Grade 11

Advanced

9 min

Lesson 7 of 12

Diffraction Gratings and Spectroscopy

Not Started

How multiple slits create sharp spectral lines, enabling the chemical analysis of stars and gases.

How do we know that a star trillions of miles away is made of Hydrogen and Helium without ever leaving Earth? The secret lies in a tiny piece of glass with thousands of invisible scratches.

Learning Objectives

1.Derive and apply the diffraction grating equation $d \sin \theta = n\lambda$ to find wavelengths or angles.
2.Calculate the slit spacing $d$ from the number of lines per millimeter ( $N$ ).
3.Explain how emission spectra act as 'chemical fingerprints' to identify elements in distant gases.

From Two Slits to Thousands

In Young's double-slit experiment, we saw light create a pattern of blurry fringes. A diffraction grating takes this to the next level by using thousands of parallel, closely spaced slits. While the principle of interference remains the same, the result is different: the 'bright spots' (maxima) become incredibly sharp and narrow. This happens because with more slits, light waves must be perfectly in phase to interfere constructively; even a tiny deviation in angle causes the waves from the thousands of slits to cancel each other out. This precision allows scientists to separate wavelengths of light that are nearly identical.

Quick Check

Why are the bright fringes produced by a diffraction grating much sharper than those produced by a double-slit setup?

Think about how many more waves are 'competing' to stay in phase.

Answer

With thousands of slits, waves only interfere constructively at very precise angles; at almost any other angle, the waves from the many slits destructively interfere and cancel out.

The Grating Equation

To locate these sharp maxima, we use the grating equation:

d \sin \theta = n\lambda

Here,

d

is the distance between adjacent slits,

\theta

is the angle from the center,

n

is the order of the maximum (

n=0, 1, 2...

), and

\lambda

is the wavelength. A critical step is finding

d

. If a grating has

N

lines per millimeter, then:

d = \frac{1}{N}

Because

\theta

depends on

\lambda

, different colors of light will appear at different angles, creating a beautiful, spread-out spectrum for each order

n

A diffraction grating has 500 lines per mm. Calculate the angle of the first-order maximum ( $n=1$ ) for red light with a wavelength of $650\text{ nm}$ .

1. Find $d$ : $d = \frac{1}{500\text{ lines/mm}} = 0.002\text{ mm} = 2.0 \times 10^{-6}\text{ m}$ . 2. Convert $\lambda$ to meters: $650\text{ nm} = 6.5 \times 10^{-7}\text{ m}$ . 3. Use the equation: $\sin \theta = \frac{n\lambda}{d} = \frac{1 \times 6.5 \times 10^{-7}}{2.0 \times 10^{-6}} = 0.325$ . 4. Calculate $\theta$ : $\theta = \arcsin(0.325) \approx 18.97^\circ$ .

Spectroscopy: The Cosmic Fingerprint

When an element like Hydrogen is heated, its electrons jump to higher energy levels and then fall back, emitting light at very specific wavelengths. This creates an emission spectrum. Because a diffraction grating spreads light out so precisely, we can measure these exact wavelengths. Since every element has a unique electron configuration, every element has a unique 'fingerprint' of spectral lines. By pointing a spectroscope (a grating-based tool) at a star, we can see which lines are present and identify the elements burning inside that star.

A technician observes a green spectral line ( $\lambda = 540\text{ nm}$ ) at an angle of $32.7^\circ$ in the second order ( $n=2$ ). How many lines per millimeter does this grating have?

1. Rearrange for $d$ : $d = \frac{n\lambda}{\sin \theta}$ . 2. Substitute values: $d = \frac{2 \times 540 \times 10^{-9}}{\sin(32.7^\circ)} = \frac{1.08 \times 10^{-6}}{0.540} = 2.0 \times 10^{-6}\text{ m}$ . 3. Convert $d$ to mm: $2.0 \times 10^{-6}\text{ m} = 0.002\text{ mm}$ . 4. Find $N$ : $N = \frac{1}{d} = \frac{1}{0.002} = 500\text{ lines/mm}$ .

Quick Check

If you use a grating with more lines per millimeter, will the light be spread out more or less?

Look at the relationship between

d

and

\sin \theta

in the equation.

Answer

More. A higher $N$ means a smaller $d$ , which results in a larger $\sin \theta$ for the same wavelength.

A grating has $600$ lines per mm. What is the highest order ( $n$ ) visible for blue light ( $\lambda = 450\text{ nm}$ )?

1. Find $d$ : $d = \frac{1}{600,000\text{ lines/m}} \approx 1.67 \times 10^{-6}\text{ m}$ . 2. Recognize the limit: The maximum possible angle is $90^\circ$ , so $\sin \theta \le 1$ . 3. Set up the inequality: $n\lambda / d \le 1 \implies n \le d / \lambda$ . 4. Calculate: $n \le \frac{1.67 \times 10^{-6}}{450 \times 10^{-9}} \approx 3.71$ . 5. Conclusion: Since $n$ must be an integer, the highest visible order is $n=3$ .

Key Takeaways

1Diffraction gratings produce sharper, more defined maxima than double slits due to the interference of thousands of light sources.
2The grating equation $d \sin \theta = n\lambda$ relates the slit spacing, angle, order of the fringe, and wavelength.
3The slit spacing $d$ is the reciprocal of the line density $N$ (lines per unit length).
4Spectroscopy uses these gratings to identify elements by their unique emission spectra, which appear as distinct colored lines.

Quick Check

Multiple Choice

What happens to the diffraction angle $\theta$ if you switch from a grating with 300 lines/mm to one with 600 lines/mm?

Multiple Choice

In a standard diffraction grating experiment with white light, which color will be found at the largest angle for the first order?

True/False

The zero-order maximum ( $n=0$ ) for white light through a grating appears as a rainbow.

What's Next?

Review Tomorrow

In 24 hours, try to sketch the setup of a diffraction grating and write down the formula $d \sin \theta = n\lambda$ from memory, explaining what each variable represents.

Practice Activity

Look up the 'Hydrogen Emission Spectrum' online. Try to calculate the angles for its four visible lines (410nm, 434nm, 486nm, 656nm) using a hypothetical grating of 600 lines/mm.

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Diffraction Gratings and Spectroscopy

Learning Objectives

From Two Slits to Thousands

The Grating Equation

Spectroscopy: The Cosmic Fingerprint

Key Takeaways

Quick Check

What's Next?

Diffraction Gratings and Spectroscopy

Learning Objectives

From Two Slits to Thousands

The Grating Equation

Spectroscopy: The Cosmic Fingerprint

Key Takeaways

Quick Check

What's Next?