Poisson Statistics

The purpose of this lab was to get an introduction to Poisson statistics, and to gain some familiarity with Matlab.  Poisson statistics is a way of representing information using a probability measurement dependent only on the mean of a set of data.  There are characteristic shapes for the Poisson distribution depending on where the mean is.  If the mean is very close to zero, it is possible to see a curve with a large vale at x=0 that rapidly drops to zero as x increases.  As the mean increases, a peak appears, but the amplitude of the peak decreases as the mean increases.  Eventually, the Poisson curve looks like the Gaussian, and in fact the Poisson curve can be used to derive the Gaussian equation.

I used a radioactive Cobalt-60 source to generate a set of data centered around a mean number of counts recorded in a time interval, or a “bin”.  Depending on how long I recorded the source, I could get a small mean or a much larger mean.  The activity of the source was recorded by a photomultiplier tube sensor, which translated each event into a current.  This current went to a pre-amplifier, which turned the current into a series of voltage pulses, which were further amplified by a delay line amplifier and displayed on an oscilloscope.  A multi-channel analyzer (MCA) counted the number of pulses it recorded in a window of time.  Thus each time the PMT recorded an event, it was translated into a voltage pulse and recorded by the MCA.

The MCA was a computer program, and after it had finished collecting a set of data I exported it into Matlab.  Using this program, I was able to find the mean number of events per time interval, and to graph this data.  I varied the time intervals, and the resultant graphs are shown below.



Figure 1:  This is the graph generated when the bin size was 10 us.  There are many bins that have no events, and a few bins that do record events. This data yielded a mean of 0.019.


This figure was created with the bin width at 1 ms.  Most bins recorded 1,2, or 3 counts, resulting in a mean of 1.9792.   The fit clearly has a more defined peak than in the previous figure, but still does not quite resemble a Gaussian.

Figure 2:  This figure was created with the bin width at 1 ms. Most bins recorded 1,2, or 3 counts, resulting in a mean of 1.9792. The fit clearly has a more defined peak than in the previous figure, but still does not quite resemble a Gaussian.

The mean of this data is much larger than either of the previous sets, yielding a fit that more closely resembles a Gaussian (although the fit in this graph is the Poisson distribution).  The bin size in this case was 550 ms.

Figure 3:  The mean of this data is much larger than either of the previous sets, yielding a fit that more closely resembles a Gaussian (although the fit in this graph is the Poisson distribution). The bin size in this case was 550 ms.

One of the main points of this experiment was to compare the Poisson distribution to other distributions.  I compared the Poisson distribution to the Gaussian and to the Binomial distributions.

It only makes sense to compare the Gaussian fit to the Poisson fit when the mean is fairly large, since the Poisson fit at low means is very different from the fit at large means.  I compared the fits at a mean of 20777.7, when the bin size was 10 s.  While there are slight differences in the fits, they both closely adhere to the data.

Comparison of the Gaussian fit to the Poisson fit.  The Gaussian is shown in red and the Poisson fit is shown in blue.

Figure 4:  Comparison of the Gaussian fit to the Poisson fit. The Gaussian is shown in red and the Poisson fit is shown in blue.  These fits, while not exactly the same, are very similar and both are good representations of this data.

I also compared the Poisson distribution to the Binomial distribution.  The Binomial distribution, unlike either the Poisson distribution or the Gaussian, deals with data that has either two responses: 0 or 1.  It was necessary, therefore, for the bins to either record no events or one event.  This was accomplished by decreasing the bin size to 5 us, so the mean was 0.0097.  Such a distribution is shown in Figure 5.  Notice that most of the bins recorded zero events, with a small number recording only one event.  No bins recorded more than one event.

This is the graph generated when the bin size was 5 us.  There are many bins that have no events, and a few bins that do record events.  This data yielded a mean of 0.0097.

Figure 5:  This is the graph generated when the bin size was 5 us. There are many bins that have no events, and a few bins that do record events. This data yielded a mean of 0.0097.

We then examined how many events we were likely to get with a bin size of 5 us.  This was accomplished by obtaining 300 sets of data (6 sets were taken at a time) and plotting the number of events in each 10000-bin set.  This was then compared to the binomial distribution.  The Binomial Distribution is simply another way to analyze this data.

Figure 6:  The Binomial Distribution.  This is a graph of the number of events recorded in 10,000 bins.  The Binomial distribution is clearly a good fit for this data.

Figure 6: The Binomial Distribution. This is a graph of the number of events recorded in 10,000 bins. The Binomial distribution (shown in blue) is clearly a good fit for this data.  The mean number of events recorded in 10,000 bins 5 us wide is around 110.

This lab was interesting because it explored a number of ways to represent the data associated with random events.  While some of these fits, like the Gaussian, I had worked with before, there were new fits such as the Binomial distribution that I had never encountered before.  This lab also taught me a lot about fitting these distributions in Matlab.  It was an interesting way to end the semester!

Optical Spectroscopy

Most hydrogen atoms have only one proton in their nucleus, but some are composed of one proton and one neutron.  These atoms are called deuteron, but are simply an isotope of hydrogen.  Neutrons are neutrally charged atoms, but have about the same mass as a proton.  Thus the ratio of the weight of a hydrogen atom to a deuteron atom  is 1:2.  I was able to measure the ratio of hydrogen mass to deuteron mass experimentally by looking at the light emitted by these atoms as electrons move to lower energy states.

The experimental setup consisted of a lamp which excited the electrons in hydrogen or deuteron atoms via an electric current.  This lamp was placed so that the light entered a darkened chamber through a slit.  By refracting the light, we were able to spatially separate light of different wavelengths.  Since an atom has defined energy states, there are specific wavelengths of light that may be emitted.  These wavelengths are different for every atom, since they depend on the atomic number and reduced mass of the particular particle.  Thus, since hydrogen and deuteron atoms have different masses, the wavelengths of light they emit will differ.  The energy levels of an atom can be described by

Screen shot 2013-04-17 at 4.04.30 PMwhere Z is the nuclear charge of the atom, α is the fine structure constant, μ is the reduced mass, c is the speed of light in a vacuum, and n is the principle quantum number in the Bohr theory.  Thus the wavelength of light emitted by a change in energy levels is

Screen shot 2013-04-17 at 4.07.10 PM

I took four spectra of these emissions around four of the peaks of deuteron.  These pictures are shown below.  Each photo has two peaks.  The larger one is from deuteron atoms, and the smaller one is from hydrogen atoms.  Because it is so hard to separate deuteron from hydrogen, each deuteron sample contained some hydrogen. These pictures have good resolution, so it is easy to estimate their centers and their widths.  The resolution was determined by the width of the slits, such that a good resolution featured two well defined, separate peaks.  I fit each spectra to two Gaussians (one for each peak), to get numerical values for the mean and the standard deviation of each peak.

Figure 1: The first transition state.

Figure 1: The first transition state.  This spectrum is not a good fit for the double Gaussian because it has plateaus at both peaks.  However, the centers of the peaks are easy to find, so it was possible to get a rough estimate for the ratio of md/mp.

Figure 2: H_beta

Figure 2:  Light emitted from electrons going from n=4 to n=2.  Again, there are clearly defined peaks that allowed me to calculate the difference in wavelength of the emissions.

Figure 3: H_delta

Figure 3: The transition from n=5 to n=2.  In order to get a clear peak, it was necessary to increase the integration times to get more data.  This happens because the n=5 is so energetically unfavorable that few electrons begin here.

Figure 4: H_gamma

Figure 4:  The transition from n=6 to n=2.  When the light emitted is from a large transition, there seems to be a smaller change in wavelength than from a smaller transition.  In calculations of md/mp, this difference also appears in the change of Aair for different Δλs.



In order to find the ratio of md/mp, I derived a relationship between the mass of a hydrogen atom and the mass of a deuteron atom.

Screen shot 2013-04-17 at 3.52.51 PM

where Aair= Δλh-Δλand Δλair is the separation of the peaks shown by my spectra.  Aair was different for each emission, since each λ depends on the energy of its energy state.  I found that my measured ratio was very close to the expected value of 2.  For the first spectrum, the fit was not very close because each peak had a plateau, but by doing a preliminary calculation I found md/mp=2.  In the other cases I was able to find more precise values.  For the n=4 to n=2 case, md/mp=2.0±0.1, for the n=5 to n=2 case, md/mp=2.03±0.05, and for the n=6 to n=2 case md/mp=1.94±0.05. This led to a weighted average of md/mp=1.97.  It is thus evident that my measurements for md/mp adhered to what was theoretically expected.

This lab was very finicky, but overall I enjoyed it.  Finding a slit width that provided me with a good resolution of the peaks was tricky, and required many tries.  However, spending this time to get a good spectra paid off in the accuracy of my calculations.  Deriving the relationship between md and mp was also a bit tricky, but I got it to work eventually!

Electron-Positron Annihilation

The purpose of this experiment was to examine the γ rays created when an electron and a positron meet.  Since the positron is the antiparticle of the electron, the two will annihilate if they come close enough to each other.  One of the main goals of the experiment was to measure γ rays that were created in the same reaction.  Since the rest energy of an electron is 511 keV, I was looking for γ rays with 511 keV of energy.  In addition, because the initial particles have no net kinetic energy, the rays were emitted in opposite directions from each other.

I first took a spectrum of the Na-22 sample to make sure I could find the peak I was interested in.  I placed the sample directly in front of a photomultiplier tube sensor.  The γ rays were absorbed by the detector, which created an electric pulse whose magnitude depended on the energy of the incoming ray.  The electric pulse was enhanced by a pre-amplifer before entering a delay-line amplifier (DLA).  I displayed the pulses generated by the pre-amplifier and by the DLA on the same oscilloscope to see how the DLA changed how the pulse was represented.  What I found was that the DLA displayed the pulse in a much easier way to understand.  The pulse was displayed as a peak whose magnitude depended on the magnitude of the pulse.  While many peaks of many different sizes were visible, there was one peak that appeared brighter than the rest.  It was brighter because more rays were detected with this particular energy, and I later found that this peak was from the gamma rays I was interested.  I attached the unipolar output from the DLA to the computer multi-channel analyzer (MCA) and got a spectrum for Na-22.  The second detector is set up exactly like the first, and I took another spectrum to ensure that it was working in the same way.

I then connected our DLA to a TSCA, or timing single channel analyzer.  The TSCA counts the number of pulses, but does not represent their different energies so long as they are large enough to be counted.  I displayed this on another oscilloscope along with the output from the DLA, and adjusted the time delay on the TSCA so that the pulses from both inputs overlapped.  Because the TSCA counted pulses with a specific range of energies, adjusting the upper- and lower-level discriminators on the DLA influenced how often the TSCA output was a pulse.  However, because most of pulses were from the 511 keV photopeak, if this pulse was in the TSCA range the output pulse appeared fairly constant.

I used the TSCA peak to find precisely the placement of the 511 keV photopeak.  By connecting the MCA gate to the DA via oscilloscope 3, I found the upper- and lower-level discriminators on the TSCA that completely blocked out the photopeak from my spectrum.  For detector one, these limits were 0.54 and 0.72, and for detector two these limits were 0.74 and 0.93.  The two are different because each setup has a different gain.

The next step was to calibrate the coincidence measurement. I wanted a way to measure when the two detectors measured events at the same time, since the emitted γ rays I was interested in were emitted simultaneously.  I attached both pre-amplifiers to a pulser so that they would receive information at the same time.  Without the pulser, setup one counted 6 events and setup two counted 12 events after 10 seconds, but with the pulser setup one counted 602 events and setup 2 counted 606 events in the same time interval.  I could measure the events that both detectors counted by connecting them both to a universal coincidence unit which would count the number of times pulses arrived together.  There was a finite width of time that the unit would count as “simultaneously,” and I found this interval by adjusting the gain of one setup and seeing the range for which pulses were counted. This eliminated all background radiation, since the probability of both detectors sensing events within the time interval was marginal.  Since this interval was twice the resolving time of my setup, I was able to calculate the resolving time as tau=1.01±0.02 us.

I then turned off the pulser and measured the radiation emitted from my Na-22 source.  Separately, my setups 1 and 2 detected 516±23 events and 570±23 counts, respectively, but together they only detected 49±7 counts.  Thus 49 counts came from annihilation events in the radioactive source; this is the information I was interested in.  I was able to use this data to calculate the real activity R of the source and the efficiency of my detectors, e1 and e2.  C, the observed count rate, depended on these parameters as well as on the rate at which each detector individually counted events (R1 and R2), and the portion of the emitted gamma rays that would be recorded by the sensor (f).  C=e1e2fR+R1R2(2τ)

In this way I calculated R=1292741=39 uCi.  This number is too high, but comparisons with others who have done this lab showed that all of us had abnormally high values for R. This indicates that there is some problem with the setup or the theory behind the calculation, but we were not able to find the source of the error.

We then calculated how the coincidence measurement depended on angle.  In theory, only when the detectors were 180 degrees apart should they have both recorded events at the same time.  In practice, however, since both detectors have a finite surface area, there is a range of angles for which events will be recorded simultaneously.  I varied the relative angles of the sensor to see whether there was a clear peak at 180 degrees where the sensors recorded the most coincident events.  There was a clear Gaussian distribution around my data, which centered at 178 degrees and had σ=6 degrees.  The Gaussian that fit my data had the equation G(x)=584e^(-(x+1.9)^2)/2(5^2))-10.5, and is shown in Figure 1.


Angle vs. Coincidence counts.  This graph clearly fits a Gaussian fit.  While I expected the peak of the graph to be centered at 0 degrees, the experimental center seems to be closer to 2 degrees.  The width of the Gaussian indicates the size of sigma because sigma is equal to the half width at half height.  If the detectors were placed farther away, the width of the peak would be smaller since they would record a smaller fraction of the emitted rays.

I liked this experiment because it was an interesting way to understand how to eliminate background radiation and focus on the parameters that you are interested in.  Most of the lab focused on calibrating the system, both in terms of understanding the signals to and from each device as well as making sure the two setups were in sync.  We were also able to use this result to find the real activity of our source and to calculate the angular distribution of emitted gamma rays.

Spatial Modes of a Laser

Lasers are often described as focusing light of a single wavelength to a single point.  The light emitted is therefore spatially and temporally coherent, because the electromagnetic waves are all in phase and all transverse positions the emissions have non-varying phase relationships.

In free space, light can exist at any wavelength and frequency.  In a laser, however, light is bounced off two opposite mirrors and creates a standing wave in the longitudinal direction.  This largely restricts the wavelengths of the light because an integer number of half-wavelengths of the light must fit in the space between the mirrors.  All lasers can be described by the number of nodes, n, m, and q, in the x, y, and z directions, respectively.

Figure 1.  Two different modes of a laser.

Figure 1. Two different modes of a laser.  The transverse mode is described by the integers n and m while the longitudinal mode is described by q.  n, m, and q all describe the number of wavelengths of light contained within the laser cavity.

It is thus possible to describe the length of the laser cavity by L=qλ/2, where q is an integer and λ is the wavelength of emitted light.  Since the wavelength is restricted by the boundary conditions, the frequency of emitted light is as well.  The wavelength and frequency of electromagnetic waves are related by c=λf, so f=qc/2L, where q is an integer, c is the speed of light, and L is the length of the cavity.

n and m describe the pattern of waving in the transverse direction.  There are two classes of transverse modes: Gauss-Hermite modes and Gauss-Laguerre modes.  Gauss-Hermite modes have a very grid-like structure, and are described by the equation

hermitewhere Hn and Hm are Hermite polynomials.  The simplest form is when the Hermite polynomials are both equal to one; this yields


This pattern is a simple dot!  The transverse intensity is a gaussian, and for this reason the pattern is called the Gauss-Hermite mode.  Gauss-Laguerre modes use Laguerre polynomials in the place of Hermite polynomials.  Laguerre polynomials are described in polar coordinates, using r and θ to describe position.  The patterns therefore have circular symmetry, and look overall circular.

After aligning the commercial laser, I observed the patterns reflected on the wall of light emitted from the laser, passing through a diverging lens and then through a Fabry-Perot interferometer.  The interferometer was composed of two lenses whose angle and separation distance I could change.  I noticed a pattern of circular rings that seemed to converge or diverge when I changed the separation distance.  As I kept increasing the separation distance, the outer rings got closer and closer together: this is because the number of half-wavelengths of light that fit in the cavity kept increasing.

I then measured this pattern using a Spectrum Analyzer connected to an oscilloscope.  One of the inputs of the oscilloscope measured the voltage applied to the analyzer, and the other input displayed the detection of light.  The voltage applied to the analyzer had a deep well pattern.  The other input mimicked the wells, but also had clear areas of peaks in between the walls of each well.

Each of these areas was made up of three little peaks.  The middle one was the highest, and the adjacent ones had the same amplitude, although it was slightly lower than that of the middle.  The number of peaks corresponded with the number of nodes in the transverse plane.  Since I saw three peaks, there were two  nodes in the cavity.

The distance between large peaks depended on the size of the crystal within the analyzer, which was a known value, and the distance between the small peaks depended on the transverse size of the laser cavity.  They were thus polarized in opposite directions, and I attempted to use a polarizer to see this phenomenon.  I used the oscilloscope to measure the distances between the peaks, and was then able to use the wavelength of the emitted light to calculate the size of the laser cavity.  I did this for the commercial laser as well as the open-cavity laser, and found widths of 2.4 cm and 0.35 cm, respectively.  When I measured the width of each laser, these numbers were very close to the actual dimensions.

If we didn’t know the length of the analyzer crystal, we could calculate it using measurable quantities in the lab.  I initially measured the width of the laser chamber using Δf1=Δf2Δt1/Δt2, where Δf1 was the frequency along the width of the chamber, Δt1 was the time between the smaller peaks as measured by the oscilloscope, Δt2 is the time between the larger peaks on the scope, and Δf2 is a property of the analyzer.  Since we also know that L1=c/2Δf1, where L1 is the length of the cavity, we find that Δf2=cΔt2/2L1Δt1.  Inserting my data into this equation, I found the analyzer to have a free spectral range of 7.56 GHz with an uncertainty of .52 GHz.  This is easily within error of the actual value of 7.4 GHz.

I then placed a camera  so that it could record the patterns created by the open-cavity laser.  By making small adjustments on the variable mirror, I could see a wide variety of patterns being recorded by the camera.  Some of these patterns had a very linear geometry while others had a circular geometry more likely associated with the Gauss-Laguerre models discussed previously.  Some of these images are shown below.

Figure 2.  These are just some of the patterns I was able to see.

Figure 2. These are just some of the patterns I was able to see from the open-cavity laser.  A camera connected to a computer recorded these images.

I then set up an image of two adjacent spots, and measured how their separation changed depending on the distance from the output of the laser to the camera.  The closest and farthest images are compared in Figure 3.

Figure 3.  The left image is at a distance of 21 cm from the edge of the laser and the image to the right is at a distance of 127 cm from the laser.  It is clear that the separation between the two spots increases with distance.

Figure 3. The left image is at a distance of 21 cm from the edge of the laser and the image to the right is at a distance of 127 cm from the laser. It is clear that the separation between the two spots increases with distance.

In an ideal laser, the separation between these two spots should remain constant, because the laser beam should not vary in diameter.  However, most lasers have a slight divergence.  By graphing separation of spots vs distance between laser and sensor, I calculated the divergence of my open-cavity laser.  The relation between these two variables was clearly linear, and I fit a line to the graph.  The graph allowed me to extrapolate the separation of the two spots at any distance from the mirror.  I then measured the reflection of the spots against two walls in order to convert my equation to units of distance rather than of distances in the computer program.  I could set up a triangle with one leg as the distance between two points and the other leg their differences in separation.  As shown in Figure 4, the angle facing the differences in separation is related to the divergence of the laser.  I calculated this divergence to be 0.28 degrees.

Figure 4.  By measuring the radius of the laser beam at two distances, we can calculate the divergence angle.

Figure 4. By measuring the radius of the laser beam at two distances, we can calculate the divergence angle.  If we use two points 2 meters apart, we find that tana=(y2-y1)/1.  In this way we can calculate the angle of divergence θ=2a.

Inside the laser cavity, there is a location z0 when the transverse waves are all aligned in a plane.  This location is the beam waist, and depends on the curvatures of the two mirrors within the laser.  Theoretically, a mirror focuses light onto a specific point, called the focus.  In reality, this focus is not precisely a point, but has a clearly defined width: this is the beam waist.  The diverging rays from the laser also converge to this point, so I extrapolated my equation to find where it would theoretically hit the x-axis: this occurred at x=-1.77186 cm, so the beam waist of my laser was 1.77186 cm from the second mirror inside the laser.

My laser was made up of one concave mirror and one planar mirror, so light was focused to the beam waist only by the concave mirror.  Thus the curvature of the mirror was equal to the distance between it and the focus.  I found this curvature to have a radius of 8.22814 cm.

I never really knew how lasers work, so this experiment was very interesting for me.  I learned a lot about how theoretical lasers differ from experimental lasers, and how some experimental phenomena are dependent on the specific properties of each laser.  The theory was sometimes difficult for me to grasp, but by the end of the experiment I had a much better understanding of the physics of lasers.

Millikan Oil Drop

Charged particles respond to electric fields.  The amount of charge a particle has,  its velocity, and the strength of the field are all major factors that affect how the particle will move.  In this experiment, I was able to observe charged oil drops and, by measuring their motion, was able to calculate the amount of charge they contained.

The oil drops I observed were put into a chamber where I could observe them through a lens.  In this chamber I could vary the magnetic field between positive and negative, and thus control where the charged drops went.  I picked one oil drop that clearly responded to the changing magnetic field, and measured how quickly it rose and fell.  With the help of a nifty excel program I could measure start time and end times, while keeping my eyes on the drop.  I did this a number of times in order to get a fairly accurate measurement.

I was then able to calculate the charge on the oil drop using the equation:

  • q is the charge on the electron
  • ρ is the density of the oil: 886 kg/m^3
  • g is the gravitational constant: 9.8 m/s^2
  • b is a constant: 8.02*10^-3
  • p is the pressure in the chamber; equal to the atmospheric pressure in the room
  • η is the density of air? inside the chamber.  I calculated this by measuring the resistance within the apparatus, which was related to the temperature inside the chamber.  The lab manual contained a chart that related temperature to the density of the air, and I used this chart to convert between them.
  • vis the velocity of the oil drop as it fell.  This was different from the rise velocity because gravity also had a slight effect on the drop’s motion.  To calculate the velocity, I measured the time it took the drop to move .5mm, and then averaged these numbers and converted to the correct units.
  • vis the velocity of the oil drop as it rose.  I calculated this in the same way as I calculated vf.
  • E is the strength of the electric field.  It is therefore related to the voltage difference between the two plates and their separation; I measured both these variables before starting my experiments.

This calculation takes into account all the forces acting on this system, including the force of the magnetic field on the charged particle as well as the effects of gravity and air resistance.  This is why there are so many parts to the equation, and it is relatively easy to derive this equation using these fundamental ideas.

I calculated that the drop I was observing contained a net charge of 6.29*10^-19 C.  Since the extra charge corresponds to the presence of extra electrons, this means I had 4 extra electrons, since each electron has a charge of 1.6*10^-19 Coulombs.

I then completed the experiment again, but varied the charge on the drop of oil.  My oil drop was initially easy to manipulate, as it had an average rise time of 4.70 s and an average fall time of 3.9 s.  Using the same equation as outlined above, I calculated the initial charge on this drop to be from 10 extra electrons.

However, when I changed the charge on the oil drop, I allowed it to gain to many excess electrons, and it started moving very quickly.  The rise time dropped to 0.62 s and the fall time dropped to 0.60 s.  Predictably, this was extremely hard for me to measure accurately, and my data thus covered a very large range.  However, I was able to graph the data on excel (see figure 1) and see that the first 15 data points were fairly accurate, while the latter half of my data is where most of the extreme errors occurred.  I therefore used only the first data points in my analysis (see figure 2)

I was able to calculate that there were 163 extra electrons on my oil drop.  However, because of the large degree of uncertainty in my time, the error in my answer is very high – about 70%.  Thus my final answer for the number of excess electrons on my oil drop was 160 ± 110 excess electrons.

The calculation of my error in this experiment was a very involved process, both because of the complexity of the equation as well as because of all the terms that contained significant amounts of error.  However, I believe that I was able to manage it well, and my resulting error is easy to believe since it fits the initial rough estimates.

One of the main points of this experiment is that charge can be quantized, since in the larger world charge comes from the presence of unbalanced protons or electrons.  In my oil drop, I could connect the extra charge to an integer multiple of the charge of an electron, and could also show that by exposing my drop to other electrons which it could pick up, I could change its charge.  In fact, it is possible (and would have been very interesting) to observe an oil drop that changed its charge by the equivalent of one electron.  Unfortunately, this is not something that I experienced, but it is certainly possible, and maybe, given enough trials, this is something that I would actually be able to see!

Relativistic beta Spectroscopy


When an atom radioactively decays, there are three main products.  The largest piece is a new element, while the others are either an electron and an antineutrino or an antielectron and a neutrino.  In this experiment, we examined elements that emitted electrons and antineutrinos.  Because of conservation of energy, the energy of the original atom is equal to the sum of the energies of the resulting particles.  Since the resulting atom has a lower rest energy than the original, the electron and the neutrino carry the difference.   Thus, it is only necessary to measure one of these particles.  Neutrinos do not react to any of the fundamental forces that we know of, so we take measurements of the electron.

Four various radioactive materials, I found the maximum kinetic energy of the electron.  By attaching a sensor to an MCA, I was able to see a range of electron energies as they showed up as channel numbers in the display.  As Figure 1 shows, the range of electron energies stops after a certain point.  This is the maximum kinetic energy, Kmax, and corresponds to when the antineutrino has no kinetic energy since the electron has all of it.  As you can see, the maximum channel number for Technitium-99 is around 800.

This is the data collected by the MCA. There is a high range of channel number which counted electrons, but after channel 1200 the number electrons sensed drops off. This maximum channel number corresponds to the maximum kinetic energy of the electrons emitted by the Technitium samples.

Because the maximum electron energy of each element is known very precisely, we can find an equation that converts channel number to kinetic energy, by fitting a curve to our data points.  Using a line of best fit (see Figure 2) between the maximum channel numbers and their known maximum kinetic energies for five different elements, I was able to get an equation that allowed me to go between the two numbers.  This equation could then be used to calculate the kinetic energy of any electron, given its channel number as calculated by this particular MCA.

Figure 2: This graph compiles the Maximum Channel Number and Maximum Kinetic Energy data for four elements: Cl-36, Tl-204, Pm-147, and C-14. The line of best fit is the equation that relates these to variables.

Since electrons are charged particles, their motion is affected by magnetic fields.  The next part of my experiment was to induce a magnetic field in the vicinity of Chlorine-36 to measure how the electrons acted.  I put a current through the chamber, and measured the corresponding electric field using a gaussmeter.  The sample I used, Chlorine-36, was placed at the opposite side of the sensor.  The only way for electrons to leave the sample and reach the sensor was if they followed a well-defined circular path; only electrons with a specific kinetic energy would turn at the correct radius to reach the sensor, as can be shown in Figure 3.  Thus, I was able to relate magnetic field strength to electron energy; I did this for various magnetic field strengths.

Figure 3: Data compiled by the MCA for Chlorine with a current of 1.999 amps providing a magnetic field. There is a clearly defined peak around Channel Number 550, indicating that electrons with this energy were the most likely to reach the sensor.

By relating the path of electrons with a specific energy to the strength of the magnetic field at that time, I found a relationship between B^2/K and K, which I was able to use to calculate the charge and rest mass of an electron.  The slope of the graph of  vs K is dependent on the constant c, the radius of the curved path, and the charge of an electron.  We could then use this number to calculate the rest mass of the electron, which was also related to the y-intercept of the graph of B^2/K vs. K, the radius of the curved path, and the constant c.

Figure 4: Graph of B^2/K vs. K for Cl-36. The slope and the y-intercept of the line of best fit can be used to calculate the charge and rest mass of the electrons used in this experiment.

This experiment was an interesting first task for me.  It contained two distinct parts that both needed to be understood to begin to comprehend the physics of beta particles.  This experiment also reaffirmed two important properties: that the beta particles emitted have a maximum kinetic energy and that moving charged particles will behave in a specific way when traveling through a magnetic field.  I used both of these big ideas to understand my results, as well as in my calculations.