Question

Use Equation 7-13 for the resolving power of a grating monochromator to estimate the theoretical minimum size of a diffraction grating that would provide a profile of an atomic absorption line at 500 nm having a line width of 0.002 nm. Assume that the grating is to be used in the first order and that it has been ruled at 2400 grooves/mm.

Short answer

Minimum theoretical size of first order grating is 104 mm.

Step-by-step solution

Step1. Given information

$$\begin{gathered} \text{Linewidth=0.002nm} \\ \lambda =500\text{nm} \end{gathered}$$

Step2.Resolving power

The resolving power is a measurement taken to ensure a clear view of an object. The ratio of wavelength and difference of wavelength or the multiplication of order of diffraction with the number of lighting blazes by radiation yields the resolving power.

Step3. Formula for resolving power

$R=\frac{\lambda }{∆\lambda }$

=$nN$

Here, $\lambda$is the wavelength and $△\lambda$is the wavelength difference, and $n$is the diffraction order, and$N$is the total number of slits.

Step4. Calculate Resolving power 

Resolving power can be calculated by using the formula given in step 3 as follows:-

$$\begin{gathered} R=\frac{500\hspace{0.17em}\text{nm}}{0.002\text{nm}} \\ =2.5\times {10}^5 \end{gathered}$$

Step5. Calculate the total number of slits

The total number of slits can be calculated by using the formula given in step3 as follows:-

Substitute 1 for n as the first-order grating is used.

$$\begin{gathered} 1.2\times {10}^5=n\times N \\ 1.2\times {10}^5=1\times N \\ N=1.2\times {10}^5 \end{gathered}$$

Step6. Calculate the theoretical minimum size of a diffraction grating 

$\text{Width}=\frac{N}{numberofgroovespermillimeter}$

Substitute $1.2\times {10}^5$for $N$, and $2400\raisebox{1ex}{$\text{grooves}$}\!\left/ \!\raisebox{-1ex}{$\text{mm}$}\right.$for number of grooves per millimeter in the above equation as follows:-

$$\begin{gathered} \text{Sizeofgrating=}\frac{2.5\times {10}^5\hspace{0.17em}\text{grooves}}{2400\raisebox{1ex}{$\text{grooves}$}\!\left/ \!\raisebox{-1ex}{$\text{mm}$}\right.} \\ \text{=104mm} \end{gathered}$$