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Zeeman Effect - Lab exercises 24 Pieter Zeeman Franziska Beyer August 2010

Zeeman Effect - Lab exercises 24 · Zeeman Effect - Lab exercises 24 Pieter Zeeman Franziska Beyer August 2010. 1 Overview and Introduction The Zeeman effect consists of the splitting

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Zeeman Effect - Lab exercises 24

Pieter Zeeman

Franziska Beyer

August 2010

1 Overview and Introduction

The Zeeman effect consists of the splitting of energy levels of atoms if they are situatedin a magnetic field. The distances between these components increases linearly with themagnetic field and can be used for the estimation of the specific charge e/m of the electron.The splitting can be observed by electronic transitions. The detection of the splitting requiresthe dispersion of the emitted light by usage of e.g. a prism. Thus the components can befound separately. The Zeeman splitting is rather small that’s why a high resolution isneeded which is realized in our case, by a Fabry-Perot spectrometer. By analyzing theproperties of the emitted photons, namely wavelength and type of polarization, one canlearn about the states of the electron before (initial) and after (final) the transition.

In the mid-nineteenth century, the widening of the atomic spectral lines was observedfor the first time. No satisfactory explanation to this broadening was found until the endof that century. Already at this time, a connection of this phenomenon to the presenceof magnetic field was proposed. In 1896, a Dutch physicist Pieter Zeeman, succeeded topartially explain the experimental results. He showed that the spectral line splitting can beclassified in, what is today known as, the normal Zeeman effect and the anomalous Zeeman

effect. While the normal Zeeman effect was in agreement with the classical theory developedby Lorentz, the anomalous Zeeman effect remained unexplained for the next thirty years.After the development of quantum theory in the early twentieth century, it turned out thatfor the understanding of the anomalous Zeeman effect the concept of quantum theory wasnecessary.

2 Experimental observation

Observation along the magnetic field vector corresponds to the longitudinal Zeeman effectand perpendicular to it to the transverse Zeeman effect, respectively, as it can be seen infigure 1.

B

π

σ

σ+

σ-transversal

longitudinal

Figure 1: Observed polarization directions in relation to the applied magnetic field, B.

The normal Zeeman effect is characterized by a triplet or doublet splitting of the spec-tral line in case of transverse or longitudinal observation, respectively. The middle line inthe triplets represents the component of the spectral line which is unaffected by the mag-netic field, the other two lines shift by the same amount to higher and lower wavelengths,respectively, due to the applied field. Depending on the observation direction, the polariza-tion of the split lines is different. In the longitudinal case, circular polarization occurs withopposite sense of rotation for the two components. Transversally, the middle component ofthe triplet is polarized parallel to the field and the other two perpendicular.

For the anomalous Zeeman effect, the splitting is more complicated, even if the shift isstill proportional and symmetrical to the applied field.

During the laboration we are going to study the normal Zeeman effect in transversal

observation direction.

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3 Classical explanation for the normal Zeeman effect

For singlet states, the spin is zero, S = 0, and the total angular momentum J is equal to theorbital angular momentum L, J = L. Magnetic moments, µ are inseparably connected tothe angular momentum, L= r×p = mervn. The interaction between the magnetic momentof an atom and the external magnetic field B causes splitting of the atomic energy levels.

We can look at separate electrons in the electron shell as point charges orbiting aroundthe nucleus. In such a case, each electron represents a loop carrying a current I. A particlewith electric charge e in a circular orbit with radius r and speed v gives a current (lookingat one loop only using Q = e and v= 2πr

t):

I =Q

t=

e

2πrv (1)

The magnetic moment, µ due to such a current loop is (negative sign due to the charge ofthe electron):

µ = Iπr2n = −e

2rvn = −gl

e

2me

L gl = 1 (2)

In an external magnetic field, the current loop experiences a torque T = µ xB andsubsequently a force with the corresponding potential energy U :

U = −µ ·B =e

2me

L ·B (3)

The angular momentum is conserved by precession, i.e. µ moves on a cone around B withthe Lamour frequency:

ωL = 2πfL =eBz

2me

(4)

The light emitted by the atom (frequency f0) superimposes with this precession frequencyas follows: v0 + vL and v0 − vL. This explains the normal Zeeman effect. Thus thepolarization of the emitted radiation is circular, looking in the direction of the field, due tothe circular motion of the precession. Perpendicular, the precession occurs as linear as weonly see its projection.

∆ml = −1, 0, +1

l = 1

l = 2

ml

2

1

0

−1

−2

1

0

−1

B 6= 0B = 0

Figure 2: Normal Zeeman effect for the transition between d and p levels.

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4 Quantum mechanical description the Zeeman effect

Emission of light is now regarded as consequence of an electronic transition from a level ofhigher energy (initial state E2) to a level of lower energy (final state E1). The transitionfrequency, f is

f =E2 − E1

h(5)

with h the Planck’s constant. The En are determined by the atomic structure. How-ever, a magnetic field (flux density B) interacts with the magnetic moments, µ of theelectrons, which thus possess additional potential energy, equation 3. Thus for an atom ina weak magnetic field, |B| < 1 T, the total energy is: E = Enl + ESO + U . Orientationand magnitude of µ in the different states are generally different. The transition frequencychanges to:

f =E02 − µ2 · B− (E01 − µ1 · B)

h(6)

The z-component of the orbital magnetic moment µLzis coupled to its angular momentum

Lz:

µLz= −gl

e

2me

Lz = −eh

2me

ml = −glµB · ml using : |L| = ml · h (7)

Equation 6 shows that f increases with increasing field and equation 7 indicates that quan-tum mechanical treatments of the angular momentum have to be taken into account.

4.1 The Quantum numbers

An electron in an atom is characterized by quantum numbers, which completely describeit’s energy {n l ml ms}:

1. principal quantum number n defines the electronic shell, n = 1 . . . nmax; where nmax

is the electron shell containing the outermost electron of that atom

2. orbital angular momentum quantum number l describes the sub shell (0 = s-orbital,1 = p-orbital, 2 = d-orbital, 3 = f -orbital, etc.), l = 0 . . . n − 1, |l| =

l(l + 1)h

3. magnetic quantum number (projection of angular momentum) ml describes the specificorbital (or ”cloud”) within that sub shell ml = −l . . . ml . . . l, total of 2l + 1 values

4. spin projection quantum number ms describes the electron spin with angular momen-tum vector s and quantum numbers s = ±1/2, |s| =

s(s + 1)h

For electrons, as for all fermions (particles with half-integer spin), the Pauli exclusionprinciple is valid: In an atom, there cannot exist a pair of electrons with an identicalquadruple of quantum numbers. The total angular momentum of an atom can be estimatedas vector sum of the individual electronic contributions. The total spin S and the totalangular momentum L are given by: S = |

ms|; L = |∑

ml|.There is a number of possibilities to combine ml and ms. Hund’s Rules helps to find

the ground state:

• completely filled sub shells do not contribute to J

• partially filled sub shells follow the Pauli exclusion principle and:

1. as much as possible electrons have parallel spin (S → max)

2. electrons occupy states to maximize L

• J = L ± S, ”+” for sub shells which are more than half filled and ”−” for less thanhalf filled sub shells

The atomic states are specified using n2S+1LJ.

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4.2 Magnetic moment

Equation 7 can be expressed for electrons by means of the Bohr magneton, µB

µB =eh

2me

= 9.274 · 10−24J/T (8)

The spin generates a magnetic moment, µS which is double of that of the orbital angularmomentum, µL:

µL = −glµBL and µS = −gsµBS gs ≈ 2 (9)

The exact free electron value for gs = 2.00232 can be obtained from quantum electrody-namics. Then the total angular momentum is: µJ = gj ·µBJ with the scaling factor gj , theLande factor, which for free atoms can take values between 1 and 2.

gj = 1 +J(J + 1) − L(L + 1) + S(S + 1)

2J(J + 1)(10)

Particular cases:

• S = 0 → J = L → gj = 1 → normal Zeeman effect

• L = 0 → J = S → gj = 2 (e.g. free electrons)

• S 6= 0, L 6= 0 → gj = 1 · · · 2

Equation 10 gives an approximative value of the Lande factor.

4.3 The general (anomalous) Zeeman effect

The general, also known as anomalous, Zeeman effect is present for atoms with non-zerototal spin. Since electronic spin can have only two values, namely + 1

2and − 1

2, all atoms

with an odd number of electrons posses a non-zero spin. The orbiting electrons in the atomare equivalent to a classical magnetic gyroscope. The torque applied by the field causes theatomic magnetic dipole to precess around B (Larmor precession). The external magneticfield therefore causes J to precess slowly about B. L and S meanwhile precess more rapidlyabout J due to the spin-orbit interaction, see Fig. 3. The speed of precession about B isproportional to the field strength.

S

LJ

B

Figure 3: Addition of angular moments.

spin-orbit interaction

The orbital and the spin magnetic moments do not interact independently with the smallexternal magnetic field (|B| < 1T). Rather, the orbital and spin magnetic moments interactwith each other in such a way to form a combined magnetic moment µj that interact withthe external field. The orbital motion of the electron about the nucleus results in a magneticfield at the location of the electron. The spin magnetic moment of the electron interactswith this field so as to couple the spin and the orbital magnetic moments together. The

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magnitude of the field seen by the electron is approximately 1 T. If the external field is higherthan the internal than the spin and the orbital magnetic moments decouple and interactindependently with the external field.

The angular momenta combine vectorial to form a total angular momentum J = L + S.J is the angular momentum with a definite projection along the z -axis , see Fig. 3. Themagnitude of J is

J(J + 1)h where the total angular momentum quantum number isdetermined by J = |L − S|, |L − S| + 1, . . . , L + S. The projection a long the z -axis of thetotal angular momentum is the quantum number mj :

Jz = Lz + Sz = (ml + ms)h = mj h (11)

The angular momentum and also the magnetic moments are quantized along the mag-netic field, which is here chosen in z-axis. So for a multi-electron atom the expression forthe energy shift or the additional potential energy in the external field is given by:

U = −µJz· Bz = gjµBmJBz (12)

Without a field, the states with different mJ have equal energy (degenerate states). In-creasing B, the states split and more transitions are possible. There exist following selection

rules ∆mJ = ±1 or 0, but not 0 → 0 if ∆J = 0 see allowed transitions in figure 4. Thedistances between neighboring energy levels amounts to µBgjBz.

1/22s1/2 1/2

−1/2

J = L + S

−1/2

1/21/2

−3/2

−1/2

1/2

3/2

mj

B 6= 0

Anomalous Zeeman-effect

J

3/2

Fine structure

3p3/2

B = 0

3p1/2

l = 0, s = 1/2

l = 1, s = 1/2

3p

2s

∆mj = −1, 0, +1( not mj = 0 → mj = 0 if ∆J = 0)

∆J = −1, 0, +1( not J = 0 → J = 0)∆L = −1, 0, +1∆S = 0

Figure 4: General or anomalous Zeeman splitting for the transition between 3p and 2slevels (e.g. Na-doublet).

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Figure 5: Ne spectrum

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5 Questions and Exercises

The questions and exercises should be prepared before the laboration takes

place. During the experiments we will discuss the answers.

1. Explain the Zeeman effect and its experimental detection. What means ”transverse”and ”longitudinal” Zeeman effect?

2. Why does the magnetic field force the magnetic dipoles of atoms to precess, insteadof aligning with the field?

3. In which cases does the classical description fail? Which observations cannot be ex-plained?

4. Why do completely filled shells not contribute to the total angular momentum J of anatom?

5. Why is a normal Zeeman effect expected for the transition 31D2 ↔ 21P1.

6. Without magnetic field we have only one transition, which gives rise to one spectralline. With magnetic field present, we get multiple transitions resulting from the split-ting of spectral lines. According to Fig. 2, can you imagine the transitions between p

and s levels and between f and d levels?

7. Compare Fig. 2 and Fig. 4. Try to explain how the presence of electronic spin changesatomic energy levels.

8. What implies setting Lande factor to unity?

9. Why is the general Zeeman effect also known as ”anomalous” Zeeman effect?

10. According to Fig. 3 and Fig. 4, explain what is spin-orbit coupling, fine structure andhow this is formed.

11. During the lab exercises you will observe certain transitions in neon, with wavelengths5852 A, 6074 A and 6164 A, Fig. 5.Calculate g for the upper (ovre niva) and the lower (undre niva) levels. Draw atransition schema for the possible transitions.Hint: The value of the quantum numbers are given in Fig. 5 and by 2S+1LJ . J arethe fine structure levels and J = |L − S|, |L − S| + 1, . . . , L + S

12. The difference in wavelengths ∆λ between Zeeman components as a function of mag-netic field is given by the relation (derive this relation):

∆λ =gjµBBz∆mJλ2

hc(13)

Our magnetic field is approximately 0.1T. Take the g values determined in the previousexercise. Calculate the approximate value of ∆λ at that field for some of the mentionedtransitions in neon, you will later observe during this lab. You should get a fewhundredths of A for ∆λ .

That also means, that the difference between spectral lines under the influence of themagnetic field will be a few hundredths of A. During the lab you will measure this differencein wavelengths, and see that the Zeeman effect exists in reality. After all, it is nice to beable to compare measurements and theoretical results. The most important contributionfrom the theory is just g-factor, which, as already shown, has a relatively simple expression(10).

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6 Measurements

1. We start with observing the light from a Na lamp. This light consists only of a smallpart in the visible region (yellow) which is composed of two close lying wavelengths,the famous Na-doublet. You will observe the doublet without the interferometer, toget an impression of the prism’s capability to separate the neighboring wavelengths(resolution).

λ(D1) = A (14)

λ(D2) = A (15)

Out of your result, you can guess the spectrometer’s resolution, which is for 6000 Aaround

A. (Keep this in mind - this belongs to the general physical properties.)

2. As can be seen from the Fig. 5, the neon spectrum contains a large quantity of lines. Inorder to be sure that we are looking at the right lines, we have to check the wavelengthscale on the spectrometer. There is no need to do any precise calibration, because wewill not determine wavelengths in neon. But to be on the safe side, you should checkthe gradation. This can be done using a Cd-lamp spectrum – for which you will findwavelength values for each lines on a paper in the lab.

3. The next step is to increase the wavelength resolution using a Fabry-Perot interferom-eter. Put it in the beam path and study its function. The Zeeman effect is difficultto observe since the splitting causes only a very small differences in wavelengths (fewhundredths of A). From exercise 1 it is clear that the resolution of the spectrometerprism is far too coarse to distinguish between the Zeeman lines. If the incoming lightis composed of multiple discrete wavelengths, they will be refracted differently by theprism. Each wavelength will form an image of the light entrance opening in the fo-cal plane of the ocular. This kind of spectrometer can give a resolution of a few A.However, by putting the interferometer in front of the collimator lens it is possibleto increase significantly the resolution of the instrument. Let’s see how Fabry-Perotinterferometer is constructed and how it works.

Fabry-Perot interferometer

A Fabry-Perot interferometer works in the same way as an interference filter. It iscomposed of two glass plates covered each with semi-reflective coating. The incominglight beam is partially transmitted through the semi-reflective mirror and partiallyreflected as shown in Fig. 6a. The separated light beams coming out of the interferom-eter differ not only in intensity but also in phase. This phase shift will cause them tointerfere with each other. If the out-coming light from the interferometer is focused bya lens the resulting interference pattern looks like shown in Fig. 6b. The interferencepattern looks as in Fig. 7, when observed through a slit.

By measuring the distances ∆R and dR between the interference lines and Zeeman

splitting lines, one can calculate the difference in wavelength between atomic transitionlines and Zeeman lines as given by the equation:

∆λ =dR

∆R·λ2

2d=

2dR

∆R + ∆R′·λ2

2d(16)

∆R and dR are measured by cross-hairs that you can observe in the ocular, moved bya micrometer screw. d is given as d = 1 cm.

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Semi−reflective mirrors

d

a) b)

Figure 6: a) Light path through interferometer; b) Interference pattern.

4. Zeeman splitting observation

The transition, which we are observing, will split into the three lines under the influenceof the magnetic field: one π-component and two σ-components. This results in threenarrow-lying circle segments after passing the interferometer. Because of the very smallsplitting, the three lines are placed very close to each other. This makes it difficultto really detect splitting and to distinguish the lines. Since the σ and π componentsare polarized perpendicularly to each other, it is possible to block one component by apolarizer and transmit the other component. By changing the polarization direction,one can switch between the two polarization directions and thus between the threelines.

R∆∆R’

b)

dR

πσ

σ

a)

Figure 7: Interference pattern of the spectral lines as seen in the eyepiece. a) Withoutmagnetic field, or with σ components blocked by the polarizer; b) Zeeman splitting in thepresence of the magnetic field. Interference pattern.

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Measurement procedure:

• Turn on the neon lamp and focus carefully on the entrance slit; adequate openingsize 0.075 mm.

• Look first at the line at 5852 A.

• Turn on the power supply to the electromagnet - check before that the outgoingvoltage is zero.

• Increase the voltage, thus the magnetic field and check if you can observe thewidening of the diffraction line – no polarizer on the beam path.

• Place the polarizer on the beam path. Turn it and convince yourself that you canfilter out either π-component or σ-components.

• Change the magnetic field, check how dR changes and observe how two neigh-boring diffraction orders can overlap with each other, i.e., dR → ∆R.

• Measure ∆R′ , ∆R and 2dR for around eight diffraction orders. The easiest is tosuccessively measure values for neighboring lying orders. Take care not to change

voltage to the magnet during the measurements, because the distance of the line

splitting is dependent on the magnetic field!

• Measure the magnetic field with a gauss-meter.

5. Finally, you can calculate the experimental values for the actual levels and comparethem to the theoretical values obtained from Landes equation. As a value of dR

∆Rwe

take 2dR∆R + ∆R′

in each order. The mean value over all measured orders is then put inthe actual formula.

Estimate relative error of the input values. Which factor gives the biggest contributionto the total error in g in equation (13)?

Try to summarize what you get out of this laboratory exercise. Which new thingshave you learnt and which previous knowledge have you improved?

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Zeeman splittingin neon

Colour Intensity Upper level Lower level

Figure 5: Ne spectrum