UNIVERSITY OF CALIFORNIA - SANTA CRUZ DEPARTMENT OF PHYSICS PHYS 133 PROFESSOR: SHER Atomic Spectra Benjamin Stahl Lab Partners: Aaron Lopez & Dillon Teal April 2, 2014 Abstract As an introduction to spectroscopy, several experiments were conducted using a diffraction grating-spectrometer, a laser, and several gas lamps. First, the spectral lines of helium were measured and then used to determine the grating constant of 3.330 ± 0.015 µm for the diffraction grating. Next, the spectral lines of hydrogen were measured and then used with the calculated value of the grating constant constant to determine the Rydberg constant, which was found to be 11064109 ± 172161 m 1. Finally, the line for a helium-neon laser was compared with the spectral lines for helium and neon. From this investigation, the wavelength of the helium-neon laser was determined to be 627.00 ± 13.91 nm. The wavelength of the helium-neon laser was found to match one of the spectral lines of neon and to lie in between the "yellow" and "red" lines of helium.
CONTENTS 1 Introduction 3 2 Methods 4 2.1 Helium & the Determination of the Slit Spacing of the Diffraction Grating................ 5 2.2 Hydrogen & the Determination of the Rydberg Constant........................... 5 2.3 Helium-Neon Laser................................................. 5 3 Results 6 3.1 Helium & the Determination of the Slit Spacing of the Diffraction Grating................ 7 3.2 Hydrogen & the Determination of the Rydberg Constant........................... 8 3.3 Helium-Neon Laser................................................. 9 4 Conclusion 10 Appendices 11 A Helium Data 11 B Hydrogen Data 12 C Laser Data 13 D Statistics 14 D.1 z-score........................................................ 14 D.2 p-value........................................................ 14 2
1 INTRODUCTION At the most fundamental level, this experiment was intended to develop a better understanding of the concept of spectroscopy and to add a practical appreciation. Spectroscopy has been, and continues to be an extremely valuable tool in science as it allows for a great deal of information to be discovered. For example the field of Astronomy has benefited immensely from spectroscopy; by looking at the spectra of distant stars and planets, astronomers can infer their chemical composition, relative speed, and many other physical characteristics. Atoms are comprised of a small, dense nucleus which is surrounded by electrons. These electrons occupy quantized energy levels, and thus if the atom is exposed to the appropriate amount of energy the electrons can be excited up to higher energy levels. The amount of time that an electron can stay at a higher energy level varies but eventually the electron will drop down to a lower energy level by releasing energy in the form of a photon. Since the levels are quantized, there are specific energies associated with moving between energy levels and thus there is a photon of specific energy released for a specific transition between energy levels. Since the energy of a photon depends on its wavelength λ, there are photons of specific wavelength released when electrons drop from one energy level to another in an atom. This relationship is shown explicitly in Equation 1.1 where E is the energy difference between two energy levels, h is Planck s constant, c is the speed of light, and λ is the wavelength of the emitted photon: E = hc (1.1) λ These specific wavelengths of light produced by electrons dropping energy levels within an atom are unique and characteristic of that atom (element) and are known as the atom s spectrum. In the case of the hydrogen atom, the Rydberg equation can be used to predict the wavelengths of light produced by an electron transitioning from one level n 1 to another n 2. This relationship is presented in Equation 1.2 where R H is the Rydberg constant (10967758 ± 1 m 1 ): ( ) 1 λ = R 1 H n2 2 1 n1 2 Gas lamps of helium, hydrogen, and neon were utilized to investigate the atomic spectra of these elements. When a sufficient voltage is applied across such a gas lamp, it excites the electrons of the atoms in the gas up to higher energy levels and when they drop back down, photons of characteristic wavelength are released which create the spectrum of the element in the gas. By using a diffraction grating with slit spacing d, the light from the gas lamp can be split into its component wavelengths, which are the wavelengths of light characteristic to that element. The angle θ at which these characteristic wavelengths are scattered from the normal to the diffraction grating can be measured using a spectrometer. The relationship between these quantities 1 is given in Equation 1.3 where m corresponds to the order of the spectral line (for example m = 1 is the first time the spectral line is seen, m = 2 is the second time, and so on as θ is increased). dsinθ = mλ (1.3) cos A laser (which is an acronym for Light Amplification by Stimulated Emission of Radiation) was also used in this experiment. In principle, lasers emit light through the same process as do gas lamps. Lasers however, trap the light and amplify it until a focused, coherent beam is formed and allowed to escape. In particular, a helium-neon laser was utilized in this experiment. Such a laser operates by exciting the helium gas in the laser, then the excited helium transfers energy to neon through collisions. When the neon drops to a lower energy state the amplification process is accomplished due to stimulated emission (where light of just the right energy causes an electron to spontaneously drop energy levels and release a photon of identical energy). Since it is the neon that facilitates the stimulated emission in this type of laser, it is expected that the wavelength of the light from the laser will correspond to a transition between the energy levels of neon, but not necessarily helium. Figure 1.1 contains an energy level diagram for a helium-neon laser. (1.2) 1 The angle is defined in Section 3, but is ultimately of little consequence to the experiments conducted 3
Figure 1.1: Energy level diagram for a helium-neon laser [1] A series of experiments were completed in order to become familiar with atomic spectra and how they can be measured. First, a lamp containing helium was used in conjunction with the known wavelengths of the spectral lines of helium to accurately determine the slit spacing of the diffraction grating used in the experimental apparatus. Next, the spectral lines of hydrogen were measured and from this data and the experimentally determined slit spacing the Rydberg constant was determined. Lastly, a neon lamp and a helium-neon laser were simultaneously viewed through the spectroscope and the spectrum from the lamp was compared to the line from the laser. This process with the laser was then repeated using a helium lamp. 2 METHODS In this experiment, a spectrometer (PASCO Scientific, Model SP-9268A) was used with a diffraction grating (with a printed value of 300 lines/mm) to take measurements of of the spectra of different gas lamps as well as observe a laser. The housing for the different lamp bulbs was put directly in front of the entrance slit of the spectrometer such that the the incident light was roughly normal to the diffraction grating. Then the spectrometer was aligned according to procedures included as a handout with the spectrometer as well as in accordance with those present in the lab manual [2]. Figure 2.1 presents a diagram with the spectrometer and its parts: Figure 2.1: The spectrometer. A: Entrance slit. B: Collimator objective. C: Diffraction grating. D: Telescope objective. E: Plane of cross-hairs. I: Grating table clamping screw. J: Grating table clamping and tangent screws. K: Telescope arm clamping and tangent screws. L: Eyepiece ring. M: Telescope focusing ring. O: Grating table leveling screws. P: Collimator and telescope leveling screws. [2] 4
The quantities to be measured were the angle φ of incident light on the grating to the grating normal and the angle θ of the diffracted beam to the undeflected beam. This angles are presented in Figure 2.2 which shows the geometry of the apparatus. Figure 2.2: The geometry of the apparatus [2] Note that the angles were measured using a Vernier scale on the spectrometer. There were two scales on the instrument, thus for each angle turned, two angles were measured and averaged (these angles will be denoted by the subscripts A and B). In addition, each angle θ to a given spectral line was measured on both sides of the center and then these two angles were averaged to get one angle for each spectral line of a given order (these angles will be denoted by the subscripts 1 and 2). 2.1 HELIUM & THE DETERMINATION OF THE SLIT SPACING OF THE DIFFRACTION GRATING The helium lamp was put into the lamp housing and the angle φ was measured by centering the scope on the central line from the helium and measuring the angle. Next, the angle θ was found by turning to a given spectral line and taking the measurements to find θ 1, then the same spectral line was found on the other side so that θ 2 could be found. This process was repeated for multiple spectral lines up through the third order (m = 3). Following methods of analysis that will be outlined in Section 3.1 this data was used to determine the grating constant d of the diffraction grating. 2.2 HYDROGEN & THE DETERMINATION OF THE RYDBERG CONSTANT The hydrogen lamp was put into the lamp housing and then the angles φ, θ 1, & θ 2 were measured in the same fashion as was outlined in Section 2.1. Following methods of analysis that will be outline in Section 3.2 this data was used to determine the Rydberg constant R H. 2.3 HELIUM-NEON LASER The neon lamp was put into the lamp housing and then the helium-neon laser was set up such that the beam from the laser hit the neon lamp at a height corresponding to roughly halfway up the entrance slit of the spectrometer. 5
Next, a piece of paper was attached at this location on the lamp so that the light from the laser was scattered into the entrance slit. When properly adjusted, this configuration allows for the spectrum of the neon lamp to to be visible in the upper half of the scope, and for the laser light to be visible in the lower half of the scope. Once such a configuration was accomplished the angle φ was measured in the same fashion as was outlined in Section 2.1. Then the angles θ 1, & θ 2 were measured for the line from the laser as well as any lines from the neon lamp that were in close proximity to the line from the laser when viewed through the spectrometer. This process was then repeated with the helium lamp instead of the neon lamp. 3 RESULTS All angles turned to were read off the scales on the instrument and then converted into an angle between the center and the spectral line according to the following rules where the subscripts A and B denote which of the two scales on the instrument were used and the subscripts 1 and 2 denote which side of the center the scope was on for a given spectral line. θ 1A = 180 o 0 raw reading θ 1B = 360 o 0 raw reading θ 2A = raw reading 180 o 0 θ 2B = raw reading (3.1) Each angle found using the vernier scale was measured in degrees ( o ) and minutes ( ). Prior to performing any rigorous analysis, these angles were converted into decimal degrees according to the relationship outlined in Equation 3.2 using the arbitary angle α: ( 1 o ) α decimal degrees = α degrees + α minutes (3.2) 60 Based on the vernier scale, which had markings corresponding to 1 minute, the uncertainty of each angular measurement on the scale was presumed to be ±0.5. In degrees this uncertainty is: ±0.0083 o. Errors were propagated using Equation 3.3 which is developed from the concepts of differential calculus where u = f (x, y,...) and x, y,... are random independent variables [2]. ( ) f 2 ( ) f 2 σ 2 u = σ 2 x x + σ 2 y +... (3.3) y As discussed in Section 2, each angle turned to (ie φ, θ 1, & θ 2 ) was measured on the two scales of the instrument (denoted by the subscripts A and B) and then averaged. Thus, for an arbitrary 2 angle α, the value is calculated as shown in Equation 3.4. α = 1 2 (α A + α B ) (3.4) The variance σ 2 α is found using Equation 3.3 where the function u = f (x, y,...) is equivalent to Equation 3.4 and the variables are α A and α B. Using the uncertainties σ αa = σ αb = ±0.0083 o results in σ 2 α = ±(3.472 10 5 ) o2. The average angle θ between the center and a spectral line was found according to Equation 3.5: θ = 1 2 (θ 1 + θ 2 ) (3.5) Where θ 1 and θ 2 are the measured angles on the left and right sides to the spectral line. Note that θ 1 and θ 2 are corrected according to φ (where φ is subtracted from one and added to the other to compensate for the grating not being perfectly normal to the incident light). 2 The arbitrary angle α is used for generality, but it can be replaced by φ as well as θ 1 & θ 2 when analyzing these quantities because the mathematical processes are identical 6
The variance σ 2 is found using Equation 3.3 where the function u = f (x, y,...) is equivalent to Equation 3.5 and θ the variables are θ 1 and θ 2. Using the variances σ 2 = σ 2 = ±(3.472 10 5 ) o2 results in σ 2 θ 1 θ 2 θ = ±(1.3761 10 5 ) o2. The angle is defined in Equation 3.6: = 1 2 (θ 1 θ 2 ) (3.6) The variance σ 2 is found using Equation 3.3 where the function u = f (x, y,...) is equivalent to Equation 3.6 and the variables are θ 1 and θ 2. Using the variances σ 2 = σ 2 = ±(3.472 10 5 ) o2 results in σ 2 θ 1 θ 2 = ±(1.3761 10 5 ) o2. Measurements were made very carefully and precisely in this experiment, which resulted in the angle being very small (a fraction of a degree) for each spectral line. Thus, the small angle approximation cos 1 is employed to simplify Equation 1.3 into the following expression presented in Equation 3.7: dsinθ = mλ (3.7) 3.1 HELIUM & THE DETERMINATION OF THE SLIT SPACING OF THE DIFFRACTION GRATING With the helium lamp in place and the apparatus properly aligned, the angle φ was turned to. Reading both scales resulted in φ A = 5 and φ B = 4. Running this data through the process outlined in Section 3 yielded: φ = 0.075 o ± 0.006 o. Next, the spectral lines were measured with the spectrometer. The raw angular measurements made with the spectrometer for helium are presented in Table A.1 in Appendix A. Next, the raw angular measurements were processed according to the methodology outlined in Section 3 in order to determine the angles θ and as well as the associated variances. The resulting quantities are presented in Table A.2 in Appendix A. To determine the grating constant d of the diffraction grating, Equation 3.7 was solved for d. The result is presented as follows in Equation 3.8: d = mλ (3.8) sinθ Using Equation 3.8 d was calculated using the order m and the angle θ which are presented in Table A.2 (in Appendix A) in conjunction with the known wavelengths of helium which are presented in Figure A.1 (in Appendix A). The variance σ 2 was found using Equation 3.3 where the function u = f (x, y,...) is equivalent to Equation 3.8. d Resulting from this analysis, the expression presented in Equation 3.9 was used to calculate σ 2 d : σ 2 d = m2 λ 2 cos 2 θ sin 4 σ 2 θ (3.9) θ The grating constant d and its variance were calculated for each spectral line of helium that was measured. The results of these calculations are presented in Table A.3 in Appendix A. A weighted average d of the grating constant was found according to Equation 3.10: n d k k=1 d = σ 2 k n 1 k=1 σ 2 k (3.10) The variance σ 2 was found according to Equation 3.11: d σ 2 d = 1 n k=1 1 σ 2 k (3.11) Using Equations 3.10 & 3.11 with the values of d and σ 2 presented in Table A.3 in Appendix A the grating constant d was determined to be 3.330 ± 0.015 µm. 7
The diffraction grating had a printed value of 300 lines/mm, which corresponds to a grating constant of 3.3 µm. Comparing the experimentally determined grating constant with the printed value according to the statistical methods outlined in Appendix D results in a z-score of 0.2 and a p-value of 0.176. 3.2 HYDROGEN & THE DETERMINATION OF THE RYDBERG CONSTANT With the hydrogen lamp in place and the apparatus properly aligned, the angle φ was turned to. Reading both scales resulted in φ A = 8 and φ B = 9. Running this data through the process outlined in Section 3 yielded: φ = 0.142 o ± 0.006 o. Next, the spectral lines were measured with the spectrometer. The raw angular measurements made with the spectrometer for helium are presented in Table B.1 in Appendix B. Next, the raw angular measurements were processed according to the methodology outlined in Section 3 in order to determine the angles θ and as well as the associated variances. The resulting quantities are presented in Table B.2 in Appendix B. To determine the Rydberg constant, Equation 1.2 was solved for R H. The result is presented as follows in Equation 3.12 ( ) R H = 1 1 λ n2 2 1 1 n1 2 (3.12) To determine the wavelengths of the observed spectral lines of hydrogen, Equation 3.7 was solved for λ. The result is presented in Equation 3.13: λ = dsinθ (3.13) m Equation 3.13 is then substituted into Equation 3.12 resulting in Equation 3.14 which is presented here: ( ) R H = m 1 dsinθ n2 2 1 1 n1 2 (3.14) Since the grating constant d was constructed as a weighted average, the expression for R H is rewritten to avoid biasing the result by weighing d again. This revision is presented in Equation 3.15 Where f (θ) = x = ( ) 1 m 1 sinθ 1. n2 2 n1 2 R H = f (θ) d (3.15) The variance σ 2 x is found using Equation 3.3 where the function u = f (x, y,...) is equivalent to x = f (θ). Resulting from this analysis, the expression presented in Equation 3.16 was used to calculate σ 2 x : ( ) σ 2 x = 1 n2 2 1 2 m 2 cos 2 θ n1 2 sin 4 θ Note that since all of the observed spectral lines of hydrogen were at wavelengths visible to the human eye, they were part of the Balmer series. In the Balmer series, all of the transitions between energy levels terminate at the second energy level (that is, n 2 = 2 for the Balmer series). The upper energy levels n 1 were determined according to the colors of the observed spectral lines [3]. The function x = f (θ) and its variance were calculated for each first order (m = 1) spectral line of hydrogen that was measured. The results of these calculations are presented in Table B.3 in Appendix B. (3.16) 8
A weighted average x = f (θ) of the function x = f (θ) was found according to Equation 3.17: n x k k=1 x = f (θ) = σ 2 k n 1 k=1 σ 2 k (3.17) The variance σ 2 was found according to Equation 3.18: x σ 2 x = 1 n k=1 1 σ 2 k (3.18) Using Equations 3.17 & 3.18 with the values of x and σ 2 x presented in Table B.3 in Appendix B the following results were found: x = 36.839 and σ 2 x = 0.302. Using the calculated values and variances of x = f (θ) and d, R H was calculated using Equation 3.15. The variance σ 2 R H was found using Equation 3.3 where the function u = f (x, y,...) is equivalent to Equation 3.15. Resulting from this analysis, the expression presented in Equation 3.19 was used to calculate σ 2 R H : σ 2 = 1 R H d 2 σ2 x + x 2 Utilizing Equations 3.15 & 3.19 with the appropriate values, the Rydberg constant was determined to be 11064109± 172161 m 1 using only the first order 3 spectral lines of hydrogen. d 4 σ2 d The Rydberg constant has an accepted value of 10967758 ± 1 m 1. Comparing the experimentally determined Rydberg constant to the accepted value according to the statistical methods outlined in Appendix D results in a z-score of -0.560 and a p-value of 0.425. (3.19) 3.3 HELIUM-NEON LASER With the neon lamp in place and the helium-neon laser properly placed so that its light would scatter off of the paper and into the entrance slit of the spectrometer as was discussed more thoroughly in Section 2.3, the angle φ was turned to. Reading both scales resulted in φ A = 8 and φ B = 9. Running this data through the process outlined in Section 3 yielded: φ = 0.142 o ± 0.006 o. Next, the line from the laser was measured with the spectrometer. Then a spectral line from the neon lamp that appeared to exactly match the line from the laser was measured with the spectrometer. Next, the neon lamp was swapped out and the helium lamp was put into the apparatus. The line from the laser appeared to be between two of the spectral lines of helium, thus both lines were measured using the spectrometer. The raw angular measurements of the laser line, the neon spectral line, and the two helium spectral lines are presented in Table C.1 in Appendix C. Next, the raw angular measurements were processed according to the methodology outlined in Section 3 in order to determine the angles θ and as well as the associated variances. The resulting quantities are presented in Table C.2 in Appendix C. Using Equation 3.13, the wavelength λ was calculated for each entry in Table C.2 using the grating constant d that was calculated in Section 3.1. In addition, the variance σ 2 was found using Equation 3.3 where the function u = f (x, y,...) is equivalent to Equation 3.13. Resulting from this analysis, the expression presented in Equation 3.20 was used to calculate σ 2 λ λ : σ 2 λ = sin2 θ m 2 σ2 d + d 2 cos 2 θ m 2 σ 2 θ (3.20) 3 There is less confidence behind the second order lines because they were much fainter, thus they were left out of the analysis 9
The wavelength and its variance were calculated for each relevant line. The results of these calculations are presented in Table C.3 in Appendix C. Thus, the wavelengths of the laser and relevant spectral lines of helium and neon were determined to be: λ laser = 627.00 ± 13.91nm λ He: yellow = 581.04 ± 13.90nm λ He: red = 658.84 ± 13.91nm λ Ne = 627.00 ± 13.91nm As expected, the wavelength of the laser falls between the wavelengths of the two spectral lines of helium that were measured. In addition, the wavelength of the laser matches that of the spectral line of neon that was measured as was expected. The emitted wavelength of helium-neon laser has a theoretically predicted value of 632.8 nm. Comparing the experimentally determined wavelength of the helium-neon laser to the theoretical value according to the statistical methods outlined in Appendix D results in a z-score of 0.417 and a p-value of 0.323. 4 CONCLUSION In this lab, a series of experiments were conducted with the goal of of better understanding and appreciating atomic spectroscopy and the way in which lasers operate. This was accomplished by investigating the spectral lines of gas lamps containing helium, hydrogen, and neon, as well as the line produced by a helium-neon laser using a diffraction grating-spectrometer. First, the spectral lines of helium were measured using the spectrometer and a helium gas lamp. These angular measurements were used in conjunction with the known wavelengths of helium spectral lines to determine the grating constant of the diffraction grating. This grating constant was found to be 3.330 ± 0.015 µm. The diffraction grating had "300 lines/mm" printed on it which corresponds to a diffraction constant of 3.3 µm. This "printed" value fits within the uncertainty on the experimental value and results in a z-score of 0.2 and a p-value of 0.176. Next, the spectral lines of hydrogen were measured using the spectrometer and a hydrogen gas lamp. These angular measurements were used in conjunction with the experimentally determined grating constant to determine the Rydberg constant to be 11064109 ± 172161 m 1. The accepted value in science for the Rydberg constant is 10967758 ± 1 m 1, which fits within the uncertainty on the experimental value and results in a z-score of -0.560 and a p-value of 0.425. The line from a helium-neon laser was observed with the spectrometer, and from these angular measurements the wavelength of the light was determined to be 627.00 ± 13.91 nm. When compared against the spectral lines of neon, it was found to match one of the lines as was predicted. Next, it was compared to the spectral lines of helium and found to to lie between the "yellow" and "red" lines. The theoretically predicted wavelength of the visible light emitted by a helium-neon laser is 632.8 nm which fits within the uncertainty on the experimental value and results in a z-score of 0.417 and a p-value of 0.323. As is evident in the results, atomic spectroscopy can be used to make high precision measurements leading to a deeper understanding of atomic behavior and the fundamental theory behind the operation of lasers. In particular, the experiments conducted in this lab allowed for the observation of characteristic wavelengths of light associated with the spectra of atoms, for the determination of the grating constant of a diffraction grating, for the experimental determination of the Rydberg constant, and for the measurement of the wavelength of a laser as well as the identification of which of its component elements has the energy level transition corresponding to the observed wavelength of the laser. 10
APPENDICES A HELIUM DATA color m θ 1A )( o )) θ 1A )(')) θ 1B )( o )) θ 1B )(')) θ 2A )( o )) θ 2A )(')) θ 2B )( o )) θ 2B )(')) purple 1 172 30 352 26 187 49 7 43 green 1 171 33 351 30 188 40 8 37 red 1 168 41 348 40 191 30 11 30 purple 2 164 47 344 48 195 30 15 30 green 2 162 58 342 55 197 35 17 30 red 2 157 0 337 3 203 35 23 37 purple 3 156 55 336 54 203 36 23 40 green 3 153 51 333 51 206 48 26 49 Table A.1: Raw angle measurements for the helium spectral lines that were investigated of order 1-3 color m θ σ 2 θ Δ σ 2 Δ purple 1 7.650 1.7361E505 0.042 1.7361E505 green 1 8.558 1.7361E505 0.008 1.7361E505 red 1 11.413 1.7361E505 0.013 1.7361E505 purple 2 15.354 1.7361E505 0.071 1.7361E505 green 2 17.300 1.7361E505 0.167 1.7361E505 red 2 23.288 1.7361E505 0.238 1.7361E505 purple 3 23.363 1.7361E505 0.196 1.7361E505 green 3 26.479 1.7361E505 0.254 1.7361E505 Table A.2: Calculated angles θ & with associated variances for helium color m λ'(a) d'(m) σ 2 d purple 1 4387.929 3.296E:06 1.046E:14 green 1 4921.931 3.307E:06 8.385E:15 red 1 6678.15 3.375E:06 4.853E:15 purple 2 4387.929 3.314E:06 2.529E:15 green 2 4921.931 3.310E:06 1.961E:15 red 2 6678.15 3.378E:06 1.070E:15 purple 3 4387.929 3.320E:06 1.025E:15 green 3 4921.931 3.312E:06 7.673E:16 Table A.3: Calculated grating constant d and its variance with the known wavelengths in angstroms (A) of the helium spectral lines taken from Figure A.1 11
Figure A.1: Known wavelengths of helium with the wavelengths relevant to this experiment identified by color [4] B HYDROGEN DATA 12
color m θ 1A )( o )) θ 1A )(')) θ 1B )( o )) θ 1B )(')) θ 2A )( o )) θ 2A )(')) θ 2B )( o )) θ 2B )(')) purple 1 172 36 352 37 187 30 7 30 green 1 171 46 351 48 188 25 8 24 red 1 168 55 348 55 191 28 11 26 green 2 163 24 343 23 196 49 16 50 red 2 157 15 337 16 203 4 23 4 Table B.1: Raw angle measurements for the hydrogen spectral lines that were investigated of order 1-2 color m θ σ 2 θ Δ σ 2 Δ purple 1 7.446 1.7361E405 0.088 1.7361E405 green 1 8.313 1.7361E405 0.046 1.7361E405 red 1 11.267 1.7361E405 0.042 1.7361E405 green 2 16.717 1.7361E405 0.033 1.7361E405 red 2 22.904 1.7361E405 0.021 1.7361E405 Table B.2: Calculated angles θ & with associated variances for hydrogen color m x'='f(θ) σ 2 x purple 1 36.746 1.373 green 1 36.890 1.107 red 1 36.852 0.594 Table B.3: Calculated values of the function x = f (θ) and its variance for each first order spectral line of hydrogen that was observed C LASER DATA source m θ 1A +( o )+ θ 1A +(')+ θ 1B +( o )+ θ 1B +(')+ θ 2A +( o )+ θ 2A +(')+ θ 2B +( o )+ θ 2B +(')+ laser 1 169 18 349 16 191 0 10 59 he+(yellow) 1 170 0 350 0 190 8 10 4 he+(red) 1 168 41 348 40 191 30 11 30 neon 1 169 18 349 16 191 0 10 59 Table C.1: Raw angular measurements of the laser and relevant spectral lines of helium and neon 13
source θ σ 2 θ Δ σ 2 Δ laser 10.854 1.7361E705 0.063 1.7361E705 he9(yellow) 10.050 1.7361E705 0.025 1.7361E705 he9(red) 11.413 1.7361E705 0.013 1.7361E705 neon 10.854 1.7361E705 0.063 1.7361E705 Table C.2: Calculated angles θ & with associated variances for the laser and relevant spectral lines of helium and neon source m λ)(m) σ 2 λ)(m 2 ) σ λ )(m) laser 1 6.270E607 1.935E616 1.391E608 he)(yellow) 1 5.810E607 1.933E616 1.390E608 he)(red) 1 6.588E607 1.936E616 1.391E608 neon 1 6.270E607 1.935E616 1.391E608 Table C.3: Calculated values of the wavelength and its variance for the laser and relevant spectral line of helium and neon D STATISTICS D.1 Z-SCORE The z-score z was calculated according to Equation D.1 where X is the quantity being compared, µ is the sample mean, and σ is the sample standard deviation: z = X µ (D.1) σ D.2 P-VALUE After calculating the z-score z it can also be useful to determine the p-value (which will be done using the same variables as in Appendix Section D.1), which is best explained with respect to Figure D.1: Figure D.1: Plot of the normal distribution, with the z-score shown on the horizontal axis 14
Graphically, the p-value is the lightly shaded area under the normal distribution between z and z. That is, the p-value is the probability of of any particular X falling between z and z. The p-value p can be analytically determined according to Equation D.2, where the function being integrated is the normal distribution: p = 1 z e z2 2 dz (D.2) 2π Applying symmetry to Equation D.2 leads to a slight simplification, resulting in Equation D.3: z p = 2 z e z2 2 dz 2π 0 (D.3) Equation D.3 was used throughout with Wolfram Alpha to compute the p-values as needed. REFERENCES [1] "Helium-neon Laser." Ksu.edu. Kansas State University, n.d. Web. 15 Mar. 2014. [2] Brown. Physics 133 Lab Manual. Winter 2014. UCSC, 2013. Print. [3] "Measured Hydrogen Spectrum." Hydrogen Spectrum. N.p., n.d. Web. 17 Mar. 2014. [4] Lide, David R., ed. Handbook of Chemistry and Physics: CRC Handbook. 75th ed. Boca Raton, FL: CRC, 1993. Print. 15