, Adam Watts

USPAS Concepts:�Transverse Motion

Pavel Snopok

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Schedule

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Describing Particles in the Accelerator

To design and operate a particle accelerator…

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We need equations of motion that combine information about

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particle distribution + beamline components (or “lattice”)

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to describe particle motion through the accelerator.

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The Problem: For a given type of particle, create an ideal system to guide particles to a final location with a desired trajectory and desired kinetic energy per particle, at the desired time (and within tolerable spreads of these quantities)

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Single-pass & repetitive systems

• Beam Transport (from point A to point B)

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• Acceleration along the way
• (single-pass acceleration)

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• (multi-pass acceleration)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Coordinate System

s: arclength as particle travels down the beamline from injector to target

x(s): side-to-side motion in the beamline

y(s): up-down motion in the beamline

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The x-y plane is a slice of the beam in time (along s); transverse plane

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You may encounter different conventions (e.g., x and z are transverse coordinates, y is the longitudinal)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle trajectory vectors

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In the horizontal plane, the coordinate “x” represents a transverse deviation from the reference “s” trajectory.

The arc length that subtends this deviation is simply a = r*θ. For small angles, a becomes identical to x, the perpendicular offset between the particle and reference trajectories. Therefore a = r*θ becomes dx = ds*theta.

small angles

small deviations

tan(θ) ≈ θ = dx/ds = x’

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle coordinate vector

• Each particle can be fully described by a 6-dimensional phase space vector
• You may see other coordinates used: (x, p_x) instead of (x, x’), (y, p_y) instead of (y, y’), (∆t, ∆E) instead of (z, z’)
• Sometimes, (u,u’) space is called trace space, (u,p_u) – phase space (“u” can be x or y)
• x’, y’ are the slopes of the trajectory at a given point
• typically calculated as�x’ = p_x/p₀ ≈ p_x/p_z
• p₀ is the total momentum of the reference particle, and p_z is a very large fraction of p₀

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x-axis

y-axis

z-axis

transverse motion

longitudinal motion

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle coordinate vector evolution: Drift

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Consider a single particle’s trajectory through a space with no magnetic fields, i.e. a “drift” of length L. Again in the small angle approximation:

tan(x’0) ≈x’0 = x/L

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle deflection: dipole electromagnet

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle deflection: dipole electromagnet

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Particles spreading out

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle deflection: quadrupole electromagnet

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Can only focus in one plane at a time.

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle deflection: quadrupole electromagnet

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Magnetic field (blue)

Force (red)

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle deflection: multiple quadrupole magnets

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Beam

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Group behavior of particle trajectories

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Now we develop the mathematics for describing the group of particle. Focusing on the statistical distributions of the particles, i.e. first and second moments, rather than tracking the trajectory of every particle.

The second moments <x^2>, <x'2>, and <xx'> are the variances (standard deviation squared) in position and angle, and the average correlation between position and angle. The second moments propagate as follows:

The first moments <x> and <x'> are the average of all the particle positions and angles. The math for propagating these moments is similar to that of the single particle:

Typically, particle angles are very small and difficult to measure. Usually, we are only able to measure the transverse beam profile at a single point using a profile monitor (like a screen). The characteristic width of the Gaussian profile is directly related to the root-mean-square (RMS) of the position distribution:

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Off-center “kick” in quadrupole magnet

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Note that a beam passing through the center (i.e < x0 >= 0) of a quadruple magnet is un-perturbed in trajectory:

However, if the beam is off-center in the quadrupole, a net angular “kick” occurs to the average beam trajectory:

Beam

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Beam transport: Strong Focusing

• F = focusing quadrupole (focusing in the horizontal plane, defocusing in the vertical)
• O = drift (empty space)
• D = defocusing quadrupole (defocusing in the horizontal plane, focusing in the vertical)

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For long-distance beam transport, or to design a stable circular accelerator, it is advantageous to use “Strong Focusing”, or “Alternating Gradient Focusing”. This is a periodic arrangement of quadrupoles that alternate polarity. This technique allows for stable transport of beam over arbitrarily-long distances without net increase in the beam size in either plane. Transverse particle oscillations due to strong focusing are known as “betatron oscillations”.

• Christofilos, N. C. (1950). "Focusing System for Ions and Electrons". US Patent No. 2,736,799.
• Courant, E. D.; Snyder, H. S. (Jan 1958). "Theory of the alternating-gradient synchrotron" (PDF). Annals of Physics. 3 (1): 1–48. Bibcode:2000AnPhy.281..360C. doi:10.1006/aphy.2000.6012

Beam

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

FODO: the basic periodic lattice cell

• F = focusing quadrupole (focusing in the horizontal plane, defocusing in the vertical)
• O = drift (empty space)
• D = defocusing quadrupole (defocusing in the horizontal plane, focusing in the vertical)

Simulation -> animation

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Phase Space and Courant-Snyder Parametrization

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Emittance

• Emittance: an important beam parameter that combines information about particle position and momentum.
• The area drawn out in phase space by particles moving through lattice is called the emittance.
• The ellipse shape is typically parameterized by (α, β, γ, ε), often referred to as Twiss parameters
• (u, u’) here can be (x, x’) or (y, y’)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Phase Space

Emittance is one of the most important beam characteristics we measure!

• Relating the statistical definition of the beam with the elliptical form of the beam trajectories:

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Normalized Emittance

• There are several definitions of emittance
• Geometric emittance relates to the physical area covered by the phase space ellipse:

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• Normalized emittance stays invariant as energy increases
• This is typically the emittance we are reporting in the control room and in papers/text:

relativistic factors

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Motion Through Accelerator Lattice: Drift (empty space)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Courant-Snyder Parameters and Strong Focusing

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Courant-Snyder Parameters and Strong Focusing

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Particle Distribution

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• The area occupied by the ellipse in phase space is called emittance
• Emittance is a very important quantity for several reasons
• For linear transformations (dipoles and quads), it is preserved
• It tends to grow due to nonlinearities and errors, and the goal is to keep this growth minimal

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Emittance Matching

• Beam parameters must be matched to the accelerator line to ensure effective emittance stays low.
• Only particles injected with positions and momenta within certain ranges will be able to propagate successfully down each beamline. Beam particle positions can be changed upstream to better match them to an upcoming beamline.

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Beam Mismatch and Emittance Growth

• Emittance is conserved along the beamline
• With nonlinearities and errors: beams with parameters that are mismatched to beamline components will filament, increasing effective area in phase space.
• So, effective emittance can only stay the same or increase from source to target.

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Filamentation

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Filamentation

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Filamentation

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Chromatic Effects

Recall that the bend strength of a dipole and focal length of a quadrupole depends on the particle's momentum. Since a beam will always have some spread in particle momenta, there are “chromatic” effects that occur from this deviation in how the magnetic fields affect the particles.

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Chromatic Effects

These chromatic effects can be summarized by a “dispersion function” in each transverse plane. These functions, along with the Courant-Snyder ellipse parameters, describe how the beam size and trajectory propagates down a beamline or along a ring as a function of either a particle's momentum or the momentum spread in a beam.

The transverse deviation of a particle is directly proportional to its momentum deviation from the reference momentum. Similarly, this same equation shows how an off-momentum beam's transverse trajectory will deviate from the design trajectory.

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The beam size increases with a spread in the particle momenta:

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Nonlinear Magnets: Sextupole

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Higher-order magnets such as the Sextupole and Octupole have a nonlinear field dependence on the transverse particle position.

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These magnets are used for more advanced beam manipulation. However, all magnets have small non-zero higher-order terms that cause non-linear effects (manufacturing or alignment errors, etc.) Sometimes these effects are not so small (Fermilab MI quadrupoles, for example)!

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Nonlinear Magnets: Octupole

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The field strength of an octupole depends on the cube of the distance from the beam pipe center. Octupoles allow for control of how the tune spread depends on the amplitude of the betatron oscillations, and can also be used to control the beam shape in beamlines.

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Parameterization of the Beam

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Beamline Components (Lattice)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Motion in the Transverse Plane. Familiar example (Simple harmonic oscillator)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Hill’s Equation

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Hill’s Equation: Solutions

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Motion Through Accelerator Lattice: Drift (empty space)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Motion Through a Quadrupole

• Magnets (quadrupoles, sextupoles, solenoids, etc.) act like lenses – bend charged particle trajectories:

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FQ

DQ

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Circular accelerator transverse equations of motion

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Circular accelerators: betatron motion and tune

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: equations of motion

• Transverse particle motion is guided by magnetic fields (E = 0):

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• For typical lattice elements (dipoles and quadrupoles), it can be transformed into Hill’s equation:

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• that has the following solution:

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: equations of motion

• Linear transverse motion of particles can be conveniently described using linear algebra:

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• (Here, once again, u represents x or y)
• We can combine both (x, x’) and (y, y’) into a single 4x4 matrix:

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• To include longitudinal effects, a third set of equations can be added.

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: beam line components

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• So what are C and S from the last slide??
• C/S are the coefficients in the solutions to our linear equations
• They are referred to as “cos-like” and “sine-like” trajectories
• Any actual trajectory is a linear combination of the “cos” and “sin” trajectories
• Physically, they represent the combined effect of all the lattice elements between the starting and ending points of the beamline
• We can use matrices to represent linear beamline components
• Direct connection to glass optics + light, except we use electromagnets + charged particles

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: matrix math refresher

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Try working out what the total M would be…

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: common lattice elements

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: thin-lens quadrupole

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: common beam line components

Drift

Used to focus and defocus the beam

Quadrupole

The matrices for bending magnets can get complicated quickly. We show the rectangular case here

Bending Magnet

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Empty beam pipe

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Nonlinear transverse dynamics: Resonant extraction

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It’s possible to take advantage of this nonlinear filamentation, as well as resonance conditions in the betatron oscillations, to slowly “spill” the beam out to the experiments.

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Emittance measurement: Multi-profile method

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Values circled in orange are simply the squared RMS beam width (or the standard deviation of the gaussian fit) of the profile at monitor A. Solving for the remaining unknowns yields:

Recall:

Goal: Determine the full beam matrix at profile monitor A.

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Emittance measurement: Quadrupole scan method

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Goal: Determine the full beam matrix just before the quadrupole.

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Emittance measurement: Quadrupole scan method

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Solving for the beam width at the profile monitor, and collecting terms as a function of the magnet strength yields:

Scanning the quadrupole strength k, plotting the resulting beam width on the profile monitor as a function of k, and fitting the resulting curve to a parabola of the form ak^2+bk+c, this system of equations can be solved to determine the full beam matrix just before the quadrupole:

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Summary

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Summary

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• Magnetic fields are used to control the transverse size and trajectory of particle beams.
• Particle trajectories can be described by position/angle vectors or statistical first moments, and distributions by statistical second moments.
• Geometric (phase space, ellipses) representations of particle distributions are also useful.
• Courant-Snyder parametrization serves as an alternate useful method for parametrizing beam distributions in circular accelerators.
• Non-linear magnets can control chromatic and higher-order beam shape effects.
• Emittance is an important figure of merit for beam distributions, and can be indirectly measured with multiple methods.

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Extra slides

(time-permitting)

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Example: A Beam Line Calculation

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Example: A Beam Line Calculation

• Consider two particle trajectories, starting at
• (x, x’ ) = (0, 0.5 mrad), and (x, x’ ) = (5 mm, 0)
• A distance 6 m later, the trajectories enter a focusing thin-lens quadrupole of focal length F = 3 m. This is followed by a defocusing quadrupole of the same focal length, a distance 1 m later
• Find the trajectories (x, x’) for each case at the exit of the second quad

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Example: A Beam Line Calculation

• Consider two particle trajectories, starting at
• (x, x’ ) = (0, 0.5 mrad), and (x, x’ ) = (5 mm, 0)
• A distance 6 m later, the trajectories enter a focusing thin-lens quadrupole of focal length F = 3 m. This is followed by a defocusing quadrupole of the same focal length, a distance 1 m later
• Find the trajectories (x, x’) for each case at the exit of the second quad

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Example: A Beam Line Calculation

• Consider two particle trajectories, starting at
• (x, x’ ) = (0, 0.5 mrad), and (x, x’ ) = (5 mm, 0)
• A distance 6 m later, the trajectories enter a focusing thin-lens quadrupole of focal length F = 3 m. This is followed by a defocusing quadrupole of the same focal length, a distance 1 m later
• Find the trajectories (x, x’) for each case at the exit of the second quad

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Example: A Beam Line Calculation

• Consider two particle trajectories, starting at
• (x, x’ ) = (0, 0.5 mrad), and (x, x’ ) = (5 mm, 0)
• A distance 6 m later, the trajectories enter a focusing thin-lens quadrupole of focal length F = 3 m. This is followed by a defocusing quadrupole of the same focal length, a distance 1 m later
• Find the trajectories (x, x’) for each case at the exit of the second quad

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Matrix Formalism: FODO calculation

Focusing

Defocusing

• Let’s define a matrix for a beam line now.
• This is a FODO lattice we saw before:
• Focusing quadrupole (F)
• Drift space (O)
• Defocusing quadrupole (D)
• Drift space (O)
• This is a very common lattice configuration, and used to keep beam sizes within a specified range (often periodic).

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Twiss parameter calculation

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USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion

Emittance Measurement

• The sigma matrix can be solved for given multiple measurements of beam size.
• There are two methods:
• multi-location (usually wire)
• quad scan
• Note, the beam sizes needs to change significantly in order for this technique to work.
• Rule of thumb is for double the beam size at the ends of the parabola vs. the minimum.

USPAS Summer 2026 · Concepts of Accelerator Science and Technology · Transverse Motion