Zeeman Effect Physics 481

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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)

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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 Lz, for l = 1,2 and 10. The numbers on the z-axis are the values of the magnetic quantum number ml. 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 E0. When B is switched “on” the level splits in a multiplet of 5 equally spaced levels.

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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.

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An Additional Complication: The Lande g Factor

Again, as an atomic electron orbits around the nucleus, it produces a magnetic moment:

μl = -(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, μs. This magnetic moment due to the spin motion is proportional to S, or,

μs = - 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:

μs = -( e/m) S

The total magnetic moment of any electron is just the sum of its orbital and spin magnetic moments:

μtotal = μl + μs = -e/2m { L + 2S }

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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 energy21.1 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

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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 Lz = z-component of L = mlħ, with ml = 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, E0 is (2l+1)-fold degenerate.When the field is switched on, the degeneracy is “lifted” and the energy E0 now becomes,

E = E0 + Δ E

Where, ΔE = - μ.B = (e/2m)L.B = (e/2m) LzB = (eħ/2m) mlB

The Green line transition of mercury at 546.1 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.

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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.

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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.

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Zeeman Effect Physics 481

Experimental Procedure

Recall that e = 1.602 x 10-19 C and me = 9.11 x 10-31 kg. The wavelength of the mercury line we will be studying is 546.1 nm (green light).

1. Measure the applied field magnitude, B.

2. Calculate the expected (theoretical) value of Δλ, the wavelength shift. Convert this expected value of Δλ in to Zeeman “energy shift” in electron volts.

3. Adjust the alignment of the lens and camera to obtain images shown above.

4. Using the software to measure the diameters of the rings. Refer to this lab’s appendix for details on how to use the software.

5. Calculate the experimental value of Δλ and compare to the theoretical value.

Appendix 1: Using the Zeeman Effect SoftwareIntroductionThe 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 ImageWith 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

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Zeeman Effect Physics 481program. 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 AdjustmentsOn 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.

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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 DataClick 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

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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 TabNow 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.