THERMAL MOTION / LINE WIDTHS

Two-Isotope Doppler Thermometer

Record separate light- and heavy-hydrogen spectra at a common temperature. Infer both Gaussian widths from photon counts, then separate thermal motion from a shared instrumental variance.

Ideal isolated effective Balmer lines, Maxwellian atoms, first-order Doppler shifts and a shared Gaussian instrument. Known atomic masses and calibrated vacuum wavelengths. No fine or hyperfine components, natural or collisional broadening, optical depth, drift or non-equilibrium discharge. This illustrative two-line inversion is not a practical hydrogen thermometry accuracy claim.

Normal incidence on a transmission grating; the first diffracted order reaches the rotating collection arm. Physical ray angles are unscaled. Fine scanning motion is tiny; sizes, bandwidth and slit widths are schematic.

CALIBRATED VACUUM WAVELENGTH / PHOTON COUNTS

Measure two widths before assigning a temperature.

Two same-temperature samplesTransmission gratingWavelength bins + counts

Two sequential scans retain 121 bins for each known isotope. Both samples share the same temperature and Gaussian instrument. Switching gas and acquisition are accelerated; no physical elapsed time is inferred.

Ready. Predict, then acquire.

Recorded bins

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Current analyser wavelength

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RAW COUNTS / BIN-INTEGRATED POISSON FIT

The lighter line spreads further.

$(\lambda-\lambda_j^{\rm ref})/\mathrm{pm},\quad N_{j,i}$

Light isotopeHeavy isotopeSaved counts

Show source expectation separately · not acquired data

Dots retain zero bins; vertical bars show the square root of observed counts. Lines integrate over each bin. The optional gold source expectation is a simulation reference.

INFER FROM THE RETAINED COUNTS

Thermal variance and instrumental blur

Inferred temperature

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Inferred instrumental width

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Width-difference heuristic error

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No saved comparison.

Bars show observed variances. When inference is usable, gray is instrumental and green is thermal.

$s_j^2,\;\widehat\sigma_I^2,\;a_j\widehat T\quad(\mathrm{pm}^2)$

MODEL CHECK / RETAINED COUNTS

Keep the discrepancies visible.

$r_i=\operatorname{sgn}(N_i-\mu_i)\sqrt{D_i}$

Poisson deviance per bin defines the squared residual; its zero-count logarithm term is zero. Outliers beyond the plotting range appear as red edge dots. This plot checks the declared profile, not historical evidence. Fit boundaries and conservative identifiability rules can withhold the derived parameter.

Inspect the last eight raw bins

Sample	$\lambda_{\rm left}/\mathrm{nm}$	$\lambda_{\rm right}/\mathrm{nm}$	$N_i$

Model, geometry and evidence

Incident light travels along the positive horizontal axis to a normal-incidence transmission grating. Its period is one micrometre, and the positive first order lies in the horizontal dispersion plane. The collection lens, slit and detector share that outgoing axis. The calibrated Gaussian bandpass is assumed, not derived from the exaggerated slit geometry.

d\sin\beta=\lambda,\quad d=1000\,\mathrm{nm},\quad m=1

Independent Poisson bins integrate the full line profile. The source budget is the ideal expected signal sum over a complete scan, apart from truncated tails; it is not an absolute radiometric calibration. Background is fitted from the same bins. Counts are synthetic, and recovery of their generating law is not an independent experimental test.

Known proton and deuteron masses plus the electron mass set the approximate atom masses. Both sources have one shared translational temperature and one shared instrumental width. Each real Balmer multiplet is collapsed to one effective line; applying this inversion directly to real hydrogen without resolving its components would bias temperature. The displayed error scale is a leading Gaussian count heuristic, not a fitted confidence interval or significance.

NIST · Atomic spectroscopy Tel Aviv · Isotope spectroscopy NIST · Doppler thermometry

Compare: infer a nuclear mass Hydrogen orbitals and transitions Historical spectral evidence

Start with the essentials

Focus question: Two masses separate heat from blur.
One-sentence intuition: Keep source assumptions separate from estimates made from records.

Core mathematical model

Thermal and instrumental variance

s_j^2=\sigma_I^2+a_jT,\qquad a_j=\frac{\lambda_j^2 k_B}{m_jc^2}

Independent Gaussian convolution adds variances. Wavelength broadening is a first-order Doppler approximation.

Two widths, two unknowns

\widehat T=\frac{s_H^2-s_D^2}{a_H-a_D},\qquad\widehat\sigma_I^2=s_D^2-a_D\widehat T

Both measured widths enter; neither source temperature nor instrumental width is supplied to this estimator.

Common difficulties

A simulation supplies its law

Typical misconceptionRecovering a source parameter independently confirms the law.

Better mental modelThese synthetic records test a measurement and inference workflow. They are not historical observations, a parameter confidence interval, or external scientific certification.

Run the experiment

01
Compare two gases
Acquire both isotope scans at the same source temperature.
What to observe: The lighter atoms have a broader thermal line.
02
Separate the blur
Increase only the instrumental width and acquire a new comparison.
What to observe: Both variances rise together; their difference carries temperature.
03
Check identifiability
Use the instrument-dominates preset.
What to observe: A small width difference can be overwhelmed by count uncertainty.