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Chapter 36: Problem 114

Two emission lines have wavelengths \(\lambda\) and \(\lambda+\Delta \lambda,\) respectively, where \(\Delta \lambda<\lambda\). Show that their angular separation \(\Delta \theta\) in a grating spectrometer is given approximately by $$\Delta \theta=\frac{\Delta \lambda}{\sqrt{(d / m)^{2}-\lambda^{2}}}$$ where \(d\) is the slit separation and \(m\) is the order at which the lines are observed. Note that the angular separation is greater in the higher orders than the lower orders.

Short Answer

Expert verified
The angular separation \( \Delta \theta \) is approximately \( \frac{\Delta \lambda}{\sqrt{(d/m)^2 - \lambda^2}} \).

Step by step solution

01

Understand the Basic Setup

In a grating spectrometer, light is diffracted by a grating with slit separation \( d \). When light of wavelength \( \lambda \) is incident on the grating, it is diffracted and forms maxima at an angle \( \theta \) given by the grating equation: \( d \sin \theta = m \lambda \), where \( m \) is the diffraction order.
02

Apply the Grating Equation

For two close wavelengths \( \lambda \) and \( \lambda+\Delta \lambda \), we have two equations: \( d \sin \theta = m \lambda \) and \( d \sin (\theta + \Delta \theta) = m (\lambda + \Delta \lambda) \). We aim to find the small angle \( \Delta \theta \), the angular separation between these two wavelengths.
03

Use Small Angle Approximation

Assume \( \Delta \theta \) is small and use the approximation \( \sin(\theta + \Delta \theta) \approx \sin \theta + \Delta \theta \cos \theta \). Substitute into the second equation to get: \( d ((\sin \theta) + (\Delta \theta \cos \theta)) = m (\lambda + \Delta \lambda) \).
04

Simplify the Equation

Replace \( d \sin \theta \) from the first equation into the expanded second equation: \( d \sin \theta + d \Delta \theta \cos \theta = m \lambda + m \Delta \lambda \). Cancelling \( d \sin \theta = m \lambda \) gives us: \( d \Delta \theta \cos \theta = m \Delta \lambda \).
05

Solve for Angular Separation

Solve the equation for \( \Delta \theta \):\[\Delta \theta = \frac{m \Delta \lambda}{d \cos \theta}\]We need to express \( \cos \theta \) in terms of known quantities. From \( d \sin \theta = m \lambda \), use \( \cos \theta = \sqrt{1-\sin^2 \theta} = \sqrt{1-(m\lambda/d)^2} \). Substitute this back into the equation for \( \Delta \theta \): \[\Delta \theta = \frac{m \Delta \lambda}{d \sqrt{1-(m\lambda/d)^2}}\]
06

Final Simplification

Recognize that \( 1-(m\lambda/d)^2 = (d/m)^2 - \lambda^2 \) under the identity \( 1-a^2 = 1 - (b/c)^2 = (c^2-a^2)/c^2 \). Therefore, express \( \Delta \theta \) as:\[\Delta \theta = \frac{\Delta \lambda}{\sqrt{(d/m)^2 - \lambda^2}}\]confirming the original approximation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Angular Separation
In a diffraction grating setup, understanding angular separation is key. It represents the angle between the two different wavelengths of light after they pass through the grating. When light with varying wavelengths passes through the grating, each wavelength is diffracted at a distinct angle. The diffraction angle for a particular wavelength of light is defined by the grating equation:
\[ d \sin \theta = m \lambda \]where:
  • \( d \) is the slit separation.
  • \( m \) is the diffraction order.
  • \( \lambda \) is the wavelength of light.
Smaller changes in wavelength result in slight angular shifts, which is what is known as angular separation \( \Delta \theta \). The expression for angular separation can be derived considering a small angle approximation, and greatly depends on the wavelength difference and the diffraction order. This separation is crucial in identifying how distinct the spectral lines will be when viewed through a spectrometer.
Grating Spectrometer
A grating spectrometer is an instrument used to analyze light spectra by spreading out light into its component wavelengths. At its core is the diffraction grating, which consists of many equally spaced slits. When light interacts with the grating, diffraction occurs, and a spectrum is formed. This allows for the precise measurement of wavelengths.
Key components and features of a grating spectrometer include:
  • A diffraction grating with known slit spacing \( d \).
  • Capability to measure light intensity at various diffraction angles \( \theta \).
  • Application of the grating equation to relate these angles to the corresponding wavelengths.
By examining the angles where light of different wavelengths are diffracted, it becomes possible to determine the composition of the light source. Higher diffraction orders provide more detailed separation of closely spaced wavelengths, enhancing resolution in spectral analysis.
Wavelength Difference
Wavelength difference, denoted \( \Delta \lambda \), represents the small variation between two close emission lines when analyzed through a spectrometer. It's imperative in spectroscopy, especially when an instrument must distinguish between nearly identical colors in the light spectrum.
The angular separation formula:\[\Delta \theta = \frac{\Delta \lambda}{\sqrt{(d / m)^{2}-\lambda^{2}}}\]shows that a smaller wavelength difference results in a reduced angular separation, making it challenging to resolve the lines. The precision of measurement in determining these small changes in wavelength helps scientists better understand the material or source of emissions being studied. The brilliance of optical devices like grating spectrometers lies in their ability to discern even these minute variations.
Diffraction Order
Diffraction order, represented by \( m \), is a critical integer in the diffraction equation. It describes the sequence in which wavelengths produce maxima as they pass through a grating. Higher orders (\( m \) > 1) correspond to secondary, tertiary, and further spectra, appearing at steeper angles.
The angular separation is significantly influenced by the diffraction order. Higher orders yield greater angular separation between wavelengths due to the linear dependency of \( m \) in the expression for \( \Delta \theta \):\[\Delta \theta = \frac{m \Delta \lambda}{d \cos \theta}\]A larger \( m \) amplifies the separation, providing better resolution of closely spaced lines in spectroscopy. Hence, optical systems that exploit higher orders can better magnify tiny wavelength differences, an advantage in precision spectroscopy and applications demanding detailed spectral information.

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Most popular questions from this chapter

An x-ray beam of a certain wavelength is incident on an \(\mathrm{NaCl}\) crystal, at \(30.0^{\circ}\) to a certain family of reflecting planes of spacing \(39.8 \mathrm{pm} .\) If the reflection from those planes is of the first order, what is the wavelength of the x rays?

Light of wavelength \(500 \mathrm{nm}\) diffracts through a slit of width \(2.00 \mu \mathrm{m}\) and onto a screen that is \(2.00 \mathrm{~m}\) away. On the screen, what is the distance between the center of the diffraction pattern and the third diffraction minimum?

A diffraction grating \(3.00 \mathrm{~cm}\) wide produces the second order at \(33.0^{\circ}\) with light of wavelength \(600 \mathrm{nm}\). What is the total number of lines on the grating?

Perhaps to confuse a predator, some tropical gyrinid beetles (whirligig beetles) are colored by optical interference that is due to scales whose alignment forms a diffraction grating (which scatters light instead of transmitting it). When the incident light rays are perpendicular to the grating, the angle between the firstorder maxima (on opposite sides of the zeroth-order maximum) is about \(26^{\circ}\) in light with a wavelength of \(550 \mathrm{nm}\). What is the grating spacing of the beetle?

A slit \(1.00 \mathrm{~mm}\) wide is illuminated by light of wavelength \(589 \mathrm{nm}\). We see a diffraction pattern on a screen \(3.00 \mathrm{~m}\) away. What is the distance between the first two diffraction minima on the same side of the central diffraction maximum?
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