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 number 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 constant, c is the speed of light, and λ1 and λ2 are the doublet 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. *+"+,)-+./(+-

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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 compared 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 using 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 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 (n0=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 1 1 the value of the wavelength had it been measured in a FIG. 5: Hydrogen Balmer Lines with Fit ( λ vs 1/4 1/n2 ): − 0 vacuum. The slope of the line of best fit is equal to the Rydberg con- The Rydberg formula (Equation 1) was applied to the stant for Hydrogen. The error bars are nearly invisible due data and graphed to find the Rydberg constant. Figure to their small size. A similar analysis was performed for Deu- 5 shows this graph for hydrogen. The fit was generated terium. using the method of least squares. The calculated value of the Rydberg constant for hy- 7 7 1 B. Sodium Fine Structure drogen was Rh =1.0966 10 0.0002 10 m− com- × ± × 7 1 pared to the expected value of Rh =1.0968 10 m− . The calculated value of the Rydberg constant× for deu- Eight peaks in the sodium spectrum were measured 7 7 1 terium was Rd =1.0970 10 0.0003 10 m− com- and calibrated. Figure 6 is a plot of the measured sodium × ± × 7 1 pared to the expected value of Rd =1.0971 10 m− . spectrum. Table 2 shows the calibrated wavelengths, dif- Both of these measurements were within one× standard ferences in wavelengths, and energy separation for each deviation of the expected value. transition that was used to determine the energy differ- 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 3 for the 3P level of sodium was ∆E = 2.1 10− 0.3 3 × ± × 10− eV. This value is within one standard deviation of 3 the expected value found of 2.1 10− eV. ×

FIG. 7: Mystery Spectrum (Intensity vs Calibrated Wave- length): Measurements were taken between 200.0 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.

To help identify the tube, several factors were consid- ered 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 FIG. 6: Sodium Spectrum (Intensity vs Measured Wave- tubes only makes four others of the same style according length): The dots mark the sodium doublets used to deter- to their website and costumer service agent: argon, kryp- mine the energy separation. Many of the unmarked lines are ton, neon and xenon. Although these factors narrowed due to the argon from the lamp. The wavelength was measured in angstroms. the the search to four elements, other elements were con- sidered to account for the possibility that Junior Lab recieved the tube from another source. The spectrum was compared to the values listed for Table 1: Energy Change in Sodium Doublets: prominent peaks of different elements listed in the CRC Trans λ1 (A)* λ2 (A)* ∆λ(A)** ∆Energy (eV) Handbook for Chemistry and Physics and the Typical 3p5s 6160.10 6166.62 6.53 (2.1 0.06) 10 3 Spectra of Oriel Spectral Calibration Lamps provided by ± × −3 3s3p 5893.96 5899.86 5.90 (2.1 0.07) 10− the company. It was noticed after some time that the ± × 3 3p4d 5683.84 5685.49 5.68 (2.2 0.08) 10− spectrum shared some basic characteristics to neon. Two ± × 3 3p6s 5149.00 5150.49 4.44 (2.1 0.09) 10− of the four lines were within one standard deviation from ± × 3 3p5d 4978.81 4983.05 4.24 (2.1 0.10) 10− the CRC values for neon, and the other two were also ± × 3 3p7s 4748.36 4752.19 3.83 (2.1 0.11) 10− close. The overall spectrum shape also matched the spec- ± × 3 3p6d 4664.96 4669.00 4.03 (2.3 0.11) 10− trum of neon. The two high intensity regions between ± × 3 330nm to 360nm and 600nm to 630nm and the low in- 3p7d 4495.06 4498.68 3.62 (2.2 0.12) 10− ± × tensity region between 460nm to 600nm are also present for neon. *Error in λ1 (A) & λ2 = 1.97 A **Error in ∆λ = 2.78A Table 2: Mystery Peaks and Known Values of Neon Peaks Measured λ (nm) CRC (nm) Oriel (nm) C. Mystery Lamp 337.5 0.2 337.8 337.0 354.0 ± 0.2 354.2 352.1 ± In addition to hydrogen, deuterium, and sodium, the 358.0 0.2 357.4 359.4 ± spectrum of an unlabeled lamp was taken in order to 607.3 0.2 607.4 607.4 ± identify which element it contains. Data were obtained from 2000.0 to 6300.0 Angstroms, and then calibrated Even though most of the spectrum of the mystery lamp using the fit created from the mercury data. Figure 7 can be explained by the contents of neon, the range be- shows the measured mystery spectrum data. The most tween the wavelengths 365nm to 450nm can not. In fact, prominent peaks and the overall shape of the spectrum this region can not be explained by any of the lamps made were used to determine which element was in the tube. by Oriel. More data and searching is required to deter- 5 mine what is causing this spectral pattern. The most using a smaller step size for measurements and obtaining probable cause would be another element besides neon more points. The peak could also have been determined in the tube. more precisely by fitting the peak using the Cauchy- Lorentz distribution with the Cauchy probably density function. V. SUMMARY The spectrum of the mystery tube resembles the spectrum of neon in the location of prominent peaks and in The measured values for the Rydberg constant for hy- overall shape. One region was not similar to neon, which drogen and deuterium, the ratio of the mass of a deu- indicates there may be another gas in the tube in addition tron to proton, and energy separation between the two to neon. Because the resemblance was not perfect, it is 3P states of sodium were found. Table 3 summarizes my recommendation that more data should be obtained these results. The four determined values were within before labeling the tube. one standard deviation of the expected value, indicating a consistency with quantum theory.

Table 3: Optical Spectroscopy Summary Measured λ Expected 7 1 7 1 Rh (1.0966 0.0002) 10 m− 1.0967 10 m− ± × 7 1 × 7 1 Acknowledgments R (1.0970 0.0003) 10 m− 1.0971 10 m− d ± × × Md/Mp 2.35 0.45 2.00 ± 3 3 ∆E (2.1 0.3) 10− eV 2.1 10− eV I would like to acknowledge Kathryn Decker French for ± × × taking data with me, Burak Alver for spending his time Most of our error was due to an imperfect calibration to teach me more about error analysis, and Sid Creutz using mercury. Two ways to reduce this error include for proof reading my paper.

Melissionos, Experiments in Modern Physics (Academic French and Taylor. An Introduction to Quantum Press, 1966). Physics (Norton, 1978). MIT Physics Department, Junior lab written report notes Georgia State University. Hyperphysic- (2007). shttp://hyperphysics.phyastr.gsu.edu/hbase/quantum/ CRC. Handbook for Chemistry and Physics.