Q15Engineering Physics

In spectroscopy, the true value of a diffraction grating is not merely its ability to bend light, but its ability to distinctly separate two very closely spaced spectral wavelengths (for example, the sodium doublet lines $D_1$ at $589.0\text{ nm}$ and $D_2$ at $589.6\text{ nm}$). This highly critical metric is known as the Resolving Power (RP).

Theoretical Definition and Formula

According to Lord Rayleigh's criterion, two spectral lines of wavelengths $\lambda$ and $(\lambda +d\lambda )$ are defined as being "just resolved" if the principal maximum of one line exactly coincides with the first minimum adjacent to the principal maximum of the other line.

Based on this geometric criterion, the resolving power is defined mathematically as the ratio of the wavelength to the smallest measurable wavelength difference. Derivations show that for a grating, this strictly depends on the order of the spectrum ($n$) and the absolute total number of illuminated slits ($N$) on the grating surface:

This formula fundamentally states that to achieve incredibly high resolution (to separate very tight spectral lines), you must either look at higher orders ($n=2,3$) or physically manufacture a wider grating with vastly more total lines ($N$).

Calculation Procedure

First, we must calculate the total number of ruling lines ($N$) present on the entire effective width of the given grating.

• Ruling density given: $5000\text{ lines per cm}$.
• Total illuminated width of the grating: $6\text{ cm}$.

Now we calculate the specific resolving power for different orders of the spectrum:

1. For the First Order Spectrum ($n=1$):

This means that in the first order, the grating can separate two wavelengths that differ by $\frac{\lambda }{30000}$. For yellow light ($\sim 600\text{ nm}$), it could resolve a difference of $0.02\text{ nm}$.

2. For the Second Order Spectrum ($n=2$):

By moving to the second order, the spectral lines are spread out twice as far, thus doubling the resolving power, allowing for the distinction of even finer atomic spectral details.