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S-2 Investigating the Fine Structure of H and D using Fabry-Perot Interferometry

Austin Yu LiuSchool of Applied and Engineering Physics, Cornell University

Ithaca, New York 14850, USA

(Dated: November 29, 2014)

The purpose of this experiment is to measure the fine-structure splitting in the H and D linesin units of wavenumber, and in so doing obtain a value for the fine structure constant. The mea-

surements are also used to determine the mass ratio of deuterium to hydrogen. The measurementsobtained give a value of the fine-structure splitting to be 0.3160.016 cm1 for hydrogen, and to be0.308 0.013 cm1 for deuterium. A value of = 7.04 7 103 from hydrogen fine-splitting anda value of = 7.00 7 103 from the deuterium fine-splitting was obtained. In addition, a valueof mD

mH= 1.85 0.38 for the mass ratio of deuterium to hydrogen was obtained. The uncertainties

above are the statistical uncertainty from the measurements.The experiment was performed using Fabry-Perot interferometry to obtain an image of the inter-

ference fringes from an ionized gas discharge. Subsequently, a charge-coupled device (CCD) camerawas used to obtain an image of the fringes for data analysis.

I. INTRODUCTION

The purpose of this report is to outline the background

and rationale for investigating the fine-structure of theH and D lines, as well as to discuss the relevant re-sults obtained from my investigation of the fine structure.The experiment seeks to determine the the fine-structuresplitting in the H and D lines in units of wavenumber.The dimensionless fine-structure constant can then becalculated from the value of the fine-structure splitting.The interference pattern of the fringes are then used todetermine the mass ratio of deuterium to hydrogen.

The experiment uses a Fabry-Perot etalon for a highspectral resolution of interference fringes. In the experi-ment, the light from a ionized gas discharge of mercury,deuterium and hydrogen is passed through a slit, allowing

for interference fringes to be produced from the etalon.The etalon consists of two thinly silvered plates that arehighly reflective. The various reflected rays then inter-fere, as shown in FIG. 2.

The fine structure arises from the relativistic correctionto the kinetic energy of the electron and the spin-orbitcoupling due to the interaction of the electrons spin withthe charge of the proton. Due to the substantial amountof Doppler broadening in the setup, we do not expect toobserve the Lamb shift [1], which is roughly an order ofmagnitude less than the shift predicted by the relativis-tic correction and spin-orbit coupling. The mass ratioof deuterium to hydrogen is obtained by considering thedifference in the Doppler broadening of the fringes of thehydrogen and deuterium spectra. However, the equip-ment used does not allow for the measurement of thetemperature or pressure in the Woods tube, which af-fects the Doppler broadening.

II. THEORETICAL BACKGROUND

As early as the 19th century, the doublet structure ofthe H lines had been noticed [2]. The first group to

do so was Michelson and Morley, who performed theirexperiment with a Michelson interferometer in 1887 [2].

In 1913, Bohr obtained an expression for the al-

lowed energies of the hydrogen atom in a somewhatserendipituous[3] manner. This result was later put onmuch firmer footing in 1924 after the Schrodinger equa-tion was proposed. Solving the Schrodinger equation forthe hydrogen atom gives the same energies as those pre-dicted by Bohr [3].

While this expression explained the large scale transi-tions where electrons transition between different princi-pal quantum number states, it failed to explain the ob-served fine structure of atomic spectra. In 1928, the finestructure was finally given a more consistent explanationby Dirac [4]. The Dirac equation, put forth in [4], allowsfor the explanation of spin as a consequence of requiring

quantum mechanics to be consistent with special relativ-ity. The fine structure splitting in hydrogen and otheratoms can then be computed.

The fine structure of hydrogen can be analyzed us-ing time-independent perturbation theory using non-relativistic quantum mechanics by treating the relativis-tic correction to the kinetic energy and the spin-orbitcoupling as a perturbation. A standard treatment isfound in, for example, Chapter 6.3 of [3]. Alternatively,it is possible to solve for the energies of the electron in ahydrogen atom more directly using the Dirac equation,which applies to spin-1/2 particles [4] [5]. This accountsfor the relativistic kinetic energy and the spin-orbit cou-pling.

Solving the Dirac equation for the case of an electronin a Columbic potential gives the following expression forthe total relativistic energy eigenvalues:

Enj =mc2

1 + 2

nj 12

+

j+ 1

2

2 2

2

1/2

(1)

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Here is the fine-structure constant given by = e2

cin CGS units. m above is the mass of the electron, butshould be replaced with the reduced mass of the electron-nucleus system to accurately determine the difference inenergies between deuterium and hydrogen. The rest massenergy of the electron has to be subtracted from this ex-pression to give the observed energy levels of the hydro-gen atom. n is the principal quantum number which

defines the principal energy levels and takes on non-negative integer values, and j is the total angular mo-mentum number, which takes on values 1/2, 3/2. Fora given n, the angular momentum quantum number lranges from 0 to n 1. Each electron has a spin quan-tum number ms which can be 1/2 or 1/2. A detailedoverview of how the Dirac equation is solved for hydrogencan be found in, for instance, [6]. A Taylor expansion ofthe equation above, followed by subtracting the rest massenergy of the electron, then gives the following expression(ignoring terms of higher order).

Enj = mc22

2

1

n2+

2

n2

n

j+ 12

3

4

(2)

The line we seek to observe involves a transition fromn = 3 to n= 2. Ignoring the second term in the expan-sion, i.e. assuming there is no fine structure, we get theenergy of a transition from n = 3 ton= 2 to be 656.3 nmfor hydrogen and 656.1 nm for deuterium. These are theH and D lines respectively. The slight difference inwavelength is due to the difference in the reduced massesof the two systems. For the n = 2 state, l = 0, 1 and

j = 1/2, 3/2, giving rise to the 2s1/2, 2p1/2 and 2p1/2states. Solving the Dirac equation gives a degeneracy inenergy between the 2s1/2, 2p1/2 states. However, thesetwo states are in fact not degenerate, with the shift be-ing termed the Lamb shift [1]. We do not expect toobserve this as it is a much smaller effect, roughly anorder of magnitude less than the splitting due to the rel-ativistic kinetic energy and spin-orbit coupling that theDirac equation accounts for [1]. The energy differencebetween the 2s1/2/2p1/2 and 2s3/2 states is found to be

E= mc24

32 . A summary of the relevant computational

steps can be found at [7]. The corresponding wavenum-ber is therefore

= 1

hcE=

mc4

32h (3)

From which we can obtain

=

32h

mc

1/4(4)

FIG. 1. Experimental Setup

FIG. 2. Fabry-Perot etalon [9]

III. DESCRIPTION OF APPARATUS AND

EXPERIMENT

A. Experimental Setup

As shown in FIG. 1, the experimental setup consists ofa Woods Tube, a vacuum pump, and heavy water(D2O)and water (H2O) samples to generate the discharge. TheWoods tube contains the electrodes of a high voltagepower supply, which is used to ionize the water molecules

in the tube, allowing for electron transitions between thedifferent energy levels which give rise to the spectrum ofhydrogen.

The mercury arc lamp is used to calibrate the Fabry-Perot etalon and also to determine the fringe broadeningdue to the instrument. The dominant contribution tofringe broadening at the eyepiece will be assumed to bethe Doppler effect. A detailed discussion of this effectand other fringe broadening effects can be found in [8].A brief discussion of the importance of this effect to theexperiment will be discussed later.

The optical setup can be seen on the right side of FIG.1 The Fabry-Perot etalon is shown in FIG. 2, which isreprinted from page 274 of [9]. The etalon consists oftwo thinly silvered planar plates and produces fringesas a result of interference between light rays undergoingmultiple reflections between the two plates. The prismshown in the diagram is used to provide spectral resolu-tion as light of different wavelengths is bent at differentangles by the prism. The Fabry-Perot etalon then pro-vides an even higher spectral resolution.

The image of the fringes can be viewed at the eyepiece,where the image can be brought into focus by adjustingthe focusing lens accordingly. At the focal plane, the eye-

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piece may be replaced by a charge-coupled device (CCD)camera. The CCD camera allows for the image of thefringes to be captured at a high resolution, allowing usto generate a plot of the intensity as a function of posi-tion.

B. Methodology and Data

The Fabry-Perot etalon was first aligned by viewingthe fringes produced by the discharge from the mercurygreen lamp and checking that the image captured by theCCD was in focus. Once the alignment of the opticalsetup was done, a discharge from the Woods tube wasobtained. This was done as follows: First the Woodstube was sealed off by closing the air valve and clamp-ing the water and heavy water valves. Next, the vacuumpump is activated until a sufficiently good vacuum in-side the Woods tube is obtained. Then, the clamp forthe water is loosened to allow water vapor to enter thetube. The corresponding valve is used to regulate the

flow of water vapor in the Woods tube. The high volt-age is then applied across the Woods tube to ionize thegas inside for a discharge. At this point, the main valve(the one leading to the vacuum pump) was also adjustedto attain an optimal discharge. After the discharge wasobtained, the fringes were viewed from the eyepiece. Asthe position of the mercury green line was known, alongwith the relative position of the mercury yellow line, theposition of the H line at the eyepiece could be deducedand the H line could be viewed accordingly through theeyepiece. Following this, the CCD camera was used tocheck that each fringe of the doublet had approximatelythe same intensity.

To switch over to the deuterium discharge and view theD line, I loosened the heavy water clamp and openedthe heavy water valve before closing off the water valveand tightening the water clamp. This was to ensure thatthe Woods tube remained in discharge throughout theswitching process. Finally, the experiment was closedoff by first unplugging the high voltage source. Next,I checked that all the water and heavy water clampsand valves were closed before the vacuum pump was un-plugged. The air valve was then opened until the tubewas repressurized. Finally, the air valve was closed off.

The .tif images captured by the CCD camera are shownin FIG. 3.

IV. DATA ANALYSIS

From the images captured by the CCD camera, weobtained a plot of relative intensity against position forthe various gas discharges. The following intensity plotswere obtained by using the NumPy sumleft routine onthe various .tif images captured by the CCD camera, thenremoving the baseline noise. This baseline noise was de-termined from an intensity plot of the mercury fringes

FIG. 3. Images captured by the CCD camera for H, D andHg discharge respectively

FIG. 4. Intensity plot of fringes from deuterium dischargewith fitted curves

by assuming the mercury fringes to be relatively widelyspaced Gaussian functions, and taking the value of theintensity between the Gaussians to be the noise.

Following this, the plots in FIG. 4, FIG. 5 and FIG.6 were fitted to a linear sum of Gaussian functions thatwere displaced along the position coordinate. All numer-ical manipulations were handled using the NumPy andSciPy libraries in Python.

For constructive interference between the rays at theetalon, the rays must satisfy

TABLE I. Table of results for deuterium

Splitting (cm1) Distribution SD0.323 16.1

16.80.310 15.7

17.20.291 14.4

16.4

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FIG. 5. Intensity plot of fringes from hydrogen discharge withfitted curves

FIG. 6. Intensity plot of fringes from mercury discharge withfitted curves

2dcos() 2t= 1

(5)

where we assume thatis a sufficiently small angle andt is the distance between the plates of the etalon. Here,

TABLE II. Table of results for hydrogen

Splitting (cm1) Distribution SD0.338 22.0

19.40.313 19.4

18.80.300 18.7

18.7

is the wavenumber, which serves as a unit of energy.For our etalon, d = 0.7000 cm. Hence we obtain a valueof 1

2d 0.714 for the wavenumber between orders.To obtain the fractional order shift, the intensity plots

for the hydrogen and deuterium discharges were fitted toGaussian curves for the three central doublets in FIG. 5and FIG. 4. From this, the relative positions of the in-tensity maxima and the relative separation between the

doublets could be obtained. The fractional order shiftwas found by finding the ratio of half the distance be-tween two fringes of the same doublet to the distancebetween doublets. A summary of the relevant data canbe found in Tables I and II, where SD is an abbrevi-ation for standard deviation. Consequently, for hydro-gen, I obtained a fractional order shift in the H line of0.222 0.011 for the shifts measured from FIG. 5. Fordeuterium, I obtained a fractional order shift in the Dline of 0.216 0.009 over the shifts measured from FIG.4. The error bars are due to statistical uncertainty. Mul-tiplying the value of the wavenumber between orders, ,by twice the fractional order shift directly gives the finestructure splitting. For hydrogen, this is found to be0.316 0.016 cm1, and for deuterium, this is found tobe 0.308 0.013 cm1. These values are in agreementwith those obtained by R.C. Williams [2], who measureda splitting of 0.319 cm1 for hydrogen and 0.321 cm1

for deuterium. The theoretical value of the splitting is0.328 cm1 [2].

To determine the fine-structure constant, we use theexpression outlined above in equation (4). Hence we ob-tain a value of = 7.04 7 103 from the splitting inhydrogen and a value of = 7.00 7 103 from thesplitting in deuterium. This is within 5 percent of thecurrently accepted value of = 7.29 103, where isin CGS units.

In order to find the mass ratio between deuterium andhydrogen, we considered the fringe broadening as a resultof the Doppler effect. Qualitatively, the Doppler effectcauses fringe broadening because the atoms have somevelocity relative to the optical setup when they emit aphoton. As the velocities are distributed about a meanvelocity of zero, the frequencies emitted are similarly dis-tributed. As a result, the intensity is also distributedabout an expected frequency corresponding to the origi-nal, unshifted frequency. This is seen as a broadening ofthe fringe.

Quantitatively, if the velocities follow a Maxwell-Boltzmann distribution about the mean velocity, the

probability distribution function for the velocity in a par-ticular direction is given by

dw

du =

eu

2

(6)

whereu is the component of the velocity in the direc-tion incident on the optical setup, = m

2kBT, and m is

the mass of the particle. The Doppler frequency shift,, is given by 0 =

uc . Hence the relative intensity as

a function of the frequency is found to be

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