Magnetic Trajectories

Transverse motion and the gyroradius

In a uniform magnetic field, resolve a particle velocity into components parallel and perpendicular to the field:

v = v_{∥} + v_{⊥}, v_{∥} ∥ B, v_{⊥} \cdot B = 0.

Only the transverse component appears in the magnetic-force magnitude. The force has magnitude $∣ q ∣ v_{⊥} B$ and remains perpendicular to the instantaneous transverse velocity. For a nonrelativistic particle, equating that magnitude to the required centripetal force gives

∣ q ∣ v_{⊥} B = \frac{m v _{⊥}^{2}}{r _{L}}, r_{L} = \frac{m v _{⊥}}{∣ q ∣ B} .

The radius $r_{L}$ is often called the gyroradius or Larmor radius. It grows with transverse momentum and shrinks with charge magnitude or field strength. A sign change of charge reverses the sense of rotation without changing the radius. The field direction fixes the orbit plane: a magnetic field along the page normal gives a circular projection in the page, whereas a field tilted in space gives a circle in the plane perpendicular to that tilted direction.

Circular orbit of a positive charge in a uniform magnetic field pointing out of the page (dot symbols). The velocity is tangent to the circle and the magnetic force points to the centre, so the orbit is traced clockwise; a charge of opposite sign circulates the other way around a circle of the same radius

r_{L} = m v_{⊥} / (∣ q ∣ B)

The force has no component along the velocity, so it does no mechanical work:

\frac{d K}{d t} = F_{B} \cdot v = q (v \times B) \cdot v = 0.

The transverse speed stays constant, and the circle is traversed at a constant angular speed. This conservation statement holds at each point of the path. Momentum direction rotates continuously while its magnitude and kinetic energy remain constant.

Angular frequency, period, and orientation.

Dividing the centripetal-force relation by $m v_{⊥}$ gives the angular frequency

ω_{c} = \frac{∣ q ∣ B}{m}, T_{c} = \frac{2 π m}{∣ q ∣ B}, f_{c} = \frac{∣ q ∣ B}{2 π m} .

In the nonrelativistic uniform-field model, neither frequency nor period depends on orbit radius or speed. A faster particle traces a larger circle in the same period. This feature permits frequency-based mass-to-charge measurements, while the radius gives a separate momentum measurement. Relativistic momentum changes both relations at high speed and is treated by replacing $m$ with the energy-dependent factor $γ m$ .

Direction questions need a defined coordinate system. Put $B$ along $+ z$ . Take a positive particle initially moving along $+ x$ . The cross product $v \times B$ points along $- y$ , so the orbit begins by turning toward negative $y$ . An electron with the same velocity turns toward positive $y$ . Apply the right-hand rule to the vector product first, then multiply by the signed charge; combining those two steps mentally is a frequent source of reversed tracks.

Helical trajectories and pitch angle

The component parallel to $B$ experiences zero magnetic force and therefore remains constant. Combining uniform parallel translation with transverse circular motion gives a helix. If $α$ is the pitch angle between velocity and field,

v_{⊥} = v sin α, v_{∥} = v cos α, p = v_{∥} T_{c} .

Here $p$ is the axial advance during one complete turn, conventionally called the pitch of the helix. The radius depends on $v sin α$ , whereas the pitch depends on $v cos α$ . A trajectory nearly parallel to the field has small radius and large pitch; a trajectory nearly perpendicular to the field approaches a circle. Both limiting cases follow from the same component equations.

Helical path of a charge in a uniform magnetic field. The velocity component along

B

is unchanged and advances the orbit by one pitch

p

each period, while the transverse component drives the circular motion of radius

r_{L}

The uniform-field helix is an ideal local model. A particle in a slowly changing magnetic field can be followed by a guiding-centre description, but gradients and curvature modify the orbit centre and may produce drifts. Collisions randomize velocity components, and an electric field parallel to the magnetic field changes particle energy. Each effect has a distinct signature in a measured track, so a helix fit should report the fitted interval and the field-uniformity range.

Reconstructing a trajectory from measurements

A cloud chamber, bubble chamber, tracking detector, or plasma diagnostic records positions rather than force vectors. In a known uniform field, fit a circle to the projection perpendicular to the field. The fitted curvature gives

p_{⊥} = ∣ q ∣ B r_{L} .

If the instrument also measures the advance along the field, the ratio of pitch to circumference gives $v_{∥} / v_{⊥}$ . A sequence of three-dimensional points can then distinguish a low-momentum tightly curved track from a high-momentum broad track. The charge sign follows from curvature only after the magnetic-field orientation and the particle travel direction have both been established.

Finite spatial resolution biases a small-radius fit because only a few detector samples span the orbit. Multiple scattering can make a high-momentum trajectory appear more curved, while energy loss changes radius along a long path. A robust analysis fits short segments, compares curvature across adjacent segments, and propagates the field-calibration uncertainty into momentum. The radius formula therefore provides the physical relation; it does not replace the measurement model.

Equations of motion and phase-space description

Take a uniform, time-independent magnetic field in the positive z direction and write the transverse velocity as $v_{⊥} = (v_{x}, v_{y})$ . The Lorentz force has no z component, so the transverse Newton equations are

m \frac{d v _{x}}{d t} = q B v_{y}, m \frac{d v _{y}}{d t} = - q B v_{x} .

With the signed cyclotron frequency $Ω = q B / m$ , differentiating either equation once gives a simple harmonic equation: $d^{2} v_{x} / d t^{2} + Ω^{2} v_{x} = 0$ , and likewise for $v_{y}$ . The two components are not independent oscillators. One is shifted by a quarter cycle relative to the other because each component's rate of change is set by the other component. A convenient phase choice is

v_{x} = v_{⊥} cos (Ω t + ϕ), v_{y} = - v_{⊥} sin (Ω t + ϕ) .

The signed frequency $Ω = q B / m$ determines the rotation direction for any field and coordinate orientation. Changing the sign of charge reverses that rotation without changing its speed or radius.

The transverse velocity components differ by one quarter cycle, which keeps $v_{⊥}$ constant. When $v_{x}$ has its positive maximum, $v_{y} = 0$ and $d v_{y} / d t = - Ω v_{x}$ . For positive charge in a positive z-directed field, the velocity turns from positive x toward negative y; negative charge reverses that rotation.

Initial transverse position and velocity determine four constants: two centre coordinates, the speed $v_{⊥}$ , and phase $ϕ$ . For example, measured values at a reference time give $x_{c} = x + v_{y} /Ω$ and $y_{c} = y - v_{x} /Ω$ , after which the phase follows from the normalized velocity components. There is no independent radial acceleration to solve for; the apparent centripetal acceleration is already contained in the coupled Cartesian equations. A numerical integrator should preserve the circular velocity locus and the fixed centre over many periods. A scheme that updates one component with a new value and the other with an old value may introduce an artificial growth or decay of speed, even though the exact magnetic force has no such effect.

Integrating the velocity components gives the transverse position in a form that makes the orbit centre explicit:

x = x_{c} + \frac{v _{⊥}}{Ω} sin (Ω t + ϕ), y = y_{c} + \frac{v _{⊥}}{Ω} cos (Ω t + ϕ) .

Equivalently, the combinations $x + v_{y} /Ω$ and $y - v_{x} /Ω$ remain constant. They are the centre coordinates $x_{c}$ and $y_{c}$ for the chosen sign convention. The geometric radius is $r_{L} = v_{⊥} /∣Ω∣ = m v_{⊥} / (∣ q ∣ B)$ . These relations separate a particle's instantaneous position from the fixed centre about which it rotates, a distinction needed when comparing tracks with different starting phases.

In velocity phase space, the point $(v_{x}, v_{y})$ moves around a circle of radius $v_{⊥}$ at angular rate $∣Ω∣$ . The phase-space circle is conserved because the magnetic force does no work: $v \cdot q (v \times B) = 0$ . Thus transverse kinetic energy, total speed, and the parallel velocity are constant in this ideal field. A correct numerical integration preserves the velocity-phase circle, guiding centre, and speed.

Relativistic motion and the limits of the cyclotron period

At relativistic speed, the momentum rather than the velocity is the natural quantity in the magnetic equation of motion:

\frac{d p}{d t} = q v \times B, p = γ m v, γ = \frac{1}{1 - v ^{2} / c ^{2}} .

A static magnetic field remains perpendicular to velocity, so it does no work and keeps $γ$ constant during an individual orbit. For motion perpendicular to a uniform field, the momentum magnitude is constant and the curvature relation becomes $p_{⊥} = ∣ q ∣ B r$ . Replacing $m v_{⊥}$ by $p_{⊥} = γ m v_{⊥}$ gives the relativistic angular frequency and period:

ω_{c} = \frac{∣ q ∣ B}{γ m}, T_{c} = \frac{2 π γ m}{∣ q ∣ B} .

The orbit radius grows with momentum, while the revolution frequency falls as the Lorentz factor grows. The familiar mass-independent nonrelativistic frequency is therefore an approximation valid only when kinetic energy is small compared with rest energy. Its failure is not a change in the magnetic force law; it follows from the increasing relation between momentum and velocity.

The momentum substitution also clarifies what remains unchanged. The transverse curvature equation still reads force equals rate of momentum-direction change, so a measured radius continues to determine transverse momentum through $p_{⊥} = ∣ q ∣ B r$ without a separate velocity estimate. What changes is the conversion from that momentum to speed and orbital rate. For a particle with a parallel velocity component, the transverse rotation frequency has the same $1/ γ$ reduction because the total energy fixes one common Lorentz factor. The particle follows a helix whose pitch is enlarged by the longer period, while its transverse radius is set by transverse momentum. These distinctions matter in track reconstruction: a large radius can represent high momentum, but timing information is needed to test the accompanying relativistic frequency shift.

Relativistic fall of the cyclotron frequency. A fixed drive frequency (dashed) stays flat, while the true revolution rate

∣ q ∣ B / (γ m)

decreases as energy and the Lorentz factor rise; the widening gap is the phase slip a fixed-frequency cyclotron must correct.

A conventional cyclotron drives particles across an accelerating gap with a fixed radio-frequency voltage. At low energy, the orbital period matches the RF period, so a particle reaches the gap when the electric field has the accelerating polarity on every turn. As energy rises, $γ$ increases, the particle takes longer to return, and its arrival phase lags the fixed RF waveform. This phase slip eventually places the particle in a weakly accelerating or decelerating part of the cycle. Increasing the magnetic field alone changes the frequency in the opposite direction only if it is programmed to compensate for the increase in $γ$ .

A synchrocyclotron makes that correction by sweeping the RF frequency downward as the orbit frequency falls. The drive is synchronized to an accelerating packet over a programmed energy range rather than held at one frequency for every radius. The method restores phase coherence for particles in the selected packet, although its repeated frequency sweep gives lower average beam current than a fixed-frequency machine. More elaborate synchrotrons also change magnetic field and RF frequency together, keeping a chosen closed orbit while momentum increases.

Measure the revolution interval, track radius, and RF arrival phase to test relativistic phase slip. Track radius in a calibrated field gives momentum, so plotting revolution frequency against $p$ tests $ω_{c} = ∣ q ∣ B / (γ m)$ . At an RF gap, phase-sensitive beam monitors compare arrival time with drive phase. A steadily drifting phase, broadened arrival-time distribution, or loss of energy gain indicates inadequate frequency programming.

Magnetic rigidity and bending measurements

A uniform magnetic field converts transverse momentum into a geometric curvature. The relation

p_{⊥} = ∣ q ∣ B r_{L}

is often written in accelerator and detector work as magnetic rigidity. A particle with large momentum requires either a large radius or a stronger field to bend by a specified amount. The relation contains no kinetic-energy approximation when $p_{⊥}$ is interpreted as relativistic momentum. It is therefore a practical bridge between a measured track shape and a particle momentum scale.

Suppose a charged particle crosses a uniform-field region of path length $ℓ$ and changes direction through a small bend angle $β$ . The circular-arc geometry gives

β = \frac{ℓ}{r _{L}} = \frac{∣ q ∣ B ℓ}{p _{⊥}} .

The field-length product $B ℓ$ sets the momentum kick. Doubling field strength, magnet length, or charge magnitude doubles the bend angle at fixed momentum. Doubling transverse momentum halves it. A real dipole magnet has fringe regions at its entrance and exit, so an experiment uses the calibrated integral $\int B d ℓ$ along the reference path rather than a nominal central field times a mechanical length.

Momentum analysis in a uniform bending field. The track enters straight, curves through the magnet gap, and leaves deflected by the bend angle

β = ∣ q ∣ \int B d ℓ / p_{⊥}

; a wider gap or stronger field bends a given momentum more.

Track measurements usually use sagitta instead of an explicitly measured bend angle. A circular arc observed between two outer measurement points separated by chord length $L$ has sagitta $s$ , the maximum distance from chord to arc. When $L$ is much smaller than the radius,

s ≃ \frac{L ^{2}}{8 r _{L}}, p_{⊥} ≃ \frac{∣ q ∣ B L ^{2}}{8 s} .

The inverse dependence on sagitta makes high-momentum measurements demanding. Doubling momentum halves the deflection and halves the small sagitta. Detector point resolution, mechanical alignment, multiple scattering, and the uncertainty in the magnetic map can then dominate the fitted curvature. A broad track with a nearly straight chord is often a less precise momentum measurement than a tightly curved low-momentum track, even when the broad track crosses more detector layers.

Sagitta of a curved track. Three measured points fix a chord of length

L

and a mid-chord offset

s ≃ L^{2} / (8 r_{L})

; because

s

falls as

1/ p_{⊥}

, nearly straight high-momentum tracks give the smallest, hardest-to-measure offset.

The sign of curvature is as valuable as the magnitude. A field-map convention specifies the direction of $B$ in the detector coordinate system. The ordered hit sequence gives the direction of travel. The Lorentz-force direction then determines the charge sign. Reversing the hit order or using a field map with the opposite polarity reverses the inferred sign. Reconstruction software should retain those conventions with the fitted parameters throughout the analysis.

An uncertainty estimate begins with the differential form of the rigidity relation. With small independent fractional uncertainties, the magnitude estimate is

(\frac{δ p _{⊥}}{p _{⊥}})^{2} ≃ (\frac{δ B}{B})^{2} + (\frac{δ r _{L}}{r _{L}})^{2} .

The radius uncertainty contains statistical point resolution and systematic alignment error. A scale error in the magnetic field shifts every momentum in the same direction; random hit errors broaden individual fits. Keeping those components separate helps a calibration team decide whether additional events, tighter alignment, or a better field survey will improve the result.

Departures from the ideal uniform orbit

The circular and helical formulas assume a static, spatially uniform magnetic field and no force except the magnetic Lorentz force. A uniform electric field changes that equation of motion. Its component parallel to velocity can change kinetic energy, while a transverse component shifts the path away from a circle. A velocity selector deliberately uses this competition. In an accelerator gap, the electric field transfers energy and the magnetic field bends the growing momentum; treating either region as a magnetic-only orbit would miss the actual energy transfer.

Collisions change a track in a different way. Elastic scattering changes direction through localized deflections, producing a sequence of circular segments with slightly different fitted centres. Ionization and radiation lower momentum, so a particle can curve more tightly later in a detector. A fit that enforces one radius over a long material path can then average away physical energy loss and underestimate the uncertainty. Segment-by-segment curvature and a material map distinguish such effects from a magnetic-field calibration error.

Radiation from accelerated charge is usually tiny for slow heavy particles, yet it sets an important high-energy limit for light particles in circular machines. Synchrotron radiation carries energy away while the magnetic force itself remains perpendicular to instantaneous velocity. The energy loss arises through the emitted electromagnetic field, not from a direct magnetic-work term. Radio-frequency systems replace that loss in storage rings, and the required power rises strongly with particle energy and with tighter bending radius.

Field nonuniformity can be handled locally when the field changes little over one gyroradius. The instantaneous force remains perpendicular to velocity, but the circle centre and radius evolve as the particle samples different field strength. The resulting guiding-centre drifts belong to a broader nonuniform-field treatment. In a uniform-field measurement, map boundaries and fringe fields should be included in the propagated trajectory; discarding them because the central region is uniform changes the effective bending integral.

Worked example (Proton helix in a 0.5 T field). A proton enters a uniform $0.500 T$ field at speed $v = 3.00 \times 1 0^{6} m s^{- 1}$ , its velocity making a $30. 0^{\circ}$ pitch angle with the field. Only the transverse component sets the radius, so resolve the speed first:

v_{⊥} = v sin 30. 0^{\circ} = 1.50 \times 1 0^{6} m s^{- 1}, v_{∥} = v cos 30. 0^{\circ} = 2.60 \times 1 0^{6} m s^{- 1} .

With $m_{p} = 1.67 \times 1 0^{- 27} kg$ and $e = 1.60 \times 1 0^{- 19} C$ ,

r_{L} = \frac{m _{p} v _{⊥}}{e B} = \frac{( 1.67 \times 1 0 ^{- 27} ) ( 1.50 \times 1 0 ^{6} )}{( 1.60 \times 1 0 ^{- 19} ) ( 0.500 )} = 3.13 \times 1 0^{- 2} m .

The period comes from the cyclotron rate, which is independent of speed and pitch angle,

T_{c} = \frac{2 π m _{p}}{e B} = 1.31 \times 1 0^{- 7} s,

and the axial advance in one turn is the pitch,

p = v_{∥} T_{c} = (2.60 \times 1 0^{6}) (1.31 \times 1 0^{- 7}) = 3.40 \times 1 0^{- 1} m .

Radius and pitch differ by more than an order of magnitude because the proton moves mostly along the field. Substituting the full speed $v$ for $v_{⊥}$ would double the radius, the most common error here. Both limits check: $p \to 0$ as $v_{∥} \to 0$ (a circle), and $r_{L} \to 0$ as $v_{⊥} \to 0$ (straight drift along the field).

Two projections of the worked proton helix. Seen along the field the orbit is a circle of radius

r_{L} = 3.13 cm

; seen across the field it advances one pitch

p = 34.0 cm

per turn. Keeping the projections separate prevents substituting the full speed for the transverse speed in

r_{L}

Suppose the measured radius is $3.13 cm$ with an uncertainty of $0.20 mm$ and the field calibration has relative uncertainty $0.30%$ . The radius term contributes about $0.64%$ to transverse-momentum uncertainty. Combining independent radius and field terms in quadrature gives a relative uncertainty near $0.71%$ . That estimate presumes the fitted circle is unbiased. A detector offset that shifts every hit in one direction can change curvature systematically and must be constrained by alignment data, charge-sign reversal, or tracks entering from the opposite side.

The proton speed here is only about one percent of the speed of light, so $γ - 1$ is roughly $5 \times 1 0^{- 5}$ . The nonrelativistic period is adequate at this precision. Repeating the same calculation for an electron at high kinetic energy requires relativistic momentum even when the observed path still resembles a circle. The visual shape of a track does not determine which momentum model is appropriate; the energy scale does.

Numerical propagation and validation

A field map sampled at discrete points gives the coupled trajectory equations equations

\frac{d r}{d t} = v, \frac{d p}{d t} = q v \times B (r) .

A time step should resolve both the cyclotron period and the spatial scale over which the magnetic field changes. In a uniform field, first propagate many periods with no electric field and compare the numerical result with an exact circle or helix. The fitted speed, radius, and orbit centre should remain constant to the intended tolerance. A trajectory that slowly spirals outward or inward is a numerical artifact unless another physical force has been included.

The Boris rotation method is widely used because it rotates momentum under a magnetic update while preserving its magnitude accurately in the absence of electric field. Runge--Kutta methods can also produce accurate short trajectories, but their step size must be checked against long-time energy drift. The choice of integrator does not remove the need for a field-map interpolation test: a coarse or discontinuous map can introduce artificial kicks even when the time integrator is stable.

Measure step-size convergence with physical outputs and the solver's internal error estimate. Halve the time step, refit curvature and arrival phase, then compare those quantities with the measurement precision. Smooth-looking coordinates can still accumulate an unacceptable phase error over many turns. In a spectrometer, a small phase error may matter little if only curvature is used; in a cyclotron, the same error can move a particle out of the accelerating RF phase.

Energy and momentum checks need the correct scope. Magnetic force alone preserves speed, yet a field map that includes fringe regions can change the direction of the parallel and transverse components as the local field direction changes. A validation log records total speed, local field magnitude, fitted radius in a uniform reference region, phase error relative to an analytic orbit, and any work done by electric fields. Those quantities separate physical departures from uniform motion from an integration failure.

Detector coordinates, projections, and uncertainty

Track curvature is measured in the plane perpendicular to the local magnetic field. In a solenoidal detector with field approximately parallel to the laboratory $z$ axis, that plane is the transverse $x$ -- $y$ plane. A helix fit returns transverse momentum from its projected circle and returns the longitudinal momentum from the advance in $z$ per turn. The total momentum is

p = p_{⊥}^{2} + p_{∥}^{2}, tan α = \frac{p _{⊥}}{p _{∥}} .

Viewing a helical track from the wrong direction can conceal its curvature. A side view along a transverse axis shows an oscillating projection whose wavelength is the pitch. A view parallel to the field shows a circle. The same three-dimensional trajectory can look nearly straight in one projection and tightly curved in another. Detector displays, fit diagnostics, and written solutions should identify the projection before quoting a radius or a bend sign.

A curvature fit minimizes the signed distance between measured hit positions and a model circle or helix. The residual pattern matters as much as the residual size. Random measurement errors produce residuals that change sign without a spatial pattern. A detector layer shifted outward produces a common residual sign at that layer. A field-scale error changes curvature consistently across tracks with different radii. Energy loss produces a gradual change in curvature with path length. These signatures guide calibration because they identify which part of the measurement model needs correction.

Point resolution is only one term in the uncertainty budget. A field map has a scale uncertainty and local interpolation uncertainty. Mechanical survey establishes the positions and rotations of detector layers. Material before or between layers causes multiple Coulomb scattering, which adds a random bend that is largest for low momentum particles. A standard estimate for the rms scattering angle in thickness $x$ is

θ_{0} ≃ \frac{13.6 MeV}{β p c} ∣ z ∣ \frac{x}{X _{0}} [1 + 0.038 ln (\frac{x}{X _{0}})],

where $X_{0}$ is radiation length. The expression estimates a distribution width; an individual event can scatter more or less strongly. Curvature fitting must allow these random changes instead of interpreting every departure from a circle as an incorrect field polarity.

In a short track with hit-position uncertainty $σ_{x}$ , the sagitta uncertainty often scales with $σ_{x}$ divided by a geometry factor. Extending the lever arm $L$ improves curvature resolution rapidly because the sagitta grows as $L^{2}$ . Adding material simply to add more layers can work against that gain when scattering becomes large. Detector design balances magnetic field, lever arm, point resolution, and material budget rather than maximizing any one quantity alone.

Calibration tracks provide a direct test. A particle species with known momentum can be sent through the detector at both charge signs or both field polarities. The fitted curvature magnitude should agree after the sign convention is reversed. Cosmic-ray tracks crossing the detector from above and below offer a complementary alignment test because the same physical path can be reconstructed in opposite directions. Disagreement between these reversals identifies an offset that a single sample of tracks would hide.

Momentum, energy, and charge-state interpretation.

Magnetic curvature measures momentum divided by charge magnitude. A measured radius alone cannot identify mass, kinetic energy, and charge state simultaneously. For nonrelativistic motion, a separate speed measurement gives

\frac{m}{∣ q ∣} = \frac{B r _{L}}{v _{⊥}} .

At relativistic energy, combine curvature with a time-of-flight, calorimetric, or Cherenkov speed measurement and use $p = γ m v$ . A mass spectrometer specializes this combination with a velocity selector and a known charge state. A tracking detector often measures momentum first and identifies particle species by additional detector responses.

Charge-changing interactions illustrate the same limitation. If an ion loses an electron in material, its charge magnitude rises and its later curvature changes even at nearly unchanged momentum. A radius change can therefore signal energy loss, a charge-state change, a field change, or a combination. Independent energy deposition, timing, and field-map data determine which explanation is physically consistent. The track geometry provides a strong constraint within the full particle-identification measurement.

Magnetic fields also establish an acceptance range. A particle with gyroradius much larger than the detector aperture bends too little to remain inside enough layers for a reliable fit. A particle with very small radius can curl repeatedly and cross the same layers many times, complicating hit association. The usable momentum range is set by detector size, field strength, readout spacing, and the material budget. Stating those limits prevents a curvature formula from being applied outside the region in which a track can actually be reconstructed.

Analysis protocol for an unknown track.

An orbit reconstruction begins with the coordinate and field conventions. Record the magnetic-field direction in the detector axes, the definition of positive curvature, the ordered direction of the hit sequence, and the unit system used by the field map. These choices determine the charge-sign result. A curvature number without an orientation convention carries only a magnitude.

Select a track segment that lies inside the mapped uniform region and contains enough separated hit positions to constrain a circle. Exclude entrance and exit fringe regions when the goal is a simple uniform-field radius. Fit the transverse projection, inspect residuals by layer and by arc length, then repeat the fit after removing one hit at a time. A radius that changes substantially when one hit is removed is dominated by leverage or an outlier; its nominal least-squares error understates the measurement fragility.

Convert the fitted radius to transverse momentum with the signed charge hypothesis kept explicit:

p_{⊥} = q r_{L} \overset{n}{^} \times B, ∣ p_{⊥} ∣ = ∣ q ∣ B r_{L} .

The cross product fixes the curvature orientation from the field direction and the charge sign. The magnitude expression supports a quick scale check. In SI units, tesla times coulomb times metre has units of momentum. In laboratory units, any numerical conversion factor must accompany its declared momentum, field, and length units; copying a factor from a different unit convention changes a track momentum by a large scale factor without visibly changing the fit.

Add longitudinal information only after the transverse fit has passed its checks. Time-of-flight gives speed through a known path length. Stereo hit positions or multiple planes give the axial advance per revolution. A calorimeter measures energy after material interactions. Combining two independent quantities exposes invalid assumptions: a radius and time-of-flight pair can reveal a charge-state change, while a radius and calorimetric energy pair can distinguish a relativistic particle from a slow high-mass particle.

The fitted covariance matrix must follow the reported momentum. Statistical uncertainty from hit resolution can be small while the total uncertainty is set by field calibration or alignment. Quote these components separately when they have different correlation structures. A common field-scale shift affects an entire dataset coherently and can move a resonance mass or beam-energy result. Point resolution broadens individual tracks and often averages down in an ensemble. A single combined error bar conceals the correction strategy.

Repeated reversal tests expose sign and alignment mistakes. Reverse the magnetic field while holding the detector geometry fixed, or compare oppositely charged particles with matched momentum. The physical curvature direction reverses. A fitted offset that remains unchanged under reversal belongs to geometry or readout; an offset that reverses with the field can arise from field-map or charge-sign conventions. This symmetry test is stronger than visual agreement with one expected arc because it changes the underlying Lorentz-force sign.

Dimensional, limiting, and physical checks.

Each result has limiting cases that should be inspected before it is used. As $B \to 0$ , the gyroradius tends to infinity and the local path becomes straight. As $v_{⊥} \to 0$ , the radius tends to zero while the particle moves along the field with finite parallel speed. As $v_{∥} \to 0$ , the pitch tends to zero and the helix becomes a circle. Reversing either charge sign or field direction reverses the rotation sense; reversing both restores it. These checks catch most sign and component errors without requiring numerical values.

The magnetic force has units of newtons:

[q v B] = C \frac{m}{s} \frac{kg}{C s} = N .

The radius has units of metres and the cyclotron frequency has units of inverse seconds. An expression for period that increases with field strength, or an orbit radius that uses total speed when a parallel component is present, fails these physical checks. A result that predicts changing speed in a static magnetic-only region has included an electric field, a collision, numerical energy drift, or an incorrect force direction.

The charge-state assumption deserves an explicit check for ions. A spectrometer may accelerate singly charged ions, yet a collision or stripping foil can produce a different charge state before the analyzing magnet. The measured curvature then corresponds to a different momentum-to-charge ratio. Isotope separation follows this momentum-to-charge dependence: two ions with equal momentum and different mass-to-charge ratios follow different radii, while two charge states of the same isotope can also follow different radii. The apparatus must establish charge state through source preparation, energy selection, or complementary detector signals.

The word uniform refers to the modeled field over the fitted orbit segment. A large instrument can contain central and fringe regions with different fields. Carry a radius from a central calibration region into a fringe region only after trajectory propagation through the measured field map. Field-integral methods handle that transition directly. The local circle remains a diagnostic, while the full path uses the actual spatial field.

Separate measurement uncertainty from model inadequacy. Random hit noise, thermal sensor drift, and finite field-probe resolution broaden a valid result and enter a covariance estimate. Electric stray fields, energy loss, charge exchange, and unmodelled material change the equation of motion. The second group requires an extended trajectory model rather than a larger error bar.

Uniform-field model boundaries in practice.

The uniform-field fit is local. State the fitted spatial interval, time interval, field map, and charge-state assumption beside every quoted radius or momentum.

departure from the local model	measurement signature	required treatment
fringe or field-gradient region	curvature changes along the path	propagate through the measured field map
electric potential or accelerating gap	speed changes across the segment	include the electric term in the Lorentz equation
gas or material interaction	kink, energy loss, or scattering tail	split the track or apply a material model
pulsed field	bend depends on arrival phase	pair the track timestamp with the field waveform
field-map tilt	bias between parallel and transverse momentum	apply alignment survey and reference-track constraints

Field extent. A particle can follow a near-circular arc in the central field and then receive additional bend in a fringe region. A central radius is a valid local diagnostic; downstream transport still requires the field integral along the complete path.
Electric contribution. Patch potentials, drift electrodes, and accelerating gaps add the electric term to the Lorentz force. Compare measured speed before and after the fit segment whenever an energy change is plausible. A changing speed excludes a magnetic-only model even when a projected arc appears circular.
Matter interaction. Collisions in gas produce random scattering. Solid material produces average ionization loss and occasional hard kinks. Fit separate uniform-field segments on either side of a kink; one circle through the kink replaces a physical event with a spurious intermediate curvature.
Time and alignment. In a pulsed magnet, match the particle arrival timestamp to the field waveform and field-probe record. Define parallel velocity relative to the local field direction, not to a mechanical ruler. A map tilt can transfer reconstructed momentum between parallel and transverse components while leaving total momentum nearly unchanged.
Report and reversal test. Record charge convention, field-map uncertainty, fit segment, coordinate axes, material corrections, and relativistic treatment. Send a reference beam of known momentum through the prescribed central arc and compare its inferred field integral with an independent probe map. Repeat after reversing field polarity. A polarity-dependent difference indicates a convention, hysteresis, or timing problem; a polarity-independent scale difference indicates the map, geometry, or reference-momentum calibration.
Period check. At nonrelativistic energy, equivalent orbital phases are separated by $2 π m / (∣ q ∣ B)$ , independent of radius. A measured radius dependence can indicate gradients, electric acceleration, relativistic momentum, or phase-pickup bias. Radius and period together test more of the model than a circular projection alone.

These checks validate a stated field region, charge state, and energy range. A track through a large instrument can cross several local models. Connect those segments with the Lorentz equation and measured boundaries before assigning one global momentum or uncertainty.

Segment acceptance record.

Geometry selection. Retain the hit coordinates, the chosen fit interval, the local field direction, and the entrance and exit boundaries. A radius fitted from hits spanning a fringe region is not interchangeable with a radius fitted wholly inside the central map region.
Field variation. Evaluate the mapped field at the hit positions and report its variation across one fitted gyroradius and along the fitted arc. The average field is sufficient only when that variation is smaller than the measurement precision required for the reported momentum.
Residual structure. Store signed residuals versus arc length and detector layer. Random signs are consistent with point resolution. A smooth drift can indicate field-scale error or energy loss; one localized displacement can indicate scattering or a shifted layer. Quote a goodness-of-fit statistic together with the residual pattern rather than accepting a circle from its visual appearance.
Independent momentum check. Convert the fitted transverse radius to $p_{⊥} = ∣ q ∣ B r_{L}$ , then compare with timing, calorimetry, or a reference beam when available. The comparison separates field-scale error from a charge-state or material correction. A radius alone measures momentum divided by charge magnitude.
Boundary handoff. Pass the fitted position, direction, covariance, charge hypothesis, and local field to the next transport region. Replacing that state by a single global radius discards the information needed to propagate through a nonuniform boundary or to combine several detector regions.

A reference-track run should use the same reconstruction window, hit selection, and field interpolation as the unknown-track analysis. Fit the reference under both field polarities and compare the inferred $p_{⊥} /∣ q ∣$ with its independently known value. Retain the signed curvature, residual distribution, and fitted period when timing is available. This control constrains field scale, coordinate handedness, and charge-sign convention simultaneously. Arc shape alone cannot establish those three calibration conditions.