4.4 Diffraction Gratings

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 $\text{H}_{\beta}$ emission line of hydrogen $\left(\lambda = 656.3 \text{nm}\right)$ is measured at $11.36 °$ using a spectrometer with a diffraction grating. What is the line spacing of this grating?

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

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 $25.0 °$ ?

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.

(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.

An opal such as that shown in the figure acts like a reflection grating with rows separated by about $8 \mu\text{m} .$ 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) 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?