The mass of charged particles – such as protons and electrons – can be investigated using mass spectrometry whereby a beam of such particles is bent by a magnetic field and the radius of curvature is measured. For chargeless particles – like neutrons – such techniques do not work and we must be more creative. In this experiment, you will use some known facts about neutrons to design and test an experiment with the goal of determining the mass of the neutron. This lab – unlike many others in this course – allows for a significant amount of creativity and will require you to measure, adapt, and iterate in pursuit of improved measurement techniques.

M. E. Anderson and R. A. Neff, "Neutron energy spectra of different size 239Pu-Be(α, n) sources", Nucl. Instr. and Meth. 99, 231 (1972). This work measures the neutron energy spectra of Pu-Be sources of various sizes.

Before coming to the lab on the first day, read through the Theory section below.

In addition, consider the following:

(a) The mass of a charged particle can be measured by sending the particle with known velocity through a perpendicular magnetic field and measuring the radius of the circular orbit. (This is called a cyclotron.) Look up the mass of the proton and the mass of the deuteron* and report the value (to as many digits as you can find). Provide a reference for your values.

*Note that the deuteron is a bound nucleus with one proton and one neutron, while deuterium is the bound state of a deuteron nucleus with an electron. The two have slightly different energies since there is some binding energy associated with the electron.

(b) If you send a charge-less particle through a magnetic field, it’s trajectory will not bend. We therefore need to think of a different method for determining the mass of the neutron. Perhaps we can infer the mass by observing the formation of a deuteron (n + p → d + γ), where γ represents the energy released (as a photon) due to the fact that the bound state of the deuteron has a lower rest mass than the sum of the neutron and proton separately.

To make an order of magnitude estimate, approximate the mass of the neutron to be equal to the mass of the proton. Determine the gamma energy as the difference in rest energies between the deuteron and two times the rest energy of the proton. We know from other evidence (e.g. the beta decay of a free neutron into a proton, an electron and an antineutrino) that that the neutron is actually slightly heavier than the proton. Does this mean the expected gamma energy is larger or smaller than your estimate?

We will use a source of high energy neutrons in the lab to perform studies on the interaction of neutrons with nuclei. You will devise a series of measurements which you will use to do the following:

• Confirm that deuterons are produced by the interaction of neutrons with protons.
• Measure the mass of the neutron.

Unlike most of the experiments in this course, you will not be provided with a detailed set of instructions on exactly what measurements to take. Instead, you will be given instructions on how to use the detector – a NaI+PMT-type which you should already be familiar with from Introductory Lab: Gamma Cross Sections – and it will be up to you to decide how to address the points listed above. As neither of the above points can be directly achieved by pointing the detector at the source, you will have to instead make measurements of indirect radiation and set up a logic based on different configurations of lead, paraffin, and carbon shielding.

It is not expected that you will figure out an optimal series of measurements on your first attempt. Progressing through the experiment will be an iterative process where you try out some ideas for collecting data, do some preliminary analysis, evaluate your results with the assistance of the instructors, refine your methods, and repeat. This iterative process is similar to how research experiments in a professional lab are performed and is the real point of this instructional experiment.

A 30 second exposure of a 5“ diameter NaI crystal adjacent to a 250 $\mu$Ci ${}^{22}$Na source. You can see the faint blue light produced by the scintillation process as gamma rays from the ${}^{22}$Na source lose energy to electrons in the crystal. If you were to sit in a completely dark room for 15 to 20 minutes to allow your eyes to dark adapt you would be able to see a soft glow from the crystal.

Our source of neutrons for this experiment is a “neutron howitzer”. The core of the howitzer consists of 80-grams of ${}^{239}$Pu mixed homogeneously with ${}^{9}$Be. The plutonium decays with the emission of 5.18 MeV alpha particles at a rate of ~$2 \times 10^{11}$ decays/sec (or, about 5 Curies). Some of these alphas in turn interact with the beryllium to produce neutrons by the process

Neutrons with energy up to almost 11 MeV are emitted from the source. Most of the alpha particles lose energy by ionization in the source before interacting, but approximately 4 out of every $10^5$ produce a neutron by the above reaction, giving a flux of ~$8 \times 10^6$ neutrons/sec.

The four ports of the howitzer should be plugged and locked when not in use and one should never look into a port or unnecessarily spend time in front of it.

Table 1: Measured neutron howitzer radiation dose rates (in mrem per hour) due to neutrons and gammas, separately

Table 2: Maximum permissible doses above background (where background rate is about 360 mrem/year)

To shield against these energetic neutrons, the source is surrounded by a thick layer (~30 cm in thickness) of paraffin, a molecular chain of hydrogen and carbon, $\textrm{CH}_2$. Recall that neutrons do not interact electromagnetically (they have zero charge), so the main method for energy loss is through collisions. Since neutrons and protons (i.e. hydrogen nuclei) have nearly the same mass, the neutron loses half its energy, on the average, for each collision with a proton. The mean free path of the neutrons in paraffin (i.e. the distance between collisions) is a few centimeters, so most neutrons which escape the paraffin volume have undergone enough collisions to lose most of their energy. When these neutrons do escape, they will have been thermalized (that is, reduced in energy to $E_{avg} = k_BT = 0.025 ~\mathrm{eV}$ ) and are no longer dangerous.

We do, however, want some safe access to the high energy neutrons, and this is achieved though the four side ports where the shielding can be removed. Each port has a plug made of lucite (another hydrocarbon material) which will thermalize or block neutrons when inserted, or allow a direct beam to escape when removed. The energy spectrum of the neutrons emerging from an open port of the source is shown in Fig. 1.

Figure 1: Experimentally-measured neutron energy spectra for two Pu-Be sources comparable in size to the one used in this experiment. (source: [1])

The cross sections for thermal neutrons can be enormous: $\sigma_{thermal} \le 10^{-16}\textrm{cm}^2$. Such values mean that thermal neutrons travel very short distances before being scattered. At higher energies, cross-sections are smaller and typically measured in barns ($1~\textrm{b} = 10^{-24}~\textrm{cm}^2$) and can penetrate to greater depths. 

The fact that the neutron has no electrical charge makes it difficult to measure its mass. The mass of charged particles and nuclei can be measured through the technique of mass spectrometry. So, if we assume that the masses of the proton and the deuteron (a deuteron is a bound state of a proton and a neutron) are known from mass spectrometry, we can find the mass of the neutron from the following interaction by which deuterons are produced,

A photon (called the capture gamma) is produced because the bound deuteron has a lower total energy than the separate neutron and proton. Using conservation of energy, if the masses of the proton and deuteron are known and the energy of the capture gamma can be measured, then the mass of the neutron can then be calculated.

If our neutron howitzer is indeed producing energetic neutrons, some of them should be producing deuterons and capture gammas by the process given in Eq. (2). A NaI+PMT detector can be used to detect and measure the energy of gamma-rays coming from the howitzer.  

Measuring the energy of a gamma is easy. The experimental challenge lies in demonstrating that the detected gammas were produced by the formation of deuterons, as opposed to being from some other unrelated source of radiation. This is the part of the experiment which we are challenging you to figure out.

For this experiment, you will make use of the fact that the size of the pulses from the PMT is proportional to the energy of the gamma which struck the NaI crystal. Pulses from the PMT are sent to a SpecTech UCS spectrometer which measures their total integrated charge and displays a histogram of pulse height sizes on the computer. You can use radioactive sources which produce gammas of known energy to calibrate the pulse height axis.

Figure 2: Neutron howitzer and the detector inside the lead “chamber”.

We will use the known energies emitted by ${}^{60}$Co as calibration references.

• Make sure that the port from the neutron howitzer is closed.
• Use lead to shield the detector from radiation produced by the howitzer.
• Place the Co-60 rod source in the chamber in front of the NaI detector and photomultiplier tube (PMT). 
• Verify that the high voltage and anode output cables are appropriately connected to the NaI detector. These cables should be routed through the wall into the next room. With the high voltage cable plugged into the power supply, turn on the voltage and set it to -1000V.
• Attach the anode output BNC cable to the pre-amp input of the pulse-height analyzer (PHA).
• Turn on the PHA and start the PHA software by double-clicking on USX on the desktop.
• Collect a spectrum. It may be necessary to adjust the coarse and/or fine gain to move all features on-screen.
• If the “Dead Time” meter reads more than ~15%, move the source further away from the detector face to lower the count rate.

It is important that you make detailed notes of how you perform this calibration in your lab notebook. It is likely that you may need to refine your calibration procedure after preliminary analysis of the data. In such a case, you will need to know exactly what you did and how you set things up. (Remember, for example, to record the distance between the howitzer and your detector cart in case either object is moved between sessions.)

• Collect a spectrum. From the Nuclear Decay Schemes, identify the energies corresponding to the two emitted gammas and identify the corresponding features on the spectrum.
• Remember that you ultimately will want to measure the centroid of a peak corresponding to a gamma of greater than 1 MeV (as determined in the Preparation Question). In order to make sure that such a peak will appear be on screen, adjust the gain so that energies of at least 3 MeV will be on screen. (You can assume that the relationship between channel and energy is linear.) Note the gain settings in your notebook.
• In addition to the two peaks, you should find a third peak at higher energy (> 2 MeV). Can you explain the origin of this peak?
  • HINT: Consider both the typical time scale of a sodium iodide crystal pulse (look back to the notes from your first lab or plug the detector output into a scope) and the lifetime of the intermediate state in the relaxation of Ni-60 to the ground state as shown on the decay scheme.
  • Once you determine the origin of this peak and know its energy, this point can be used as a third calibration point.
• In the drop-down menus, select “Three-Point Calibration” and use the values of the energy and channel to calibrate the x-axis. Verify that the peaks now appear with the correct energies.

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.

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

NOTE: It does not hurt to do the in-software 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.

We can obtain additional calibration points from Na-22 and Cs-137. With these additional points, we can fit the data to a linear function and properly incorporate all of our uncertainties. For now, simply collect the additional Na-22 and Cs-137 spectra, record the full energy peak centroid values, and identify the known energies that these peaks correspond to. You may later fit the peaks to Gaussians, or you may identify the peaks “by-eye”; either is OK so long as you properly estimate uncertainties.

Do not adjust the gain between calibration points!

From the peak centers for all three samples, you can perform a calibration to extract a function converting channel to energy, $E(ch)$. However, your dominant uncertainties are in the channel (peak centroid), so we must first fit $ch(E)$ and then invert the function (and the associated uncertainties).

At this point you now have the information you need to measure the mass of the neutron. It is up to you to decide how to use the detector and available shielding materials. Here are some things to consider.

In order to measure the mass of the neutron we are making the following assumptions:

1. That energetic neutrons from the source are interacting with protons in the paraffin shielding material, producing capture gammas as described in the theory section above.
2. That the rest masses of the proton and deuteron are already known from mass spectrometry. (You can look these values up in the literature.)

The experiment then boils down to identifying the capture gamma and measuring its energy. 

The detector can be used to measure the spectrum of the radiation coming from the howitzer with various shielding configurations. Once the full energy peak of the capture gamma has been identified, measuring its energy is straight-forward, assuming the detector has been calibrated as described in Sec. 3. The tricky part is demonstrating that the full energy peak is from the capture gamma which requires a bit of detective work. This task can be accomplished using the following:

• Given assumption #1 above, the howitzer should be a source of capture gammas, regardless of whether or not the ports are open or closed. So, with the ports closed, the howitzer should be a strong gamma source with very little neutron radiation escaping.
• When a port is open, it will emit a collimated beam of energetic neutrons in addition to the capture gammas.
• Lead bricks are provided for use in shielding the detector. Lead is more effective at attenuating gammas than neutrons. (Measure the ratio of gamma attenuation to neutron attenuation for the thickness of lead shielding that you choose to use.)
• Paraffin (a material containing both protons in the form of hydrogen atoms, and carbon nuclei) blocks are provided.
• Blocks of graphite (a form of pure carbon) are provided.

You will be expected to make measurements which not only demonstrate that capture gammas are being produced in the howitzer, but to quantify the result in terms of a statistical significance as well. As you work through this experiment do not forget to consult with the lab instructors who will challenge your results. Keep in mind that this is intended to be an iterative process where your analysis of initial measurements leads to new ideas and a better measurement. This experiment is more about learning the process of thinking through how to approach and improve a measurement than it is about obtaining the mass of the neutron.

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.

Your final report 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 determining peak locations, 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.