the anomalous zeeman splitting of the sodium 3p states

The Anomalous Zeeman The Anomalous Zeeman

Splitting of the Sodium 3P Splitting of the Sodium 3P

StatesStates

David GaleyDavid Galey

Lindsay StanceuLindsay Stanceu

PrasenjitPrasenjit BoseBose

April 5, 2010April 5, 2010

Advanced Optics Laboratory

ObjectivesObjectives

�� Calibrate Calibrate FabryFabry--Perot interferometerPerot interferometer

�� Determine the Zeeman splitting of the 3P Determine the Zeeman splitting of the 3P

energy state of the Sodium atomenergy state of the Sodium atom

�� Determine effective nuclear charge of Determine effective nuclear charge of

sodium atomsodium atom

�� Determine strength of internal magnetic Determine strength of internal magnetic

fieldfield

Zeeman Effect Zeeman Effect -- Historical OriginHistorical Origin�� Zeeman Effect is named after Dutch Zeeman Effect is named after Dutch

Physicist, Pieter Zeeman.Physicist, Pieter Zeeman.

�� The experimental Evidence of this The experimental Evidence of this effect was published in:effect was published in:

i)i) P. Zeeman, "P. Zeeman, "The Effect of The Effect of MagnetisationMagnetisation on the Nature of Light on the Nature of Light Emitted by a SubstanceEmitted by a Substance" " Nature 55: Nature 55: 347. (1897)347. (1897)

ii)ii) P. Zeeman, "P. Zeeman, "Doubles and triplets in Doubles and triplets in the spectrum produced by external the spectrum produced by external magnetic forcesmagnetic forces". ". Phil. Mag. 44: 55. Phil. Mag. 44: 55. (1897)(1897)

�� He obtained the Nobel Prize for He obtained the Nobel Prize for Physics in 1902.Physics in 1902.

Dr. Pieter Zeeman

Zeeman EffectZeeman Effect

�� Zeeman Effect is the splitting of spectral Zeeman Effect is the splitting of spectral

lines into multiple lines in the presence lines into multiple lines in the presence

of a static magnetic fieldof a static magnetic field

Modern Uses:Modern Uses:

�� Nuclear magnetic Resonance Nuclear magnetic Resonance

SpectroscopySpectroscopy

�� ElectronElectron--spin resonance spectroscopyspin resonance spectroscopy

�� Magnetic Resonance Imaging Magnetic Resonance Imaging

�� MMöössbauerssbauer spectroscopyspectroscopy

Sodium DoubletSodium Doublet

�� A sodium Yellow doublet transition A sodium Yellow doublet transition happens because of transition from 3p to happens because of transition from 3p to 3s transition.3s transition.

�� The 3p level is split into states with total The 3p level is split into states with total angular momentum (j=angular momentum (j=l+sl+s) of j=3/2 and ) of j=3/2 and j=1/2 by the magnetic energy of the j=1/2 by the magnetic energy of the electron spin in the presence of the electron spin in the presence of the internal magnetic field caused by the internal magnetic field caused by the orbital motionorbital motion..

�� In the case of the sodium doublet, the In the case of the sodium doublet, the difference in energy for the 3pdifference in energy for the 3p3/23/2 and and 3p3p1/21/2 comes from a change of 1 unit in comes from a change of 1 unit in the spin orientation with the orbital part the spin orientation with the orbital part presumed to be the same. presumed to be the same.

Sodium Doublet ContinuedSodium Doublet Continued

Energy Difference

For Sodium doublet, ∆ j = 1 as there is

only spin shift from 1/2 to -1/2 or vice-

versa (m remains same)

g = Gyromagnetic ratio = 2.002319304386

B = magnetic field produced by the

Nucleus when observed from the electron

reference frame (Classical View point)

TheThe FabryFabry--PerotPerot InterferometerInterferometer

Path Difference for bright fringe:

λθ mdn f ==∆ ))(cos(2

Fringes result from interference

between multiple reflected beams, with

bright fringes corresponding to

constructive interference and dark

fringes to destructive interference

TheoryTheory

2211 λλ mmOPL ==∆

2211 )1()( λλ ++=+=∆ nmnmOPL

2 Fringe Patterns from Wavelength Components

of light source coincide when:

Next Coincidence occurs at:

Where n = number of fringes

between coincidences

n

221

λλλλ =−=∆

TheoryTheory

1

2

λ

dn =

Number of fringes is can be related to the mirror

displacement by:

dd 22

2

2

2

1 λλλ ≈≈∆

Therefore, the

difference in

wavelengths is

approximately:

However, using average wavelength gives a more

accurate result:

d

avg

2

2λλ =∆

(1)

(2)

Since the average requires

both wavelengths to be

known, with only 1

wavelength, equation 1 can

be used to get a first

approximation. The results

can then be used with

equation 2 to get a more

accurate value.

Theory QuestionTheory Question

�� For reflectivity of R=0.85,For reflectivity of R=0.85,

Finesse: F 31.192

)( 2/1

==Fπ

Minimum Wavelength Increment:

nmnm

nm

dnFinesse f

0564.0159149*1*2*31.19

)995.588(

2*)(

22

0min0 ===∆

λλ

11.151)85.01( 2

85.0*4

)1( 2

4=

−=

−=

R

RFCoefficient of

Finesse:

Resolving Power: R nmnm

nm 4

min0

0 10*04.10564.0

995.588

)(==

∆=

λ

λ

Free Spectral Range: nmFinesseFSR 00564.0*31.19)(*)( min00 == λλ

nmFSR 089.1)( 0 =λ

Theory QuestionTheory Question

Hz319.5)( min0 =∆υ

m

smc

FSR

FSR 9

8

0

010*089.1

/10*3

)()(

−==

λυ

Minimum Frequency Increment:

m

smc9

8

0

min010*0564.0

/10*3

min)()(

−=

∆=∆

λυ

Free Spectral Frequency Range:

Hz17

min0 10*755.2)( =∆υ

Experimental SetupExperimental Setup

Laser Source

Mirrors

Micrometer

Diverging lens

Telescope

Mirrors

Experimental SetupExperimental Setup

Telescope

Movable mirror

Stationary mirror

Micrometer

Mirror adjustment bar

Collector lens

Procedure Procedure –– CalibrationCalibration

�� Assemble Assemble FabryFabry--Perot interferometer with Perot interferometer with

the mirrors close, but not touchingthe mirrors close, but not touching

�� Record the value on the micrometerRecord the value on the micrometer

�� Rotate micrometer and count the number Rotate micrometer and count the number

of fringes that passof fringes that pass

�� Record the micrometer reading every 50 Record the micrometer reading every 50

fringes until 500 fringes have passedfringes until 500 fringes have passed


�� Convert fringe count to mirror Convert fringe count to mirror

displacement and plot mirror displacement displacement and plot mirror displacement

vs. micrometer displacementvs. micrometer displacement

�� Slope of graph is the calibration factor, the Slope of graph is the calibration factor, the

amount the mirror moves per tic on the amount the mirror moves per tic on the

micrometermicrometer

Experimental Setup Experimental Setup -- SodiumSodium

Sodium lamp

Filter

Interferometer

(same setup as laser)

Precision leveling device

(Handbook of

Mathematical Functions)

Procedure Procedure –– Sodium DoubletSodium Doublet

�� Replace laser source with sodium sourceReplace laser source with sodium source

�� Sodium produces 2 sets of fringe patternsSodium produces 2 sets of fringe patterns

�� Adjust mirror separation until 2 patterns are coincident Adjust mirror separation until 2 patterns are coincident (each fringe is made up of 2 closely spaced lines)(each fringe is made up of 2 closely spaced lines)

�� Rotate micrometer in both directions and record position Rotate micrometer in both directions and record position where the 2 close lines start to blur, average the two where the 2 close lines start to blur, average the two numbers to determine the position of coincidencenumbers to determine the position of coincidence

�� Rotate micrometer further and find another position of Rotate micrometer further and find another position of coincidencecoincidence

�� Repeat for 6+ coincidencesRepeat for 6+ coincidences


�� Convert micrometer displacements to mirror Convert micrometer displacements to mirror

displacements using previous calibration factordisplacements using previous calibration factor

�� Plot mirror displacement vs. coincidence numberPlot mirror displacement vs. coincidence number

�� Slope is d, the distance the mirror must move Slope is d, the distance the mirror must move

between two coincidencesbetween two coincidences

�� Calculate the energy difference between the two Calculate the energy difference between the two

3p states3p states

�� Using that, calculate the effective nuclear charge Using that, calculate the effective nuclear charge

and the magnetic fieldand the magnetic field

Results Results -- CalibrationCalibration

2

*WavelengthtFringeCounlacementMirrorDisp =

Results Results -- CalibrationCalibration

mEnFactorCalibratio µ)399809.394853.1( −±=

Results Results –– Sodium Double (Trial 1)Sodium Double (Trial 1)

2

2Re1ReRe

adingCircularadingCircularadingAvgCirc

+=

75.558Re −= adingAvgCirccementCircDispla

nFactorCalibratiocementCircDisplalaceMirrorDisp *=

Results Results –– Sodium Doublet (Trial 1)Sodium Doublet (Trial 1)

md µ)625.0541.284( ±=


λ1theo 589.592nm:= ∆λ 1

λ1theo2

2 d⋅0.611nm⋅=:=

∆λ ∆λ ∆λ 1←

λ2 λ1theo ∆λ−←

λavg

λ1theo λ2+( )2

←

∆λλavg

2

2 d⋅←

i 1 100..∈for

:=

∆λ 0.6102088333000636nm⋅= λ2 λ1theo ∆λ− 588.982nm⋅=:=

∆λ theo 0.597nm:=

%err∆λ

∆λ ∆λ theo−( ) 100⋅

∆λ theo

2.213=:=%err λ2

λ2 λ2theo−( ) 100⋅

λ2theo

2.243− 103−

×=:=

λ 2theo 588.995 nm:=


∆Eh c⋅ ∆λ⋅( )

λavg2

3.4905756734853055 1022−

× J=:=

∆E ev∆E

1.602 1019−

J⋅

2.179 103−

×=:=∆E evtheo .0021:=

%err∆E ev ∆E evtheo−( ) 100⋅

∆E evtheo

3.756=:=

n 3:= l 1:=

Zeff

∆E ev n3

⋅ l⋅ l 1+( )⋅

7.24 104−

⋅

12.748=:=

ms1

2:= e 1.6021764610

19−C⋅:=

hb 6.595 1016−

⋅ s:= me 9.109381881031−

kg⋅:=

B∆E ev

2 hb⋅ ms⋅ e⋅

me⋅ 18.784T=:=

%err BB 18T−( ) 100⋅

18T4.358=:=

Internal magnetic field:

Effective nuclear charge:

Energy separation between states:

Results Results –– Sodium Double (Trial 2)Sodium Double (Trial 2)

2

2Re1ReRe

adingCircularadingCircularadingAvgCirc

+=

75.558Re −= adingAvgCirccementCircDispla

nFactorCalibratiocementCircDisplalaceMirrorDisp *=


md µ)8471.9722.283( ±=


λ1theo 589.592nm:=

∆λ ∆λ ∆λ 1←

λ2 λ1theo ∆λ−←

λavg

λ1theo λ2+( )2

←

∆λλavg

2

2 d⋅←

i 1 100..∈for

:=

∆λ theo 0.597nm:=

∆λ 1

λ1theo2

2 d⋅0.613nm⋅=:=

∆λ 0.6119681211422627 nm⋅=

%err∆λ

∆λ ∆λ theo−( ) 100⋅

∆λ theo

2.507=:=

λ2 λ1theo ∆λ− 588.98nm⋅=:=

%err λ2

λ 2 λ 2theo−( ) 100⋅

λ 2theo

2.541− 103−

×=:=


∆E evtheo .0021:=

∆Eh c⋅ ∆λ⋅( )

λavg2

3.5006497732207364 1022−

× J=:=

∆E ev∆E

1.602 1019−

J⋅

2.185 103−

×=:=


∆E evtheo

4.056=:=

n 3:= l 1:=

ms1

2:= e 1.6021764610

19−C⋅:=

hb 6.595 1016−

⋅ s:= me 9.109381881031−

kg⋅:=

Zeff

∆E ev n3

⋅ l⋅ l 1+( )⋅

7.24 104−

⋅

12.766=:=

B∆E ev

2 hb⋅ ms⋅ e⋅

me⋅ 18.839 T=:=

%errBB 18T−( ) 100⋅

18T4.659=:=

Internal magnetic field:

Effective nuclear charge:

Energy separation between states:

ConclusionsConclusions

�� Wavelength separationWavelength separation

�� Energy differenceEnergy difference

∆λ 0.6102088333000636 nm⋅=

%err∆λ

∆λ ∆λ theo−( ) 100⋅

∆λ theo

2.213=:=

∆λ 0.6119681211422627 nm⋅=

%err∆λ

∆λ ∆λ theo−( ) 100⋅

∆λ theo

2.507=:=

∆Eh c⋅ ∆λ⋅( )

λavg2

3.4905756734853055 1022−

× J=:=


∆E evtheo

3.756=:=

∆E ev∆E

1.602 1019−

J⋅

2.179 103−

×=:=

∆Eh c⋅ ∆λ⋅( )

λavg2

3.5006497732207364 1022−

× J=:=

∆E ev∆E

1.602 1019−

J⋅

2.185 103−

×=:=


∆E evtheo

4.056=:=

ConclusionsConclusions

�� Effective nuclear chargeEffective nuclear charge

�� Magnetic fieldMagnetic field

Zeff

∆E ev n3

⋅ l⋅ l 1+( )⋅

7.24 104−

⋅

12.748=:= Zeff

∆E ev n3

⋅ l⋅ l 1+( )⋅

7.24 104−

⋅

12.766=:=

B∆E ev

2 hb⋅ ms⋅ e⋅

me⋅ 18.784T=:=

%err BB 18T−( ) 100⋅

18T4.358=:=

B∆E ev

2 hb⋅ ms⋅ e⋅

me⋅ 18.839 T=:=

%errBB 18T−( ) 100⋅

18T4.659=:=

Conclusions/ObservationsConclusions/Observations

�� Interferometer is extremely difficult to align and keep Interferometer is extremely difficult to align and keep aligned, slightest bump throws the whole thing off, had to aligned, slightest bump throws the whole thing off, had to rere--align using laseralign using laser

�� When rotating micrometer to increase mirror separation, When rotating micrometer to increase mirror separation, spring pulling mirror back did not always do so smoothly spring pulling mirror back did not always do so smoothly or had to be manually pushed backor had to be manually pushed back

�� Fringes became narrower as mirror separation Fringes became narrower as mirror separation decreased, made it very hard to see if there was decreased, made it very hard to see if there was coincidence or notcoincidence or not

�� Results ended up being very accurate despite problems, Results ended up being very accurate despite problems, because once it was aligned and your eyes were used to because once it was aligned and your eyes were used to seeing the fringes, they were extremely clear and could seeing the fringes, they were extremely clear and could easily be readeasily be read

Sources of ErrorSources of Error

�� EyestrainEyestrain�� Too much staring at fringes in one day and they start to be hardToo much staring at fringes in one day and they start to be harder to focus oner to focus on

�� RandomRandom

�� Measurement limitationsMeasurement limitations�� Micrometer precision was limitedMicrometer precision was limited

�� RandomRandom

�� Minor misalignmentMinor misalignment�� Mirrors might not have been perfectly parallel, could throw readMirrors might not have been perfectly parallel, could throw readings off slightlyings off slightly

�� RandomRandom

�� Resolution of deviceResolution of device�� Though we used the range of positions where we could separate thThough we used the range of positions where we could separate the two closely e two closely

spaced lines as our coincidence range, we could not resolve the spaced lines as our coincidence range, we could not resolve the fringes enough fringes enough to be able to identify the exact position of maximum coincidenceto be able to identify the exact position of maximum coincidence

�� Narrow fringes at smaller mirror separation made position of coiNarrow fringes at smaller mirror separation made position of coincidence even ncidence even harder to seeharder to see

�� Systematic Systematic –– device limitationsdevice limitations

�� Random Random –– human vision limitationshuman vision limitations

ReferencesReferences

�� Advanced Optics Laboratory manualAdvanced Optics Laboratory manual

�� Lecture notesLecture notes

�� http://hyperphysics.phyhttp://hyperphysics.phy--astr.gsu.edu/Hbase/quantum/sodzee.htmlastr.gsu.edu/Hbase/quantum/sodzee.html

�� BeiserBeiser, Arthur. Concepts of Modern Physics, 6, Arthur. Concepts of Modern Physics, 6thth

ed.ed.

�� http://wyant.optics.arizona.edu/MultipleBeamIntehttp://wyant.optics.arizona.edu/MultipleBeamInterference/MultipleBeamInterferenceNotes.htmlrference/MultipleBeamInterferenceNotes.html

�� http://fabryperot.oamp.fr/FabryPerot/jsp/more.jsphttp://fabryperot.oamp.fr/FabryPerot/jsp/more.jsp%3Bjsessionid=5FA6263719B44278227A10115%3Bjsessionid=5FA6263719B44278227A1011578BE6CE78BE6CE

the anomalous zeeman splitting of the sodium 3p states

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