Radioactive nuclei spontaneously decay. The decay of an unstable nucleus
is a quantum process. The probability that a given nucleus will decay in
the next time interval Δt is independent of the history of the nucleus. The
decay process is entirely random, and it is impossible to predict when a
particular nucleus will decay. The decay constant λ of a nucleus is its
decay probability per unit time. The probability that the nucleus will
decay in the next small time interval Δt is λΔt.

Starting with
a large number N_{0} of radioactive nuclei at t = 0, we find that the
number still present at time t is well approximated by a function representing
exponential decay.

N(t) = N_{0}exp(-λt).

The mean lifetime of the nuclei is given by τ = 1/λ,
and the half-life is given by t_{½} = τ ln2 =
ln2/λ.

Open a Microsoft Word document to keep a log of your procedures, results and discussions.

**Experiment 1:**

In this experiment you will simulate the decay of radioactive nuclei by rolling dice. You will start the experiment standing upright, representing a radioactive nucleus before its decay. You will sit down when the nucleus you are representing decays. Whether the nucleus you represent decays or not, will depend on the roll of a die.

Experimental procedure:

Before the experiment starts, download the linked
spreadsheet and open it so you can enter data.

In this experiment **all students** have to participate. Your instructor will be the conductor.

- All students stand up. Enter the number of students standing into the spreadsheet.
- When instructed, each student rolls a die. Students rolling a "1" sit down. Enter the number of students still standing into the spreadsheet.
- When instructed, each student still standing rolls a die. Students rolling a "1" sit down. Enter the number of students still standing into the spreadsheet.
- Repeat for 20 rolls.
- To get better statistics, repeat the whole experiment for a second time.

Data analysis:

- Add the results of the two trials.
- Plot the number of students still standing (sum of trial 1 and trial 2) versus the number of rolls.
- Add a trendline to your graph. Choose the exponential type and
display the equation on the chart.

The trendline fits your data with a function y = A exp(-bx). It gives you values for the constants A and b. - Does this trendline produce a good fit? Describe your data and the fit. Paste a copy of the plot with trendline into your log.
- In this experiment time is measured in units of "number of rolls". What is your decay constant or probability of decay per roll from the trendline fit.
- What would you have expected for this decay probability per roll? Do the two values agree?
- What is your half-life in units of "number of rolls"? What does "half-life" mean in the context of this experiment?

How does this experiment simulate nuclear decay? Discuss this with your group members.

**Exploration 1:**

Some nuclear reactions do not occur spontaneously, but require external sources of energy, in the form of "collisions" with outside particles. "Activation energy" has to be provided before a much larger amount of "reaction energy" is released. Nuclear fission can be "activated" when a slow neutron collides with a fissionable nucleus.

You will use an on-line simulation from the
University of Colorado PhET group to explore nuclear fission.

Link to the simulation
http://phet.colorado.edu/en/simulation/nuclear-fission.

Click the Fission: One Nucleus tab.

- Before the gun is fired, is the material stable?
- What type of "bullet" does the gun fire?
- What happens to the nucleus when it is hit?

Click the Chain Reaction tab.

- Add some uranium-238. Is uranium-238 “fissionable”? How does firing the gun on a uranium-238 nucleus change it? (Note you can aim the gun.)
- Reset the sim and add ~50 fissionable uranium-235 nuclei. Fire the gun. Describe what happens and why it happens.
- Naturally occurring levels of the U-235 isotope are about 0.72%, with the majority being U-238. Round the level up to 1% U-235 (one atom of U-235 and 99 atoms of U-238). Use the simulation to find out if naturally derived uranium is able to start a chain reaction,
- Use the simulation to find a minimum ratio of U-235 to U-238 that can start a chain reaction? Compare your mixture to “weapons-grade” enriched uranium (about 80%-85% U-235).
- Use the simulation to make a nuclear weapon. What conditions are needed? (Check the box “containment vessel”, and determine the level of enrichment needed.)

Click the Nuclear Reactor tab.

- What is needed to start the nuclear reactor?
- What does adjusting the control rods accomplish?
- Without the control rods in position, what happens?

Watch the linked video. Is this a good analogies of a nuclear chain reaction? If mousetraps and Ping-Pong balls are used to illustrate a fission chain reaction, what do each represent?

**Exploration 2:**

Use an on-line
simulation from the University of Colorado PhET group to explore a simplified
version of NMR and MRI.

Link to the simulation:
http://phet.colorado.edu/en/simulation/mri

Click "Run Now!" or "Download".

Start with the simplified NMR simulation.

- Familiarize yourself with the interface.
- Adjust the magnetic field to 2 T.
- Power up the radio wave source.
- Find the resonance frequency for hydrogen atoms and verify the g = B/f (MHz/T) given in the notes and below.
- Find the resonance frequency for sodium atoms and verify the g (MHz/T) for sodium atoms given in the notes.
- Find g (MHz/T) for sulfur atoms and the unknown atoms. You may want to increase the field to 3 T.

Nuclei | g (MHz/T) from notes | g = f/B from your measurements |
---|---|---|

^{1}H |
42.58 | |

^{23}Na |
11.27 | |

S | ||

unknown |

Paste your table into your word document and briefly discuss your results.

Switch to the MRI simulation.

- Familiarize yourself with the interface.
- Set the main field to 2 T, power up the radio source, and find the resonance frequency. It should be very close to the resonance frequency for hydrogen you found above. Record the resonance frequency.
- Add a tumor. Adjust the resonance frequency slightly to produce the strongest signal from the tumor. Record the tumor resonance frequency. Is there a shift?
- Remove the tumor, but add a horizontal and a vertical gradient field of
0.6 T. The magnetic field now is no longer uniform, but is a function of
position.

Slowly bring up the frequency from 10 MHz and 130 MHz and observe that MRI signals are only generated in selected regions of the head. What are the approximate resonance frequencies for the upper left and the lower right portion of the head? Record your values.