Physics 316. Experiment A3: Adjustment of the Spectrometer and Measurement of Prism Angle

Transcription

1 Experiment : Adjustment of the Spectrometer and Measurement of Prism Angle Readings: Hecht: Optics 4 th edition: pp G.S.Monk: Light App. IV R.A.Sawyer: Experimental Spectroscopy R.S.Loughurst: Geometrical and Physical Optics p K.A.MacFayden: Physics Laboratory Handbook p. 62 The spectrometer consists of a collimator, a telescope, a prism table and an angle measuring scale with two verniers. The collimator is designed to produce an accurately parallel beam of light that is then deviated and dispersed by a prism or grating, and then examined through the telescope. There are numerous adjustments possible - optically (collimator and telescope) and mechanically (optic axes of these two tubes plus tilt and rotation of prism or grating). This experiment will introduce you to the careful adjustments needed, to the use of the instrument for some precise measurements, and to some problems in the analysis of your data. Figure 1 1 Revised 2/2/10

2 Apparatus: Spectrometer with Gaussian eyepiece White light source Spectral lamp Flashlight Prism I. Adjustment of the Spectrometer Before measurements, we first need to adjust the spectrometer. Below we follow the outlined procedures to aligh the spectrometer so the following conditions will be obtained: 1. Cross-hairs in focus. 2. The telescope must be focused for parallel light and the crosshairs must be at the secondary focus of the telescope objective. 3. Adjust the telescope and collimator for parallel light and to align their optical axes. 4. Make the optical axes of telescope and collimator perpendicular to the instrument (mechanical) axis. 5. Make the refracting prism faces parallel to the instrument (mechanical) axis. 6. Make the image of the slit sharp and coincident with cross-hairs (no parallax) and centered for all conditions (direct, reflected, and refracted light). Optical Alignments The voltage range of the DC power supply used to supply power to the lamp for the Gaussian eyepiece should be set the 6 volt range. Turn on the light in the Gaussin eyepiece. Be careful when increasing the voltage, if you increase the voltage well beyond the 6 volt scale max marker the bulb can burn out. Focus the eyepiece on the cross hairs by rotating the cross-hairs to see an X. You may find it easiest to set the cross-hairs diagonally, rather than vertical/horizontal. If you choose the latter alignment the vertical wire often disappears into the bright slit image, but with the diagonal alignment you can easily locate the intersection of the wires against the slit image. Next, focus the telescope without changing, the focus of the eyepiece on the cross-hairs. This can be done approximately, by focusing a distant object so that its image coincides with the cross-hairs (with the light turned off). 2 Revised 2/2/10

3 Figure 2 Illuminate the Gaussian eyepiece from the side. The telescope must now be made approximately normal to a plane reflecting surface on the prism table. (You can use one face of the prism.) The light must be placed so that a beam entering the Gaussian eyepiece from the side is reflected from the prism face and back through the telescope objective. Tilt or rotate the prism face slightly until a reflected circle (or most of a circle) of light is seen. Some trial and error may be required. When the reflected circle of light is reasonably centered, a fuzzy reflected set of cross-hairs should be seen unless the telescope is very far out of focus. (Note: if a 60 prism with three polished surfaces is used, several sets of cross-hairs may be seen. The extra cross-hair images. arise from internal reflections in the prism.) The telescope can now be focused (being careful not to change the focusing of the eyepiece on the x-wires). Focus the collimator With the prism removed and keeping the eyepiece and telescope focusing fixed, close the collimator slit to make it as narrow as possible, and then focus the collimator to produce the sharpest image. Rotate the telescope so it points 180 from the collimator. When fully adjusted, the spectrometer should produce an image that coincides with the cross-hairs in the telescope, and shows no parallax between that image and the cross-hairs. Alternative method of focusing (Schuster s method) This is a method that is fast to use (once you know how to do it). (Reference: Longhurst.) Set the instrument with approximate focusing, and with a prism on the turntable. Set the collimator, prism and telescope so that the light falling on the prism is at an angle of incidence slightly greater than that for minimum deviation. Turn the prism on the table so that the angle of incidence increases, and then adjust the telescope only, to produce the sharpest image. Then, keeping the telescope position fixed, rotate the turntable with the prism so that the images move across the field of view, halt momentarily at the position of minimum deviation and then start moving back. You will now have an angle of incidence less than that for minimum deviation. This time adjust only the collimator, to improve the focus. Repeat these steps, focusing the telescope and collimator alternately, until the focusing converges. Very few iterations should be needed. 3 Revised 2/2/10

4 Mechanical alignment of the spectrometer a) Make the telescope and collimator parallel The collimator slit should now be turned horizontal (if not already so). Adjust either the telescope or the collimator leveling screws to bring into coincidence the horizontal slit image and the intersection of the cross-hairs. The telescope and collimator axes are now parallel, but not necessarily perpendicular to the mechanical instrument axis, as illustrated in the Figure 3. Figure 3 b) Make one prism face normal to the mechanical instrument axis. Move the telescope so that it makes approximately a right angle with the collimator axis. Set one face of the prism to reflect light from the collimator into the telescope. Make the slit image coincide with the cross-hairs by tilting the prism face. This face of the prism is now parallel to the mechanical axis. If the collimator is tipped up from the normal, by, say 2 degrees, the telescope which was parallel to the collimator must be down 2 degrees. The prism face bisects this vertical angle as well as the horizontal angle between collimator and telescope, and hence is normal to the instrument axis. c) Make the telescope axis normal to the adjusted prism face. Without altering the level adjustment of the prism table, rotate the table so that a prism face is normal to the telescope. Using the Gaussian eyepiece, adjust the tilt of the telescope so that the telescope is now perpendicular to the prism face. The adjustment of the telescope is now complete. d) Again make the collimator and telescope axes parallel. Adjust the collimator-axis to make it parallel to the axis of the telescope. Return to (b) above, with the telescope viewing the collimator at right angles via the prism. Adjust the collimator until coincidence between the slit and cross-hairs is obtained. Now the collimator and telescope are both adjusted: they are parallel to each other, and the telescope is normal to the instrument axis. 4 Revised 2/2/10

5 Figure 4 e) Make the collimator slit vertical. The slit may be rotated to a vertical position with reasonable accuracy by eye or else by a reflection method after the prism is adjusted, provided the prism is a 60 one with 3 polished sides. The spectrometer is now in proper adjustment. II. Measurement of Prism Angle Without changing the focusing of either the collimator or the telescope, rotate the prism table so that the prism is symmetrically located with respect to the beam from the collimator (as shown in Figure 5A). This alignment is not critical. Lock the prism turntable so that it cannot rotate once you start making your measurements. 5 Revised 2/2/10

6 Figure 5 Locate the left and right reflected beam, in turn, through the telescope, and check that the slit image is vertically centered. If all of the previous adjustments were correctly made, the images should be centered; if not, then some adjustment must be made in the prism table (using the three leveling screws on the turntable). The angle between the two reflected beams is twice the prism angle. Note the positions of these reflected beams, viewing them alternately left-right-left... until you have three sets of readings. You should find that the readings repeat to one or two arc minutes. If your readings do not agree that well, repeat so that you have at least 5 sets of readings. For all readings, read both verniers. You may find small differences between the two verniers. The prism angle (and its standard deviation) can then be calculated. 6 Revised 2/2/10

7 III. Measurement of the refractive index. Rotate the prism turntable to one of the positions shown in the diagram (see Figure 5B, above), so that you can see the dispersed line spectrum, each line being an image of the collimator slit. Locating the deviated spectrum is best done by eye, rather than through the viewing telescope. With the telescope well out of the way, move your eye around in the general area where you expect to find the deviated spectrum. You should be able to find the circular outline of the collimator lens, then the spectrum in the center of that lens. Keep your eye in that position, while moving the telescope into place. Rotate the prism turn-table so as to minimize the angle of deviation and locate the position of minimum deviation. For best adjustment, tighten the locking screw beneath the telescope and use the tangent screw for the final setting. Note readings on both verniers. Take readings on at least five spectral lines. (What kind of spectrum do you see using the white light source?) Then rotate the prism table to the other setting (deviation on the other side of the straight-through line), and repeat the measurements. This time, the positions of minimum deviation will be inverted from those you have previously seen, i.e. inverted left to right or vice versa. Be sure you re-adjust the prism table and the telescope position for minimum deviation for each spectral line in turn: the prism-table position is different for minimum deviation for each line. Note that most of our spectrometers have vernier scales with 1/3º divisions on the main scale and 40 divisions on the vernier. You thus have 1/40 of 1/3 of a degree = ½ (60 arcminutes = 1 degree) reading accuracy, but check each instrument. You may find it easiest to read these scales with the aid of a hand lens, and with a flashlight to provide oblique illumination. It can be shown that the position of minimum deviation is one where the incident and emergent rays are symmetrical about the angle of the prism, and we can then derive an expression for the refractive index of the prism sin A + D $ # 2 % n = sin A $ # 2 % where A = angle of the prism and D = angle of minimum deviation. Since n, the refractive index, is a function of wavelength, so the value of D will depend on wavelength. The relation between refractive index and wavelength can be well represented by Cauchy s formula. n = A + B 2 + C 4 7 Revised 2/2/10

8 where A, B and C are the Cauchy constants. Replace white light source with mercury lamp. Measure the angle of minimum deviation for at least five spectral lines. The Mercury lamps will give lines between deep violet and yellow; some of the Hg lamps have Cadmium and will give a red line. If your Hg lamp does not show a red line, then use the Cd lamp for measuring the refractive index in the red. Without the red line, your measurements will cover a small range of wavelengths and the results will be very sensitive to measurement errors and rounding off in calculations. IV. Calculations: We will assume the wavelengths are known for the lines that you use; see the next page or check wavelength tables and use at least 6 significant figures. The Cauchy formula for each wavelength then is: n = A + B 2 + C 4 With at least five wavelengths, you will have more equations than the number of unknowns (A, B, C). To determine A, B and C using a least-squares method, refer to class notes, Taylor (Chapter 8) or Bevington. Determine the values of the Cauchy constants. Use your values for these constants to compute the refractive index for each line, and compare the calculated and measured values. Plot a graph of refractive index vs. wavelength. This will be curved, and its smoothness is a useful guide for the consistency of your data; an error in measurement will show up very quickly. Indeed, it will show up better on a graph of refractive index vs. 1/λ 2, which effectively drops the fourth-power term from the Cauchy formula. Plot this n vs. 1/λ 2 graph, as a check, and calculate the least-squares line for n = A + B/λ 2. From the Cauchy formula, one can derive: dn d = # 2B # 4C 3 5 and then proceed to the expression for the resolving power of the prism: d = 2L # dn% $ d & sin # A % $ 2 & where A = prism angle and L = length of prism face. Here dλ is the wavelength separation of two lines that can just be separated by a prism of refractive index n and with L the length of the prism face. 8 Revised 2/2/10

9 Questions: Based on your values, do you expect the Na doublet to be resolved? Check by using a Na lamp. Do you see the two lines? (The approximate wavelengths are 5890 and 5896Å) The Abbe number is a measure of the dispersive power of glass, and it varies from one type of glass to another. The Abbe number is defined as V d = n d 1 n F n C where the n s are the refractive index values at standard wavelengths, as listed in Hecht p From your measurements, and your calculated values for the Cauchy constants, derive values of the n s for your prism and calculate V d. Using the diagram on pg. 271 of Hecht identify the type of glass used in the prism. Errors: From your measurements, calculate the standard deviations on the observed values of the refractive index values. You will need to consider the errors in the determinations of A and D, and their propagation through the n values. Calculation of the errors for all of the Cauchy constants is complicated and should be omitted, but you can calculate the errors for A and B from the leastsquares straight-line fit: n = A + B/λ 2 From sin A + D $ # 2 % n = sin A $ # 2 % you can calculate n A and n D and proceed to calculate dn. Remember da and dd are in radian. Reference Wavelengths (Å) Mercury Cadmium Sodium Neon Wavelength Wavelength Wavelength (laser) Revised 2/2/10

10 NOTE: Several lamps exhibit 'mystery' spectral lines that are probably due to impurities but which we have been unable to identify. Ask an instructor to help you select lines which are unambiguously identifiable. 10 Revised 2/2/10