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Significance

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.

Check Your Understanding If the line spacing of a diffraction grating d is not precisely known, we can use a light source with a well-determined wavelength to measure it. Suppose the first-order constructive fringe of the H β emission line of hydrogen ( λ = 656.3 nm ) is measured at 11.36 ° using a spectrometer with a diffraction grating. What is the line spacing of this grating?

3.332 × 10 −6 m or 300 lines per millimeter

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Take the same simulation we used for double-slit diffraction and try increasing the number of slits from N = 2 to N = 3 , 4 , 5.. . . The primary peaks become sharper, and the secondary peaks become less and less pronounced. By the time you reach the maximum number of N = 20 , the system is behaving much like a diffraction grating.

Summary

  • A diffraction grating consists of a large number of evenly spaced parallel slits that produce an interference pattern similar to but sharper than that of a double slit.
  • Constructive interference occurs when d sin θ = m λ for m = 0 , ± 1 , ± 2 , ... , where d is the distance between the slits, θ is the angle relative to the incident direction, and m is the order of the interference.

Problems

A diffraction grating has 2000 lines per centimeter. At what angle will the first-order maximum be for 520-nm-wavelength green light?

5.97 °

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Find the angle for the third-order maximum for 580-nm-wavelength yellow light falling on a difraction grating having 1500 lines per centimeter.

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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 25.0 ° ?

8.99 × 10 3

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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 60.0 ° ?

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Calculate the wavelength of light that has its second-order maximum at 45.0 ° when falling on a diffraction grating that has 5000 lines per centimeter.

707 nm

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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 24.2 ° , 25.7 ° , 29.1 ° , and 41.0 ° when projected on a diffraction grating having 10,000 lines per centimeter?

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(a) What do the four angles in the preceding 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.

a. 11.8 ° , 12.5 ° , 14.1 ° , 19.2 ° ; b. 24.2 ° , 25.7 ° , 29.1 ° , 41.0 ° ; 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.

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What is the spacing between structures in a feather that acts as a reflection grating, giving that they produce a first-order maximum for 525-nm light at a 30.0 ° angle?

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An opal such as that shown in [link] 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?

a. using λ = 700 nm, θ = 5 .0 ° ; b. using λ = 460 nm, θ = 3 .3 °

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At what angle does a diffraction grating produce a second-order maximum for light having a first-order maximum at 20.0 ° ?

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(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. 26,300 lines/cm; b. yes; c. no

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(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 line per centimeter a diffraction grating can have and produce a complete second-order spectrum for visible light?

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The analysis shown 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? ( Hin t : The distance between adjacent fringes is Δ y = x λ / d , assuming the slit separation d is comparable to λ . )

Figure shows two vertical lines, grating on the left and screen on the right separated by a line of length x, perpendicular to them both. There are two slits in the grating, a distance d apart. A line at an angle theta to x meets the screen at point delta y equal to x lambda by d.

1.13 × 10 −2 m

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Questions & Answers

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unpolarized light refers to a wave collection which has an equal distribution of electric field orientations for all directions
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In a grating, the angle of diffraction for second order maximum is 30°.When light of wavelength 5*10^-10cm is used. Calculate the number of lines per cm of the grating.
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Practice Key Terms 1

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Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
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