Beam Diagnostics Bench

Why can a tiny spot hide a divergent beam? A screen measures position, so a small spot can still contain a wide range of particle angles. Changing a quadrupole reveals several position–angle projections. Calibrated screen widths then recover the projected phase-space area, provided resolution, clipping and a changing beam do not hide it.

Read four-electrode signals, independent bunch charge and screen pixels. Scan a finite quadrupole to recover projected emittance and Twiss parameters.

A spot is one projection

Drag the source, magnet or camera dial. Keep a scan to acquire new standards and pixels.

Entrance width scale

\beta/\mathrm{m}

1.5

Position–angle correlation

\alpha

1.2

Source steering

x_0/\mathrm{mm}

0

Quadrupole scan span

f_k

1

Optical blur and weak tails

Optical blur

\sigma_{\mathrm{opt}}/\mathrm{mm}

0.15

Weak-tail fraction

f_h

0

Acquiring independent standards and a screen scan…

Acquisition and model limits

Charge command

Q/\mathrm{pC}

Acquisition seed

A linear hard-edge proton beamline with a rectangular absorbing bore. Constant beam moments, ideal linear-cut electrodes and uniform affine imaging are assumptions. Reference failures withhold affected inference; old raw records stay unchanged.

Model assumptions

Synthetic proton diagnostics with independently acquired electrical, charge, position, imaging and magnetic standards. The finite hard-edge quadrupole transports paraxial, fixed-momentum particles through a rectangular absorbing bore. Seven fit exposures, two held-out strengths and a repeat test a stationary second-moment model. Pixel-centre moments retain signed dark-subtracted weights; resolution comes from independent finite-width beams. Statistical standard uncertainty includes shared reference fits and pixel-noise covariance; deterministic pixel-bin bounds are reported separately in the area interval. Geometric RMS emittance has no extra pi factor; normalization assumes known protons and certified rigidity. The linear field omits fringe fields, aberrations, dispersion, coupling forces and collective dynamics; a conservative perveance screen limits the space-charge budget. Screen clipping, unqualified collection, calibration-domain failures and unresolved area withhold affected inference while retaining raw data. Finite checks cannot certify absence of arbitrarily weak invisible tails. The displayed electrode law assumes ideal linear-cut plates, the CT a calibrated finite gate, and the camera uniform affine imaging. Hardware noise and dimensions are teaching assumptions. Transverse size is magnified, markers represent the distribution and motion is slowed.

Focus question: How can equally small spots hide different divergence?
One-sentence intuition: Projected phase-space area depends on position spread, angular spread and their correlation.

Core mathematical model

Normalized electrode difference

d_x=\frac{A_R-A_L}{A_R+A_L}

Independent electrical and mechanical standards calibrate ideal linear-cut electrodes to a centroid. A separate charge ring supplies absolute bunch charge.

The screen is a transported projection

\sigma_s^2=R_{11}^2\Sigma_{11}+2R_{11}R_{12}\Sigma_{12}+R_{12}^2\Sigma_{22}

The matrix includes the finite quadrupole and drift. Each strength has a newly acquired image; held-out strengths are excluded from the fit.

Projected RMS emittance

\varepsilon_{\mathrm{rms}}=\sqrt{\Sigma_{11}\Sigma_{22}-\Sigma_{12}^2}

The moments are centred on the measured beam. Statistical standard uncertainty and deterministic pixel-bin bounds are separate; unresolved area is withheld.

Common difficulties

Waist and area

Typical misconceptionA smaller bright spot always means a better beam.

Better mental modelParticles can share a narrow waist and leave at very different angles. Emittance includes that directional spread.

Common reference errors

Typical misconceptionMore photographs remove every calibration error.

Better mental modelA shared scale or reference affects the entire scan. Repetition does not average that common error away.

Run the experiment

01
Watch a scan
Compare the default screen images and width curves.
What to observe: One beam makes different spots as the field changes. Horizontal and vertical focusing act in opposite directions.
02
Compare equal spots
Choose Same spot, different angles.
What to observe: One projection can agree even when the full scan curves separate. The missing information is angular spread.
03
Blur the optics
Drag the camera dial, then keep a scan.
What to observe: Independent reference images are acquired again with the unknown images. Their measured widths support the resolution correction.
04
Steer off the screen
Drag the source or choose the lost-beam demonstration.
What to observe: Electrodes reveal the displacement. A mismatch between independent charge and screen light withholds full-beam area and preserves the pixels.

Beam Diagnostics Bench

Acquired pixels and independent readouts

From one spot to positions and angles

Core mathematical model

Normalized electrode difference

The screen is a transported projection

Projected RMS emittance

Common difficulties

Waist and area

Common reference errors

Run the experiment

Watch a scan

Compare equal spots

Blur the optics

Steer off the screen

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