RADIOACTIVE DECAY - MEASUREMENT OF HALF-LIFE

Transcription

1 MP9 OBJECT 17 RADIOACTIVE DECAY - MEASUREMENT OF HALF-LIFE The object of this experiment is to measure the half-life of the beta decay of Indium-116. THEORY Reference: Section 29.3, College Physics, Serway and Vuille The decay rate, or activity, R, of a radioactive isotope is the rate of change of the number of nuclei of the radioactive isotope, i.e. the number of decays occurring per unit time: R = N or, more precisely, R = t MP9.1 dn (1) dt R depends on a property of the radioactive isotope called the decay constant, λ, (a quantity with units of inverse time) and on the remaining number of nuclei of the radioactive isotope. Combining (1) and (2) yields R = λn (2) dn = λn (3) dt which is an equation that can only be solved using calculus. The solution to (3) is that the abundance, N, of a radioactive isotope decreases with time from its initial abundance, N 0, according to the equation N = N 0 e λt (4) The half-life, T ½, is defined as the time needed for ½ of the original sample to undergo radioactive decay. Evaluating equation (4) at a time of one half-life gives: ½N 0 = N 0 e λt ½ (5) Taking the natural logarithm of both sides and solving for T ½, λn T ½ = = (6) λ λ In this experiment, the beta particles (electrons) emitted as the Indium decays are detected by a Geiger-Muller tube and recorded as a count. The Geiger tube will also record counts due to background radiation from the air, the ground, the building, etc. These background counts, C b, must be subtracted from the total counts, C, to obtain the counts due to the source alone, C. i.e. C = C C b (7) Note that this equation for C is only appropriate if equal time intervals are used to measure C b and C. If the time intervals are not equal then the appropriate equation is C = C C b t tb where t b is the time interval used to measure the background radiation and t is the time interval used to measure the total counts.

2 18 MP9.2 The number of counts from the source, C, recorded in any time interval, is directly proportional to the activity, R, which in turn is directly proportional to N (equation (2)). Therefore C = C 0 e λt (8) where C 0 = initial source counts (at time t = 0). Taking the natural logarithm of both sides of equation (8) yields ln(c ) = λt + ln(c 0 ) (9) Thus a graph of ln(c ) versus t should be linear and have a slope of λ. Therefore, the decay constant, λ, = slope and, as before, T ½ = λn 2 λ (6) PROCEDURE 1. Set the counter knob to 10 minute counts and set the voltage knob to the value indicated on your machine. Press the POWER button. 2. Press the RESET button and then the COUNT button to take a 10 minute background count with no source present. At the end of 10 minutes, the apparatus will cease registering additional counts and the STOP light will be illuminated. Record the displayed value as C b and reset the counter. Checkpoint 1 ask the TA to check your measurement of the background. 3. Place the Indium source in the top shelf below the Geiger tube. 4. Simultaneously start the clock timer (for measuring elapsed times) and press the COUNT button to make a 10 minute count. 5. After recording each count, reset the counter and start the next count at the starting times specified in Table 1. ANALYSIS 1. Calculate the error in each count as the square root of the count. ( δ C = C ) 2. Calculate the ected count (equation (7)). Calculate the error in the ected count. 3. Calculate the natural logarithm of the ected count, ln(c ), and its error: δ (ln(c )) = δc C Checkpoint 2 ask the TA to check your first row of calculated values. 4. While you are collecting data, plot ln(c ) versus starting time, t, in minutes on the fly. 5. Draw, in light pencil, your proposed min/max slope lines on your graph. Checkpoint 3 ask the TA to check your proposed min/max lines. 6. Calculate the min/max slopes, the average slope and its error, and λ (= slope) and δλ. 7. Calculate the half-life of Indium-116 using equation (6).

3 19 Checkpoint 4 ask the TA to check your calculated half-life. MP9.3 CONCLUSION Discuss the significance of the shape of your graph. Checkpoint 5 ask the TA to join your discussion of the Conclusion. SOURCES OF ERROR Consider separately the effects of short-lived and long-lived contaminants. Checkpoint 6 ask the TA to join your discussion of the Sources of Error.

4 20 MP9 RADIOACTIVE DECAY - MEASUREMENT OF HALF-LIFE DATA & RESULTS Indium-116 Half-life Data using 10 minute counts Background Count (C b ) = ± Starting Time, t (min) Count, C δ C Corrected Count C δ (C ) ln(c ) δ (ln(c ))

5 Determination of Half-life Natural Logarithm of Corrected Counts, ln(c ) Starting Time, t (min)