In this experiment, we will detect the arrival of short-lived, cosmic ray muons and measure the time difference between their arrival and subsequent decay. From the distribution of these time differences, we wish to deduce the mean lifetime of the muon and use this lifetime to determine the Fermi coupling constant, the fundamental constant characterizing the weak force.

Before coming to the lab on the first day, read through the Theory section below. Look online for information about:

• What a muon is
• How they are created in the Earth's atmosphere
• What (approximately) their lifetime is.

You will use an oil drum filled with mineral oil as your detector. See what you can find online about liquid organic scintillator detectors and similarities/differences with the NaI+PMT type detectors you used in the Gamma Cross Sections experiment.

In this experiment you will study the decay of cosmic ray muons and measure the mean lifetime of this unstable particle. Specifically, your goals for this experiment include the following:

• to use and understand the properties of an organic scintillator detector;
• to understand the electronics necessary to make energy discrimination and timing measurements on the pulses produced by the detector;
• to use a time-to-amplitude converter to build-up a histogram of muon decay time intervals;
• to analyze this histogram and extract the mean lifetime of the muon; and
• to use the mean lifetime to calculate the Fermi coupling constant, a fundamental constant characterizing the weak interaction.

The muon (symbolized by $\mu$ or $\mu^-$ ) is an unstable lepton particle with charge ${}^-\mathrm{e}$ which decays into an electron, a neutrino and an antineutrino. The antimuon (also called the positive muon and symbolized by $\mu^+$) has charge ${}^+\mathrm{e}$ and similarly decays into a positron (also known as the antielectron, e+), a neutrino and an antineutrino. Both processes can be symbolically expressed as

Consider a large ensemble of unstable particles – either muons (or antimuons) like those just discussed or the more familiar radioactive nuclei that you have worked with elsewhere in this course. We can measure number of decays per second within this ensemble as a function of time and find that – since the count rate is proportional to the number of remaining unstable particles – the decay rate falls exponentially as time increases,

Here, $\tau$ is the time for the decay rate to fall to $\frac{1}{\mathrm{e}}$ of its initial value. We call $\tau$ the mean lifetime since $\tau$ is the mean (or expectation) value of the exponential distribution, $e^{-t/\tau}$. (Interestingly, $\tau$ is also the  standard deviation of such a distribution.) Alternately, one can instead consider an individual unstable particle. In this case, Eq. (2) may be interpreted as the relative probability for the decay of that individual particle as a function of time,

Here $\tau$ has the same meaning as the mean (or expected) lifetime of the particle. In this experiment, we will collect a large number of muon decays in order to determine the probability function. While one measurement of a muon decay cannot tell us much, a large number of measured decay times is expected to yield a distribution of values as described by Eq. (3).

As muons are short-lived, we do not have a ready, in-lab source. Instead, we will use muons which are produced in the earth's upper atmosphere (10 km to 20 km above sea level) by the impact of cosmic rays (primarily high-energy protons) with air nuclei. These cosmic showers include both muons and antimuons (in approximately equal number), as well as other charged particles and gamma rays. Muons and antimuons have identical lifetimes of about $\tau$  = 2.2 μs and possess masses of $m_μ = 105.7$ MeV/c${}^2$, 200 times more than the mass of the electron, $m_e$ = 0.511 MeV/c${}^2$. Calculated classically, a typical muon produced in the upper atmosphere, moving at nearly  the speed of light, would travel only about 1/2 mile before decaying. Therefore almost no muons would reach the earth’s surface. However, relativistic time dilation effectively lengthens the lifetime, (or shortens distance) enabling a more significant number of muons to reach the surface of the earth.

Muons and their decay electrons are charged particles. If they have kinetic energy they will deposit energy in any bulk matter, including scintillator material, through the Coulomb interaction. Some of that deposited energy will produce a flash of light in the scintillator. So, if a high energy muon enters a scintillator, it will produce a flash of light. In addition, if the muon decays while in the scintillator, its decay electron will produce a second flash. Measuring the distribution of times between the two flashes for a large number of decays enables us to determine the mean lifetime of the muon. It should be noted that, for muons which decay while moving in the scintillator, the two flashes of light are smeared together as one continuous flash. On the other hand, for muons which stop in the scintillator and later decay, two discrete flashes will be produced. The time between muon flashes and electron flashes may have any value greater than or equal to zero. We shall model the distribution of times by Eq. (2).

The decay of the muon is one of the few processes in nature which involves only the weak interaction.  The other three forces are the electromagnetic, strong and gravitational forces.  As no other forces are involved, the lifetime of the muon (along with its mass) is a direct measurement of the “strength” of the weak interaction. This strength can be characterized in a few different ways, but we will concentrate on a quantity known as the Fermi coupling constant, $G_F$, given (approximately) by

where $m_\mu c^2 = 105.7 \mathrm{\;MeV}$  is the rest energy of the muon and $\tau$  is the measured muon lifetime. The literature value (according to the National Institute of Standards and Technology) is $G_F/ (\hbar c)^3 = 1.1663787(6) \times 10^{-5} \mathrm{\;GeV}^{-2}$  can be expressed in terms of electron volts as $\hbar = 6.582119569\times 10^{−16} \mathrm{\;eV\cdot s}$

Deriving this quantity is beyond the scope of this course, but in the next (optional) section, let us at least give some context to the meaning of this parameter. 

The goal for the first day is to understand the apparatus, get it working and calibrate it. You will then leave it running between the first and second days of the lab. On the second day you will save the time distribution from the PHA and check the calibration.

Figure 1: Components of the experimental setup.

 Here we give a brief description of the components of the experiment shown in Figure 1.

Pulses generated by the PMT come primarily from the following sources:

• scintillation light from muons moving through the scintillator;
• scintillation light from decay electrons moving in the scintillator;
• scintillation light from particles passing through, but not decaying in, the scintillator;
• noise pulses, originating in the PMT, due to electrons ejected from the PMT photo-cathode by thermal excitation.

We want to measure the time between muons and their associated decay electron; the pulses from non-decaying muons and the noise both represent background pulses. It is particularly important to minimize the number of noise pulses which we detect, since they are far more frequent than the muon and electron pulses. Fortunately, the noise pulses are generally smaller than the muon or electron pulses. Thus, the discriminator module can be set to reject pulses with an amplitude smaller than the threshold voltage, but allow larger (muon and electron) pulses to be processed.

Set up the electronics as shown in Fig. 2.

Figure 2: Electronics setup for detecting muon decays in the scintillator.

The discriminator threshold is adjusted by a rather delicate potentiometer. For this reason, we will leave it fixed at about -30 mV and instead adjust the PMT high voltage which affects the amplification of all pulses in the PMT. We will look for a voltage setting where noise pulses fall below the threshold, but true signals make it over.

Check that the discriminator threshold is set appropriately. To do so, use a digital voltmeter to measure the voltage between the small white jack on the discriminator and ground. (An unpainted screw or the chassis of the NIM bin will make a suitable ground.) Note that this jack reads 10x the actual threshold voltage, so (for example) a discriminator setting of -10 mV would be registered as -100 mV on the multimeter.

NOTEBOOK: Record the measured discriminator threshold voltage.

NOTEBOOK: Sketch the noise pulses to scale. Note the range of voltages. Will the noise pulses get over the threshold?

NOTEBOOK: Sketch the larger pulses to scale. Note the range of voltages. Will the larger, less frequent pulses get over the threshold?

Search for muon and decay electron pairs by setting your time axis to an appropriately long window. (What time scale should you choose?) Use the “persist” feature of the oscilloscope, located under “display”. With this feature turned on, each trigger event of the scope will remain displayed for some time allowing you to watch a history of events build up. With persist set to infinity, look for an event at t = 0 (a muon) followed by another event (the decay electron) a few microseconds later. 

NOTEBOOK: Sketch these pulse pairs. Do more pairs seem to appear at small time separations or long time separations? Do all second pulses make it over the threshold? Does every start pulse have an accompanying second pulse?

Next, use a BNC tee to look at both the input and output of the discriminator at the same time (on channels 1 and 2 of the scope).

NOTEBOOK: Sketch the discriminator output pulses to scale. Does a discriminator output occur only when the input surpasses threshold? Are all the discriminator outputs identical?

We wish now to find the optimal PMT operating voltage. This is a voltage which amplifies the desired (muon and electron) pulses above threshold, while keeping the noise pulses below threshold. However, because the muon and decay events deposit a range of energies (leading to pulses with a range of amplitudes), this cross-over is not sharp. You will need to make a judgement call based on the trade off between higher count rate (good) and more noise (bad).

While looking at the output of the PMT amplifier, vary the PMT high voltage until you see the noise pulses approaching the -30 mV threshold, but not exceeding it. Look again at the muon and electron pulses to see how they've changed as well. Continue adjusting until you feel you have the best compromise between true pulses over threshold and noise pulses below.

NOTEBOOK: Record your final PMT voltage. 

To measure the time between pulses, we will use a time to amplitude converter (TAC). The TAC takes two inputs (a start and a stop) and produces one output whose amplitude is proportional to the time difference between the inputs (so long as that time is less than or equal to the range setting on the TAC).

Note that, in our system, we have just one PMT (with only one signal cable) to detect all the muon and electron pulses. Somehow, we must split the chain of PMT pulses to provide a start and stop the TAC. We do that splitting by using two outputs of the discriminator, and running one output (the Start branch) through a delay of about 50 ns as shown in Fig. 2. The start/stop sequence is illustrated in Fig. 3. 

CONSIDER: Referring to Fig. 3, consider what time differences you would be measuring if you simply omitted the delay box or put the delay box in the Stop branch. Why is the configuration we use the only correct one?

The time measured by the TAC is therefore the time between the muon and its decay electron, minus 50 ns. If no second pulse arrives within the range time, the TAC resets automatically.

CONSIDER: Your ultimate goal is to measure a distribution of times between muon detection and electron detection. How will this 50 ns offset affect the distribution, assuming it is exponential as predicted by Eq. (3)?

Figure 3: The effect of the 50 ns delay on the pulse sequence.

Set the TAC range to 20 μs and make sure the “gate in” switch is set to “off”. Use the BNC tee to look at either the start or stop input on channel 1 and the TAC output on channel 2 of the oscilloscope. Using persist, look for pulse pairs that produce outputs. Do the output amplitudes vary as expected with the time between pulses?

NOTEBOOK: Sketch the TAC input and output pulses to show the general behavior.
CONSIDER: If the maximum TAC output amplitude corresponds to a time delay equal to the TAC range, what is the maximum output amplitude? (You may not directly observe any pairs with a time delay that long. Assume a linear relationship between time and amplitude, and extrapolate.)

We may now measure the output rate from the discriminator using the scaler and timer to determine how many muon and electron events we are detecting. Because all discriminator outputs are identical, we cannot tell which counts are due to muons and which are due to electrons, but we can assume that muon events far outnumber electron events. Most muons do not come to rest in the detector and therefore produce only one flash of light, not two.

• Use the discriminator output as the scaler input. (Note the polarity of the output signal and chose your input port appropriately.)
• Set the timer to run for a short period of time, perhaps 30 seconds.
  • Note that the timer has two dials (call them a and b) which set a time window $T = a \times 10^{b-1}$s. For example, setting the dial to a = 2 and b = 4 gives $T = 2 \times 10^3 \mathrm{s} = 2000 \mathrm{s}$.
• Push “reset” to reset the timer and “start” to begin counting. The scaler should stop automatically.
• Calculate the average rate and uncertainty.

If your PMT voltage is set appropriately, you should measure a count rate of a few tens of Hertz.

CONSIDER: In order to determine whether your rate is reasonable, perform the following back-of-the-envelope estimation:

The typical flux of cosmic ray muons at sea level (coming from all angles in the sky above) is a little more than 1 muon/cm${}^2$/min. Assume that all muons which pass through the detector deposit enough energy to get over threshold, but that most (say, maybe 99%) do not decay while inside the detector. Using this information and the dimensions of the scintillator above in Sec. 3.1, estimate the expected rate of muon pulses out of the discriminator. Is this estimation of the same order of magnitude as your measured rate?

As we are interested in pulse pairs (muon followed by electron), our TAC output rate will be much lower. Assuming that 1% of muons which enter the tank decay, what is your predicted average time between TAC outputs?

The TAC is set to trigger whenever a stop pulse is received within the TAC range window. Even if the average time between pulses is larger than this range, there is some probability that two uncorrelated pulses (e.g. two separate muons rather than a muon and an electron) will accidentally serve as the start and stop within that time window. The probability that a second muon appears within a differential time window $dt$ some time $t$ after the first muon is governed by Poisson Statistics, and has the form

where $R$ is the overall (singles) rate of muons, measured above in Sec. 3.4.1. The total accidental rate for starting and stopping with uncorrelated muons within TAC time range $T$ is therefore given by

In the case where the TAC range is much smaller than average time between muons ($T \ll 1/R$), we can approximate this by using $e^x \approx 1 + x + \dots$ as

(Notice that this form is exactly like the form seen in other experiments (e.g. Wave-Particle Duality,  for accidental coincidences within a time interval $\tau$  for two detectors with singles rates $R_1$ and $R_2$; namely, $R_{acc} = R_1R_2\tau$.)

NOTEBOOK: Calculate the predicted accidental rate assuming T = 20 μs.

We know that the TAC produces an output voltage which is proportional to the time between the start and stop pulses, but in order to calibrate the PHA horizontal axis in units of time, we need to supply it with pulse pairs whose separation we can control and measure. In that way, we can connect a known time delay to a known channel on the PHA.

To create these pulse pairs, we will use a special function generator called a double pulser. The double pulser will create a steady stream of identical pulse pairs (representing a muon followed by an electron) so that the TAC produces a stream of identical outputs.

We wish to set up apparatus as shown in Fig. 4, but we will do so in steps (below).

Figure 4: Arrangement for time-calibration of the TAC/PHA system

The double pulser has many knobs and options; consider all numerical values displayed on the box as approximate only. Note the following settings:

• The “Pulse Mode” switch changes the output from single (SGL) to double (DBL) pulses. Make sure this is set to DBL.
• “Pulse Width” changes the pulse width. We want this width to be to the muon pulses you measured directly from the PMT.
• “Amplitude” changes the pulse amplitude. We want this to be large enough to get over the discriminator threshold, but not so large that distortions in the pulse shape cause multiple discriminator outputs.
• “Pulse Delay” changes the time between pulse pairs. We will vary this time over the same values as the TAC range settings.
• “Repetition Rate” changes the repetition rate. We may set this to be higher than the natural muon rate in order to collect data faster, but do not want to set it too high. It is possible to set the time between one pair of pulses and the next to be smaller than the time between the first and second pulse within a pair leading to nonsense.

In order to properly set up and verify the settings, carefully walk through the following steps:

• Now insert a BNC tee at the double pulser output. One side of the tee should still go the oscilloscope, while the other should go to the discriminator input.
• Notice that the amplitude of the signal on the scope should drop in voltage; this is normal, since the tee now sets up a voltage divider with different impedances on the two legs.

• Remove the discriminator output from the oscilloscope and connect the two discriminator outputs as shown in Fig. 4 above. (One output should go to the TAC stop. One output should go to the delay box, and then the TAC start.)
• Verify that the TAC range is set to 20 μs.
• Plug the TAC output into Channel 2 of the oscilloscope, but this time do not use the 50 Ω terminator. (The terminator is necessary when the signal is short in duration compared to the time it takes for a reflection to travel the length of the wire and back. The TAC output is considerably longer and the terminator will serve only to attenuate the signal at the scope.)
• Set the scales on the oscilloscope so that you can see both the pulses from the double pulser (on Ch 1) and the TAC output (on Ch 2).
• Adjust the pulse delay and verify that the TAC output increases (decreases) as the time delay increases (decreases).
• Check to see that the TAC output has a maximum voltage of ~ 10 V when the time separation is ~ 20 μs.

NOTEBOOK: Produce a table of time and channel for the double pulses. Estimate uncertainties on both quantities. Check to make sure that the relationship between the two is linear by doing a quick sketch of the data in your notebook.

We are now ready to collect a spectrum of muon decay times. We will run this data collection from the end of Day 1 to the start of Day 2 in order to collect enough data for reasonable statistics.

• Return to the setup shown in Fig. 1.
• Clear all previous regions of interest from the spectrum and create a new region of interest from the first to last channel.
• Begin collecting data. The region of interest will record all counts. 
• If all is well, you should observe a slow build-up of pulses on the PHA.

CONSIDER:  From Eq. (3), what is the most likely time for a decay? Is that prediction consistent with what you observe after a few minutes of data collection?

NOTEBOOK: Is the average rate of counts on the PHA that you observe after a few minutes consistent with your calculation above in Sec. 3.4.1 for the expected rate for muon and electron pairs?

Check with a TA or staff member to make sure everything looks OK. Leave the system to run until the next lab period. The computer has been set so that it will not go into sleep mode or restart for updates.

When you return on the second day, you should see a clear exponential decay on the PHA histogram. Stop the collection and save your data. It is recommended that you save both a *.spu file and a *.tsv file. The spu format is only readable by the Spectech software while the tsv format is a text file which is readable by most plotting programs.

After saving, recalibrate to check the system for drifts. If you find that your calibration has changed since Day 1, determine a way to correct for it.

The PHA distribution you have collected is binned according to pulse height channel. We wish, however, to ultimately extract the lifetime in units of microseconds. To do so, we need to determine the calibration equation between channel and time.

• Plot your time and channel values from the double pulser. You should see a linear relationship.
• Fit this data to an appropriate function. Depending on which variable has the dominant uncertainties (and therefore should be plotted along the y-axis), you may fit either channel as a function of time or time as a function of channel.
• One could in principle use these best-fit values in the fit function to convert the channel axis of the PHA decay time distribution to time (in microseconds). However, leave the data in terms of channel for the fit and convert only the final lifetime from channel to time.

ANALYSIS: Provide a plot and fit of your calibration data. Report your fit function and parameters (with uncertainties) and describe how this information could be used to convert the x-axis of the muon decay plot from channel to time (but do not perform the conversion).

NOTE: It seems natural to convert from channel to time as soon as possible and to do the exponential fit in terms of time rather than channel. However, we encounter a problem in how to properly propagate uncertainties. Namely, while the uncertainty in the calibration is purely derived from statistical uncertainties such that each channel ultimately has some uncertainty in time (and that the magnitude of this uncertainty varies from channel-to-channel), the value we want to extract – the muon lifetime, i.e. time constant of the exponential – is a property only of time intervals, not of absolute times. Therefore, even though the calibration uncertainty arises (mostly) as the result of statistical fluctuations, its contribution to the muon lifetime uncertainty is ultimately systematic.

Therefore, propagating the calibration uncertainties into _x-_error bars on the histogram and including these x-error bars in the fit is not the correct approach. Instead, we should work directly with the measured variable (channel) and determine statistical uncertainties only from the measured counts. Then, after a fit value has been extracted, we can combine the statistical (from fit) and systematic (from calibration) uncertainties together to arrive at the final uncertainty.

Replot your PHA data and fit the data to the form:

where $ch$ is the PHA channel, $\tau$ is the time constant for muon decay, $N_0$ correlates to the number of events, and $B$ is related to the accidental count rate. If you are having issues with your fit not converging due to success code 3, see this.

ANALYSIS:  Show a plot of the distribution giving counts versus channel. Fit this data and extract the mean lifetime (in units of channel) with uncertainties. Then, using the calibration formula determined above, convert the lifetime to time units and propagate the uncertainties from both the fit and calibration. (Use only the slope of the calibration formula in your conversion, not the y-intercept.)

ANALYSIS: Should all the data (from channel 0 to 512) be included? Consider adjusting the fit region by cutting data at small or late times. What justification do you have for such cuts?

ANALYSIS: Why do we need to include a background term, B? Is the background truly a constant? If not, why are we justified in approximating it as such? (See 

Sec 3.4.2.)

The background of your fit has physical meaning. We can make a comparison between the total number of background counts in your data to the number which is expected from the accidental coincidence rate.

ANALYSIS: Your fit provides you with a background term B which represents the number of background counts present in each of your data points. To calculate the total number of background counts in your data, multiply this value by the number of data points (i.e., the number of channels). To convert this count to a rate, divide by the total live time of your data collection. This is your experimental accidental count rate.

Recall that in Sec 3.4.2 we made a prediction for the accidental count rate. Using Eq. (14) again, (along with the muons singles rate and the time window of your PHA), calculate the expected accidental count rate. (Note that your PHA has a smaller time range than the TAC, so T < 20 μs for this calculation.)

Do your experimental and accidental rates agree (within uncertainty)?

Typical measurements of τ yield slightly low values for an interesting physical reason. Muons arrive at the detector in approximately equal numbers of positively and negatively charged varieties. The only decay path for the positively charged muons is the one studied here: decay into electrons as in Eq. (1). Negatively charged muons, however, can also mimic an electron and be captured by an atom within the scintillator. Some of these bound muons eventually may interact to convert a proton in the nucleus into a neutron,

One can model the probability that a negative muon is captured in this way by computing an absorption time constant $\tau_{abs}$ . The decay which you observe in your histogram is a composite of these two processes such that your measured time constant is an effective time constant $\tau_{meas}$  given by

For the scintillator used in this experiment, the absorption time constant is much bigger than the free muon decay time constant τ leading to a measured time constant about 4% lower than the free muon lifetime. If we wished to measure the lifetime of the muon without this capture effect, we would need a flux of purely positive muons. One could achieve such a flux, for example, by separating the muons produced via collisions in a particle accelerator into positive and negative beams with a magnetic field. 

REPORT Rubric Item #5: Include both your uncorrected and corrected (scaled up) estimates of the muon lifetime in your report.

The pulse height analyzer builds up a histogram of decay times during the course of the experiment. Because the overall number of counts is small, we may find many bins have N < 20 where the normal assumption that $dN = \sqrt{N}$  breaks down. (This is especially true if we have used a conversion gain of 1024 or 2048, which will lead to twice or four times as many bins as for 512.) It may therefore be useful to re-bin the data by adding together two or more adjacent channels (and shifting the time to the corresponding average of those channels). This trade-off gives you better statistical uncertainties (since each channel contains more counts), but worse time resolution (since you have more space in time between points). If you collected data with a large number of channels, you may want to explore how re-binning affects your results. This process is entirely optional and you are not required to do this for the report. Below we provide some simple code that can be inserted into your python script that you can play with. Insert this code after you have read in your data from file, but before you attempt to plot or fit. (This code assumes that you read in your channel and count data as chtemp and Ntemp, respectively. It will then produce two final arrays, ch and temp, which can be used as normal.)

Python code (courtesy of Caner Nazaroglu):

#Assume the channel data is initially in an array "chtemp" and the count data is in an array "Ntemp"
 
binsize = 2 #This sets the number of channels that will be put into a new bin
numdatapoints = int(binsize*(len(Ntemp) / binsize)) #This determines the NEW number of total channels
N = np.sum((Ntemp[0:numdatapoints]).reshape(-1,binsize), axis=1) #This creates the new N array
ch = np.mean((chtemp[0:numdatapoints]).reshape(-1,binsize), axis=1) #This creates the new ch array
 
#NOTE: If the total number of original channels cannot be divided evenly by "binsize", the program
# will drop the leftover (remainder) channels at the end.

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 experiment your final result will be your measured value for the life time of the muon. Your conclusions would be your evaluation of how well your measured value did or did not match other measured values and a discussion of anything you may have discovered about how the results depend on experimental factors.

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 calibrated the PHA.
• How you assessed, quantified and propagated uncertainties should be discussed and made clear including examples of how calculations were done where appropriate.
• Stability of the experiment over the two days of data collection.
• Possible corrections to the measured life time.

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. The rubric is not a format for your analysis, you are not expected 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 analysis writeup. For example you will be presenting data in your discussion of the calibration, your discussion of the distribution of event times, and likely in your final results. 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.