Experiment 8: The Grating Spectrometer OBJECTIVES A diffraction grating is a way to measure the wave properties of light. The grating produces a diffraction pattern that depends on the wavelength of the light that passes through it. In this experiment you observe the diffraction pattern produced by light from excited Mercury atoms. This allows you to determine the emussion spectrum of Mercury. The objectives of this experiment are as follows: 1. To observe the emission spectrum of Mercury using a diffraction grating 2. To measure the angles to various Lines in the diffraction pattern 3. To calculate some of the wavelengths of light emitted by Mercury THEORY A diffraction grating consists of a surface with regularly-spaced parallel slits through which light can pass. Light passing through different slits travels different distances from the grating to the screen (or spectrometer telescope), as shown in figure 8.1. Normal ! Grating d Incoming Incoming light lLight Figure 8.1 Diffraction angles for light passing through a diffraction grating When the light reaches the screen there can be constructive mterference, which produces a bright line, or destructive interference, m which case no light 1s observed. Consider two slits with a separation distance d. The light passing through the slits is observed at an angle 6 with respect to the normal to the diffraction grating. As shown in figure 8.1, the ray to the left travels an extra distance D. D is calculable using trigonometry, as shown in equation 8.1. Extra distance traveled (light) D = d sin6 (8.1 Constructive interference occurs when the path difference for the two rays 1s exactly an integer number of wavelengths, as shown in equation 8.2. Constructive Interference D=nA=dsm6, 8.2) Here * risaninteger (0,1,2, - * 0,1s the corresponding angle at which constructive interference occurs * ulis the path difference * If the gratings have N lines per mmsod = lTNnm = (1/N)x107m The angle for constructive interference depends on the wavelength A, so by measuring the angle you can determine the wavelength of the light. You observe light from Mercury atoms, which is a mixture of different wavelengths (colors). You measure the angle for constructive interference from a grating for which we know the line spacing 4 and use equation 8.2 to calculate the wavelength A, of the light. The telescope 1s initially set perpendicular to the grating (68 = 0) where all wavelengths of the beam constructively interfere, which appears as a bright white line. To observe the separated colors the spectrum, swing the telescope away from 6 = 0 until you observe separate color lines from violet to red, which comprise the n = 1 angular position for the lines of the spectrum. The angular position of the telescope for the first appearance of a given color line to the left of the 1utial position 1s a poor estiumate of 6, for that spectrum line. To get a more accurate estimate, also measure the angular position of the telescope for the first appearance of same color line to the right of the initial position. Half of the difference between the angular positions of these two observations 1s a good estimate of 6, for that color. Measuring the angular position to the lett and then to the right of the perpendicular direction corrects for any error in positioning the grating perpendicular to the collimator beam. This process must be repeated for each color line in each spectrum, #=1 and #=2, which correspond to the first order and second order constructive mterference patterns of the light. The #=2 spectrum 1s more spaced out than the #=1 spectrum. ACCEPTED VALUES In one fringe, you can typically make out four distinct color bands, sumilar to the spectrum shown in tigure 8.2. 45.80 40.08 39.75 44.70 Figure 8.2 A typical diffraction pattern for one fringe from the light emitted by Mercury A Mercury lamp emits light at specific wavelengths. The colors and wavelengths of the strongest visible lines in the Mercury spectrum are: Color Wavelength (nm) Intensity Violet 404.6 moderate 407.8 weak blue-violet 435.8 strong green 546.1 strong vellow 5710 strong 579.0 strong APPARATUS * Spectrometer * Diffraction grating * mercury arc lamp The grating 1s clamped in a rotating table so that it is perpendicular to the collimator which focuses the light from the lamps. The telescope s initially set perpendicular to the grating (8 = 0) where all wavelengths (colors) of the beam constructively interfere. The goniometer angular scales display different initial angles through the magnifying glasses, neither of which may be equal to zero, as shown m figure 8.3. Make sure you choose one of the goniometer angular scales to use for the entire experiment. We will call the angle measured on the goniometer scale @. For each line you will measure the angle twice, once to the right (@) and once to the left (r). The deflection angle is half the absolute value of the difference, sl 9=|'