1. Title and aim
Wavelength of light using a diffraction grating. Determine the wavelength of monochromatic light from the angles of successive diffraction orders produced by a transmission grating at normal incidence.
2. Abstract
Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of wavelength of 549.972 nm, differing by 0.005% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.
3. Apparatus and fixed settings
- Transmission grating and holder: Provide parallel equally spaced grooves with density 300 lines per millimetre; hold the grating perpendicular to the incident beam.
- Monochromatic source and collimator: Send a narrow, approximately parallel beam of one wavelength toward the grating.
- Spectrometer telescope and angular scale: Centre left and right images of the same spectral line and determine their angular separation.
- Levelled rotating table: Keep the grating vertical and permit alignment of its normal with the incoming beam.
- Normal incidence; grating density = 300 lines/mm; spacing d = (10⁻³/300) m.
- Fixed monochromatic model wavelength = 550 nm.
- Symmetric ±k readings; θ is half their angular separation, rounded to 0.01°.
4. Theory and graph choice
A diffraction grating turns wave interference into an angular ruler. Select an order, read its simulated diffraction angle, and plot sin(theta) against order. Use the slope and known groove spacing to recover wavelength. The virtual instrument represents one monochromatic spectral line and a 300-lines-per-millimetre grating.
The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units dimensionless per order.
5. Method
- Focus the eyepiece crosshairs, focus the telescope for a distant image and adjust the collimator for a sharp narrow slit. Level the grating and set its normal along the incident beam.
- Read the specified groove density and convert 300 lines/mm into d = 10⁻³/300 m. Identify the central undeviated image as order zero.
- Select a positive order from 1 to 5 in the virtual apparatus. In a physical spectrometer locate the same monochromatic line in the corresponding orders on both sides.
- Centre the left and right images and compute half their angular separation after accounting for any scale wrap. The simulator returns this reduced angle directly, rounded to 0.01°. Record it before changing order.
- Take readings for orders 1, 2, 3, 4 and 5. Calculate sin(theta) using the degree setting or convert each angle into radians first. Fit sin(theta) = s m + b.
- Multiply the slope by grating spacing, then convert metres to nanometres by multiplying by 10⁹. Check an individual row with lambda = d sin(theta)/m and discuss calibration, alignment and model omissions.
6. Observations and calculation table
All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.
| No. | Diffraction order k (order) | Half-separation angle θ (°) | x (SI) | y (SI) | Fitted y (SI) | Residual (SI) |
|---|---|---|---|---|---|---|
| 1 | 1 | 9.50 | 1.00000 | 0.16505 | 0.16503 | 1.3191e-05 |
| 2 | 2 | 19.27 | 2.00000 | 0.33002 | 0.33003 | -5.8443e-06 |
| 3 | 3 | 29.67 | 3.00000 | 0.49500 | 0.49502 | -1.3837e-05 |
| 4 | 4 | 41.30 | 4.00000 | 0.66000 | 0.66001 | -7.5592e-06 |
| 5 | 5 | 55.59 | 5.00000 | 0.82501 | 0.82500 | 1.4049e-05 |
Download the sample observations (CSV) ↓
7. Worked calculation for one observation
8. Graph, gradient and final calculation
Dots show the sample observations. The line is the ordinary least-squares fit.
Σ(x − x̄)² = 10.00000
Σ(x − x̄)(y − ȳ) = 1.64992
m = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = 0.16499160
b = ȳ − mx̄ = 4.2810e-05
d = (10⁻³ m)/300 = 3.3333e-06 m; λ = md = 0.16499160 × d = 549.9720 nm
9. Uncertainty and comparison
Percentage difference = |estimate − model value| / |model value| × 100 = 0.0051%. Percentage difference is not experimental uncertainty.
For n = 5 observations, Σresidual² = 6.5414e-10; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 4.6695e-06. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 1.5487e-05.
Propagating only the relevant fit coefficient gives a fit-only standard error of 0.01557 nm. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.
Angular resolution, line-density calibration, alignment and normal incidence affect wavelength. Symmetric angle readings reduce a shared zero offset but do not remove all alignment errors. sin(θ) has angle-dependent uncertainty. The slope error excludes uncertainty in grating spacing.
10. Result and discussion
Model comparison value: 550.000 nm
The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.
11. Precautions and limitations
- Keep the grating normal to the incident beam and the grooves vertical.
- Identify the same line and the same order on the left and right; approach telescope settings consistently.
- Use the full sine function and check the calculator's angle mode.
- Do not look into a direct laser beam or its specular reflection; a real optical source and viewing method require the laboratory's approved setup.
This deterministic model assumes a perfectly aligned transmission grating, monochromatic parallel illumination and exact groove spacing. It computes peak positions and rounds the reduced angle to 0.01°. It does not predict intensity, blaze efficiency, missing orders, slit profiles, spectral overlap, aperture clipping or telescope resolution. Availability of orders 1 to 5 in this lesson does not guarantee their practical visibility in every apparatus. The ordinary fit omits angle-dependent vertical weights and shared grating-calibration uncertainty. The generator uses wavelength 550 nm and density 300 lines/mm, giving d = 10⁻³/300 m. For orders 1 to 5, theta = arcsin(m lambda/d) is rounded to 0.01 degrees. The stored y value is sin(theta_deg × pi/180), x is the integer order and the estimated wavelength is fitted gradient times d. The fifth-order sine is 0.825, so it is geometrically permitted. The reference wavelength is a chosen teaching parameter rather than an identification of a particular atomic emission line. The displayed raw angle represents half a symmetric left/right separation; independent telescope readings are not generated.
12. Conclusion
The worked example illustrates how to obtain wavelength from measurements, graph variables and a fitted coefficient. The estimate is 549.972 nm. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.