---
title: "Atomic Magnetic-Moment Comparator Interactive Lab | Sandbox Physics"
description: "How can you measure the electron's spin factor without knowing the magnetic field? Compare two atomic levels in the same field."
canonical_url: "https://sandboxphysics.com/labs/atomic-magnetic-moment-comparator/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/atomic-magnetic-moment-comparator/"
last_modified: "2026-10-27"
---

Quantum evidence / Magnetic moment

# Atomic Magnetic-Moment Comparator

**How can you measure the electron's spin factor without knowing the magnetic field?** Compare two atomic levels in the same field. Each one shifts in proportion to its own magnetic factor, so find both resonance frequencies and divide one by the other. The unknown field cancels, and the ratio that remains fixes the electron spin factor.

Put two ideal sets of orbital and spin levels in the same magnetic field. Count atoms in selected states, find both resonances, and separate the electron spin factor from a field drift that shifts both.

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

The atom is an illustrative doublet with no nuclear spin, orbital angular momentum one and spin one half. Its orbital factor is fixed at one, the Zeeman shifts are linear, and the transfer response is a calibrated, incoherent Lorentzian. The model does not compute a real gallium hyperfine spectrum, coherent pulses, trajectories in a field gradient, atomic-structure corrections or QED. The approach is inspired by comparisons of magnetic factors, but it does not refit the Kusch–Foley data.

Atoms pass through ideal state preparation, a transverse RF interaction in a shared vertical field and state selection, then reach an atom counter. The second state selection is measured afterward on the same apparatus. No calculated deflection trajectories are shown.

Observe and keep the record

## Compare the two resonances.

At each coordinate, each branch records a state-selected count and a separate whole-beam reference. All four exposures are independent Poisson observations. Scan playback is sped up.

Ready. Predict, then acquire.

State-selected count rates per second

### A field shift or a factor shift?

Applied RF · first branch

$0$

Applied RF · second branch

$0$

$x/\mathrm{kHz},\quad N/\tau\;(\mathrm{s^{-1}})$

Teal: first branch. Gold: second branch. Points: acquired rates. Fine curves: source expectation. Purple dashes: inference from records. Gray: saved observations. Click the plot to move the probe.

Inference without the source parameters

### Infer field and spin factor together.

Inferred spin factor

$\text{—}$

Inferred shared field

$\text{—}$

Spin-factor-two deviance / degree

$\text{—}$

First resonance · estimate

$\text{—}$

Second resonance · estimate

$\text{—}$

Fit deviance / degree

$\text{—}$

$r_i=(N_i-\widehat\mu_i)/\sqrt{\widehat\mu_i}$

Each center and its source rate are profiled from the joint signal/reference likelihood. Both centers are searched over the full displayed interval. At least sixteen distinct coordinates are required. Deviance per degree is descriptive; no confidence interval or discovery significance is reported.

Compare with a saved run

Latest six raw observations

| # | Offset / kHz | First / ref | Second / ref |
| --- | --- | --- | --- |

The CSV includes the raw observations, seed, units and the source and instrument settings. Source settings are simulator metadata and are not used in the inverse estimate. One saved run lasts for this page session.

The model and the evidence behind it

### What this measurement leaves out.

$$
\begin{aligned}P_J&=\frac{0.4}{1+4(\nu-\nu_J)^2/w^2}\\N_J&\sim\operatorname{Poisson}[\tau(S_JP_J+b)]\end{aligned}
$$

Known transfer contrast 0.4, width and background 8 per second; independently unknown source rates. Linear LS coupling with no nuclear spin, hyperfine mixing, isotope shifts, state lifetime or atomic-structure corrections. The two electronic manifolds and selectors are idealized; this is not calibrated gallium. A small input spin anomaly is supplied, not calculated from QED. The fixed coarse field only labels RF coordinates.

[Steck · LS / Zeeman conventions](https://steck.us/alkalidata/) · [Kusch–Foley 1948 · author abstract](https://journals.aps.org/pr/abstract/10.1103/PhysRev.74.250) · historical method only; no original data refit.

[Read the historical evidence](https://sandboxphysics.com/stories/quantum-evidence/kusch-foley-electron-magnetic-moment/) [Measure an oil-drop charge](https://sandboxphysics.com/labs/oil-drop-charge/)

Physics tutorial

## Atomic Magnetic-Moment Comparator

**Background** Put two ideal sets of orbital and spin levels in the same magnetic field. Count atoms in selected states, find both resonances, and separate the electron spin factor from a field drift that shifts both.

**Why it matters** You infer a constant from recorded observations, using an instrument response that has been calibrated.

Start with the essentials

- **Focus question**: How can you measure the electron's spin factor without knowing the magnetic field?
- **One-sentence intuition**: Keep what the simulation assumes separate from what you estimate from the recorded data.

### Core mathematical model

#### Landé factors of the two levels

$$
\begin{aligned}g_{1/2}&=\frac{4-g_S}{3}\\g_{3/2}&=\frac{2+g_S}{3}\\\nu_J&=g_J\frac{\mu_B B}{h}\end{aligned}
$$

The factors are written as positive magnitudes, with the orbital factor equal to one. The electron magnetic moment points opposite to the angular momentum. Nuclear spin is left out.

#### The calibrated frequency coordinate of the scan

$$
\begin{aligned}\nu_J&=g_J^{(0)}(\nu_c+x)\\\nu_c&=\frac{\mu_B}{h}(0.5\,\mathrm{mT})\\g_J^{(0)}&\in\{2/3,4/3\}\end{aligned}
$$

The scan coordinate maps to frequency through a known relation. It does not use the true source field or the spin factor, which are unknown. Each branch is driven at its own physical RF frequency.

#### Canceling the shared field

$$
\begin{aligned}r&=\widehat\nu_{3/2}/\widehat\nu_{1/2}\\\widehat g_S&=\frac{4r-2}{1+r}\\\widehat B&=\frac{h(\widehat\nu_{1/2}+\widehat\nu_{3/2})}{2\mu_B}\end{aligned}
$$

Each coordinate has four Poisson exposures. Fit both resonance centers and their independent source rates to these counts, then solve for the spin factor and field from the two center frequencies.

### Common difficulties

#### The simulation already contains the law

**Typical misconception** If I recover the parameter the simulation used, I have independently confirmed the law.

**Better mental model** These simulated records test the method of measuring and inferring. They are not historical observations, they are not a confidence interval for the parameter, and they do not count as outside scientific certification.

### Run the experiment

1. 01

   #### Find the two resonance centers

   Scan both state selections over the same coarse frequency range.

   **What to observe:** On the scaled axis, the two peaks depend on the spin factor in different ways.

2. 02

   #### Move the field

   Save the run and choose the Field drift preset.

   **What to observe:** Both centers move, but the inferred spin factor stays the same within sampling scatter.

3. 03

   #### Compare with spin factor fixed at two

   Choose the Spin factor two preset and acquire a new scan.

   **What to observe:** The alternative model fits only the shared field and the two rates. Its deviance describes the fit; it is not a discovery significance.

Physical conclusion

The atom is an illustrative doublet with no nuclear spin, orbital angular momentum one and spin one half. Its orbital factor is fixed at one, the Zeeman shifts are linear, and the transfer response is a calibrated, incoherent Lorentzian. The model does not compute a real gallium hyperfine spectrum, coherent pulses, trajectories in a field gradient, atomic-structure corrections or QED. The approach is inspired by comparisons of magnetic factors, but it does not refit the Kusch–Foley data.

## Related experiments and articles

Quantum evidence · Reading path

### Fields, spins and precision spectra

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How Kusch and Foley found the electron’s extra magnetism**](https://sandboxphysics.com/stories/quantum-evidence/kusch-foley-electron-magnetic-moment/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

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