---
title: "Hydrogen–Deuterium Isotope Spectrometer Interactive Lab | Sandbox Physics"
description: "The nucleus moves too. Resolve two ideal Balmer lines with a calibrated scanning grating analyser. Retain every photon-count bin, fit their separation and infer a continuous nuclear-mass ratio."
canonical_url: "https://sandboxphysics.com/labs/hydrogen-deuterium-isotope-spectrometer/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/hydrogen-deuterium-isotope-spectrometer/"
last_modified: "2026-10-18"
---

FINITE NUCLEAR MASS / LINE POSITIONS

# Hydrogen–Deuterium Isotope Spectrometer

Resolve two ideal Balmer lines with a calibrated scanning grating analyser. Retain every photon-count bin, fit their separation and infer a continuous nuclear-mass ratio.

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

Nonrelativistic Coulomb reduced masses; each Balmer multiplet is replaced by a single effective line with a common Gaussian instrumental width. No thermal, natural, fine or hyperfine structure, molecular bands, air dispersion or real grating aberrations. Continuously tunable heavy mass; synthetic counts are not historical isotope discovery data.

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

## Count both peaks before assigning a mass.

Mixed isotope source Transmission grating Wavelength bins + counts

One scan retains 181 equal-width bins. A common fitted Gaussian response acts on two ideal effective Balmer lines. Changing the upper level changes the vacuum wavelength and isotope separation.

Ready. Predict, then acquire.

Recorded bins

$\text{—}$

Current analyser wavelength

$\text{—}$

RAW COUNTS / BIN-INTEGRATED POISSON FIT

### A small shift records a moving nucleus.

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

Retained counts Fitted doublet Saved 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

### One displacement. A continuous mass.

Inferred heavy/light nuclear mass

$\text{—}$

Fitted wavelength separation

$\text{—}$

Fitted common width

$\text{—}$

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 / 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.

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](https://physics.nist.gov/Pubs/AtSpec/AtSpec.PDF) [Tel Aviv · Isotope spectroscopy](https://www.tau.ac.il/~phchlab/exp-hydrogen-setup.html) [NIST · Doppler thermometry](https://www.nist.gov/programs-projects/deployable-doppler-thermometry)

[Next: infer a temperature](https://sandboxphysics.com/labs/two-isotope-doppler-thermometer/) [Hydrogen orbitals and transitions](https://sandboxphysics.com/labs/hydrogen-orbitals-spectra/) [Historical spectral evidence](https://sandboxphysics.com/stories/quantum-evidence/atomic-spectra-stationary-states/)

Physics tutorial

## Hydrogen–Deuterium Isotope Spectrometer

**Background** Resolve two ideal Balmer lines with a calibrated scanning grating analyser. Retain every photon-count bin, fit their separation and infer a continuous nuclear-mass ratio.

**Why it matters** A constant is inferred through a calibrated response and retained observations.

Start with the essentials

- **Focus question**: The nucleus moves too.
- **One-sentence intuition**: Keep source assumptions separate from estimates made from records.

### Core mathematical model

#### Reduced mass

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

Vacuum wavelengths from the nonrelativistic Coulomb model. Fine structure is omitted.

#### Invert the separation

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

The fitted light-line center and positive separation enter the inversion; the source mass and mixture do not.

### Common difficulties

#### A simulation supplies its law

**Typical misconception** Recovering a source parameter independently confirms the law.

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

### Run the experiment

1. 01

   #### Predict the sign

   Increase the heavy nuclear mass before acquiring.

   **What to observe:** The heavy line moves to shorter wavelength.

2. 02

   #### Retain the scan

   Acquire all bins, including zeros. Save the run and export CSV.

   **What to observe:** Moving the analyser probe leaves the recorded spectrum intact.

3. 03

   #### Lose resolution

   Compare the blended-pair preset with the resolved pair.

   **What to observe:** A low residual alone does not make two masses identifiable.

Physical conclusion

Nonrelativistic Coulomb reduced masses; each Balmer multiplet is replaced by a single effective line with a common Gaussian instrumental width. No thermal, natural, fine or hyperfine structure, molecular bands, air dispersion or real grating aberrations. Continuously tunable heavy mass; synthetic counts are not historical isotope discovery data.

Quantum & fields · Starter path

## Start from a single measurement

Predict outcomes and collect counts, building from one qubit to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare it with the frequency of acquired 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)