Radio-Frequency Linear Accelerator

Why do longer drift tubes help faster particles keep the beat? A drift tube shields the particle while the field turns back toward acceleration. Faster particles travel farther during each cycle, so later tubes must be longer. Timing can tighten when early particles gain less energy and late particles gain more; the opposite field slope makes the arrival spread grow.

Drag the tube-length adjuster and follow ions through an alternating field. Compare bunches with similar energy gain but different arrival spread, retaining gap events and energy observations.

Arrive when the field can help

Mean exit energy

2400

u=1

Measured energy gain

400

u=1

Observed exit-time spread

2

Within the design window

32

Gap-voltage amplitude

150\,V

RF phase shift

0\,{}^\circ

Tube-length scale

1

Source-energy spread

5\,\mathrm{eV}

Source-phase spread

10\,{}^\circ

Gap width

5\,\mathrm{mm}

Hold the gold adjuster and drag sideways. Tube lengths and live observations change before release. Focus the canvas and use arrow keys for the same adjustment.

Source, timing and model limits

Number of drift tubes

5

Source-spread controls are Gaussian scales before truncation; draws are bounded at three scales. Plots and arrival widths describe the actual finite sample.

The known reference is a singly charged 40-u ion at 2000 eV. All gaps share one 5-MHz field phase: Alvarez zero mode. Tube lengths are designed once for the reference particle; changing the source never redesigns them. Each gap contains a prescribed uniform axial electric field and takes finite time to cross. Radial cutaway sizes and ion markers are enlarged; cavity electrodynamics, transverse focusing, space charge and wall losses are omitted.

Thirty-two independently tagged, dilute exposures are separated by 100 RF cycles. The animation folds out those known whole cycles and overlays their ideal forward paths on the same programme clock, slowed two million times. It does not align each ion to its unknown launch time. Arrival spread is an observed detected-event RMS; useful counts refer to a fixed design phase/energy window, not total transmission or cooling.

Timing pickups and the probe transfer response are ideal teaching detectors with explicit counter/ADC cells. Independent voltage, position and clock standards, withheld checks and a species certificate support raw-only reconstruction. First-order shared reference covariance and observation cells propagate to outputs; means also include finite-source sampling. Phase or acceptance can be withheld independently of supported flight energy. Nonrelativistic readout is restricted to 100–10000 eV for the known species.

Model assumptions

Three to five shielded drift tubes use Alvarez zero mode: all prescribed uniform axial gap fields share one 5-MHz phase. The known design reference is a singly charged 40-u ion at 2000 eV; tube lengths are designed once from its trajectory. Finite-gap work and time are continuous, not instantaneous kicks. Thirty-two independent dilute exposures are separated by 100 RF cycles. Replay folds out known whole programme cycles, overlays ideal forward paths on the common programme clock and slows them two million times. Axial coordinates retain metres; radial cutaways and ion markers are enlarged. Cavity electrodynamics, transverse focusing, space charge, wall losses and real detector bandwidth are omitted. Timing pickups and probe response are ideal teaching assumptions. Independent voltage, position and clock standards, a species certificate and heldout checks support raw-only reconstruction with first-order shared-reference and observation-cell covariance. Flight-energy readout is restricted to 100–10000 eV for the known species. Unresolved phase retains supported flight energy. Useful counts describe only a fixed design energy/phase window, including ambiguous membership. Observed arrival spread has no instrumental deconvolution and does not establish cooling or total transmission efficiency. Each capture has an independent virtual clock origin and UTC validity anchor; controls/reset preserve saved history.

Observed run

Timing pickups give the energy points. The phase plot is centred on the design phase; dashed guides mark its useful window. Gap work comes from adjacent flight energies.

Grey bars show source arrivals; cyan bars show detected exits. Each uses its own observed mean, without per-ion launch alignment. Bins span at least one calibrated clock tick.

Individual events and finite-gap checks

Scroll sideways, or focus the table and use the arrow keys, to read every column.

Exposure	Tube	Energy / eV	Standard uncertainty / eV	Gap work / eV	Field check

Raw counter/ADC observations and independent standards

Each run is a new virtual acquisition with its own clock origin and UTC validity anchor. Parent links preserve comparison history, not continuous occupancy of one instrument. JSON includes separately labelled ideal forward diagnostics; CSV retains only raw observations. Imported unknown data are never regenerated from settings.

Import JSON / CSV

Scroll sideways, or focus the table and use the arrow keys, to read every column.

Channel	Role	Reference value	Standard uncertainty	Raw code	Actual clock interval

Scroll sideways, or focus the table and use the arrow keys, to read every column.

Exposure	Gate start / tick	Gate end / tick	Channel	Event / tick

Focus question: Why must tube lengths grow with speed, and why must the field phase also be chosen?
One-sentence intuition: Zero-mode gaps share one phase. The reference crosses successive gap centres one full RF period apart. On a rising accelerating field, earlier particles gain less and later particles gain more, allowing arrival times to tighten.

Core mathematical model

Work through a finite gap

\Delta K=q\int_{z_a}^{z_b}E_z\!\left(z,t(z)\right)\,dz

Positive ions travel forward along the axis. The field changes continuously during their crossing, so peak gap voltage need not equal the actual energy increment per charge.

Zero-mode synchronism

\Delta t_{\mathrm{centres}}=1/f,\qquad d\simeq v/f

Reference flight time between neighbouring gap centres is one full period. The length estimate assumes a narrow gap and little speed change. Actual tube lengths come from the finite-gap reference trajectory, not the half-period rule for opposite-phase gaps.

Flight-energy observation

v=\frac{\Delta z}{\Delta t},\qquad K=\tfrac12 mv^2

The two timing planes lie within one shielded tube. Distance and time have independent standards and heldout checks; an independent species certificate supplies mass and charge. Energy conversion stays inside the declared nonrelativistic domain.

Weak-field transit-time factor

T=\frac{\sin(\pi f g/v)}{\pi f g/v}

This signed factor multiplies the peak voltage and centre-phase cosine only for a uniform gap with nearly constant crossing velocity. Actual trajectories use the continuous gap solution. A negative factor also changes the phase interpretation.

Common difficulties

Does a longer tube provide more accelerating work?

Typical misconceptionThe ion accelerates throughout the metal tube.

Better mental modelAn ideal tube shields its interior field. The ion drifts there while the field reverses; exposed gaps do the work. Tube length controls arrival time.

Does a shorter pulse prove cooling?

Typical misconceptionNarrow arrival times guarantee a narrower energy distribution.

Better mental modelLongitudinal bunching exchanges timing and energy spread. Both coordinates matter; this apparatus does not infer temperature or cooling rate.

Run the experiment

01
Shorten the tubes
Hold the gold adjuster and drag left, or focus the canvas and press the left arrow.
What to observe: The ions reach subsequent gaps earlier and can encounter a decelerating field. The live geometry and observations change before release.
02
Compare similar energy gains
Compare “Catch the rising field” with “Same gain, growing timing spread”.
What to observe: Their mean energies can be similar while their arrival spreads differ. The field slope changes how timing errors feed back through energy gain.
03
Widen the gap
Choose “A gap takes time to cross” and open the event and finite-gap checks.
What to observe: The field rotates further during each crossing. Work cannot be obtained by simply adding peak voltages. Measured after-minus-before flight energies still test the continuous model.
04
Keep a run before changing it
Acquire a run, change phase and select the earlier run; export and reimport its raw CSV.
What to observe: Earlier counter events, encoder codes and standards remain unchanged. The apparatus retains the current setup, while the selected data are reconstructed from their own original channels.

Radio-Frequency Linear Accelerator

Cross each gap at a useful time

Core mathematical model

Work through a finite gap

Zero-mode synchronism

Flight-energy observation

Weak-field transit-time factor

Common difficulties

Does a longer tube provide more accelerating work?

Does a shorter pulse prove cooling?

Run the experiment

Shorten the tubes

Compare similar energy gains

Widen the gap

Keep a run before changing it

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