Atomic emission spectra experiment Contents 1 Overview 1 2 Equipment 1 3 Measuring the grating spacing using the sodium D-lines 4 4 Measurement of hydrogen lines and the Rydberg Constant 5 5 Measurement of mercury lines 6 6 Measurement of helium lines 6 A Review of Interference and Diffraction 6 1 Overview In this experiment, we will use a grating spectrometer to measure the emission spectrum of hydrogen, mercury and helium gas. We will measure the Rydberg constant, and identify as many atomic transitions as possible. Remember that you will be graded on the clarity and completeness of your report, the accuracy of your experimental data and calculations, and the insightfulness of your conclusions. 2 Equipment 1. Spectrometer (Sargent-Welch, Model S-75903-80) 2. Diffraction grating (6000 lines per centimeter, black block says down on one side) 3. Hydrogen, mercury and helium discharge tubes 4. Discharge tube power supply (5kV, 10 ma) 5. High intensity mercury and sodium discharge lamp 1
Figure 1: Clockwise from left: spectrometer, discharge tube power supply with hydrogen tube, spare tube box, spectrometer light shields, and diffraction grating. 2
Spectrometer components The collimator is an optical device used to direct a narrow beam of parallel light at the prism (or grating). It is rigidly attached to the base and is factory adjusted to have its optical axis perpendicular to the central vertical axis of the instrument. At the front of the collimator is an adjustable slit designed to operate smoothly and retain good parallelism of jaws at all slit openings. The telescope is the optical instrument used to view the emergent light. It is rigidly attached to a bracket supported on the central shaft in such a way that its optical axis is always at the same plane as the optical axis of the collimator, yet it can be rotated about the axis of the prism table and clamped independently. It contains a focusing eyepiece and a built in cross hair. The eyepiece tube is provided with a small opening into which light may be passed to illuminate the cross-hairs. This is particularly useful for viewing faint spectra when the background setting is too dark to make the cross-hair setting precise. The cross-hairs can be illuminated through a small hole above the eyepiece. The prism table is used to hold the prism or grating in alignment with the telescope and collimator. It can be rotated through 360 degrees or locked in position with the locking screw beneath the table. Its top surface is parallel to the optical axis of the telescope and collimator. A vertical post and spring hold the prism or grating rigidly on the table. An especially designed light shield is provided to eliminate stray light. The shield consists of an inner sleeve and an outer hood. The degree scale and the angstrom scales are mounted on the base. The outermost scale is the master angstrom scale. It is rigidly attached to the base and is immovable. The outer scale on the rotatable center dial is the vernier for the master angstrom scale. One side of the vernier is labeled for use with the prism while the other side is labeled for use with the grating. The innermost scale is the degree scale. Spectrometer setup The spectrometer is a precision instrument and rather expensive; please treat it with care. Note that the movable parts that can rotate, such as the grating table and the telescope arm, have clamping screws. DO NOT FORCIBLY MOVE ONE OF THESE PARTS WITH THE CLAMP SCREW TIGHT. IF ANY PART THAT YOU WANT TO TURN RESISTS MOVEMENT, ASK THE INSTRUCTOR FOR HELP. 1. Adjust the telescope for parallel rays (a) Point the telescope at some distant object through an open window or at the far wall of a large room (b) Focus the eyepiece on the cross hairs in the telescope by first withdrawing the eyepiece and then slowly pushing it inward with a slight turning motion until the cross hairs appear in sharp focus. 3
(c) Move the telescope sleeve out or in until the object selected is in clear focus (d) Test for parallax between cross hairs and distant object by moving the head slightly to left and right while sighting through the telescope. If any relative motion appears between the two, repeat the previous two steps until no parallax can be observed. The telescope is now in proper adjustment and should remain so. However it is good practice to recheck the adjustment periodically by repeating the above procedure. 2. Adjust the collimator for parallel rays (a) Having first adjusted the telescope, rotate the telescope arm until the telescope and collimator tubes are aligned end to end. (b) Direct the instrument toward a light wall or white paper. Open the adjustable slit wide to admit plenty of light. (c) Sight on the slit through the telescope and adjust the slit tube in and out of the collimator tube until the slit edges are in sharp focus and parallax between cross hairs an d slit image has been eliminated. The collimator is not adjusted. In any subsequent applications, if the instrument appears to be out of focus, first readjust the eyepiece to obtain sharp focus on the cross hairs. Then, if the slit is out of focus or parallax exists, repeat all steps in the given order above. 3 Measuring the grating spacing using the sodium D-lines The diffraction grating supplied has a nominal number of lines per inch of 15,000 (or 6000 per centimeter), but this should not be considered precise. One always finds the grating spacing, d, from a known optical wavelength because this is much more accurate than the control of grating manufacture. We will use the bright doublet known as the sodium D-lines, with wavelengths λ 1 = 588.996nm and λ 2 = 589.593nm. First, make the room as dark as possible. Turn of the overhead lights. Use a small pen light for use in reading your notes and the dials of the spectrometer. This will allow your eyes to remain adjusted to the darkness. Use the sodium lamp as the light source for the spectrometer. After focusing the spectrometer, adjust the slit width until the sodium yellow line is resolved. Adjust the position of the tube until the lines are as brilliant as possible. Record, as shown in Tab.1, the angular settings of the sodium d-line. You should use the vernier scale so as to read the angle to within a tenth of a degree. Also record the order number, m, of each measured yellow line and the angular position of the central maxima. Use this data to compute the grating spacing. If the deviations of the line to the left and the right of the central maximum differ by more than a degree, readjust the orientation of the grating until they are made more nearly 4
order angular m position 2θ m θ m sin θ m m λ mλ d 1 (+) 1 ( ) 2 (+) 2 ( ) Table 1: Data table used to compute grating spacing. equal. If the diffracted lines to the right of the central maximum are much lower or higher in the field of view than the lines to the left, the table holding the grating is not level. This condition can be corrected by using the leveling screws. What is the diffraction grating spacing? How would your pattern look different if you had used a shorter wavelength light, such as blue light? Would this change your measured values of the slit spacing and slit width? Finally, identify the atomic transition that gives rise to the sodium D-lines. Why is this a doublet (as opposed to, say, a singlet or a triplet.) 4 Measurement of hydrogen lines and the Rydberg Constant Place a hydrogen discharge tube in the power supply and the diffraction grating on the table in the spectrometer. Place the entrance slit of the spectrometer as close as possible to the hydrogen tube without touching. Find the position of the spectrometer for which the hydrogen lines are as brilliant as possible. Record the angular positions, to the left and right of the central image, of all the hydrogen lines that are visible, in both first and second orders. These lines should include a red one, a blue-green one and a violet one. A far-violet one may also be visible, but the far-violet line is very faint and requires good eyes as well as optimum adjustment of the slit width, and reduction of background light, to perceive it at all. These lines are listed in reverse order as seen when going left or right from the central maximum (i.e. you will first see the violet, then blue-green, and then red). Record the angular positions of the lines you can see from both orders as shown in Tab.2 From your data and the grating spacing which you determined from the sodium D-lines in your previous experiment, determine the wavelength of each of the observed hydrogen lines. Plot your values of 1/λ against 1/n 2. What shape do you expect for this graph? Does your data fit this? Make use of the Rydberg formula ( 1 1 λ = R n 2 1 ) 1 n 2 (1) 2 where n 1 and n 2 are the various n values of the terms, to analyze your graph. From the slope of the graph, deduce the value of the Rydberg constant, R, and compare it with the accepted value. From the intercept, compute the series limit. Compare this to the Balmer 5
color angular order position 2θ m θ m sin θ m m d d sin θ m λ 1/λ 1/n 1/n 2 (+) ( ) (+) ( ) (+) ( ) (+) ( ) Table 2: Example data table used to compute hydrogen spectrum. series limit. Also, from your measurements of the Rydberg constant, determine the value of Planck s constant, and compare it with the accepted value. 5 Measurement of mercury lines Now use a mercury discharge tube, or a high intensity mercury source, and identify as many of the spectral lines in the mercury spectrum as possible. Be as clear as possible. Do you see all of the lines you would expect in the visible region? Why or why not? You may use Fig. 75 in Herzberg s book[1]. 6 Measurement of helium lines Now use the helium discharge tube. Identify as many of the spectral lines in the helium spectrum as possible. Be as clear as possible. Do you see all of the lines you would expect in the visible region? Why or why not? You may use Fig. 27 in Herzberg s book. A Review of Interference and Diffraction The interference and diffraction of light may be understood using Huygens Principle and the Principle of Superposition. Huygens principle states that each point on a primary wavefront acts as a source of spherical secondary waves. The envelope formed by these spherical secondary waves make up the wave front at any later time. This suggests that if waves of light strikes a barrier with two tiny pinholes, the pinholes will act as two spherical point sources of waves, as shown in Fig. 2. If instead two long but 6
very narrow slits were cut in the barrier, the slits would act as two cylindrical sources of waves. Figure 2: Light falling on a barrier with two holes The Principle of Superposition states that the amplitude of a wave at a particular position in space is the algebraic sum of the amplitudes of the waves arriving at that position from different sources. In the case of waves of light passing through a barrier with two slits, the slits act as two sources of waves which may interfere with one another to produce regions of constructive or destructive interference. Constructive interference occurs at positions where the two light beams are in phase with one another that is, where the difference in path length of the beams is an integral multiple of the light s wavelength. In terms of the angle θ, shown in Fig. 3, this occurs when the following formula is satisfied. d sin θ m = mλ for m = 0, ±1, ±2... (interference maxima) (2) Here, d is the spacing between the slits, m is the order of the interference maximum, and θ m is the angle at which the m th order maximum occurs. On the other hand, interference minima occur where the difference in path length of the beams is a half-integral multiple of the light s wavelength, i.e. where d sin θ m = (m + 1/2)λ for m = 0, ±1, ±2... (interference minima) (3) For small θ, we can use the approximation sin θ = θ. The above formula for interference maxima thus predicts that for angles measured in radians, maxima occur at the angles θ m given by θ = m λ d (4) 7
Figure 3: Path of light passing through two narrow slits. The difference in path length is d sin θ. Figure 4: Light intensity as a function of angle θ for small θ for the case of two infinitesimally narrow slits separated by distance d. 8
The intensity of light as a function of angle for two infinitesimally narrow slits is depicted in Fig. 4. In the case that there are more than two infinitesimally narrow slits, the interference pattern can appear more complicated. This is because we must add together the light beams from each slit. When there are three or more slits, we must distinguish between principle maxima and secondary maxima. Principle maxima occur when the light beams from all the slits are in phase. Principle maxima appear as very bright lines. Secondary maxima occur when the light beams from not all of the slits are in phase. Secondary maxima appear as fainter lines. As before, minima occur when there is complete destructive interference between the beams from all the slits. It can be proved fairly easily, for example, that for the case of three infinitesimally narrow slits separated by distances d, the first minimum appears when d sin θ = λ/3, the second minimum appears when d sin θ = 2λ/3, and the first order principle maximum occurs when d sin θ = λ. As the number of slits increase, the secondary maxima become fainter and the principle maxima become narrower. A diffraction grating is a very dense series of slits, perhaps a few hundred slits per inch. The principal maxima for a diffraction grating occur when d sin θ m = mλ for m = 0, ±1, ±2... (diffraction grating maxima) (5) where d is the slit spacing and m is the order of the maximum. In practice, the intensity pattern for two slits in a barrier seldom appears as depicted in Fig. 4. This is because there is no such thing as an infinitesimally narrow slit. Slits have some finite width, D, so the light passing through the slit does not in fact act as a point (or line) source of waves. Consider, for instance, light passing through a long rectangular single slit of width D, as shown in Fig. 5. According to Huygens principle, each position on the wave front acts as a cylindrical source of waves. These sources interfere with one another so as to produce a single slit interference pattern called a diffraction pattern, shown in Fig. 6. The location of the first diffraction minimum may be found by considering in turn all pairs of sources in the slit which are separated by a distance of D/2, as shown in Fig. 6. Each pair of sources create destructive interference at an angle θ is such that D/2 sin θ = (m + 1/2)λ. More generally, diffraction minima occur when the following equation is satisfied D sin θ m = mλ for m = ±1, ±2... (diffraction minima) (6) Here, D is the slit width, m is an integer which labels the first, second, etc. dark fringe which occurs at an angle θ m, and λ is the wavelength of the light. Now reconsider the actual light pattern which occurs when there are two real (not infinitesimally narrow) slits of width D separated by a distance d. The intensity pattern is just the double slit interference pattern modulated by the single slit diffraction pattern, as depicted in Fig. 7. In the case that the barrier has three or more slits of finite width, there will also be secondary maxima between the principle maxima. 9
Figure 5: Light passing through a single slit of width D. Each position on the wave front in the slit acts as a source of secondary waves, according to Huygens Principle. Figure 6: Light intensity as a function of angle θ for a single slit of width D. Figure 7: Light intensity as a function of angle θ for two slit of width D separated by distance d. 10
References [1] G. Herzberg. Atomic spectra and atomic structure. New York: Dover, 1945, 1945. 11