Investigating the inverse square law for a radioactive source

Marc Wierzbitzki Physics HL Hockerill Anglo-European College00-0815-083 Internal Assessment Experiment Portfolio IB Session May 2012

Investigating the inverse square law for a radioactive source

Aim:This experiment aims to examine the inverse square law for a radioactive γ -source (Cobalt-60).

Construction:

Equipment used:- Geiger counter- γ -Source- 2 rulers- Stop watch

Equipment Use3 stands and clamps The stands and clamps will be used to hold

the Geiger counter, the aluminium plate and the gamma source. This allows to easily adjust the set up, which will be necessary since the distance will be changed.

γ -source (Cobalt-60) Cobalt-60 is not a pure gamma source - it also emits beta particles. It was used for this experiment nevertheless, as a thin aluminium plate can absorb beta particles. This means that only gamma radiation will pass through to the Geiger counter.

Aluminium plate Used to absorb any beta radiation coming from the source. The plate used in this experiment was ~0.5cm (±0.1cm) thick.

2 rulers The rulers were used to measure the distance from the source to the Geiger counter.

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The above diagram and the photos show the exact set up that was used for this experiment. Stands were used because of two simple reasons. Firstly, they allowed changing the distance between the source and the Geiger counter easily and secondly it allowed changing the design of the experiment without having to touch the radioactive source. The gamma source that was used in this experiment is Cobalt-60. As the source also emittedβ-radiation, an aluminium plate was set up between the source and the Geiger counter, since it absorbs β-particles, whereas the γ -radiation can pass through it.

Method:

Once the experiment has been set up as instructed above, the first readings can be taken. Firstly, a pure measurement of the background radiation is needed. The Geiger counter should be switched on for about 10 minutes, with no radioactive source nearby. After those 10 minutes, the counter showed 259 ionizations, which means that there were about 26 ionizations per minute. This will be important for the data processing later on. Then, the Cobalt-60 should be placed in one of the clamps. Here it is advisable to use modelling clay to make sure that the gamma source is safe and cannot fall down. The clamps holding the source and the one holding the Geiger counter should then be adjusted until they are at the same height and so that they directly face each other. Lastly, before the first reading can be taken, the (shortest) distance between them has to be measured (see diagram above).Now the first readings can be taken. The readings that were taken range from a distance of 5.5cm±0.5cm to 15.5cm±0.5cm. It was not possible to measure the ionizations per minute for a smaller distance due to the chosen set up. The stands simply could not be brought closer to each other because of the ruler in between them and their bases also have a certain width. After the first test-readings were taken, we found out that at a distance greater than 15.5cm±0.5cm the Geiger counter showed a reading that was only slightly bigger than the background radiation. This meant that the gamma-source had a

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range of about 15.5cm±0.5cm and at a larger distance the equipment used could not measure its effects anymore. However, it was still possible to reach a conclusion whether the gamma-source follows the inverse square law or not because the range (5.5cm±0.5cm to 15.5cm±0.5cm) was big enough (~3r as shown in figure 1). In order to reduce errors and also because radioactivity is a random process, it was decided to take three readings for each distance. One reading lasts one minute and counts the ionizations that happened within that time. After three readings have been taken, an average has to be calculated, from which the background radiation of 26 ionizations per minute has to be subtracted. With this data, a graph can be plotted showing the average ionizations per minute against the distance between the source and the Geiger counter. If the radioactive source follows an inverse square law, we should then see a negative exponential curve. The inverse square law is explained in the diagram below:

Results:

Distance from

source to Geiger

counter

Ionizations per minute

1 2 3 AverageLess

background radiation

d (in cm)±0.5cm

5.5 109 160 144 138 1126.5 122 117 125 121 957.5 92 101 98 94 689.5 80 56 83 73 47

11.5 54 59 56 56 3013.5 48 62 47 52 26

Figure 1: Illustration explaining the inverse square law.

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15.5 52 33 29 38 12

Graphs:

4 6 8 10 12 14 16 180

20

40

60

80

100

120f(x) = 3902.71691524984 x^-2.00911931568747

Average ionizations per minute compared to the distance from the source to the

Geiger counter

Distance from source to Geiger counter (in cm)

Ave

rage

ion

izat

ion

s p

er m

inu

te

1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 30

1

2

3

4

5

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f(x) = − 3.28546204411153 x + 10.9200673394613

f(x) = − 1.25241371092187 x + 6.27214552010848

f(x) = − 2.00911940863242 x + 8.26942849636861

Ln(Ionizations per minute)/Ln(distance)

Ln(distance)

Ln(I

oniz

atio

ns

per

min

ute

)

max

min

The above graph proves that the experiment followed an inverse square law, just as expected. This can be seen by the fact that the gradient of [Ln(Ionizations per minute)/Ln(distance)] is ~2.00. ±1.29.

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Error analysis:The error in the ionizations per minute was calculated by finding the biggest deviation from the mean and then expressing this as a percentage (~37%). However, this only considers that radioactivity is a random process, but other possible errors such as the error in the Geiger counter were not included. The error in the distance was chosen to be ±0.5cm, as the construction chosen for the experiment was probably not the best possible one. It should be optimized for future experiments, which should decrease the error, as it is quite big.

Possible ways to improve the experiment:One problem that was faced during the experiment was that the Geiger counter did not work properly. It was not working at all at first (due to a broken wire), but we were then able to fix it. However, we do not know whether we repaired it properly or if it still was not working accurately, causing a systematic error. This possible error and the general inaccuracy of the Geiger counter were not considered, but when redoing the experiment, it should be paid attention that the equipment is working without any interference. It was also found that the last few distances were far too big, as the background radiation was bigger than the actual number of ionizations per minute caused by the gamma source. This means that there could be a very big inaccuracy. To avoid this, measurements should only be taken for small distances where the number of ionizations is significantly bigger than the background radiation. Alternatively, a stronger source could be used. Furthermore, it should always be taken into account that radioactive decay is a random process, which can only be generalized if enough measurements are taken. This, however, was not really the case in this experiment. Therefore, a more accurate reading of the background radiation should be taken. It is proposed to leave the Geiger counter switched on (with no radioactive sources nearby) for at least 20 minutes, from which the average ionizations per minute can then be calculated. The error in the ionizations per minute for the readings definitely has to be decreased, as it is fairly big (~37%). This could be done by doing the experiment with more readings for each distance, preferably 10. But, it is also important to note that a perfect reading cannot be achieved, since radioactive decay is a random process. Lastly, the error in the distance between the Geiger counter and the radioactive source definitely has to be decreased. The only way to do this is by improving the set up. The clamps and stands used were quite shaky and this did not allow for an accurate measurement. Better equipment should therefore be used. It is also a good idea to use the rulers as shown on the photos above and tape them onto the desk so that they cannot move.

Conclusion:

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Overall, despite all these errors and inaccuracies, we still got a very satisfying result and were able to prove that the experiment follows the inverse square law (gradient on the second graph is ~2). There are still some issues that have to be addressed, as discussed in the improvements section above, which should then lead to an even more accurate result.

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Investigating the inverse square law for a radioactive source