(h) the condition for maxima d sin θ = nλ, where d is the grating spacing

The idea

For a diffraction grating, constructive interference produces maxima when the path difference between light from adjacent slits equals a whole number of wavelengths. This condition is d sin θ = nλ, where d is the grating spacing (distance between adjacent slit centres), θ is the angle from the zero-order direction, n is the order (0, 1, 2, ...) and λ is the wavelength. The zero-order maximum (n = 0) occurs at θ = 0° for all wavelengths. Higher orders spread out more, so the grating separates wavelengths, which is why it is used in spectrometry. The maximum possible order is limited because sin θ cannot exceed 1.

What earns the marks

• Constructive interference occurs when the path difference between adjacent slits is a whole number of wavelengths.
• The condition for maxima is d sin θ = nλ, where d is the grating spacing.
• n is the order of the maximum and takes integer values 0, 1, 2, ...
• The zero-order maximum occurs at θ = 0° for all wavelengths.
• The maximum observable order is limited by sin θ ≤ 1, so n ≤ d / λ.

Common mistakes and how to put them right

• Treating d as the slit width rather than the distance between adjacent slit centres.

• Using degrees in the sine calculation when the calculator is set to radians, or vice versa.

• Forgetting that n must be an integer.

  Put it right: Non-integer values do not give maxima.

• Assuming the zero-order maximum occurs at different angles for different wavelengths.

  Put it right: It occurs at θ = 0° for all wavelengths.

Exam tips

• Check that your calculator is in the correct angle mode before evaluating sin θ.
• Rearrange d sin θ = nλ carefully: to find θ use sin θ = nλ / d, then take the inverse sine.
• Remember that the maximum order is the largest integer n for which nλ / d ≤ 1.