In this lab the student
will become familiar with the use of the Geiger-Muller tube for the detection of nuclear radiation and be
introduced to the statistics of nuclear counting.
Sources include natural background radiation as well as radiation from weak sources of $\alpha$, $\beta$, and $\gamma$
radiation. The student will become familiar with units of measure and with important figures of merit
related nuclear radiation. Moreover, we explore the distribution of counts of actual decay events in 2 different limits which give rise to our research questions. Rutherford [1] describes these sorts of experiments as follows (here focusing on $\alpha$ decay),
In counting the $\alpha$ emitted from radioactive substance either by the scintillation or electric method, it is observed that, while the average number of particles from a steady source is nearly constant, when a large number is counted, the number appearing in a given short interval is subject to wide fluctuations. These variations are especially noticeable when only a few scintillations appear per minute. For example, during a considerable interval it may happen that no
$\alpha$ particles appears; then follows a group of $\alpha$ particles in rapid succession; then an occasional $\alpha$ particle, and so on. It is of importance to settle whether these variations in distribution are in agreement with the laws of probability, i.e., whether the distribution of $\alpha$ particles on an average are expelled at random both in regard to space and time. [my emphasis] It might be conceived, for example, that the emission of an $\alpha$ particle might precipitate the disintegration of neighboring atoms, and so lead to a distribution of $\alpha$ particles at variance with the simple probability law.
Our research questions are these:
how do the counts per counting interval vary if as Rutherford describes above, the average number of counts is not steadily decreasing during the total counting interval?
What probability distribution function best describes the variation in counts per sampling interval, a Poisson distribution or a Gaussian distribution? The radioactive nuclide used for this question is $^{210}_{84}Po$.
how do the counts vary per counting interval if the average number of counts is systematically varying during the total counting interval? What law of random decay best describes the systematic change in counts per interval in time? The radioactive nuclide used for this question is $^{137m}_{56} Ba.$
The Poisson distribution function arises from what are called Bernoulli trials, that is, from random events that can have only 2 outcomes, namely that the thing happened, or it didn't, whatever the thing is. In this case, the event is whether a nucleus decays or not after a time $t$. If we observe the sample for $N_t$ consecutive counting intervals $\Delta t$, composing a long time $T = N_t \Delta t$, where $N_t \gg 1$, then the probability of recording $n$ counts in any one counting interval ($\Delta t$) is \begin{equation}
P(n) = \frac{e^{-\mu}\mu^n}{n!}, \end{equation}
and where $\mu$ is the true average count per counting interval, best estimated for a given data set by $\overline{n} = N/N_t$, where $N$ is the total number of recorded counts. The distribution of the number of counting intervals in which
a given count $n$ is expected is then $N_t*P(n)$. It is the case that for such a distribution, there is a necessary connection between the mean value and its variance. The function $P(n)$ is defined on the integers, and is often plotted as a histogram, frequency of a given count, vs. counts. The ordinate records how often a given count occurs among the counting intervals, and the abscissa simply enumerates the count itself from 0 to some integer larger than the largest count. We mention in passing that standard deviation of a Poisson distribution function from the mean $\overline{n}$ is $\sqrt{\overline{n}}$. This has implications for the experimental uncertainty to be ascribed to the number $n$ recorded for any given interval.
On the other hand, Gaussian distribution functions may or may not have to do with random variables, and could in principle model deterministic outcomes such as is the case for the Maxwell-Boltzmann distribution of velocities in a classical gas, where deterministic trajectories exist, but because of the sheer enormity of trajectories that must be modeled, one makes general arguments that the inevitable distribution of velocities must follow given certain constraints, such as thermal equilibrium. In such a case there is no conceptual connection between mean values and their variance. One could easily, for example, have a very cold beam of atoms, or hot still air, each nicely described by Gaussians. To model counts in a counting interval with a Gaussian, one writes \begin{equation}
G(n) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{ -(n-\overline{n})^2/(2\sigma^2) },
\end{equation}
and one has to separately model the mean and the standard deviation. But in the limit that $\overline{n}$ is sufficiently large, the distributions become indistinguishable. This is called the Central Limit Theorem.
Finally, if the total counting interval exceeds, significantly, the time it takes for the probability that a given nuclide will decay to reach 1/2, the rate at which decays occur in a sample itself decays exponentially. After a 'long time', we expect the count rate to relax to the ambient background count rate, \begin{equation}
R(t) = R_0 e^{-\lambda t} + R_b,
\end{equation}
where $\lambda$ is rate constant characteristic of the nuclide.
The method we use to count particle emitted in nuclear decay events is the so-called electric method Rutherford refers to above, a gas filled cylindrical tube, well shielded except at its face, with a high voltage pin coaxial with the tube. As an energetic particle traverses the volume, ionization events rapidly occur, and the strong electric field inside the tube causes an electron avelanche to the pin, producing a current pulse which is registered electronically as a count. A species of such a device is called the Geiger-Muller tube (GMt), and we are using one for our experiments, pictured below.
We will use the GMt to obtain counts from 3 sources for analysis:
background radiation
an 'old' source of $\alpha$ particles, i.e. a sample of $^{210}_{84}Po$ prepared more than a year ago,
a newer source of $\alpha$ particles, i.e. a sample of $^{210}_{84}Po$ prepared about a year ago,
Figure 1. The GMt (a) is positioned directly above the source of radioactivity placed in a plastic holder, a few cm. away. An electronic
timer (b) coordinates the counting, and is hardwired to a pc, running software (c) that creates a spreadsheet (a .tsv file) of the counts. The manual for the hardware and the software are found on our public course website (ST260 Manual [2]). A Geiger-Muller tube (d) is also shown, simplified. In common tubes, the anode and cathode are coaxial, with the cathode being a central wire. A current avalanche arising from ionization is collected and forms a single count.
Notes for data taking and analysis:
Prepare the GMt for counting experiments. Set the GMt voltage to 770V using the analog knob on the timer itself (figure 1b) so that the tube operates near the so-called plateau regime (see figure 34 in the ST260 manual).
Set the time (duration) of the bins to 1sec, and 'runs' to 1000. Look for the Preset menu item, and set 'time' and 'number of runs', and note the green diamond....in a moment you'll click on that the start the counting.
Figure 2. Presets for time (duration of $\Delta t$) and number of counting intervals.
without any source placed under the GMt in the shelves, start counting (see above about the little green box) and begin measuring background counts.
Before completing your writeup and abstract, compare the background count rate in Bq with an estimate of the count rate in your body from an estimate of its $^{14} _6 C$ content in Bq. Compare the internal dose of beta radiation this constitutes in Sv, and compare this with the annual background dose (see table Q15.1 in Moore) From these calculations and comparisons, express your opinion about whether the radiation level in the room is above, equal to, or below the natural background radiation level given in the Moore. State any assumptions you may have to make to arrive at an opinion.
Copy the .tsv file generated by the software to a handy location for further analysis. In your lab notebook, record the chosen file name, what's in it, where it's recorded, and be sure share with members of the team.
Obtain a sealed source of 'old' $^{210}_{84}Po$, and place on the second level, holding it with a little plastic tray (with a finite potential well confining potential, haha, but you'll see how useful the indentation is...), using a little washer that lifts it 2mm from the bottom of the tray. But before placing it in the little plastic tray on its back, record all the information on the top of the source. Be complete.
Take a 1000 runs with the time counting interval set to 1sec. Copy the .tsv file generated by the software to a handy location for further analysis. In your lab notebook, record the chosen file name, what's in it, etc., etc.
Repeat in detail with a newer source (including the writing down of all the information on the back of the source, etc., etc.,), but this time without the washer
Use Fitteia to model the distribution of counts for all 3 runs, both with a Poisson distribution and a Gaussian distribution, comparing, quantitatively, the goodness of the fit: this means obtaining the reduced $\chi_{\nu}^2$, that is, $\chi_{\nu}^2 = \chi^2/(N-M)$, where $N$ is the number of data points used in the fitting, and $M$ is the number of adjustable parameters. Notice the way in which the factorial must be coded in C. Actually C doesn't have the factorial function exactly, and Pedro Sebastião himself wrote a factorial function for us so that we could model Poisson distributions using Fitteia. Thanks Pedro!!!
Figure 3. Histogram of variation in number of counts for a Po-210 source, measured by Rutherford et al. [1], as modeled using a Poisson Distribution in Fitteia. This is standard Fitteia output; note the modeling function.
Note also that there is a Gaussian function that you can cut and past into the
Function buffer, (see 'Function and Parameters (Specific function's library)', esp. GAUSS1, and a bit you could use is, verbatim
y = a/sqrt(3.14159265)*exp(-pow((x-f0)/b,2.0)),
using $a$, $b$ and $f_o$, as fitting parameters. If you use something like this, or whatever you use, be careful to write down in your lab notebook how the function used for the numerical fitting maps in detail to equation 2. The same applies to all fitting excercises (and to the Poisson fitting above, and to the nuclear decay fitting in task #2).
Which probability distribution best fits the histogram? A Poisson distribution, or a Gaussian distribution? Discuss. Evaluate. Coming to a defensible conclusion and evaluation of this question is one of the principle results of these experiments.
8.3 Task #2: measurement of $t_{1/2}$ of $^{137m}_{56} Ba$
Figure 4. One of the decays exhibited by Cs 137 is the $\beta^-$ decay is to the metastable state Ba 137m which decays 90 % through a .66 Mev $\gamma$ to the stable state $^{137}_{56} Ba$.
What's different this time is that our sample will be freshly prepared at the moment of use. Dr. Skelton will prepare the sample using a weak solution of hydrochloric acid. The fluid removes Ba-137 from Cs-137, suspending Barium isotopes in solution. Just 5-10 drops of the 'eluting' solution will do. The drops are placed into metal cup (planchet) which is the then set into the plastic sample holder and the counting cacophony begins. The energy level diagram shown below indicates the principle decays, particles, energies, and half-lives. The known half-life (in the rest frame of the nucleus :) is 2.55 minutes. At the end of our work, the student will demonstrate an analysis of counts vs. time measurement that is to be compared with the known half-life. This is one of the principal results of this experiment.
In more detail,
Set the HV to 770V as we did last time.
Set the time of the bins to 10 sec, number of runs to 300, and begin measuring background counts, while waiting the freshly prepared source. One wants at least 250 runs once the planchet has been received. This is a design choice. Consider if it is a good design choice, given that $t_{1/2} = 2.55 min.$
Place the planchet on the 2nd shelf, finish recording counts.
Copy the .tsv file generated, uniquely named, etc., etc.,
Using Fitteia in the usual way (do please record your procedure for preparing files for analysis in detail!), using equation 3. Note: you'll have to truncate the first part of your data, up till the counts 'go crazy up'. You will know what this means. Does the measured half-life agree with the known half-life within error? Explain! Use significant digits appropriately (note that the Fitteia report does not...it's not wrong, but it is extremely unhelpful, and nonstandard in format).
The decay shown above in figure 4, or rather one of them,
\begin{equation} ^{137m}_{56} Ba \longrightarrow ^{137}_{56} Ba + \gamma, \end{equation}
produces a $\gamma$ that only inefficiently ionizes the gass in the GMt. A fraction of the time (about 10 \%), an internal conversion event occurs and an electron is ejected from the atom, one from the inner or lowest atomic energy states, which is then followed by a cascade of x-ray transitions. The internal conversion electron is much more efficient at ionizing the gas in the Gmt. Before turning in your report explain why the energetic electron is more efficient than the $\gamma$ at ionizing the gas.
Do you expect the background counts before and after to be the same? Carefully consider. Be sure to study the SpecTech manual for this particular experiment [3]. If the background count exceeds that measured earlier in the day, what might account for it?
Now, the abstract.
References:
Rutherford, Geiger, and Bateman, 'The Probability Variations in the Distribution of $\alpha$ Particles', Phil. Mag.20, 698, (1910).
