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Note to 8.13 Students 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 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, 7 7 1 7 7 1 Rh =1.0966 10 0.0002 10 m− and Rd=1.0970 10 0.0003 10 m− . Also, the mass ratio between× the deuteron± and× proton was calculated to× be 2±.35 0.45.× The energy separation 3 ± 3 between the sodium doublet peaks was measured, ∆E =2.1 10− 0.3 10− 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 transitions between a higher energy level and nf = 2 are known as the Balmer series. The measurements of spectra of hydrogen and other By knowing this relationship between an atom’s rest one electron atoms were essential building blocks for mass and its spectra, the mass of the nucleus can calcu- modern atomic theory. In 1885, Johann Balmer noticed lated from its spectral data. In this lab, the measured a relationship between the emitted wavelengths of light spectra of hydrogen and deuterium and the known mass from hydrogen. Around five years later, Johannes Ryd- of a hydrogen nucleus (Mp = 1amu) is used to calcu- berg built offBalmer’s work by generalizing a relation- late the mass of a deuteron. From the Rydberg formula ship for the emitted wavelengths of light for particles (Equation 1), the mass of the deuteron is given by other than hydrogen. In 1913, Niels Bohr published a theory describing the quantized nature of the atom, in- MeλH spired by and incorporating Rydberg’s work. M = (2) d M λ ∆λ ∆λM During the development of quantum mechanics a more e H − − e accurate theory of the atom was created which could ex- where Md is the mass of the deuteron, Me is the mass plain other properties of emission spectra. These the- of an electron, λH is the measured hydrogen line, and ories take into account not only an electron’s principal ∆λ is the difference between the measured hydrogen and quantum number, but also its angular momentum and deuterium line. its spin, which can be used to explain spectral properties like doublets. B. Sodium Fine Structure II. THEORY Within the same principal energy level in an atom, electrons can have different amounts of energy depend- A. The Rydberg Formula Related to Hydrogen ing upon their angular quantum number (l). This num- ber gives information about the shape of the electron’s When an electron moves from a higher energy state to orbit and is often represented by the letters and num- a lower energy state, a photon of a specific wavelength bers s=0, p=1, d=2, f=3. The energy difference between is emitted. The Rydberg formula relates the change in these states is caused by the spin of the electron interact- energy levels of this transitioning electron to the wave- ing with the atom’s internal magnetic field. If the spin length of the emitted photon for a given particle: is parallel to the orbital dipole, the energy state will be lower than if it is anti-paralell. In this lab we calculated 1 1 the energy splitting in the sodium atom between the two = µR ( ) (1) 3P states using the equation λ ∞ 1/n2 1/n2 f − 0 where λ is the wavelength, µ = Mnucleus/(Mnucleus + hc hc 7 1 ∆E = (3) Melectrons) is the rest mass, R =1.097373 10 m− λ − λ ∞ × 1 2 is the Rydberg constant, and n0 and nf are the initial and final energy states of the electron. For hydrogen, the where ∆E is the difference in energy, h is Plank’s con- stant, c is the speed of light, and λ1 and λ2 are the dou- blet wavelengths. Due to the conservation of angular momentum, when ∗Electronic address: [email protected] a photon is emitted, it possesses a quantized, non-zero 2 value of angular momentum. This means that the angu- monochromator was 1800 grooves/mm, the entrance and lar momentum quantum number can only change by 1, exit slit widths were 10.0 micrometers, and the PMT so the transitions that can occur are restricted. Figure± voltage was kept at 900 Volts. 1 shows the allowed transitions between energy states in The step size of the monochromator could be varied the sodium atom. to obtain! di"ff#erent$%&' levels()&'*+"+,)-+ of detail. For this./ experiment,(+- most measurements were taken using a step size of 0.05 Angstroms. *+"+,)-+./(+- 45$('23$( 8+",/9&'*$--+-'; @)+(+.13($03$&-' 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 In order to account for the systematic error in the op- FIG. 1: Energy Levels in Sodium. Allowed transitions be- tics of our setup, the emission spectrum of an Oriel 65130 tween energy states in sodium are represented by lines con- necting the energy levels involved in the transition. The num- mercury lamp was measured. The measured spectral bers are equal to the corresponding wavelength emitted in the lines for mercury were extracted by locating the wave- transition from higher to lower energy. lengths 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 be- III. EXPERIMENTAL SETUP tween 2974.85 and 7037.35 Angstroms, which were com- pared to the established values listed in the CRC Hand- book for Chemistry and Physics. A fit was then created A. Apparatus from this comparison to interpolate for other wavelengths not yet measured. Many different fits were tried includ- The spectrum of different gasses were measured us- ing linear through seventh degree polynomials, sine, and ing a research grade monochromator. The path of light exponential. The Harttman method, a technique tradi- through the monochromator is represented in Figure 2. tionally used to account for systematic error in optical Wavelengths from the incoming light are separated using systems involving prisms, was also tested. The fit which spherical concave mirrors and a reflection grating. Cer- minimized the difference between the measured and es- tain wavelengths will be directed toward the next concave tablished values was mirror, depending on the orientation of the grating. The light that is directed toward the second concave 6 2 10 3 mirror is focused onto the exit slit and then measured by y = 51+1.033x 8.165 10− x +7.708 10− x (4) a photomultiplier tube (PMT). The reading is then sent − − × × to the computer where the measurement is displayed in where x represents the measured lines and y represents LabVIEW. The path of light through the monochromator the expected lines. Figure 3 plots the measured versus is represented in Figure 2. The grating density in the expected lines, showing the obtained data and calculated 3 2 fit. The reduced chi squared of the fit was χr = 0.99998, with degrees of freedom ν = 7 which leads to a proba- bility of P=0.57. The standard error of the fit was σ = 1.97. 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. 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. IV. RESULTS AND DISCUSSION A. Balmer Series for Hydrogen and Deuterium Isotope Shift In order to calculate the Rydberg constants for hydro- gen and deuterium, measurements of the six Balmer lines were obtained for both lamps.
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