Cambridge Notes, Past Papers, Revision Questions

Cambridge A-Level Physics 9702 – Diffraction Grating

Diffraction Grating

1. Principle of a Diffraction Grating

A diffraction grating consists of a large number of equally spaced parallel slits or rulings. When monochromatic light falls on the grating, each slit acts as a source of secondary wavelets. The superposition of these wavelets produces constructive interference at specific angles, giving bright diffraction orders.

2. Grating Equation

The condition for constructive interference (bright fringe) is

\$d\sin\theta = n\lambda\$

where

\$d\$ – grating spacing (distance between adjacent slits)

\$\theta\$ – diffraction angle measured from the normal to the grating

\$n\$ – order of the spectrum (integer: \$0, \pm1, \pm2,\dots\$)

\$\lambda\$ – wavelength of the light

3. Determining the Wavelength

Place the grating so that the incident light is normal to its surface (incident angle \$i = 0^\circ\$).

Observe the diffracted spots on a screen or detector and measure the angle \$\theta\$ for a chosen order \$n\$.

Calculate the grating spacing \$d\$ from the specification (see Table 1).

Re‑arrange the grating equation to solve for the wavelength:
\$\lambda = \frac{d\sin\theta}{n}\$

4. Example Calculation

Suppose a grating has 600 lines mm$^{-1}$ and the first‑order (\$n=1\$) bright spot is observed at \$\theta = 20^\circ\$. The wavelength is:

\$d = \frac{1}{600\ \text{mm}^{-1}} = 1.67\times10^{-6}\ \text{m}\$

\$\$\lambda = \frac{(1.67\times10^{-6}\,\text{m})\sin20^\circ}{1}

\approx 5.71\times10^{-7}\,\text{m}

= 571\ \text{nm}\$\$

5. Practical Considerations

Order selection: Higher orders give larger angles, improving resolution but may overlap with other wavelengths.

Grating spacing: Finer gratings (more lines mm$^{-1}$) increase angular dispersion.

Alignment: The incident beam must be perpendicular to the grating to avoid systematic errors in \$\theta\$.

Measurement of \$\theta\$: Use a calibrated protractor or the angular scale of a spectrometer; record both left‑ and right‑hand angles and average to reduce random error.

6. Sources of Error

Source of Error	Effect on \$\lambda\$	Mitigation
Incorrect grating spacing \$d\$	Systematic shift of all calculated wavelengths	Use manufacturer‑provided value; verify with a known spectral line
Mis‑alignment of incident beam	Measured \$\theta\$ deviates from true value	Ensure normal incidence; check with a reference mirror
Parallax when reading angles	Random scatter in \$\theta\$ values	View scale from directly above; take multiple readings
Overlap of higher orders	Incorrect identification of order \$n\$	Use a filter or limit to low orders; compare with known spectrum

7. Typical Grating Specifications

Lines per mm	Grating spacing \$d\$ (µm)	Typical use
300	3.33	Low‑resolution spectroscopy, visible range
600	1.67	Medium resolution, visible and near‑UV
1200	0.833	High resolution, detailed line identification

8. Summary

The diffraction grating provides a simple, quantitative method for determining the wavelength of light. By measuring the diffraction angle \$\theta\$ for a known order \$n\$ and using the grating equation, the wavelength \$\lambda\$ can be calculated. Accuracy depends on precise knowledge of the grating spacing, careful alignment, and accurate angle measurement.

Suggested diagram: Light incident normally on a diffraction grating with rays diffracted into orders \$n = 0, \pm1, \pm2\$, showing angle \$\theta\$ measured from the normal.

describe the use of a diffraction grating to determine the wavelength of light (the structure and use of the spectrometer are not included)