---
title: "Momentum Microscopy: Time-of-Flight Cube Interactive Lab | Sandbox Physics"
description: "When does time-of-flight counting distort the band map in a momentum microscope? When the arrival times stop meaning what you assume."
canonical_url: "https://sandboxphysics.com/labs/momentum-tof/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/momentum-tof/"
last_modified: "2026-10-27"
---

E57 · Photoelectron microscopy

# Momentum Microscopy: Time-of-Flight Cube

**When does time-of-flight counting distort the band map in a momentum microscope?** When the arrival times stop meaning what you assume. If pulses come too close together, a slow electron can't be assigned to the right pulse. A short time window cuts off the latest arrivals. A wrong time zero throws off every energy. A second electron in the same pulse can go uncounted. Pushing the rate higher helps only while every arrival time can still be read correctly.

A stronger field pulls more electrons through the extractor opening. Slow them down in the drift tube, then compare the recorded arrival times and energy slices to see where timing and counting distort the band map.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

The band is synthetic: isotropic and parabolic, with its bottom at −0.5 eV, occupation fixed at 40 K and a Gaussian emission linewidth of 0.012 eV. A uniform planar extraction gap feeds an ideal calibrated relay and a 25 by 25 detector. The gap is 0.5 mm long with a 20 micrometer opening radius; a Fermi-level electron starts at 20 eV. These are assigned teaching values. The positive extractor voltage is the selected field magnitude times the gap. Axial acceleration conserves transverse momentum and adds the corresponding potential-energy drop to kinetic energy. The lateral displacement at the opening determines transmission. Zero field means a field-free gap with ballistic divergence, unlike real zero-surface-field lenses that retain focusing fields elsewhere. A circular momentum aperture limits the acceptance and lowers the transmission relative to a fixed 0.55 inverse-angstrom reference range. The pulse charge is normalized over the fixed source-energy interval from minus 0.8 to plus 0.05 eV before the extractor and momentum aperture; neither loss is renormalized away. The low-energy field-free drift is 0.5 m long, and the flight time follows the nonrelativistic energy. The drift energy is the kinetic energy of a Fermi-level electron after retardation. Arrival time includes the extraction gap as well as the field-free drift. Energy conversion solves that monotone clock relation with the recorded optical settings and measured momentum pixel. Fixed transit delays in the ideal relay are included in the 25 ns trigger offset; the offset you supply changes only the processing. The energy range of each time bin is integrated between its boundaries, including the energy-to-time Jacobian, a Gaussian timing response and an assumed space-charge shift and width. The time grid runs over 200–900 ns in 1 ns bins, and the electronic gate opens at 200 ns. The single-hit detector records only the earliest accepted electron in each pulse, over the whole detector. This detector model is deliberately restrictive; it is not the specification of a multi-hit DLD. Poisson arrival probabilities and a conditional multinomial split keep the counts within the number of pulses. Sparse binomial draws are exact, and central draws use a bounded, rounded Gaussian approximation. If the pulse period is shorter than the latest modeled arrival or the open gate, the acquisition is ambiguous and is rejected, so nothing is recorded. This rejection is a teaching choice; wrapped spectra are not simulated. Counts lost outside the window and counts lost to pile-up are reported separately. Calibration, energy slices and momentum cuts all reuse the same stored cube, and converted counts that fall out of range show up in the represented-count readout. The momentum images are gate sums of recorded events. The energy FWHM is a local linear estimate from timing, with bin-width and assumed space-charge contributions; it is not a calibrated instrument resolution. Fringing fields, field emission, electrical breakdown, angular aberrations, matrix elements, dispersion with perpendicular momentum, a Coulomb solver, detector dark counts and damage are left out. Extraction trajectories use the same constant-acceleration model. In that gap, transverse distances have an additional factor of 16 schematic magnification relative to axial distances. Displayed rays are Fermi-level references; the cube includes the full emission-energy distribution. Nonrelativistic motion is retained up to the maximum 5 keV gap energy gain. The downstream optics are schematic, and particle animation is not a physical clock.

Physics tutorial

## Parallel acquisition still has a clock and a count limit

**Background** The cathode lens projects an image of the electrons' momenta. A time-resolving detector adds the arrival time, so each event carries two momentum coordinates and one clock reading.

**Why it matters** Slowing the electrons down lengthens the flight and makes the timing more sensitive to energy, but it needs enough time between photon pulses. When pulses are crowded, single-hit counting favors electrons that arrive early.

Start with the essentials

- **Focus question**: When does time-of-flight counting distort the band map in a momentum microscope?
- **One-sentence intuition**: Calibrate and slice one recorded cube. Pulse overlap, a clipped gate and pile-up are different failures, and each needs a different setting to fix.

### Core mathematical model

#### Acceleration through the extraction gap

$$
\begin{aligned}a_z&=eF/m_e\\t_g&=\frac{2d}{v_{z0}+\sqrt{v_{z0}^2+2a_zd}}\\r_g&=v_\perp t_g\le R\end{aligned}
$$

The field accelerates the electron axially. Its transverse momentum stays fixed, so a shorter crossing time lowers the lateral displacement at the opening.

#### Flight time through the field-free drift

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

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

#### Energy calibration with both flight sections

$$
\begin{aligned}t-t_0&=t_g(E,k_\perp,F)+t_f(E)\\t_f(E)&=L\sqrt{\frac{m_e}{2e(E_d+E)}}\end{aligned}
$$

The recorded optics and momentum pixel define a monotone relation. Solve it for energy; changing the supplied trigger offset reprocesses the same events.

#### Single-hit detector efficiency

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

Only the earliest admitted event in each pulse is recorded, so the efficiency depends on the mean number of admitted electrons in the open gate.

### Common difficulties

#### Not every electron becomes an event

**Typical misconception** Every emitted electron is recorded as an event.

**Better mental model** Pile-up and a clipped time gate both lose electrons. Calibrate and slice the same recorded cube, and treat pulse overlap, gate clipping and pile-up as different failures that need different fixes.

### Run the experiment

1. 01

   #### Open the extraction cone

   Choose Weak extraction clips the cone. Raise Surface extraction field, then return to Count a clean parallel cube. Drag the gold aperture edge and compare it with changing Retarded Fermi-level drift energy.

   **What to observe:** The extractor and aperture remove momentum regions before counting. Drift energy changes the downstream clock; calibration does not recover intercepted electrons.

2. 02

   #### Reduce pile-up

   Compare Count a clean parallel cube with Same mean rate with crowded pulses.

   **What to observe:** When each pulse is crowded, the single-hit detector loses more electrons and the arrival spectrum is distorted.

3. 03

   #### Slice the recorded cube

   Change the Energy-slice center, the Momentum-cut row and the Supplied trigger offset.

   **What to observe:** The converted views change, while the raw arrival histogram and the detector totals stay the same.

4. 04

   #### Check the flight window

   Try Overlapping pulse periods and Close the gate too early, then return to Count a clean parallel cube and press Check target.

   **What to observe:** With overlapping pulses, the ambiguous acquisition is rejected. Closing the gate early removes the late, low-energy electrons.

Physical conclusion

Faster acquisition helps only when the timestamps and recorded probabilities can still be interpreted.

Fields & matter · Starter path

## From a field to an electron microscope

Map an electric field and steer a charged particle. Then find the electron's wavelength and use electron beams to see a crystal and a surface.

[Start here → **Make an Electric Field Visible** Move the probe around a dipole and check that the field always points downhill on the potential landscape. Observe · 8 min](https://sandboxphysics.com/labs/electric-field/)

1. [1 Make an Electric Field Visible](https://sandboxphysics.com/labs/electric-field/)
2. [2 A Charge Bending through Fields](https://sandboxphysics.com/labs/charged-particle/)
3. [3 Electron Wavelength & Voltage](https://sandboxphysics.com/labs/electron-wavelength/)
4. [4 Electron Diffraction & Reciprocal Space](https://sandboxphysics.com/labs/electron-diffraction/)
5. [5 TEM Bright Field, Dark Field & SAED](https://sandboxphysics.com/labs/tem-mode-selector/)
6. [6 SEM Secondary-Electron Topography](https://sandboxphysics.com/labs/sem-topography/)

[View the full path →](https://sandboxphysics.com/topics/fields-and-matter/#starter-path)