Suggestions from Dylan for the Gamma lab report.
GammaRayAbsorptionDiscussionTopics.pdf
Friday, February 17, 2017
Tuesday, February 14, 2017
Gamma Lab Uncertainty
Working through these questions will help you figure out the correct error bars for lead absorption data points in the gamma ray lab.
1. In lab you confirmed that the source emits gamma rays according to a poisson distribution \[ P(k) = \frac{\lambda^k}{k!} e^{-\lambda} , \] where \(P(k)\) is the probability of measuring \(k\) counts.
(a) Show that for the poisson distribution, the average value is \(\lambda\) and the uncertainty is \(\sigma=\sqrt{\lambda}\).
(b) Show that if you measure \(N\) total counts in some time interval, the uncertainty in the measured total number of counts is \(\sigma_N = \sqrt{N}\).
2. Suppose you measure counts over a time interval \(T\) seconds long, and observe \(N\) total counts. Your measured count rate is then \(n=N/T\) counts per second. Assuming \(\sigma_N = \sqrt{N}\), show that the uncertainty in the count rate is \(\sigma_n = \frac{\sqrt{n}}{\sqrt{T}} \). (Hint: There are two ways to show this. One is by averaging over \(T\) one-second intervals. The other is by using the propagation of error formula.)
3. Let \(n=n_m-n_b\) be the difference between a measured count rate and a previously measured background count rate. You have already determined values for \(\sigma_{n_m}\) and \(\sigma_{n_b}\). Show that \(\sigma_n = \sqrt{\sigma_{n_m}^2 + \sigma_{n_b}^2 } \). When your measured count rate is small, the background uncertainty will be a significant contribution to the error bar.
4. Assume that you have measured count rates \(n_1, n_2\) and calculated uncertainties \(\sigma_{n_1},\sigma_{n_2}\) at two different thicknesses of lead. Let \( f = - \ln \frac{n_2}{n_1} \). Show that \[ \sigma_f = \sqrt{ \left( \! \frac{\sigma_{n_1}}{n_1} \!\! \right)^{\! 2} + \left( \! \frac{\sigma_{n_2}}{n_2} \!\! \right)^{\! 2} } \; . \] This is the error bar that should appear in your plot, where the uncertainties on the right-hand side take into account the previous questions.
1. In lab you confirmed that the source emits gamma rays according to a poisson distribution \[ P(k) = \frac{\lambda^k}{k!} e^{-\lambda} , \] where \(P(k)\) is the probability of measuring \(k\) counts.
(a) Show that for the poisson distribution, the average value is \(\lambda\) and the uncertainty is \(\sigma=\sqrt{\lambda}\).
(b) Show that if you measure \(N\) total counts in some time interval, the uncertainty in the measured total number of counts is \(\sigma_N = \sqrt{N}\).
2. Suppose you measure counts over a time interval \(T\) seconds long, and observe \(N\) total counts. Your measured count rate is then \(n=N/T\) counts per second. Assuming \(\sigma_N = \sqrt{N}\), show that the uncertainty in the count rate is \(\sigma_n = \frac{\sqrt{n}}{\sqrt{T}} \). (Hint: There are two ways to show this. One is by averaging over \(T\) one-second intervals. The other is by using the propagation of error formula.)
3. Let \(n=n_m-n_b\) be the difference between a measured count rate and a previously measured background count rate. You have already determined values for \(\sigma_{n_m}\) and \(\sigma_{n_b}\). Show that \(\sigma_n = \sqrt{\sigma_{n_m}^2 + \sigma_{n_b}^2 } \). When your measured count rate is small, the background uncertainty will be a significant contribution to the error bar.
4. Assume that you have measured count rates \(n_1, n_2\) and calculated uncertainties \(\sigma_{n_1},\sigma_{n_2}\) at two different thicknesses of lead. Let \( f = - \ln \frac{n_2}{n_1} \). Show that \[ \sigma_f = \sqrt{ \left( \! \frac{\sigma_{n_1}}{n_1} \!\! \right)^{\! 2} + \left( \! \frac{\sigma_{n_2}}{n_2} \!\! \right)^{\! 2} } \; . \] This is the error bar that should appear in your plot, where the uncertainties on the right-hand side take into account the previous questions.
Monday, February 6, 2017
2nd Homework dues date.
If you have midterm tomorrow (Tuesday), then you can hand in your 2nd homework on Thursday instead of tomorrow.
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