P19: INTRODUCTORY PHYSICS III Experiment IV: Atomic Spectra and the Bohr model Department of Physics and Astronomy Dartmouth College 6127 Wilder Laboratory Hanover, NH 03755 USA Overview In this lab, we will study the light emitted by different atoms. We will see spectral line emission and identify patterns in them which are characteristic of the different atoms. We will measure wavelengths and compare the corresponding energies to what atomic models predict, including the Bohr model of one-electron atoms. 1. Introduction Around 1816-1817, the Bavarian physicist Joseph von Fraunhofer developed the experimental setups to separate light into its different colors. He is usually credited for being the first person to note that the light emitted by the Sun was not composed of a continuous sequence of colors, but that dark lines are present in it ( Fraunhofer lines, which are absorption lines). He went further and invented the diffraction grating, which is equivalent to a N-slit interference apparatus. During the years of 1859-1860, by way of combustion of different materials, Gustav Kirchoff and Robert Bunsen presented evidence that different chemical elements were associated with a different pattern of color lines ( flame spectra ); they, too, used diffraction gratings. These emission lines are the reverse of the aborption spectrum, with the lines occuring exactly where the dark Fraunhofer lines appear in the corresponding absorption spectrum. In 1885, the German school teacher Johan Balmer was able to study the spectrum of hydrogen with an electric arc lamp, and found a simple empirical formula that could reproduce the pattern of emitted visible wavelengths [1]. This prompted Johannes Rydberg to seek similar empirical formulas for other atoms. For the hydrogen spectrum, Rydberg wrote Balmer s formula in the form ( 1 1 λ = R 1 ) n 2 n,n 0 are integers. (1) n 2 0 The constant R that appears in H and other one-electron atoms is called Rydberg s constant. For the hydrogen Balmer series, n 0 = 2. Based on the experiments, in 1913 Niels Bohr developed a simple phenomenological explanation of the spectrum of hydrogen, which we will deal with in this laboratory. The Bohr model was not the end of the story, and the quantum theory of the hydrogen atom was developed in 1926 simultaneously by Wolfgang Pauli and Paul Dirac. It introduced some small corrections to even the Bohr model energy levels, so that the measured hydrogen wavelenghts are a little off from what the Bohr model and the associated Rydberg formula predict (less than 0.1%) Dated: Oct 2012 HRM, partially based on Astro 3 Lab (2009), P17 (2003), and P19 (2010 - L. Motta).
Figure 1: Sketch of the apparatus used. 2. Apparatus The apparatus of this lab consists of a light source, a diffraction grating and a microscope (observation arm) on an angular scale. The light source is a spectral lamp with interchangeable bulbs, each bulb containing one specific gas, as labeled (including hydrogen (H), helium (He), neon (Ne), mercury (Hg), nitrogen gas (N 2 ), and others). An applied high voltage creates enough free electrons and ions in the gas to establish a current; the moving charged particles collisionally excite the neutral atom s electrons to different levels. The light of the spectral bulb is shielded by a large pane except for a small aperture, and the adjustable entrance slit (see Figure 1) additionally selects a geometrically narrow portion of the light flux. As you recall from the discussion of light interference (interference lab), after the grating one can observe a maximum of intensity at an angle θ for a wavelength λ given by mλ = d sinθ, where d is a length constant of the grating (the effective slit-to-slit distance) and m is an integer. In this lab, it is enough to observe only the first series of maxima, at m = 1, which are typically the strongest: λ = d sinθ (2) In a high-quality grating, d is very small so that the angles are large, which is another reason why m = 2 might not be observed. Gratings act as interference devices with N slits, with N large. The effect of this is to make the maxima narrower and brighter than the double-slit case, and as such, gratings are ideally suited to measure wavelengths exactly. The observation device is a microscope with which the light can be studied at different angles, with the angles being measured with an accurate vernier scale. The microscope can be aligned with the overall optical axis defined by spectral lamp and grating, to see the direct spectral bulb emission (that would be m = 0, θ = 0, central line ; no wavelength measurement here). The color spectrum then occurs symmetrically to either side of this axis. For accurate measurements, crosshair structures are in the optical path inside the microscope, and parts of the crosshair can be illuminated through a lamp attached to a switchable power source. 2
3. Procedure 1. Overview. In order to measure different wavelengths with the apparatus we will need to get the constant d, and to calibrate the offset of our angle θ scale. This corresponds to identifying what wavelength is measured at a given θ. For this purpose we will use the known spectral lines of the hydrogen (H) atom. The only lines in the visible range are part of the Balmer series, and are easy to identify. Each apparatus will have its own calibration curve. To assure that we made no mistakes, and everything is consistent, we will use our calibration to measure the spectral lines of mercury (Hg), and then compare our results with other equipment. If our measurements of Hg turn out correct, we will then measure the visible wavelengths of helium (He), and qualitatively observe neon (Ne) and nitrogen (N 2 ). 3.1. Reading off angles In case you are unsure how to read the angles off the Vernier scale, in degrees and minutes (1 minute = 1/60 degree), here is how: 1. The angle scale on the base of the spectrometer is divided into two parts, one moving with the microscope, the other fixed to the base. The complete circle scale contains 360 degree markings. Each degree is divided into two parts each representing thirty minutes of a degree. The small scale is divided into thirty equal divisions each representing one minute of a degree. A magnifying glass with a lamp is provided on the spectrometer to assist you in reading the scales. The scales are read in the following way. 2. Determine in which degree division on the degree scale the zero line on the small scale is falling. This gives you the whole degree of the angle being measured. 3. Determine whether the zero line falls beyond the minor tick mark on the degree scale, in which case you need to add 30 minutes to the value determined in the next step. 4. Determine which line on the small scale exactly lines up with a line on the degree scale. This number from the small scale is then your minute measurement. 5. All together, this gives an angle in degrees and minutes, to the nearest minute. When computing angle differences, make sure to take the 360 degree periodicity of angles into account when necessary, to get the correct differences. Before plugging into a sine function of a calculator, you will also have to convert degree + minutes into decimal degrees. 3.2. Set up Make sure that your apparatus is set up correctly. First verify that the arms of the spectrometer are roughly aligned with the horizontal. Place the H spectral lamp in its place and try to get the maximum intensity to pass through the diffraction grating. For this it is helpful to move the microscope away and simply look directly into the entrance arm and then move the lamp until one can see a bright line. Turn the observation arm to the center, look through the microscope, and further fine-tune the position of the observation arm (or perhaps the lamp) until you see a bright central line. By the side of the entrance slit, there is a screw that controls the width of the slit, so you may have to open the slit if you cannot see the central line. You can change focus by moving the eyepiece in and out of the observation arm. There is a green crosshair that you can switch on to improve your ability of centering the line in the microscope. Once you have seen that your central line is in focus and visible, turn the observation arm into one direction and try to observe the Balmer emission lines of H. You will be able to see a sequence of lines in order like the second panel of Fig. A.2, with colors/wavelengths as listed in Table 1. Since we will be measuring the angle at which each line appears, you should try to make the width of the entrance slit small so that the lines are as thin as possible without making them invisible. After contemplating the structure of H-Balmer lines, now we proceed to calibrate our equipment. 3
Color Wavelength Apparent Angle 1 Angle 2 (nm) brightness ( ) ( ) (H δ ) Violet 410.174 (H γ ) Violet 434.046 (H β ) Blue 486.136 (H α ) Red 656.285 Table 1: The visible lines of H. The data are taken from the NIST database, Basic Atomic Spectroscopic Data, http://www.nist.gov/pml/data/handbook/index.cfm 3.3. Calibration When you move the observation arm away from the center, you should be able to see a sequence of colored lines. The sequence you observe in order of appearance from the closest to the center is tabulated in Table 1. Note that H δ is challenging to see and requires very good conditions (dark room; aperture not too narrow; spectral tube lined up optimally). Fill in Tab. 1 with your data. Do a sequence of measurements with the microscope to the right of the center position (the optical axis made by bulb and entrance aperture). When using the cross hairs, always put them consistently onto one edge of the emission line, for example the one on the blue side (least angle). Do another sequence of measurements (Angle 2) to the left of the center position. In the apparent brightness column note whether the line is strong, medium, weak, or very weak. This could help you later to check that your lines were identified correctly, because one can verify the brightness of lines in the NIST database. Obtain constant offset. When complete, take the average of the two angles in each row, which gives you an estimate of the angle of the center direction. Do an average of the averages to obtain the (bestestimate) master offset angle, the angle of the center direction that needs to be subtracted from every angle measurement of your apparatus to get θ. Obtain your calibration by finding the adequate linear coefficient d that will provide you the function λ(θ) for the apparatus (Equation 2). For this, calculate the ratio λ/sinθ for each row in your Table 1 and do a statistical average. In your writeup, come up with the error estimate for d. Does your value of d match the label on your grating? With the function λ(θ) one can translate measurements of angles into measurements of wavelengths. In the course of this calculation, since your errors for each measurement are in θ, you will need to propagate the errors to wavelengths: ( σλ λ ) 2 ( σd ) 2 ( σθ ) 2 = + (3) d tanθ where σ 2 θ must be in radians. Test your calibration to make sure that no misidentification occurred. For this task, use the Hg lamp. The spectrum of Hg is also shown in Fig. A.2. Table 2 contains a sample list of bright lines of Hg; make your own measurements of the lines of Hg, convert them to wavelengths, and check whether your wavelengths match most of those of Table 2. If yes, then you are confident of your calibration. 3.4. Helium spectrum Now that you have measured accurately already the spectra of two elements, obtain the wavelengths of all lines of helium that you see. At least you should be able to see strong and medium lines: one red, one yellow, two teal-greenish, two cyan, one blue, one violet. For a challenge, there can be additional violet ones seen if conditions are optimal. Also, beyond the red line, another red line can be seen if conditions are super-optimal. A word about red: The associated angles for red can be large enough that in the microscope, a 4
Color Wavelength (nm) Measured angle ( ) λ(x) Violet 1 404.6565 Violet 2 435.8335 Green 1 491.6068 Green 2 546.0750 Yellow/orange 576.9610 Table 2: The visible brightest lines of Hg - partial list. Source: As in Table 1. distractingly bright background can appear: one sees the reflection of light off the wall because that direction is beyond the wooden shade of the spectral lamp. In this case, do create your own shade with your hand (but do not block the optical path from grating to microsope, of course). 3.5. Other gases: mostly qualitative observations There are other lamps available in the lab. Consider the case of neon, and compare to H, He and Hg. Measure a few prominent wavelength. Consider next molecular nitrogen, N 2, and compare to H, He and Ne and Hg. Clearly, N 2 contains more transitions. Based on the phenomenology of Bohr and the orbitals structure of N 2, can you give a possible qualitative explanation to why? You can explore some relationships with equations you expect to change from H to N 2, however you don t have to show a quantitative definite prediction of the N 2 spectrum. Measure one or two of the sharp lines in the spectrum, and later in the writeup try to identify them. Another famed spectrum is that of sodium, also available in the lab. 4. Lab report 4.1. Bohr model of one-electron atoms Write up your objectives and the description and physics of the apparatus. Give your λ - θ relationship with all the propagated errors, and state how the offset angle with its error was obtained, and what they are. For the hydrogen atom: In hindsight, with the calibration available, comment on whether you were able to consistently and accurately measure the Balmer lines. Calculate the wavelengths for the Balmer series from the Bohr model, and compare to the NIST values and to your measurements, both of Table 1. Discuss any agreement or disagreement. This requires the identification of the statistical error in your determination of λ, and a comparison of the theoretical value with experiment taking the error into account. The singly ionized helium (a helium nucleus = 2 protons + 2 neutrons, surrounded by one single electron) can also be described well by the Bohr model. Figure A.3 shows the wavelengths, in Ångstrom, of transitions in this ion. In the lab, did you see any of these lines in the He spectra? In other words, is He + present or not when the helium spectral bulb is in operation? For this ion, with the wavelength data from Figure A.3, calculate the energy levels E n for n = 2, 3, and 4. Then, compare these results with what the Bohr model predicts for this Z = 2 ion. Determine E 1. Compare the spectrum of Na to H, even if you did not use the sodium bulb in the lab. What features do you expect them to share and why? [Hint: what are the optically active electrons?]. 4.2. More complicated levels of multi-electron atoms Even for the simplest atom with more than one electron, namely, the helium atom, the Bohr model does no longer seem to hold, but some ideas about the transition spectra are still valid. For the energy levels of helium, and hence the helium lines you observed in the lab, not only the attraction between proton and electrons needs to be taken into account but also the mutual repulsion of the two 5
electrons. On top of that, electrons have spin (an intrinsic angular momentum characteristic), and the energy levels depend on whether the two electron spins are anti-parallel, or whether they are parallel. The first case is called parahelium, the second orthohelium. The first case is also called singlet (total spin S = 0), the parallel case triplet (S = 1). It turns out that one of the two electrons always remains at the lowest energy level, and the transitions you saw in the lab occur by the other electron changing from an excited state to a lower state. There are a few rules for such transitions, for example a transition cannot convert a parahelium into an orthohelium or vice versa, meaning that photon transitions between two different total spin states do not occur. Figure A.4 shows the energy levels of the helium atom (for the one electron that does not remain in the ground state); each energy level (horizontal bar) is labeled with a number and a letter, with the number being the principal quantum number n, and the letter expressing the orbital angular momentum state l of the electron (s: l = 0; p: l = 1; d: l = 2; f: l = 3; note that l < n must hold). Convert all the helium lines that you measured in the lab, into energy differences in ev. With these numbers, identify which line is which transition, and add this transition information to your helium line measurement table. Using Figure A.4 is probably too inaccurate to be useful (reading energies off the figure), so it is probably more useful to draw your own version of it with the help of http://physics.nist.gov/physrefdata/handbook/tables/heliumtable5.htm. Explanation of the table: First column starts with 1s, signifying that one electron is in the ground state; the following two symbols then describe the second electron. Second column: The number up and left of the capital letter distinguishes between singlet (1) and triplet (3) state. Fourth column: these numbers can be converted into ev by multiplying with hc. Do not worry about multiple entries for the same energy level due to a very fine splitting (J), and take a suitable average. The upshot of this identification is that you have observed lines both from parahelium and orthohelium (they are both produced in the bulb), and that once separated out, the lines still follow a Bohr-like energy level idea, yet now with differences introduced by the orbital angular momentum l. [1] Thomas, N. C., The early history of spectroscopy, Journal of Chemical Education, 68, 631, doi: 10.1021/ed068p631 (1991) [2] Eisberg and Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles, John Wiley and Sons (1985) [3] National Institute of Standards and Technology, Atomic Spectra Database, http://www.nist.gov. Appendix A. Supporting Figures 6
Figure A.2: Visible spectra of Sun, hydrogen, and other elements. For illustration only; do not rely on the indicated wavelengths. 7
Figure A.3: The photon emission transistions of singly ionized helium, i.e. the He + ion. Figure A.4: The neutral helium atom energy level diagram. 8