Lab 12

Nuclear decay and MRI

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 N₀ 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) = N₀exp(-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.

Exploration 1:

You will use an on-line simulation from the University of Colorado PhET group to study alpha decay
Link to the simulation: http://phet.colorado.edu/en/simulations/alpha-decay

Explore the interface!

• You can Pause the simulation and then use Step to incrementally analyze.
• After the Polonium nucleus decays to Lead, click Reset to start over with a new Polonium nucleus.

Click the "Single Atom" tab.

• Watch the Polonium-211 nucleus until it decays.  Click "Reset Nucleus" and watch it again.  Repeat this at least 15 more times.
• Inspect the Decay Time Chart on the top of the screen.  It displays the decay times of all the nuclei you observed. 
• Do you observe a pattern?  Can you predict the decay time for the next nucleus?  Watch what happens in the bottom-half of the display.

Click the "Multiple Atoms" tab.  Quickly empty the Bucket of Polonium by rapidly clicking the "Add 10" button until the bucket is empty.

• Observe the decay of the nuclei and inspect the Decay Time Chart on the top of the screen.
• Click "Reset All Nuclei".  Do you observe the same pattern on the Decay Time Chart as in your previous trial?   Repeat this experiment at least 5 more times to be sure of your answer. 
• Describe any similarities and/or differences in pattern on the Decay Time Chart.

Answer the following questions.

• The half-life of the Polonium-211 nucleus is approximately 500 ms.  The nucleus decays by emitting an alpha particle.  How does this alpha particle make it out of the nucleus.  (Watch what happens in the bottom-half of the display in "Single Atom" mode.)
• After the Polonium-211 nucleus has decayed, a lead-207 nucleus is left behind.  Why does lead-207 not decay by emitting an alpha particle?
• The half-life of the Polonium-211 nucleus is approximately 500 ms.  What do you observe in "Multiple Atoms" mode, when you monitor approximately 100 nuclei?
• In "Single Atom" mode switch to a custom nucleus.  What can you do in the bottom-half of the display to build a nucleus with a half-life of a billion years?

Exploration 2:

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/simulations/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 3:

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/simulations/mri 

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 = f/B (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.

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 0.06 T.  The magnetic field now is no longer uniform, but is a function of position.
  Slowly bring up the frequency from 10 MHz and 110 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.