Question Solved1 Answer 2. Counting statistics and propagation of errors. You have a Cs-133 gamma ray source #1 that was produced on Sept. 1, 2008, and another Cs-133 gamma ray source #2 that was produced on Sept. 1, 2020. You had the sources accurately calibrated when you bought them, so that you know that each had an initial activity of 1.000±0.001 microCuries. Cs-133 decays with 2. Counting statistics and propagation of errors. You have a Cs-133 gamma ray source #1 that was produced on Sept. 1, 2008, and another Cs-133 gamma ray source #2 that was produced on Sept. 1, 2020. You had the sources accurately calibrated when you bought them, so that you know that each had an initial activity of 1.000±0.001 microCuries. Cs-133 decays with an emission of a 662 keV gamma ray and has a half-life of about 30 years. That is, each source emits gamma rays at a rate -In 21-to t-to 1/2 R(t) = Re 11/2 = R₁ 2 (1) where R is the emission rate at the time the source was produced, t is the time, to is the time at which the source was produced, and t₁/2 is the half-life. You decide to see if you can measure the Cs-137 lifetime by comparing the count rate of the gamma rays from the two sources with the same gamma ray detector. You do this by alternately positioning one source and then the other in front of the detector, and you are able to make this change in such a way that the detection efficiency is exactly the same for the two sources. You find that the average count rate for a 1 microCurie source is 170 counts per second. (The count rate is nR, and is less than the decay rate because n is less than 1.) You carry out your measurements on Sept. 8, 2020. a) Suppose you count gamma rays from each source for the same length of time. How long would you need to count for in order to determine the Cs-137 half-life to a relative accuracy of 10%? b) You have a systematic uncertainty in your measurement of the half-life from your uncertainty in the initial activity level of each source. How large is that uncertainty in 11/2? How long would you need to count for in order to make your statistical uncertainty in the half-life equal to your systematic uncertainty?

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Transcribed Image Text: 2. Counting statistics and propagation of errors. You have a Cs-133 gamma ray source #1 that was produced on Sept. 1, 2008, and another Cs-133 gamma ray source #2 that was produced on Sept. 1, 2020. You had the sources accurately calibrated when you bought them, so that you know that each had an initial activity of 1.000±0.001 microCuries. Cs-133 decays with an emission of a 662 keV gamma ray and has a half-life of about 30 years. That is, each source emits gamma rays at a rate -In 21-to t-to 1/2 R(t) = Re 11/2 = R₁ 2 (1) where R is the emission rate at the time the source was produced, t is the time, to is the time at which the source was produced, and t₁/2 is the half-life. You decide to see if you can measure the Cs-137 lifetime by comparing the count rate of the gamma rays from the two sources with the same gamma ray detector. You do this by alternately positioning one source and then the other in front of the detector, and you are able to make this change in such a way that the detection efficiency is exactly the same for the two sources. You find that the average count rate for a 1 microCurie source is 170 counts per second. (The count rate is nR, and is less than the decay rate because n is less than 1.) You carry out your measurements on Sept. 8, 2020. a) Suppose you count gamma rays from each source for the same length of time. How long would you need to count for in order to determine the Cs-137 half-life to a relative accuracy of 10%? b) You have a systematic uncertainty in your measurement of the half-life from your uncertainty in the initial activity level of each source. How large is that uncertainty in 11/2? How long would you need to count for in order to make your statistical uncertainty in the half-life equal to your systematic uncertainty?
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Transcribed Image Text: 2. Counting statistics and propagation of errors. You have a Cs-133 gamma ray source #1 that was produced on Sept. 1, 2008, and another Cs-133 gamma ray source #2 that was produced on Sept. 1, 2020. You had the sources accurately calibrated when you bought them, so that you know that each had an initial activity of 1.000±0.001 microCuries. Cs-133 decays with an emission of a 662 keV gamma ray and has a half-life of about 30 years. That is, each source emits gamma rays at a rate -In 21-to t-to 1/2 R(t) = Re 11/2 = R₁ 2 (1) where R is the emission rate at the time the source was produced, t is the time, to is the time at which the source was produced, and t₁/2 is the half-life. You decide to see if you can measure the Cs-137 lifetime by comparing the count rate of the gamma rays from the two sources with the same gamma ray detector. You do this by alternately positioning one source and then the other in front of the detector, and you are able to make this change in such a way that the detection efficiency is exactly the same for the two sources. You find that the average count rate for a 1 microCurie source is 170 counts per second. (The count rate is nR, and is less than the decay rate because n is less than 1.) You carry out your measurements on Sept. 8, 2020. a) Suppose you count gamma rays from each source for the same length of time. How long would you need to count for in order to determine the Cs-137 half-life to a relative accuracy of 10%? b) You have a systematic uncertainty in your measurement of the half-life from your uncertainty in the initial activity level of each source. How large is that uncertainty in 11/2? How long would you need to count for in order to make your statistical uncertainty in the half-life equal to your systematic uncertainty?
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