Question

A 5.000-cm-wide diffraction grating with 200 slits is used to resolve two closely spaced lines (a doublet) in a spectrum. The doublet consists of two wavelengths, \(\lambda_{\mathrm{a}}=629.8 \mathrm{nm}\) and \(\lambda_{\mathrm{b}}=630.2 \mathrm{nm} .\) The light illuminates the entire grating at normal incidence. Calculate to four significant digits the angles \(\theta_{1 \mathrm{a}}\) and \(\theta_{\mathrm{lb}}\) with respect to the normal at which the first-order diffracted beams for the two wavelengths, \(\lambda_{\mathrm{a}}\) and \(\lambda_{\mathrm{b}}\), respectively, will be reflected from the grating. Note that this is not \(0^{\circ} !\) What order of diffraction is required to resolve these two lines using this grating?

Short answer

Answer: The first-order diffracted beams for the given wavelengths will be reflected at approximately \(8.3256^{\circ}\) and \(8.3316^{\circ}\). The order of diffraction required to resolve these two lines is 4.

Step-by-step solution

Identify the given values

We are given the following values: - Grating width = 5.000 cm - Number of slits = 200 - Wavelength a (\(\lambda_{a}\)) = 629.8 nm - Wavelength b (\(\lambda_{b}\)) = 630.2 nm

Calculate the slit spacing (d)

Using the given width of the grating and the number of slits, we can calculate the slit spacing (d) as follows: \(d = \frac{\text{Grating width}}{\text{Number of slits}} = \frac{5.000\,\text{cm}}{200} = 0.025\,\text{cm}\) Then, convert to meters: \(d = 0.025\, \text{cm}\times\frac{1\, \text{m}}{100\, \text{cm}} = 2.5\times10^{-4}\, \text{m}\)

Use the diffraction grating equation

The diffraction grating equation is given by: \( \sin\theta_m=m\frac{\lambda}{d} \) where, - m is the order of diffraction; - \(\lambda\) is the wavelength of light; - d is the slit spacing; - \(\theta_m\) is the angle of diffraction. We will solve for the angles \(\theta_{1a}\) and \(\theta_{1b}\) for the two given wavelengths at the first order (m=1).

Calculate angles for the first-order diffracted beams

Calculate the angle \(\theta_{1a}\) for \(\lambda_{a}\): \(\sin\theta_{1a} = \frac{1*\lambda_{a}}{d}\) \(\theta_{1a} = \sin^{-1}\frac{1*629.8\times10^{-9}\, \text{m}}{2.5\times10^{-4}\, \text{m}}\) \(\theta_{1a} = 0.14529\,\text{rad}\) Convert radians to degrees: \(\theta_{1a} = 0.14529\, \text{rad}\times\frac{180^{\circ}}{\pi} \approx 8.3256^{\circ}\) Now, calculate the angle \(\theta_{1b}\) for \(\lambda_{b}\): \(\sin\theta_{1b} = \frac{1*\lambda_{b}}{d}\) \(\theta_{1b} = \sin^{-1}\frac{1*630.2\times10^{-9}\, \text{m}}{2.5\times10^{-4}\, \text{m}}\) \(\theta_{1b} = 0.14542\,\text{rad}\) Convert radians to degrees: \(\theta_{1b} = 0.14542\, \text{rad}\times\frac{180^{\circ}}{\pi} \approx 8.3316^{\circ}\)

Calculate the order of diffraction required to resolve the doublet lines

To determine the order of diffraction required to resolve the two lines, we use the resolution equation for a diffraction grating: \(R = \frac{\lambda}{\Delta\lambda} = mN\) where, - R is the resolution; - \(\lambda\) is the average wavelength of the doublet; - \(\Delta\lambda\) is the difference in wavelengths; - m is the order of diffraction; - N is the number of slits. We already know N=200. We calculate the average wavelength and the difference in wavelengths: \(\lambda_{\text{avg}} = \frac{\lambda_{a} + \lambda_{b}}{2} = \frac{629.8 + 630.2}{2}\, \text{nm} = 630\, \text{nm}\) \(\Delta\lambda = \lambda_{b} - \lambda_{a} = 630.2 - 629.8 = 0.4\, \text{nm}\) Now, we solve for m: \(\frac{630\,\text{nm}}{0.4\,\text{nm}} = m*200\) \(m \approx 3.15\) Since m must be an integer, we find the smallest integer larger than 3.15, which is 4.

Write the final answer

The angles \(\theta_{1a}\) and \(\theta_{1b}\) for the first-order diffracted beams for the two wavelengths are approximately \(8.3256^{\circ}\) and \(8.3316^{\circ}\), respectively. The order of diffraction required to resolve these two lines using this grating is 4.

Key concepts

Wavelength Resolution

Understanding wavelength resolution is crucial when working with diffraction gratings. Wavelength resolution refers to the ability of a diffraction grating to distinguish between two closely spaced wavelengths, such as the doublet in our textbook example with wavelengths \(\lambda_a=629.8\,\mathrm{nm}\) and \(\lambda_b=630.2\,\mathrm{nm}\).

Improving resolution can be achieved by either increasing the number of slits \(N\) on the grating or by observing higher orders of diffraction \(m\), which enhance the differentiating capability of the grating. This is crucial in applications like spectroscopy, where identifying small differences in wavelength can determine the chemical composition or other characteristics of a substance.

The mathematical expression for resolving power \(R\) of a grating is given by \(R = \frac{\lambda}{\Delta\lambda} = mN\), where \(\Delta\lambda\) represents the smallest difference in wavelength that can be resolved. To provide context for students, if we increase the number of slits \(N\) while keeping the same grating width, the slit spacing \(d\) becomes smaller, leading to a greater separation of diffraction orders and thus a higher resolution.

Angle of Diffraction

The angle of diffraction is the angle at which light is scattered after passing through a diffraction grating. The angle is precisely determined by the diffraction grating equation \(\sin\theta_m=m\frac{\lambda}{d}\) and is dependent on the slit spacing \(d\), the order of diffraction \(m\), and the wavelength of the incident light \(\lambda\).

In our example, the angles of diffraction were calculated for the first-order beams for wavelengths \(\lambda_a\) and \(\lambda_b\) using the diffraction grating equation. This calculation is fundamental for students who are studying the properties of light and its interaction with materials. It illustrates how slight changes in wavelength can lead to measurable differences in the angle of diffraction, allowing for detailed analysis of light's spectral composition.

For practical implementations, consider that high-precision measurements are required when dealing with such small angular differences, especially in scientific research and optical engineering where detecting and measuring diffraction angles becomes pivotal for accurate data.

Order of Diffraction

The order of diffraction, denoted by \(m\), signifies the series or 'level' of the diffraction pattern produced by the grating. It's a whole number that shows how many wavelengths of phase difference between the light from adjacent slits are causing constructive interference. For instance, the first-order diffraction (\(m=1\)) occurs when this phase difference is exactly one wavelength, whereas second-order diffraction (\(m=2\)) occurs at two wavelengths, and so on.

In our exercise, we calculated that an order of 4 is required to resolve the two wavelengths, \(\lambda_a\) and \(\lambda_b\), using the grating. This concept is pivotal for students to comprehend as it demonstrates the role of coherent light and the importance of interference in studying various diffraction phenomena. With higher orders of diffraction, the separation between the diffracted beams increases, which enhances resolution - but this also decreases the intensity of the diffracted light, which is a critical consideration in experiments and technological applications.