Optical pumping is an experimental technique by which specially prepared photons are used to repeatedly excite an ensemble of Rb atoms in a way which drives the atoms into a specific atomic energy state from which the atom can no longer be excited by the photons. Once you learn how the technique works and how to apply it to Rb atoms in a vapor cell you will use it as a tool to make measurements of the energy of the Zeeman splitting as a function of magnetic field. You will be able to test the predictions of the quantum mechanical model for Zeeman splitting for a range of magnetic fields which covers both the small field approximation and larger fields at which larger order terms become important.

Since the zero-field crossing resonances where pumping occurs in Rb are very sensitive to local magnetic fields, such setups can be used to measure minute magnetic fields. Previously SQUID (Superconducting QUantum Interference Device) magnetometers were best for such measurements, but they tend to be bulkier or require extensive cooling. (See Moving magnetoencephalography towards real-world applications with a wearable system.)

Optically pumped Rubidium is also of interest in the field of quantum optics. The German group Laboratoire Kastler Brossel have published papers on storing and retrieving images in Rb85, with a long-term goal of creating a memory for a quantum communications network. (See Spatially addressable readout and erasure of an image in a gradient echo memory.)

More broadly, the dependency of the polarization of atomic emission spectra on local magnetic field is being used by astronomers to gather information about the conditions on the surface of stars. (See Discovery of Ground-state Absorption Line Polarization and Sub-Gauss Magnetic Field in the Post-AGB Binary System 89 Her.)

Before your first meeting with your TA, your group needs to do some background research. You want to go into your first group meeting with the TA understanding:

What is Optical Pumping and how does it work for Rubidium?

• How is Optical Pumping achieved.
• What does the atomic energy level structure look like for the D1 line of Rb, both with and without Zeeman Splitting.
• What is the Zeeman Effect and why is Zeeman splitting necessary to optically pump Rb?
• Which energy state does the electron occupy when the atom has been pumped? Why can the electron not absorb another photon?
• What happens to the intensity of the light passing through the vapor cell as the atoms are being pumped?
• Would optical pumping work if we used left circularly polarized photons instead of right circularly polarized?
• What does “depumping” mean?
• What does a quarter waveplate do?

By the end of the prelab meeting you want to make sure that you have a clear understanding of the topics listed above. You also want understand what you need to do in order to perform the experiment and have a plan for how you will get started when you first get to lab.

Another helpful thing to think about is what are you measuring? Specifically what are you directly measuring in the lab with the tools at hand. Hint, you are not directly measuring energies or magnetic fields.

This lab will help you explore your understanding of electricity and magnetism, as well as the basic quantum mechanics of single electron atomic transitions.

Atoms may occupy only discrete energy states, and if the atoms are in thermal equilibrium, the relative numbers of atoms occupying each state is given by the Boltzmann distribution. If we shine light of the appropriate energy and polarization on those atoms, some atoms will be excited to a higher energy state. If the atoms have no easy escape path from this excited state, then the population of atoms will no longer be Boltzmann-distributed. Such an inverted population is described as optically pumped.

In this experiment we will look at the behavior of rubidium-87 (Rb-87) atoms in a vapor cell, illuminated by light from a rubidium lamp. The rubidium atoms have an intrinsic net magnetic dipole moment and therefore will react to an applied magnetic field and exhibit energy level splitting due to the Zeeman effect. These atoms also posses angular momentum and can transition from energy level to energy level only through interactions with light that conserve the total angular momentum of the atom and absorbed/emitted photon system.

Simply stated, optical pumping is a process by which an atom is caused to repeatedly absorb and emit photons so that the atom eventually ends up in an energy state which does not allow it to continue absorbing photons.

In this lab you will optically pump an ensemble of Rb atoms by placing them in a precisely controlled magnetic field and illuminating them with 794.8 nm photons which have been right circularly-polarized. Absorption of the right circularly-polarized photons drives the Rb atoms into a pumped state which cannot then absorb a right circularly-polarized photon. A photodetector is used to record the intensity of the light which passes through the Rb vapor. As more of the atoms are driven into the pumped state, fewer photons are absorbed by the gas and thus more light reaches the photodetector. Changes in the opacity of the vapor are correlated with the degree of pumping which has occurred.

Understanding the details of how optical pumping works in this case is analogous to the quantum description of the hydrogen atom and requires a conceptual understanding of the following:

• Quantum numbers associated with atomic energy states such as $n$, $l$, $s$ and $m_{s}$; and
• Electromagnetic transitions between atomic energy states and their associated selection rules; and
• Behavior of magnetic dipoles in an external magnetic field; and
• Hyperfine and Zeeman effects.

In the next section we describe the optical pumping process in more detail.

Details

Rubidium is a hydrogen-like atom in the sense that its ground state has just a single electron in the outermost shell. Thus. the structure of the energy states (outside the full inner shell) is very similar to that of hydrogen which also has only one electron in its outermost shell. Figure 1 below shows the energy level structure for the ground and first two excited states of Rb.

Figure 1: The electron energy states of Rb87 and Rb85.

This diagram uses spectroscopic notation to denote the values of the quantum numbers of the various energy states. As a refresher, the relevant quantum numbers are:

• $n$ - The principle quantum number denoting which orbital the electron is in.
• $L$ - The orbital angular momentum of the electron.
• $S$ - The spin angular momentum of the electron.
• $J$ - The sum of the orbital and spin angular momenta of the electron: J = L + S.
• $F$ - The hyperfine quantum number which is the sum of the electron momentum $J$ and the nuclear angular momentum $I$: F = J + I.
• $m_{F}$ - The magnetic quantum number which denotes the orientation of $F$ with respect to the quantization axis defined by an external magnetic field $B$.

As indicated on this diagram, transitions between the first excited state $^{2}P_{1/2}$ and the ground state $^{2}S_{1/2}$ absorb (in the case of excitation) or emit (in the case of de-excitation) a photon with a wavelength $\lambda$ = 794.8 nm. Likewise transitions between the ground and second excited states involve photon wavelengths of wavelength 780.0 nm.

For an ensemble of Rb atoms at 50 $^{\circ}$C, the number of atoms in the ground and first excited states at any given moment is determined by the Boltzmann distribution, such that there will be more atoms in the ground state than are in the first excited state; this is the equilibrium state of the ensemble. This equilibrium state is the result of a balance between physical processes which drive atoms from the ground state into an excited state (such as the thermal motion of the atoms in the gas) and processes which cause excited atoms to de-excite (such as the natural lifetime of the excited state and thermal interactions).

Referring back to Fig 1, we see that both the ground and first excited states split into additional energy sub-states. Rubidium nuclei have an intrinsic spin (and corresponding magnetic moment) which interacts with the electron's magnetic moment resulting in two hyperfine states. If the atom is in an external magnetic field, (such as the Earth's magnetic field, for example), there will be additional splitting of the hyperfine states due to the interaction of the electron magnetic moment with the external field. This is known as the Zeeman effect.

Thus, in the presence of an external magnetic field, electronic transitions between the ground and first excited states are in fact transitions from the different Zeeman levels of the excited and ground states. Although there are a lot of possible combinations of transitions between ground and first excited state Zeeman levels, many of these combinations are “forbidden”. In this context, saying that a particular transition is “forbidden” simply means that the probability of that transition occurring is 0 or very near 0. You have probably encountered these quantum selection rules for electromagnetic transitions, summarized below:

$\Delta L = 0, \pm 1$ (but not $0\rightarrow 0$)

$\Delta S = 0$

$\Delta F = 0, \pm 1$

$m_{F} = 0, \pm 1$

As an example, let's walk through one possible sequence of transitions for the case where the light being absorbed is right circularly-polarized. In Fig. 2 above, we show a sequence of five hypothetical electronic transitions events – (a), (b), (c), (d), and (f) – and discuss each in detail below.

Figure 2: A sequence of four hypothetical transitions

(a): For purposes of this example, we pick an arbitrary starting point where the atom is in the $^{2}S_{1/2}(f = 1, m_{f}=0)$ ground state. This atom absorbs a right circularly-polarized photon. (Right circularly-polarized is another way of saying the photon carries +1 unit of angular momentum, so upon absorption the $m_{f}$ quantum number must increase by +1.) In this example then, the $F$ quantum number increases by 1, and since $m_{f}$ must increase by 1, this puts the atom in the $^{2}P_{1/2}(f = 2, m_{f}=0)$ state.

(b): When the atom de-excites, it is not required to lose +1 unit of angular momentum; that requirement only exists for absorption because of the circular polarization of the incident light. Instead, the atom can lose 0, +1 or -1 unit of angular momentum (which corresponds to the possible amounts of angular momentum that a photon can carry with it.) Therefore, the atom could potentially decay to any one of three possible states: $^{2}S_{1/2}(f = 2, m_{f}=0, 1, \textrm{ or } 2)$. As an example, let's suppose the atom decays to the $m_{f} = 1$ state.

(c): If the atom now absorbs another right circularly-polarized photon, it would only be able to be excited to the $^{2}P_{1/2}(f = 2, m_{f}=2)$ state because of the requirement that $\Delta m_{f} = +1$.

(d): At this point the atom can de-excite to more than one possible ground state. One possibility is that it ends up in the $^{2}S_{1/2}(f = 2, m_{f} = 2)$ state.

(f): The atom is now pumped because it cannot absorb another right circularly-polarized photon since that would require it to go to a $m_{f} = 3$ state (of which there are none). Therefore, unless some other process drives the atom into some other ground state (such as thermal agitation or stimulated emission by an RF frequency photon), the atom is stuck in the pumped state.

The sequence of excitations and de-excitations described above is just one of many possible permutations of allowed transitions. Not all atoms in the ensemble will end up in the pumped state, but some will end up there. The rate at which atoms are driven into the pump state is in part determined by the intensity of the light source. At the same time energy exchanges due to random thermal collisions between atoms will tend to de-pump atoms out of the pumped state. Equilibrium is reached when the rate of pumping equals the rate of depumping.

For Rb-87, the relation between the applied magnetic field and energy splitting is

where $\mu_B = e\hbar/2m_e \approx 5.7883 \times 10^{-9}$ eV/G is the Bohr magneton. The Landé g-factor, $g_f$, relates the atom's magnetic dipole moment to its quantum numbers, and is given by

Here, $g_j$ relates the electron's contribution to the total magnetic moment and is given by

The energy level shifts are therefore given by

Note that this energy difference represents the shift up or down from the un-split (hyperfine) level characterized by quantum number $g_f$.

In this section, we describe the apparatus and help you obtain an initial signal. In the image above, you see (from left to right) the main apparatus, the control console and digital scope, a function generator, a DC power supply, a couple digital multimeters, some compasses, and a computer.

The main apparatus consists of a Rb vapor cell in a temperature-controlled oven situated in the center of a pair of Helmholtz coils, all sitting atop an optical rail. Mounted to the optical rail, we have the following (from left to right):

• An amplified photodiode detector;
• a focusing lens;
• the vapor cell inside the Helmholtz coil assembly;
• a quarter-waveplate;
• a linear polarizer;
• a D1 transition filter;
• a collimating lens; and
• a rubidium lamp light source.

The light source uses a radio frequency (RF) oscillator to excite Rb atoms, thereby producing light at wavelengths needed to pump the Rb atoms in our vapor cell. Light from the source is collimated before passing through the filter which is tuned to only pass light from the D1 transition.

The filtered photons then pass through a linear polarizer and a quarter-waveplate whose optical axes are oriented such that light which emerges is right circularly-polarized. A quarter-waveplate will convert linearly polarized light into circularly polarized light. (If you send unpolarized light through a quarter-waveplate, you will get a mixture of left and right circularly-polarized light out. By placing a linear polarizer in front of the quarter waveplate, and setting the correct angle between the optical axes of the two, you can produce an output beam ranging from pure right circular polarization to pure left circular polarization (or anything in-between).) To produce a beam of right circularly-polarized photons, the linear polarizer should be set to an angle of 45° and the quarter waveplate should be set to 0° in their holders. If you do not remember how polarizers and quarter waveplates work, Google them to refresh your memory.

The vapor cell sits in the middle of a pair of orthogonal Helmholtz coils; one produces a vertical magnetic field when current is passed through it and the other produces a horizontal magnetic field when energized. These coils are used to produce the horizontal and vertical magnetic fields that define the net magnetic field in which the Rb atoms reside. If you do not recall how Helmholtz coils work, Wikipedia has a nice description.

The vertical coil frame contain $N = 20$ turns on each of the two coils. (Recall that a Helmholtz coil is actually a pair of coils, so the use of the word “coil” can get confusing.) The wire has been wound 5 across and 4 layers deep. According to the manufacturer, the coil radius is about $4.61 \pm 0.01$ inches. The spacing of the coils should match their radius.

With this information, you can calculate the magnitude of the magnetic field created by this coil if you know the current which is flowing through the wires. One of the functions of the TeachSpin control console is to control this current.

The horizontal coil frame actually contains two separate pairs of coils; the wire of the second coil is wound directly atop the first coil. This arrangement allows us to create two independently-controllable magnetic fields along the horizontal axis. We call one set of these coils the “Horizontal Sweep Coil” and the other the “Horizontal Field Coil.”

These coils contain $N = 154$ turns on each of the two coils of the pair. The wire has been wound 11 across and 14 layers deep. According to the manufacturer, the coil radius is about $6.22 \pm 0.01$ inches. The spacing of the coils should match their radius.

These coils will be used to create a steady magnetic field along the horizontal axis, similar to how the vertical Helmholtz coil is used. The TeachSpin control console can be used to set the current in this coil.

The top layer of wires which you can see make up the “Horizontal Sweep Coil”. This coil contains $N = 11$ turns on each of two coils of the pair. The wire has been wound 11 across in a single layer. According to the manufacturer the coil radius is about $6.46 \pm 0.01$ inches.

This Helmholtz coil is used to create a magnetic field which sweeps periodically over a range in a sawtooth pattern. This small time varying magnetic field component is central to how the TeachSpin apparatus achieves depumping (as you will soon see).

The TeachSpin control console is set up to allow you to measure the voltages and currents to the different coils so that you can then calculate the magnetic fields being generated.

For the Vertical Coil, there is a pair of test points across a 1$\Omega$ monitor resistor which is in series with the coil wire. Thus, by measuring the voltage drop across the 1$\Omega$ resistor, you can get the current flowing through the coil using Ohm's law.

For the Horizontal Sweep Coil, getting the current is a bit trickier. Since this voltage (and therefore the current) changes in time, we view it on the scope. However, the actual value of the voltage which is sent to the scope via the Recorder Output is amplified for viewing on a scope. Thus, the voltage you read on the scope is not the same as the voltage going to the sweep coils.

You can determine the conversion factor to go from scope voltage to coil current by either using the test points across the 1$\Omega$ monitor resistor on the front panel next to the sweep coil output jacks, or by reconnecting the coil wires at the output jacks to run through an ammeter to directly measure the current. In either case, you want to set the Sweep Range to zero, then set the Start voltage to several different values from its minimum to maximum. For each Start voltage setting, you can measure both the voltage of the signal at the scope and the actual current in the coils. This will give you all the information you need to determine the linear relationship between scope voltage and coil current.

The TeachSpin optical pumping control console is pictured above. Controls on the console are grouped based on function and consist of the following (from left to right):

• RF Amplifier - Not used (and therefore not highlighted in the photo).
• Cell Heater/Controller - Controls and indicates the temperature of the vapor cell. It automatically starts when the power switch is turned on and requires no user input. The temperature display will stabilize at 50°C in about 10 minutes.
• Magnetic Field Modulation - Not used (and therefore not highlighted in the photo.)
• Horizontal Magnetic Field Sweep - These controls are used to set up a time-varying current to the horizontal Helmholtz coils. Note that the BNC output labeled Recorder Output on the lower panel is connected to channel 1 of the scope and is used to monitor the current going to the coils.
  • The Range control allows you to set the range over which the current to these coils will sweep.
  • The Start control sets the starting current of the sweep. In other words, the Start control can be used to move the whole sweep Range up and down.
• Vertical Magnetic Field - This 10-turn potentiometer is used to set the current going to the vertical Helmholtz coils.
• Horizontal Magnetic Field - This 10-turn potentiometer sets a constant current to the horizontal Helmholtz coil. This current is in addition to any current being provided by the Horizontal Magnetic Field Sweep controls.
• Detector Amplifier - The output from the photodetector on the optical rail comes here. The user can amplify and add an offset voltage to the detector signal, and filter out high frequency noise. The final signal is then sent to channel 2 on the scope.

Start by turning on the control console; the power switch is located on the upper left (as viewed from the front of the apparatus) corner on the back of the unit.

It will take approximately 20 minutes for the Rb vapor cell to warm up to its operating temperature of 50ºC. Until the temperature stabilizes the transparency of the vapor cell will not be constant because the vapor pressure is changing. While you wait for the system to warm up you should make sure that everything is set to the default state.

• Gain = 2
• Gain Multiplier = 1
• Time Constant = 10ms
• Meter Multiplier = 1
• Offset Fine = Set to its midpoint
• DC Offset = Anything is fine for now.

• Horizontal Magnetic Field = 0

• Vertical Magnetic Field = 0

• Time = 1s
• Start/Reset = Reset
• Single/Continuous = Continuous
• Range = Max (Full CW)
• Start Field = 0 (Full CCW)
• Recorder Offset = 0 (Full CW)
• Magnetic Field Modulation
  • Amplitude = 0
  • Start Field / MOD = Start Field
  • Input = No Connection

• Do not make any adjustments.

• Gain = 0
• Input RF Modulation = No Connection
• Output = No Connection
• Input = No Connection

One of the main goals of this course is to become comfortable sitting down to a new experiment (with new apparatus) and understanding how things work (and what your signals mean). As a third year physics major you should now know enough physics to start making sense of what is going on with your experimental apparatus.

Conceptually, you are using the controls on the console to do the following two things:

1. Create magnetic fields along the horizontal axis of the optical rail and along the vertical axis by sending current to the Helmholtz coils.
2. Amplify and filter the signal from the photodetector so that you can view it conveniently on the scope.

(That's pretty much it. It looks more complicated, but at the basic level those are the only two things.)

Our initial goal is to use magnetic fields generated by the the Helmholtz coils to cancel out the ambient magnetic field present in the lab where the Rb vapor cell is located. We then want to use those same coils to create an additional magnetic field (whose properties we know and can manipulate) and watch how the intensity of the light passing through the vapor cell changes as we change that known field.

There are four 10-turn potentiometers which control the currents to the horizontal and vertical coils. Turn all of them fully counter-clockwise to zero. In the Horizontal Magnetic Field Sweep control section, there are two toggle switches. Flip both down so that they are in the Reset and Continuous settings. There should now be no external magnetic fields being generated by the coils and the Rb atoms are now sitting in the ambient magnetic field of the basement lab.

Optical pumping is very sensitive to the presence of external magnetic fields. You want to make sure that there are no magnets near the apparatus. You might be surprised by how many common items contain magnets. (For example, your cell phone likely has some strong magnets in it.) There should be a compass on the table. Take it and move it around the apparatus, looking for any unexpected deflections of the needle which would indicate the presence of something magnetic. Don't forget to take your cell phone out of your pocket.

We can only control the field along two axes – vertical and horizontal along the axis of the optical rail – using the Helmholtz coils. To completely zero the magnetic field in the vapor cell, you will also want to rotate the whole apparatus so that the optical rail is aligned with the horizontal component of the ambient field in the room. Do a rough alignment using the compass; there is no need to be super-precise as we will fine tune this alignment in a moment.

Now that you have roughly aligned the optical axis of the apparatus with the horizontal field in the room, use the Vertical Magnetic Field control on the console to cancel out the vertical component of the ambient field. There is a magnet mounted in a gimbal which you can use as a 3-D compass. Hold it near the vapor cell, inside the volume enclosed by the vertical coils. Now turn up the current to the vertical coils and watch how the magnet behaves. You can easily turn up the vertical field enough to reverse it. Play with the current to the vertical coils while observing the effect on the 3-D magnet and convince yourself that you understand what is happening. Then, set the current to create roughly zero field along the vertical axis. You will fine tune this cancellation in a later step.

At this point, you should be familiar with the concept of optical pumping and comfortable setting up and using the apparatus to observe the depumping signal while sweeping $B_{net}$ through the zero field condition.

Now we want you to extend your knowledge of the apparatus and the technique to include RF depumping which can be used to directly measure the energy of the Zeeman splitting for a given $B_{net}$. Recall that there is one more set of coils on the apparatus which we have not yet used. These are the RF coils which are connected to the output of a function generator. By sending an oscillating current to these coils, we produce an oscillating magnetic field which is orthogonal to the sweep field generated in the horizontal coils. When the energy of the RF photons, $E=\hbar \nu$, matches the energy difference between Zeeman energy levels, they will drive electronic transitions between the Zeeman states. If the gas is in the pumped state when we turn on this RF field, electrons which are in the pumped state will be driven into a different state and gas will be depumped.

To begin, set up the apparatus so that it is sweeping through the zero field condition and so that the zero field depumping signal occurs at approximately the mid-point of the horizontal field sweep. Set the Horizontal Sweep Range to its maximum value so that the Rb atoms are seeing the largest range of $B_{net}$ possible.

Now turn on the function generator and set it to produce a 40 kHz sine wave with an amplitude of ~0.5 V peak-to-peak. Turn on the output to channel 1 which should already be connected to the RF coils on the apparatus. The signal from the detector should now show additional depumping dips on both sides of the zero field point.

Spend a bit of time making sense of what you are seeing on the scope. Keeping in mind that the vapor cell we are using contains natural Rb which is a mixture of Rb-85 and R-87b, see if you can understand the following:

• Can you explain why you see the number of RF depumping dips which appear when the RF coils are energized?
• What is the significance of where along the horizontal sweep these RF depumping signals appear?
• Increase and decrease the RF frequency and see if you can explain how the RF depumping signals change in response.
• Increase and decrease the amplitude of the RF. Do you understand how the signals respond?

To explore the Zeeman splitting at high magnetic fields, you will need to use the external power supply to power a set of concentric Helmholtz coils within the apparatus. The process goes roughly as follows:

1. Set up your system such that you can observe the RF depumping dips
2. Turn on the external power supply, but keep the voltage near zero.
3. Increase the power from the external supply, which will shift the depumping dips on the scope
4. Choose one of the two dips to follow, and increase the RF depumping frequency to better center the signal on the screen.
5. Repeat steps 3 & 4 to track the behavior to stronger fields.

We expect that, at some point, we will observe the depumping dip start to broaden and split when in a sufficiently strong B field.

Since there will be other groups working on the apparatus, it is your responsibility to ensure that everything in the lab is in order when the next group arrives. The room should be tidy, and everything should be either put away or reset to it's default. If the lab room was in disarray when you arrived, you are still responsible for leaving it in the appropriate state for the next group.

Here are some general tips on things to check before you leave.

• Dial all vertical and horizontal field coil currents to zero.
• Set the sweep amplitude to 100%.
• Set the sweep offset to zero.
• Disconnect any function generators, multimeters, or external power supplies which you may have been using.
• Disconnect the scope from the apparatus and turn it off.
• Close any applications on the computer and don't forget to log out of any accounts which you may have opened.
• The computer may be left on; it will go into sleep mode.
• Reorient the apparatus to be square to the room.
• Place the compasses on the table next to the apparatus.

The analysis is not a lab report, rather it is all of the data reduction, number crunching, calculations, curve fitting, error propagation etc. which is necessary for you to establish your final conclusions. Think of it as being more like an extended homework set where you have to show how you got your final results.

Three days after your analysis submission your group will have a meeting with the TA to go over your analysis and make sure you are prepared to write your final report.

Your graded analysis will be returned along with your graded final report.

Shortly after your analysis is due your group will meet with the TA to discuss the overall analysis and make clear what needs to go into your final report. Note that this meeting is not for the purpose of discussing your grade on the analysis, you will receive the grade on the analysis along with the graded final report. Instead this is an opportunity for the TA to have reviewed your analysis to identify where you may have short comings or misconceptions in your understanding of the experiment with the goal of improving what goes into your final report. It is also an opportunity for you to make sure that you understand what your TA is looking for in your report.

Your analysis, like your reports, should be submitted as a single PDF. It is not expected that you will write narrative descriptions as you will in your final report. For the analysis it is acceptable to organize it into sections with one or two brief sentences of description. Things should be put in a sensible order so that the TA can follow what you are doing. For example plots, fits and calculations related to your energy calibration should be grouped together into a section, and that section should be placed before you apply the calibration to your data. For cases such as fitting and extracting peak locations for all of your scattering data it is sufficient to show one representative plot of a fit to the data along with a table containing all of the values. Scans or photographs of calculations done on paper or in your lab notebook are acceptable but absolutely MUST be clear and readable.

The final report is due three days after your scheduled analysis meeting with the TA.

Your final report will be evaluated based on the following rubric. The rubric is not a format for your report; you are not expected, for example, to have a specific section on Data Handling or Presentation of Data. Elements of the different rubric categories will appear at different points through out your report writeup. Your writeup of your analysis should be structured in a way that is clear and readable, there should be a logic to the flow of it.

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.

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