210 PHYWE Systeme GmbH & Co. KG · D-37070 GöttingenLaboratory Experiments Physics

Physical Structure of Matter Physics of the Electron

5.1.10-05 Zeeman effect

Principle:The “Zeeman effect” is the splittingup of the spectral lines of atomswithin a magnetic field. The simplestis the splitting up of one spectral lineinto three components called the“normal Zeeman effect”. The normalZeeman effect is studied using acadmium spectral lamp as a speci-men. The cadmium lamp is submittedto different magnetic flux densitiesand the splitting up of the red cad-mium line (643.8 nm) is investigatedusing a Fabry-Perot interferometer.The evaluation of the results leads toa fairly precise value for Bohr’smagneton.

Tasks:1. Using the Fabry-Perot interferom-

eter and a selfmade telescope thesplitting up of the central line intotwo �-lines is measured in wavenumbers as a function of themagnetic flux density.

Screenshot of software used to measure the diameters of the interferencerings as captured by the CCD-Camera.

2. From the results of point 1. a valuefor Bohr’s magneton is evaluated.

3. The light emitted within the direc-tion of the magnetic field is qual-itatively investigated.

Fabry-Perot interferometer 09050.02 1Cadmium lamp for Zeeman effect 09050.01 1Electromagnet without pole shoes 06480.01 1Pole pieces, drilled, conical 06480.03 1Rotating table for heavy loads 02077.00 1Power supply for spectral lamps 13662.97 1Variable transformer, 25 V AC/20 V DC, 12 A 13531.93 1Capacitor, electrolytic, 22000 µF 06211.00 1Digital multimeter 07134.00 1Optical profile-bench, l = 1000 mm 08282.00 1Base for opt. profile-bench, adjust. 08284.00 2Slide mount for opt. profile-bench, h = 30 mm 08286.01 5Slide mount for opt. profile-bench, h = 80 mm* 08286.02 2Lens holder 08012.00 4Lens, mounted, f = +50 mm 08020.01 2Lens, mounted, f = +300 mm 08023.01 1Iris diaphragm 08045.00 1Polarising filter, on stem 08610.00 1Polarization specimen, mica 08664.00 1Connecting cord, l = 25 cm, 32 A, red 07360.01 1Connecting cord, l = 25 cm, 32 A, blue 07360.04 1Connecting cord, l = 50 cm, 32 A, red 07361.01 1Connecting cord, l = 50 cm, 32 A, blue 07361.04 1Connecting cord, l = 75 cm, 32 A, red 07362.01 1Connecting cord, l = 100 cm, 32 A, red 07363.01 1Connecting cord, l = 100 cm, 32 A, blue 07363.04 1CDC-Camera for PC incl. measurement software* 88037.00 1

PC with USB interface, Windows 98SE/Windows Me/Windows 2000/Windows XP

*Alternative to CCD-Camera incl. measurement software, two slide mounts,h = 80 mm for classical version of the Zeeman Effect:Slide mount for optical profile-bench, 08286.00 1Sliding device, horizontal 08713.00 1

What you need:

What you can learn about …

� Bohr’s atomic model� Quantisation of energy levels� Electron spin� Bohr’s magneton� Interference of

electromagnetic waves� Fabry-Perot interferometer

Swinging arm 08256.00 1Plate holder with tension spring 08288.00 1Screen, with aperture and scale 08340.00 1Slide mount for opt. profile-bench, h = 80 mm 08286.02 1

Complete Equipment Set, Manual on CD-ROM includedZeeman effect P2511005

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Related topicsBohr’s atomic model, quantisation of energy levels, electronspin, Bohr’s magneton, interference of electromagneticwaves, Fabry-Perot interferometer.

PrincipleThe “Zeeman effect” is the splitting up of the spectral lines ofatoms within a magnetic field. The simplest is the splitting upof one spectral line into three components called the “normalZeeman effect”. The normal Zeeman effect is studied using acadmium spectral lamp as a specimen. The cadmium lamp issubmitted to different magnetic flux densities and the splittingup of the red cadmium line (643.8 nm) is investigated using aFabry-Perot interferometer. The evaluation of the results leadsto a fairly precise value for Bohr’s magneton.

EquipmentFabry-Perot interferometer 09050.02 1Cadmium lamp for Zeeman effect 09050.01 1Electromagnet without pole shoes 06480.01 1Pole pieces, drilled, conical 06480.03 1Rotating table for heavy loads 02077.00 1Power supply for spectral lamps 13662.97 1Variable transformer, 25 V AC/20 V DC, 12 A 13531.93 1Capacitor, electrolytic, 22000 mF 06211.00 1Digital multimeter 07134.00 1Optical profile-bench, l = 1000 mm 08282.00 1Base for opt. profile-bench, adjust. 08284.00 2Slide mount for opt. profile-bench, h = 30 mm 08286.01 5Slide mount for opt. profile-bench, h = 80 mm* 08286.02 2Lens holder 08012.00 4Lens, mounted, f = +50 mm 08020.01 2

Lens, mounted, f = +300 mm 08023.01 1Iris diaphragm 08045.00 1Polarising filter, on stem 08610.00 1Polarization specimen, mica 08664.00 1Connecting cord, l = 25 cm, 32 A, red 07360.01 1Connecting cord, l = 25 cm, 32 A, blue 07360.04 1Connecting cord, l = 50 cm, 32 A, red 07361.01 1Connecting cord, l = 50 cm, 32 A, blue 07361.04 1Connecting cord, l = 75 cm, 32 A, red 07362.01 1Connecting cord, l = 100 cm, 32 A, red 07363.01 1Connecting cord, l = 100 cm, 32 A, blue 07363.04 1CDC-Camera for PC incl. measurement software* 88037.00 1PC with USB interface, Windows 98SE/Windows Me/Win-dows 2000/Windows XP*Alternative to CCD-Camera incl. measurement software, twoslide mounts, h = 80 mm for classical version of the ZeemanEffect:Slide mount for optical profile-bench, 08286.00 1Sliding device, horizontal 08713.00 1Swinging arm 08256.00 1Plate holder with tension spring 08288.00 1Screen, with aperture and scale 08340.00 1Slide mount for opt. profile-bench, h = 80 mm 08286.02 1

Tasks1. Using the Fabry-Perot interferometer, a self-made tele-

scope, a CCD-camera and measurement software, thesplitting up of the central line into two s-lines is measuredin wave numbers as a function of the magnetic flux density. In the classical version where the CCD-Camera is not avail-able, a screen with scale and a sliding device are used tomeasure the splitting.

Fig.1: Experimental set-up for the Zeeman effect.

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2. From the results of point 1. a value for Bohr’s magneton isevaluated.

3. The light emitted within the direction of the magnetic field isqualitatively investigated.

Set-upThe electromagnet is put on the rotating table for heavy loadsand mounted with the two pole-shoes with holes in such away that a gap large enough for the Cd-lamp (9-11 mm)remains for the Cd-lamp. The pole-shoes have to be welltightened in such a way that they cannot move later on whenthe magnetic flux is established. The Cd-lamp is inserted intothe gap without touching the pole-shoes and connected to thepower supply for spectral lamps. The coils of the electromag-net are connected in parallel and via an ammeter connectedto the variable power supply of up to 20 VDC, 12 A. A capac-itor of 22000 mF is in parallel to the power output to smooth-en the DC-voltage.

The optical bench for investigation of the line splitting carriesthe following elements (their approximate position in cm isgiven in brackets):

(80) CDC-Camera(73) L3 = +50 mm(68) Screen with scale (only in classical version)(45) Analyser(39) L2 = +300 mm(33) Fabry-Perot Etalon(25) L1 = +50 mm(20) Iris diaphragm(20) Drilled pole-shoesCd-spectral lamp on rotating table

The iris diaphragm is eliminated for initial adjustment and forthe observation of the longitudinal Zeeman effect. Duringobservation of the transverse Zeeman effect the iris dia-phragm is illuminated by the Cd-lamp and such it acts as thelight source. The lens L1 and a lens of f = 100 mm, incorpo-rated in the étalon, create a nearly parallel light beam whichthe Fabry-Perot étalon needs for a proper interference pattern.

The etalon contains a removable colour filter that lets the redcadmium line at 643.8 nm pass. The lens L2 produces aninterference pattern of rings which can be observed throughL3. The ring diameters can be measured using the CCD-cam-era and the software supplied with it. In the classical versionthe interference pattern is produced within the plane of thescreen with a scale mounted on a slide mount which can lat-terally be displaced with a precision of 1/100th of a millimeter.The measurement here can be done for instance, by system-atic displacement of the slash representing the „0“ of thescale.

The initial adjustment is done in the following way:The rotating table with electromagnet, pole-shoes and Cd-lamp already mounted is adjusted so that the center of theholes in the pole-shoes lies about 28 cm above the table. Theoptical bench with all elements (except iris diaphragm andCCD-camera) mounted, is then moved closer to the electro-magnet in such a way that one of the outlet holes of the pole-shoes coincides with the previous position of the irisdiaphragm. L1 is then adjusted so that the outlet hole is with-in the focal plane of it. All other optical elements of Fig. 2. aresubsequently readjusted with respect to their height corre-

Fig. 2: Arrangement of the optical components.

Fig.1b: Set-up for the classical version of the experiment.

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spondingly. The current of the coils is set for some time to 8 A(increase in light intensity of the Cd-lamp !) and the ring inter-ference pattern in axial direction is observed through L3 by theeye. The pattern must be centered and sharp which is even-tually achieved by a last, slight movement of the étalon (to theright or to the left) and by displacement of L2 (vertically andhorizontally).

Finally the CCD-camera with the 8 mm lens attached ismounted to the optical bench and adjusted in horizontal andvertical position as well as in tilt and focus until a clear pictureof the ring pattern is visible on the computer screen. Forinstallation and use of the camera and software please refer tothe manual supplied with the camera.

In the classical version the screen with scale is shifted in a waythat the slash representing the „0“ of the scale is clearly seencoinciding, for instance, with the center of the fairly brightinner ring. The scale itself must be able to move horizontallyalong the diameter of the ring pattern. (set-up see Fig. 1b)

Hint: best results are achieved when the experiment is carriedout in a darkened room.

The electromagnet is now turned by 90°, the iris diaphragm isinserted and the analyzer turned until the π-line (explanationfollows) disappears completely and the two s-lines appearclearly visible.

Remark: For later evaluations the calibration curve of the mag-netic flux density versus the coil current has to betraced previously. This can be done if a teslameter isavailable. Otherwise the results of Fig. 3 must beused. The curve of Fig. 3 was traced by measuringthe flux density in the center of the gap in theabsence of the Cd-lamp. For the evaluations thesecenter-values were increased by 3.5% to account forthe non-uniform flux distribution within the gap.

Fig. 3: Magnetic flux density B in the center of gap without theCd-lamp (gap width: 9 mm) as a function of twice thecoil current.

TheoryAs early as 1862, Faraday investigated whether the spectrumof coloured flames changes under the influence of a magnet-ic field, but without success. It was not until 1885 that Fievezfrom Belgium was able to demonstrate an effect, but it wasforgotten and only rediscovered 11 years later by theDutchman Zeeman, who studied it together with Lorentz.

This experiment, which was of importance to the developmentof the theory of the atomic shell, can now be carried out withmodern equipment in the students’ experiment laboratory.

The splitting of the Cd-spectral line l = 643.8 nm into threelines, the so-called Lorentz triplets, occurs since the Cd-atomrepresents a singlet system of total spin S = 0. In the absenceof a magnetic field there is only one possible D � P transitionof 643.8 nm, as indicated by Fig. 4.

In the presence of a magnetic field the associated energy lev-els split into 2 L + 1 components. Radiating transitionsbetween these components are possible, provided that theselection rules

%ML = +1; %ML = 0; %ML = –1

are taken into account. In this case, therefore, there are a totalof nine permitted transitions. These nine transitions can begrouped into three groups of three transitions each, where alltransitions in a group have the same energy and hence thesame wavelength. Therefore, only three lines will be visible.

Fig. 4: Splitting up of the components in the magnetic fieldand permitted transitions.

The first group where %ML = –1 gives a s-line the light ofwhich is polarized vertically to the magnetic field. The middlegroup %ML = 0 gives a p-line. This light is polarized parallel tothe direction of the field. The last group where %ML = +1 givesa s-line the light of which is again polarized vertically to themagnetic field.

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In the absence of the analyser all three lines can be seensimultaneously. Each ring which was observed in the absenceof a magnetic field is split into three rings when a magneticfield is applied. Inserting the analyser the two s-lines can beobserved exclusively if the analyser is in the vertical position,while only the p-line appears if the analyser is turned into itshorizontal position (transverse Zeeman effect). Turning theelectromagnet by 90° the light coming from the spectral lampparallel to the direction of the field can also be studied sincethe pole-shoes have been drilled. It can be shown that thislight is circular polarized light. Whatever the position of theanalyser may be, each of the rings seen without a magneticfield is now permanently split into two rings in the presence ofa magnetic field (longitudinal Zeeman effect). Fig. 5 summariz-es the facts.

Fig. 5: Longitudinal and transvese Zeeman effet.

Turning the electromagnet back for the observation of the twos-lines of the transverse Zeeman effect it is easy to see thatthe size of the splitting increases with increasing magneticfield strength. For a quantitative measurement of this splittingin terms of number of wavelengths, a Fabry-Perot interferome-ter is used, the functioning of which may briefly be explained.

The Fabry-Perot étalon has a resolution of approximately300000. That means that a wavelength change of approxi-mately 0.002 nm can still be detected.

Fig. 6: Reflected and transmitted rays at the parallel surfaces(1) and (2) of the étalon. The étalon spacing is t.

The étalon consists of two parallel flat glass plates coated onthe inner surface with a partially reflecting layer. Let us consid-er the two partially transmitting surfaces (1) and (2) in Fig. 6seperated by a distance t. An incoming ray forming an angle 6with the normal to the plates will be split into the rays AB, CD,EF, etc. the path difference between the wave fronts of twoadjacent rays (for example, AB and CD) is

E = BC + CK

where, obviously, BK is normal to CD. With

CK = BC cos 2 6 and BC cos 6 = t

we obtain

E = BCK = BC (1 + cos 26)= 2 BC cos2 6= 2 t cos 6

and for a constructive interference to occur one mustdemand:

nl = 2 t cos 6

where n is an integer. If the refractive index of the mediumbetween the plates is m q 1, the equation still has to be modi-fied in the following way:

nl = 2 mt cos 6 (1)

Equation (1) is the basic interferometer equation. Let the par-allel rays B, D, F, etc. be brought to a focus by the use of a lensof focal length f as shown in Fig. 7.

Fig. 7: Focusing of the light emerging from a Fabry-Perotétalon. Light entering the étalon at an angle 6 isfocused onto a ring of radius r = f6 where ƒ is the focallength of the lens.

Then, when 6 fulfills equation (1), bright rings will appear inthe focal plane, their radius being given by

rn = ftan 6n � ƒ6n (2)

for small values 6n, e.g. rays nearly parallel to the optical axis.

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Since

cos 6n = n0 cos 60

with

we finally obtain

or (3)

If 6n is to correspond to a brigth fringe, n must be an integer.However, n0, which gives the interference at the center (cos 6= 1 or 6 = 0 in equation [1] ), is in general not an integer. If n1is the interference order of the first ring, clearly n1 < n0 sincen1 = n0 cos 6n1

. We then let

n1 = n0 – e ; 0 < e < 1

where n1 is the closest integer to n0 (smaller than N0). Thus,we have in general for the p-th ring of the pattern, as meas-ured from the center out,

np = (n0 – e) – (p – 1) (4)

Combining equation (4) with equations (2) and (3), we obtainfor the radii of the rings, substituting rp for rnp

,

(5)

we note that the difference between the squares of the radii ofadjacent rings is a constant:

(6)

e can be determined graphically plotting r2p versus p and

extrapolating to r2p = 0.

Now, if there are two components of a spectral line (splittingof one central line into two components) with wavelengths laand lb, which are very close to one another, they will havefractional orders at the center ea and eb:

where n1,a, n1,b is the interference order of the first ring. Hence,if the rings do not overlap by a whole order n1,a = n1,b and thedifference in wave numbers between the two components is

simply

(7)

Furthermore, using equations (5) and (6), we get

(8)

Applying equation (8) to the components a and b, yields

and

By substituting these fractional orders into equation (7), weget for the difference of the wave numbers:

(9)

From equation (6) it is clear that the difference between thesquares of the radii of component a,

is equal to (within a very small part) the same difference forcomponent b

Hence,

whatever the value of p may be. Similarly, all values

must be equal, regardless of p and their average can be takenas may be done for the different %-values. With E and % asaverage values we get for the difference of the wave numbersof the components a and b, anticipating m = 1,

(10)

Equation (10)* gives evidence of the fact that %O> does notdepend on the dimensions used in measuring the radii of thering system nor on the amplification of the interference pat-tern.

¢ nS �12t

d

¢

da,bp�1,p � rp�1,a

2 � rp�1,b2

¢ap�1,p � ¢b

p�1,p

¢bp�1,p � rp�1,b

2 � rp,b2 �

2f2

n0,b

¢ap�1,p � rp�1,a

2 � rp,a2 �

2f2

n0,a

¢n �1

2mt a

rp�1,a2

r2p�1,a � r2

p,a

�rp�1,b

2

r2p�1,b � r2

p,b

b

rp�1,b2

r2p�1,b � r2

p,b

� p � eb

rp�1,a2

r2p�1,a � r2

p,a

� p � ea

rp�1,a2

r2p�1 � rp

2 � p � e

¢n � na � nb �ea � eb

2mt

eb �2mt

la� n1,b � 2mt nb � n1,b

ea �2mt

la� n1,a � 2mt na � n1,a

rp�12 � rp

2 �2f2

n0

rp � B2f2

n0 · 21p � 1 2 � e

™n � B2 1n0 � n 2

n0

n � n0 a1 �™n

2

2b

n0 �2 mt

l

� n0 a1 � 2 sin2 ™n

2 b

n �2mtl

* Melissinos, Adv. Exp. in modern Physics

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Measurement and Evaluation1. Provided the ring pattern has been properly established asexplained in the section “set-up” above, the radii of the ringshave to be measured at different magnetic flux densities. Thenit is possible by using equation (10) to determine the corre-sponding difference in wave numbers ∆n.We proceed in two steps: first we take pictures of the ring pat-terns at different coil currents/magnetic field intensities. Thenin a second step the ring diameters in these pictures are mea-sured.To get a life picture from the camera go to the <File> menuand chose the entry <Capture Window>. In the capture win-dow the settings regarding e.g. contrast, brightness and satu-ration of the image can be optimized via the menu you get towhen choosing <Video Capture Filter> from the <Option>menu.When satisfied with the image quality and a certain coil cur-rent (magnetic field) is established, the picture is captured bychoosing <Still Image> from the <Capture> menu. Thisaction also closes the capture window and the pictureappears in the main window of the application. At this stage itis advisable to write the value of the coil current at which thepicture was taken into it by using the <Text> tool. This pre-vents a mix-up later on.The above procedure is repeated using different magneticfields for instance, with coil currents of 5 A, 6 A, 8 A and 10 A. Once these pictures have been collected, we proceed to mea-sure the radii of the rings choosing <Circle> from the<Measure> menu. By dragging the mouse across the picture,a circle is drawn. Fit this circle in size and position as good aspossible to the innermost ring. You will see that radius, areaand perimeter of the circle will be displayed in a little box andin a table below the picture (compare Fig. 8). What we aremainly interested in is the radius of the circle, this is r1,a. Notethat the units (mm, mm, cm) are of no importance in this exper-iment, that means no calibration of the camera has to be per-formed. Proceed to draw and fit circles to as many sets ofrings as are visible in the picture, this will give you; r1,b; r2,a;r2,b; r3,a…. Do the same with the other pictures captured. In the classical version without the CCD-camera a set of radiiof rings is determined in the following way:

The slash of the scale „0“ is shifted horizontally along a diam-eter through the ring pattern until it coincides, for instance,with the fourth ring to the left. A magnetic field correspondingto a coil current of lets say 4 A is established and the splittingof the rings observed. The analyzer is put into the verticalposition so that only the two s-lines appear. The „0“ slash isnow adjusted to coincide perfectly with the outer ring of thetwo rings, into which the fourth ring has split. The first readingon the socket of the sliding mount is taken. The „0“ slash isthen moved from left to right through all the rings. The lastreading is taken when the „0“ slash coincides with the outerring of the fourth ring to the right. The last reading minus thefirst reading divided by two then provides the radius r4,b.Evaluating the previous readings in a similar way leads to thefollowing radii:

I = 4 [A]: r4,b ; r4,a ; r3,b ; r3,a ; r2,b ; r2,a ; r1,b ; r1,a

Further sets of radii are received when repeating the proce-dure, for instance, for coil currents of 5 A, 6 A, 8 A and 10 A.Using the slide mount, all readings are done in “mm“ with aprecision of 1/100th of a mm. Still, the dimension used is notsignificant since it cancels out when evaluating due toequation (10).

Now the following square array can be formed for each set ofradii measured, regardless if they are measured with the CCD-camera and the software or the classical way:

The mean values % and E are calculated here in the followingway:

*

The étalon spacing is t = 3 · 10-3 [m].

Equation (10) was used to calculate the difference in wavenumbers of the two s-lines as a function of the magnetic fluxdensity and the coil current respectively. The following tablesummarizes the results:

d �14

a4

p�1 da,b

p

¢ �14

a2

p�11¢a

2p,2p�1 � ¢b2p,2p�1 2

component ring number

1 2 3 4

a r21,a %a

2,1 r22,a %a

3,2 r23,a %a

4,3 r24,a

Ea1,b Ea

2,b Ea

3,b Ea

4,b

b r21,b %b

2,1 r22,b %b

3,2 r23,b %b

4,3 r24,b

¢n

Fig. 8: Screenshot of software used to measure the radii of theinterference rings.

I [A] B [mT] %O> [m-1]

4 417 43.0

5 527 52.2

6 638 59.0

8 810 75.4

10 911 83.6

* Note that not every available %-value can be used. Only alternate ones avoidloss of inormation.

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2. The difference in wave numbers of one of the T-lines withrespect to the central lines is %O> /2. For the radiating electronsthis means, for instance, a change in energy of

%E = EL,ML– EL–1,ML–1

= (11)

On the other hand the change in energy %E is proportional tothe magnetic flux density B. The factor of proportionalitybetween %E and B is mB, Bohr’s magneton.

%E = mBB (12)

Combining equations (11) and (12) results in an expressionfor mB :

mB = (13)

In Fig. 9 %O> /2 was plotted versus the magnetic flux density B.From the regression-line we find a mean value for

and its standard deviation. Hence,

mB =

= (9.06 ± 0.46) 10-24

The literature value for Bohr’s magneton is:

mB,Lit. = (9.273) 10-24

3. The electromagnet is turned by 90° to observe the longitu-dinal Zeeman effect. In the presence of a magnetic field (a coilcurrent of 8 A is recommended) each of the rings is alwayssplit up into two, whatever the position of the analyser may be.

A l/4-plate is generally used to convert linear into ellipticalpolarized light. In this experiment the l/4-plate is used in theopposite way. Namely, by means of the l/4-plate, insertedmanually between L2 and the analyser, the light of the longitu-dinal Zeeman effect is investigated. If the optic axis of the l/4-plate coincides with the vertical, it is observed that one ringdisappears if the analyser includes an angle of +45° with thevertical while the other ring disappears for a position of –45°.That means that the light of the longitudinal Zeeman effect ispolarized in a circular (opposed way).

JT

JT

hc a¢n 2Bb

¢n 2B

hc ¢n 2B

hc ¢n2

Fig. 9: Zeeman splitting of spectral line l = 643.8 nm as a function of Flux density B.

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