EXPERIMENT 13 THE PRISM SPECTROMETER OBJECTIVES: 1) Learn the theory of the prism spectrometer, and be able to explain the functions of its various components. 2) Observe the spectrum of a mercury discharge lamp and record the angle of deviation for the spectral lines. 3) Determine the index of refraction of a glass prism for various wavelengths. 4) Use the calibrated prism to measure wavelengths from a second discharge lamp. 5) Observe color sensation caused by light of particular wavelengths. INTRODUCTION: When a beam of light is transmitted from air to glass, the ray is bent according to Snell's law sinθair = nsinθglass (1) where the angles are measured from the surface normal (the line perpendicular to the surface) and n is the index of refraction of the glass. The index of refraction is a dimension-less number and is a measure of how strongly the medium bends light. The greater n is, the more the light is bent. The index of refraction of air is 1. For glass, n varies from 1.3 to 1.8, depending on the type of glass and on the wavelength of the light. Figure 1 White light is made up of all the colors of the rainbow - red, yellow, green, blue, and violet. Different colors correspond to different wavelengths. Human eyes are sensitive to light with wavelengths in the range 390 nm (violet) to 750 nm (red) (1 nm = nanometer = 10-9 m).
300nm 400 500 600 700 wavelength(nm) ultraviolet violet indigo blue green yellow red Range of human vision Figure 2 infrared Glass has a greater index of refraction at shorter wavelengths, that is, it bends blue light more than red light. So a prism can be used to disperse white light into its component colors. white red blue n Figure 3 blue red wavelengthλ In this experiment, we will use a prism spectrometer to measure the dispersion angle of various wavelengths. From the measurements, we will make a graph of the index of refraction vs. wavelength. The form of the curve of index of refraction as a function of wavelength, known as the Cauchy formula, is B n= A+ (2) λ 2 As a light source, we will use a mercury lamp, which emits light at several discrete wavelengths. The device we are using is called a prism spectrometer because, once the prism is calibrated, it can be used to measure the wavelengths of the lines in the spectra produced by various atoms. The spectra contain bright lines at particular wavelengths, which correspond to light emitted during the transition between different energy states of the atoms. You see distinct lines because the atoms exist only in distinct, quantized energy states. Trying to explain the data from such experiments the existence and pattern of sharp spectral lines led to the development of quantum mechanics. When a ray of light is refracted by a prism, the angle between the incoming and outgoing rays is called the angle of deviation (β). For a given prism and a given wavelength, the value of β depends on the angle between the incoming ray and the surface of the prism. β is minimum when the angles of the incoming and outgoing rays make equal angles with the
prism surfaces. In this special symmetric case, the prism's index of refraction (n) is related to β and the apex angle of the prism (α) by α + β sin 2 n = (3) α sin 2 The uncertainty of n is found by evaluating δ n= n( β + δβ) n( β) or: α + β + δβ α + β sin sin 2 2 δ n = (4) α α sin sin 2 2 You will not have to worry about evaluating this ugly looking formula; it is already programmed in your Excel spreadsheet. The prisms that we will use all have α = 60 (exactly, we assume). Figure 4 There exist extensive tables of the line spectra of many elements. In the first part of the experiment, you will be using the known spectrum of mercury to calibrate your prism spectrometer. As a result, you have measured the curve of index of refraction as a function of wavelength. So if you measure a new line of a spectrum, you can calculate the index of refraction and use your curve to look up the wavelength for the new line. This process is used in identifying the elements present in unknown samples, such as the atmospheres of distant stars. The element helium, used to inflate birthday balloons, was first discovered by observing the atmosphere of a nearby the star, the sun (helios is Greek for sun). In the last part of the experiment you will have the opportunity to measure the spectrum of a gas in this fashion.
PROCEDURE 1. Become familiar with the spectrometer a) Identify each component: the large table, the prism table, the collimator, and the telescope (see Figure 5). b) Note the clamping screws and the fine adjustment screws for the telescope and the black table. Note the clamping screw for the prism table. c) Note how to adjust the telescope focus and the eyepiece. d) Note how to adjust the slit focusing in the collimator tube. Note how the slit width can be adjusted and how the slit orientation can be rotated. Figure 5 2. Practice reading the angle from a precise protractor scale on the rim of the black table. Use the Vernier scale with the little magnifying glass to read the angle to the nearest arc minute. (1 arcmin = 1' = 1 degree.) The following is an example: 60
0 10 20 30 40 50 Figure 6 In this example, the zero line of the Vernier scale (the upper scale) is between 40.5 and 41, so the angle is somewhere between 40 30' and 41. The Vernier scale tells exactly where in between. Look along the Vernier for the line that exactly lines up with the line below it. In this case, it's the 17' line. So the angle is 40 47', which we get by adding 17' to 40 30'. Before using this angle in equation (2), we must convert it to decimal degrees: 40 + (47/60) degrees = 40.78. 3. Align the spectrometer In order to correctly measure angles with the spectrometer, we must first align it. To do so, use the following steps: a) Crosshair focus: Do not put the prism on the prism table yet. Look through the eyepiece of the telescope. You should see crosshairs which you will use to accurately locate the spectral lines. Rotate the eyepiece to focus the crosshairs. b) Telescope focus: There are two knobs under the large table. If you are facing the prism spectrometer, on the side opposite the collimator, loosen the right hand knob (BLUE) under the large table this allows you to rotate the telescope. Turn the telescope so that it is not pointing at the collimator, but instead is aimed at something as far away from you in the room as possible. Loosen the thumb screw on top of the telescope and slide the eyepiece in/out until the distant object comes clearly into focus. Tighten the thumbscrew. After this adjustment, you should not adjust the focus of the telescope again. c) Telescope alignment: Place a mercury vapor lamp in front of the slit at the end of the collimator. Look through the collimator, you should see the slit. If you do not, the slit is most likely closed use the (BLACK) screw at the far end of the collimator to open/adjust the width of the slit. Now, move your light source around until the slit as seen through the collimator appears the brightest. Rotate the telescope until it is pointing directly at the collimator. By looking through the telescope, you should be able to line up the slit with the crosshair. By locking down the telescope (tightening the knob from part b) and using the fine adjustment you should be able to do this very accurately. The fine adjust knob is at the bottom of the telescope s support on its right hand side. If the slit does not have very crisp edges when looking through the telescope, loosen the thumbscrew on top of the collimator and slide it in/out to focus
it it s best to start with the far end pushed all the way in. Also, if the slit is not vertical, adjust it at this time and then tighten the thumbscrew on top of the collimator. Once you have a nice thin, well-focused slit with your crosshairs centered on it and your telescope locked down, you are ready to align your angular scales. d) Angle adjustment: The outer rim of the large table is marked in 0.5 increments. The inner half of the large table has two Vernier scales divided into 30 one minute increments we ll only use the scale on the right hand side of the telescope. Loosen the left hand screw (WHITE) under the large table this unlocks the outer rim of the large table. Rotate the outer rim and align the zero degree mark on the outer rim with the zero minute mark on the inner Vernier scale. Use a magnifying glass to assist in this alignment. Lock the outer rim so it will no longer rotate. e) Prism placement: Place the prism on the prism table. Recall, light is bent toward the base of a prism, so it should be placed on the table so that its gray plastic holder makes a C shape if you are looking at it from the telescope side. Loosen the prism table locking screw located below the prism table on its right hand side. Now, find mercury s bright line spectrum without the telescope. To do this, you should look through the prism with your eye at approximately the same location as the telescope is in Figure 5. Your eye must be at the same elevation as the collimator and the telescope. Next, rotate the prism table until you can see the bright line spectrum. You should notice the spectrum is inside a black circle. You are seeing the light coming out of the collimator and being bent through the prism. Once you have found the spectral lines, unlock the telescope by loosening the BLUE thumbscrew and rotate the telescope to point at the spectral lines. Observe the spectral lines through the telescope. If your spectral lines are difficult to see, too wide or not in focus, you may have to adjust the focus on the collimator or the slit width. f) Angle of minimum deviation: Align the crosshairs with the green spectral line. To accurately place the crosshairs, use the fine adjust knob. While looking at the green spectral line, rotate the prism table so that the green line moves to the right. If you continue to rotate the table, the green line will stop moving to the right, reverse direction and move to the left. Rotate the prism table to the orientation which corresponds to the point at which the green line stops moving to the right, but has not started moving back to the left. Align the crosshairs with the center of the green line. The spectrometer is now set at minimum deviation for the green line. The angle should be 51 degrees 30 ±10 minutes. If it is not, you may not have aligned the scales correctly and you need to repeat steps d, e and f from above. g) Estimate the uncertainty in your measured value of the angle of minimum deviation. Enter the value in minutes in your Excel spreadsheet and use Excel to convert this value to degrees. h) Record the angle of minimum deviation for the green line in the Excel spreadsheet. 4. Measure the angle of deviation for each of the spectral lines of the Mercury lamp. The wavelengths and colors of the spectral lines are given in the Excel Spreadsheet. While
making measurements, unclamp and rotate the prism table to check that the prism is oriented for minimum angle of deviation for the red, green, and blue lines. When measuring very closely spaced lines, like the double yellow lines, make the slit very narrow and check the focus. Make the slit wider when measuring dim lines. 5. Calculate the quantity of 1/λ 2. 6. Use Excel to calculate the index of refraction (n) for each spectral line using equation (3). The prism apex angle α is 60. Keep four digits after the decimal point. 7. Make a graph of n vs. 1/λ 2. Fit your graph with a best fit line including the uncertainties in the slope and intercept. 8. Observe the spectral lines of another emission lamp. For any three bright lines in the spectrum, record the perceived color and intensity, measure the angle of deviation (β), and calculate the index of refraction (n). Now look back to your graph of n vs. 1/λ 2. This graph told us how light of a certain wavelength was bent by our prism. Since we are using the same prism in this part of the experiment, this data can be used to help us. We can (and have) calculated a slope and a y-intercept from this graph. We know the equation of a straight line and we can compare that equation to Cauchy s equation which reads: n = A + B/λ 2. Using this information, we can calculate 1/λ 2 from the slope, y- intercept and the index of refraction (n).
Questions 1. From the equation of your best fit line, what is your experimental value of A from Cauchy s equation? 2. From the equation of your best fit line, what is your experimental value of B from Cauchy s equation? 3. Using Cauchy s equation, what are the wavelengths of three of your unknown λ δn+ δa δb spectral lines and their uncertainties? δλ = + Include a sample 2 n A B calculation of λ and δλ for one of your three wavelengths.
4. Obtain the known wavelengths for the second emission lamp from your TA. Discuss the consistency of your experimental values for three wavelengths with the established values. If any are not consistent, suggest a possible reason for the discrepancy.