3. Absorption of Beta Particles

3.1. Background

The transmission of beta particles (electrons and positrons) emitted by a source through material shows a quite different behavior than that of mono energetic electrons. The electron energy spectrum of most beta sources, e.g. the \(^{90}Sr\) source, is continuous with an intensity maximum at about 0.7 MeV and a maximum electron energy of 2.2 MeV. High energy electrons loose energy in materials through (a) inelastic collisions with the atoms of the material ionizing its atoms and (b) through radiation of electromagnetic radiation (Bremsstrahlung). Calculating the transmission rate for electrons is very complicated making it necessary to measure it as a function of material thickness. This is important if one needs to design shielding for highly active material.

Often it is found that the experimental result can be described by a simple exponential function:

(3.1)\[T = \frac{I}{I_0} = e^{-\alpha x}\]

where \(x\) is the material thickness (in appropriate units), \(\alpha\) the absorption coefficient and \(T\) the transmission. If \(x\) is given in meters then \(\frac{1}{\alpha}\) represents the thickness at which the transmission falls to a value of \(1/e\) to about 37%. Another characteristic thickness would that where \(T = 0.5\) which would be called the half-thickness \(x_{1/2}\). Find and expression for \(x_{1/2}\) as a function of \(\alpha\).

3.2. Experiment

Use a \(^{90}Sr\) source, a Geiger-Mueller (GM) tube and the corresponding electronics. Connect the coaxial cable of the GM tube to the input of the counter and set the time interval to 60 seconds.(THE END WINDOW IS VERY FRAGILE. DO NOT TOUCH IT.) As absorber material you will use index cards.

  1. Determine the detector plateau. The GM tube together with the counting electronics starts to work only above a certain minimal voltage which needs to be determined experimentally. To do this set the counting time to 10 seconds and mount the source as close to the detector as possible (do not damage the thin entrance window!). If you use a small detector (about a diameter of 1”) start with a voltage of 320 V. If you have a bigger detector you need to start at about 450 - 500 V. If the system does not show any counts, increase the voltage by 20 V and try again. Once you observe counts record the number of counts obtained for this voltage, increase the voltage by 20 V and record the number of counts again. Repeat this process until at some point the number of counts do basically not change anymore. You have now reached a counting plateau. Plot the counts as a function of detector voltage. Select as an operating voltage a value where the counts do not vary anymore. (Typical values for the small detector are 420 V and for the large detectors 740V).

  2. Place the source in such a way that it is as close as possible to the detector but that you can insert all index cards without having to move the detector.

  3. Determine the background rate by counting about 200 events without a source. Note the time it took to reach that many counts. (What is the uncertainty in the background rate ?)

  4. Put the source back and without any cards, count until you have about 200 counts and record the time.

  5. Place 10 index cards between the source and the detector. And count again until you have about 100 - 200 counts and record the time.

  6. Continue to add index cards until you measure only background events. In this case all your electrons have been absorbed.

  7. Take more data in between to get a set of about 10 measurements, covering the range of index cards up to your maximum.

3.3. Analysis

  1. For each data point, subtract the expected number of background events from the measured events and calculate the final rate and its error. When calculating the final rate make sure you include the contribution from the back ground using error propagation.

  2. Plot the natural logarithm of the rate (and its corresponding error) as a function the number of index cards.

  3. You should see a straight line, indicating that the expression in (3.1) is indeed a reasonable description.

  4. From the slope determine \(\alpha\) and \(x_{1/2}\) (including their errors) and check that these values are consistent with your measurement.