THE INDEX OF REFRACTION OF GASSES using a Michelson
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1. Whenever you change gasses completely purge the previous gas from the system by flushing the gas cell several times with the new gas You will know that the old gas has been completely removed when your fringe counts converge to some small range of values When you are finished collecting data unplug the vacuum pump fill the gas cell with air close the gate valves on top of the two gas bottles and turn off the lasers Check the temperature of the room again 19 15 11 11 2015 DATA ANALYSIS The two Helium Neon laser vacuum wavelengths are 543 52 nm green and 632 99 nm red and they correspond to transitions in Neon out of a state 20 66 eV above its ground state to a state 18 38 eV or 18 70 eV above the ground state respectively Neon atoms are excited to the 20 66 eV state by collisions with excited Helium atoms the He atoms are excited by an electric discharge through the gas From your measured fringe count data determine the index of refraction at both laser wavelengths for each of the gasses you tested see prelab problem 1 Use the ideal gas law P Nk T gt N P T where N is the number density of particles in the gas to calculate the indices of refraction with uncertainties at 0 C and 760 Torr Present you results as a table of values for n 1 Use your results and your consideration of prelab problem 3 to calculate the average number of oscillation electrons n and the wavelength corresponding to the oscilla2. 19 3 These oscillating charges will radiate another plane electromagnetic wave of the same frequency and in the same direction as the wave as well as in the opposite direction Feynman s Lectures on Physics Volume I section 30 7 gives an excellent derivation of the radiation from such a sheet of oscillating charge if we have electrons oscillating with the phasor x the electric field phasor of the induced wave will be cf Feynman eqn 30 18 E 2 0 joxa e i 19 4 oC where z is the distance from the sheet and o is the surface number density of the oscillating electrons in the sheet Using 19 3 in the limit that Q gt the sum of the original and induced wave phasors leaving the sheet is 2 Elo z E z E z 1 jE mE 19 5 c 2em o 0 Since the electron surface density in the thin sheet is small we know that Enna lt E Thus the expression in the brackets in 19 5 represents a small phase delay in the wave after passing through the sheet j jo _ 1_ jdg e sincedd 1 19 6 If the sheet thickness is dz the surface density of the electrons in the sheet is given by the volume number density of the molecules N and the number of oscillating electrons molecule Ne SO O Nn dz If another sheet of molecules is added immediately following the first the wave will be further delayed and so on for additional sheets The phase delay per thickness of the combined finite volume of mol
3. Millikan at U Chicago Michelson s interferometer as we implement it here works as follows 1 The coherent source emits light of a well defined wavelength The light is focused into a plane wave beam i e the source image is at co by the collimator 2 The beam enters the beam splitter so that half of its energy passes through to the reference path and the other half is reflected into the signal path 3 The reference path and signal path beams are each reflected by their respective mirrors and returned to the beam splitter 4 The beam splitter sends half of the energy from each returning beam toward a viewing screen and the rest of the energy is sent back toward the light source 5 The waves returning from each path interfere causing the viewing screen to be bright or dark depending on the relative phases of the two waves at the screen 19 6 11 11 2015 If one or both of the mirrors is not quite normal to its incoming beam then the path length of the light from the beam splitter to the mirror and back will vary depending on the lateral position of the ray away from the beam centerline Thus a slight tilt of one mirror will produce an interference pattern at the screen of bright and dark fringes as shown in Figure 2 As the length of one of the two paths or arms of the interferometer changes the relative phase of the light from the two returning beams will change causing the interference pattern to shift If
4. atom is proportional to the field strength Barom Yaron E 19 1 19 2 11 11 2015 In 19 1 Baon is the average dipole moment induced by the field E averaged over the various possible relative orientations of the field and atom and y is very nearly constant for small external fields A similar relationship would hold for the average polarization of a simple molecule subject to a weak external electric field Because the electrons are so much lighter than the nucleus they are the ones that mainly change position as a result of the field not the nucleus In fact because they are much more weakly bound by the nuclear Coulomb force it is reasonable to assume that the displacements of a few outer valence electrons make the dominant contribution to the induced dipole moment 19 1 so that Pum Ne OX 19 2 where ne is the number of electrons in the atom or molecule mainly contributing to the generation of the dipole moment and x is their average displacement relative to the nucleus which is opposite to the direction of E since they are negatively charged The restoring force from the Coulomb attraction of the nucleus grows with an electron s small displacement and this force is very nearly proportional to it leading to the linear relationship 19 1 This implies that if the external field were to suddenly vanish the electrons displacement x could oscillate with simple harmonic motion at some resonant frequency o and
5. operation of the valves Figure 4 2 the initial placement and orientation of the fringes around the photodetector using the interferometer mirror adjustments Figure 8 3 the operation of the counter and the correct setting of the counter trigger threshold Figure 6 and Figure 7 4 the red laser flip mirror to change between laser colors Figure 3 Data Collection A data point consists of a count of the number of fringes through which the interference pattern shifts as you slowly fill or evacuate the gas cell You repeatedly fill and evacuate the cell with one gas use air at first save helium for last recording fringe counts You repeat this process for both the red and green lasers and for each of the gasses available It is important that you accurately interpolate each fringe count value Figure 8 the fringe count is not an integer Get sufficient data for each gas and each color to get a good measurement with uncertainty of the mean fringe count When using gas from a cylinder slowly fill the balloon to the proper volume using the reservoir fill valve attached to the gas cylinder pressure regulator With this valve closed slightly opening the gas cell fill valve will slowly transfer the gas into the cell At equilibrium there should still be some gas in the balloon but its material should be limp and have clearly visible folds so that you know that the pressure in the cell is exactly the same as the ambient room air pressure
6. the fringes shift by one fringe spacing then the round trip path length difference between the two arms has changed by one wavelength of the light Figure 2 A typical interference fringe pattern from the Michelson interferometer The source is a helium neon HeNe laser tuned to emit 543 nm light If the fringe pattern shifts left or right by a distance equal to the fringe spacing then the difference in the round trip path lengths between the signal and reference beams has changed by one wavelength In this experiment the two arms of the Michelson interferometer are each approximately 10 inches long The signal path of the interferometer contains a chamber which can be filled with various gasses or pumped to vacuum the gas cell referred to in Figure 3 and Figure 4 The changing index of refraction of the medium in the cylinder will change the total number of wavelengths of light in the beam traveling along the signal path thus causing the fringe pattern to shift The total shift of the pattern as a gas is evacuated from or released into the cell will allow you to accurately determine the gas s index of refraction Two lasers red and green are available so that the index may be determined at two different wavelengths 19 7 11 11 2015 _ gt Y gt a yX 4 N Red Laser Flip Mirror f Gas Cell N Reference Path 7 rar Signal Path IE hs I eam Stop Mirror 3 Red Laser Flip Mirror up a pF C
7. 19 i THE INDEX OF REFRACTION OF GASSES using a Michelson Interferometer INTRODUCTION THEORY ORIGIN OF THE INDEX OF REFRACTION OF A MATERIAL MEDIUM THE APPARATUS The Michelson interferometer Gas handling system Fringe counting electronics PRELAB PROBLEMS PROCEDURE Initial Setup Data Collection DATA ANALYSIS 11 11 2015 A A A 12 13 13 14 15 19 ii 11 11 2015 19 1 11 11 2015 THE INDEX OF REFRACTION OF GASSES using a Michelson Interferometer INTRODUCTION An electromagnetic wave light propagating past an atom or molecule will induce an oscillating electric polarization of the particle at the same frequency as that of the light This electric dipole oscillation will in turn generate electromagnetic radiation which modifies the original wave If the light is propagating through a gas of atoms or molecules then the induced radiation by these particles will manifest itself as a slight reduction in the phase velocity of the wave as it passes through the gas This effect is the origin of the index of refraction of a transparent material medium In this experiment you will use a very sensitive instrument the Michelson Interferometer to measure the small change in phase velocity of light passing through various gasses The index of refraction depends not only on the composition of the gas but also on the frequency of the light so you will perform measurements using both a red laser and a green lase
8. Timer used to count the number of dark fringes passing the photodetector The proper settings of its controls are shown in this photo its user manual is available here http www sophphx caltech edu Lab_ Equipment Fluke_7261a manual pdf The Channel A input is used for the signal from the photodetector whereas the Channel B input is terminated to ground The Reset button lower left is used to zero the count between measurements A voltmeter is attached to a rear panel monitor port of the counter which shows the trigger level voltage With the Channel A Atten set to X10 as shown the actual trigger level is 10 times greater that the monitor voltage 19 11 11 11 2015 Fringe Motion Figure 8 Accurately determining the actual wavelength shift starting from the photodetector electronics count data demands some experimenter savvy and thoughtfulness The goal should be to determine a wavelength count with a resolution of 0 1 wavelength The left photo shows a desirable starting fringe pattern with the photodetector centered between dark fringes The right photo shows a possible ending fringe pattern In this case the counting electronics by counting the passage of dark fringes will give a count which is effectively rounded to the nearest whole number For the situation shown in the photo with the fringe motion past the detector having been from right to left the fringes stopped moving before the detector quite reached the center between t
9. ecules is therefore d dz and after traveling a finite distance z through such a medium the wave s phasor will be E z E 0 exp i 2 42 i 19 7 ae E 0 exp tfi it A a 19 4 11 11 2015 Note that the spatial phase variation of the plane wave described by 19 7 is that of a wave traveling with a reduced phase velocity c c n The factor n is called the index of refraction of the medium and for a tenuous gas and our simple theory it is given by the formula 2 Nne 2em o 0 a n le An excellent much more thorough derivation of 19 8 is given in Feynman chapter 31 Let s express 19 8 in terms of vacuum wavelengths and the classical electron radius 2nc el E n 2 82x10 m 4e mc 19 9 Nnr 1 1 i n l z 2m A A THE APPARATUS The Michelson interferometer Let s use 19 9 to estimate the order of magnitude of n 1 for air near standard density The number density of the molecules in this case would be 1 mole 22 4 liter 2 7 x 10 m Thus A 80 nm in the ultraviolet A 630 nm red light ne 4 1 Caer 0 Si Nnr 1 1 2 7x10 x4x2 8x10 E CNA ae aa ata S A Se 1 eo E F EETA So the speed of light should be decreased by only about 0 03 in air vs vacuum a small change indeed Even worse the difference in the index of refraction for green light 543nm vs that for red light is only about 2x10 Accurately measuring
10. h is several inches so that a typical fringe count is well over 100 except for helium It can be tedious to count so many fringes especially since the count data should be repeated a few times to improve the accuracy of the measurement A photodetector and associated counting electronics are therefore added to the apparatus to help the experimenter accurately obtain count data The following figures Figure 6 to Figure 8 on page 10 11 briefly describe the electronics and how to properly set up and use the system for accurate fringe counting This clever design is due to Don Skelton the long time Caltech undergraduate physics laboratory manager who retired in 2001 19 10 11 11 2015 Ny Photodetector Amplifier Oscilloscope Amplifier Filter V Increment count as signal rises through threshold Counter Figure 6 left fringe counting electronics The detector is constructed from a silicon photodiode whose output is amplified and filtered to remove noise before input to the oscilloscope and counter right the light intensity from the interferometer causes the photodiode signal to vary sinusoidally as fringes pass across the detector The counter s trigger threshold should be set so that the fringe count is incremented whenever the intensity has begun rising following the passage of a dark fringe as illustrated by the dashed line and the arrow in the right hand graphic Figure 7 The Fluke 7261A Counter
11. he gas in the balloon and the ambient room air surrounding it Thus when the balloon s rubber fabric is relaxed and folds in the fabric are evident then the gas pressure within the balloon is equal to the ambient room air pressure This will also apply to the gas within the interferometer gas cell if the gas fill valve is open and the vacuum valve is closed Pressure Gauge Figure 5 Diagram of the gas handling system controls used to introduce one of the bottled gases CO or helium to the gas Gas Cell Reservoir cell of the interferometer The reservoir fill Gas Gell Fill Valve Fill Valve from Gas valve is used to slightly inflate the balloon Bottle After exhausting the gas cell s contents using the vacuum pump gas from the balloon is expanded through the gas cell fill Gas Cell valve to slowly bring the gas cell up to Vacuum Ae atmospheric pressure If some small amount of excess gas remains in the balloon after the fill valve is opened then to Vacuum the pressure in the cell will equal the Pump ambient room air pressure Balloon Reservoir Differential Pressure Gauge Fringe counting electronics Because the relative difference in n 1 for green and red light is typically less than 1 the number of fringes which must be counted as the interferometer s gas cell is emptied or filled must be fairly large to obtain a reasonable estimate of this difference Thus the gas cell s lengt
12. his system of equations is aC d Cy y a a c Cy 19 10 a a a a Assuming the uncertainties in 2 and r 2 82x10 m may be neglected then the only remaining uncertainties would be in the n 1 values at the two laser wavelengths and the air density determination N k T P Note further that the solution for Ay 1 x does not depend on 27 Nr but only on the laser wavelengths and the values of n and m Should n 1 of a dilute gas be additive that is should n 1 for air be given by a weighted sum of the values of n 1 for N and O neglecting the trace gasses in air 19 13 11 11 2015 PROCEDURE Initial Setup Don t touch or blow on any of the optical surfaces Activate both lasers by turning the keys on their power supplies It may take a few minutes for the green laser to turn on the red laser should turn on after a few seconds If one of the lasers is not emitting light even after several minutes ask your TA for assistance Check for spots of stray laser light on the walls of the room Don t let a laser beam hit you in the eyes Make sure you record the temperature and pressure of the air in the room so that you may accurately calculate the number density of the molecules in the gas cell when filled Use calipers to accurately measure the length of the metal gas cell don t include the thicknesses of the end windows measure just the metal bit see Figure 3 on page 7 Rev
13. hose shown in the photo The right hand photo shows the valves used to control gas flow into and out of the interferometer gas cell A small vacuum pump on the floor behind the gas bottles is used to evacuate the interferometer gas cell The pump should be turned on at the beginning of the experiment and remain active throughout the data collection By opening and closing the vacuum valve shown in the right hand photo in Figure 4 you can control the connection of the gas cell to the pump The fill valve shown in the same photo is used the control the release of gas into the cell The pressure gauge displays the gas cell s pressure relative to ambient atmospheric pressure called the cell s gauge pressure In the photo it shows that the gas cell is near vacuum which would be at about 29 25 inch Hg on the gauge since ambient air pressure in the lab is usually about 743 Torr mm Hg A flexible hose connects the fill valve to the source of gas for the cell It is left open to the room air when air is the desired gas otherwise it is connected to the outlet of the balloon fill valve 19 9 11 11 2015 connected to the desired gas bottle s pressure regulator The flexible hose has a tee fitting which connects a balloon gas reservoir to the fill system The balloon has folds in its rubber fabric so if the folds are not pulled tight by the volume of gas within it then the rubber is relaxed and does not support a pressure difference between t
14. iew Figure 4 and Figure 5 The gas cell fill tube should be open to the ambient room air if necessary disconnect it from any compressed gas bottle Ensure that the gas cell vacuum valve is fully closed clockwise and that the gas cell fill valve is open counterclockwise and then plug in the power to the vacuum pump You should leave the vacuum pump on for the remainder of the experiment Don t let any part of your body or your clothing get near the vacuum pump The drive belt and pulleys can easily remove your fingers or other body parts that you probably want to keep If something falls on the floor near the pump unplug the power to the pump before reaching down to retrieve it Don t completely close the door to the lab room Let fresh air enter the room so that you have oxygen to breathe not just the fumes from the vacuum pump and the CO gas you will exhaust into the room Ensure that the fringe counting electronics is activated and operating properly Your TA can help you get everything turned on If necessary adjust the kinematic mounts of the signal path and 19 14 11 11 2015 reference path mirrors until the interference fringes are clearly visible and are conveniently spaced and aligned with the reference lines on the projection screen Figure 8 Conduct a few trial runs by evacuating and filling the gas cell while checking that the electronics system is properly counting fringes Familiarize yourself with 1 the
15. ollimator and Ng a Sa Cell Beam expander d x e e e OEE Signal Path B Beam Splitter f Mirror Figure 3 Two views of the interferometer used in the experiment The two HeNe laser wavelengths are 543nm green and 633nm red The mirror used to inject the red laser light into the apparatus flips up or down to select the laser color The paths of a central ray are indicated by the dashed lines As the gas cell is filled or emptied the phase length of the signal path changes causing the fringe pattern to shift The lower photo shows where the length of the gas cell should be measured 19 8 11 11 2015 Gas handling system The gasses you will use in this experiment are ambient room air carbon dioxide and helium Figure 4 shows the arrangement of the major components of the gas handling system Fill Pressure Regulator Fill Valve 2 l Balloon d si Fill Valve To Gas Cell i TO Fill Gas Gas Cell CO2 Bottle Vacuum Valve 4 To Vacuum Pump Figure 4 The gas handling components The vacuum pump is on the floor underneath the bench it should be left running for the duration of the experiment The balloon serves as a gas reservoir and also ensures that the gas cell pressure when filled is the same as the ambient air pressure in the room Air from the room s atmosphere carbon dioxide and helium are the available gasses The He and CO2 bottles positions may be interchanged from t
16. r You will use your data to investigate the reasonableness of a simple theory of the atomic origin of the index of refraction and your lab work will give you a bit of experience with modern optical components and techniques of interferometry THEORY ORIGIN OF THE INDEX OF REFRACTION OF A MATERIAL MEDIUM An isolated atom will arrange its electron charge distribution so that the nucleus experiences no net electric field otherwise the nucleus would experience a net electrostatic force causing it change its position relative to the electron distribution and the total energy of the system is minimized In the presence of an external electric field however the original configuration of nucleus and electrons is no longer the equilibrium state since the electrons and the nucleus are subject to externally generated electrostatic forces in opposite directions Thus the nucleus and the electron charge distribution must shift position relative to each other to find a new equilibrium configuration and the electron charge distribution changes shape because the outer electrons feel less Coulomb attraction by the nucleus than do the inner electrons As a result of the shift to a new equilibrium configuration in the presence of the applied field the atom experiences a slight separation of its positive and negative charges so that it acquires an electric dipole moment For relatively small fields applied to a thin gas of atoms the induced dipole moment per
17. such small changes in the speed of light could be a daunting task were we not able to take advantage of the fact that light is a wave A change in phase velocity means that light of a given frequency has a slightly different wavelength and we can very accurately measure this wavelength shift by monitoring the change in the interference pattern between a signal wave whose wavelength is changing and a reference wave whose wavelength remains constant The Michelson Interferometer accomplishes this task 19 5 11 11 2015 Coherent Light Source Collimator C_ gt Mirror Figure 1 A modern version of the Michelson Interferometer A plane wave with a well defined wavelength is produced by the source and collimator It is split and sent down two orthogonal paths by the beam splitter The mirrors reflect the light back toward the beam splitter where it is recombined and sent toward the screen as well as back toward the source Interference fringes are displayed on the screen A schematic representation of the interferometer is shown in Figure 1 This design was invented by Albert Michelson in about 1880 He with Edward Morley used an improved design in 1887 to conduct the famous Michelson Morley experiment which along with other evidence inspired Albert Einstein to develop the theory of special relativity 25 years later Michelson was the first American to win the Nobel Prize in physics and was also the Ph D research advisor of Robert
18. tor resonant frequency A for each of the gasses you tested Your helium data may not be precise enough to see an index of refraction difference between the two laser wavelengths but you should be able to see an effect for both air and CO Does n seem to be related to the total number of electrons in the molecule Does the energy Ey hc 4 seem to be related to the ionization energy of the molecule N2 15 6 eV O2 12 1 eV CO2 13 8 eV He 24 6 eV
19. with some quality factor Q In the presence of a passing light wave with frequency o the electrons will experience an oscillating electric field E which will drive their average displacement at that same frequency since the wavelengths of visible light are 1000 s of times larger than the size of an atom or simple molecule the electrons experience is that of an oscillating field and not a passing wave Thus we have a situation analogous to the driven harmonic oscillator the series RLC circuit of Experiment 2 and the displacement x t can be written as the real part of x o Ee or where the complex phasor x is given by ofh A 2 19 3 mo QM O o You should compare this expression to equation 11 of Experiment 2 page 2 6 As we know from that experiment or a careful examination of 19 3 if lt 1 n Q q for large Q and n gt 3 then x is very nearly real This happens to be the case for visible light and the gasses used in this experiment where is in the ultraviolet and Q is very large Thus from 19 2 Basm oscillates in phase with E t and with an amplitude 1 0 Assume that a plane wave traveling in a direction normal to a thin tenuous sheet of molecules passes through it The oscillating electric field E t of the wave at the position of the sheet will be in the plane of the sheet and will induce oscillations of the electrons in the molecules of the 19 3 11 11 2015 sheet according to
20. wo dark fringes Therefore a count of N from the electronics should be correctly interpolated to N 1 0 7 with an uncertainty of approximately 0 1 fringe 19 12 11 11 2015 PRELAB PROBLEMS 1 If the length of the gas cell is L the index of refraction of the gas is n and the laser vacuum wavelength is Aw then what is the difference in the number of wavelengths of a light ray passing through the cell when it evacuated and when it is filled with the gas What then is the difference in wavelengths for a round trip along the signal path of the interferometer How will this difference relate to the number of interference fringes which pass the photodetector when the gas cell is evacuated or filled with the gas If the interference fringe count for the green laser A 543 52 nm is 181 4 as air slowly fills the gas cell from vacuum to room pressure and the length of the cell is 7 354 inches then what is the measured n 1 If the ambient air temperature and pressure are 22 2 C and 740 Torr for this data then what would be the expected n 1 of air at 0 C and 760 Torr vac Equation 19 9 may be expressed in the form 1 ax x c y where algae y n 0 20 1 and a n 1 c gt Nr A For each gas you would have two such equations one for each laser wavelength with unknowns x from which we get 4 and y from which we get n and coefficients a ci a and c2 Show that the solutions for x and y of t
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