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Thermodynamics

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Width of Principal Maxima

The direction of the n^th principal maximum is given by

(a + b) sin θn = nλ                                  (i)

Let θ_n + dθ and θ_n – dθ give the directions of the first secondary minima on the two sides of the n^th primary maxima.

Then,

(a + b) sin [θ_n ± dθ] = nλ ± λ/N              (ii)

where N is the total number of lines on the grating surface.

Dividing (ii) by (i)

Expanding this equation

For small values of dθ; cos dθ = 1 and sin dθ = dθ.

∴ 1 ± cot θ_n.dθ = 1 ± (1/Nn)                         (iv)

cot θ_n.dθ = (1/Nn)

In equation (iv), dθ refers to half the angular width of the principal maximum. The half width dθ is (i) inversely proportional to N the total number of lines and (ii) inversely proportional to n cot θ_n. The value of n cot θ_n is more for higher orders because the increase in the value of cot θ_n is less than the increase in the order. Thus, the half width of the principal maximum is less for higher orders. Also, the larger the number of lines on the grating surface, the smaller is the value of dθ. Further, the value of θ_n is higher for longer wavelengths and hence the spectral lines are more sharp towards the violet than the red end of the spectrum.