Hydrogen molecules / Magnetic deflection

Molecular-Beam Proton Magnetic Moment

How can a beam of hydrogen molecules measure the proton's magnetic moment when the molecules' rotation is magnetic too? By allowing for the known rotational part. A field gradient pushes each molecule sideways according to its magnetic moment. Para hydrogen has none and flies straight; ortho hydrogen splits into components mixing the two protons' moment with the rotational one. Slow molecules bend more, so every speed is timed. With that part and the ortho share known, the spread of hits gives the proton moment.

Send hydrogen molecules through the magnet, alternating between field off and field on, and record each molecule's upstream flight time and where it lands on the detector. Then infer an effective proton moment from the measured velocity distribution and a stated rotational contribution.

Prepare, deflect, then infer the moment

How can a beam of hydrogen molecules measure the proton's magnetic moment when the molecules' rotation is magnetic too?

Calibrated transverse field gradient

G/(\mathrm{T}\,\mathrm{m}^{-1})

2200

Source mean speed · excluded from fit

\bar v/(\mathrm{m}\,\mathrm{s}^{-1})

1000

Lognormal speed spread · excluded from fit

s_{\ln v}

0.18

Source transverse standard deviation · fitted from reference

\sigma_y/\mathrm{mm}

0.1

Calibrated ortho fraction · rotation one

f_o

0.75

Source effective proton moment · excluded from fit

\mu_p/\mu_N

2.793

Triggered molecules per field setting

N_{\rm trig}

1200

Inspect retained event · no acquisition change

i

0

Inspection keeps every record. Acquisition changes clear current data; a saved comparison remains in memory.

The source prepares hydrogen in its vibrational ground state: para hydrogen with rotation zero and ortho hydrogen with rotation one. The field is strong enough that the nuclear and rotational projections are uncoupled. The ortho fraction and the rotational moment are known, the force comes from a constant gradient in the paraxial limit, and the reference beam is Gaussian. Molecular shielding, diamagnetism, higher rotational states, spin-rotation coupling, collisions, fringe fields and calibration uncertainty are left out. The events are synthetic; they are neither Stern’s historical data nor a precision measurement of the free proton.

Beam travels left to right; positive magnetic moment deflects upward. Transverse displacement is enlarged 56 times relative to the longitudinal scale. Colored paths are source expectations at the mean speed, not recorded trajectories.

Synthetic records, all observations kept

Switch the field on and off and keep every trigger

Ready to collect simulated data.

Upstream timing triggers alternate field-off / field-on. Detector misses retain a trigger and flight time with an empty position. The known gradient, ortho fraction and rotational moment are calibration inputs; source moment, mean speed and beam width are excluded from inference.

Raw observations

Record a reference beam before inferring the moment.

$y/\mathrm{mm},\quad N_i$

Green: field off. Brown: field on. The detector bins have equal width, and empty bins show zero. Gray dashes show saved field-on counts. Missed events stay in the CSV and in the censored likelihood. The plot shows these counts, not an ideal detector image.

Inference using stated calibrations

Estimate only what the data can resolve.

Inferred effective proton moment

\text{—}

Fitted field-off center and width

\text{—}

Recorded detector acceptance

\text{—}

Second-moment cross-check · full acceptance only

\text{—}

Fixed rotational contribution

\text{—}

Inspected flight time and position

\text{—}

Kept events with a profile likelihood

Separate proton and rotational moments.

$\mu_p/\mu_N,\quad 2(\ell_{\max}-\ell)$

The likelihood mixes nine ortho magnetic components and a para reference, using each flight time. Center and width are fixed by field-off records; the profile omits calibration uncertainty and is not a confidence interval. Values above forty are clipped visibly at the top.

Check the records

Each flight time stays paired with its deflection.

$t_{\rm flight}/\mathrm{\mu s},\quad y/\mathrm{mm}$

First 600 registered events; all triggers are retained in the export. Slower molecules deflect more. No source mean speed is used in the estimator.

Inspect the last eight raw records

Flight times are in microseconds; detector positions are in millimetres. Missed positions remain blank in CSV.

Index	Field on	Flight	Position	Detected

Model, apparatus and primary evidence

\begin{aligned}\frac{\mu_y}{\mu_N}&=2\frac{\mu_p}{\mu_N}m_I+g_Rm_N\\m_I,m_N&\in\{-1,0,1\}\\g_R&=0.88291\end{aligned}

In a strong field, ortho hydrogen splits into nine components with equal populations. In this model, para molecules with rotation zero have no permanent moment.

\begin{aligned}F_y&=\mu_yG\\y-y_0&=\frac{\mu_yGL(D+L/2)}{M v^2}\\K(v)&=\frac{\mu_NGL(D+L/2)}{M v^2}\end{aligned}

The magnet is one tenth of a metre long. The molecules then drift half a metre to a detector that faces the beam.

v=\frac{\ell}{t_{\rm flight}},\quad \ell=0.10\,\mathrm m

Two timing gates upstream measure the speed of every triggered molecule, including those that miss the detector. The source speeds follow an illustrative lognormal distribution, not the Maxwell law of an effusive beam.

The two timing gates are upstream of the magnet and ten centimetres apart. The detector is half a metre beyond the ten-centimetre magnet and accepts positions within four millimetres of the axis. All event flight times are ideal; timing uncertainty is omitted. Known prepared ortho fraction is not inferred from the profile. The reported positive magnitude cannot determine the sign of the proton moment from a symmetric unpolarized beam.

\begin{aligned}z&=\frac{y-y_0}{K(v)}\\V&=\operatorname{Var}(z)-\left\langle\frac{\sigma_y^2}{K(v)^2}\right\rangle\\V&=f_o\left[\frac83\left(\frac{\mu_p}{\mu_N}\right)^2+\frac23g_R^2\right]\end{aligned}

A separate second-moment cross-check is shown only when every detector event is accepted. Finite acceptance enters the likelihood through Gaussian tail probability for each upstream trigger. No diamagnetic or shielding correction is applied; the inferred parameter is effective within this molecular model.

Stern · Molecular rays Barnes, Bray & Ramsey · Rotational moment NIST · CODATA

Single-Electron Penning Trap & Spin Readout Quantum evidence collection

Start with the essentials

Focus question: How can a beam of hydrogen molecules measure the proton's magnetic moment when the molecules' rotation is magnetic too?
One-sentence intuition: Keep what the simulation assumes about the source apart from what you estimate from the recorded data.

Core mathematical model

Molecular magnetic moment

\begin{aligned}\frac{\mu_y}{\mu_N}&=2\frac{\mu_p}{\mu_N}m_I+g_Rm_N\\m_I,m_N&\in\{-1,0,1\}\\g_R&=0.88291\end{aligned}

In a strong field, ortho hydrogen splits into nine components with equal populations. In this model, para molecules with rotation zero have no permanent moment.

Deflection by the field gradient

\begin{aligned}F_y&=\mu_yG\\y-y_0&=\frac{\mu_yGL(D+L/2)}{M v^2}\\K(v)&=\frac{\mu_NGL(D+L/2)}{M v^2}\end{aligned}

The magnet is one tenth of a metre long. The molecules then drift half a metre to a detector that faces the beam.

Speed from the flight time

v=\frac{\ell}{t_{\rm flight}},\quad \ell=0.10\,\mathrm m

Common difficulties

A simulation already contains its law

Typical misconceptionGetting a source parameter back out of the data independently confirms the law.

Better mental modelThe synthetic data test a measurement and inference procedure. They are not historical observations, they give no confidence interval for the parameter, and they are not external scientific certification.