In a particle physics experiment to determine the mean...

In a particle physics experiment to determine the mean lifetime of muons, the muons enter a scintillator and decay. Students have sampled the individual lifetimes of muons decaying within a time interval between $1 \mu \mathrm{s}$ and $10 \mu \mathrm{s}$ after being stopped in the scintillator. It is assumed that the muons obey the radioactive decay law $R=R_{0} e^{-t / \tau}$ where $R_{0}$ is the unknown activity at $t=0$ and $R$ is the activity (counts $/ \mu \mathrm{s}$ ) at time $t .$ Here is their data: Time $(\mu \mathrm{s})$ $\begin{array}{llllll}1.5 & 2.5 & 3.5 & 4.5 & 5.5 & 6.5\end{array}$ $R(t)$ $\begin{array}{llllll}55 & 35 & 23 & 18 & 12 & 5\end{array}$ Make a graph of $\ln \left(R / R_{0}\right)$ versus time $t(\mu \mathrm{s}),$ and from the best fit of the graph to a straight line find the mean life $\tau$. The accepted value of the mean life of the muon is $\tau=2.19703 \mu \mathrm{s} \pm 0.00004 \mu \mathrm{s} .$ What is the percentage error of their result from the accepted value?

In a particle physics experiment to determine the mean lifetime of muons, the muons enter a scintillator and decay. Students have sampled the individual lifetimes of muons decaying within a time interval between $1 \mu \mathrm{s}$ and $10 \mu \mathrm{s}$ after being stopped in the scintillator. It is assumed that the muons obey the radioactive decay law $R=R_{0} e^{-t / \tau}$ where $R_{0}$ is the unknown activity at $t=0$ and $R$ is the activity (counts $/ \mu \mathrm{s}$ ) at time $t .$ Here is their data:
Time $(\mu \mathrm{s})$
$\begin{array}{llllll}1.5 & 2.5 & 3.5 & 4.5 & 5.5 & 6.5\end{array}$ $R(t)$ $\begin{array}{llllll}55 & 35 & 23 & 18 & 12 & 5\end{array}$
Make a graph of $\ln \left(R / R_{0}\right)$ versus time $t(\mu \mathrm{s}),$ and from the best fit of the graph to a straight line find the mean life $\tau$. The accepted value of the mean life of the muon is $\tau=2.19703 \mu \mathrm{s} \pm 0.00004 \mu \mathrm{s} .$ What is the percentage error
of their result from the accepted value?

Key Concepts

Exponential Decay Law

This law describes processes in which the rate of change of a quantity is proportional to its current value, leading to a function that decays exponentially over time. It is fundamental in radioactive decay, where the number of undecayed particles decreases at a rate proportional to the number present.

Logarithmic Linearization

Taking the natural logarithm of an exponential decay equation transforms it into a linear form. This technique simplifies the analysis of experimental data by converting the exponential relationship into a straight line, where the slope and intercept have clear physical interpretations.

Mean Lifetime

The mean lifetime (commonly denoted by ?) is the average time a particle exists before decaying. It is an intrinsic property of unstable particles and is central to understanding their behavior in experiments involving radioactive decay.

Linear Regression

This statistical method is used to fit a straight line to a set of experimental data points. In the context of decay measurements, linear regression helps in estimating the slope of the line obtained after the logarithmic transformation, from which the mean lifetime can be derived.

Percentage Error

Percentage error quantifies the accuracy of an experimental result by comparing the measured value with a known accepted value. It is calculated as the absolute difference between the experimental value and the accepted value, divided by the accepted value, multiplied by 100, providing a clear measure of the deviation in percentage terms.

Transcript

00:01 So in this problem, we're given data of time points and a number of events are.

00:09 So we're talking about the mean lifetime of milan's decaying.

00:16 So this is actually a numerical problem.

00:19 The problem actually wants you to make a graph using a computer.

00:23 So i've included this image where i've plotted data points against time where time is in microseconds.

00:31 So we're given this decay law.

00:35 So at time equals zero, the r values r not, it's going to work out to be a little over 200, something like that.

00:47 Because at 1 .5 microseconds is 55, and then it's just what ends up working out to roughly.

00:59 This tau constant is known to be 2 .197, and some more decimal places microseconds, to a fairly high precision.

01:11 So what this problem is doing, we have these data points.

01:15 We want to find tau approximately from the data points, and then figure out how far off that is from the real value.

01:28 So actually, i'll just put.

01:31 The r values here so it's 55 35 23 18 12 5.

01:46 Number of vents is decreasing as we want to decay.

01:52 So the log value of r is something you calculate from these r values.

02:00 So that's 4 .007, 3 .553 .5 .3 .1 .3 .1.

02:16 352 .890, 2 .485, 1 .60.

02:26 So the problem is telling you the graph log versus log of r versus time.

02:36 It tells you to scale it r over r not, but the solution key in the textbook does it as just log r.

02:49 So i did it the way the textbook has it.

02:52 So it wants you to fit, do a best fit line.

02:58 So, you know, if you're doing the computer problem, it's going to want you to put an actual line in the graph, but it's going to look something like that.

03:10 So the point of this is that the slope of this decay is defined by the value tau.

03:22 So so why is that? so we have this decay law up here.

03:33 So hold on.

03:37 So r divided by r0 equals e minus t, tau...

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