UNIVERSITY OF CALIFORNIA - SANTA CRUZ

DEPARTMENT OF PHYSICS

PHYS 133PROFESSOR: SHER

Atomic Spectra

Benjamin StahlLab 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 determinethe grating constant of 3.330 ± 0.015 µm for the diffraction grating. Next, the spectral lines of hydrogen weremeasured and then used with the calculated value of the grating constant constant to determine the Rydbergconstant, which was found to be 11064109±172161 m−1. Finally, the line for a helium-neon laser was comparedwith the spectral lines for helium and neon. From this investigation, the wavelength of the helium-neon laser wasdetermined to be 627.00± 13.91 nm. The wavelength of the helium-neon laser was found to match one of thespectral lines of neon and to lie in between the "yellow" and "red" lines of helium.

CONTENTS

1 Introduction 3

2 Methods 42.1 Helium & the Determination of the Slit Spacing of the Diffraction Grating . . . . . . . . . . . . . . . . 52.2 Hydrogen & the Determination of the Rydberg Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3 Helium-Neon Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 Results 63.1 Helium & the Determination of the Slit Spacing of the Diffraction Grating . . . . . . . . . . . . . . . . 73.2 Hydrogen & the Determination of the Rydberg Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Helium-Neon Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Conclusion 10

Appendices 11

A Helium Data 11

B Hydrogen Data 12

C Laser Data 13

D Statistics 14D.1 z-score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14D.2 p-value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2

1 INTRODUCTION

At the most fundamental level, this experiment was intended to develop a better understanding of the conceptof spectroscopy and to add a practical appreciation. Spectroscopy has been, and continues to be an extremelyvaluable tool in science as it allows for a great deal of information to be discovered. For example the field ofAstronomy 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 quan-tized energy levels, and thus if the atom is exposed to the appropriate amount of energy the electrons can beexcited up to higher energy levels. The amount of time that an electron can stay at a higher energy level varies buteventually the electron will drop down to a lower energy level by releasing energy in the form of a photon. Sincethe levels are quantized, there are specific energies associated with moving between energy levels and thus thereis a photon of specific energy released for a specific transition between energy levels. Since the energy of a pho-ton depends on its wavelength λ, there are photons of specific wavelength released when electrons drop from oneenergy level to another in an atom. This relationship is shown explicitly in Equation 1.1 where 4E is the energydifference between two energy levels, h is Planck’s constant, c is the speed of light, and λ is the wavelength of theemitted photon:

4E = hc

λ(1.1)

These specific wavelengths of light produced by electrons dropping energy levels within an atom are unique andcharacteristic 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 producedby an electron transitioning from one level n1 to another n2. This relationship is presented in Equation 1.2 whereRH is the Rydberg constant (10967758±1 m−1):

1

λ= RH

(1

n22

− 1

n21

)(1.2)

Gas lamps of helium, hydrogen, and neon were utilized to investigate the atomic spectra of these elements. Whena sufficient voltage is applied across such a gas lamp, it excites the electrons of the atoms in the gas up to higherenergy levels and when they drop back down, photons of characteristic wavelength are released which create thespectrum 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 wave-lengths, which are the wavelengths of light characteristic to that element. The angle θ at which these characteristicwavelengths are scattered from the normal to the diffraction grating can be measured using a spectrometer. Therelationship between these quantities1 is given in Equation 1.3 where m corresponds to the order of the spec-tral 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 isincreased).

d si nθ = mλ

cos4 (1.3)

A laser (which is an acronym for Light Amplification by Stimulated Emission of Radiation) was also used in thisexperiment. In principle, lasers emit light through the same process as do gas lamps. Lasers however, trap thelight and amplify it until a focused, coherent beam is formed and allowed to escape. In particular, a helium-neonlaser was utilized in this experiment. Such a laser operates by exciting the helium gas in the laser, then the excitedhelium transfers energy to neon through collisions. When the neon drops to a lower energy state the amplificationprocess is accomplished due to stimulated emission (where light of just the right energy causes an electron tospontaneously drop energy levels and release a photon of identical energy). Since it is the neon that facilitatesthe stimulated emission in this type of laser, it is expected that the wavelength of the light from the laser willcorrespond to a transition between the energy levels of neon, but not necessarily helium. Figure 1.1 contains anenergy level diagram for a helium-neon laser.

1The angle 4 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 mea-sured. First, a lamp containing helium was used in conjunction with the known wavelengths of the spectral lines ofhelium 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 spacingthe Rydberg constant was determined. Lastly, a neon lamp and a helium-neon laser were simultaneously viewedthrough the spectroscope and the spectrum from the lamp was compared to the line from the laser. This processwith 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 (witha printed value of 300 lines/mm) to take measurements of of the spectra of different gas lamps as well as observea laser. The housing for the different lamp bulbs was put directly in front of the entrance slit of the spectrometersuch that the the incident light was roughly normal to the diffraction grating. Then the spectrometer was alignedaccording to procedures included as a handout with the spectrometer as well as in accordance with those presentin 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 ob-jective. 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 levelingscrews. 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 theangle θ of the diffracted beam to the undeflected beam. This angles are presented in Figure 2.2 which shows thegeometry 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 theinstrument, thus for each angle turned, two angles were measured and averaged (these angles will be denotedby the subscripts A and B). In addition, each angle θ to a given spectral line was measured on both sides of thecenter and then these two angles were averaged to get one angle for each spectral line of a given order (these angleswill 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 thecentral 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 θ1, then thesame spectral line was found on the other side so that θ2 could be found. This process was repeated for multiplespectral 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 gratingconstant 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, θ1, & θ2 were measured in the samefashion 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 Rydbergconstant RH .

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 fromthe 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 intothe entrance slit. When properly adjusted, this configuration allows for the spectrum of the neon lamp to to bevisible in the upper half of the scope, and for the laser light to be visible in the lower half of the scope. Once sucha configuration was accomplished the angle φ was measured in the same fashion as was outlined in Section 2.1.Then the angles θ1, & θ2 were measured for the line from the laser as well as any lines from the neon lamp thatwere 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 centerand the spectral line according to the following rules where the subscripts A and B denote which of the two scaleson the instrument were used and the subscripts 1 and 2 denote which side of the center the scope was on for agiven spectral line.

θ1A = 180o0′− raw reading

θ1B = 360o0′− raw reading

θ2A = raw reading−180o0′

θ2B = raw reading

(3.1)

Each angle found using the vernier scale was measured in degrees (o) and minutes (′). Prior to performing any rig-orous analysis, these angles were converted into decimal degrees according to the relationship outlined in Equa-tion 3.2 using the arbitary angle α:

αdecimal degrees =αdegrees +αminutes

(1o

60′

)(3.2)

Based on the vernier scale, which had markings corresponding to 1 minute, the uncertainty of each angular mea-surement 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 whereu = f (x, y, . . . ) and x, y, . . . are random independent variables [2].

σ2u =

(∂ f

∂x

)2

σ2x +

(∂ f

∂y

)2

σ2y + . . . (3.3)

As discussed in Section 2, each angle turned to (ie φ, θ1, & θ2) was measured on the two scales of the instrument(denoted by the subscripts A and B) and then averaged. Thus, for an arbitrary2 angle α, the value is calculated asshown in Equation 3.4.

α= 1

2(αA +αB ) (3.4)

The variance σ2α is found using Equation 3.3 where the function u = f (x, y, . . . ) is equivalent to Equation 3.4 and

the variables are αA and αB . Using the uncertainties σαA =σαB =±0.0083o

results in σ2α =±(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(θ1 +θ2) (3.5)

Where θ1 and θ2 are the measured angles on the left and right sides to the spectral line. Note that θ1 and θ2 arecorrected according toφ (whereφ is subtracted from one and added to the other to compensate for the grating notbeing perfectly normal to the incident light).

2The arbitrary angle α is used for generality, but it can be replaced by φ as well as θ1 & θ2 when analyzing these quantities because themathematical processes are identical

6

The variance σ2θ

is found using Equation 3.3 where the function u = f (x, y, . . . ) is equivalent to Equation 3.5 and

the variables are θ1 and θ2. Using the variances σ2θ1

=σ2θ2

=±(3.472×10−5)o2results in σ2

θ=±(1.3761×10−5)o2

.

The angle 4 is defined in Equation 3.6:

4= 1

2(θ1 −θ2) (3.6)

The variance σ24 is found using Equation 3.3 where the function u = f (x, y, . . . ) is equivalent to Equation 3.6 and

the variables are θ1 and θ2. Using the variances σ2θ1

=σ2θ2

=±(3.472×10−5)o2results in σ2

4 =±(1.3761×10−5)o2.

Measurements were made very carefully and precisely in this experiment, which resulted in the angle 4 being verysmall (a fraction of a degree) for each spectral line. Thus, the small angle approximation cos4≈ 1 is employed tosimplify Equation 1.3 into the following expression presented in Equation 3.7:

d si nθ = 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 scalesresulted in φA = 5′ and φB = 4′. Running this data through the process outlined in Section 3 yielded: φ= 0.075o ±0.006o .

Next, the spectral lines were measured with the spectrometer. The raw angular measurements made with thespectrometer 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 orderto determine the angles θ and 4 as well as the associated variances. The resulting quantities are presented inTable 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 presentedas follows in Equation 3.8:

d = mλ

si nθ(3.8)

Using Equation 3.8 d was calculated using the order m and the angle θ which are presented in Table A.2 (in Ap-pendix A) in conjunction with the known wavelengths of helium which are presented in Figure A.1 (in Appendix A).

The variance σ2d was found using Equation 3.3 where the function u = f (x, y, . . . ) is equivalent to Equation 3.8.

Resulting from this analysis, the expression presented in Equation 3.9 was used to calculate σ2d :

σ2d = m2λ2cos2θ

si n4θσ2θ (3.9)

The grating constant d and its variance were calculated for each spectral line of helium that was measured. Theresults 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:

d =∑n

k=1dk

σ2k∑n

k=11σ2

k

(3.10)

The variance σ2d

was found according to Equation 3.11:

σ2d= 1∑n

k=11σ2

k

(3.11)

Using Equations 3.10 & 3.11 with the values of d and σ2d presented in Table A.3 in Appendix A the grating constant

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 statisticalmethods 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 bothscales resulted in φA = 8′ and φB = 9′. Running this data through the process outlined in Section 3 yielded: φ =0.142o ±0.006o .

Next, the spectral lines were measured with the spectrometer. The raw angular measurements made with thespectrometer 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 orderto determine the angles θ and 4 as well as the associated variances. The resulting quantities are presented inTable B.2 in Appendix B.

To determine the Rydberg constant, Equation 1.2 was solved for RH . The result is presented as follows in Equa-tion 3.12

RH = 1

λ

(1

n22

− 1

n21

)−1

(3.12)

To determine the wavelengths of the observed spectral lines of hydrogen, Equation 3.7 was solved for λ. The resultis presented in Equation 3.13:

λ= d si nθ

m(3.13)

Equation 3.13 is then substituted into Equation 3.12 resulting in Equation 3.14 which is presented here:

RH = m

d si nθ

(1

n22

− 1

n21

)−1

(3.14)

Since the grating constant d was constructed as a weighted average, the expression for RH is rewritten to avoidbiasing the result by weighing d again. This revision is presented in Equation 3.15

RH = f (θ)

d(3.15)

Where f (θ) = x = msi nθ

(1

n22− 1

n21

)−1

.

The variance σ2x 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 σ2x :

σ2x =

(1

n22

− 1

n21

)−2m2cos2θ

si n4θ(3.16)

Note that since all of the observed spectral lines of hydrogen were at wavelengths visible to the human eye, theywere part of the Balmer series. In the Balmer series, all of the transitions between energy levels terminate at thesecond energy level (that is, n2 = 2 for the Balmer series). The upper energy levels n1 were determined accordingto 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 thatwas measured. The results of these calculations are presented in Table B.3 in Appendix B.

8

A weighted average x = f (θ) of the function x = f (θ) was found according to Equation 3.17:

x = f (θ) =∑n

k=1xk

σ2k∑n

k=11σ2

k

(3.17)

The variance σ2x

was found according to Equation 3.18:

σ2x = 1∑n

k=11σ2

k

(3.18)

Using Equations 3.17 & 3.18 with the values of x and σ2x presented in Table B.3 in Appendix B the following results

were found: x = 36.839 and σ2x= 0.302.

Using the calculated values and variances of x = f (θ) and d , RH was calculated using Equation 3.15.

The variance σ2RH

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 σ2RH

:

σ2RH

= 1

d2σ

2x +

x2

d4σ

2d

(3.19)

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 order3 spectral lines of hydrogen.

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 statisticalmethods outlined in Appendix D results in a z-score of -0.560 and a p-value of 0.425.

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 thepaper and into the entrance slit of the spectrometer as was discussed more thoroughly in Section 2.3, the angle φwas turned to. Reading both scales resulted inφA = 8′ andφB = 9′. Running this data through the process outlinedin Section 3 yielded: φ= 0.142o ±0.006o .

Next, the line from the laser was measured with the spectrometer. Then a spectral line from the neon lamp thatappeared 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 laserappeared 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 arepresented in Table C.1 in Appendix C.

Next, the raw angular measurements were processed according to the methodology outlined in Section 3 in orderto determine the angles θ and 4 as well as the associated variances. The resulting quantities are presented inTable 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 thatwas calculated in Section 3.1.

In addition, the variance σ2λ

was found using Equation 3.3 where the function u = f (x, y, . . . ) is equivalent to Equa-tion 3.13. Resulting from this analysis, the expression presented in Equation 3.20 was used to calculate σ2

λ:

σ2λ =

si n2θ

m2 σ2d + d 2cos2θ

m2 σ2θ (3.20)

3There 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 pre-sented 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:

λlaser = 627.00±13.91nm

λHe: yellow = 581.04±13.90nm

λHe: red = 658.84±13.91nm

λNe = 627.00±13.91nm

As expected, the wavelength of the laser falls between the wavelengths of the two spectral lines of helium that weremeasured.

In addition, the wavelength of the laser matches that of the spectral line of neon that was measured as was ex-pected.

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 accordingto 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 appreciatingatomic spectroscopy and the way in which lasers operate. This was accomplished by investigating the spectrallines of gas lamps containing helium, hydrogen, and neon, as well as the line produced by a helium-neon laserusing a diffraction grating-spectrometer.

First, the spectral lines of helium were measured using the spectrometer and a helium gas lamp. These angularmeasurements were used in conjunction with the known wavelengths of helium spectral lines to determine thegrating constant of the diffraction grating. This grating constant was found to be 3.330±0.015 µm. The diffractiongrating 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 an-gular measurements were used in conjunction with the experimentally determined grating constant to determinethe Rydberg constant to be 11064109± 172161 m−1. The accepted value in science for the Rydberg constant is10967758± 1 m−1, which fits within the uncertainty on the experimental value and results in a z-score of -0.560and a p-value of 0.425.

The line from a helium-neon laser was observed with the spectrometer, and from these angular measurementsthe wavelength of the light was determined to be 627.00±13.91 nm. When compared against the spectral lines ofneon, it was found to match one of the lines as was predicted. Next, it was compared to the spectral lines of heliumand found to to lie between the "yellow" and "red" lines. The theoretically predicted wavelength of the visible lightemitted by a helium-neon laser is 632.8 nm which fits within the uncertainty on the experimental value and resultsin 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 adeeper 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 associatedwith the spectra of atoms, for the determination of the grating constant of a diffraction grating, for the experimen-tal determination of the Rydberg constant, and for the measurement of the wavelength of a laser as well as theidentification of which of its component elements has the energy level transition corresponding to the observedwavelength of the laser.

10

APPENDICES

A HELIUM DATA

color mθ1A)(o))(raw)

θ1A)('))(raw)

θ1B)(o))(raw)

θ1B)('))(raw)

θ2A)(o))(raw)

θ2A)('))(raw)

θ2B)(o))(raw)

θ2B)('))(raw)

purple 1 172 30 352 26 187 49 7 43green 1 171 33 351 30 188 40 8 37red 1 168 41 348 40 191 30 11 30

purple 2 164 47 344 48 195 30 15 30green 2 162 58 342 55 197 35 17 30red 2 157 0 337 3 203 35 23 37

purple 3 156 55 336 54 203 36 23 40green 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 σ2θ Δ σ2

Δ

purple 1 7.650 1.7361E505 0.042 1.7361E505green 1 8.558 1.7361E505 0.008 1.7361E505red 1 11.413 1.7361E505 0.013 1.7361E505

purple 2 15.354 1.7361E505 0.071 1.7361E505green 2 17.300 1.7361E505 0.167 1.7361E505red 2 23.288 1.7361E505 0.238 1.7361E505

purple 3 23.363 1.7361E505 0.196 1.7361E505green 3 26.479 1.7361E505 0.254 1.7361E505

Table A.2: Calculated angles θ & 4 with associated variances for helium

color m λ'(A) d'(m) σ2d

purple 1 4387.929 3.296E:06 1.046E:14green 1 4921.931 3.307E:06 8.385E:15red 1 6678.15 3.375E:06 4.853E:15

purple 2 4387.929 3.314E:06 2.529E:15green 2 4921.931 3.310E:06 1.961E:15red 2 6678.15 3.378E:06 1.070E:15

purple 3 4387.929 3.320E:06 1.025E:15green 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 thehelium 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 mθ1A)(o))(raw)

θ1A)('))(raw)

θ1B)(o))(raw)

θ1B)('))(raw)

θ2A)(o))(raw)

θ2A)('))(raw)

θ2B)(o))(raw)

θ2B)('))(raw)

purple 1 172 36 352 37 187 30 7 30green 1 171 46 351 48 188 25 8 24red 1 168 55 348 55 191 28 11 26green 2 163 24 343 23 196 49 16 50red 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 σ2θ Δ σ2

Δ

purple 1 7.446 1.7361E405 0.088 1.7361E405green 1 8.313 1.7361E405 0.046 1.7361E405red 1 11.267 1.7361E405 0.042 1.7361E405green 2 16.717 1.7361E405 0.033 1.7361E405red 2 22.904 1.7361E405 0.021 1.7361E405

Table B.2: Calculated angles θ & 4 with associated variances for hydrogen

color m x'='f(θ) σ2x

purple 1 36.746 1.373green 1 36.890 1.107red 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 hydrogenthat was observed

C LASER DATA

source mθ1A+(o)+(raw)

θ1A+(')+(raw)

θ1B+(o)+(raw)

θ1B+(')+(raw)

θ2A+(o)+(raw)

θ2A+(')+(raw)

θ2B+(o)+(raw)

θ2B+(')+(raw)

laser 1 169 18 349 16 191 0 10 59he+(yellow) 1 170 0 350 0 190 8 10 4he+(red) 1 168 41 348 40 191 30 11 30neon 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 σ2θ Δ σ2

Δ

laser 10.854 1.7361E705 0.063 1.7361E705he9(yellow) 10.050 1.7361E705 0.025 1.7361E705he9(red) 11.413 1.7361E705 0.013 1.7361E705neon 10.854 1.7361E705 0.063 1.7361E705

Table C.2: Calculated angles θ & 4 with associated variances for the laser and relevant spectral lines of helium andneon

source m λ)(m) σ2λ)(m2) σλ)(m)

laser 1 6.270E607 1.935E616 1.391E608he)(yellow) 1 5.810E607 1.933E616 1.390E608he)(red) 1 6.588E607 1.936E616 1.391E608neon 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 heliumand 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 samplemean, and σ is the sample standard deviation:

z = X −µσ

(D.1)

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 samevariables 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

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Graphically, the p-value is the lightly shaded area under the normal distribution between −z and z. That is, thep-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 isthe normal distribution:

p = 1p2π

∫ z

−ze−

z22 d z (D.2)

Applying symmetry to Equation D.2 leads to a slight simplification, resulting in Equation D.3:

p = 2p2π

∫ z

0e−

z22 d z (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.

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