Direct tests of special relativity are rare in the undergraduate laboratory as they often require large and expensive particle accelerators. In this experiment, we will use a low-tech – but very accurate – gamma detection technique to measure both the kinetic energy and momentum of electrons produced during Compton scattering. In addition to “discovering” the energy-momentum dispersion relation for electrons, we will measure the electron's rest mass in a few different ways and compare this value to literature.

• [1] [P. L. Jolivette and N. Rouze, "Compton scattering, the electron mass, and relativity: A laboratory experiment," Am. J. Phys. **62** (3), Mar. 1994. This paper outlines a nearly identical undergraduate experiment and also discusses a Monte Carlo simulation of how Compton scattering events deposit energy in gamma detectors.
• [2] |P. J. Riggs, "A Comparison of Kinetic Energy and Momentum in Special Relativity and Classical Mechanics", The Physics Teacher 54, February 2016, 80.

This paper manipulates equations for kinetic energy and momentum in order to present exactly how (and where) the classical mechanics gives way to special relativity.

Before the first day of lab, read the Theory section below carefully to understand the measurement technique we will use in this lab.

In addition, complete the following:

Consider the NaI+PMT gamma detectors you have used in other labs in this course.

(a) Draw (by-hand, with an appropriate energy axis) the spectrum you expect to see when the detector is pointed towards a Na-22 button source. (Feel free to refer back to sketches you made in your notebook for the Gamma Cross Sections experiment.) Identify the origin of each feature you include in the spectrum.

(b) Are the features of this spectrum infinitely sharp? What causes the peaks (and other features) to spread out and have finite width? (Is it due to the source? Is it due to the detector?)

(c) Sketch a hypothetical spectrum with a perfect source and a perfect detector and comment on what is different from the NaI+PMT spectrum you drew above.

In this experiment you will study the energy-momentum relation – the dispersion relation – for free electrons over an energy range from about 300 keV to almost 2 MeV. Specifically, the goals for this experiment include the following:

• to observe and interpret gamma spectra produced by a germanium detector;
• to measure the energy of decay gammas from radioactive sources and to measure the resultant electron kinetic energy in the case of 180º Compton scattering inside the detector;
• to use kinematic arguments to compute the scattered electron momentum;
• to empirically “discover” the special relativity dispersion relation through plots of electron kinetic energy and momentum;
• to study both kinetic energy and momentum individually as functions of velocity; and
• to determine the rest mass of the electron.

A dispersion relation is an equation which connects the energy of a particle to its momentum using only properties of the particle and fundamental constants. Though you may not have heard them termed this way before, you already know several dispersion relations.

For example, the classical kinetic energy of a non-relativistic particle of mass $m$ is 

where $T$  is the kinetic energy and $p$ is its momentum. Or, consider the energy-momentum relationship for a photon, 

where $E$ is the energy of the photon, $p$ is its momentum and $c$ is the speed of light. The challenge in this experiment is to discover the proper energy-momentum relation for electrons which holds not just in the non-relativistic range (like that of Eq. (1) above), but over a wider range of energies well into the relativistic regime. 

A straightforward way to do this experiment would be to use an electron accelerator:

• Accelerate an electron through a potential difference to give it a known kinetic energy. This electron energy can be continuously varied by changing the voltage in order to access a range of values limited only by the range of achievable voltages.
• Bend the energetic electron by passing it through a constant magnetic field perpendicular to its trajectory. By measuring the cyclotron radius, one can independently measure the momentum.

While an accelerator like that would be great, it would be large, expensive and difficult to maintain. Instead, let us propose a different (read “smaller, cheaper and easier to maintain”) experiment that makes use of apparatus you already have some familiarity with.

Consider a gamma detector made from a crystal of germanium with one end held at a high voltage. (In previous experiments you have used similar PMT-coupled NaI(Tl) crystals as detectors, but as we will see later, this germanium setup gives better energy resolution.) If we bombard this material with photons of fixed energies, some of the electrons in the solid will cause the photons to Compton scatter. By measuring features of the resulting gamma spectrum and using momentum and energy conservation, we can independently identify the kinetic energy and momentum of the electrons.

While this method is simple, we do not have a continuously tunable electron energy like we would with an electron accelerator; we are limited here to discrete values of electron energies, since we use only discrete photon energies as input. Nonetheless, we can still cover a range of electron measurements from about 300 keV to almost 2 MeV.

Figure 1: Compton scattering at θ = 180º.

Consider the electron-photon collision (i.e. Compton scatter) shown in Fig. 1. Compared with scattering through an angle less than 180°, this “head-on” collision imparts the maximum possible momentum (and energy) to the electron, and the kinematics can be easily worked out as a 1-dimensional problem. When such a scatter occurs inside a gamma detector, we can measure the two following quantities:

• the incident photon energy, $E_\gamma$, from the full energy photopeak, and
• the scattered electron's kinetic energy, $T$, from the energy of the Compton edge (since Compton scatters at $\theta \lt 180^\circ$ will impart less energy and therefore appear at lower energy values in the plateau).

While we measure the electron's kinetic energy directly, we do not measure the momentum. Let's see if we can use the kinematic formulation to derive a relationship that allows us to extract this quantity.

Momentum conservation gives (in the lab frame)

(where $p$ is the electron momentum and $p^\prime_\gamma$ are the initial and final gamma momenta respectively), while energy conservation gives

(where $E_\gamma$ and $E^\prime_\gamma$ refer to the photon energies before and after the collision, and $E_e$ and $E^\prime_e$ refer to the electron). We can collapse the difference between the initial and final electron energies into the acquired kinetic energy

such that Eq. (4) becomes

Recalling that we only measure $T$ and $E_\gamma$, let us use the dispersion relation appropriate for photons – Eq (2) above – to eliminate the photon momentum. Plugging this into Eq. (3) we find

Multiplying Eq. (7) by $c$ and adding it to Eq. (6), we finally have

Eq. (8) gives the electron's momentum in terms of incident gamma energy and the electron's kinetic energy, $E_\gamma$  and { $T$. It is not, however, the desired energy-momentum relation because of the presence of $E_\gamma$, which is not a property of the electron. Nevertheless, Eq. (8) enables us to calculate values of the electron’s momentum p from the measured quantities, $E_\gamma$ and $T$, suggesting construction of the following table:

It simply remains to fill in the table with measured values and find the dispersion relation – the mathematical expression which relates the values of the $T$ column to the values of the $pc$ column using only fundamental constants and properties of the electron.

You are provided with the following set of gamma-emitting sources:

• Sodium-22
• Cobalt-57
• Cobalt-60
• Barium-133
• Cesium-137
• Europium-152
• Bismuth-207

Additionally, you have metal foils or disks which can be “activated” by thermal neutrons to produce the following short-lived radioactive nuclides:

• Indium-116
• Aluminum-28

Ask a TA or a member of the lab staff to help you place the indium foils and aluminum disks in the neutron howitzer so that they can be bombarded with neutrons for at least one hour prior to use. The resulting foils will then have enough radioactive material to give you reasonable count rates before decaying after a few hours.

Consult the Nuclear Decay Schemes to identify the most prominent gamma emissions for each source.

In this experiment, precise measurements of gamma energies and Compton edges are needed. We will use an intrinsic high-purity germanium detector for this purpose which has better energy resolution (but worse timing resolution) than the sodium-iodide [NaI(Tl)] type you used in the Gamma Cross Sections experiment.

Gamma radiation enters the detector and Compton scatters with electrons (or, eventually, undergoes the photoelectric effect), producing electron-hole pairs. This separated charge is collected by the application of a high voltage. The quantity of charge collected is proportional to the energy deposited by the gamma in the crystal. A charge-sensitive pre-amplifier produces pulse heights proportional to the collected charge, an amplifier amplifies the signal, and a pulse height analyzer (PHA) digitizes the pulse heights and displays a histogram of the energies.

NOTEBOOK: Look at the output from the amplifier on the oscilloscope. Sketch (to scale and with values) a typical pulse shape. It may be necessary to place a radioactive source in front of the detector to generate a high enough count rate to observe pulses.

It is necessary to understand the features of the spectrum in order to make measurements intelligently. Obtain a spectrum of Cs-137. It has the simplest spectrum of the sources provided.

Figure 3: Features of the energy spectrum of Cs-137.

A sketch of the spectrum of Cs-137 appears in Fig. 3 and the features are as follows:

1. The Compton shelf is due to gamma particles Compton scattering one or more times from electrons in the detector, but leaving the detector before all their energy is deposited.
2. The Compton edge is the maximum energy which can be deposited in the detector by a single Compton scatter. The maximum energy is deposited when the incoming gamma scatters through 180º, as in Fig. 2. Notice that the count rate of the shelf rises just before the edge.
3. The region between the Compton edge and full energy photopeak Is due to multiple Compton scatters in the detector which are_not_ followed by the absorption of the scattered photon via the photoelectric effect.
4. The full energy photopeak is due to the full energy transfer from the gamma to the detector. This process usually requires one or more Compton scatters, but the final interaction must be absorption of the photon by photoelectric effect.
5. The dashed lines show smearing of the energies caused by finite detector resolution. Resolution is influenced by statistical fluctuations in the detector and the quality of the electronics used to process the pulses.

For each photon within each source you study in this experiment, you will need to extract two values: the energy of the incoming photon, _E_γ (given at point IV in Fig. 3), and the kinetic energy given to an electron in a 180º scatter, T (which is the Compton edge energy, given at point II in Fig. 3). Note that there is some uncertainty in each of these measurements (the spreading described by dashed lines) and this should be quantified when recording values.

Vary the amplifier gain and observe the shift in the Cs-137 peak position. Using the known energy of this peak, adjust until you know that the highest peak we wish to study – the 2.112 MeV peak of In-116 – will be displayed near the highest channel number, but still on-screen. Once set, the gain should not be changed.

NOTEBOOK: Record the final gain settings. Sketch the Cs-137 spectrum (to scale) and identify the features (see previous section) along with channel number location and approximate count rate scale.

We wish to relate channel number on the PHA to energy deposited in the detector. One can calibrate in-software or calibrate after the fact during the analysis.

The Spectrum Techniques UCS-20 (or UCS-30) PHA software has a two- or three-point calibration feature which allows the x-axis of the display to change from channel number to energy. To calibrate in this way, one needs to know the true energy and the corresponding channel location for two (or three) gamma spectrum features. In our case, we will use the location of the 0.122 MeV full energy peak of Co-57 and the 1.7697 MeV full energy peak of Bi-207. This gives us points at both the high and low end of our energy spectrum.

NOTEBOOK: Collect a PHA spectrum for Co-57 and Bi-207 and sketch each in your notebook. Identify the two peaks in question and record the peak centroids (with uncertainties). While the software cannot incorporate the peak uncertainties into its calibration, you may wish to comment on these when discussing the calibration in your report.

Once the peak positions are known, select “Settings: Energy Calibrate: 2 point” from the drop down menu. Follow the software instructions and make sure to enter values of energies (in keV) with the full number of digits available from the decay schemes.

CHECK:  Will the 2.1 MeV peak of indium be on-screen? (If you plan to collect the optional background spectrum (see Sec. 3.6), make sure that the gain is set such that the 2.6 MeV Tl-208 line will be on scale as well.)

The in-software calibration method described above has several limitations.

• It does not incorporate the uncertainties on the measured peak channel positions.
• It does not explicitly provide the fit function and cannot produce uncertainties on the fit parameters determined.
• One can only use two or three calibration points.

Therefore, it is preferable to collect data in raw channel number and then, at home, do a proper calibration plot to convert from channel to energy.

NOTE: It does not hurt to do the two-point calibration described above even if you plan to do a better calibration later. When exporting the data in the *.tsv format, both the channel number and calibrated energy values are saved. In this way, you may use the rough calibration values from the software as a guide while in lab, but do the proper calibration when producing final plots and analysis.

As you collect data, you will find that several full energy peaks can be identified, but their Compton edges cannot. These points (which include the Co-57 0.122 MeV and Bi-207 1.7697 MeV gammas used in the software calibration) are useless for determining the dispersion relation (both $E_\gamma$ and $T$ are required to calculate momentum), but useful for calibration; all told, you may be able to find 4-6 peaks which are suitable for use in the calibration.

IMPORTANT: To avoid circular logic, a gamma can be used EITHER as part of the calibration OR as part of the dispersion relation analysis; it cannot be used in both.

More information on how to use these points for a post-experiment calibration is given below in the Analysis section.

Proceed through each radioactive source (repeating the sources used above, if necessary), measuring all usable gamma energies and, when possible, the energies of their associated Compton edges. You will need to refer to the Nuclear Decay Schemes to identify all the features for each radioactive source and to find their relative intensities.

NOTEBOOK: For each spectrum, make a sketch (to scale) and identify features. Record full energy peak positions $E_\gamma$ and Compton edge positions $T$ in a table like the one suggested at the end of the Theory section. It is suggested that you record both values in terms of raw channel and energy (as determined by the software calibration). Estimate uncertainties in all your measurements.

|Identifying which Compton edge corresponds to which gamma can be tricky! You may wish to use the Compton scattering formula

to check your guesses, recalling that the Compton edge corresponds to the electron kinetic energy for a photon scattered through $180^\circ : T = E_\gamma - E^\prime_\gamma \;(\theta = 180^\circ)$. Likewise, separating small peaks from weak Compton edges can also be confusing. Look for asymmetry in Compton edges and for the characteristic break in the slope shown in Fig. 3. CAUTION: While you may use the Compton formula as a “check”, you must still measure each edge. Using calculated values for the edge is circular logic as the Compton formula can be derived from special relativity… which is what we're trying to test by measuring Compton edges!

Save your spectra in both the *.tsv and *.spu formats (for use at home and in-lab respectively).

NOTEBOOK: Record the filenames for each spectrum or clearly describe your naming scheme in your notebook.

In addition there following are two optional spectra you can collect which are quite interesting:

• The Eu-152 sample has many, many gamma emissions with overlapping peaks and edges. You aren't expected to be able to extract any peak and Compton edge pairs, but you might be able to identify peaks for use as calibration points.
• There are several naturally-occuring radioactive isotopes which may be present in small quantities around the room (principally in the masonry of the building). If you collect a long background spectrum (e.g. between Day 1 and Day 2), you may be able to find peaks associated with K-40, Bi-214, Tl-208, Pb-212, or Pb-214. The most promising gammas for seeing both peak and edges are the 1460.85 keV line from K-40 and the 2614.5 keV line from Tl-208.

Above, we have implied that you can estimate the peak center and uncertainty in the center (and therefore the photon energy and uncertainty in energy) by eye. One can alternately fit the peaks to a Gaussian function (with appropriate background) and allow the fit routine to identify the center and uncertainty. Is this an appropriate method?

While peak fitting has the potential to yield smaller uncertainties (it is difficult, for example, to claim an uncertainty of much less than one channel by eye), it also can be skewed heavily by poor background modeling (especially when peaks overlap or peaks and Compton edges intersect). The germanium detector that we use naturally produces very narrow peaks (especially when compared to the NaI(Tl) detectors used elsewhere in this course) and the count rates for most sources are high enough that clear features can be collected without waiting too long. Therefore, the added work of doing Gaussian fits – and added systematic bias due to background issues –  may not be worth it. Use your judgement about how to estimate peak positions and consult with your TA if you have questions. 

When you are finished with the experiment, you should turn the bias voltage to the detector to zero, but leave the NIM box (the electronics box housing the power supply module) on. Just as the detector can be damaged by quickly turning the voltage on, it can also be damaged by quickly turning it off.

Crystal gamma detectors typically have a linear relationship between the energy deposited by the photon and the amount of charge (or the height of the signal pulse) read out. We will therefore assume a generic linear model,

where $E$  is the energy and { $ch$  is the channel number. However, because we take energy to be exact and have uncertainty only on channel, it is instead proper to fit the inverse function

(where $A^\prime = 1/A$  and $B^\prime = -B/A$ ) so that our error bars are on the y-axis variable. 

ANALYSIS: Plot $ch$ versus $E$ and fit to Eq. (10b). Do not round literature values for the calibration energies; use as many digits as you can.

This fit yields a conversion from energy to channel, but practically we need the conversion from channel to energy of Eq. (10a).

ANALYSIS: Using $A = 1/A^\prime$ and $B = -B^\prime/A^\prime$, convert your best-fit values $A^\prime$ and $B^\prime$ (with uncertainties) into parameters $A$ and $B$ (with uncertainties).

Equation (10a) can now be used to convert all channel axis into energy axis. (Use this calibration method for final plots instead of the two-point in-software calibration done in lab.)

We now recognize that there are two contributions to the uncertainty in the measured quantities $E_\gamma$  and $T$ :

• The first is the uncertainty which comes from determining the peak center or Compton edge position.
• The second is the uncertainty which comes from converting channel to energy.

The first of these uncertainties is statistical (and will differ for all points), while the second is systematic (and cause a related error to all points). Depending on how you estimated these values, either one error may dominate over the other (meaning you may neglect the smaller one) or both will be comparable (meaning you may need to account for both in your calculations).

ANALYSIS: Discuss the magnitude of these two contributions to the energy uncertainty and justify your choice to include or neglect one or both errors. If you include both errors, discuss how they are incorporated into your analysis.

We now want to look at our data and investigate the relationship between momentum and energy for electrons. Let us try to “discover” the special relativity form which is appropriate even at high energies.

Recall that the energy-momentum relation for a non-relativistic mass $m_{nr}$ is given by

This can be rearranged to give the mass 

which we normally believe to be a constant.

ANALYSIS: Plot $p^2/2T$ (with uncertainties) versus $T$. You should find that it is not constant as classical mechanics predicts by Eq. (12). What functional form does the data actually appear to take? Fit the data to this form and extract the fit parameters with uncertainties. Make a plausible identification of each fit parameter in terms of integer or fractional numbers or fundamental constants.

A NOTE ON UNITS: The SI unit for energy is J = kg m2/s2, for momentum it is kg m/s, and for mass it is kg. However, we have measured energies in units of electron volts, eV. It is possible to convert eV to joules and arrive at the SI units, but it is preferable to keep the eV as part of our units for all three quantities. To do this, we can introduce the unit system common in high energy physics (where special relativity is de rigueur) where energy has unit eV, momentum has unit eV/c and mass has unit eV/c2. In this system, the pesky factors of the speed of light get rolled into the unit and do not need to be handled numerically.

Consider the following example. Suppose we measure a full energy peak energy $E_\gamma$ = 573.6 keV and a corresponding Compton edge at $T$ = 392.0 keV. Using Eq. (8), we find that this corresponds to $pc = 2E_\gamma -T$ = 755.2 keV, or an electron momentum of $p$ = 755.2 keV/c. When used with Eq. (12), we find that this data point yields a value $p^2/2T$ = (755.2 keV/c)2/(2*392.0 keV) = 727.5 keV/c${}^2$. This has units of mass (as expected), and could be converted to kilograms if desired (which it is not). CAUTION: You must always be mindful about propagating independent uncertainties. As $p$ depends on both $T$ and $E_\gamma$, the variables $p$ and $T$ are NOT independent and therefore the uncertainties in $p$ and $T$ are NOT independent. In order to calculate the uncertainty in the quantity $p^2/2T$, you must first rewrite it in terms of the independent variables as $\left(2E_\gamma -T\right)^2\big/\left(2Tc^2\right)$ and propagate so that the result is in terms of uncertainty in $E_\gamma$ and $T$ only. From your fit, you now have an empirical relation

where $g(T)$ is a function of $T$ with coefficients that only involve numbers and fundamental constants. We now have the dispersion relation!  To test whether you have made the correct identification, let us now compare this form to the known special relativity dispersion relation

where $E = T+m_0c^2$ is the full energy (kinetic plus rest energy). Rewriting in terms only of $T$, this becomes

ANALYSIS: From your fit function, Eq. (13), can you rearrange to find the relativistic dispersion relation, Eq. (15)?

We expect that this relation should reduce to the classical energy-momentum relation, Eq. (11), in the non-relativistic limit.

ANALYSIS: Does your fit function $g(T)$ reduce to a constant $m_0$ in the limit of small kinetic energies, $T \ll 2m_0c^2$? Can you recover the classical kinetic equation relation, Eq. (11), from the relativistic relation, Eq. (15), in the same limit?

Now that we have empirically determined the dispersion relation, we can manipulate this equation to show different interesting behaviors that arise as a consequence of special relativity. In the sections which follow, we do not necessarily derive anything new, but by plotting this relation in different ways, hope to show new phenomena.

Classical mechanics predicted that a plot of Eq. (12) would yield a constant equal to the mass of the electron, but we found that this was not true. Since we now know the correct dispersion relation, we can rearrange Eq. (15) to solve for the rest mass (or rest energy) and again test the constant mass prediction. We find

ANALYSIS: Compute the rest energy $m_0c^2$  according to Eq. (16) for each $T$ and $E_\gamma$ pair and calculate the weighted mean and weighted uncertainty (standard deviation in the mean). Plot the rest energy values versus $T$  and include a horizontal line indicating the average. Again, be sure to express the uncertainties in terms of independent variables $E_\gamma$ and $T$.

In addition to the dispersion relation of Eq. (14), special relativity also predicts individual relationships between energy and velocity and between momentum and velocity. If we define the reduced velocity $\beta = v/c$, we can express everything in terms of the relativistic mass,

which we see reduces to $m_0$ at small velocity $\beta \ll 1$) and diverges at large velocity $\beta \rightarrow 1$ ).  Momentum can be expressed in the normal form $p = mv$, which becomes

Energy is given by Einstein's $E = mc^2$, or rewriting in terms of just the kinetic energy,

We are able to compute$\beta$  from our measured quantities as

ANALYSIS: Make the following two separate plots of your data: $p$  versus $\beta$ and $T$ versus $\beta$. You do not need to fit these data to a function; instead overlay the expected forms – Eq. (18) and (19) – using a dotted or dashed line on top of the data. Comment on the behavior in the limit as $\beta \rightarrow 1$.

Eq. (20) is a bit hairy, so we'll be nice and provide the formula for the uncertainty in $\beta$ for you. (See? We're not so mean!) The uncertainty in velocity $\Delta\beta$  is given (after much algebraic simplification) by

where $\Delta T$ is the uncertainty in the kinetic energy $T$ and $\Delta E_\gamma$ is the uncertainty in the full energy peak energy $E_\gamma$.

Finally, we can plot the electron dispersion relation – $T$ as a function of $p$. The non-relativistic prediction is the well-known form

whereas we now know that the relativistic prediction is $T = E-m_0c^2$ or

ANALYSIS: Plot data for $T$ versus $p$. You do not need to fit these data to a function; instead overlay the two prediction curves – Eq. (22) and (23) – using dotted or dashed lines on top of the data. Do any of your data fall in a region where both predictions appear to plausibly hold?

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 showing where the classical definitions of energy and momentum begin to deviate from the data, and how a relativistic treatment of the problem resolves these deviations. Through out the wiki you will find Analysis notes at various points which suggest plotting certain quantities against one another. Following these notes will guide you through the more significant aspects of a complete analysis. They are not a check list for an analysis writeup.

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 estimated peak locations and their uncertainties.
• How uncertainties were propagated through calculations.
• How Compton edges were identified, measured and appropriate uncertainties arrived at.
• Difficulties associated with identifying various parts of the PHA spectra.
• How the energy scale of the PHA was calibrated, how uncertainties in this calibration were estimated and propagated through calculations.

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.