Experiment 10. Zeeman Effect. Introduction. Zeeman Effect Physics 244

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1 Experiment 10 Zeeman Effect Introduction You are familiar with Atomic Spectra, especially the H-atom energy spectrum. Atoms emit or absorb energies in packets, or quanta which are photons. The orbital motion of electrons in an atom, gives rise to tiny magnetic dipoles, each having a magnetic moment, μ = (e/2m) L, Where, e and m are charge and rest mass of an electron and L is its orbital angular momentum. Since the charge on an electron is negative, μ and L are anti-parallel. You know that L has (2l+1) Page 10-1

2 orientations in space. Thus μ too has (2l+1) orientations in space. This is shown in the diagram below: Fig. shows the allowed values of L z, for l = 1,2 and 10. The numbers on the z-axis are the values of the magnetic quantum number m l. The figures are drawn to different scales. In the absence of an external magnetic field, there is spherical symmetry to the electron distributions (circular orbits of electrons revolving round the nucleus). If we apply an external magnetic field, however, this spherical symmetry gets broken because the magnetic field gives a unique direction, say, along the z-axis, which gives rise to the Zeeman Effect. The figure below shows the Zeeman Effect. In the absence of field B, an atomic level with l = 2 (and no spin moment) becomes 5-fold degenerate with energy E 0. When B is switched on the level splits in a multiplet of 5 equally spaced levels. Page 10-2

3 In the picture below, a transition between an excited state of Helium (l = 1) and the ground state (l = 0) is depicted with field off and on. With B on, there are 3 distinct transitions possible due to the Zeeman splitting and hence 3 distinct spectral lines (triplet) as shown, with different frequencies. Page 10-3

4 An Additional Complication: The Lande g Factor Again, as an atomic electron orbits around the nucleus, it produces a magnetic moment: μ l = -(e/2m) L An atomic electron also has a spin angular momentum S, and so we would expect it to also have a spin magnetic moment, μ s. This magnetic moment due to the spin motion is proportional to S, or, μ s = - g S, g is a constant called the spin gyromagnetic ratio, or the Lande g-factor. In the case of orbital motion, the gyromagnetic ratio is seen to be e/2m. The spin g-factor will not have necessarily this same value, since it would depend upon the distribution of charge and mass within the electron. Experiment shows that the spin gyromagnetic ratio is e/m, or just twice the value of the orbital ratio (orbital g-factor), e/2m. That is: μ s = -( e/m) S The total magnetic moment of any electron is just the sum of its orbital and spin magnetic moments: μ total = μ l + μ s = -e/2m { L + 2S } Page 10-4

5 For historical reasons, the splitting of levels and spectral lines due to the various orbital m values (ignoring the effect of spin) is called the normal Zeeman effect. In cases where spin does contribute, we have the anomalous Zeeman effect. For example, the splitting of the He line (transition from l=1 to l=0 level shown in the bottom figure on the previous page) of energy 21.0 ev in to a triplet is an example of normal Zeeman effect, since spin plays no part in it. On the other hand the Zeeman pattern that you are studying for the 546 nm transition in mercury is an example of the anomalous Zeeman effect, where spin does play an important role. Consequently, we do need to take into account the g-factors for the two levels involved in the transition. The anomalous Zeeman effect in a mercury atom By applying an external magnetic field B, one can change an atom s energy level by an amount equal to the work done by the field B in rotating these tiny magnetic dipoles until they become parallel to the field B. The work done is W= - μ.b This implies that the energies of the photons emitted or absorbed by the atom will change by putting an atom in a magnetic field; we change its energy spectrum! This effect was first observed by the Netherlands physicist Pieter Zeeman, who received the 1902 Nobel Prize in Physics. Zeeman s is one of the landmark experiments in Modern Physics and it brings in to focus quantum nature of atomic and subatomic world. Not only is the energy of such bound systems quantized (shows spectrum ), but the angular momentum has a set of possible orientations in space the space is also quantized. Keeping things simple for the moment, in order to appreciate this very important and somewhat abstract concept of space quantization, let us consider He-atom: there are 2 electrons and the spins cancel out. These are known as singlet states. For He, one of the electrons has zero orbital angular momentum and so the magnetic moment μ= - (e/2m)l is due Page 10-5

6 to the orbital motion of the second electron alone. For this second electron, the angular momentum quantum number, l, can take the values, l = 0, 1, 2, 3, L has (2l+1) possible different orientations, corresponding to (2l+1)-values of L z = z-component of L = m l ħ, with m l = l, l-1, -l. In the absence of the magnetic field B, the energy is same for all these states with different m-values. The energy level, say, E 0 is (2l+1)-fold degenerate. When the field is switched on, the degeneracy is lifted and the energy E 0 now becomes, E = E 0 + Δ E Where, ΔE = - μ.b = (e/2m)l.b = (e/2m) L z B = (eħ/2m) m l B The Green line transition of mercury at nm splits in to nine transitions (see figures below) when the field is turned on. The experiment is performed with the magnetic field normal to the direction of propagation of light (Transverse position), as shown in the diagram at the beginning. In particular the pi-component Zeeman splitting will be studied it is the most intense pattern of lines in the Zeeman spectrum. Page 10-6

7 The figures below show the appearance of the patterns observed on the monitor for viewing transverse to the magnetic field in various states. Fig a: The first figure shows the initial appearance of pattern with no imposed magnetic field. Fig b: The magnet power supply is now turned on, starting with lowest field. The polarizer mounted behind the lens is now rotated until a pattern shown in Fig. (b) is observed. This shows the full range of Zeeman split lines emitted in the transverse direction. Page 10-7

8 Fig c: Rotation of the polarizer is continued until the triple fringes shown in Fig (c) are seen. These are the strong π-polarized components that are used for measurements. Fig d: Rotating polarizer further by 90 degrees yields the pattern shown in Fig (d). These fringes arise from the σ-polarized components and are noticeably less intense. Page 10-8

9 Experimental and Theoretical Formula Derivations The figure below shows a schematic of the ray paths in the Fabry-Perot etalon and the focused image at the camera. If the separation of the mirrors in d, the optical path difference between 2 adjacent light rays at an angle θ is, Δ = 2d cos θ For constructive interference with light of wavelength λ, Δ = kλ, and so Page 10-9

10 Δ = 2d cos θ = kλ Near the center of the pattern, cos θ ~ 1. The order of the interference k is highest at the center of the pattern, and subsequent rings going outward are k-1, k-2, and so on. From the figure, clearly, the diameter of an interference ring for small value of θ at the center of the pattern is, D = 2fθ So for the kth order ring, we get, 2d ( 1 D k 2 /8f 2 ) = kλ If the spectral line under study at frequency ν is produced by an electron transition between two energy levels E 1 and E 2, we have, hν = E 2 E 1 Page 10-10

11 Theoretically, when the magnetic field B is applied, the additional energy ΔE shifts the energy levels E 1 and E 2 to E 1 +ΔE 1 and E 2 +ΔE 2. The new frequency ν is : hν = (E 2 +ΔE 2 ) (E 1 +ΔE 1 ) The frequency shift Δν of the line is: Δν = ν-ν = (1/h) (ΔE 1 ΔE 2 ) Since λ = c/ν, Δν = - (c/λ 2 )Δλ, and ΔE = (eħ/2m) MgB, where M is the magnetic quantum number and g is the Lande factor, we can now write, Δλ = (-λ 2 /c) Δν = (M 2 g 2 -M 1 g 1 ).(λ 2 /4πc).(e/m).B Page 10-11

12 Here M 1,M 2 & g 1, g 2 are the magnetic quantum numbers & g-factors of the upper (3S1) and lower (5P2) states, between which the transition leading to the green line (546.1nm) takes place (see fig depicting relevant transition in mercury). Table above gives the Lande g-factors for π-polarized ΔM = 0 transitions which lead to the observed triplet fringe set. Lande g-factor for upper state (n=7, j=1) = 2 and lower state (n=6, j=2) = 3/2. M values (π-component) for the two states are 1,0,-1. Using these values g and M, we can now write the expression for the theoretical change in wavelength: 2 Δλ = (½).(λ 2 /4πc).(e/m)B *theoretical expression Experimentally, it can be shown that, 2 2 Δλ = ( λ 2 /2d ). (D k 2 -D 2 k /Dk-1 2 D 2 k ) *experimental expression where d = separation of Fabry-Perot mirrors, D k = Diameter of the kth ring, D k = diameter of k th ring, D k-1 = diameter of (k-1)th ring of the interference pattern. Page 10-12

13 Procedure Recall that e = x C and m e = 9.11 x kg. The wavelength of the mercury line we will be studying is nm (green light). 1. Your TA will use a Hall probe to measure the applied field magnitude, B. 2. Calculate the expected (theoretical) value of Δλ, the wavelength shift. Show your work: Page 10-13

14 3. Convert this expected value of Δλ in to Zeeman energy shift in electron volts. Careful: ΔE hc/δλ! Show your work: 4. You will be using software to analyze the images collected by your TA using the experimental apparatus. For the most part, this software should be fairly intuitive. However, you may need help at some point, so please refer to this lab s appendix for details on how to use the software. Use the Zeeman Effect software on the computer to measure the diameters of the appropriate rings (7 total see the analysis tab in the software for the labeling conventions for the relevant rings), and place these diameter values (measure in # of pixels) in the appropriate boxes on the analysis tab. 5. Once you have filled in the boxes, the software will automatically calculate the experimental value of Δλ for you, as given by the experimental expression on the previous page. You may have noticed that the expression only involves 3 values D k, D k, and D k-1, even though you just measured 7 diameters. The 7 values are all used to acquire more experimentally precise values of these 3 diameters. This is done behindthe-scenes by the software, according to the formulas on the analysis tab. 6. Compare the values of Δλ here: Δλ experimental = Δλ theoretical = Percent Difference = Page 10-14

15 Question 1 The SI units of magnetic moment are ampere.meter 2. Verify that μ = -(e/2m) L has the correct units. Question 2 Explain: The application of the magnetic field in the z-direction destroys the spherical symmetry in atoms and lifts the degeneracy of an atomic energy level. Question 3 How does the Zeeman effect study bring out the rather abstract concept of space quantization? Explain by a suitable diagram showing the orientation of the angular momentum vector in space in a magnetic field. Page 10-15

16 Question 4 As you go away from the center of the Zeeman pattern, the order of the circular fringes (the k-value) decreases explain briefly. Question 5 Write about 10 lines explaining the features of the software developed for you to analyze the data. Page 10-16

17 Appendix 1: Using the Zeeman Effect Software Introduction The program we will be using to analyze the data from the Zeeman Effect apparatus is currently in beta. The program consists of three separate pages, or tabs. The three pages can be accessed by clicking on the name of the page along to top edge of the program interface. They are listed as Fit Data, Analysis, and Data. The first page will be where you do the bulk of your work. It is here that you will load a JPG image of the rings and determine the diameters of the rings in arbitrary units (in this case, number of pixels). The second page will assist you with determining the change in wavelength using the data you determined on the first page. The third and final page is simply a list of the raw data you have collected, and can be ignored. As the program is still in its beta, be sure to follow these operating instructions closely. Should something go wrong, know that you cannot in any way hurt the program itself, and simply closing and reopening the program will return you to the initial state. Opening the Image With the program open, first select the JPG you wish to analyze by clicking on the folder icon to the right of the text box on the Fit Data page. This will open the browser, where you can locate and select the image. With the image path loaded into the File Path text box, you can run the program. To run the program, click the right arrow located on the top of the program window just below the menu selection items. This will be the left most button just below 'Edit'. With the program running, you should see an image appear in the center of the program. If it does not, stop the program (by clicking the stop button on that same row as the run button), reset the file path, and try again. Image Adjustments On the far left side of the image screen there are four buttons. These are the magnify, select, pan, and cursor buttons. With the magnify button selected, you can click anywhere on the image and the image will center and magnify about that point. The select button will allow you to select different circles you have drawn, and will be used later. The pan button allows you to grab and drag the image. Finally the cursor button will allow you to place markers on the image. The markers you place will define the points that the program will fit a circle to, which is how the graph will be analyzed. Page 10-17

18 Before we start drawing circles, we should discuss the buttons on the right side of the Fit Data page. Starting from the bottom and working our way up, we see a filter selection box with the colors Red, Green, and Blue. The wavelength of the emission for this particular transition lies in the green, so we will want to filter out the red and blue. Do this by clicking on the appropriate buttons. Notice that if you filter all three you see nothing, and if you only let red or blue pass through you see only noise. Above the filter selection box is the maximize contrast button, along with the max brightness. Turning this on may help to make the darks darker and the brights brighter. The Zoom to Fit button will resize the image so that you can see it in its entirety. Above this button is the readout for the Diameter of the circle selected in pixels. (We are only concerned with the relative diameters between rings, so the units of the diameter are not important, as long as they are always the same). Next up is the button to remove this circle, which will remove the selected circle from the list. Remove last point is useful if you place a point and then realize that the point is not where you want it to be. Continuing upward we have the Show/Hide Selected Circle button, which may be useful in determining the quality of the circle fit. The drop down list labeled Circle will list the circles you are working with. When you begin there will be only one circle. When you have defined this circle pressing the button to the right labeled Next Circle will allow you to set the current circle and begin defining the next. At the very top are two color selection boxes, which, if you like, will let you to define the colors for circles you have placed and/or the one you are currently creating. Fitting the Circular Data Click on the analysis tab; on this page you ll see an image of the ring structure along with a number of data entry boxes. Take note of the rings of interest. Back on the Fit Data tab, you should have already set the filters and contrast to a convenient setting. Magnify and pan your image so that you can see the first set important rings clearly. Now select the cursor tool (the bottom one which looks like crosshairs). With this tool selected, begin clicking on the image where you can best see one of the rings clearly. Every click you make will leave a dot on the image with the color you have defined for the selected circle. Once you have four dots placed, a circle will appear. Be sure that this circle best represents the ring by adding dots around the ring. Try to keep your dots evenly spaced. With practice, it will likely Page 10-18

19 take no more than 4 to 6 dots to define the circle properly. If you place a bad dot, click the Remove Last Point button. Once your first ring is fitted, press the next circle button. You should see the previous circle change color. Now you can start placing points on the second ring, just as you had the first. Continue this process until you have fit all of the rings of interest (7 total). The Analysis Tab Now that you ve done all the fitting, you can change to your select tool and select individual rings; the diameter will appear in the box to the right. You will use these values on the analysis page. When you insert numbers into the appropriate data entry boxes, you will see the value for Δλ update. Once you are certain that you have the correct diameters from your analysis on the first page entered here, and that the constants are correct, you can take your resulting value for the change in wavelength and finish the calculations on your own. The mathematical formula used in the derivation of the change in wavelength is listed, as well as the derivation of the formula. Page 10-19