Note to 8.13 students:

Transcription

1 Note to 8.13 students: Feel free to look at this paper for some suggestions about the lab, but please reference/acknowledge me as if you had read my report or spoken to me in person. Also note that this is only one way to do the lab and data analysis, and there are nearly an infinite number of other things to do that would be better. I made some mistakes doing this lab. Here are a couple I found (and some more tips): Use a smaller step and get more precise data than what we had. Then fit a curve, don t just pick out a peak. The Hg calibration is actually something like a sin wave instead of a cubic. Also don t follow the old version of Melissionos word for word. They use an optical system with a prism oppose to a diffraction grating. We took data strategically so we could use unweighted fits The mystery tube was actually N, whoops my bad. We later found a drawer full of tubes and compared the colors. It was obviously N. Again whoops, it was our first experiment okay?! 1

2 Optical Spectroscopy Rachel Bowens-Rubin MIT Department of Physics (Dated: July 12, 2010) The spectra of mercury, hydrogen, deuterium, sodium, and an unidentified mystery tube were measured using a monochromator to determine diffrent properties of their spectra. The spectrum of mercury was used to calibrate for error due to optics in the monochronomator. The measurements of the Balmer series lines were used to determine the Rydberg constants for hydrogen and deuterium, R h = ± m 1 and R d = ± m 1. Also, the mass ratio between the deuteron and proton was calculated to be 2.35 ± The energy separation between the sodium doublet peaks was measured, E = ± ev. The prominent lines and the overall shape of the spectrum of the mystery tube were used to identify that the tube contains neon. Overall, the results were consistent with quantum mechanical theories. I. INTRODUCTION The measurements of spectra of hydrogen and other one electron atoms were essential building blocks for modern atomic theory. In 1885, Johann Balmer noticed a relationship between the emitted wavelengths of light from hydrogen. Around five years later, Johannes Rydberg built off Balmer s work by generalizing a relationship for the emitted wavelengths of light for particles other than hydrogen. In 1913, Niels Bohr published a theory describing the quantized nature of the atom, inspired by and incorporating Rydberg s work. During the development of quantum mechanics a more accurate theory of the atom was created which could explain other properties of emission spectra. These theories take into account not only an electron s principal quantum number, but also its angular momentum and its spin, which can be used to explain spectral properties like doublets. II. THEORY A. The Rydberg Formula Related to Hydrogen When an electron moves from a higher energy state to a lower energy state, a photon of a specific wavelength is emitted. The Rydberg formula relates the change in energy levels of this transitioning electron to the wavelength of the emitted photon for a given particle: 1 λ = µr 1 ( 1/n 2 f ) (1) 1/n2 0 where λ is the wavelength, µ = M nucleus /(M nucleus + M electrons ) is the rest mass, R = m 1 is the Rydberg constant, and n 0 and n f are the initial and final energy states of the electron. For hydrogen, the Electronic address: rbowru@mit.edu transitions between a higher energy level and n f = 2 are known as the Balmer series. By knowing this relationship between an atom s rest mass and its spectra, the mass of the nucleus can calculated from its spectral data. In this lab, the measured spectra of hydrogen and deuterium and the known mass of a hydrogen nucleus (M p = 1amu) is used to calculate the mass of a deuteron. From the Rydberg formula (Equation 1), the mass of the deuteron is given by M d = M e λ H M e λ H λ λm e (2) where M d is the mass of the deuteron, M e is the mass of an electron, λ H is the measured hydrogen line, and λ is the difference between the measured hydrogen and deuterium line. B. Sodium Fine Structure Within the same principal energy level in an atom, electrons can have different amounts of energy depending upon their angular quantum number (l). This number gives information about the shape of the electron s orbit and is often represented by the letters and numbers s=0, p=1, d=2, f=3. The energy difference between these states is caused by the spin of the electron interacting with the atom s internal magnetic field. If the spin is parallel to the orbital dipole, the energy state will be lower than if it is anti-paralell. In this lab we calculated the energy splitting in the sodium atom between the two 3P states using the equation E = hc λ 1 hc λ 2 (3) where E is the difference in energy, h is Plank s constant, c is the speed of light, and λ 1 and λ 2 are the doublet wavelengths. Due to the conservation of angular momentum, when a photon is emitted, it possesses a quantized, non-zero

3 2 value of angular momentum. This means that the angular momentum quantum number can only change by ±1, so the transitions that can occur are restricted. Figure 1 shows the allowed transitions between energy states in the sodium atom. monochromator was 1800 grooves/mm, the entrance and exit slit widths were 10.0 micrometers, and the PMT voltage was kept at 900 Volts. The step size of the monochromator could be varied to obtain different levels of detail. For this experiment, most measurements were taken using a step size of 0.05 Angstroms. *+"+,)-+./(+- 45$('23$( A1B& 6-/($"7!"01('23$( ' <$7)('$"'+='./">'?/9&3&"7()#!! 8+",/9&'*$--+-': FIG. 2: Inside the Monochromator: The instrument works by first focusing the light from a lamp onto the input slit. The light passes through the input slit and hits a concave mirror where it is collimated. The collimated light is reflected and dispersed after it hits a grating, which can be turned at different angles. In this figure, the grating is turned a large amount, causing the longer wavelength light to be reflected into the photomultiplier tube. B. Mercury Calibration FIG. 1: Energy Levels in Sodium. Allowed transitions between energy states in sodium are represented by lines connecting the energy levels involved in the transition. The numbers are equal to the corresponding wavelength emitted in the transition from higher to lower energy. III. EXPERIMENTAL SETUP A. Apparatus The spectrum of different gasses were measured using a research grade monochromator. The path of light through the monochromator is represented in Figure 2. Wavelengths from the incoming light are separated using spherical concave mirrors and a reflection grating. Certain wavelengths will be directed toward the next concave mirror, depending on the orientation of the grating. The light that is directed toward the second concave mirror is focused onto the exit slit and then measured by a photomultiplier tube (PMT). The reading is then sent to the computer where the measurement is displayed in LabVIEW. The path of light through the monochromator is represented in Figure 2. The grating density in the In order to account for the systematic error in the optics of our setup, the emission spectrum of an Oriel mercury lamp was measured. The measured spectral lines for mercury were extracted by locating the wavelengths of maximum intensity. The measurement error in our determination of the peak was equal to one-half of our step size. Eleven mercury lines were measured in the range between and Angstroms, which were compared to the established values listed in the CRC Handbook for Chemistry and Physics. A fit was then created from this comparison to interpolate for other wavelengths not yet measured. Many different fits were tried including linear through seventh degree polynomials, sine, and exponential. The Harttman method, a technique traditionally used to account for systematic error in optical systems involving prisms, was also tested. The fit which minimized the difference between the measured and established values was y = x x x 3 (4) where x represents the measured lines and y represents the expected lines. Figure 3 plots the measured versus expected lines, showing the obtained data and calculated

4 3 fit. The reduced chi squared of the fit was χ 2 r = , with degrees of freedom ν = 7 which leads to a probability of P=0.57. The standard error of the fit was σ = FIG. 3: Mercury Data with Fit (Expected Wavelength vs Measured Wavelength): The data is plotted by the blue points, and the calculated fit is red. The error bars are nearly invisible due to their small size. FIG. 4: Example Peak from Deuterium Lamp (Intensity vs Measured Wavelength): The peak on left is the delta line measured from Deuterium, and the peak on the right is the same line from the hydrogen in the tube. The calibrated wavelength is measured in Angstroms. IV. RESULTS AND DISCUSSION A. Balmer Series for Hydrogen and Deuterium Isotope Shift In order to calculate the Rydberg constants for hydrogen and deuterium, measurements of the six Balmer lines were obtained for both lamps. Figure 4 shows an example of the raw data obtained for the Delta line (n 0 =6) using the deuterium lamp. Once the data were obtained, they were calibrated using the cubic fit obtained from the mercury data (Equation 4). The calibrated data was then multiplied by the refractive index of air to obtain the value of the wavelength had it been measured in a vacuum. The Rydberg formula (Equation 1) was applied to the data and graphed to find the Rydberg constant. Figure 5 shows this graph for hydrogen. The fit was generated using the method of least squares. The calculated value of the Rydberg constant for hydrogen was R h = ± m 1 compared to the expected value of R h = m 1. The calculated value of the Rydberg constant for deuterium was R d = ± m 1 compared to the expected value of R d = m 1. Both of these measurements were within one standard deviation of the expected value. FIG. 5: Hydrogen Balmer Lines with Fit ( 1 vs 1 ): λ 1/4 1/n 2 0 The slope of the line of best fit is equal to the Rydberg constant for Hydrogen. The error bars are nearly invisible due to their small size. A similar analysis was performed for Deuterium. B. Sodium Fine Structure Eight peaks in the sodium spectrum were measured and calibrated. Figure 6 is a plot of the measured sodium spectrum. Table 2 shows the calibrated wavelengths, differences in wavelengths, and energy separation for each transition that was used to determine the energy differ-

5 4 ence between the two 3P states. In addition to the sodium peaks, many extra lines were measured in the spectral data. These extra lines closely matched the CRC values for argon. Overall, the measured value for the difference in energy for the 3P level of sodium was E = ± ev. This value is within one standard deviation of the expected value found of ev. FIG. 7: Mystery Spectrum (Intensity vs Calibrated Wavelength): Measurements were taken between and 630.0nm of an unknown lamp in order to identify its contents. The mystery spectrum has two regions of higher intensity within the data set. In each of these higher intensity regions, the most pronounced lines were found and used to compare to known emission spectra. FIG. 6: Sodium Spectrum (Intensity vs Measured Wavelength): The dots mark the sodium doublets used to determine the energy separation. Many of the unmarked lines are due to the argon from the lamp. The wavelength was measured in angstroms. Table 1: Energy Change in Sodium Doublets: Trans λ 1 (A)* λ 2 (A)* λ(a)** Energy (ev) 3p5s (2.1 ± 0.06) s3p (2.1 ± 0.07) p4d (2.2 ± 0.08) p6s (2.1 ± 0.09) p5d (2.1 ± 0.10) p7s (2.1 ± 0.11) p6d (2.3 ± 0.11) p7d (2.2 ± 0.12) 10 3 *Error in λ 1 (A) & λ 2 = 1.97 A **Error in λ = 2.78A C. Mystery Lamp In addition to hydrogen, deuterium, and sodium, the spectrum of an unlabeled lamp was taken in order to identify which element it contains. Data were obtained from to Angstroms, and then calibrated using the fit created from the mercury data. Figure 7 shows the measured mystery spectrum data. The most prominent peaks and the overall shape of the spectrum were used to determine which element was in the tube. To help identify the tube, several factors were considered to narrow the search. First, the visible color of the mystery tube was similiar to the color of hydrogen and deuterium but was slightly redder. Second, the company from which Junior Lab orders the mercury calibration tubes only makes four others of the same style according to their website and costumer service agent: argon, krypton, neon and xenon. Although these factors narrowed the the search to four elements, other elements were considered to account for the possibility that Junior Lab recieved the tube from another source. The spectrum was compared to the values listed for prominent peaks of different elements listed in the CRC Handbook for Chemistry and Physics and the Typical Spectra of Oriel Spectral Calibration Lamps provided by the company. It was noticed after some time that the spectrum shared some basic characteristics to neon. Two of the four lines were within one standard deviation from the CRC values for neon, and the other two were also close. The overall spectrum shape also matched the spectrum of neon. The two high intensity regions between 330nm to 360nm and 600nm to 630nm and the low intensity region between 460nm to 600nm are also present for neon. Table 2: Mystery Peaks and Known Values of Neon Peaks Measured λ (nm) CRC (nm) Oriel (nm) ± ± ± ± Even though most of the spectrum of the mystery lamp can be explained by the contents of neon, the range between the wavelengths 365nm to 450nm can not. In fact, this region can not be explained by any of the lamps made by Oriel. More data and searching is required to deter-

6 5 mine what is causing this spectral pattern. The most probable cause would be another element besides neon in the tube. V. SUMMARY The measured values for the Rydberg constant for hydrogen and deuterium, the ratio of the mass of a deutron to proton, and energy separation between the two 3P states of sodium were found. Table 3 summarizes these results. The four determined values were within one standard deviation of the expected value, indicating a consistency with quantum theory. Table 3: Optical Spectroscopy Summary Measured λ Expected R h ( ± ) 10 7 m m 1 R d ( ± ) 10 7 m m 1 M d /M p 2.35 ± E (2.1 ± 0.3) 10 3 ev ev Most of our error was due to an imperfect calibration using mercury. Two ways to reduce this error include using a smaller step size for measurements and obtaining more points. The peak could also have been determined more precisely by fitting the peak using the Cauchy- Lorentz distribution with the Cauchy probably density function. The spectrum of the mystery tube resembles the spectrum of neon in the location of prominent peaks and in overall shape. One region was not similar to neon, which indicates there may be another gas in the tube in addition to neon. Because the resemblance was not perfect, it is my recommendation that more data should be obtained before labeling the tube. Acknowledgments I would like to acknowledge Kathryn Decker French for taking data with me, Burak Alver for spending his time to teach me more about error analysis, and Sid Creutz for proof reading my paper. Melissionos, Experiments in Modern Physics (Academic Press, 1966). MIT Physics Department, Junior lab written report notes (2007). CRC. Handbook for Chemistry and Physics. French and Taylor. An Introduction to Quantum Physics (Norton, 1978). Georgia State University. Hyperphysicshttp://hyperphysics.phyastr.gsu.edu/hbase/quantum/