Chapter 4 Dispersion of Glass 4.1 Introduction This experiment will develop skills in choosing a suitable fit for data and plotting the resulting curve. Curve fitting will count for a big chunk of the marks in this lab, so don t try to avoid it! 4.2 Apparatus For this experiment you will need the following pieces of equipment: Spectrometer, Bubble Level, 60 Prism, Front Surface Mirror, small Allen wrench, screwdriver, Sodium Lamp, Hydrogen, Helium and Mercury Geissler Tubes with Power Supply, Variable Intensity White Light Source, assorted stands and clamps.
4-2 Dispersion of Glass 4.3 Theory In this experiment we shall use the angle of minimum deviation to measure the index of refraction of glass as a function of wavelength. When a ray of light is obliquely incident on the interface separating two transparent media of different indices of refraction, its direction changes. The amount of deviation is described by Snell s Law: n 1 sin θ i = n 2 sin θ r (4.1) where n 1 is the index of refraction of medium 1, θ i is the angle of incidence (measured from the normal to the interface), n 2 is the index of refraction of medium 2, and θ r is the angle of refraction. For most transparent media, the angle of refraction increases with increasing frequency (for a fixed angle of incidence) thus, blue light is refracted through a much larger angle than red light. Since the definition of the index of refraction is: n = c (4.2) v where c is the speed of light in vacuum and v is the speed of light in a medium, and since the index of refraction is always greater than or equal to 1 for any medium, it is clear that v is less than or equal to c. The speed of light is related to frequency and wavelength by: c = νλ (4.3) and the frequency does not depend on the medium through which the light travels. Thus it follows that the wavelength of light must change when it enters a material medium. Furthermore, the index of refraction of a material is not a constant, but depends upon the frequency of the incident light. From measurements of the index of refraction of a medium as a function of wavelength, we can construct a dispersion curve by plotting a graph of n(λ vac ) vs λ vac, where λ vac refers to the vacuum wavelength of the incident light. (For the purposes of this experiment, we shall assume that the wavelengths in air are identical to the wavelengths in vacuum). The dispersion of the medium at a particular wavelength λ 0 is defined to be the slope of the dispersion curve at that wavelength: D = ( dn dλ ) λ 0 (4.4)
4.4 Procedure 4-3 Figure 4.1: Path of Light Through a Prism in Air Referring to Figure 4.1, when a ray of monochromatic light passes through the prism, it is deviated through some angle δ. The value of this angle depends on the angle of incidence at the first surface of the prism. When the angle of incidence φ i is equal to the angle of emergence φ e, the angle of deviation δ is a minimum, called the angle of minimum deviation δ min. The index of refraction of the glass, n, can then be found from: n = n 1 sin[(α + δ min )/2] sin(α/2) (4.5) where α is the prism angle. 4.4 Procedure 4.4.1 Aligning the Spectrometer See the appropriate section on the course web site. 4.4.2 Measurement of Prism Angle Place the prism on the prism table with the prism angle pointing toward the collimator as in Figure 4.2. Place the white light source in front of the collimator. Use a narrow collimator slit in the following procedure. Note that the slit image can be seen reflected in the prism faces. Adjust the telescope
4-4 Dispersion of Glass Figure 4.2: Orientation of Prism to Find Prism Angle so that the slit image is aligned with the cross hairs, read the vernier scale (30 part vernier every half degree; thus readings are in degrees and minutes), and record this reading (the telescope angle). Be as careful as possible when taking measurements, and be sure to record values to the nearest minute of angle. Sloppy measurements will give you meaningless results. Repeat this for the other prism face and enter your readings into Table 4.1. Repeat these measurements 3 times. The prism angle is one half the difference between the two vernier readings. From the three series of measurements recorded in Table 4.1, determine the prism angle α for each measurement, and calculate the average value of the prism angle. Include measurement uncertainties in your final answer. 4.4.3 Measuring the Angle of Minimum Deviation Place the sodium vapour lamp in front of the collimator slit. Rotate the prism 90 so that now light will be refracted through it on the way to the telescope. Observe the spectral lines and reduce the slit width until they are sharp and narrow. Rotate the prism table as shown in Figure 4.3 until the angle of minimum deviation is achieved for one of the Sodium D lines
4.4 Procedure 4-5 Trial # Left Reading Right Reading Prism Angle α ( ) ( ) ( ) 1 2 3 Table 4.1: Calculation of Prism Angle Figure 4.3: Orientation of Prism to Find Prism Angle
4-6 Dispersion of Glass (2 lines separated by 6 Å; these may be too close together to be clearly resolved if so, set the cross hairs to the center of the doublet and use the mean wavelength of 5893Å). Record the vernier reading in Table 4.2. Repeat this measurement three times. If you have any difficulty, see the laboratory demonstrator. Now rotate the prism through 180, repeat the above measurements, and average the results. Q1:How do you determine the uncertainty in the average of a set of numbers? The difference between the vernier readings from each side is twice the angle of minimum deviation. Repeat using the Hydrogen Geissler tube and the four visible lines of the Hydrogen spectrum (red, blue and two violet). Repeat using the Mercury Geissler tube and as many as possible of the following lines of the Hg spectrum: two close yellow lines separated by 20Å; two greens; a blue green; and a blue (this is a triplet use the center of the image). Repeat using the Helium Geissler tube and any 4 visible lines. 4.5 Analysis 4.5.1 Plotting the Dispersion Data Using the angles of minimum deviation for each wavelength measured, along with their uncertainties, and the average value of the prism angle α, along with its uncertainty calculated above, determine the index of refraction at each wavelength, along with the uncertainties, and place them in Table 4.2. Plot a graph of n vs λ. If your graph does not suggest a smooth, monotonically decreasing function, go back and check your data. Bad data will make your results meaningless. 4.5.2 Curve Fitting This is the most important part of this experiment. shortcuts in this section. Don t try and take
4.5 Analysis 4-7 Colour Wavelength Left Dev. Right Dev. δ min n Å ( ) ( ) ( ) H (r) 6563 Na (y) 5893 Hg (y) 5790 Hg (y) 5770 Hg (g) 5461 Hg (g) 5015 H (b-g) 4861 Hg (b-v) 4358 H (v) 4300 H (v) 4101 Hg (v) 4077 Hg (v) 4047 He (y)? He (g)? He (b-g)? He (b-v)? Table 4.2: n vs Wavelength b blue; g green; r red; v violet; y yellow (colours are approximate.)
4-8 Dispersion of Glass For each of the following fits, do the fit and determine A, B, and C (if applicable). Make a table of these results including the SSE for each fit. Determine which of the following functional forms best approximates your dispersion curve: n = A + B λ n = A + B λ 2 n = A + 1 λ B n = A + B λ + C λ 2 n = A + B λ 2 + C λ 4 n = A + B λ C Note that two of the above fits are non-linear, and will force you to use the optimizer. You ll also need to come up with reasonable starting values for the optimizer. This will take time, but that s why this part of the lab will carry a lot of marks. 4.5.3 Extracting Information from the Graph After finding the best fit to the data, draw the best fitting smooth curve through the data points. From your dispersion curve, determine the index of refraction of the glass for the Fraunhofer F, C and D lines whose wavelengths are 4861Å, 6563Åand 5893Å, respectively. Using this information, calculate the dispersive power W of the prism from W = n F n C (4.6) n D 1 Determine the unknown wavelengths of the 4 Helium lines from your dispersion curve, using the solve for spreadsheet function and determine uncertainties appropriately. Compare your results with the accepted values found in the C.R.C. Handbook.