A grating has 600 lines mm⁻¹ and illuminated width 12.0 mm. It receives wavelengths 589.0 nm and 589.6 nm. Determine the first-order diffraction angle for 589.0 nm, the number of illuminated slits, and the resolving power required and available in first order. Decide whether the lines are resolved, find the minimum order that meets the resolving criterion, and check whether that order can physically exist.

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First-order angle ≈ 20.7°. N = 7200 slits. Required resolving power ≈ 982; first-order available = 7200, so the lines are resolved. The minimum order is 1 and it exists because mλ/d < 1. Line spacing is d = 1/(600 mm⁻¹) = 1.667 × 10⁻⁶ m. First-order grating equation gives sinθ = λ/d = 0.3534, hence θ = 20.7°. The illuminated number of slits is N = (600 lines mm⁻¹)(12.0 mm) = 7200. Required resolving power is λ/Δλ ≈ 589.0/0.6 = 982. A grating's ideal resolving power in order m is mN. First order provides 7200, greater than 982, so the Rayleigh criterion predicts that the two wavelengths are resolved. The smallest integer order satisfying mN ≥ 982 is m = 1. An order exists only if mλ/d ≤ 1. For m = 1 the ratio is 0.353, so it is physical. Real resolution can be worse because finite slit width, aberrations, detector sampling and incomplete illumination reduce contrast.

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