UNITED SCIENTIFIC SUPPLIES, INC. OPERATION AND EXPERIMENT GUIDE

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1 UNITED SCIENTIFIC SUPPLIES, INC. OPERATION AND EXPERIMENT GUIDE ZEEMAN EFFECT APPARATUS 1. Introduction This apparatus was designed for modern physics labs at universities and colleges. It demonstrates the influence of a magnetic field on light emitted in a gas discharge, the Zeeman effect, and shows the quantum nature of light and the behavior of electrons in atoms emitting light. A mercury discharge lamp is placed between the pole pieces of a powerful electromagnet and an interference filter selects the green emission line at nm. The resulting light beam is passed to a high resolution, fixed separation Fabry- Pérot étalon whose interference pattern is captured by a CCD camera and passed to a computer, where special software analyzes the changes observed as the magnetic field is varied. Light can be observed both parallel and transverse to the magnetic field, and filters allow the polarization states of the emitted lines to be investigated. Students can collect data and analyze it to determine the light wavelength shift Δλ, as well as e/m for bound electrons in an atom. 2. Specifications Lamp: Electromagnet: Fabry-Pérot Etalon: Mercury discharge lamp, slim design, powered A.C., 1500Vp-p, 5W Observed emission line: 3 S1 (6s7s) -> 5 P2 (6s7s) at λ=546.1 nm Built-in current-controlled power supply, 90V DC, A, with analog ammeter Magnet rotatable 90 for transverse or longitudinal viewing Maximum magnetic field at lamp position: B=1.3 T Aperture: 40 mm Separation of quartz plates: d=2.0 mm High reflection bandwidth: 100 nm Central wavelength: nm

2 Interference Filter: Collimating Lens: Central wavelength: nm Transmission bandwidth: 10 nm Peak transmission: 50% Aperture: 37 mm Focal length: 110 mm Environmental: Operating limits: 0-40 C ( F), relative humidity 85% Choose a location free from vibration or electromagnetic interference Power Input: 110VAC/60 Hz, 190W Care of Apparatus The Zeeman Effect Apparatus should be used in a dry clean room with good ventilation. Do not touch any optical part with your hand. To remove dust or dirt, use a photographic lens brush, or blow the dust off with canned air. Do not use compressed air from an airline as this frequently contains oil droplets, which will contaminate the surface of the optical parts. Stubborn dirt may be removed by gently cleaning the surface with a cotton ball soaked in a 1:1 mixture of denatured alcohol and ether. To prolong the life of the mercury lamp avoid turning it on and off frequently. When applying the magnetic field, increase the current slowly and avoid leaving the potentiometer in the zone marked by yellow dots for long periods. This will avoid excessive heat generation in the magnet. 3. Basic Theory 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. For a given L, there could be (2l+1) orientations in space. Thus μ too has (2l+1) orientations in space. This is shown in the diagram below: It 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

3 the Zeeman effect. The figure below shows the Zeeman effect. In the absence of the magnetic 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. 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. An Additional Complication: The Lande g Factor Again, as an atomic electron orbits around the nucleus, it produces a magnetic moment: μ l = - g (e/2m) L where g is a constant called the gyromagnetic ratio, or the Lande g-factor for orbital angular momentum. An atomic electron also has an intrinsic angular momentum spin (S), and so we would expect it to also have a magnetic moment, μ s, This magnetic moment due to the spin motion is proportional to S, or, μ s = - g s (e/2m) S where g s is a constant called the spin gyromagnetic ratio, or the Lande g-factor for spin. In the case of orbital motion, the gyromagnetic ratio is 1. Experiment shows that the spin gyromagnetic ratio is 2. The total magnetic moment of any electron is just the sum of its orbital and spin magnetic moments: μtotal = μ l + μ s = -e/2m {L + 2S} For historical reasons, the splitting of levels and spectral lines due to the various orbital m l values 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 21.1 ev into 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 transitions 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

4 work done by the field B in rotating these tiny magnetic dipoles until they become parallel to the field B. The work is W= - μb This implies that the energies of the photons emitted or absorbed by the atom will change in a magnetic field. The Netherlands physicist Pieter Zeeman, who received the 1902 Nobel Prize in Physics, first observed this effect. Zeeman s is one of the landmark experiments in Modern Physics and it brings in to focus quantum nature of atomic and subatomic world. The energy of such bound systems is quantized. The angular momentum has a set of possible orientations 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 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=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 E 0 is (2l+1)-fold degenerate. When the field is on, the degeneracy is lifted and the energy becomes, E = E0 + Δ E where ΔE = - μb = (e/2m) LzB = (eħ/2m) m l B When a light-emitting atom is subjected to a magnetic field, its emission lines are split into multiple components at slightly shifted wavelengths. The atomic magnetic moments along a given axis are quantized, and assume defined orientations when subjected to a magnetic field. In these orientations, the atoms bound electrons have different energies than in the zero-field state, the amount of the difference depending on which of the several allowed orientations an electron assumes. Each energy level is therefore now split into several sub-levels, and transitions between levels which were all of the same energy at zero field regardless of the electron orientation have now become a series of different, orientation-dependent transitions at slightly differing energies. This is experienced as a splitting of the observed spectral line on imposing a magnetic field. The transitions which can be observed in a magnetic field are governed by the selection rule for the magnetic quantum number M, namely that for allowed transitions ΔM = 0 or ±1, and also by the state of polarization of the emitted light. Linearly polarized light is observed in a direction transverse to the magnetic field, while circularly polarized light is emitted parallel to the field. In this apparatus, the spectral line observed is the green line of mercury at nm, which splits into nine components (Figure 1). The distribution of the observable lines in a magnetic field is shown in Table 1. ΔM Perpendicular to Field +1 Linearly polarized σ 0 Linearly + polarized π -1 Linearly polarized σ - Parallel to Field Circularly polarized σ No light + Circularly polarized σ - Table 1 Figure 1

5 When the lines are observed perpendicular to the magnetic field direction, all nine components will be visible. They include three π-polarized lines and six σ-polarized ones. The polarizations arise from the relative orientations of the magnetic field and the vibration directions of the emitted light, as illustrated by Figure 2. Figure 2 Figure 3 The three π lines are located in the center of the pattern and are the most intense. The six σ lines are weaker and located on the outside of the pattern. Figure 3 illustrates this arrangement. Overlapping of the closely spaced lines makes observation and measurement difficult. However, since all the lines are linearly polarized, a polarizing filter can be used to block the six σ lines, leaving the three central π lines more clearly visible for measurement. Rotation of the filter enables the existence and polarization state of the six σ lines to be confirmed. 4. The Zeeman Effect Apparatus Figure 4 shows the arrangement of the components of the Zeeman Effect Apparatus. Figure 4 1 Magnet with Mercury Lamp 6 Fabry-Pérot Etalon 2 Magnet Power Supply 7 CCD Camera 3 Focusing Lens 8 Optical Bench with Riders 4 Polarization Filter 9 Computer 5 Interference Filter The mercury lamp is mounted in a support at the center of the poles of the electromagnet (1), whose yoke can be rotated 90 on top of the magnet power supply (2), which also energizes the mercury lamp.

6 Light from the lamp is focused by a lens (3) and passes successively through a rotatable polarization filter (4) and an interference filter (5), which selects the green line at 546.1nm, before falling on a Fabry-Pérot étalon (6). The étalon generates an interference pattern of concentric rings, which is observed by a CCD camera (7). The optical elements (3) through (7) are mounted on an optical bench (8) for easy adjustment; careful alignment of the optical arrangement significantly improves the interference pattern and the accuracy of the results. The signal from the CCD camera is passed to the computer (9). Setting Up (Some parts already done) Mount the four riders onto the optical bench and fit the lens and polarization filter, étalon, interference filter, and CCD camera onto the riders as shown in Figure 4. Position the riders on the bench with the support rods at approximately the following separations: Front of magnet yoke base polarizing filter: 15 cm Polarizing filter - interference filter: 6 cm Interference filter étalon support table: 10 cm Étalon support table - CCD camera 15 cm Carefully adjust the heights of the polarization filter, interference filter, and CCD camera so that their centers are aligned with the center of the mercury lamp and clamp them in place. Fix the adjusted heights with the positioning rings on the support rods (see the left figure). This allows the optical elements to be easily removed and replaced accurately. Place the Fabry-Pérot étalon on the support table, oriented with the three plate adjusting screws on the lamp side and the single tilt adjustment screw on the camera side. Make sure the étalon is centered on the optical axis and square to it. Do not adjust the plate adjustment screws. Optimizing the Interference Pattern The procedure for determining Δλ and e/m involves measuring the diameters of concentric circles which differ only slightly. Precise adjustment of the optical setup to yield properly centered and undistorted circular fringes is important for obtaining an accurate result. To obtain a good interference pattern, attention should be paid to the following factors: Even illumination of the Fabry-Pérot étalon. Careful positioning of the various optical elements so they are centered along the optical axis of the setup and correctly aligned about their vertical axes. Precise parallelism of the Fabry-Pérot étalon mirrors. Optimal focus and aperture of the camera. Aligning the setup: Check that the centers of the lamp, lens, filter, and étalon are collinear and that each element is perpendicular to the optical axis. Adjust the position of the lens along the bench to maximize the size of the illuminated area of the étalon. View the étalon from the rear and adjust the position and height of the end of the bench so that the interference pattern is centered in the étalon mirror.

7 Adjusting the camera: Mount the camera onto the optical bench, connect it to the computer and start the EOS Utility software (click on Camera settings/remote shooting, check the settings are 1, F5.6, 100, click on Live view shoot ). The Fabry-Pérot fringes are located at infinity, so the camera focus ring should be adjusted for distant viewing. The camera can now be moved along the bench as close to the back of the étalon as possible to exclude disturbing ambient light and increase the contrast in the monitor image. Optimize the image by adjusting the focus and aperture rings as necessary. Viewing the patterns Figure 5 shows the appearance of the patterns observed on the monitor for viewing transverse to the magnetic field in various states. Figure 5a shows the initial appearance of the pattern with no imposed magnetic field. Turn the potentiometer on the magnet power supply all the way down, press the MAGNET button Rotate the potentiometer clockwise to increase the magnet current until a visible splitting of the circular fringes is seen on the monitor. Note: Do not run the magnet with the potentiometer in the yellow zone for long periods of time to avoid overheating the magnet coils. Rotate the polarizer mounted behind the lens until a pattern like Figure 5b is observed. This shows the full range of Zeeman- split lines emitted in the transverse direction. Continue to rotate the polarizer until the triple fringes shown in Figure 5c are seen. These are the strong π-polarized components that will be used for measurement. Rotating the polarization filter by a further 90 yields the pattern shown in Figure 5d. These fringes arise from the σ-polarized components and are noticeably less intense. Figure 5a Figure 5b Figure 5c Figure 5d

8 Saving the transverse pattern for analysis With the magnet set up for transverse viewing, adjust the polarizer to obtain a clear pattern of only the triple fringe set of the π-polarized emission as in Figure 5c. Adjust the magnet current to 1.3A to yield clearly separated fringes. The EOS software allows you to save the captured pattern in a directory named after the date on the desktop of the computer 5. Analysis Figure 6 Figure 6 shows a schematic of the ray paths in the Fabry-Pérot étalon and the focused image at the camera. The medium between the two mirrors is air with a refractive index of 1, so if the separation of the mirrors is d, the optical path difference between two adjacent light beams at an angle θ is Δ = 2dcosθ. For constructive interference with light of wavelength λ, Δ = kλ, and so Δ = 2dcosθ = kλ At the center of the pattern, θ = 0 and cosθ = 1. Since cosθ decreases with increasing angle from the center, the order of interference k is highest at the center of the pattern, and subsequent rings going outward are k-1, k-2, etc. The rays exiting the étalon are parallel and when they are focused at a distance f from the camera lens, the incident angle θ is preserved. The diameter D of an interference ring is 2f.tanθ (Figure 10) and for small values of θ at the center of the pattern, tanθ = θ so θ = D/2f. Approximating for cos θ at small angles, we obtain for the kth order ring 2d(1 - Dk 2 /8f 2 ) = kλ (1) If the spectral line under study at frequency v is produced by an electron transition between two energy levels E2 and E1, we have hv = E2 - E1 In an external magnetic field B, the potential energy of magnetic dipole moment is: E = Mg (eh/4πm) B (2) where e and m are the charge and mass of the electron, h is Planck s constant, M is the magnetic quantum number and g is the Landé factor. The additional potential energy shifts the energy levels E1 and E2 to E1 +ΔE1 and E2 +ΔE2 so the new frequency of the transition, v is obtained from The shift Δv in the frequency of the line is hv = (E2 +ΔE2) - (E1 +ΔE1) Δv = v - v = (1/h)(ΔE1 - ΔE2 )

9 Substituting for ΔE from equation (2) and converting from frequency to wavelength: Δλ = (-λ 2 /c)δv = (M 2g 2 - M 1 g 1 ) (λ 2 /4πc) (e/m) B (3) From equation (1) we obtain for the wavelength shift in terms of the kth order ring: Δλ = λ - λ = (Dk 2 - Dk 2 ) (d/4f 2 k) (4) The effective value of f cannot easily be determined. To eliminate f, we introduce the (k-1) th order ring. From equation (1): Subtracting (1) from (5) 2d(1 - Dk-1 2 /8f 2 ) = (k - 1)λ (5) d/[4f 2 (Dk Dk 2 )] = λ (6) Solving equation (6) for f 2 and substituting into equation (4): Δλ = (λ/k) (Dk 2 - Dk 2 )/(D k D k 2 ) (7) For the rings close to the center, where cos θ 1, k 2d/λ. Substituting into (7) gives: Δλ = (λ 2 /2d) (Dk 2 - Dk 2 )/(Dk Dk 2 ) (8) Combining equations (3) and (8) and rearranging also gives an expression for e/m in terms of measurable quantities: e/m = [2πc/(dB)] [1/(M 2g 2-M 1g 1)](Dk 2 - Dk 2 )/(Dk Dk 2 ) (9) Analysis is performed on the π-polarized triplet fringe set that corresponds to the transitions for which ΔM = 0. Table 2 details the magnetic quantum numbers and Landé factors for these transitions: Upper State, 3 S1 Lower State, 5 P2 Δ M g 2 3/2 n/a 2 3/2-1/2 Mg /2 +1/2 Table 2 Comparing the values of Δ(Mg) for these transitions with the corresponding term in equation (3), it is clear that the wavelength of the central fringe remains unshifted while the outer two are displaced symmetrically by an amount 1/2 (λ 2 /4πc) (e/m) B. This allows us to use the diameters of two central lines in adjacent orders as Dk and Dk-1, and the diameters of the outer lines as Dk in equation (8) and (9) to determine Δλ and e/m.

10 Appendix 1: Using the Zeeman Effect Software Introduction The Labview program Zeeman consists of two separate pages, or tabs. They are listed as Fit Data, and Data. The first page Fit Data 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 Data will list the radius of the rings using the data on the first page. 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. Image Adjustments On the far left side of the image screen there are four buttons. These are 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. 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

11 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 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). Zeeman Effect Outline 1. Purpose: A) Demonstrate Zeeman effect B) Use Zeeman effect to measure Δλ and e/m for electron. 2. Temporarily remove the camera. Turn on the magnet power supply. If needed move the optical bench so small Mercury light source is in line with optical bench. Look through the Fabry-Perot interferometer. Move you head around until you see a ring pattern. Adjust the vertical and horizontal knobs and move interferometer toward or away from the wall until ring pattern is close to centered. Adjust the height of the polaroid/lens combination for brightest image. 3. Put the camera in place. Turn on the camera and adjust camera and optical parts so you are seeing the ring pattern. Some additional adjustment may be needed later. 4. Connect cable from computer into camera. Click on camera icon on program display. You should see a fuzzy pattern. Check the settings are. 5. Adjust camera and optical parts as needed to try to get 3-4 complete rings. This involves a certain amount of trial and error. 6. Once you have the ring pattern save the image (jpeg). Turn on the magnet. Observe what happens as you increase the field. Save image for maximum current (~1.5A). Return the current to lowest setting. Turn the polarizer to 0 deg. Observe and describe what happens as you increase the current. Which component are you seeing? Save image for maximum current. Observe and describe what happens when you change the polarizer to 90 deg. Save image for maximum current. 7. The rings you see with polarizer at 90 deg are actually made up of 3 rings (pi components). Once you have a good image save it. You need to have enough of the smallest ring so that the center of the circle you will draw is in the display area. Do not touch the terminals the tube is attached to. High voltage needed to operate tube. The field will be around 1.3 Tesla. Turn down magnet current all the way. 8. Analyze the image as described above. Estimate Δλ and e/m and compare to book value. 9. Repeat step 8 three times. Take the average and standard deviation. 10. Don t forget to copy your images for your report.