E57 · Photoelectron microscopy

Momentum Microscopy: Time-of-Flight Cube

Record transverse momentum and arrival time in one exposure. Convert stored timestamps into energy, inspect slices of the same data cube, and tune drift energy and pulse rate. Detect pulse overlap, clipped time windows, calibration errors and single-hit pile-up.

Pulse identification $\text{—}$

Total recorded cube events $\text{—}$

Events in inspected energy slice $\text{—}$

Single-hit loss within open gate $\text{—}$

Momentum-aperture transmission $\text{—}$

Expected distribution in open gate $\text{—}$

Pulse period $\text{—}$

Latest modeled arrival with timing margin $\text{—}$

Local energy FWHM estimate $\text{—}$

Accepted rate before counting losses $\text{—}$

Recorded event rate $\text{—}$

Counts represented in converted energy range $\text{—}$

Experiment target $\text{—}$

Predict · tune · inspect

When does parallel counting distort the band map?

Retarded Fermi-level drift energy

E_d

5\,\mathrm{eV}

Photon pulse repetition

f_{rep}

1\,\mathrm{MHz}

Electrons per pulse before aperture

\mu

1

Acquisition exposure

t_{exp}

0.2\,\mathrm{s}

Timing response standard deviation

\sigma_t

0.1\,\mathrm{ns}

Electronic gate end

t_{end}

700\,\mathrm{ns}

Supplied trigger offset

t_0

25\,\mathrm{ns}

Energy-slice center

E_s

-0.105\,\mathrm{eV}

Momentum-cut row

k_y

0\,\mathrm{\mathrm{\mathring A}^{-1}}

Circular momentum acceptance

k_{max}

0.5\,\mathrm{\mathrm{\mathring A}^{-1}}

Target: momentum acceptance at least 0.4 inverse angstrom; isolated pulses; more than 10,000 events; gate coverage above 99 percent; expected pile-up loss below 10 percent; local energy width below 0.04 eV. Calibrate the known 25 ns trigger offset to within 0.5 ns.

Synthetic isotropic parabolic band with bottom −0.5 eV, fixed 40 K band occupation and 0.012 eV Gaussian emission linewidth. A cathode lens and ideal calibrated relay map both transverse momenta to a 25 by 25 detector. A circular momentum aperture limits acceptance and reduces transmission relative to a fixed 0.55 inverse-angstrom reference horizon. Pulse charge is specified before this aperture. The low-energy field-free drift length is 0.5 m; flight time follows nonrelativistic energy. Drift energy is the kinetic energy of a Fermi-level electron after retardation. The true fixed trigger offset is 25 ns; the user-supplied offset changes processing only. Integrating time-bin energy boundaries includes the energy-to-time Jacobian, Gaussian timing response and assumed space-charge shift and width. The time grid is 200–900 ns with 1 ns bins, and the electronic gate starts at 200 ns. The single-hit detector records only the earliest accepted electron in each pulse across the entire detector. It is a deliberately restrictive detector model, not a multi-hit DLD specification. Poisson arrival probabilities and conditional multinomial allocation give pulse-bounded counts; sparse binomial draws are exact, central draws use bounded rounded Gaussian approximations. A pulse period shorter than the modeled latest arrival or the open gate is rejected as ambiguous and yields no acquisition. This is a fail-closed teaching policy; wrapped spectra are not simulated. Lost window counts and pile-up loss are reported separately. Calibration, energy slice and momentum cut reuse the identical stored cube; out-of-range converted counts are reported by the represented-count readout. Displayed momentum images are gate sums of recorded events. Energy FWHM is a local linear timing estimate with bin-width and assumed space-charge contributions; it is not a calibrated instrument resolution. No real lens fields, angular aberrations, matrix elements, kz dispersion, Coulomb solver, detector dark counts or damage. Geometry is schematic and particle animation is not a physical clock.

Start with the essentials

Focus question: When does parallel counting distort the band map?
One-sentence intuition: Calibrate and slice the same recorded cube. Pulse overlap, gate clipping and pile-up are different failures and require different controls.

Core mathematical model

Field-free electron flight

t_f=L\sqrt{\frac{m_e}{2eE_d}}

Energy is in eV and flight length is in metres. The displayed clock also contains a trigger offset.

Timestamp energy conversion

E=\frac{m_e L^2}{2e(t-t_0)^2}-E_d

Time is in seconds here. The supplied trigger offset changes conversion without changing counts.

Single-hit saturation

\eta=\frac{1-e^{-\mu_g}}{\mu_g}

The earliest admitted event is recorded; the mean admitted count in the open gate controls efficiency.

Common difficulties

A recorded fraction needs a model

Typical misconceptionAll emitted electrons become recorded events.

Better mental modelCalibrate and slice the same recorded cube. Pulse overlap, gate clipping and pile-up are different failures and require different controls.

Run the experiment

01
Reduce pile-up
Compare clean pulses with crowded pulses at the same mean incoming rate.
What to observe: The single-hit detector loses more electrons and distorts the arrival spectrum when each pulse is crowded.
02
Slice the acquired cube
Change the energy slice, momentum row and supplied trigger offset.
What to observe: Converted views change while the raw arrival histogram and collector totals remain identical.
03
Check the flight window
Try overlapping pulse periods, close the gate early, then restore the clean preset and check the target.
What to observe: Overlap rejects an ambiguous acquisition; early gating removes late low-energy events.