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.

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.

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.

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.

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 (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!

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

where 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

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.  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 m_d/m_p.

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: 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:  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 m_d/m_p, this difference also appears in the change of A_air for different Δλs.

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

where A_air= Δλ_h-Δλ_∞ and Δλ_air is the separation of the peaks shown by my spectra.  A_air 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 m_d/m_p=2.  In the other cases I was able to find more precise values.  For the n=4 to n=2 case, m_d/m_p=2.0±0.1, for the n=5 to n=2 case, m_d/m_p=2.03±0.05, and for the n=6 to n=2 case m_d/m_p=1.94±0.05. This led to a weighted average of m_d/m_p=1.97.  It is thus evident that my measurements for m_d/m_p 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 m_d and m_p was also a bit tricky, but I got it to work eventually!