First, we will become familiar with the characteristics of the Geiger-Müller (GM) tube, including the properties of threshold, starting and operating voltages, GM plateau region and plateau slope. Then we will use the GM tube to perform a simple statistical experiment for the purpose of studying some of the techniques of statistical analysis. Finally, we will measure the half-life of an excited state of the $^{137}\mathrm{Ba}$ nucleus.

The basic GM circuit diagram is shown in Figure 1. The power supply and counter are one unit. The GM tube itself consists of a cylindrical outer shell which serves as a cathode and is grounded. In the center it has a coaxially mounted wire serving as anode. The cylinder is filled with an inert gas (typically Argon) at low pressure plus a small amount of “quenching” gas (e.g. ethyl alcohol).

A DC power supply provides a voltage across the tube in the range of a few hundred to over a thousand Volts depending on the requirements of the particular tube. Caution: The front window of the tube is very thin and delicate and should not be touched.

Radiation entering the tube can ionize some of the gas atoms, thereby freeing electrons which are accelerated towards the central wire by the electric field and gaining enough energy to cause further ionization. The result is a “cascade” or “avalanche” effect which produces a current pulse through the tube. This current pulse causes a voltage pulse across resistor $R$, which is used to trigger a counting device that records the number of pulses. We are thus able to count the number of ionizing particles entering the tube. There is, however, no information about the initial particle energy. An oscilloscope is connected to the system to allow us to visually observe the pulses from the tube.

The purpose of the quenching additive to the gas is to effectively absorb UV-photons emitted from the electrodes when the ions produced in the multiplication process impact on the electrodes. Such photons may otherwise liberate secondary electrons (via the photo-electric effect) which may initiate the avalanche process all over again, thereby leading to catastrophic breakdown of the tube (i.e. a spark).

- Place a radioactive source near the GM tube. Note the operating voltage listed on the tube. Make an exploratory study of the effect of turning up the voltage across the tube.
**Do not exceed the operating voltage by more than about 100 V!**Note the appearance of the pulses on the oscilloscope and the behavior of the counter. - Systematically increase the voltage in steps of about 50 V, starting at 500 V. Decrease or increase the step size as necessary to see the shape of the curve you are graphing. For each voltage setting, record and plot the counting rate (counts/second or Hz) and the pulse height as seen on the oscilloscope. Determine the GM ``threshold voltage'', i.e. the voltage at the beginning of the GM plateau. Also measure the slope of the plateau (increase in counting rate per 100 V).
- Arrange a source to give about 5-6 counts per 10 sec interval (perhaps room background will give you sufficient counting rate). Record the number of counts $N$ per 30 sec interval for as many intervals as you can. Plot the number of counts per 30 sec interval as the abscissa and the number of times that number of counts occurred as the ordinate of a graph.
- Determine the mean number of counts $\langle N \rangle$ and the standard deviation $\sigma$. The result should approximate the “Poisson” distribution, for which the probability $P$ for observing $N$ counts in a given time interval is given by

Compare your graph with this expression. Also, compare your results with Figure 2.

We will now measure the half-life of an excited state of the $^{137}Ba$ nucleus. Samples are prepared from a radioactive isotope of $^{137}Cs$. This isotope has too many neutrons to be stable; it decays by $\beta$-decay into $^{137}Ba^{*}$ . This daughter nucleus is produced in an excited state, indicated by the asterisk ($^{*}$). The excited state subsequently decays into the ground state of $^{137}Ba$ with a half-life of the excited state of 2.6 min. The decay of the excited state is detected by measuring the emitted $\gamma$-rays of 0.66 MeV. This energy corresponds exactly to the energy difference between the excited state in $^{137}Ba^{*}$ and the ground state.

The $\gamma$-ray emission rate $R(t)=-\frac{dN(t)}{dt}$, where $N(t)$ is the number of $^{137}Ba^{*}$ nuclei present at time $t$. The emission rate is proportional to $N(t)$:

where the proportionality factor $\lambda$ is defined as the decay constant. Integration of the above expression leads to

where $N_{0}$ is the number of excited nuclei present at $t=0$. The half-life $t_{1⁄2}$ is defined as the time it takes for the activity to be reduced by half; thus, at $t=t_{1⁄2}$ , $N=\frac{N_{0}}{2}$. Derive the relationship between $\lambda$ and $t_{1⁄2}$ . Show that at any time $t+t_{1⁄2}$ , there will remain only one-half of the excited nuclei that were present at time $t$!

- Adjust the high voltage of the Geiger-Müller tube to a value in the plateau region that you found previously (400-600 V usually). Measure the room background for about 10 minutes. You will need this information later to correct your counting measurements of $^{137}Ba^{*}$.
- Use a freshly prepared (!) sample of $^{137}Ba^{*}$ , place it under the Geiger counter, and measure the decay rate $\Delta N$ in some suitable time interval $\Delta t$ as a function of the elapsed time. Continue until the count rate is comparable with the background rate over the same time interval.
- Correct your data for the background and plot the logarithm of the corrected count rate versus the time $t$. You should obtain a straight line. Don't forget to put proper error bars on your data points. From the slope of the straight line you can obtain the half-life and the decay constant (including the errors). Compare with the accepted values.