Physics 390, Lab 5: Diffraction and Optical Spectroscopy (modified from materials by KJ Park & Stephen Gregory)

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1 Physics 390, Lab 5: Diffraction and Optical Spectroscopy (modified from materials by KJ Park & Stephen Gregory) Second year lab 2, Winter, 2008 Reality provides us with facts so romantic that imagination itself could add nothing to them. Jules Verne. What is the meaning of it all, Mr. Holmes? Ah. I have no data. I cannot tell, he said. Conan Doyle The Adventures of the Copper Beeches. Arthur Hi there! what s your spectrum (overheard as a pickup line by Physics Lab Instructor) Introduction Because light behaves macroscopically as a wave (when averaged over millions and millions of photons), it diffracts when encountering regularly spaced scatterers such as grooves or lines in a diffraction grating. By diffract, we mean that light waves appear to bend when encountering obstacles whose size is on the order of the wavelength of the light under consideration. It is perhaps better to think of light from a source as forcing the atoms or molecules of the diffraction centers (grooves, lines, etc.) into vibration, whereupon they re-radiate the original light in many directions. If the path length from successive diffraction centers to an observation point is an integer multiple of the wavelength, constructive and destructive interference will occur and bright and dark diffraction patterns will be observed. The spacing between bright parts of the pattern will depend upon the wavelength of the light and so, for example, longer wavelength light will appear to bend more when encountering a diffraction grating. By measuring how light from a particular source is bent upon passing through a diffraction grating, one can learn interesting things about the source. This branch of science is called spectrometry. The optical spectrometer enables us to study the emission and absorption of light by atoms (among many other things). Measurements of optical spectra at the end of the 19 th and beginning of the 20 th Century lead to the creation of atomic physics and the understanding of atomic behavior in terms of quantum theory. In more automated versions the grating spectrometer is still one of the most important laboratory instruments. We shall use a grating spectrometer to measure the positions of lines in the emission spectrum of mercury, hydrogen and sodium. These lines correspond to the discrete energy levels of the electrons in the atoms, and measurements of these lines directly test our understanding of the atomic model. Hydrogen, a two body system, is the only atomic spectrum which can be analytically solved in quantum mechanics. Even before the invention of quantum mechanics, Bohr had discovered a certain pattern in the spectrum and had proposed a set of discrete energy levels with a certain geometric relationship (the Bohr model). Other elements, like Sodium, exhibit some characteristics of this single-electron spectrum owing to their single valence electron outside fully-populated inner shells. 1

2 Experiment Goals Establish spacing of optical spectrometer diffraction grating. Using a mercury gas discharge source, measure the angle of diffraction of the zero-th order for the green line. Use the measured angle for the green line and the known wavelength of the green line to determine the spacing between lines of the diffraction grating. Measure diffraction angles for the Balmer series of hydrogen Use your setup and a hydrogen gas source to determine the wavelengths of the Balmer series (of lines) for hydrogen. Use the diffraction grating spacing determined above. Compare your values for the Balmer series to those stated in textbooks. Use your Balmer series wavelengths to determine the Rydberg constant for hydrogen. Fine structure in sodium Using a sodium lamp, observe the three primary wavelengths of the sodium spectrum. Measure the fine structure splitting of these primary lines Relate this structure to the model of the sodium atom. Background Light Sources Our light sources are gas-discharge tubes. These consist of a glass tube with a metal electrode at each end. The tubes contain low-pressure gases of various types. When a high voltage is applied between the electrodes the gas ionizes (the atoms split into electrons and positive ions). These charged particles are accelerated in the electric field and collide with atoms, causing these to enter a higher energy excited state. (Another way of looking at this process is that the ions are carrying a current which heats up the atoms, creating more ions and also exciting the atoms.) When the atoms return to their original states, they emit photons of particular wavelengths. The set of wavelengths which are emitted the spectrum of the atoms is characteristic of the atoms and can be used to identify them. In fact, a standard way of identifying materials is to burn them in a flame and observe the emission spectrum. The Spectrometer The spectrometer consists of three basic elements, the collimator, the diffraction grating and the telescope. First, some of the light from the tubes is collected by the collimator: Collimator The light to be analyzed enters the collimator through a narrow slit whose position can be adjusted to put it at the focal point of the collimator lens. The light leaving the collimator should therefore be a parallel beam, which ensures that all the light from the slit strikes the diffraction grating at the same angle of incidence. This is necessary if a sharp slit image is to be formed. 2

3 Think about the best method you can use to ensure that the light leaving the collimator really IS a parallel beam. The parallel light beam now should enter the diffraction grating perpendicularly: Diffraction Grating Diffraction gratings are made by scribing closely spaced grooves on glass or some other substrate. The phenomenon of diffraction is one of many manifestations of interference in physics. The incoming beam hits the grooves and scatters in all directions. However, because the initial, unscattered, wavefront had a common phase, the scattered light will interfere with itself in such a way as to produce maxima and minima as a function of angle (usually measured from the perpendicular to the exit face of the grating. θ d Fig. 1. Diffraction Geometry Many text books contain the theory of the diffraction grating. The basic result is that maxima in the diffraction pattern are found for angles satisfying the relationship d sin θ = n λ where d is the spacing between grooves, θ is the angle between the perpendicular and the diffracted beam, λ is the wavelength and n is the order of the diffraction where n = 0, 1, 2, etc. There is considerably more to the general theory of the diffraction grating. In fact, it is nowadays presented in terms of Fourier Transforms. However, the above expression is quite correct and useful for present purposes. The light from the gas-discharge tube is not monochromatic. There will therefore be an angle which gives a maximum, i.e. a bright image of the slit which we will call a line, for each wavelength present in the spectrum of the gas. This means that there will be a characteristic set of lines. Further, this set will repeat as the angle, θ, increases because the phase relationship for a maximum repeats for each increment of 2π. Each repeat is called an order. Telescope The telescope can be rotated to collect the diffracted light at very precisely measured angles. With the telescope focused at infinity and positioned at an angle to collect the light of a particular color, a precise image of the collimator slit can be seen. For example, when the 3

4 telescope is at one angle of rotation, the viewer might see a red image of the slit, at another angle a green image, and so on. By rotating the telescope, the slit images corresponding to each constituent color can be viewed and the angle of diffraction for each image can be measured. This measurement is aided by cross-hairs in the telescope. Operating the Spectrometer In order to save you some time, the main alignment of the spectrometer will be performed at the start of the week before you come to lab. You should only need to adjust the focus of the collimator and telescope to obtain reasonable measurements. To gain a better appreciation of what is involved, the full alignment procedure is described in the appendix. The attached material from the manufacturer will be referred to as the MANUAL. 1) Read the Equipment page of the MANUAL. 2) Identify the lock screws for the telescope and the spectrometer table. Practice rotating the two bases coarsely by hand (release lock screws) and using the fine-adjustment screws when the lock screws are engaged. 3) NOTE: The scales on the spectrometer are in degrees, minutes and seconds. This is ridiculous (in the opinion of Steve Gregory), but gives you some of the feel of what it would have been like to be some guy in a starched shirt and suit sitting at the lab bench at the end of the 19 th Century. Well, anyway, we can convert to decimal values in order to do the calculations. Reading the Vernier Scales(with the magnifying glass): To read the angle, first find where the zero point of the vernier scale aligns with the degree plate and record the value. If the zero point is between two lines, use the smaller value. In Figure 8 the zero point on the vernier scale is between the 172 o 20' mark and the 172 o 40' mark on the degree plate, so the recorded value is 172 o 20'. Now use the magnifying glass to find the line on the vernier scale that aligns most closely with any line on the degree plate. In the figure, this is the line corresponding to a measurement of 12'30'' of arc. Add this value to the reading recorded Figure 2. Reading the Vernier Scales above to get the correct measure ment to within 30 seconds of arc: that is, 172 o 20' + 12'30" = 172 o 32'30". 4

5 4) When analyzing a light source, angles of diffraction are measured using the telescope vernier. In general, we will measure the difference between angles of diffraction on either side of the undeflected position and divide by two, but it is useful to establish a vernier reading for the undeflected beam (see Figure 8 of the MANUAL). To obtain a vernier reading for the undeflected beam, first you would align the vertical cross-hair of the telescope with the fixed edge of the slit image for the undeflected beam. Then read the vernier scale. This is the zero point reading, Figure 3. Measuring an Angle of Diffraction. The rotational position of the spectrometer table can be measured with the same accuracy using the spectrometer table vernier. Note: The telescope and the spectrometer table each have two vernier scales, which are exactly 180 o apart. Unless you use the same vernier scale for both the initial and final readings, you will need to add (or subtract) 180 o from your result. 6) Take turns to practice reading the vernier for arbitrary angular settings in darkened room with a flashlight or table light. You are to read verniers to 0.5 minutes. (1 minute = 1' = 60 seconds = 60'') Checking the Focusing You only have to make minor adjustments for your eyes. Remember not to change the focus of the collimator or of the telescope by large amounts. To get the cross-hairs into focus (this is independent of the actual focussing of the telescope) move the eyepiece in and out. Do not make any adjustment to the leveling screws under the telescope and the collimator. If you suspect that you need to realign, please see your instructor. 1) If the grating is present, remove it. Illuminate the slit with a mercury discharge source, placed about 1cm in front of the slit. Line up the collimator and telescope and look in the telescope. The slit image and the cross hairs should show no parallax. (i.e. if you move your head side to side, the slit image should not move with respect to the cross hairs.) If you notice some parallax, the following sequence of fine adjustments should accomplish the final alignment: Remember! These are minute adjustments! 2) Try a small clock-wise adjustment of the collimator focus. See if the parallax is decreased or increased. If decreased, go to 3; otherwise, make a counter-clock-wise adjustment. 3) When the parallax has been decreased, the slit image should be slightly fuzzy. Adjust the telescope focus, followed by cross hair focus. 4) Recheck the parallax. Repeat 2 3 until the parallax is not there. 5

6 Install the diffraction grating: 1) Study the MANUAL section, `Using the Diffraction Grating.' 2) Steps 1 3 are done for you already. In step 4, insert the grating in an orientation in which it is wider than it is tall. Make sure that the tag tape faces the source slit. Can you think of a reason why this is important. What would be the consequences of installing the grating with the tag tape facing the telescope? 3) Plug in the power plug for the Gaussian Eyepiece. 4) Follow steps 1 16 of the MANUAL section, `Appendix.' Again, complete this task without leveling adjustments for the collimator and the telescope. When using the Gaussian eyepiece note that the reflected image of the crosshairs is actually dark, because it is a shadow. 5. Make sure that the Mercury vapor discharge tube is approximately 1-4 cm in front of the slit and in line with the collimator axis. Follow steps 6 9 of the MANUAL section, `Using the Diffraction Grating.' Make sure that the Mercury green is symmetrically located with respect to the central maximum. If not, make a tiny rotation of the spectrometer table to achieve this. (See the last paragraph of the MANUAL section, 'Appendix.' [You will have to take three sets of measurements and average the results for the three angular positions. You will want to repeat with the table angular position changed by increments of approximately 1 degree. You should converge toward the symmetric situation and should be satisfied if you are within several minutes.] Accuracy of angular position measurements: The spectrometer verniers have a precision of 30 seconds. You should determine your measurement errors by setting the telescope vertical cross hair to the fixed edge of the illuminated slit. [Some people prefer to line up the cross hair to the middle of slit, whose width has been adjusted slightly wider than the cross hair width. But this tends to make lines very dim.] Make five repeated settings of the cross hairs on the slit and measure the angles. Calculate the standard deviation. Measurements and Analysis 1) Determination of the grating characteristics using a standard source: The mercury discharge tube will be used as a standard reference source. Determine the diffraction angle for the green line of mercury by sighting the line on both sides of the undeflected position and dividing by two. You should make 5 measurements of the zero diffraction and five of the first order diffraction positions. (This means resighting each time!) 2) Measure the Positions of the Lines in the Hydrogen Spectrum: Now switch to the hydrogen discharge tube in order to measure the position of as many of the lines as you can see in the first order and in the second order. Use the method of taking data on both sides of the zero point and dividing by two. You only need one measurement of each line. 3) Measure the Splitting of the Lines in the Sodium Spectrum: Switch the discharge tube for a sodium lamp. You should be able to see three spectral lines (red, yellow and green) which are each split into closely-spaced pairs. Measure the wavelengths of 6

7 these lines in both the first and second order if possible. We actually care more about the splitting than the actual position of the lines. Since there are many lines here to measure, you probably don t need to take a full five readings for each line, but try to estimate how accurately you can determine the splitting by comparing results left vs. right, or first order vs. second order. Take the grating and put it away in the grating box when you are done. Mercury/Hydrogen Analysis: 1. Calculate the mean and the standard deviation of the multiple readings taken for the mercury green line. 2. Calculate the slit spacing of the grating from the mercury line measurement. The Handbook of Physics lists the wavelength of the mercury green as nm. 3. The visible part of the Hydrogen spectrum is called the Balmer series. The Bohr model predicts that each electron orbital has an energy proportional to n 2. The visible lines are transitions between these energy levels. Use your data for the Balmer series of Hydrogen to obtain a value for the Rydberg constant, R, in the formula 1! = RZ " n _ 1 % $ '; R = x 10 # 1 n2 2 & where n 1 = 2, n 2 = 3,4,5, which refer to the red, blue and violet lines (do you know what Z is?) Discuss your uncertainties on measuring R. Sodium Analysis: The Sodium atom (Z=11) has a single valence electron outside 10 electrons which completely fill the first and second inner shells. This makes the Sodium spectrum somewhat analogous to the single-electron Hydrogen spectrum. The fine structure splitting can be understook by looking a the energy level diagram shown in Figure 4, where the states have been separated by their angular momentum. Photons can only be emitted between states where angular momentum changes by at least one unit. This leads to the transitions shown, which are all in the visible. Other allowed transitions are in the ultraviolet or infrared Figure 4: Selected Sodium atomic levels The splitting in the optical spectra can be explained if the 2 P energy levels are split by the spin-orbit coupling between the valence electron and the magnetic field produced by the rest of the atom. The 2 P levels actually consist of J=1/2 and J=3/2 states with different total angular momenta, and the energy levels of these two states in the presence of a magnetic field are slightly different. The ration of this energy splitting to the gross energy of the 2 P (n=4) state was originally measured to be α 2 = (1/137) 2 or the fine structure constant. A similar splitting is not seen in Hydrogen due to higher-order QED corrections called the 7 _ 1 m 7

8 Lamb Shift but it is much too small for us to resolve. 1. Measure the wavelength of the splitting observed in the three sets of visible Sodium lines. Include uncertainties (statistical and systematic) which you have estimated. 2. Calculate the energy difference associated with each set of lines. Recall that the energy of a photon is given by E=hν=hc/λ. 3. Is your data (with uncertainties) consistent with the hypothesis described above? Why or why not? Questions & Discussion: Here are some items for you to discuss in your report: The mercury green is often used as a reference, because its wavelength is extremely stable even when the discharge is not very good. The fact that we referenced the Balmer lines to this mercury green should make the result quite reliable. How might this process take care of systematic errors of the spectrometer? Where is your major source of error? All these wavelengths are vacuum values. How does the refractive index of air affect the diffraction pattern? How about the refractive index of the glass substrate in which the grating is sandwiched? 8

9 9 Second year lab 2, Winter, 2008

10 10 Second year lab 2, Winter, 2008

11 11 Second year lab 2, Winter, 2008