This is the second lab of a three-lab sequence on radioactivity. Using the measurement techniques learned last week, you will determine the half-lives of two radioactive isotopes of silver.

By the end of this lab, students will have…

- …bombarded stable silver atoms with neutrons to produce radioactive isotopes.
- …used a digital counter to record count events.
- …determined the half-lives of two radioactive isotopes of silver.

Unlike previous labs, you will not be coming up with your own hypothesis and prediction. Instead, everyone will be trying to answer the same question: **What are the half-lives of silver-108 and silver-110?**

The TA will introduce you to the new digital counter and software, and will later show you the neutron howitzer which can be used to activate stable silver to produce the two radioactive isotopes. As the lab progresses, you will sometimes receive explicit directions and sometimes be allowed to make your own choices.

At the end of the period, there will be a *brief* class discussion of results. Remember to keep track of your work in your **lab notebook**. Click on the template below to begin.

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

$N = N_0\bigg(\frac{1}{2}\bigg)^{\frac{t}{t_{1/2}}}$, |

where $t_{1/2}$ is the half-life. Put another way, if you start with $N_0$ particles, it will take time $t = t_{1/2}$ for half of them to decay away. (After time $t = 2t_{1/2}$, only $N = N_0(1/2)^2 = N_0/4$ will remain. After time $t = 3t_{1/2}$, only $N = N_0(1/2)^3 = N_0/8$ will remain. And so on.) Different radioactive isotopes have different half-lives.

As discussed last week, 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 $\Delta R = \sqrt{N}/t$.

The neutron howitzer provides a large flux of neutrons from a very active plutonium-beryllium (Pu-Be) source. The process is two-fold.

First, a long-lived radioactive isotope of plutonium (half-life 24,000 years) decays into uranium via alpha emission as

$^{239}_{~94}\textrm{Pu} \rightarrow ^{235}_{~92}\textrm{U} + ^{4}_{2}\alpha.$ |

Next, the beryllium absorbs the high energy alpha particles to become carbon plus an extra neutron:

$^{9}_{4}\textrm{Be} +^{4}_{2}\alpha \rightarrow ^{12}_{~6}\textrm{C} + ^{1}_{0}\textrm{n}.$ |

These neutrons leave the source with very high kinetic energy.

To protect ourselves against the neutrons (which can be damaging to human tissue if the exposure is prolonged), the plutonium-beryllium neutron source is surrounded by a shielding of paraffin (a type of wax made from hydrogen and carbon). Unlike beta and gamma radiation (which is best shielded by high-Z materials), neutrons are best shielded by low-Z materials like plastic, wax, or water. Neutrons are slowed down by collisions with protons (hydrogen nuclei) in the paraffin, so a thick layer of paraffin between the neutron source and the outer walls of the howitzer is necessary to reduce the number of high-energy neutrons emitted into the room. Any neutrons which escape have been slowed down to very low energies and are harmless.

We can use the howitzer in order to change a stable isotope of an element into another (potentially unstable) isotope through a process called “neutron activation.” A stable isotope is irradiated with neutrons until the isotope absorbs that neutron, changing the mass number by one unit:

$^{A}_{Z}\textrm{X} +^{1}_{0}n \rightarrow ^{A+1}_{~Z}\textrm{X}.$ |

In many cases, the resulting isotope is unstable and produces radiation as it decays.

Matter may be made radioactive if it is irradiated with neutrons. This process is very important in chemical analyses of trace elements, since very small quantities can be identified through their radioactive decay modes.

Naturally occurring silver (Ag) is a mixture of two stable (i.e., not radioactive) isotopes, $^{107}_{47}\textrm{Ag}$ and $^{109}_{47}\textrm{Ag}$. If silver is bombarded with neutrons, two activation processes are therefore possible. The first process produces $^{110}_{47}\textrm{Ag}$,

$^{109}_{47}\textrm{Ag} + ^{1}_{0}n \rightarrow ^{110}_{47}\textrm{Ag}$, |

which decays by emitting an electron ($\beta^{-1}$) and a neutrino ($\nu$) as

$^{110}_{47}\textrm{Ag} \rightarrow ^{110}_{48}\textrm{Cd} + \beta^{-1} +\nu$. |

This decay has a very short half-life of less than one minute.

The second activation produces $^{108}_{47}\textrm{Ag}$,

$^{107}_{47}\textrm{Ag} + ^{1}_{0}n \rightarrow ^{108}_{47}\textrm{Ag},$ |

which also decays by emitting an electron and a neutrino as

$^{108}_{47}\textrm{Ag} \rightarrow ^{108}_{48}\textrm{Cd} + \beta^{-1} +\nu.$ |

This decay has a slightly longer half-life of a few minutes.

Use what you learned from the last lab and half-lives to determine the half-lives of silver.

*Consider:* You will receive a single piece of silver foil. Use this period as a test period to determine how you want to use the detector to take accurate data over time and determine which group member is responsible for what aspect of the experiment. Timing is crucial.

*Consider*: What techniques will you incorporate from your first measurement of the silver? What methods have you taken to improve your measurements? Do you trust your measurements? What uncertainties are present in the data? Quantify these uncertainties.

*Consider*: How can you determine the half-life of your sample? How well does your data fit this curve? How effective was your method of taking data?

*Consider*: How does the half-life you calculated compare to other groups? How does it compare to the known values? Would it be useful to retake your data?

After you have completed a set of runs, you will want to plot the results in *Logger Pro* to see what the data look like.

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. **When recording your fit values, think carefully about what the units on these values are and make sure to record them in your notebook!**

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.

**Remember to log out of all your accounts after you submit!**