Department of Physics, Colorado State University PH 425 Advanced Physics Laboratory The Zeeman Effect. 1 Introduction. 2 Origin of the Zeeman Effect

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1 Department of Physics, Colorado State University PH 425 Advanced Physics Laboratory The Zeeman Effect (a) CAUTION: Do not look directly at the mercury light source. It is contained in a quartz tube. The ultraviolet radiation is not absorbed in the quartz and could damage your eyes. (b) Notice the exposed terminals on the magnet. The power supply voltage is high enough, and the current capacity is such, that you should be cautious. (c) Be sure that cooling water is flowing through the magnet before turning on the magnet power supply. (d) The Gaussmeter probe can be damaged easily. Handle it with care. 1 Introduction Michael Faraday was the first to attempt to observe an effect of magnetic fields on atomic spectra. In 1862 he looked at sodium flames, but could see no change when a strong magnetic field was applied. The effect was too small for his spectrometer to resolve. Several others tried with improved spectrometers, but it was not until 1896 that Pieter Zeeman finally succeeded. His experiments with sodium in magnetic fields of the order of 1 T showed that the spectral lines at and nm (the sodium D lines) split into four and six components respectively when the magnetic field was applied. Zeeman developed a classical theory based on an oscillating charge in a magnetic field. According to classical theory, an oscillating charge emits radiation at the frequency of oscillation, resulting in a single spectral line. Zeeman added a magnetic force to the three-dimensional harmonic oscillator equation and showed that the application of a magnetic field should result in the single spectral line being split into three components. Some spectral lines did indeed split into three components in a magnetic field, and this phenomenon came to be known as the normal Zeeman effect. Most lines, however, split into more than three components and such lines were said to exhibit the anomalous Zeeman effect. The full explanation of the Zeeman effect had to await the development of quantum mechanics and the discovery of electron spin. The explanation of the Zeeman effect was one of the early triumphs of quantum mechanics. Today the terms normal and anomalous do not accurately describe our understanding of the Zeeman effect since the anomalous effect is understood as well as the normal effect, but the historical terms remain in common usage. 2 Origin of the Zeeman Effect The Zeeman effect occurs due to the interaction of the net magnetic moment µ of an atom with an external magnetic field B. The operator H 1 representing the interaction is given by H 1 = µ B. (1) For ordinary magnetic fields this interaction is relatively small and may be treated as a perturbation. It is necessary to express µ in terms of angular momentum operators. According to quantum mechanics, each electron i has a spin angular momentum operator s i and an orbital angular momentum operator l i. In the LS coupling scheme (see books on quantum mechanics for more details) the net magnetic moment of a multielectron atom may be found by (i) adding vectorially the magnetic moments due to the orbital motion of 1

2 each electron to find the total moment due to orbital angular momentum, (ii) adding vectorially the magnetic moments due to the spin of each electron to find the total moment due to spin and (iii) adding the total spin moment and total orbital moment to get the total magnetic moment of the atom, µ = 1 e l i e s i = 1 e (L + 2S) 2 m m 2 m i i µ = 1 e (J + S) (2) 2 m Here, e represents the magnitude of the charge on the electron, m is the mass of the electron, the sum is over all the electrons of the atom, L is the total orbital angular momentum operator, S is the total spin angular momentum operator, and J = L + S represents the total angular momentum operator for the atom. Substituting Eq. 2 into Eq. 1 gives H 1 = e (J + S) B. (3) 2m The rigorous quantum mechanical treatment of this problem is too lengthy to present here; however, a simple physical approximation leads to the correct result. Due to spin-orbit coupling S and L are not constant; rather, each precesses about J. In the absence of a magnetic field, J is constant. When a magnetic field is applied J will precess about B. For fields which are not too strong the precession of L and S about J will be much faster than the precession of J about B. Thus, in Eq. 3 for S we use S avg, the projection of S along J, S avg = S J J. (4) J2 We evaluate the term S J as follows. Using L 2 = L L = (J S) (J S) = J 2 + S 2 2S J gives S J = (J 2 + S 2 L 2 )/2. Then, J + S J + S avg = J + J2 + S 2 L 2 2J 2 J. (5) Substituting Eq. 5 into Eq. 3 and using J B = J z B gives H 1 = eb (1 + J2 + S 2 L 2 ) 2m 2J 2 J z (6) To calculate the first-order shift of the atomic energy levels due to the Zeeman effect it is only necessary to calculate the expectation value of H 1. This calculation is easy because the expectation values of the operators in Eq. 6 are known. They are J 2 = j( j + 1) 2, S 2 = s(s + 1) 2, L 2 = l(l + 1) 2, and J z = m j. and Here l, s, and j are integers or half-integers characterizing the orbital, spin, and total angular momentum respectively, and m j = j, j 1,... j. The shift of the energy levels giving rise to the Zeeman effect becomes ( ) e E j = g j Bm j (7) 2m where g j = 1 + j( j + 1) + s(s + 1) l(l + 1) 2 j( j + 1) is called the Landé g-factor. The quantity e /2m is called the Bohr magneton µ B and has the value J/T. Equations 7 and 8 are the basic relations used to interpret the Zeeman effect. Each different combination of (n, j, l, s, m j ) represents a different state for the electrons in an atom (n is the principal quantum number). For full shells j, l, and s all equal zero. Generally, the different states have 2 (8)

3 different energies. In the absence of a magnetic field, states with the same (n, j, l, s) but different m j have the same energy. Because there are 2 j + 1 possible m j values for each j, Eq. 7 shows that the application of a magnetic field will split a single energy level into 2 j + 1 different levels. Each state is specified by a spectroscopic symbol, such as 4f 5d 3 H 5. The first term gives the one-electron states (4f 5d) of the valence electrons; the second term gives s, l and j according to spectroscopic notation. The superscript is equal to 2s + 1, the capital letters S, P, D, F, G, H etc. refer to l = 0, 1, 2, 3, 4, 5 etc., and the subscript is equal to j. Thus, in the example given there are two electrons outside a closed shell and these combine to give s = 1, l = 5, and j = 5. Figure 1: Energy level diagram illustrating the normal Zeeman effect. The vertical dashed and solid lines represent transitions with m j = 0 and m j = ±1 respectively. We now have the background needed to understand the Zeeman effect. Two examples will be given before considering the experiment to be performed. Figure 1 illustrates part of the energy level diagram of an atom which exhibits the normal Zeeman effect. The left-hand part of the figure shows the situation in the absence of a magnetic field. The 1 D 2 state of the atom lies higher in energy than the 1 P 1 state. If the atom is excited a transition from the higher to lower state may occur with the photon energy hν = hc/λ = E. Only one wavelength is observed. If a magnetic field is applied the energy levels split. For this example, both states haveg j = 1. (You should verify that this is the case before proceeding further.) The 1 D 2 level splits into five levels with a level separation of µ B B, while the 1 P 1 state splits into three levels with the same level separation. (You should also verify these statements before proceeding further.) If the atom is excited there may be transitions from the upper levels to the lower levels. However, a selection rule applies. Only those transitions in which m j changes by 0 or ±1 are allowed. The dotted lines in Fig. 1 correspond to transitions with m j = 0. The photon energy for these transitions is the same as in zero magnetic field. The three transitions illustrated on the left correspond to m j = +1. The photon energy for all three transitions is the same, and is µ B B less than the zero field case. Finally, the three transitions on the right correspond to m j = 1 and all three have a photon energy which is µ B B higher than the zero field case. Of course, you should work all of this out yourself. You can see that the zero field line splits into three components with the application of a magnetic field, and this is usually called the normal Zeeman effect. Figure 2 illustrates part of an energy level diagram for an atom which exhibits the anomalous Zeeman 3

4 effect. In this case a 3 D 2 state and a 3 P 1 state are shown. You should verify that the g j values are as given on the right. The energy level separation of the upper states is 7µ B B/6 while that of the lower states is 3µ B B/2. The photon energy shifts, relative to the zero field transition, are indicated at the bottom of the diagram and range from 9µ B B/6 to 9µ B B/6. Of course you should verify that this is true. Notice that the original single line in zero field is now split into nine lines! This situation illustrates the anomalous Zeeman effect. What is the crucial difference between Figs. 1 and 2 which gives nine lines in one case, but only three in the other? Figure 2: Energy level diagram illustrating the anomalous Zeeman effect. 3 Mercury Spectrum In this experiment we will study the Zeeman effect in mercury. The electron configuration of mercury is the ground state is 1s 2 2s 2 2p 6 3s 2 3p 6 3d 10 4s 2 4p 6 4d 10 5s 2 5p 6 4f 14 5d 10 6s 2. These are all filled subshells, thus the ground state of mercury has zero spin and zero orbital angular momentum. The excited levels involved in the optical spectrum consist of one of the 6s electrons being raised to a higher energy state, for example 6p. The two electrons (the remaining 6s electron and the electron in the higher energy state) can then give a total spin of the atom of either 0 or 1. Various combinations of spin, orbital angular momentum, and total angular momentum are possible. These combinations are designated by terms such as 1 S 0, 3 P 2, 3 P 1. The lowest energy states in mercury are shown in Fig. 3. The strongest visible and ultraviolet spectral lines are also indicated. The wavelengths of the transitions are indicated in nm. The most convenient line to work with in this experiment is the nm transition from 6s6d 1 D 2 to 6s6p 1 P 1. With the spectrometer used in this experiment it will not be possible to resolve the and the nearby nm lines. Fortunately, the nm line is quite weak. Most of the light you see at 579 nm is due to the nm line. Therefore, you can ignore the nm line in your analysis. Before coming to lab you should figure out how many spectral components there will be in the nm line when a magnetic field is turned on, and how large the splitting will be in a magnetic field of about 1 T. 4

5 Figure 3: Energy level diagram of mercury (not to scale, splittings are expanded for clarity). Wavelengths of several spectral lines are given in nm. Make a diagram similar to Figs. 1 and 2. 4 Experimental Apparatus and Procedure The experimental arrangement is shown in Fig. 4. A lens is used to focus light from the mercury lamp onto an adjustable spectrometer slit. A collimator lens focuses an image of the slit further down the optical train. The light then passes through the Fabry-Pérot interferometer and then into the spectrometer, where the light is split into its spectral lines by a prism. The image of the slit is inside the focussing tube, where it can be examined by either the eyepiece or the video camera. Figure 4: Experimental apparatus for the Zeeman effect experiment. Begin by inserting the eyepiece and looking for the various spectral lines. The large dial on the spectrometer can be turned to center the desired line in the eyepiece. Then the eyepiece can be swapped out for the camera. It will help if you open the slit until the yellow lines are almost overlapping. The wavelength shifts due to the Zeeman effect are quite small, and cannot be resolved by the spectrometer alone. Thus a highresolution Fabry-Pérot interferometer is used. It consists of two precisely parallel reflecting surfaces, as 5

6 shown in Fig. 5. You will probably want to consult additional reference material, such as an optics textbook, in order to understand the Fabry-Pérot interferometer better. Figure 5: Fabry-Pérot interferometer. The various reflected beams emerging from the right side interfere to produce the patterns illustrated in Fig. 6. If the light entering the Fabry-Pérot interferometer in Fig. 5 is of a single wavelength λ, rings of interference maxima will be seen in the emerging light at angles θ given by 2L cos θ = mλ where m is an integer. L is the optical path length of the interferometer (the actual length times the index of refraction). How sharp the rings appear depends on the reflectivity R of the reflecting surfaces and the quality of the collimator. When the magnetic field is turned off (one wavelength case) you should see through the eyepiece a vertical slice of the circular pattern shown in Fig. 6a. a b c Figure 6: Fabry-Pérot interference patterns. The solid and dashed lines represent two different wavelengths. (a) One wavelength; (b) two wavelengths with ν = (c/2l)( m + 1/2); (c) two wavelengths with ν = (c/2l) m, where m is an integer. With two or more wavelengths entering the Fabry-Pérot interferometer, you should see a superposition of the two or more ring patterns, as in Fig. 6b. When the two ring patterns happen to coincide, as in Fig. 6c, you should get rings that are as sharp as when you only had one wavelength. This coincidence provides a convenient means to measure the separation of two closely spaced spectral lines. The ring patterns for two wavelengths λ and λ coincide when 2L cos θ = mλ and 2L cos θ = m λ (9) where m and m are both integers. Let m = m + m and λ = λ + λ. Using Eq. 9 we have mλ = (m + m)(λ + λ). Solving for λ and neglecting the small m λ term we have λ λ 2 = m mλ. (10) Differentiating the fundamental relation ν = c/λ gives ν = c λ λ 2. (11) 6

7 Substituting mλ from Eq. 9 into Eq. 10, then combining Eqs. 10 and 11 gives ν = c m, (12) 2L where we have used the small angle approximation and set cos θ = 1. Equation 12 is the basic relation to be used with the Fabry-Pérot interferometer. Remember that Eq. 12 applies when the interference rings for the two spectral lines passing through the spectrometer coincide, yielding a sharp pattern as in Fig. 6c. Remember also that m is an integer. For the present case 2L = cm. The resolution of the Fabry-Pérot interferometer used in the present experiment may not be sufficient to easily resolve the lines as they split in a magnetic field. At zero field the pattern should look something like Fig. 6a. As the field is increased the lines may become blurry. When you reach the condition shown in Fig. 6b the lines may become sharper with twice the original number of fringes. As the field is increased further such that Eq. 12 is satisfied the rings should become sharp again with the original number of components. The best quantitative results can probably be obtained by looking for field values where the rings appear sharpest. 5 Analysis and Things to Consider 1. Observe the effect of the magnetic field on the nm line (the yellow line farthest left) in mercury. Make quantitative measurements of the magnetic fields where the Fabry-Pérot interference fringes become sharp. What is the value of m in each case? 2. Use your results to calculate a value of e/m for the electron, as Zeeman did. 3. Observe the nm line with a polarizer. Try it with the polarizer parallel, and perpendicular to the magnetic field. What do you observe. 4. Consider other spectral lines, in particular the green nm line. What do you expect for this line? What is observed? Which lines show the normal Zeeman effect and which show the anomalous Zeeman effect? 6 References 1. R. T. Weidner and R. L. Sells, Elementary Modern Physics, 3rd ed., Allyn and Bacon, pp F. K. Richtmeyer and E. H. Kennard, Introduction to Modern Physics, McGraw-Hill, 1947, pp. 75 and F. A. Jenkins and H. E. White, Fundamentals of Optics, 4th ed., McGraw-Hill, 1976, p TEC/WFP 8/85 RGL 1/97 Revised 12/06 by Stuart Field 7