The Geiger Counter

Andrew Mark Allen - 05370299

January 16th, 2012

Abstract

In this experiment, a Geiger counter, an amplifier and bias unit, and a counting unit were used to measure the half-life of a sample of radioactive Iridium, and two C^{14} samples were used to measure the dead-time of the Geiger Counter.

Aims

• To determine the half-life of an irradiated sample of indium.

• To determine the dead-time of the Geiger counter.

Introduction and Theory

Radioactive Decay

The spontaneous decay of radioactive atoms is given by the equation

\frac{dN}{dt} = λN

where N is the number of undecayed nuclei and λ is a constant parameter called the decay constant. Solving this equation for N gives us

N(t) = N_{0}e^{−λt}

where N_{0} is the number of nuclei present at time t = 0. This describes an exponential decay, and since the rate R ∝ N(t),

log R ∝ −λ

The half life of a radioactive sample, that is, the time for which it takes a radioactive sample to decay to the point where only half as many of the original radioactive atoms are present, is given by the formula

t_{1/2} = \frac{ln 2}{λ}

The Geiger Counter

A Geiger counter is a device that measures ionising radiation. It consists of a tube, the Geiger-Müller Tube (GM Tube), filled with an inert gas such as Argon at low pressure. The tube is connected to an electric circuit such that the surface acts as a cathode while a central wire acts as an anode (see diagram). The tube is open on one end but for a thin mica window, which is thin enough to allow alpha particles along with beta and gamma radiation through.

When a photon or particle enters the tube it ionises some of the particles of gas so that the resulting ions and free electrons are accelerated towards the anode and cathode respectively. On the way they will cause collisions with more particles creating further ionisations, with the resulting cascade referred to as an electron avalanche. The resulting pulse of electricity can be amplified by the electric circuit to create an audible pulse, or can be counted with an electronic circuit.

However, there is not a one to one correspondence between the interactions in the tube and the resulting pulses. There is a period of time between a gas particle being hit by radiation and the count being registered during which any additional interactions will not be recorded as additional counts. This time period is called the dead time. The effect of this dead time becomes more pronounced as the count rate increases.

Calculating Dead Time

The dead time for the GM Tube can be calculated if one has two samples of radioactive sources, by recording the individual count rates and the combined count rate, as described here. To avoid confusion, n will always refer to the true count rate, i.e. what would be recorded if the dead time of the tube was zero, while m variables refer to the measured count rate, a number in all instances read off the machine. A combined rate refers to the rate corresponding to the situation where the Geiger counter is exposed to the two sources at the same time.

Let n_{1} and n_{2} stand for the true count rates of each of the two radioactive sources, and let n = n_{1}+n_{2} be the combined count rate. Let τ be the dead time, and let m_{1} and m_{2} be the measured count rates, with the combined measured count rate m ≠ m_{1} + m_{2}.

It should be seen that during a measurement mτ is the total dead time, and thus nmτ are the number of readings lost during this time. Then,

n_{1} = m_{1} + n_{1}m_{1}τ

or rearranging,

n_{1} = \frac{m_{1}}{1 − m_{1}τ}

Since n = n_{1} + n_{2},

\frac{m_{1}}{1 − m_{1}τ} + \frac{m_{2}}{1 − m_{2}τ} = \frac{m}{1 − mτ}

Cross-multiplying and cancelling terms gives us

mm_{1}m_{2}τ^{2} − 2m_{1}m_{2}τ + (m_{1} + m_{2} + m)

and solving this quadratic gives us the equation for the dead time:

When measuring

Experimental Method

Background Radiation

To measure the background radiation the counter was set to record for two minutes, and the Geiger counter was allowed to run with no radioactive source.

Half-Life

To measure the half life of a radioactive source, the radioactive source (Indium) was placed in the Geiger counter, and the counter was set to record for a minute. Then, the count was recorded at five minute intervals over the course of an hour. The value for the half-life could then be computed from a graph of the Log of the rate versus time.

0.1 Dead time

To find the dead time, readings were taken with a component containing two sources of radioactive C^{14}. Three readings of two minutes each were recorded, in the first case with one of the sources covered, in the second with the other covered and the first uncovered, and finally with both uncovered. The dead time could then be computed as outlined in the Theory section.

Results and Analysis

Discussion and Conclusion

The experimental results for the travelling microscope experiments are shown in Figure ?? and Figure ??

The experimental results for the travelling microscope experiments are shown in Figure ?? and Figure ??

Figure 1: Plot of y^{2} against ring number 4 Figure 2: Plot of y^{2} against ring number

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