Thermal motion / Line widths

Two-Isotope Doppler Thermometer

How can two hydrogen isotopes give a gas temperature when the spectrometer also blurs the lines? Compare their line widths. At the same temperature the lighter hydrogen atoms move faster, so thermal motion broadens their line more than deuterium's, while the instrument blurs both equally. The difference of the two squared widths then depends only on temperature, and either line gives the instrument's share. If the instrument blur dominates, that small difference drowns in counting noise.

Record separate spectra of light and heavy hydrogen at the same temperature. Infer both Gaussian line widths from the photon counts, then split them into a thermal part and an instrument variance that both lines share.

The model uses ideal, isolated effective Balmer lines, atoms with a Maxwell velocity distribution, first-order Doppler shifts and one Gaussian instrument response for both lines. Atomic masses and the calibrated vacuum wavelengths are known. Fine and hyperfine components, natural and collisional broadening, optical depth, drift and non-equilibrium discharges are left out. The two-line inversion is an illustration and makes no claim about the accuracy of practical hydrogen thermometry.

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.

Photon counts against calibrated vacuum wavelength

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 with a bin-integrated Poisson fit

The lighter isotope gives a broader line.

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

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 recorded 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 against the kept 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: How can two hydrogen isotopes give a gas temperature when the spectrometer also blurs the lines?
One-sentence intuition: Keep what the source is assumed to do separate from what you estimate from the recorded counts.

Core mathematical model

Thermal and instrument widths add as variances

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

Convolving two independent Gaussians adds their variances. The thermal broadening in wavelength uses the 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

The estimate uses both measured widths. It is given neither the source temperature nor the instrument width.

Common difficulties

The simulation starts from the law

Typical misconceptionRecovering a source parameter independently confirms the law.

Better mental modelThese synthetic records test how the measurement and the inference work. They are not historical observations, they do not give a confidence interval for the parameter, and they are not outside 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 instrument blur
Increase only the source instrumental standard deviation and acquire again for comparison.
What to observe: Both variances rise by the same amount, and their difference still carries the temperature.
03
Check whether the temperature can still be found
Choose the Instrument dominates preset.
What to observe: A small difference between the two widths can be swamped by the counting uncertainty.

Two-Isotope Doppler Thermometer

Measure two widths before assigning a temperature.

The lighter isotope gives a broader line.

Thermal variance and instrumental blur

Keep the discrepancies visible.