A gamma source's count rate is expected to follow

PHYSICS · Subjective · Fermi University Physics · Scientific Practice & Experimental Methods · Uncertainty in measurement · Data analysis and model fitting in the laboratory

Computational Physics, Ch Seeing the answer: plots that can be trusted

Problem

A gamma source's count rate is expected to follow A(t)=A0e−λtA(t) = A_0 e^{-\lambda t}. Two readings, each with its statistical uncertainty, are taken: at t1=0 st_1 = 0\,\mathrm s, A1=(2000±40)A_1 = (2000\pm40) counts s−1\mathrm{s^{-1}}; at t2=92.1 st_2 = 92.1\,\mathrm s, A2=(20.0±1.5)A_2 = (20.0\pm1.5) counts s−1\mathrm{s^{-1}}. This is a drop of two decades in AA, satisfying the two-decade rule for claiming an exponent or a rate. Two rules are supplied here, because this chapter does not build the propagation of uncertainties: the uncertainty of a logarithm is σlog⁡10A=σA/(Aln⁡10)\sigma_{\log_{10}A} = \sigma_A/(A\ln10), and the contributions of two independent readings combine in quadrature. (a) Find the slope of log⁡10A\log_{10}A against tt, with its uncertainty. (b) Convert the slope to the decay constant λ\lambda, with its uncertainty, using log⁡10A=log⁡10A0−λt/ln⁡10\log_{10}A = \log_{10}A_0 - \lambda t/\ln10.

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