Measuring the Planck Constant with LEDs: A Regression and Uncertainty-Propagation Study

What this study does. The Planck constant h is no longer a number to be looked up: since the 2019 SI redefinition it is fixed by definition at 6.62607015 × 10⁻³⁴ J s, and the kilogram is built from it. That changes what measuring h means for an undergraduate — it is a reproduction of the definition, and the interesting quantity is the uncertainty budget: how close can a bench of LEDs, two multimeters and a spreadsheet get, and where does the error live?

Six LED types spanning the visible spectrum (roughly 465–635 nm) are driven from a DC supply through a series resistor. For each LED the threshold voltage — where the diode first conducts a small defined current and visibly emits in the dark box — is found under a pre-declared criterion, five times, for a repeatability standard deviation. Because a photon of wavelength λ carries energy hc/λ, the model is simply

V_threshold = (hc/e) · (1/λ)

so the slope of threshold voltage against inverse wavelength equals hc/e, and h = slope × e/c. The regression's standard error is the statistical uncertainty; an explicit systematic budget — datasheet wavelength tolerance, criterion offset, meter accuracy — is itemised and propagated in quadrature GUM-style, with a ±10 nm wavelength sensitivity analysis showing which LED dominates. The result h ± u(h) is compared with the exact SI value; agreement within combined uncertainty, and the percent difference, is the verdict — stated honestly, with no rounding toward the textbook answer.

The Planck constant anchors the modern SI, yet in most college labs it is 'measured' once, with a phototube of uncertain health, and reported as a bare number compared loosely with the textbook value. Students never meet the part that makes a measurement scientific: a declared criterion, repetition statistics, and an uncertainty budget that says where the error actually lives.

At the bench level the LED method is usually run with the threshold judged by eye under room light, one reading per LED, and no record of the LEDs' wavelength tolerance — which can exceed the statistical spread by an order of magnitude. The result is a value of h that cannot be defended, because nothing about it is quantified.

This project therefore measures threshold voltages of six LEDs under a pre-declared, repeatable criterion with five repetitions per LED, extracts h by least-squares regression against inverse wavelength, and builds a complete GUM-style uncertainty budget — statistical and systematic — with a sensitivity analysis of the wavelength term. The deliverable is not 'the value of h' but a demonstration that an undergraduate bench can reproduce a defining constant of the SI and say, honestly, how well.

Abstract

This single-member study reproduces the Planck constant with consumer LEDs. Six LED colours between roughly 465 and 635 nm are driven from a DC supply through a series resistor inside a dark box; each threshold voltage is measured under a pre-declared conduction-current criterion, five repetitions each. Fitting threshold voltage against inverse wavelength gives hc/e from the slope, hence h = slope × e/c. The statistical uncertainty is the regression standard error; the systematic budget — wavelength tolerance, criterion offset, meter accuracy — is propagated in quadrature following the GUM approach, with a ±10 nm shift sensitivity analysis finding the dominant term. The final h ± u(h) is compared against the exact post-2019 SI value, the difference discussed in units of combined uncertainty.

Introduction

The photoelectric effect gave h its historical meaning: light of frequency ν carries energy quanta hν. A vacuum phototube version is the classic undergraduate experiment, but phototubes are fragile and often aged. LEDs invert the physics: an electron crossing a forward-biased junction emits a photon of energy close to eV_threshold, so the onset of emission across several colours extracts h from a circuit costing a few hundred rupees.

Existing knowledge and the gap

The LED method appears in teaching literature and college labs, but student write-ups usually share three defects: the threshold is judged 'by eye' with no declared criterion, so two readings are not comparable; repetitions are not taken, so there is nothing to quote as repeatability; and the result is a bare number with no uncertainty budget at all. The gap filled here is methodological: a pre-registered criterion, repetition statistics, an itemised GUM-style budget, and a sensitivity analysis of the dominant wavelength term.

Proposed study

1. Characterise the six LEDs: nominal peak wavelength from datasheets, series resistance, dark-box geometry fixed and photographed.
2. Write the threshold criterion before taking data: the LED voltage at which the reference-resistor current first reaches a defined small value with visible emission as the supply ramps.
3. Measure V_threshold for every LED, five repetitions on separated runs, logging meter readings raw with ambient and polarity checks.
4. Fit V_threshold vs 1/λ by least squares in Python; extract slope, standard error, R² and residuals.
5. Build the uncertainty budget: statistical (slope SE), systematic (λ tolerance, criterion offset, meter specs), combined in quadrature; run the ±10 nm shift sensitivity analysis.
6. Report h ± u(h), the percent difference from the exact SI value, and the agreement statement in units of combined uncertainty.

Feasibility

• Technical: the circuit is a first-year electricity experiment; the rigour lives in the criterion, repetition and budget, all within 8 h/week.
• Economic: LEDs, breadboard, resistors and patch leads cost under ₹400; two college multimeters are borrowed, not bought.
• Operational: the lab in-charge signs the plan for low-voltage bench work. The risks — junction warm-up drift and datasheet peak-λ tolerance — are managed by warm-up time and the sensitivity analysis; a photodiode comparator, if available, replaces the eye in the criterion.

In scope

• Six LED types, one threshold criterion, five repetitions per LED, one regression over six points.
• Statistical uncertainty from repetition and regression; systematic budget for wavelength, criterion offset, meters and resistor.
• Sensitivity analysis of the wavelength term; final comparison against the exact SI value of h.

Out of scope

• A vacuum phototube / stopping-potential experiment for direct comparison — a separate rig and a separate project.
• Spectroradiometer measurement of the LEDs' true peak wavelengths — treated as the dominant systematic and handled by sensitivity analysis; listed under future scope.
• Temperature-controlled junction studies and full diode-equation modelling beyond the threshold model.
• Automated computerised data acquisition — manual multimeter readings are part of the budget's honesty.

Research design: a precision-measurement laboratory study with a pre-declared protocol, repetition-based statistics, and a GUM-style uncertainty budget. Individual study, 12 weeks at 8 h/week.

Phase 1 — Apparatus characterisation (Weeks 1–2). The six LEDs are catalogued from datasheets: nominal peak wavelength and its tolerance, forward current rating, viewing angle. The rig is assembled on a breadboard inside a light-tight box: DC supply → series resistor (1 kΩ, measured with a meter, value logged) → LED, with the first multimeter across the LED and the second across the resistor to read current as V_R/R. Geometry fixed and photographed; meter specifications (accuracy class) copied into the budget sheet.

Phase 2 — Criterion and shakedown (Weeks 3–4). The threshold criterion is written before data collection: as the supply ramps up slowly, the threshold is the LED voltage at which the reference-resistor current first reaches a defined small value (e.g., 50 µA) with visible emission confirmed in the closed dark box. Two shakedown LEDs (not in the final six) test the criterion: ramp-up vs ramp-down hysteresis, warm-up drift over 10 minutes, and ambient-light leak checks. Warm-up time and ramp protocol are fixed from these trials.

Phase 3 — Data collection (Weeks 5–8). Each of the six LEDs is measured on five separated runs (different days where possible), each repetition a full ramp with the criterion applied; raw meter readings are entered the same day with run order, temperature note and any anomaly. One reversed-polarity check per LED confirms no glow below threshold; one repetition per LED is repeated by a second reader to quantify observer error.

Phase 4 — Analysis (Weeks 9–10). Mean ± SD per LED; regression of V_threshold on 1/λ in Python (SciPy linregress), reporting slope s, standard error u_s, R², and a residual plot inspected for curvature. h = s·e/c with e and c taken as exact. Uncertainty budget: statistical component u_stat = u_s·e/c; systematic components itemised — wavelength tolerance (via the sensitivity analysis: re-fit with each λ shifted ±10 nm and take the half-range of resulting h), criterion offset (difference between the 50 µA criterion and an alternative 100 µA criterion on one LED), meter accuracy (from specifications, propagated through V = V_LED and I = V_R/R) — combined in quadrature: u(h) = √(u_stat² + u_sys²).

Phase 5 — Verdict and report (Weeks 11–12). h ± u(h) compared with the exact value; percent difference; agreement statement ('consistent within k = 2 combined uncertainty' or the honest opposite); residual discussion; report, slides, viva.

Timeline: W1–2 apparatus; W3–4 criterion and shakedown; W5–8 data; W9–10 analysis and budget; W11–12 report and viva.

The measurement is a chain from ramp supply to regression, and every number in the final budget is traceable to a logged raw reading. The workflow below is the report's methods spine.

flowchart TD
  A["Catalogue 6 LEDs: peak wavelength and tolerance from datasheets"] --> B["Build rig: supply, series resistor, dark box, two meters"]
  B --> C["Shakedown: hysteresis, warm-up drift, light-leak checks"]
  C --> D{"Criterion declared and frozen?"}
  D -->|Yes| E["5 repetitions per LED: ramp, read V and I at threshold"]
  D -->|No| F["Revise criterion, repeat shakedown"]
  E --> G["Mean +/- SD per LED; second-reader check"]
  G --> H["Least-squares fit: V vs 1/lambda; slope, SE, residuals"]
  H --> I{"Residuals linear and small?"}
  I -->|Yes| J["GUM budget: statistics + lambda tolerance + criterion + meters"]
  I -->|No| K["Inspect model, record curvature as limitation"]
  J --> L["Sensitivity: +/-10 nm wavelength shift, re-fit"]
  L --> M["h +/- u(h); compare with exact SI value"]

Data organisation

One row per repetition: LED code, datasheet λ and tolerance, run number, date, warm-up minutes, V_LED at threshold, V_R at threshold, computed I, observer, and notes. The slope, budget and sensitivity numbers all regenerate from this table; the regression is scripted once (SciPy linregress) and one full hand calculation of the fit for two LEDs is shown in the appendix so the examiner can verify the arithmetic.

Core calculations (abridged)

Model:       V_threshold = (h c / e) * (1 / lambda)
Fit:         slope s from least squares; u_s = standard error of slope
Statistical: u_stat(h) = (e/c) * u_s
Systematic:  u_sys(h) = sqrt(u_lambda^2 + u_criterion^2 + u_meter^2)
             u_lambda from +/-10 nm shift-and-refit half-range
Combined:    u(h) = sqrt(u_stat(h)^2 + u_sys(h)^2)
Verdict:     (h_meas - h_SI) quoted with u(h), percent difference

Results tables to fill (no numbers are pre-filled)

• Spectrometer-measured wavelengths on the actual LEDs, replacing datasheet values and collapsing the dominant systematic term.
• Photodiode-based criterion with a transimpedance amplifier, turning the threshold into an electronic rather than visual judgement.
• Temperature study: threshold voltage versus junction temperature on one LED, connecting the offset to semiconductor physics.
• Phototube comparison: borrowing the college's photoelectric apparatus for the same h by stopping potential, and comparing the two budgets.
• Weighted regression using each LED's full uncertainty as its weight, and a bootstrap alternative for the slope distribution.
• A department bench-notes booklet that turns this rig into a standing uncertainty-analysis practical for juniors.