In this Computing in Science Education (CSE) exercise we mainly use the computer's number crunching capability to quickly plot the total disintegration rate from from genetically dependent mother-daughter nuclei. To be more realistic we will also optionally add a constant "background" to the plots. In this way the plot will be comparable to what you actually would observe from a detector measuring observing radiative events caused by disintegrations from either the mother nucleus, the daughter nucleus or sporadic events from the general background radiation.

There are two principal routes to calculate the disintegration rate from the mother and daughter nuclei: Either you use the analytic solutions to the equations, or you can solve the differential equation representing the mother-daughter genetic dependence. The first one is the easiest and most practical; the second will train you in numerical methods for solving differential equations. In both cases you will be able to graphically investigate the dynamics in mother-daughter relationship and the principles behind a radionuclide generator if you add plotting capabilities to your program.

Follow this link for a general description of Computers in Education (CiE) exercises.

## Learning Outcome

In addition to getting more experienced in solving differential equations numerically, the student will get the following learning outcomes in NRC (Nuclear and Radiochemistry) from this exercise:

## Guides and material for students

## Guides and material for teachers

## Development History and Contact Person

This exercise was conceived by Jon Petter Omtvedt (UiO) and used for teaching in Nuclear Chemistry at University of Oslo since many years. It was included here at NucWik as part of the CINCH project. Håkon Beckstrøm (UiO) added some comments and experiences from using it for teaching how to solve differential equations by numerical methods.

If you have comments, suggestions, examples of programs (in any language) or anything else relevant, please write in the comment page (use the NucWik page commenting tool, but you must be a registered user) or send your feed-back to NucWik-post@kjemi.uio.no. The teaching material for this particular CSE exercise is managed and updated by: Jon Petter Omtvedt. Feel free to contact me directly if you want to discuss this exercise or need help implementing it.

There are two principal routes to calculate the disintegration rate from the mother and daughter nuclei: Either you use the analytic solutions to the equations, or you can solve the differential equation representing the mother-daughter genetic dependence. The first one is the easiest and most practical; the second will train you in numerical methods for solving differential equations. In both cases you will be able to graphically investigate the dynamics in mother-daughter relationship and the principles behind a radionuclide generator if you add plotting capabilities to your program.

Follow this link for a general description of Computers in Education (CiE) exercises.

- Practical experience in handling decay equations, providing a more in-depth understanding of what they represent.
- Improved understanding of genetically dependent nuclei, in particular for transient and secular equilibrium and the conditions that will lead to such equilibria.
- A good understanding of the dynamics of a radionuclide generator.

- Student guide to this exercise
**Theory and Background**- The numerical solution of differential equations will not be explained here but you (the teacher) should point your students to the right course material. A good introduction to one basic (but good) method - Euler's method - is available from Wikipedia (as per October 2015): https://en.wikipedia.org/wiki/Euler_method. Euler's method is suggested used in the Student Guide above, together with the analytical solution.
- You (the teacher) should point out relevant pages in your course book to your students. In some much used books this would be:
- Loveland et al. "Modern Nuclear Chemistry": Chap. 3: Radioactive Decay Kinetics
- Choppin et al: "Radiochemistry and Nuclear Chemistry" (fourth edition): Chap 5: Unstable and Radioactive Decay, sub-chapters 5.11-5.16.
- Lieser: "Nuclear and Radiochemistry: Fundamentals and Applications" (second edition): Chapter 4. Radioactive Decay

- NucWik pages with relevant theory explanations:

- Once the student has the program up and running, it should be rather straight forward to walk him/her through relevant examples:
- Short-lived mother, long-lived daughter (no equilibrium)
- Short-lived mother, short-lived daughter (no equilibrium)
- Long-lived mother, short-lived daughter (transient equilibrium, T1/2(mother) >> T1/2(daughter)
- Long-lived mother, short-lived daughter (secular equilibrium, T1/2(mother > 104 x T1/2(daughter))
- Examples of operation of a radionuclide generator, e.g. the 99Mo -> 99mTc generator. How much activity will you get after one, three and 10 daughter half-lives? What happens if you milk the generator two times in a row?

- Readymade source code and executables (intended to enable teachers considering to use this exercise to evaluate it more easily):
- LabView example: [ Source code (VI) ] [ Executable ] [ Wiring Diagram ] [ Running VI Screen Shot ]

- Tips for how this CSE exercise or parts of it could be used in your teaching:
- If you already have the program, or if you write it yourself, you can allow the students to play with it to learn about mother-daughter relationships even if they are not able or do not have time to write the program themselves. But then it would not be CSE, though...
- Alternatively, you can use the analytically derived solution in spreadsheet type programs like MicroSoft Excel or even (much) better data handling programs like Origin from OriginLab
- This CSE exercise (or the alternatives mentioned above) can be used as preparation for laboratory exercises where mother-daughter relationships are being used or investigated. Examples are:
- Laboratory exercises with 99mTc generators: [ Liquid-liquid Extraction and Measurement of 99mTc ]
- Laboratory exercises with uranium daughters: [ 234mPa Radionuclide Generator (Method 1) ] [ 234mPa Radionuclide Generator (Method 2) ]
- Other relevant laboratory exercises: [ Determination of the 238U/234U Ratio ]
- You should also search NucWik for Laboratory exercises added after this was last updated (searching the tags is the best way, provided those that entered the exercise added the relevant tags to their pages).

- Compilation of feedback from teachers and students. Here we assemble feed-back from teachers and students that have performed the exercise. Please help us to make this a long and useful list!

If you have comments, suggestions, examples of programs (in any language) or anything else relevant, please write in the comment page (use the NucWik page commenting tool, but you must be a registered user) or send your feed-back to NucWik-post@kjemi.uio.no. The teaching material for this particular CSE exercise is managed and updated by: Jon Petter Omtvedt. Feel free to contact me directly if you want to discuss this exercise or need help implementing it.