In this experiment you will study the Zeeman effect by looking at the green line transition ($\lambda = 546.1\,\mathrm{nm}$) of mercury under an applied magnetic field. Specifically, your goals for this experiment include the following:

• to learn the purpose and operation of a Fabry-Pérot interferometer and to produce interference ring patterns for the the green line transition of mercury;
• to observe the splitting of rings under the application of a magnetic field and to understand the physical reasons for this effect;
• to understand the relationships between interference rings and, from that information, measure wavelength (or energy) differences between pairs of rings as a function of applied magnetic field;
• to compare these energy differences to those predicted by the quantum model of electron energy levels and splitting due to the Zeeman effect (for the case of $\Delta m_j = 0$ transitions);
• to observe and understand relative transition rates for different Zeeman components and again compare against quantum predictions; and
• to measure the magnitude of the Bohr magneton. 

Atomic mercury (Hg) has 80 electrons with a ground state configuration

1s2; 2s2, 2p6; 3s2, 3p6, 3d10; 4s2, 4p6, 4d10, 4f14; 5s2, 5p6, 5d10; 6s2.

Figure 1: Energy level diagram for the green line of Mercury in the absence of any external magnetic field.

The energy of an atomic state, such as the $\rm{}\left(6s7s\right){}^3S_1$  state, can be calculated from the solution of the Schrödinger equation. These solutions are functions of the quantum numbers summarized below along with their physical interpretations:

• $n$ - The primary quantum number, also referred to as the radial quantum number; n comes from the radial solution of the Schrödinger equation and is related to the Coulomb potential; $n \ge 1$ (integer steps).
• $\ell$ - The azimuthal quantum number; $\ell$ is related to the orbital angular momentum of the electron(s) about the nucleus, $L^2 = \hbar^2 \ell (\ell +1)$; $0 \le \ell \le n-1$ (integer steps).
• $m_\ell$ - The magnetic quantum number; $m_\ell$ is related to the orientation of the angular momentum with respect to the z-axis, $L_Z = m_\ell \hbar$; $-\ell \le m_\ell \le \ell$ (integer steps).
• $s$ - The spin quantum number: $s$ is related to the intrinsic spin (angular momentum) of the electron, $S^2 =\hbar^2 s(s+1)$; for electrons, $s = 1/2$.
• $m_s$ - The spin projection quantum number: $m_s$ is related to the orientation of the spin with respect to the z-axis, $S_Z = m_s \hbar$; for electrons, $-s \le m_s \le s$ (integer steps).

The total angular momentum of the atom $\left ( \overrightarrow J \right )$ is given by the vector sum of the total orbital angular momentum $\left ( \overrightarrow L \right )$ and the total spin $\left ( \overrightarrow S \right )$. This leads to the following additional quantum numbers:

• $j$ - The total angular momentum quantum number: $j$ is related to the total angular momentum of the electron(s) about the nucleus, $J^2 = \hbar^2j(j+1)$; $ | \ell - s | \le j \le \ell + s$ (integer steps).
• $m_j$ - The total projection quantum number: $m_j$ is related to the orientation the total angular momentum with respect to the z-axis, $J_Z = m_j\hbar$; $m_j = m_l+m_s$ , $-j\le m_j\le j$ (integer steps).

In the absence of an external magnetic field, the energy of an atomic state of mercury is dependent on the Coulomb potential and total angular momentum which are defined by the quantum numbers $n$  and $j$. The orientation of the angular momentum vector has no effect on the energy of the state, so the state is said to be degenerate with respect to the quantum number $m_j$. Thus we can express the energy of the state as

If we now place the mercury atom in an external magnetic field $\overrightarrow{B}$, there is an additional energy term due to the interaction between $\overrightarrow{B}$ and the magnetic moment $\overrightarrow{\mu}$ of the atom as given by the relation

The magnetic moment of the atom is related to the angular motion of the outer shell electrons and has two contributions.

Figure 2: The magnetic moment for a loop of current I enclosing area A.

The first is due to the orbital angular momentum, $\overrightarrow L$. Any loop of current which encloses an area gives rise to a magnetic dipole normal to the plane of the loop with magnitude equal to the area of the loop times the current, $\mu = IA$. (See Fig. 2.) In the case of an electron in a circular orbit, we have a current

where $e$ is the charge of the electron, $T$ is the orbital period and $\omega$  is the angular frequency. This frequency is related to the linear velocity $v$  and the orbit radius $r$ by $v = \omega r$, and the orbit has an area $A = \pi r^2$. Therefore, the magnetic moment becomes

If we now note that the orbital momentum is $\overrightarrow L = m_e(\overrightarrow r \times \overrightarrow v)$ we finally have

The second contribution is due to the intrinsic spin of the electron, $\overrightarrow{S}$. The form of the magnetic moment due to spin can be derived within quantum mechanics as

Eqs. (5) and (6) can be combined, using the proper procedures for addition of angular momentum, to give the total magnetic moment of a state as

where $g$ is the Landé-g factor and $\overrightarrow J$ is the total angular momentum. The Landé-g factor can be calculated from knowledge of the quantum numbers of a given energy state according to

Combining Eqs. (2) and (7), and defining the $z$-axis to be parallel to the direction of $\overrightarrow B$, we get

If we replace $J_Z$ with its quantum mechanical expectation value, we can write

where $\mu_0 = e\hbar/2m_e$ is the Bohr magneton. The total energy of an atomic state is now given by

Eq. (10) shows that the additional energy $E_m$ due to the interaction of the magnetic moment with an external magnetic field depends on the value of the quantum number $m_j$. Since $m_j$ can take on only integer-spaced values in the range $\pm\;j$ , we see that the energy states depicted in Fig. 1 will be split into multiple states in the presence of an external magnetic field as shown in Fig. 3. The vertical arrows in Fig. 3 indicate the allowed electric dipole transitions between the $\rm{}^3S_1$ and $\rm{}^3P_2$ states.

Figure 3: Energy level diagram showing Zeeman splitting of the energy states for the mercury green line.

Note that transitions are not allowed between any arbitrary pair of atomic states. Only those transitions which obey the following selection rules are observed:

$\Delta s=0$ $\Delta \ell = \pm 1$ $\Delta j = 0^*,\pm 1$ $\Delta m_s = 0$ $\Delta m_j=0,\pm 1$ (*where the transition from $j=0 \rightarrow j=0$ is not allowed). We see that in the presence of an external field the single green line splits into 9 transitions having different energies, and therefore different wavelengths, due to the different Landé-g factors.

A Fabry-Pérot interferometer is a pair of parallel plates with highly reflective surfaces. Consider a ray of light incident on the Fabry-Pérot at angle$\theta$ . This ray will be partly transmitted and partly reflected as it strikes each surface, ultimately producing a number of parallel lines of output as shown in Fig. 4. Each subsequent output travels an additional path length of $\delta = 2D \cos \theta$ relative to the previous output ray.

Figure 4: Focusing of the light from a Fabry-Pérot interferometer showing how the angle $\theta$ of the incident light is mapped to a radius $r$ in the focal plane of the lens.  Note that this is representing only a single frequency $\lambda$ of light.

If we place a lens after the Fabry-Pérot, we can focus these parallel rays onto the 2-D focal plane of the lens, located at distance $f$ in front of the interferometer. Rays produced at most angles will destructively interfere as each parallel ray will be out of phase with the next, but when the angle satisfies

there will be constructive interference. We find bright concentric rings with radii

as each successive constructive condition is met (using the small angle approximation $\tan\theta\approx\theta$ ). The Fabry-Pérot can be used to make sensitive wavelength measurements. When the plate separation is chosen appropriately, small changes in wavelength (for example, due to a shift in transition energy due to the Zeeman shift) are amplified into large changes in ring radius. By measuring ring radii as a function of applied magnetic field around the mercury lamp source, we can recover the magnitude of the Zeeman energy shifts.

Rearranging Eq. (12) above and using the trigonometric identity $\cos\theta=1-2\sin^2(\theta/2)$, we have

where

Using the small angle approximation $\sin\theta\approx\theta$  and inserting Eq. (13), this becomes

or, rearranging,

At this point, let us note a few things:

• Identifying the diffraction order n of a particular ring is impossible from looking at a picture alone. In fact, $n$ is in general quite large.
• However, we do know that adjacent rings represent adjacent integer $n$ values and that $n$ decreases as we move out from the center.
• The center of the bullseye is not, in general, a maximum, so $n_0$ is not an integer. Instead it is some fraction of a maximum,

where $n_1$ is the n-value of the first visible ring and $0 \le\epsilon\le 1$.

In order to better organize our identifications, let us make the following substitution. Instead of dealing with diffraction order n, let us introduce ring order $p$ where e.g. $p$  = “1, 2, 3…” counts the ring “first, second, third,…” ring from the center. This $p$ can be related to $n$ via

Substituting this in above, we are left with

Suppose now that we turn on a magnetic field around our mercury lamp such that the light source emits not at one wavelength $\lambda$, but at several, $\lambda_i$. If these wavelengths are closely spaced, then rings of the same order will likewise be clustered. Let us make the identifications a, b, and c, for the inner, middle and outer rings at each order. (See Fig. 5.)

Figure 5: Fabry-Pérot ring patterns in the image plane of the CCD and the associated intensity profile taken across the thin horizontal slice illustrated by the dotted box. (a) The pattern that results when monochronomatic light is incident on the Fabry-Pérot. (b) The pattern that results when light consisting of three closely spaced wavelengths is incident on the Fabry-Pérot.

As each ring now has represents light of a slightly different wavelength, the fractional order of the bullseye center will change such that for the ith wavelength

Comparing the fractional orders for two adjacent wavelengths, we find

where we have used $n_{1i} = n_{1j}$ (which is true when the shift in wavelength is small enough that rings of different order do not overlap). Rearranging and using $E = hc/\lambda$, we have

Thus, by measuring the difference between the fractional orders of two rings, we can determine the difference in the photon energies which produced them.

When an electron transitions from a higher energy quantum state (with angular momentum quantum numbers $j_0$  and $m_{j,0}$ ) to a lower energy state (with angular momentum quantum numbers $\left ( j_f , m_{j,f} \right )$, the photon is not emitted randomly, but instead obeys certain angular distributions which preserve the conserved quantum numbers of the atom plus photon system. For the green line transitions of mercury that we are observing here, we have two types of electric dipole $(\Delta\ell=1)$  distributions; the first is the so-called “$\pi$ ” distribution when $\Delta m_j=0$  and the second is the so-called “$\sigma$ ” distribution, when $\Delta m_j=\pm 1$ . We illustrate the two distributions in Fig. 6.

Figure 6: Angular radiation intensity of the possible electric dipole transitions for the green line of mercury. (The magnetic field points along the z-direction.)

The $\Delta m_j =0$  distribution (upper figure) has an intensity which is proportional to $\sin^2\theta$  with respect to the magnetic field axis, z:

• When we view the radiation along a direction perpendicular to the B-axis, the radiation is emitted with linear polarization parallel to $\overrightarrow{B}$. 
• When we view the radiation along a direction parallel to the B-axis, the intensity drops to zero.

The $\Delta m_j = \pm 1$  distribution (lower figure) has an intensity proportional to $1+\cos^2\theta$  with respect to the magnetic field axis, z: 

• When we view the radiation along a direction perpendicular to the B-axis, the radiation is emitted with linear polarization perpendicular to $\overrightarrow{B}$.
• When we view the radiation along a direction parallel to the B-axis, the radiation is circularly polarized with a handedness dependent on the sign of $\Delta m_j$.

The emitted light has an angular distribution and polarization that reflects the change in the oscillating electric dipole moment of the orbiting electron during an atomic transition. For example, there is no change in this dipole moment for the $\Delta m_j =0$  transitions observed along the B-axis, so there is no emitted intensity in this direction for this type of transition. Similarly, changes in the electron angular momentum around the B-axis for $\Delta m_j = \pm 1$  transitions produce circularly polarized emitted waves. Since the direction perpendicular to the B-axis is perpendicular to the angular momentum quantization axis, the polarizations in this direction essentially depend only changes in the overall shape of the state and are consequently linear.

Though we see from Fig. 2 that there are 9 possible transitions, they are not equally likely to occur. In order to understand the relative likelihood of each, let us sketch a symmetry argument. During this discussion, refer to Fig. 7 where the transitions begin from one of the states in the upper energy level $ \left ( g=2 ; m_j = -1,0,+1 \right )$ and end in one of the states of the lower energy level $\left ( g=3/2 ; m_j = -2,-1,0,+1,+2\right )$.

Figure 7: Schematic representation of the transition rate symmetries in the decay from the upper ($g=2$) to lower ($g=3/2$) energy states with appropriate $m_j$ values labeled.

First, let us identify and label the different transition rates. 

1. Consider transitions which begin from the $g=2, m_j=-1$  state (the upper left state in Fig. 7). There are three possible decays and we will label the rate of each transition as 
   • $a$ (ending on $g=3/2,m_j=-2$ ), 
   • $b$ (ending on $g=3/2,m_j=0$ ), and 
   • $c$ (ending on $g=3/2,m_j=-1$ )
2. Consider transitions which begin from the $g=2, m_j = +1$  state (the upper right state in Fig. 7). These rates should be the symmetric counter parts to those considered in 1 above. (That is to say, for example, that the rate from $m_j =-1$ to $m_j = -2$ should be the same as the rate from $m_j = +1$  to $m_j = +2$  since the only change is in the orientation (direction) of $m_j$ , not in the value.) Therefore, we identify the rates again as 
   • $a$ (ending on $g=3/2,m_j=+2$ ), 
   • $b$ (ending on $g=3/2,m_j=0$ ), and 
   • $c$ (ending on $g=3/2,m_j=+1$ )
3. Consider transitions which begin from the $g=2, m_j=0$  state (the middle state in Fig. 7). These transition rates may be different from the other two, but again should show symmetry about the $m$-axis. Let us label the rates
   • $d$ (ending on $g=3/2,m_j=0$ ) and 
   • $e$ (ending on $g=3/2,m_j=-1~\textrm{or}~m_j = +1$ )

Next, we note that the total decay rate out of a given state should be the same for each of the three initial states. Therefore we have (summing from left to right)

$(a+c+b)=(e+d+e)=(b+c+a)$ Likewise, the total rate into a particular state should be the same for all five final states. Thus,

$a = (c+e)=(b+d+b)=(e+c)=a$ Putting these together, we find the following relations:

• $a=6b$,
• $c = e = 3b$,
• $d = 4b$.

Therefore, the ratio of the ring intensities (outer:middle:inner) is as follows:

• $\Delta m_j =0$  shows 3:4:3,
• $\Delta m_j =+1$  shows 6:3:1,
• $\Delta m_j =-1$  shows 1:3:6.

You may look for qualitative agreement to these predictions in your data.

To assist you with the apparatus we have made some tutorial videos on different aspects of the experiment.

Fig. 8 shows the main features of the experimental setup. The light source is a quartz bulb containing 198Hg.  We excite the mercury with a radio frequency (RF) oscillator. The oscillator is powered by a 500V power supply.

Figure 8: Schematic of the experimental setup

You will use this setup to measure the magnitude of the energy difference,$\Delta E$, between adjacent transition lines as a function of applied magnetic field, $\overrightarrow B$, for the green line $\left[\rm{}(6s7p){}^3S_1 \rightarrow (6s6p){}^3P_2\right]$ in mercury. From these data you will determine the value of the Bohr magneton, $\Delta E$ can be determined from the CCD images of the Fabry-Pérot output. $\overrightarrow B$ will be measured directly using a Hall effect gaussmeter.

A quarter-wave plate is a birefringent material with differing refractive indices in perpendicular directions to the incoming light beam. It is made so that entering, in-phase, orthogonal waves exit the plate 1/4 wavelength out of phase. (See Fig. 9). In this way, a quarter-wave plate changes plane-polarized light into circularly-polarized light.

Figure 9: Schematic of a quarter-wave plate showing the phase retardation along one axis.

Note that reversing the light path of Fig. 9 will change circularly-polarized light into plane-polarized light. The direction of the exiting plane polarization depends on the handedness of the entering circularly polarized light. Thus, we can filter out one handedness of circular polarization by following the quarter-wave plate with a linear polarizer placed in an appropriate orientation.

The mercury atoms in the lamp are excited by a radio frequency oscillator. To light the lamp, you need to provide a voltage to the oscillator and you will need to provide an initial “excitement” to jumpstart to the oscillation process.

You will use the Tesla coil to jumpstart the mercury bulb. While wearing UV protection goggles, bring the tip of the Tesla coil near the mercury lamp; the bulb should light easily. (You do not need to touch the Tesla coil to the bulb. If it fails to light, ask one of the lab staff for assistance.)

Once the mercury lamp is lit, replace the cover of the box. It is now safe to remove your goggles until you need to look at the light directly again.

We wish to produce a ring pattern as discussed above using the $\lambda = 546.1 \;\mathrm{nm}$ green line from the mercury lamp. Getting a clear image captured by the CCD camera requires a careful alignment and optimization. This alignment is CRUCIAL for collecting usable data in the sections which follow. Let us proceed in steps.

Begin by setting up the equipment and optimizing light intensity.

1. Rotate the magnet turntable so the magnetic field is perpendicular to the optical bench and center the plastic slit of the mercury lamp housing to send the light straight down the optical bench. Make sure that both pole pieces (the handles sticking out from each magnet pole) are firmly seated; they should not wobble or move easily and the handles should be sticking straight out, not drooping.
2. Adjust the heights of all the optical components to be the same as the height of the center of the mercury light source. Note that the magnet pole plugs give a good center reference. (An adjustable height guide – which looks like a pointer on a post – should be available in the room and may be useful here.)
3. Slide the CCD camera far enough to the right so that there is room for your head between the Fabry-Pérot interferometer and the camera. Remove the lens from the Fabry-Pérot and look through the interferometer at the mercury light source. (Wear the goggles!) Adjust the condenser lens position until you find the brightest and most uniform illumination of the field of view.

Next, we will roughly align and focus the Fabry-Pérot interferometer by eye. This process is tedious, but important. Proceed slowly and have patience.

1. Start with the magnetic field off and remove any polarizing filters or quarter-wave plates from the optical path.
2. There are three screws on the Fabry-Pérot which squeeze the three spacer legs and change their lengths slightly. Good adjustment is achieved when the lengths of these three legs are equal. Mechanical relaxation and thermal expansion/contraction mean that the Fabry-Pérot will likely not be properly aligned when you begin, even if no one has touched it since the last group.
3. As the three legs become approximately the same length, the Fabry-Pérot produces a set of concentric interference rings which can be seen with the naked eye. Loosen the screws to all three legs so that you feel the tension relax, and then slowly tighten each slightly (making small changes to one, then the next, then the next, etc.) until this ring pattern emerges.
4. Now, you want to further refine the alignment so that the three legs are even closer in length. To judge for proper Fabry-Pérot adjustment,look at the ring pattern and move your whole head (not just your eyes) along one of the three diameters of the Fabry-Pérot passing through each of the pressure screws. If the ring diameters change as you move your head, this screw must be turned in the appropriate direction to minimize the ring diameter change. Repeat the head motion and screw adjustment along the other two diameters. It will probably be necessary to repeat the adjustment cycle several times in order to converge on a good Fabry-Pérot adjustment.

At this point, we can further refine the alignment by viewing the image on the computer and making live adjustments.

1. Replace the lens on the front of the Fabry-Pérot by securing it with the rubber band.
2. Make sure that the camera is connected to a USB port on the computer and that the interference filter is mounted to the front of the camera. (This filter is designed to only pass a narrow wavelength band around 546 nm, the green line of mercury.)
3. Start the IC Capture 2.3 program by double clicking on its desktop icon. From the File menu select Load Profile and navigate to the file “IC Capture Zeeman Config” on the desktop to load it.
4. IC Capture should now be displaying a realtime image from the camera. Move the CCD camera closer to the Fabry-Pérot along the optical rail until a ring pattern appears on the screen.
5. Initially, only some of the ring pattern may be visible. It may be necessary to adjust the display size of the CCD image on the screen or to use the slider bars to adjust what portion of the image is displayed. Additionally, it may be necessary to adjust the height and/or the left/right position of the camera to bring the center of the ring pattern on screen.
6. Secure the camera to the optical rail and make fine adjustments with the focus knob.

Figure 10: A typical CCD image of the output of the roughly-aligned Fabry-Pérot. This image is for the green line of mercury with no applied external field.

Now, the ring image on the screen should be visible, but perhaps not perfect. (A typical roughly aligned image is shown in Fig. 10). Continue making adjustments either to the three Fabry-Pérot pressure screws or to the camera focus until your rings are as crisp and well defined as possible.

• General fuzziness may be an indication that the focus needs adjusting.
• An image which is sharp on one side, but fuzzy on the another may be an indication that the plates of the Fabry-Pérot are not completely parallel yet. Look especially for asymmetry in the sharpness along the three axes of the pressure screws: up-down and the two diagonals.

NOTEBOOK: Include comments in your notebook detailing the process of aligning the Fabry-Pérot by eye and refining that image by using the CCD camera.

As your alignment improves, you may need to play with the image display properties to further improve the quality. Select Properties from the Device menu. You can use the Gain and Exposure controls to adjust the brightness of the image.

• Increasing the Gain brightens the image by amplifying the electronic signal from the CCD pixels. This amplification also adds to the amount of noise present in the data.
• Increasing the Exposure time also brightens the image by collecting light over a longer interval of time. This results in less noise present in the image, but increases the time lag between successive updates which makes it more difficult to focus the camera.
• During focusing and fine tuning, it is recommended that you set the Exposure to about 0.1 s to obtain a reasonable update rate, and adjust the Gain as high as necessary to make the image bright enough to see.     

Once you have a well focused ring pattern, turn on the electromagnet power supply and increase B until the rings appear to defocus. What you are observing is the splitting of the green line into nine closely spaced components.

• Place the linear polarizer between the collimating lens and the Fabry-Pérot.
• Watch the CCD image on the screen as you rotate the polarizer. Orient the polarizer so that only the three $\Delta m_j=0$  transitions are seen which occurs when the polarizer is set to pass only horizontally polarized light. (You can also set the polarizer to pass only horizontally polarized light by observing the effect of the polarizer on light reflected from the floor and utilizing Brewster's Angle.)
• Note that it may be necessary to tweak the alignment of the Fabry-Pérot in order to enhance the contrast between the rings when the splitting is small.

At the end of this focusing process, you should obtain images of the same quality as those seen in Fig. 11.

Figure 11: Typical CCD images for a well-aligned Fabry-Pérot along with associated profile plots. The image on the left is with $B = 0$ while the image on the right is for$B>0$.

NOTEBOOK: Include comments in your notebook detailing the optimization of the CCD images. Note gain and exposure settings. 

Make sure that you are setup to view the $\Delta m_j=0$ component of the light emitted perpendicular to $\vec B$. We want to use the CCD to capture images of the Fabry-Pérot ring pattern for a range of magnetic field values including $B=0$. A practical upper limit for $B$ is when the splitting is large enough that components of adjacent rings begin to overlap. You should easily be able to take data for 5 or 6 different values of the magnetic field.

The magnitude of the magnetic field is related to the current supplied. At the end of the experiment, we will measure the magnetic field directly with a Hall effect gaussmeter, but for now we will record the current for each image saved so that we may reproduce the field later. Rather than trust the coarse current gauge on the front of the electromagnet power supply, we will perform a finer measurement by passing the current across a precision 0.1 Ω resistor and measuring the voltage drop. (Recall from Ohm's law that $I = V/R$).

When you are ready to capture an image for analysis turn down the Gain (to minimize electronic noise) and increase the Exposure time (to achieve the necessary brightness). Usually an Exposure time of about several seconds is sufficient, though you should use your judgment and save images under different conditions to compare. When you are ready to save a file, select Snap Image from the Capture menu, then save the image as a *.TIFF file.

NOTEBOOK: For each image, record the current setting (which you will later connect to a magnetic field measurement) and note the filename.

You will analyze the data in ImageJ; start the application by double-clicking on its icon on the desktop. From the File menu, select Open and navigate to the TIFF file you saved earlier. We want to ultimately extract the radii of the rings from this image, so we will use ImageJ to produce an intensity profile of the pixels in a narrow rectangular area running across the image. Use the rectangular selection tool to select a horizontal slice that runs through the center of the ring pattern. Keep the vertical dimension of the rectangle small so as to minimize the effect of the curvature of the rings. 

From the Analyze menu select Plot Profile. The intensity profile will appear in a new window. The x-axis of the profile is pixel column number with zero being the left most column of pixels. The y-axis of the profile is the pixel intensity averaged over all pixels in a column delimited by the upper and lower bounds of the rectangle you used to define the slice.

When you move the cursor over the profile plot, the program displays the $x$ and $y$ values where the cursor sits.

For each CCD image create and save an intensity profile. If you did a good job of tuning the Fabry-Pérot you should be able to resolve splitting in at least 3 or 4 rings for each value of $B$.

NOTEBOOK: For each plot profile, record the filename of the exported data and (optionally) make notes on the features seen.

As discussed earlier there are nine allowed transitions for Zeeman splitting of the green line of mercury. However, not all these transitions are equally probable. The differing probabilities give rise to different intensities among the rings.

For the $\Delta m_j=0$  transitions you should be able to see a clear distinction in the intensity profile for the three different components in each interference order. 

NOTEBOOK: Comment on the observed relative ring intensities for the $\Delta m_j=0$  case. Do they (qualitatively) obey the predicted ratio from Sec. 2.5?

Once we have finished collecting CCD images, we now need to measure the magnetic field strength at each current setting used in the experiment. We will measure these values by sweeping the magnet through the same set of current values and directly measuring B between the poles of the electromagnet using a Hall effect gaussmeter. The gaussmeter can be calibrated using a set of magnets of known strength available in the lab. Ask a TA or staff member for access.

To measure B values, do the following:

1. Ask a TA or one of the lab staff to remove the mercury lamp from between the poles of the magnet. (Make sure the lamp power supply is turned off!)
2. Beginning at zero field and slowly increasing the current, set the electromagnet to one of the current values for which you saved a CCD image.
3. Measure B with the gaussmeter and estimate an uncertainty. Be mindful of how you place the probe (the flat face must be perpendicular to the field for an accurate reading) and where you place it (try to measure as close to where the bulb was located as possible). The variation of the field strength over this volume is a source of systematic uncertainty in your final result.
4. Repeat steps 2 and 3 for each current used in the experiment. If at anytime you overshoot the desired current, ramp the magnet back down to zero and slowly increase back to the desired value in order to avoid hysteresis effects.

NOTEBOOK: Record calibration values and record measured field values at each current setting. Note your measurement technique and estimate uncertainties.

One of the parameters you will need for your analysis is the spacing of the Fabry-Pérot plates. Since the Fabry-Pérot plates are fragile and easily damaged, the spacing has already been measured for you and is written, along with the appropriate uncertainty, on the top on the plate holder.

NOTEBOOK: Record the plate spacing and uncertainty in your notebook.

Ultimately, we wish to show that both the initial and final energy levels corresponding to the green line transition split under the presence of a magnetic field according to Eq. (10). We cannot, however, view the energy levels directly, but instead only observe the shift in transition energy (wavelength). Therefore, we will study this energy shift in the emitted photon as a function of magnetic field B, where the energy shift is computed from Eq. (23) based on measurements of the fractional orders $\epsilon_i$. From these energy differences, one can compare the observed transition energies to the predicted transition energies based on the quantum mechanical theory of Zeeman splitting.

Extracting the values of $\epsilon_i$  is a multi-step process. 

First, you must extract radius values from your saved plot profiles. 

• Take each saved plot profile data and re-plot the points.
• Carefully examine each ring order $p$ and each peak within that ring order (abc) to identify the position (in pixels).
• Using the left-right symmetry of the plot profile, identify the ring center as the midpoint between corresponding peaks on either side.
• From the peak position and center, compute the radius of each peak (in pixels).

Recall that the ring radii obey the relation given in Eq. (20), repeated here for convenience,

Therefore, a plot of $r_p^2$ versus $p$ for each photon wavelength (inner, middle or outer ring) and for each magnetic field will yield a straight line with an intercept related to the fractional order $\epsilon$. Note, however, that this equation assumes that the radii are measured directly where the image forms at a focal length f from the front lens of the interferometer. While original versions of this experiment were performed by capturing that image on a photographic plate, we instead have captured the image on a CCD camera and saved a digital file. Our radii will therefore be measured in pixels rather than millimeters, and the correspondence between the true dimensions in the focal plane of the lens and the digital dimensions of the final image depend on properties of the camera and on resolution settings of our software.

Is all lost? No. Our pixel radius is simply related to true radius by a scale factor, $r' = Ar_{true}$, and inserting this into Eq. (24) leaves the equation functionally unchanged. In fact, we can roll all of the prefactor terms into a single generic coefficient and fit to the form

where the slope is $C$ and the intercept is $C(\epsilon -1)$.

After plotting and fitting $r_p^2$ versus $p$ for each photon wavelength (inner, middle, outer rings) and each magnetic field, we now have a $\epsilon_i$ as a function of $B$ for each ring. We now want to use this fractional order to compute energy differences between adjacent rings and test the Zeeman prediction.

• At each magnetic field, compute the difference in fractional order $\epsilon$ between adjacent rings (inner to middle and middle to outer) from Eq. (23).
• Plot these energy differences versus magnetic field $B$  and fit to the appropriate function.
• From this fit, determine the Bohr magneton and verify the Zeeman effect prediction.

NOTE: If appropriate, display both $x$- and $y$-error bars on your plots, but make clear whether both $x$- and $y$-uncertainties are used in the fit. (The fitting routines introduced in Python Tutorial only include the $y$-uncertainties in the fit by default; if you believe that it is important to include the $x$-error as well, use the method outlined in Section 3 of the Least Squares Fitting page. This should be considered optional.)

Your written analysis that you submit to be graded should be built around your final conclusions. Everything in your analysis should support your final result and conclusions. For this lab you want to show whether or not your data are in agreement with the theoretical predictions for Zeeman splitting of the green line of Hg.

You need to make clear things you did, decisions you made in the lab which are important to understanding how you arrived at your results and conclusions. This might include:

• How you extracted the ring radii from the images.
• How you identified, quantified and propagated uncertainties in the measured quantities.
• What you are using as a basis for comparison. I.e. you could choose to compare your results to either the Lande g factor or the Bohr magneton, or maybe some other parameter.
• How you calibrated and measured the magnetic fields at which you took data.

The above list is not intended to be complete, nor should it be treated as a checklist of what should go into your written analysis. Your analysis needs to make clear to the reader what your results and conclusions are, show how your data support those conclusions, demonstrate how you processed the data, etc.

For this quarter we are focusing on developing your skills in data analysis and drawing appropriate conclusions from your data. Your analysis should focus on these things. You should not include sections on the apparatus, background theory, historical significance, and things like this. This is not to say that these things are unimportant, they are just not part of a report on your analysis and results.

Your analysis will be evaluated based on the following rubric. Each item below is graded on a 0-4 point scale:

• 4 – Good (A): completes all listed tasks and provides appropriate context; thinks carefully about data and analysis; addresses all concerns raised by the results (where appropriate).
• 3 – Adequate (B): misses one or more minor element or lacks appropriate context; leaves a problem or ambiguity unaddressed; does not present analysis clearly enough.
• 2 – Needs improvement (C): omits or mishandles one or more item which renders the analysis fundamentally incorrect or incomplete; presents results in an incorrect or unclear way.
• 1 – Inadequate (D): omits or mishandles multiple items or treats them at an insufficient level; presentation is overall muddled or inaccurate; flaws in logic or process.
• 0 – Missing (F): omits all elements or makes no meaningful attempt.

All rubric items carry the same weight. The final grade for the analysis will be assigned based on the average (on a 4.0 scale) over all rubric items.