The Anomalous Zeeman Splitting Of The Sodium 3P States

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Advanced Optics LaboratoryThe Anomalous ZeemanSplitting of the Sodium 3PStatesDavid GaleyLindsay StanceuPrasenjit BoseApril 5, 2010

ObjectivesCalibrate Fabry-Perot interferometer Determine the Zeeman splitting of the 3Penergy state of the Sodium atom Determine effective nuclear charge ofsodium atom Determine strength of internal magneticfield

Zeeman Effect - Historical Origin i)ii) Zeeman Effect is named after DutchPhysicist, Pieter Zeeman.The experimental Evidence of thiseffect was published in:P. Zeeman, "The Effect ofMagnetisation on the Nature of LightEmitted by a Substance" Nature 55:347. (1897)P. Zeeman, "Doubles and triplets inthe spectrum produced by externalmagnetic forces". Phil. Mag. 44: 55.(1897)He obtained the Nobel Prize forPhysics in 1902.Dr. Pieter Zeeman

Zeeman Effect Zeeman Effect is the splitting of spectrallines into multiple lines in the presenceof a static magnetic fieldModern Uses:Nuclear magnetic ResonanceSpectroscopyElectron-spin resonance spectroscopyMagnetic Resonance ImagingMössbauer spectroscopy

Sodium Doublet A sodium Yellow doublet transitionhappens because of transition from 3p to3s transition.The 3p level is split into states with totalangular momentum (j l s) of j 3/2 andj 1/2 by the magnetic energy of theelectron spin in the presence of theinternal magnetic field caused by theorbital motion.In the case of the sodium doublet, thedifference in energy for the 3p3/2 and3p1/2 comes from a change of 1 unit inthe spin orientation with the orbital partpresumed to be the same.

Sodium Doublet ContinuedEnergy DifferenceFor Sodium doublet, j 1 as there isonly spin shift from 1/2 to -1/2 or viceversa (m remains same)g Gyromagnetic ratio 2.002319304386B magnetic field produced by theNucleus when observed from the electronreference frame (Classical View point)

The Fabry-Perot InterferometerPath Difference for bright fringe: 2n f d (cos(θ )) mλFringes result from interferencebetween multiple reflected beams, withbright fringes corresponding toconstructive interference and darkfringes to destructive interference

Theory2 Fringe Patterns from Wavelength Componentsof light source coincide when: OPL m1λ1 m2 λ2Next Coincidence occurs at: OPL (m1 n)λ1 (m2 n 1)λ2Where n number of fringesbetween coincidences λ λ1 λ2 λ2n

TheoryNumber of fringes is can be related to the mirrordisplacement by:n Therefore, thedifference inwavelengths isapproximately: λ 21λ2d2dλ1 22λ(1)2dHowever, using average wavelength gives a moreaccurate result:2 λ λavg2d(2)Since the average requiresboth wavelengths to beknown, with only 1wavelength, equation 1 canbe used to get a firstapproximation. The resultscan then be used withequation 2 to get a moreaccurate value.

Theory Question For reflectivity of R 0.85,4R4 * 0.85F 151.11Coefficient of2 (1 0.85) 2(1 R)Finesse:Finesse: F π ( F )1/ 22 19.31Minimum Wavelength Increment:λ20(588.995nm) 2( λ0 ) min Finesse * 2n f d19.31* 2 *1*159149nm 0.0564nmFree Spectral Range: (λ0 ) FSR Finesse * (λ0 ) min 19.31* 0.00564nm(λ0 ) FSR 1.089nmResolving Power: R 588.995nmλ0 1.04 *10 4 nm( λ0 ) min0.0564nm

Theory QuestionMinimum Frequency Increment:( υ 0 ) minc3 *108 m / s ( λ0 ) min 0.0564 *10 9 m( υ0 ) min 5.319 HzFree Spectral Frequency Range:(υ0 ) FSRc3 *108 m / s (λ0 ) FSR 1.089 *10 9 m( υ 0 ) min 2.755 *1017 Hz

Experimental SetupMirrorsLaser SourceDiverging lensMicrometerTelescopeMirrors

Experimental SetupMovable mirrorCollector lensTelescopeStationary mirrorMirror adjustment barMicrometer

Procedure – CalibrationAssemble Fabry-Perot interferometer withthe mirrors close, but not touching Record the value on the micrometer Rotate micrometer and count the numberof fringes that pass Record the micrometer reading every 50fringes until 500 fringes have passed

Procedure – Calibration

Procedure – CalibrationConvert fringe count to mirrordisplacement and plot mirror displacementvs. micrometer displacement Slope of graph is the calibration factor, theamount the mirror moves per tic on themicrometer

Experimental Setup - SodiumSodium lampFilterPrecision leveling device(Handbook ofMathematical Functions)Interferometer(same setup as laser)

Procedure – Sodium Doublet Replace laser source with sodium sourceSodium produces 2 sets of fringe patternsAdjust mirror separation until 2 patterns are coincident(each fringe is made up of 2 closely spaced lines)Rotate micrometer in both directions and record positionwhere the 2 close lines start to blur, average the twonumbers to determine the position of coincidenceRotate micrometer further and find another position ofcoincidenceRepeat for 6 coincidences

Procedure – Sodium Doublet

Procedure – Sodium Doublet

Procedure – Sodium Doublet

Procedure – Sodium Doublet Convert micrometer displacements to mirrordisplacements using previous calibration factorPlot mirror displacement vs. coincidence numberSlope is d, the distance the mirror must movebetween two coincidencesCalculate the energy difference between the two3p statesUsing that, calculate the effective nuclear chargeand the magnetic field

Results - CalibrationFringeCoun t *WavelengthMirrorDisp lacement 2

Results - CalibrationCalibratio nFactor (1 .94853 3.99809 E 3) µm

Results – Sodium Double (Trial 1)Circular Re ading1 Circular Re ading 2AvgCirc Re ading 2CircDisplacement AvgCirc Re ading 558.75MirrorDisp lace CircDispla cement * Calibratio nFactor

Results – Sodium Doublet (Trial 1)d (284.541 0.625) µm

Results – Sodium Doublet (Trial 1)λ 1theo : 589.592nm λ 1 : λ 1theo2 d2 0.611 nm λ : λ λ 1 for i 1 . 100 λ 2 λ 1theo λ λ λ( 1theo 2) λ avg 2 2 λ avg λ 2 d λ 2 : λ 1theo λ 588.982nm λ 0.6102088333000636 nm λ theo : 0.597nm%err λ : ( λλ 2theo : 588.995 nm) λ theo 100 λ theo 2.213%err λ2 : (λ 2 λ 2theo ) 100λ 2theo 2.243 10 3

Results – Sodium Doublet (Trial 1)Energy separation between states: E : ( h c λ )λ avg1.602 10 2.179 101ms : 2 E evtheo : .0021J( E ev E evtheo ) 100 E evtheo 16 3.756h b : 6.595 10s 19e : 1.6021764610 31me : 9.1093818810 Ckg E ev B : me 18.784T2 h b ms e Effective nuclear charge:l : 13Z eff : Internal magnetic field: 3 19n : 3J2 E E ev : %err : 22 3.4905756734853055 10 E ev n l ( l 1) 47.24 10 12.748%err B : ( B 18T ) 10018T 4.358

Results – Sodium Double (Trial 2)Circular Re ading1 Circular Re ading 2AvgCirc Re ading 2CircDispla cement AvgCirc Re ading 558 .75MirrorDisp lace CircDispla cement * Calibratio nFactor

Results – Sodium Doublet (Trial 2)d (283.722 9.8471) µm

Results – Sodium Doublet (Trial 2)λ 1theo : 589.592nm λ 1 : λ 1theo2 d2 0.613 nm λ : λ λ 1 fori 1.100 λ 2 λ 1theo λ λ λ() 1theo2 λ avg 2 2 λ avg λ 2 d λ 0.6119681211422627 nm λ theo : 0.597nm%err λ : ( λ) λ theo 100 λ theo 2.507λ 2 : λ 1theo λ 588.98nm %err λ2 : (λ 2 λ 2theo ) 100λ 2theo 2.541 10 3

Results – Sodium Doublet (Trial 2)Energy separation between states: E : ( h c λ )λ avg2 E E ev : 19( 2.185 101ms : 2) E evtheo 16 4.056Effective nuclear charge:l : 13Z eff : Internal magnetic field: E evtheo : .0021J E ev E evtheo 100n : 3J 31.602 10%err : 22 3.5006497732207364 10 E ev n l ( l 1)7.24 10 4 12.766h b : 6.595 10s 19e : 1.6021764610 31me : 9.1093818810 E evB : 2 h b ms e%errB : kg me 18.839 T ( B 18T) 10018TC 4.659

Conclusions Wavelength separation λ 0.6119681211422627 nm λ 0.6102088333000636 nm%err λ : ( λ) λ theo 100 λ theo%err λ : 2.213( λ) λ theo 100 λ theo 2.507Energy difference E : ( h c λ )λ avg E ev : 2J E : ( h c λ )λ avg E 3 191.602 10%err : 22 3.4905756734853055 10 2.179 102 E ev : E evtheo E 3.756%err : J 3 191.602 10J( E ev E evtheo ) 100 22 3.5006497732207364 10 2.185 10J( E ev E evtheo ) 100 E evtheo 4.056

Conclusions Effective nuclear charge3Zeff : E ev n l ( l 1) 43 12.748 E ev n l ( l 1)Z eff : 7.24 107.24 10 4 12.766Magnetic field E ev B : me 18.839 T2 h b ms e E ev B : me 18.784T2 h b ms e %err B : ( B 18T ) 10018T 4.358%errB : ( B 18T) 10018T 4.659

Conclusions/Observations Interferometer is extremely difficult to align and keepaligned, slightest bump throws the whole thing off, had tore-align using laserWhen rotating micrometer to increase mirror separation,spring pulling mirror back did not always do so smoothlyor had to be manually pushed backFringes became narrower as mirror separationdecreased, made it very hard to see if there wascoincidence or notResults ended up being very accurate despite problems,because once it was aligned and your eyes were used toseeing the fringes, they were extremely clear and couldeasily be read

Sources of Error Eyestrain Measurement limitations Micrometer precision was limitedRandomMinor misalignment Too much staring at fringes in one day and they start to be harderharder to focus onRandomMirrors might not have been perfectly parallel, could throw readingsreadings off slightlyRandomResolution of device Though we used the range of positions where we could separate thethe two closelyspaced lines as our coincidence range, we could not resolve the fringes enoughto be able to identify the exact position of maximum coincidenceNarrow fringes at smaller mirror separation made position of coincidencecoincidence evenharder to seeSystematic – device limitationsRandom – human vision limitations

References Advanced Optics Laboratory manualLecture ntum/sodzee.htmlBeiser, Arthur. Concepts of Modern Physics, sessionid 5FA6263719B44278227A1011578BE6CE

Sodium Doublet A sodium Yellow doublet transition happens because of transition from 3p to 3s transition. The 3p level is split into states with total angular momentum (j l s ) of j 3/2 and j 1/2 by the magnetic energy of the electron spin in the presence of the internal magnetic field caused by the orbital motion .

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