# Diffraction Grating Rainbow

The dispersion and ability to the disconnect of diffraction grating. Thanks for your attention! .,when you take a keen look at a CD or DVD the closely spaced tracks on a CD or DVD act as a diffraction grating to form the familiar rainbow pattern.

FREE K-12 standards-aligned STEM

curriculum for educators everywhere!

Find more at **TeachEngineering.org**.

### Quick Look

**Grade Level:** 8 (7-9)

**Time Required:**30 minutes

**Expendable Cost/Group:** US $0.00

**Group Size:** 1

**Activity Dependency:** None

**Subject Areas:** Earth and Space

NGSS Performance Expectations:

MS-PS4-2 |

### Summary

Students are introduced to different ways of displaying visual spectra, including colored 'barcode' spectra, like those produced by a diffraction grating, and line plots displaying intensity versus color, or wavelength. Students learn that a diffraction grating acts like a prism, bending light into its component colors.*This engineering curriculum aligns to Next Generation Science Standards (NGSS).*

### Engineering Connection

Understanding graphs and plots is crucial to engineering, as engineering in all fields is driven by the need to obtain scientific data. Engineering methods are constantly improved and new types of engineering are created based upon the types of data needed to advance science.

### Learning Objectives

After this activity, students should be able to:

- Explain that light from different sources, when passed through a prism or diffraction grating, can be separated into component colors.
- Explain the basic tools engineers use to view spectra.
- Match a 'barcode' spectrum with its corresponding line plot.

### Educational Standards Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*, a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics; within type by subtype, then by grade, *etc*.

Each *TeachEngineering* lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards.

All 100,000+ K-12 STEM standards covered in *TeachEngineering* are collected, maintained and packaged by the *Achievement Standards Network (ASN)*, a project of *D2L* (www.achievementstandards.org).

In the ASN, standards are hierarchically structured: first by source; *e.g.*, by state; within source by type; *e.g.*, science or mathematics; within type by subtype, then by grade, *etc*.

###### NGSS: Next Generation Science Standards - Science

NGSS Performance Expectation | ||
---|---|---|

MS-PS4-2. Develop and use a model to describe that waves are reflected, absorbed, or transmitted through various materials. (Grades 6 - 8) Do you agree with this alignment? Thanks for your feedback! | ||

Click to view other curriculum aligned to this Performance Expectation | ||

This activity focuses on the following Three Dimensional Learning aspects of NGSS: | ||

Science & Engineering Practices | Disciplinary Core Ideas | Crosscutting Concepts |

Develop and use a model to describe phenomena. Alignment agreement: Thanks for your feedback! | A sound wave needs a medium through which it is transmitted. Alignment agreement: Thanks for your feedback! When light shines on an object, it is reflected, absorbed, or transmitted through the object, depending on the object's material and the frequency (color) of the light.Alignment agreement: Thanks for your feedback! The path that light travels can be traced as straight lines, except at surfaces between different transparent materials (e.g., air and water, air and glass) where the light path bends.Alignment agreement: Thanks for your feedback! A wave model of light is useful for explaining brightness, color, and the frequency-dependent bending of light at a surface between media.Alignment agreement: Thanks for your feedback! However, because light can travel through space, it cannot be a matter wave, like sound or water waves.Alignment agreement: Thanks for your feedback! | Structures can be designed to serve particular functions by taking into account properties of different materials, and how materials can be shaped and used. Alignment agreement: Thanks for your feedback! |

###### Common Core State Standards - Math

- Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. (Grade 8) More Details
Do you agree with this alignment? Thanks for your feedback!

- Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (Grades 9 - 12) More Details
Do you agree with this alignment? Thanks for your feedback!

###### International Technology and Engineering Educators Association - Technology

- New products and systems can be developed to solve problems or to help do things that could not be done without the help of technology. (Grades 6 - 8) More Details
Do you agree with this alignment? Thanks for your feedback!

###### State Standards

###### Colorado - Science

- Develop and design a scientific investigation regarding absorption, reflection, and refraction of light (Grade 8) More Details
Do you agree with this alignment? Thanks for your feedback!

### Materials List

Each student needs:

- 1 copy of Graphing the Rainbow Worksheet
- 1 pencil

### Worksheets and Attachments

Visit [www.teachengineering.org/activities/view/cub_spect_activity2] to print or download.### Pre-Req Knowledge

Students should be familiar with line graphing methods and understand that graphs can be used to represent physical data. Students should have some understanding of the nature of light, i.e., rainbows are formed with light, light can be different colors, etc.

### Introduction/Motivation

How is a rainbow formed? It is created by light, and that light comes from the Sun. When the light passes through water droplets in the clouds, we can see the colors that we cannot usually see with our eyes. All light makes a pattern, and today we will be exploring the patterns that are hidden in light that we cannot normally see unless we have special tools to see them. Any light source, whether it is a light bulb, a computer monitor, a star, or a planet — when passed through a prism or a diffraction grating — displays a unique pattern of bright and dark stripes called *spectra *(the plural of 'spectrum').

Prisms and diffraction gratings are tools we can use to see these patterns. Instrumentation developed by engineers can also measure exactly how bright each color is, since this is a difficult thing to do with our eyes. The instrumentation assigns a number value to brightness that can be plotted on a graph in which the x-axis position (horizontal) represents color and the y-axis (vertical) represents brightness, or intensity.

Engineers develop instrumentation based upon the properties of light. Engineers create instrumentation to see spectral patterns of light and study the patterns to improve and develop new instrumentation. They usually use diffraction gratings. They also study the processes and the types of light that create specific spectral patterns. Engineers studying space science are interested in answering questions about the composition of planetary atmospheres, planetary moons, stars and gasses within the solar system and universe.

### Procedure

Background

White light, like that produced by an incandescent light bulb (with electricity passing through it), is composed of all of the colors of light in the rainbow combined. To our eyes, it simply looks white. A diffraction grating (or a prism) acts to break the light into its component colors. Certain colors 'bend' more than others through the grating or prism, which is why the colors line up like a rainbow.

Light passing through a cool gas produces what is called an absorption pattern when seen through a diffraction grating or a prism, and dark lines appear in the continuous spectrum. The dark lines are actually created by the gas absorbing the energy of the light. We can identify the gas based on the distinctive pattern of lines that appear in the spectrum. Conversely, an emission spectrum is seen as bright lines against a dark background and is produced by a hot gas emitting photons. Again, the pattern the gas creates is dependent on the type of gas. A particular hot gas shows emission lines in the exact same places that the same cool gas shows absorption lines. The pattern does not change, but whether you see an absorption or emission pattern through the grating does change depending whether the gas is hot or cool. The resulting spectrum has a characteristic pattern of light and dark that, when analyzed, reveals the composition of the light source.

When light passes through a material (including gases and liquids), many things may happen to it:

**Absorption**

As previously described, the light may be absorbed. This happens when the light photons (or 'packages') hit an object, and it's molecules are shaken up by the light. This causes the material to become hotter.

**Reflection**

Reflection occurs when light bounces off of a material. The types (wavelengths) of light reflected depend on the composition of the material.

**Diffraction**

Diffraction occurs when light is bend and spread around an object or a slit.

**Scatter**

Scattering occurs when light bounces of an object in a variety of direction. The sky is blue due to light scattering. The higher energy (shorter wavelength) light (violet and blue) tends to be scattered by the atmosphere, while the longer wavelength light (yellow and red) does not. Therefore blue and violet and scattered, and the sky appears blue (more blue than violet because human's eyes are more sensitive to blue).

**Refraction**

Refraction occurs when light changes direction when passing through an object. The different wavelengths are changed at different rates and allows colors to be separated The previous examples of a glass prism and rain drops are examples of refraction.

Before the Activity

- Complete the 'Patterns and Fingerprints' activity.
- Make copies of the Graphing the Rainbow Worksheet
- (optional) Make an overhead transparency of page two of the worksheet.

With the Students

- Hand out the worksheets.
- Using the worksheet as a guide, demonstrate how a visible light pattern can be graphically represented in different ways. It may help to show an overhead projection of page two of the worksheet.
- Walk around the room, talking to individual students about the graphs from the worksheet.
- Ask students questions about what they see in the plots and why the pictures correspond to a specific graph on the worksheet.

### Vocabulary/Definitions

absorption spectrum: Dark lines that appear against the continuous spectrum seen through a spectrograph.

continuous spectrum: The rainbow that white light is composed in which each color is equally bright.

diffraction: When light bends around an obstacle or through a small opening like those in a diffraction grating.

diffraction grating: Usually a piece of film covered with very thin, parallel grooves.

emission spectrum: Bright lines that appear through the spectrograph against a dark background.

light source: Any object that produces light.

spectrograph: A device that allows one to see a spectrum; it usually has a prism or diffraction grating inside.

spectrum (plural: spectra): The pattern light produces when passed through a prism or diffraction grating, as seen through a spectrograph.

### Assessment

Pre-Lesson Assessment

*Class Discussion:* Ask students what they can tell you about light. Probe them to find out what they already know and understand.

Activity Embedded Assessment

*Class Discussion*: Ask students why they think light forms rainbows or patterns when passed through a prism or diffraction grating. (Note: The bending of light through a prism does not have to do with varying speeds of the colors! All colors travel at the same speed.)

*Worksheet*: Have students complete the activity worksheet; review their answers to gauge their mastery of the subject.

Post-Activity Assessment

*Think-Pair-Share*: Ask students to discuss with a peer what steps engineers take before designing instruments that study light. Randomly select groups to share. Discuss ideas as a class.

*Graphing*: Graphing and plotting are tools that all engineers use. Graphing and plotting real-world situations allow engineers to analyze whether a tool is working, how to design an effective tool, and whether the tool can be used to create software for looking at data (just like the data in this activity). Ask students to graphically represent a real-world situation, such as driving at a certain speed, coming immediately to a complete stop at some point, and then resuming that same speed. Another example might be a plot representing the descending from the top of a flight of stairs to arrive at some distance at the bottom (a distance vs. time plot). In this instance, have students represent what it would look like if they stopped on a stair for a very long time. This establishes whether they understand how graphing spectra is a representation of a real situation that occurs with light (as opposed to motion, distance or some other variable). It also establishes whether students can apply what they have learned in a different context. Ask students to come up with their own 'real-world' graphs, and ask volunteers to explain their graphs to the class.

### Troubleshooting Tips

Colorblind and vision-impaired children will have difficulty with this activity. Students with corrective lenses will not have difficulty. Pair colorblind and blind students with other students to assist them.

### Activity Extensions

Continue the spectroscopy unit by completing the associated, 'Using Spectral Data to Explore Saturn and Titan' activities.

### References

Fisher, Diane. 'Taking Apart the Light.' 'The Technology Teacher.' March 2002.

### Copyright

© 2007 by Regents of the University of Colorado.### Contributors

Laboratory for Atmospheric and Space Physics, University of Colorado at Boulder### Supporting Program

Laboratory for Atmospheric and Space Physics (LASP), University of Colorado Boulder### Acknowledgements

The contents of this digital library curriculum were developed under a grant from the Fund for the Improvement of Postsecondary Education (FIPSE), U.S. Department of Education, and National Science Foundation GK-12 grant no 0338326. However, these contents do not necessarily represent the policies of the Department of Education or National Science Foundation, and you should not assume endorsement by the federal government.

Last modified: April 30, 2021

Find more at TeachEngineering.org

### Learning Objectives

By the end of this section, you will be able to:

- Discuss the pattern obtained from diffraction grating.
- Explain diffraction grating effects.

Figure 1. The colors reflected by this compact disc vary with angle and are not caused by pigments. Colors such as these are direct evidence of the wave character of light. (credit: Infopro, Wikimedia Commons)

An interesting thing happens if you pass light through a large number of evenly spaced parallel slits, called a *diffraction grating*. An interference pattern is created that is very similar to the one formed by a double slit (see Figure 2). A diffraction grating can be manufactured by scratching glass with a sharp tool in a number of precisely positioned parallel lines, with the untouched regions acting like slits. These can be photographically mass produced rather cheaply. Diffraction gratings work both for transmission of light, as in Figure 2, and for reflection of light, as on butterfly wings and the Australian opal in Figure 3 or the CD in Figure 1. In addition to their use as novelty items, diffraction gratings are commonly used for spectroscopic dispersion and analysis of light. What makes them particularly useful is the fact that they form a sharper pattern than double slits do. That is, their bright regions are narrower and brighter, while their dark regions are darker. Figure 4 shows idealized graphs demonstrating the sharper pattern. Natural diffraction gratings occur in the feathers of certain birds. Tiny, finger-like structures in regular patterns act as reflection gratings, producing constructive interference that gives the feathers colors not solely due to their pigmentation. This is called iridescence.

Figure 1. A diffraction grating is a large number of evenly spaced parallel slits. (a) Light passing through is diffracted in a pattern similar to a double slit, with bright regions at various angles. (b) The pattern obtained for white light incident on a grating. The central maximum is white, and the higher-order maxima disperse white light into a rainbow of colors.

Figure 2. (a) This Australian opal and (b) the butterfly wings have rows of reflectors that act like reflection gratings, reflecting different colors at different angles. (credits: (a) Opals-On-Black.com, via Flickr (b) whologwhy, Flickr)

Figure 4. Idealized graphs of the intensity of light passing through a double slit (a) and a diffraction grating (b) for monochromatic light. Maxima can be produced at the same angles, but those for the diffraction grating are narrower and hence sharper. The maxima become narrower and the regions between darker as the number of slits is increased.

The analysis of a diffraction grating is very similar to that for a double slit (see Figure 5). As we know from our discussion of double slits in Young’s Double Slit Experiment, light is diffracted by each slit and spreads out after passing through. Rays traveling in the same direction (at an angle *θ* relative to the incident direction) are shown in Figure 5. Each of these rays travels a different distance to a common point on a screen far away. The rays start in phase, and they can be in or out of phase when they reach a screen, depending on the difference in the path lengths traveled.

As seen in Figure 5, each ray travels a distance *d* sin *θ* different from that of its neighbor, where *d* is the distance between slits. If this distance equals an integral number of wavelengths, the rays all arrive in phase, and constructive interference (a maximum) is obtained. Thus, the condition necessary to obtain *constructive interference for a diffraction grating* is *d* sin *θ = mλ, *for *m* = 0, 1, −1, 2, −2, . . . (constructive) where *d* is the distance between slits in the grating, *λ* is the wavelength of light, and *m* is the order of the maximum. Note that this is exactly the same equation as for double slits separated by *d*. However, the slits are usually closer in diffraction gratings than in double slits, producing fewer maxima at larger angles.

In Figure 5, we see a diffraction grating showing light rays from each slit traveling in the same direction. Each ray travels a different distance to reach a common point on a screen (not shown). Each ray travels a distance *d* sin *θ* different from that of its neighbor.

Where are diffraction gratings used? Diffraction gratings are key components of monochromators used, for example, in optical imaging of particular wavelengths from biological or medical samples. A diffraction grating can be chosen to specifically analyze a wavelength emitted by molecules in diseased cells in a biopsy sample or to help excite strategic molecules in the sample with a selected frequency of light. Another vital use is in optical fiber technologies where fibers are designed to provide optimum performance at specific wavelengths. A range of diffraction gratings are available for selecting specific wavelengths for such use.

### Take-Home Experiment: Rainbows on a CD

The spacing *d* of the grooves in a CD or DVD can be well determined by using a laser and the equation *d* sin *θ = mλ, *for*m* = 0, 1, −1, 2, −2, . . . . However, we can still make a good estimate of this spacing by using white light and the rainbow of colors that comes from the interference. Reflect sunlight from a CD onto a wall and use your best judgment of the location of a strongly diffracted color to find the separation *d*.

### Example 1. Calculating Typical Diffraction Grating Effects

Diffraction gratings with 10,000 lines per centimeter are readily available. Suppose you have one, and you send a beam of white light through it to a screen 2.00 m away.

- Find the angles for the first-order diffraction of the shortest and longest wavelengths of visible light (380 and 760 nm).
- What is the distance between the ends of the rainbow of visible light produced on the screen for first-order interference? (See Figure 6.)

Figure 6. The diffraction grating considered in this example produces a rainbow of colors on a screen a distance from the grating. The distances along the screen are measured perpendicular to the x-direction. In other words, the rainbow pattern extends out of the page.

#### Strategy

The angles can be found using the equation *d* sin *θ = mλ (*for*m* = 0, 1, −1, 2, −2, . . . ) once a value for the slit spacing *d* has been determined. Since there are 10,000 lines per centimeter, each line is separated by 1/10,000 of a centimeter. Once the angles are found, the distances along the screen can be found using simple trigonometry.

#### Solution for Part 1

The distance between slits is [latex]d=frac{1text{ cm}}{10,000}=1.00times10^{-4}text{ cm}[/latex] or 1.00 × 10^{−6} m. Let us call the two angles *θ*_{V} for violet (380 nm) and *θ*_{R} for red (760 nm). Solving the equation *d* sin θ_{V}* = mλ* for sin θ_{V}, [latex]sintheta_{text{V}}=frac{mlambda_{text{V}}}{d}[/latex], where *m *= 1 for first order and *λ*_{V} = 380 nm = 3.80 × 10^{−7} m. Substituting these values gives

[latex]displaystylesintheta_{text{V}}=frac{3.80times10^{-7}text{ m}}{1.00times10^{-6}text{ m}}=0.380[/latex]

Thus the angle *θ*_{V} is *θ*_{V} = sin^{−1} 0.380 = 22.33º.

Similarly,

[latex]displaystylesintheta_{text{R}}=frac{7.60times10^{-7}text{ m}}{1.00times10^{-6}text{ m}}[/latex]

Thus the angle *θ*_{R} is *θ*_{R} = sin^{−1} 0.760 = 49.46º.

Notice that in both equations, we reported the results of these intermediate calculations to four significant figures to use with the calculation in Part 2.

#### Solution for Part 2

The distances on the screen are labeled *y*_{V} and *y*_{R} in Figure 6. Noting that [latex]tantheta=frac{y}{x}[/latex], we can solve for *y*_{V} and *y*_{R}. That is, *y*_{V} = *x *tan *θ*_{V} = (2.00 m)(tan 22.33º) = 0.815 m and *y*_{R} = *x *tan *θ*_{R} = (2.00 m)(tan 49.46º) = 2.338 m.

The distance between them is therefore *y*_{R} − *y*_{V} = 1.52 m.

#### Discussion

### Diffraction Grating Rainbow Light

The large distance between the red and violet ends of the rainbow produced from the white light indicates the potential this diffraction grating has as a spectroscopic tool. The more it can spread out the wavelengths (greater dispersion), the more detail can be seen in a spectrum. This depends on the quality of the diffraction grating—it must be very precisely made in addition to having closely spaced lines.

## Section Summary

A diffraction grating is a large collection of evenly spaced parallel slits that produces an interference pattern similar to but sharper than that of a double slit.

There is constructive interference for a diffraction grating when d sin *θ *= *mλ* (for *m* = 0 , 1, –1, 2, –2, …), where *d* is the distance between slits in the grating, *λ* is the wavelength of light, and* m *is the order of the maximum.

### Conceptual Questions

- What is the advantage of a diffraction grating over a double slit in dispersing light into a spectrum?
- What are the advantages of a diffraction grating over a prism in dispersing light for spectral analysis?
- Can the lines in a diffraction grating be too close together to be useful as a spectroscopic tool for visible light? If so, what type of EM radiation would the grating be suitable for? Explain.
- If a beam of white light passes through a diffraction grating with vertical lines, the light is dispersed into rainbow colors on the right and left. If a glass prism disperses white light to the right into a rainbow, how does the sequence of colors compare with that produced on the right by a diffraction grating?
- Suppose pure-wavelength light falls on a diffraction grating. What happens to the interference pattern if the same light falls on a grating that has more lines per centimeter? What happens to the interference pattern if a longer-wavelength light falls on the same grating? Explain how these two effects are consistent in terms of the relationship of wavelength to the distance between slits.
- Suppose a feather appears green but has no green pigment. Explain in terms of diffraction.
- It is possible that there is no minimum in the interference pattern of a single slit. Explain why. Is the same true of double slits and diffraction gratings?

### Problems & Exercises

- A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nm-wavelength green light?
- Find the angle for the third-order maximum for 580-nm-wavelength yellow light falling on a diffraction grating having 1500 lines per centimeter.
- How many lines per centimeter are there on a diffraction grating that gives a first-order maximum for 470-nm blue light at an angle of 25.0º?
- What is the distance between lines on a diffraction grating that produces a second-order maximum for 760-nm red light at an angle of 60.0º?
- Calculate the wavelength of light that has its second-order maximum at 45.0º when falling on a diffraction grating that has 5000 lines per centimeter.
- An electric current through hydrogen gas produces several distinct wavelengths of visible light. What are the wavelengths of the hydrogen spectrum, if they form first-order maxima at angles of 24.2º, 25.7º, 29.1º, and 41.0º when projected on a diffraction grating having 10,000 lines per centimeter?
- (a) What do the four angles in the above problem become if a 5000-line-per-centimeter diffraction grating is used? (b) Using this grating, what would the angles be for the second-order maxima? (c) Discuss the relationship between integral reductions in lines per centimeter and the new angles of various order maxima.
- What is the maximum number of lines per centimeter a diffraction grating can have and produce a complete first-order spectrum for visible light?
- The yellow light from a sodium vapor lamp seems to be of pure wavelength, but it produces two first-order maxima at 36.093º and 36.129º when projected on a 10,000 line per centimeter diffraction grating. What are the two wavelengths to an accuracy of 0.1 nm?
- What is the spacing between structures in a feather that acts as a reflection grating, given that they produce a first-order maximum for 525-nm light at a 30.0º angle?
- Structures on a bird feather act like a reflection grating having 8000 lines per centimeter. What is the angle of the first-order maximum for 600-nm light?
- An opal such as that shown in Figure 2 acts like a reflection grating with rows separated by about 8 μm. If the opal is illuminated normally, (a) at what angle will red light be seen and (b) at what angle will blue light be seen?
- At what angle does a diffraction grating produces a second-order maximum for light having a first-order maximum at 20.0º?
- Show that a diffraction grating cannot produce a second-order maximum for a given wavelength of light unless the first-order maximum is at an angle less than 30.0º.
- If a diffraction grating produces a first-order maximum for the shortest wavelength of visible light at 30.0º, at what angle will the first-order maximum be for the longest wavelength of visible light?
- (a) Find the maximum number of lines per centimeter a diffraction grating can have and produce a maximum for the smallest wavelength of visible light. (b) Would such a grating be useful for ultraviolet spectra? (c) For infrared spectra?
- (a) Show that a 30,000-line-per-centimeter grating will not produce a maximum for visible light. (b) What is the longest wavelength for which it does produce a first-order maximum? (c) What is the greatest number of lines per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light?
- A He–Ne laser beam is reflected from the surface of a CD onto a wall. The brightest spot is the reflected beam at an angle equal to the angle of incidence. However, fringes are also observed. If the wall is 1.50 m from the CD, and the first fringe is 0.600 m from the central maximum, what is the spacing of grooves on the CD?
- The analysis shown in the figure below also applies to diffraction gratings with lines separated by a distance
*d*. What is the distance between fringes produced by a diffraction grating having 125 lines per centimeter for 600-nm light, if the screen is 1.50 m away?Figure 6. The distance between adjacent fringes is [latex]Delta y=frac{xlambda}{d}[/latex], assuming the slit separation d is large compared with λ.

**Unreasonable Results.**Red light of wavelength of 700 nm falls on a double slit separated by 400 nm. (a) At what angle is the first-order maximum in the diffraction pattern? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?**Unreasonable Results.**(a) What visible wavelength has its fourth-order maximum at an angle of 25.0º when projected on a 25,000-line-per-centimeter diffraction grating? (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?**Construct Your Own Problem.**Consider a spectrometer based on a diffraction grating. Construct a problem in which you calculate the distance between two wavelengths of electromagnetic radiation in your spectrometer. Among the things to be considered are the wavelengths you wish to be able to distinguish, the number of lines per meter on the diffraction grating, and the distance from the grating to the screen or detector. Discuss the practicality of the device in terms of being able to discern between wavelengths of interest.

## Glossary

### Diffraction Grating Rainbow Loom

**constructive interference for a diffraction grating:** occurs when the condition *d* sin *θ* = *mλ* (form = 0,1,–1,2,–2, . . .) is satisfied, where d is the distance between slits in the grating, λ is the wavelength of light, and m is the order of the maximum

**diffraction grating:** a large number of evenly spaced parallel slits

### Selected Solution to Problems & Exercises

1. 5.97º

3. 8.99 × 10^{3}

5. 707 nm

7. (a) 11.8º,12.5º,14.1º,19.2º; (b) 24.2º,25.7º,29.1º,41.0º; (c) Decreasing the number of lines per centimeter by a factor of *x* means that the angle for the *x*-order maximum is the same as the original angle for the first-order maximum.

9. 589.1 nm and 589.6 nm

11. 28.7º

13. 43.2º

15. 90.0º

17. (a) The longest wavelength is 333.3 nm, which is not visible; (b) 333 nm (UV); (c) 6.58 × 10^{3} cm

19. 1.13 × 10^{−2} m

21. (a) 42.3 nm; (b) Not a visible wavelength. The number of slits in this diffraction grating is too large. Etching in integrated circuits can be done to a resolution of 50 nm, so slit separations of 400 nm are at the limit of what we can do today. This line spacing is too small to produce diffraction of light.