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8.4 The diffraction grating

Syllabus
9702–2028–2029
Topic
8.4
Level
AS

Use grating spacing, order and angle to test every possible maximum

Each grating slit diffracts light; waves from many equally spaced coherent slits overlap. A principal maximum forms when the path difference between adjacent slits is an integer number of wavelengths.

dsinθ=nλn=0,±1,±2,…θismeasuredfromthegratingnormal/central(n=0)direction.d sinθ=nλ n=0, ±1, ±2, … θ is measured from the grating normal/central (n=0) direction.

ForNlinespermetre:d=1/NForNlinespermm:d=10−3/Nmetres600linesmm−1⇒d=1.67×10−6mFor N lines per metre: d=1/N For N lines per mm: d=10⁻³/N metres 600 lines mm⁻¹ ⇒ d=1.67×10⁻⁶ m

Orders appear symmetrically at +θ and −θ. If the question gives the angle between opposite +n and −n maxima, each diffraction angle is half that total angle.

Because∣sinθ∣≤1:nλ≤dnmax=floor(d/λ)Longerλgiveslargerθforthesamenbutusuallyfewerpossibleorders.Because |sinθ|≤1: nλ≤d n_max=floor(d/λ) Longer λ gives larger θ for the same n but usually fewer possible orders.

n labels the order, not the number of slits. The central maximum is n=0 and cannot determine λ from d sinθ=nλ because both sides are then zero.

Determine wavelength from calibrated non-zero grating orders

Direct monochromatic light normally onto a grating of known line density, identify the central maximum, then locate a labelled non-zero order on both sides using a screen or spectrometer.

Convert line density N to spacing d=1/N in metres per line. Record the order n; higher orders can improve angular sensitivity but must be bright, separated and physically allowed.

Measuredirectionsα+andα−for+nand−n:θ=(∣α+−α−∣)/2λ=dsinθ/nMeasure directions α₊ and α₋ for +n and −n: θ=(|α₊−α₋|)/2 λ=d sinθ/n

Align the incident beam with the normal, use a narrow beam/slit and read angles without parallax. Repeat, average symmetric pairs and, where possible, calculate λ from several orders to check consistency.

Angular uncertainty matters most when θ is small; use the highest clear allowed order and symmetric readings to enlarge the measured separation and reduce zero/alignment bias. Quote λ with justified significant figures.

Do not use the angle from the grating plane, confuse lines-per-length with spacing, or use the full +n-to−n angle as θ. The essential observation is at least one non-zero order.