Synchrotron and Storage Ring

How can defocusing help confine a beam, and how do pickups guide an orbit correction? A magnet changes the position and angle seen by the next one. Suitable alternating strengths and spacing keep both transverse oscillations bounded. Turn-by-turn pickups and small measured steering steps then reveal how to correct the orbit.

Track a proton bunch through FODO magnets. Calibrate pickups, measure tune aliases and steering response, then observe a corrected orbit.

Model assumptions

A four-cell fixed-momentum linear paraxial ring transports a finite particle ensemble and second moments through hard-edge elements. Reference momentum is 100 MeV/c. Each scan platform independently reloads the source distribution after an ideal 20ms stabilized-current qualification; coherent motion has no damping. Pickups use a calibrated near-axis linear electrode response for a fixed bunch shape, with per-sample support; low-velocity proton electromagnetic frequency response is not solved. Uncertainty is conditional on the prescribed response and excludes omitted electromagnetic systematics. Envelope is a model prediction and electrodes measure the centroid. Turn sampling retains tune aliases. Uncertainty support is a finite-noise first-order approximation. Longitudinal RF, nonlinear multipoles, space charge and collective effects are omitted. Particle markers are enlarged and replay slowed. Radiation and beamlines belong to a separate instrument.

A pickup sees every passage

Pickup

Measured horizontal positionMeasured vertical position

Model reference · second moments

Transported widths are model predictions; electrodes measure the centroid.

Momentum changes the mean orbit

Independent momentum preparations; the withheld point checks the fitted slope.

Measured steering response and a fresh check

Immutable experiment history

Import JSON / CSV

Eight immutable records; raw codes, clocks and calibration remain exportable.

Inspect individual raw gates

Turn / pickup	Actual tick	R / T / L / B · ADC	Position support

Physical and calibration scope

A four-cell14.4m proton ring uses thick quadrupoles and45° sector bends. Every platform reloads the same initial distribution. Field plateaus have an ideal20ms qualification; there is no damping to a hidden closed orbit. The pickup is a prescribed near-axis linear response for a fixed bunch shape, calibrated by voltage and position standards. Its low-velocity electromagnetic frequency response is not solved. Position applies only within supported signal and wire ranges. Nonlinear fields, radiation, space charge and collective motion are outside this model.

All gate times and heldouts are retained. Independent ADC noise can average down; common voltage, wire, field and momentum references remain correlated. Output uncertainty is a first-order approximation; calibration heldouts use an explicitly counted comparison family.

Focus question: Which ring settings confine the bunch, and which steering changes do the recorded pickups support?
One-sentence intuition: Position and angle must travel together. The orbit estimate comes from calibrated electrodes and measured scans; the propagated beam width is a separate model prediction.

Core mathematical model

The two planes respond differently

\begin{aligned}x''+(h^2+k)x&=h\delta\\y''-ky&=0,\quad\delta=(p-p_0)/p_0\end{aligned}

The positive horizontal coordinate points outward and vertical points upward. Positive k focuses horizontally. Sector bends have curvature h. Only first-order transverse and momentum terms are retained.

Particle and beam matrix share transport

z_{j+1}=M_jz_j+d_j\delta,\qquad\Sigma_{j+1}=M_j\Sigma_jM_j^{\mathsf T}

The finite particle group and its covariance use the same thick-element map before losses. The homogeneous transverse matrix preserves the symplectic form. After clipping, surviving-particle moments are computed separately.

Calibrate position before averaging turns

r_x=\frac{V_R-V_L}{V_R+V_L},\qquad x=K_xr_x+x_{\rm off}

Independent voltage and wire-position standards determine the gains, pedestal, position scale and offset. Every electrode and position is checked against its own supported range. A low unresolved sum gives no position.

A sampled tune has aliases

x_n=c+a\cos(2\pi qn)+b\sin(2\pi qn),\qquad q\leftrightarrow1-q

The constant c separates the average orbit from undamped motion. Actual turn indices are retained. Once-per-turn real data cannot select an integer tune; every-other-turn data creates more aliases.

Measure the response and observe again

D_i=\frac{\partial\bar x_i}{\partial\delta},\qquad R_{ij}=\frac{\partial\bar x_i}{\partial\theta_j},\qquad\Delta\bar x\simeq R\Delta\theta

Independent momentum preparations measure dispersion. Positive and negative measured steering steps determine response columns, with a separate mixed-step check. A rank-supported solve is executed and tested with a fresh exposure. Shared calibration errors remain correlated.

Common difficulties

A narrow spot is not a whole-ring condition

Typical misconceptionThe smallest beam spot means the whole ring is stable.

Better mental modelPosition and angle evolve together. A crossing spot may expand downstream; stability concerns repeated transport around the complete lattice.

More turns do not erase a common zero

Typical misconceptionAveraging many turns removes all position error.

Better mental modelIndependent pickup noise can average down. The same calibration zero affects every turn and remains in the orbit and command uncertainty.

Run the experiment

01
Predict the missing family
Compare Stable FODO and Missing defocusing family. Inspect the quadrupole view and the physical loss marker.
What to observe: Both coordinates stay bounded in the stable lattice. Removing one family lets an excursion reach the physical pipe. The raw observations already taken remain available.
02
Separate an orbit from oscillation
Compare the baseline turn trace with Momentum scan. Select the independent negative and positive momentum platforms in the raw table.
What to observe: A fitted constant estimates the orbit without adding damping. The slope uses measured momentum readbacks and a withheld preparation, not a hidden model dispersion.
03
Use a measured adjustment
Choose Measure and correct, then compare before and newly observed residuals with the measured response map. Drag the first corrector control and acquire a fresh scan.
What to observe: The controller uses the scanned response and actual actuator limits. A new exposure decides whether the proposed correction improved the observed orbit. Reset starts another experiment and keeps saved history.
04
Keep an unresolved observation
Compare Every other turn and Weak pickup signal. Export JSON or CSV, import it and select the same record.
What to observe: Missing turns preserve their indices and extra aliases. Weak signals retain codes while unsupported position or response stays unreported. Rebuilding does not replace the archived signals.