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 of energy levels of atoms if they are situated in a magnetic field. The distances between these components increases linearly with the magnetic 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 requires the dispersion of the emitted light by usage of e.g. a prism. Thus the components can be found separately. The Zeeman splitting is rather small that s why a high resolution is needed which is realized in our case, by a Fabry-Perot spectrometer. By analyzing the properties of the emitted photons, namely wavelength and type of polarization, one can learn 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 observed for the first time. No satisfactory explanation to this broadening was found until the end of that century. Already at this time, a connection of this phenomenon to the presence of magnetic field was proposed. In 1896, a Dutch physicist Pieter Zeeman, succeeded to partially explain the experimental results. He showed that the spectral line splitting can be classified 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 developed by 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 that for the understanding of the anomalous Zeeman effect the concept of quantum theory was necessary. 2 Experimental observation Observation along the magnetic field vector corresponds to the longitudinal Zeeman effect and perpendicular to it to the transverse Zeeman effect, respectively, as it can be seen in figure 1. B σ π transversal σ + longitudinal σ- 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 spectral line in case of transverse or longitudinal observation, respectively. The middle line in the triplets represents the component of the spectral line which is unaffected by the magnetic 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 polarization of the split lines is different. In the longitudinal case, circular polarization occurs with opposite sense of rotation for the two components. Transversally, the middle component of the 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 is still proportional and symmetrical to the applied field. During the laboration we are going to study the normal Zeeman effect in transversal observation direction. 1

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 the orbital angular momentum L, J = L. Magnetic moments, µ are inseparably connected to the angular momentum, L= r p = m e rvˆn. The interaction between the magnetic moment of 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 around the nucleus. In such a case, each electron represents a loop carrying a current I. A particle with electric charge e in a circular orbit with radius r and speed v gives a current (looking at one loop only using Q = e and v= 2πr t ): I = Q t = e 2πr v (1) The magnetic moment, µ due to such a current loop is (negative sign due to the charge of the electron): µ = Iπr 2ˆn = e 2 rvˆn = g e l L g l = 1 (2) 2m e In an external magnetic field, the current loop experiences a torque T = µxb and subsequently a force with the corresponding potential energy U: U = µ B = e 2m e L B (3) The angular momentum is conserved by precession, i.e. µ moves on a cone around B with the Lamour frequency: ω L = 2πf L = eb z 2m e (4) The light emitted by the atom (frequency f 0 ) superimposes with this precession frequency as follows: v 0 + v L and v 0 v L. This explains the normal Zeeman effect. Thus the polarization of the emitted radiation is circular, looking in the direction of the field, due to the circular motion of the precession. Perpendicular, the precession occurs as linear as we only see its projection. l = 2 B = 0 B 0 m l 2 1 0 1 2 1 l = 1 0 1 m l = 1, 0, +1 Figure 2: Normal Zeeman effect for the transition between d and p levels. 2

4 Quantum mechanical description the Zeeman effect Emission of light is now regarded as consequence of an electronic transition from a level of higher energy (initial state E 2 ) to a level of lower energy (final state E 1 ). The transition frequency, f is f = E 2 E 1 (5) h with h the Planck s constant. The E n are determined by the atomic structure. However, a magnetic field (flux density B) interacts with the magnetic moments, µ of the electrons, which thus possess additional potential energy, equation 3. Thus for an atom in a weak magnetic field, B < 1 T, the total energy is: E = E nl + E SO + U. Orientation and magnitude of µ in the different states are generally different. The transition frequency changes to: f = E 02 µ 2 B (E 01 µ 1 B) (6) h The z-component of the orbital magnetic moment µ Lz is coupled to its angular momentum L z : e µ Lz = g l L z = e h m l = g l µ B m l using : L = m l h (7) 2m e 2m e Equation 6 shows that f increases with increasing field and equation 7 indicates that quantum 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 describe it s energy {n l m l m s }: 1. principal quantum number n defines the electronic shell, n = 1...n max ; where n max 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) m l describes the specific orbital (or cloud ) within that sub shell m l = l... m l... l, total of 2l + 1 values 4. spin projection quantum number m s describes the electron spin with angular momentum 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 exclusion principle is valid: In an atom, there cannot exist a pair of electrons with an identical quadruple of quantum numbers. The total angular momentum of an atom can be estimated as vector sum of the individual electronic contributions. The total spin S and the total angular momentum L are given by: S = m s ; L = m l. There is a number of possibilities to combine m l and m s. 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 than half filled sub shells The atomic states are specified using n 2S+1 L J. 3

4.2 Magnetic moment Equation 7 can be expressed for electrons by means of the Bohr magneton, µ B µ B = e h 2m e = 9.274 10 24 J/T (8) The spin generates a magnetic moment, µ S which is double of that of the orbital angular momentum, µ L : µ L = g l µ B L and µ S = g s µ B S g s 2 (9) The exact free electron value for g s = 2.00232 can be obtained from quantum electrodynamics. Then the total angular momentum is: µ J = g j µ B J with the scaling factor g j, the Landé factor, which for free atoms can take values between 1 and 2. Particular cases: g j = 1 + J(J + 1) L(L + 1) + S(S + 1) 2J(J + 1) (10) S = 0 J = L g j = 1 normal Zeeman effect L = 0 J = S g j = 2 (e.g. free electrons) S 0, L 0 g j = 1 2 Equation 10 gives an approximative value of the Landé factor. 4.3 The general (anomalous) Zeeman effect The general, also known as anomalous, Zeeman effect is present for atoms with non-zero total spin. Since electronic spin can have only two values, namely + 1 2 and 1 2, all atoms with an odd number of electrons posses a non-zero spin. The orbiting electrons in the atom are equivalent to a classical magnetic gyroscope. The torque applied by the field causes the atomic magnetic dipole to precess around B (Larmor precession). The external magnetic field therefore causes J to precess slowly about B. L and S meanwhile precess more rapidly about J due to the spin-orbit interaction, see Fig. 3. The speed of precession about B is proportional to the field strength. B L J S Figure 3: Addition of angular moments. spin-orbit interaction The orbital and the spin magnetic moments do not interact independently with the small external magnetic field ( B < 1T). Rather, the orbital and spin magnetic moments interact with each other in such a way to form a combined magnetic moment µ j that interact with the external field. The orbital motion of the electron about the nucleus results in a magnetic field at the location of the electron. The spin magnetic moment of the electron interacts with this field so as to couple the spin and the orbital magnetic moments together. The 4

magnitude of the field seen by the electron is approximately 1 T. If the external field is higher than the internal than the spin and the orbital magnetic moments decouple and interact independently 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. The magnitude of J is J(J + 1) h where the total angular momentum quantum number is determined by J = L S, L S + 1,...,L + S. The projection a long the z-axis of the total angular momentum is the quantum number m j : J z = L z + S z = (m l + m s ) h = m j h (11) The angular momentum and also the magnetic moments are quantized along the magnetic field, which is here chosen in z-axis. So for a multi-electron atom the expression for the energy shift or the additional potential energy in the external field is given by: U = µ Jz B z = g j µ B m J B z (12) Without a field, the states with different m J have equal energy (degenerate states). Increasing B, the states split and more transitions are possible. There exist following selection rules m J = ±1 or 0, but not 0 0 if J = 0 see allowed transitions in figure 4. The distances between neighboring energy levels amounts to µ B g j B z. Fine structure 3 p 3/2 B = 0 J 3/2 B 0 Anomalous Zeeman-effect m j 3/2 1/2 3p 1/2 3/2 l = 1, s = 1/2 3 p 1/2 1/2 1/2 1/2 J = L + S l = 0, s = 1/2 2s 2 s 1/2 1/2 1/2 1/2 S = 0 L = 1,0, +1 J = 1,0, +1( not J = 0 J = 0) m j = 1,0, +1( not m j = 0 m j = 0 if J = 0) Figure 4: General or anomalous Zeeman splitting for the transition between 3p and 2s levels (e.g. Na-doublet). 5

Figure 5: Ne spectrum 6

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, instead of aligning with the field? 3. In which cases does the classical description fail? Which observations cannot be explained? 4. Why do completely filled shells not contribute to the total angular momentum J of an atom? 5. Why is a normal Zeeman effect expected for the transition 3 1 D 2 2 1 P 1. 6. Without magnetic field we have only one transition, which gives rise to one spectral line. With magnetic field present, we get multiple transitions resulting from the splitting 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 changes atomic energy levels. 8. What implies setting Landé 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 and how this is formed. 11. During the lab exercises you will observe certain transitions in neon, with wavelengths 5852 Å, 6074 Å and 6164 Å, Fig. 5. Calculate g for the upper (övre nivå) and the lower (undre nivå) levels. Draw a transition schema for the possible transitions. Hint: The value of the quantum numbers are given in Fig. 5 and by 2S+1 L J. J are the 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 magnetic field is given by the relation (derive this relation): λ = g jµ B B z m J λ 2 hc Our magnetic field is approximately 0.1 T. Take the g values determined in the previous exercise. Calculate the approximate value of λ at that field for some of the mentioned transitions in neon, you will later observe during this lab. You should get a few hundredths of Å for λ. That also means, that the difference between spectral lines under the influence of the magnetic field will be a few hundredths of Å. During the lab you will measure this difference in wavelengths, and see that the Zeeman effect exists in reality. After all, it is nice to be able to compare measurements and theoretical results. The most important contribution from the theory is just g-factor, which, as already shown, has a relatively simple expression (10). (13) 7

6 Measurements 1. We start with observing the light from a Na lamp. This light consists only of a small part 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, to get an impression of the prism s capability to separate the neighboring wavelengths (resolution). λ(d 1 ) = Å (14) λ(d 2 ) = Å (15) Out of your result, you can guess the spectrometer s resolution, which is for 6000 Å around Å. (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. In order to be sure that we are looking at the right lines, we have to check the wavelength scale on the spectrometer. There is no need to do any precise calibration, because we will not determine wavelengths in neon. But to be on the safe side, you should check the gradation. This can be done using a Cd-lamp spectrum for which you will find wavelength values for each lines on a paper in the lab. 3. The next step is to increase the wavelength resolution using a Fabry-Perot interferometer. Put it in the beam path and study its function. The Zeeman effect is difficult to observe since the splitting causes only a very small differences in wavelengths (few hundredths of Å). From exercise 1 it is clear that the resolution of the spectrometer prism is far too coarse to distinguish between the Zeeman lines. If the incoming light is composed of multiple discrete wavelengths, they will be refracted differently by the prism. Each wavelength will form an image of the light entrance opening in the focal plane of the ocular. This kind of spectrometer can give a resolution of a few Å. However, by putting the interferometer in front of the collimator lens it is possible to increase significantly the resolution of the instrument. Let s see how Fabry-Perot interferometer is constructed and how it works. Fabry-Perot interferometer A Fabry-Perot interferometer works in the same way as an interference filter. It is composed of two glass plates covered each with semi-reflective coating. The incoming light beam is partially transmitted through the semi-reflective mirror and partially reflected as shown in Fig. 6a. The separated light beams coming out of the interferometer differ not only in intensity but also in phase. This phase shift will cause them to interfere with each other. If the out-coming light from the interferometer is focused by a lens the resulting interference pattern looks like shown in Fig. 6b. The interference pattern 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 transition lines and Zeeman lines as given by the equation: λ = dr R λ2 2d = 2dR R + R λ2 2d R and dr are measured by cross-hairs that you can observe in the ocular, moved by a micrometer screw. d is given as d = 1 cm. (16) 8

d Semi reflective mirrors 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 influence of the magnetic field: one π-component and two σ-components. This results in three narrow-lying circle segments after passing the interferometer. Because of the very small splitting, the three lines are placed very close to each other. This makes it difficult to really detect splitting and to distinguish the lines. Since the σ and π components are polarized perpendicularly to each other, it is possible to block one component by a polarizer and transmit the other component. By changing the polarization direction, one can switch between the two polarization directions and thus between the three lines. σ π σ dr R R a) b) Figure 7: Interference pattern of the spectral lines as seen in the eyepiece. a) Without magnetic field, or with σ components blocked by the polarizer; b) Zeeman splitting in the presence of the magnetic field. Interference pattern. 9

Measurement procedure: Turn on the neon lamp and focus carefully on the entrance slit; adequate opening size 0.075 mm. Look first at the line at 5852 Å. Turn on the power supply to the electromagnet - check before that the outgoing voltage is zero. Increase the voltage, thus the magnetic field and check if you can observe the widening of the diffraction line no polarizer on the beam path. Place the polarizer on the beam path. Turn it and convince yourself that you can filter out either π-component or σ-components. Change the magnetic field, check how dr changes and observe how two neighboring diffraction orders can overlap with each other, i.e., dr R. Measure R, R and 2dR for around eight diffraction orders. The easiest is to successively 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 compare them to the theoretical values obtained from Landés equation. As a value of dr R we 2dR take R + R in each order. The mean value over all measured orders is then put in the actual formula. Estimate relative error of the input values. Which factor gives the biggest contribution to the total error in g in equation (13)? Try to summarize what you get out of this laboratory exercise. Which new things have you learnt and which previous knowledge have you improved? 10

Zeeman splitting in neon Colour Intensity Upper level Figure 5: Ne spectrum Lower level