## L-6 SPECTROMETERInstructor's Notes
This experiment can be performed in one three hour lab period if part (1) of Experiment A is done by the instructor before class, and the students are instructed (by a demonstration) how to perform the prism table leveling.
- Gaertner-Peck Spectrometer
- Cloth dust cover/light shield for spectrometer.
- Gauss eyepiece and illuminator, with power supply.
- Equilateral glass prism, or right angle prism.
- Spectrum tube power supply, with 3800 ohm rheostat if needed.
- Spectrum tubes (Geissler tubes): Mercury, Helium, Argon, etc.
- Desk lamp, or other low intensity white light source.
- Holder or padded tray for the prism table when it is removed from the spectrometer.
1. See that the Gauss eyepiece illuminator is well lined-up with the hole in the eyepiece wall, for brightest crosshair image. Do this by removing the eyelens assembly (with illuminator attached) from the telescope tube. Then look into it backwards (at the light emerging, which would normally go toward the telescope objective). Look at an angle slightly downward, to see if the round hole seems uniformly illuminated with a brighter spot at its center. If the bright spot is not centered, loosen the two screws and carefully rotate the eyelens with respect to the illuminator until the brightest spot is centered in the hole. Sometimes it may help to rotate the bulb socket at the bottom of the illuminator, if the bulb is skewed, as so many cheaply-produced bulbs are these days. 2. Clean the prisms with lens tissue moistened with alcohol, to remove fingerprints. 3. Check for free and smooth motion of the prism table. If the instrument has not been used for
a while the lubricant on the shaft has "set." Wipe off the old lubricant with an oil-soaked cloth,
then re-apply lubricant with a 4. Clean dust from the entire instrument. Clean dust and grit from the surface of the main scale. 5. Check the collimation of the telescope, using the Gauss eyepiece and illuminator. Adjust if necessary. 6. Parallax between slit and cross hairs must be eliminated to ensure that light from any point of the slit emerges from the collimator as a parallel beam. The slit drawtube has a clamp, which may be loosened with a small screwdriver. In handling the tube, be careful not to damage the slit jaws. They are sharp and accurately machined. Never attempt to clean them yourself, consult the instructor if cleaning seems necessary. The slit width is adjusted by rotating the knurled screw. In this instrument the slit is spring loaded so that the jaws cannot be forcibly closed, but on some other instruments one must be very careful when closing the jaws. 7. Leave the telescope arm's clamping screw loosened.
1. Point out that the spectrometer is a precision instrument, capable of 1% accuracy in measurement of angles, and better than that in measuring the angles required in this experiment. 2. Show the students how to lift or carry the spectrometer. First tighten the telescope screw finger-tight, then lift the spectrometer by its base only, NEVER by the telescope or collimator arms. 3. Remind students that precision instruments should never be forced in any way. Clamping screws should be turned finger-tight only, never forcibly. Clamping screws facilitate use of the fine adjustment screws. The clamping screw associated with a particular adjustment should always be loosened before making that adjustment, for the screw can score (scratch or gouge) the precisely machined bearing surfaces, making future adjustments difficult. 4. Point out where the cross hairs are located. Warn against pushing anything down the tube that would break them. 5. Explain and demonstrate the auto-collimation procedure. Emphasize that once the cross hairs are in the correct position, the eyelens may be adjusted to any position that the user finds comfortable, and may be readjusted at any time without affecting the precision of measurements. 6. Demonstrate the prism table leveling procedure. The reason this is tricky is because the prism has several degrees of freedom, and these interact. If you attempt this by trial and error, you soon become frustrated, for one adjustment can disrupt a previous one. The philosophy behind a good procedure is to place the prism on the table in such a way that adjustments can't interact with previous adjustments. This is a nice example of engineering "constraint" problems, which one encounters frequently in science. 7. Warn the students against touching the 5000 volt terminals of the spectrum tube power supply. The current is quite low and unlikely to kill anyone, but the experience is sufficiently unpleasant that I have never seen a student deliberately repeat it. 8. Explain how the control rheostat is connected. The spectrum tubes should be operated only brightly enough to see and measure the desired lines. Operating them at high intensity shortens their life. 9. It is helpful to manually demonstrate the process of finding minimum deviation. Ask students to watch the prism as you rotate it in one direction while you verbally describe what you see: "The spectrum is moving to the right; it's slowing down; now turning around; now moving to the left." This simple demonstration is actually quite effective in getting the idea across. This can be set up with a prism and a beam of light projected through the prism and onto a wall. 10. Set up a spectrometer to show the mercury spectrum. Let each student look at it. Emphasize that if the two yellow lines can't be resolved, then either the slit is too wide, or the instrument is out of focus.
Since minimum deviation is so easy to obtain (after it is done once) I now ask students to
measure each spectral line at
Many laboratory manuals suggest that students measure the telescope angle when looking
straight through at the slit, implying that this reading is to be used as a reference "zero." This is, in fact what students usually will do if not better instructed. Most laboratory manuals do not point out that it's much better to measure each line's deviation both left and right, then take the difference of these and divide by two to obtain the deviation. In fact, there's no place in this experiment where one would ever have to, or want to, use the "zero" angle position. We have occasionally used lab manuals that explicitly suggest measuring each spectral line's deviation both left and right. What do the students do? They subtract the zero reading from each deviation position, then average the two deviation angles! Apparently our discussion of the data-cancelation in the "method of differences" when we did the free-fall experiment had absolutely no transference! In the same spirit, I
Over the years we have recommended various methods of spectrometer alignment, and have used manuals that sometimes recommend particular methods. Some manuals are silent on this matter! * The * Valasek's book recommends the same method, but with a two-sided mirror, both sides being silvered! * Wall and Levine's manual (Appendix J) describe an initial approximate alignment using
a spirit level! The level is placed successively on the main scale plate, the prism table,
and finally the telescope and collimator. They remark that "complete adjustment of the
spectrometer is a long and arduous task for the novice." They then describe a "better"
procedure, which, as presented, * Skolil and Smith give quite complete alignment instructions (Appendix I-K, p. 173). Their method is based on sound principles, but suffers from having a step in which the student must turn two screws equal and opposite amounts! This is mathematically correct, but not an easy task, for there are no visual clues to guide the experimenter to do this precisely. Their procedure also requires the student to rotate the prism on the table 180° in one of the steps. * Wagner's book
Can one use the best ideas from these sources to devise an optimally short and efficient method? At least there are certain things we'd like to avoid:
The latest (April 21, 1991) version I've come up with requires There's a small price to pay. The telescope axis alignment is carried out with light reflected from the prism faces at an incident angle of about 60°. Palmer recommends 45°, which is better, though one wonders why he didn't suggest a smaller value, consistent with the constraints of the instrument. The best choice of all is 0°, obtainable with the accessory plate. Is a 60° incident angle good enough for the purpose? At 0° a tilt error of θ in the prism face causes the reflected beam to be high or low by 2θ. At 45° it is off by θ. At 60° it is off by 2θ/3. My experience in making this adjustment is that this is indeed good enough, compared with other error sources. I find the error no larger than the error due to "play" or "wobble" in the spectrometer itself. The real virtue of my method is that it doesn't require resetting the prism table during the adjustments. Play in the prism table can cause enough uncertainty to confuse the adjustment process and waste the experimenter's time trying to make adjustments finer than necessary. If one wants better, one can swing the telescope all the way, as close as it will go to the collimator, then rotate the prism table to view the reflected slit image from one face. Then do the same on the other side. This adds two rotations of the prism table, then a third one to relocate the prism for measurement of the prism apex angle. That's not really much extra complication. To summarize: The best strategy for a given instrument depends on whether that instrument has a Gauss eyepiece. If it does not, the collimation must be done by focusing the telescope on a distant object, or looking into an already collimated beam from an auxiliary collimator. The tools available include:
Skolil, Lester L. and Louis E. Smith, Jr. Wall, Clifford N. and Raphael B. Levine. Wagner, Albert F. Valasek, Joseph. Palmer, C. Harvey.
The (American Optical) prisms we use are flint glass. Minimum deviation for the mercury green line is 51.5°. The prism angle is 60°. The index of refraction of the prism is 1.653. [The deviation is 57°, giving n = 1.67 for the newer prisms.] A 1° error in the prism angle and in the deviation angle causes an error of 0.045 in the index of refraction. The student may be tempted to express this as a percent (3%) but this is inappropriate for a measure which arbitrarily references to a non-zero value (n of vacuum is 1). If the student doesn't appreciate this point, suggest another example that is more familiar: temperature. Does it really make sense to express error in Celsius temperature as a percent? A one degree error would then be 1% of a 100° temperature, and 100% of a 1° temperature! What if the temperatures were converted to Fahrenheit? In some cases a percent error is meaningful for temperatures measured on the Kelvin scale, but even in that case one must consider the matter very carefully. It is not practical to derive a standard form error equation for this experiment without calculus. Non-calculus students would be better advised to insert the errors into the equation for index of refraction and recalculate the whole equation. A common mistake in the calculus method of calculating the error is to forget that the deviation angle, d, is in radians.
why they were curved. You can figure this out by
considering the geometry of rays passing through the
spectrometer, originating from various points on the slit, say
from top, bottom, or middle of the slit. (a) Explain why the
lines are curved. (b) Show how you can tell which way they
should curve (i.e., in the diagram: is side A the red end of the
spectrum, or is it side B?). (c) What shape is the curve that
geometric theory would predict? Is it a circle, ellipse,
parabola, or something else? You can't tall this just by the
visual appearance of the lines.
(2*) The determinate error equation for Eq. 1 is: Δn = ΔD {cos[(A+D)/2]/(2sin[A/2}} + ΔA (n/2){cot[(A+D)/2] - cot[A/2]} Derive this result. See next page. (3) What absolute error in the index of refraction results from an error of one minute of arc in the deviation angle? Assume glass of index 1.7 and a wavelength of 5461 Ångstroms.
(4) Under the same assumptions as question (2), what absolute error results from a one minute of arc error in the prism angle?
[The error analysis to justify these results will be found below.] (5) What are the special advantages of using minimum deviation, rather than using a larger deviation where the spectral lines are spread apart more?
Some
DERIVATION OF THE ERROR EQUATION: First, consider the case where the error is predominantly in the deviation angle D, the error in the prism angle A being negligible. n + Δn = sin [(A+D+ΔD)/2]/[sin [A/2] . where ΔD is a determinate error in D. Separate the (A+D)/2 from the ΔD/2 and then use the well known trig identity on the numerator. sin[(A+D+ΔD)/2] = sin[(A+D)/2] cos [D/2] + cos[(A+D)/2]sin[ΔD/2] so n + Δn = {sin[(A+D)/2] + (ΔD/2)cos[(A+D)/2]}/sin[A/2] n + Δn = sin[(A+D)/2]/sin[A/2] + [ΔD/2]{cos[(A+D)/2]/sin[A/2]} so therefore: Δn = [ΔD/2]{cos[A+D]/2}/[sin[(A+D)/2] A = 60° and D = 57° for n = 1.65 flint glass. For these values, this would be (ΔD/2)(0.5225/0.5) = 0.522 (ΔD). If the error in D is about 0.1° = 0.02 radian, then the error in n is 0.01. Now consider the case of an error in the prism angle, A. sin[(A+D+ΔA)/2] = sin[(A+D)/2] cos [ΔA/2] + cos[(A+D)/2] sin[ΔA/2] n+Δn = {sin[(A+D)/2] + (ΔA/2)cos[(A+D)/2]}/{sin[A/2] + ΔA/{2cos[A/2]} This looks like a mess at first, but notice that it is in the form where the "error in the numerator" and the "error in the denominator" explicitly show, so since the fractional error in the quotient is the fractional error in the numerator minus the fractional error in the denominator Δn/n = [ΔA/2]{cos[(A+D)/2]/sin[(A+D)/2] - [ΔA/2][cos[A/2]/sin[A/2] However, the fractional error in n is not a meaningful way to express the error in n so this should be converted to the absolute error in n: Δn = n(ΔA/2){cot[(A+D)/2] - cot[A/2]} This goes a lot easier with calculus! Since both A and D are about 60°, this can be done in one's head: Δn = (ΔA/2) 1.65 (89.02 - 88.09) = (ΔA/2) 1.65 (0.93) = 0.767 ΔA. Now if ΔA is, say 0.1, that's about 0.02 radian, for an error of 0.015 in n. [Actually, the error in A will be less than this.] One could now put the two error equations together into one grand-slam complete error equation! Δn = ΔD [cos[(A+D)/2]/[2(sin[A/2]} + ΔA (n/2)[{cot[(A+D)/2] - cot[A/2]} If the student consistently measured angles both left and right, taking their difference and dividing by two, the error in n should be about half of the estimates above, because the error in A and D would be half as much as if they were obtained from measurements on only one side of "zero."
Question 1: Why are the spectral lines curved? Question 2: Error due to 1 error in deviation angle?
Question 3: Error due to 1° error in prism angle?
4 and 5. Values for D and n.
6. The uncertainty in n is ± 0.02 if scale reading errors are assumed to be 0.5°. [A very conservative estimate.] If errors are smaller, one can assume linearity of the error propagation.
7. Derivation of formula for part (4). [A = a/2.] 8. Graph of deviation angle vs. wavelength. 9. Did the student 10. Did the student measure lines from a second spectral source? Is there a clear statement of how well the experimental values agree with handbook values? 11. Grating data compared with the prism data. 12. Value of grating constant. 2.54 x 10
Here's a BASIC program to quickly check the results: 10 'SPECTROM.BAS, by Donald Simanek. Calculates n for prism. 40 PI = 3.1415927 45 SCREEN 2:CLS:LOCATE 1,1,1:'visible cursor 50 PRINT "INDEX OF REFRACTION OF A PRISM, by Donald Simanek 51 PRINT 52 PRINT " sin [(A+D)/2]" 53 PRINT "n = " 54 PRINT " sin [A/2]" 55 PRINT 60 PRINT "Enter degrees and minutes separately, as prompted. 70 PRINT "If you enter degrees in decimal form, just hit 'enter' at minutes prompt. 80 PRINT:PRINT "Prism angle, A: degrees [default: 60]"; 90 INPUT 95 IF A=0 THEN A=60 :'Sets default, and traps for division by zero. 100 PRINT "Minutes"; 110 INPUT AM 120 A = A + AM/60 130 IF A=0 THEN A=60 140 PRINT:PRINT "Minimum deviation angle, D: degrees"; 150 INPUT D 160 PRINT "Minutes"; 170 INPUT DM 180 D = D + DM/60 190 N = SIN(PI*(A+D)/360)/SIN(PI*A/360) 200 PRINT:PRINT "A = ";A;", D = ";D 210 PRINT "The index of refraction, n = ";N 220 PRINT:INPUT "Another calculation [Y/n]";R$ 230 IF R$="N" OR R$="n" THEN 300 ELSE 45 300 SYSTEM Text and line drawings © by Donald E. Simanek, 1994, 2004.
## Comments on spectrometer experiment reports.Results should appear in the results section. Don't make the reader look elsewhere. You need to indicate Hardly anyone indicated In many cases, where results were shown for several spectral lines, the errors were obviously larger than your stated error, which should make you suspicious. Many people didn't Some students showed a graph of index of refraction vs. wavelength. That's good. But is A common blunder was to use telescope position readings in place of deviation values. This suggests that some students didn't understand the equation, or how the spectrometer worked. Some people gave a `refractive index for the prism' but didn't indicate what color light it was for. The prisms you used this year were flint glass, with refractive index of about 1.65 for the 5461Å green line of the Mercury spectrum. The deviation angles aren't worthy of listing as Reread the chapter 9 on lab report writing in the red book. Personal comments about how much you learned or how much you liked the experiment are not appropriate in a report. See also the document on technical report writing from Rensalaer University linked on my home page. I also have similar documents from Penn State, if you'd like to consult them. The prism equation for minimum deviation is not called Snell's Law. It is derived from Snell's Law. Willebrord van Roijen Snell (1591-1656) was a Dutch mathematician. Though Ptolemy knew about refraction, his law of refraction was only valid for small angles to the normal. Snell's discovery of the correct refraction law did not become well known, and when Descartes published it later he didn't credit the source. Johannes Kepler seems to have independently discovered it as well. Several people had identical or nearly identical reports. Since I cannot determine which of these was the author, or whether it was a group effort, I divided the score by 2. Partners work together in lab and will therefore have identical data. Partners may study together, but when it comes to writing the report, do it alone, in your own style and your own words. You should do your own calculations, and draw your own conclusions as well, not influenced by anyone else.
February 28, 1996 1. Spectral lines formed by a diffraction grating are curved when seen in the grating spectrometer, but a lot less than in the prism spectrometer. This is due to refraction through the thickness of the protective glass plates the replica grating is mounted between, but not caused by the grating itself. |