(1+\delta)\) explicit)
Nonlinear kinematics	Preserved (full \(p_z\) square root)	Not included (linearized \(x' = p_x/(1+\delta)\))
Longitudinal \(R_{56}\)	Approximate	Exact (analytical formula)
Nonlinear path length	Preserved	Not included
Symplecticity	Strictly symplectic	Symplectic (matrix is symplectic)
Computation speed	Slower (\(N \times 3\) DKD steps)	Fast (matrix multiplication)

MKM Implementation Details

MKM solves the exact solution of the linearized equation. For a pure \(K_1\) quadrupole, the u plane (focusing) uses sin/cos matrices, the v plane (defocusing) uses sinh/cosh matrices, with equivalent strength \(K = K_1 \chi / (1+\delta)\) computed per particle.

For \(K_1 + K_{1s}\) combined quadrupoles, rotational diagonalization is used:

\[\theta = \frac{1}{2}\arctan\frac{-K_{1s}}{K_1}, \quad K_{\text{eff}} = \sqrt{K_1^2 + K_{1s}^2}\]

After rotating to the principal axis frame, the matrix is applied, then rotated back. Key property: \(\theta\) is independent of \(\delta\) (\(K_1\) and \(K_{1s}\) scale proportionally by \(\chi/(1+\delta)\), so the ratio is unchanged), therefore \(\theta\), \(\cos\theta\), and \(\sin\theta\) can be precomputed once in __init__.

Special cases:

Physical State	\(K_1\)	\(K_{1s}\)	\(\theta\)
Normal quadrupole	\(K_{\text{eff}}\)	0	0 (no rotation)
Pure skew quadrupole	0	\(K_{\text{eff}}\)	\(\pi/4\) (rotate 45°)
Combined quadrupole	Nonzero	Nonzero	\(\frac{1}{2}\arctan(-K_{1s}/K_1)\)

For pure \(K_1 + K_{1s}\) (no higher-order multipoles), the matrix is exact for any slice length \(\Delta s\), so num_slice = 1 is sufficient. Multiple slices are only needed in the future when supporting \(K_2\) / \(K_{2s}\) multipole kicks.

MKM Limitations

MKM linearizes the kinematics: \(x' = p_x / (1+\delta)\), rather than the exact \(x' = p_x / p_z\). Expanding:

\[\frac{p_x}{p_z} = \frac{p_x}{1+\delta}\left(1 + \frac{p_x^2 + p_y^2}{2(1+\delta)^2} + \cdots\right)\]

MKM retains only the first term; the lost higher-order terms lead to:

• Amplitude-dependent tune shift (geometric nonlinearity): from \(p_x^3\) and similar terms; zero in MKM

• Higher-order chromaticity (\(Q''\) and above): from \(\delta \cdot p_x^2\) cross terms; lost in MKM

• Nonlinear path length: contributions of \(p_x^2\), \(p_x^4\), etc. to \(\Delta z\); lost in MKM

For typical storage ring parameters (\(p_x \sim 10^{-4}\)), the lost effects are on the order of \(10^{-8}\) per magnet, but may be amplified under full-ring accumulation and multi-turn effects. If studying nonlinear beam dynamics problems such as dynamic aperture and tune footprint, the DKD-exact model should be used.

Slice Count and Integrator Recommendations

Model	Recommended Slices	Recommended Integrator	Description
mat-kick-mat	1	— ¹	Matrix is exact for any \(\Delta s\); 1 slice is sufficient. Fast, suitable for linear optics calculations
drift-kick-drift-exact	Needs testing ²	yoshida4	Slice count needs to be determined through convergence testing

¹ The MKM model does not use the integrator parameter.

² DKD-exact slice count selection recommendations:

• Slice count should be determined through convergence testing: compare tune and chromaticity under different slice counts to confirm convergence

• For HIAF-BRing (quadrupole length approximately 1 m, \(K_1\) approximately 0.2), a slice count of 5 with yoshida4 integrator is recommended

• Insufficient slice count will lead to \(O(1/N^2)\) error in linear chromaticity; amplitude-dependent tune shift will have larger deviations

• The yoshida4 integrator has per-slice error \(O(\Delta s^5)\) and global error \(O(\Delta s^4)\), with precision far superior to uniform

Interface Parameters

Property	JSON key	Type	Unit	Description
s	s (m)	float	m	Longitudinal position of the element in the beamline
length	length (m)	float	m	Element length (must be \(\ge 0\); \(= 0\) for thin lens)
name	name	str		Element name
k1l	k1l	float	\(\text{m}^{-1}\)	Normal quadrupole integrated strength \(K_{1L}\), default 0
k1sl	k1sl	float	\(\text{m}^{-1}\)	Skew quadrupole integrated strength \(K_{1sL}\), default 0
model	model	str		Physical model, options: adaptive (default mat-kick-mat), drift-kick-drift-exact, mat-kick-mat
num_slice	num slices	int		Number of slices, default 1 (effective only for thick lens)
integrator	integrator	str		Integrator, options: adaptive (default uniform), uniform, yoshida4
aperture_type	aperture type	str		Aperture type, default off
aperture_value	aperture value	list		Aperture parameter values, default []

Usage Examples

Thick Lens Normal Quadrupole

{
    "QF1": {
        "S (m)": 10.0,
        "Command": "Quadrupole",
        "Length (m)": 0.5,
        "K1L": 0.2,
        "Num Slices": 5,
        "Integrator": "yoshida4",
        "Aperture Type": "off"
    }
}

Focusing quadrupole (\(K_{1L} > 0\)), length 0.5 m, 5 slices, 4th-order symplectic integration.

MKM Model Quadrupole

{
    "QF1": {
        "S (m)": 10.0,
        "Command": "Quadrupole",
        "Length (m)": 0.5,
        "K1L": 0.2,
        "Model": "mat-kick-mat",
        "Num Slices": 1,
        "Aperture Type": "off"
    }
}

MKM model, exact linear transport, 1 slice is sufficient. Faster than DKD-exact, suitable for linear optics calculations.

Thin Lens Quadrupole

{
    "QF2": {
        "S (m)": 20.0,
        "Command": "Quadrupole",
        "Length (m)": 0.0,
        "K1L": 0.3,
        "Aperture Type": "off"
    }
}

Zero-length quadrupole, applying only the \(K_{1L}\) thin lens kick, no body tracking, no chromaticity effects.

Defocusing Quadrupole

{
    "QD1": {
        "S (m)": 30.0,
        "Command": "Quadrupole",
        "Length (m)": 0.4,
        "K1L": -0.15,
        "Num Slices": 1,
        "Integrator": "uniform",
        "Aperture Type": "off"
    }
}

Defocusing quadrupole (\(K_{1L} < 0\)), horizontal defocusing, vertical focusing.

Skew Quadrupole

{
    "QS1": {
        "S (m)": 40.0,
        "Command": "Quadrupole",
        "Length (m)": 0.3,
        "K1L": 0.0,
        "K1SL": 0.1,
        "Num Slices": 1,
        "Integrator": "uniform",
        "Aperture Type": "off"
    }
}

Pure skew quadrupole (\(K_{1L} = 0\), \(K_{1sL} \neq 0\)), producing \(x\)-\(y\) coupling.

Normal + Skew Quadrupole Combination

{
    "QFS1": {
        "S (m)": 50.0,
        "Command": "Quadrupole",
        "Length (m)": 0.5,
        "K1L": 0.2,
        "K1SL": 0.05,
        "Num Slices": 3,
        "Integrator": "yoshida4",
        "Aperture Type": "circle",
        "Aperture Value": [0.04]
    }
}

Combined quadrupole with both normal and skew components (simulating installation rotation error), with a circular aperture check.

Equivalent Representation with Rotation Angle

A normal quadrupole with \(K_{1L} = 0.2\) rotated by angle \(\theta = 0.01\) rad is equivalent to:

\[K_{1L}' = K_{1L} \cos 2\theta \approx 0.2 \times 0.9998 = 0.19996\]

\[K_{1sL}' = K_{1L} \sin 2\theta \approx 0.2 \times 0.02 = 0.004\]

{
    "QF_rot": {
        "S (m)": 60.0,
        "Command": "Quadrupole",
        "Length (m)": 0.5,
        "K1L": 0.19996,
        "K1SL": 0.004,
        "Num Slices": 1,
        "Integrator": "uniform"
    }
}

Application Scenarios

• Linear focusing: Alternately arrange focusing (F) and defocusing (D) quadrupoles in a FODO structure to achieve transverse beam confinement

• Chromaticity correction: Utilize the natural chromaticity effect of quadrupoles, compensating chromaticity by adjusting sextupoles

• Coupling correction: Use skew quadrupoles (k1sl) to control \(x\)-\(y\) coupling and correct installation errors

• Tune adjustment: Change the working point (tune) by adjusting quadrupole strength, tuning the beam to the optimal working region

• Dispersion matching: Place quadrupoles after bending magnets to control the evolution of the dispersion function \(\eta(s)\)

• Beam transport lines: Use quadrupoles in injection and extraction lines to focus the beam and control the beam envelope

References

• Xsuite Physics Guide, Sec 1.10.3 (exact drift), Sec 1.10.5 (quadrupole)

• Xsuite source code: xtrack/beam_elements/elements_src/quadrupole.h, track_magnet.h, track_magnet_kick.h, track_magnet_drift.h

• Yoshida, H., “Construction of higher order symplectic integrators”, Phys. Lett. A 150 (1990)

• MAD-X Physics Manual: quadrupole field and linear transport