The minimum number of lines needed in the grating to resolve the doublet in the second-order spectrum is approximately 766.
How can you calculate the minimum number of lines needed in a grating?
Rayleigh's criterion states that two wavelengths can be considered resolved if the peak of one diffraction maximum coincides with the first minimum of the other. In terms of angular separation, this translates to:
θ_min = λ / (m * d)
where:
d is the grating spacing (distance between two consecutive lines)
To find the minimum resolvable angular separation, we need the smallest difference in wavelength that the grating should discern. This is the difference between the central wavelengths of the doublets:
λ min = λ₂ - λ₁ = 506.9 nm - 506.2 nm = 0.7 nm
θ min = 0.7 nm / (2 * d)
0.7 nm / (2 * d) = λ / (d * N)
d = λ * N / (0.7 nm * 2)
We want the minimum resolvable angular separation to be equal to the diffraction limit, so substitute θ_min back into the original Rayleigh's criterion equation:
λ_min = λ / (m * d)
N = λ / (λ_min * m * d)
N = λ / (λ_min * m * (λ * N / (0.7 nm * 2)))
N = 2 * λ² / (λ_min * m * 0.7 nm)
N = 2 * (506.2 nm)² / (0.7 nm * 2 * 2) ≈ 766 lines
Therefore, the minimum number of lines needed in the grating to resolve the doublet in the second-order spectrum is approximately 766.