In this final lab of a three-lab sequence on radioactivity, you will perform indium-116 “dating” to determine the age of a sample (in an analogy to the more common carbon-14 dating method). You will have to use what you've learned about radioactivity and half-life measurements in the previous two labs to come up with a dating technique and to estimate your uncertainty in the “time of death” of your indium sample.

Before we start today's lab, we are asking all students to complete a short (<5 minute) survey in which you will have a chance to provide feedback on your TA. Your answers are anonymous and will not affect your grade in any way. You may access the survey from your personal computer, a lab computer, or your phone.

If you cannot or do not want to complete the survey now, you may complete it at home. The survey will remain open until Saturday, November 23 at 5:00 pm.

By the end of this lab, students will have…

• …developed a dating technique,
• …estimated the “time of death” for their sample, and
• …estimated (and justified) an uncertainty on that time.

This week your TA will provide your group with a sample of indium-116 which “died” (i.e., was removed from the neutron howitzer) at some unknown time in the past. Your job is to use what you've learned in the past two labs to determine the “time of death” for your sample.

At your disposal, you have the following (if you need it):

• A geiger-mueller tube and digital counter with STX software.
• The Logger Pro software for plotting and fitting.
• Access to one additional sample of irradiated indium which can be removed from the howitzer at whatever time you would like.

It will be up to you to determine what quantities to measure and how to use that information to estimate the age of your sample. You will want to estimate the uncertainties on all quantities (and use them to estimate a final uncertainty on your “time of death”).

When you have your final result, report your value by writing it on the board. At the end of class, everyone will discuss their methods and the group will come to a consensus on a class value.

Remember to keep track of your work in your group's lab notebook. Click on the template below to begin.

The uncertainty in the measured number of counts from a radioactive source $N$ is given by $\sqrt{N}$. When calculating a count rate, $R = N/t$, the uncertainty in rate is therefore $\Delta R = \sqrt{N}/t$.

If you have a collection of $N_0$ identical radioactive particles at time $t = 0$, then the number of particles $N$ some time later will be

where $t_{1/2}$ is the half-life.

The neutron howitzer provides a large flux of neutrons which can be used in order to change a stable isotope of an element into another (potentially unstable) isotope through a process called “neutron activation.”

Naturally occurring indium (In) is almost entirely composed of indium-115. When bombarded with neutrons, this isotope can be activated to produce indium-116 as

Because of the inherent randomness of this activation process, two identical foils placed in the howitzer will not necessarily produce an equal number of indium-116 atoms. Past studies using these foils and this howitzer suggest that there may be as much as 15% variation in count rate between foils as measured immediately after removing from the howitzer.

Indium-116 is unstable and decays by emitting an electron ($\beta^{-1}$) and a neutrino ($\nu$) to become tin (Sn) as

Last week while you were studying the half-life of silver isotopes, your TA measured the half-life of indium-116. Did you record the experimentally-determined half-life they revealed at the end of lab?

Use what you learned in the past two labs to determine the “time of death” of your indium sample.

Consider: When you receive your “dead” sample of indium, make a note of the number or mass marked on it (so you don't confuse it with any other indium foils later in the lab). Plan how you will determine the “time of death” and make whatever measurements you want with this source.

Consider: After you have discussed with your group, explain your plan to your TA. Make any preparations needed for your experiment and ask to retrieve your fresh indium foil from the howitzer (if desired). Consider what measurements can be repeated, and perform these (if desired).

Consider: Determine the “time of death” of your sample and estimate an uncertainty on this time. (If you think it will be useful, the Logger Pro software can be used for plotting and fitting.) Report your final value to the class by writing it on the board.

Consider: How does your “time of death” compare with other values from the class? What could be some reasons for discrepancy (if any is found)? Based on a class discussion, what is the best class-wide estimate for the “time of death” and the corresponding uncertainty?

At the end of the lab, you will need to record your final conclusions (about 1 or 2 paragraphs) in your lab report summing up the important results and take-away points from your experiment. Remember that you should only draw conclusions which are supported by the data, so be ready to back up any statements you make!

When you're finished, save your file as a PDF and upload it to Canvas. (Only one student needs to submit the report, but make sure everyone's name is on it!) If you make a mistake, you can re-submit, but work done after the end of the lab period will not be accepted.

The instructions below are repeated from last week's lab. Depending on your chosen technique, you may or may not want to plot or fit data.

To import your data from STX…

• From the Edit menu in STX, select Copy.
• Open Logger Pro, and click on the upper left cell in the Data Set. From the Edit menu, select Paste.
• The paste results in extra data that we do not need. To remove…
  • Delete the rows above the first line of data by clicking on each row number hitting the delete key.
  • Delete columns titled y, column2 and column3 by clicking on the column headers and hitting the delete key.
• We now need to reformat the data we’ve got.
  • Double-click on the header of the first remaining column. A new screen should pop up, and under Column Definition find the Name field and type Run. In the Short Nm field, type n.
  • Repeat for the second remaining column and type the name Counts, the short name N, and units counts.
• Next, we want to add two new columns.
  • From the Data menu, select New Calculated Column…
    • For Name, type Time.
    • For Units, type sec.
    • For Short Nm, type t.
    • In the Expression box, type the equation converting from run number to time. For example, if each run interval was 10 seconds, then the equation would be “Run”*10. (Note that column names are written inside quotation marks.)
  • Again, from the Data menu, select New Calculated Column…
    • For Name, type Uncertainty.
    • For Short Name, type dN.
    • For Units, type counts.
    • In the Expression box, type sqrt(“Counts”).
• Finally, we want to let Graphical Analysis know that the “Uncertainty” column represents the error bars on the “Counts”.
  • Double-click on the “Counts” header.
  • Under Options, click the box for Error Bar Calculations.
  • Click Use Column and select “Uncertainty” from the drop-down menu.

To plot the data…

• Double-click on the graph area.
• Under Graph Options, type a title for the plot in the Title field.
• Under Axes Options, select “Counts” from the list for Y-Axis Columns, Choose “Autoscale” (not “Autoscale Larger”) from the menu for Scaling.
• For X-Axis Column, select “Time (sec)”. Set Scaling to “Autoscale”.
• Click Done to update the plot.

To fit a plot of your data to a functional form…

• Click on the plot, then select Curve Fit… from the Analyze menu.
• Under the General Equation list, select Define Function. We want to fit our data to the half-life equation, $N = N_0(1/2)^{(t/t_{1/2})}$ so type in the form A*(1/2)^(t/B).
• Make sure that Automatic is selected under Fit Type.
• Click Try Fit.

If the fit does not work, try to adjust the initial guesses for the fit parameters in the Coefficients box. Think about the physical interpretation of the parameters A and B in your fit in order to decide what good initial guesses might be. Click Try Fit again.

If you have included the error bars in your plot, the fit routine will use those uncertainties when determining the best-fit values.