Finite nuclear mass / Line positions

Hydrogen–Deuterium Isotope Spectrometer

Why do hydrogen and deuterium emit their spectral lines at slightly different wavelengths? Because the nucleus is not infinitely heavy. Electron and nucleus both circle their common center of mass, so the levels depend on a reduced mass slightly below the electron mass. Deuterium's heavier nucleus moves less, its levels spread a little wider, and each line shifts to shorter wavelength by about 0.03 percent. With the lines resolved, that gap gives the ratio of the two nuclear masses.

Separate two ideal Balmer lines with a calibrated scanning grating analyser. Keep every bin of photon counts, fit the spacing between the lines and infer the nuclear-mass ratio, which can take any value.

Line positions use nonrelativistic Coulomb energies with the reduced mass. Each Balmer multiplet is replaced by one effective line, and both lines share the same Gaussian instrument width. Thermal and natural broadening, fine and hyperfine structure, molecular bands, air dispersion and real grating aberrations are left out. The heavy mass can be tuned continuously, and the synthetic counts are not the historical data from the isotope's discovery.

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

Count both peaks before assigning a mass.

Mixed isotope sourceTransmission gratingWavelength bins + counts

One scan keeps 181 equal-width bins. Both ideal effective Balmer lines are blurred by the same fitted Gaussian response. Changing the upper level changes the vacuum wavelength and the isotope separation.

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

A small shift shows that the nucleus moves too.

$(\lambda-\lambda_H^{\rm ref})/\mathrm{pm},\quad N_i$

Retained countsFitted doubletSaved 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

One displacement. A continuous mass.

Inferred heavy/light nuclear mass

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Fitted wavelength separation

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Fitted common width

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No saved comparison.

Green: fitted light component. Brown: fitted heavy component.

$\lambda-\lambda_H^{\rm ref}\;(\mathrm{pm}),\quad \widehat N_{j,i}$

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.

The lighter nucleus is a known proton. The heavy nucleus is continuously tunable near deuterium; the estimator does not impose its expected ratio. Both ideal unbroadened lines have the same fitted instrumental width. Thermal broadening is omitted in this protocol.

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

Next: infer a temperature Hydrogen orbitals and transitions Historical spectral evidence

Start with the essentials

Focus question: Why do hydrogen and deuterium emit their spectral lines at slightly different wavelengths?
One-sentence intuition: Keep what the source is assumed to do separate from what you estimate from the recorded counts.

Core mathematical model

Line positions with the reduced mass

R_M=\frac{R_\infty}{1+m_e/M},\qquad\lambda_M^{-1}=R_M\left(\frac14-\frac1{n^2}\right)

The formula gives vacuum wavelengths from the nonrelativistic Coulomb model, without fine structure.

Mass ratio from the line spacing

r=\frac{M_D}{M_H}=\left[1-\frac{\Delta\lambda}{\lambda_H}\frac{1+\eta}{\eta}\right]^{-1},\quad\eta=\frac{m_e}{M_H}

Only the fitted center of the light line and the positive line spacing go into the inversion. The source mass and the isotope mixture do not.

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
Predict which way the line moves
Raise the source heavy/light nuclear mass before you acquire.
What to observe: The heavy line moves to shorter wavelength.
02
Keep the whole scan
Acquire every wavelength bin, including those with zero counts. Save the run and export the CSV.
What to observe: Moving the analyser probe does not change the recorded spectrum.
03
Lose resolution
Compare the Blended pair preset with Resolved pair.
What to observe: A small residual alone does not mean the two masses can be told apart.

Hydrogen–Deuterium Isotope Spectrometer

Count both peaks before assigning a mass.

A small shift shows that the nucleus moves too.

One displacement. A continuous mass.

Keep the discrepancies visible.