Nuclear Physics

When the nucleus of an atom has an imbalance between protons and neutrons, it becomes unstable. To regain stability, the nucleus undergoes disintegration, releasing energy and subatomic particles in the process. This phenomenon is called radioactivity.

If the disintegration occurs naturally without external influence, it is referred to as natural or spontaneous radioactivity. On the other hand, if the process is triggered by human intervention (e.g., by bombarding the nucleus with particles), it is called induced or artificial radioactivity.

Note that, the words “decay” and “disintegration” will be used interchangeably in subsequent discussions.

The Decay Law

This law describes the relationship between the rate of decay of a radioactive sample and the amount of it present at any time. It is stated as follows:

The rate of disintegration/decay of a given nuclide at any time is directly proportional to the number of nuclei N of the nuclide present.

This means that, as the amount (N) gets smaller over time, the rate of decay also gets smaller (slower), because there are fewer nuclei available to decay.

This statement is stated mathematically as follows:

− dN___ dt ∝ N − dN___ dt = λN , where the negative sign (-) indicates that the nuclei reduce in number as time passes.

Applying calculus to the above equation, N(t) = N₀ e−λt

Note that: "N(t)" means N is a function of time t, not N × t .

Where t is the duration or period of the decay, N is the amount of the sample not yet decayed at the end of time t, N₀ is the initial amount of the sample not decayed, at the beginning of the time t, λ is called the decay constant.

Half-life The half-life of a radioactive nuclide or radioisotope is defined as the time interval during which half of a given number of radioactive nuclei decay.

In one half-life, half of the original nuclei will decay. In the second half-life, half of those remaining will decay, leaving one-fourth (a quarter) of the original number.

Figure 4.12: Diagram to show the number of undecayed nuclei in a sample after every half life.

Figure 4.13: A graph showing the percentage of the initial number of undecayed nuclei remaining after every half life.

Half-life is unique to a radioactive element. It is different for different elements;

some half lives of a few seconds where others have a half life of hundreds of thousands of years.

Mathematical statement of half-life:

N = N₀ e−λt e^(λt)= N₀_ N From the definition, let the time taken for N₀ to decay to half be T₁__ 2 and N = N₀__ 2 e^(λT1)__ 2 = N₀____ (N₀__ 2 ) e^(λT1)__ 2 = 2 λ T₁__ 2 = ln 2 T₁__ 2 = 1n 2____ λ = 0.693_____ λ Generally, the age or time elapsed (t) of any sample/specimen can be found using the formula; t = ln(N₀__ N )______ λ .

Activity 4.9 Group Discussion on Radioactivity Concepts

Objective: To enhance understanding of key concepts related to radioactivity through collaborative discussion and clarification of ideas among peers.

Materials Needed

1. A comfortable space for group discussion

2. Whiteboard or flip chart

3. Markers or pens

4. Notebook or digital document for taking notes

5. Phones or other internet-enabled devices for conducting research What to do

1. Organise yourselves into small groups of not more than four-six. This size will allow everyone to participate actively in the discussion.

2. Establish some basic ground rules for the discussion to ensure a respectful and productive environment:

a. Listen actively to each other.

b. Allow everyone a chance to speak.

c. Stay on topic and respect differing opinions.

3. Before starting the discussion, take a few minutes to review key concepts related to radioactivity that you have learned previously. This may include terms like radioactive decay, half-life, and applications of radiocarbon dating.

4. Begin the discussion by addressing the first guided question:

a. “What is radioactive decay?”

Encourage each group member to share their understanding of the concept. Discuss different types of radioactive decay (e.g., alpha decay, beta decay) and their significance. Use internet research to help you if you are stuck.

5. After discussing radioactive decay, move on to the next guided question:

a. “How do we measure half-life?”

Discuss how half-life is defined and its importance in understanding radioactive materials. Share examples of how half-life is used in various fields, such as archaeology and medicine.

6. Continue the discussion with additional guided questions such as:

a. “What factors influence the rate of radioactive decay?”

b. “How does radiocarbon dating work?”

c. “What are some limitations of radiocarbon dating?”

7. As you discuss these topics, encourage group members to ask questions for clarification or further exploration. If someone is unsure about a concept, invite others to provide explanations or examples.

8. Designate one person in the group to take notes during the discussion.

These notes can include key points, definitions, and any questions that arose during the conversation.

9. At the end of the discussion, take a few minutes to summarise the key points that were discussed. Each member can share one important takeaway from the conversation.

Activity 4.10 Visualising radioactive decay Watch the following video / animation to reinforce the concept of radioactive decay: click here

Figure 4.14: QR link to video

Activity 4.11 Computer Simulation of Radiocarbon Dating Using PhET Objective: To understand the principles of radiocarbon dating by using the PhET simulation that models the decay of carbon-14 over time, allowing you to observe how the decay process works and its implications for dating organic materials.

Materials Needed

1. A computer or tablet with internet access

2. Access to the PhET Interactive Simulations website: PhET Radioactive Dating Game

3. Notebook or digital document for recording data and observations What to do

1. Open a web browser and navigate to the PhET Interactive Simulations website. Click on the Radioactive Dating Game simulation.

Figure 4.15: PhET radioactive dating game simulation

2. Spend a few minutes exploring the interface of the simulation. You will see options to select different isotopes, including carbon-14, and how they decay over time.

3. In the simulation, choose Carbon-14 as your isotope for dating. This option will allow you to model how carbon-14 decays over time.

4. Use the simulation to drag a probe to an object (e.g., a piece of wood or bone) that you want to date. The simulation will show you the percentage of carbon-14 remaining in that object.

5. Based on the percentage of carbon-14 remaining, use the slider or input box to estimate the age of the object. Type your estimated age into the provided box.

6. Click on “Check Estimate” to see if your estimated age is correct. The simulation will provide feedback on your estimate.

7. Create a table in your notebook to record your observations about different objects and their estimated ages based on remaining carbon-14:

Object Percentage of Carbon-14

Remaining Estimated

Age (Years) Correct Age

(Years) Wood Bone

Sample 3

8. Try dating various objects by repeating steps 4 through 7 for different samples available in the simulation. Record your findings in your table.

9. After completing several estimates, analyse your data:

a. How does the percentage of remaining carbon-14 relate to the estimated age?

b. Discuss any patterns you notice regarding how much carbon-14 remains in older versus younger samples.

Activity 4.12 Carbon dating research Objective: Having completed Activity 4.10, research and summarise the use of carbon dating in a field of your choice.

Methodology

1. Using the internet or textbook resources, find choose a field of dating which interests you. This could be, for example, the dating of ancient artefacts or of human remains.

2. Summarise onto a poster or fact-sheet how carbon dating is used in this context, including any notable or interesting discoveries that you come across in your research.

Expected outcomes: You should allow your peers access to your poster for them to read and learn, or could alternatively present your poster to your peers.

Activity 4.13 Dice Simulation of Radioactive Decay

Objective: To model the process of radioactive decay using dice, allowing you to visualise the decay process, calculate half-life, and understand the randomness of radioactive decay.

Materials Needed

1. 100 six-sided dice (each die represents a nucleus in a sample of a radioactive substance)

2. A notebook or digital document for recording data

3. A graphing tool (graph paper or digital graphing software)

4. A calculator (optional for calculations)

Figure 4.16: Dice

What to do

1. Start with 100 dice. Each die represents a nucleus in a sample of a radioactive substance.

2. For this simulation, define a rule for decay: if a die shows a 6, it is considered “decayed.” All other numbers (1-5) represent undecayed nuclei.

3. Roll all 100 dice at once. After rolling, determine which dice will be considered “decayed” based on your rule.

4. Count the number of dice that show a 6 and remove those from your total.

Record the number of remaining “undecayed” dice in your notebook.

5. Roll the remaining undecayed dice again. Apply the same rule to determine which dice are “decayed.” Record the new count of undecayed dice after each roll.

6. Continue this process for 10–15 rolls, documenting the number of undecayed dice after each roll.

7. Use the following table format to record your results:

Roll Number Initial Dice Decayed Dice Remaining Undecayed Dice 1 100 2 3 4 5 6 7 8 9 10

8. After completing your rolls, create a plot with the number of rolls on the horizontal axis and the number of remaining undecayed dice on the y-axis. This visual representation will help you see how the number of undecayed nuclei decreases over time.

9. Determine when approximately half of your original number of dice remain undecayed. For example, if you started with 10 dice, identify when there are about 5 remaining. Note how many rolls it took to reach this point; this is an estimate of the half-life for your simulated decay process.

10. Reflect on how this simulation illustrates the randomness of radioactive decay:

a. Discuss how each roll represents an independent event and how some nuclei decay while others do not.

b. Consider how this randomness relates to real-world radioactive substances.

c. Talk about the concept of decay constant and how it relates to the probability of decay in actual radioactive materials.

Activity 4.14 Popcorn to simulate radioactive decay Objective: To model the process of radioactive decay using popcorn, allowing you to visualise the decay process and understand the randomness of radioactive decay.

Materials needed

1. 100 un-popped popcorn kernels

2. A saucepan with a small amount of oil in the bottom

3. A clear lid, or sieve to use as a lid

4. A phone or video camera

5. A stopwatch or wristwatch

6. A notebook and pen

7. Graph paper

Figure 4.17: Popcorn in a pan with a sieve lid, Savvysurf.co.uk What to do

1. Set up the phone or camera to record the content of the pan with the stopwatch or wristwatch in the shot.

2. Add the un-popped kernels to the pan, turn on the heat, add the lid and press record on the camera.

3. Record the kernels popping, shaking the pan occasionally to stop the kernels from sticking and burning, ensuring that the ticking stopwatch can be seen on the screen.

4. Turn off the heat and stop the recording when all the kernels have popped.

5. Re-play the video in slow motion, making a note of how many un-popped kernels remain every 10 seconds. Record this data into a table.

6. Plot onto graph paper the number of un-popped kernels (y axis) against time (x axis).

7. Compare the shape of the graph to a radioactive decay curve and write a conclusion as to the similarities and differences; how well does popcorn model radioactive decay?

Activity 4.15 Calculating Half-life

Materials Needed

1. Worksheet

2. Calculator

3. Pen/pencil

4. Study the worked examples below carefully before attempting the example questions that follow

Worked Example 1

The decay constant of a radioactive substance is 0.001 s⁻¹. Calculate its half- life.

Step-by-Step Solution

Step 1: Recall the formula for half-life T₁__ 2 = 1n2____ λ

Step 2: Substitute the given values T₁__ 2 = 1n2/0.001 T ¹__ 2 = 1 n 2/0.001

Step 3: Calculate to obtain the answer T₁__ 2 = 693 s

Worked Example 2:

A radioactive substance has a half-life of 10 years. If the initial mass of the substance is 80 grams, how much remains after 30 years?

Step-by-Step Solution

Step 1: Identify the known values

• Initial mass N₀=80 g

• Half-life T₁__ 2 = 10 years

• Time (t) = 30 years

Step 2: Calculate the decay constant T₁__ 2 = 1n2____ λ λ = 1n2/T₁_ ₂ λ = 1n2/10 λ = 0.069314718056 per year

Step 3: Introduce the decay law formula N = N₀ e−λt

Step 4: Substitute the known values N = 80 e−(0.069314718056×30)

Step 5: Calculate to obtain the answer N = 80 × 1/8 N = 10 grams After 30 years, 10 grams of the substance remains.

Worked Example 3

A radioactive sample decays from 100 grams to 25 grams in 12 years. What is the half-life of the sample?

Step-by-Step Solution

Step 1: Identify the known values Initial mass N₀ = 100 grams Remaining amount N = 25 grams Time elapsed t = 12 years

Step 2: find the decay constant as follows

a. Introduce the decay law formula and rearrange it to make the decay constant the subject λ = 1n(ᴺ⁰__ _(N) )______ t

b. Substitute the values λ = 1n(100/25 )_______ 12

c. Calculate to obtain the decay constant λ = 0.1155

Step 3: calculate half-life using the half-life formula

a. Introduce the half-life formula T₁__ 2 = 1 n2____ λ

b. Substitute the known and calculated values T₁__ 2 = 1n2/0.1155

c. Calculate to obtain the answer T₁__ 2 = 0.693/0.1155 = 6 years

Worked Example 4

A fossil is found to have 25% of its original C-14 remaining. How old is the fossil? (T₁__ 2 = 5730years ) Step-by-Step Solution

Step 1: Identify the known values

• Remaining percentage N/N₀ = 25/100 = 1/4

• Half-life T₁__ 2 = 5,730 years

Step 2: Calculate the decay constant λ = 1n2/T₁_ ₂ = 1n2/5730 = 1.2097 × 10⁻⁴per year

Step 3: Introduce the decay law formula and rearrange N = N₀ e−λt t = 1n(N₀__ N )______ λ

Step 4: Substitute the known and calculated values t = 1n(4)__________ 1.2097 × 10⁻⁴Step 5: Calculate to obtain your answer t = 11, 460 years Practice Problems Now, using the worked example as a guide, solve the following problems individually or in groups.

1. A radioactive substance has a half-life of 12 years. What is its decay constant?

2. A sample of radioactive substance has an initial amount of 150 g. If its half- life is 4 years, how much will remain after 10 years.

3. A sample decays to 10 grams after 6 years. If the half-life is 3 years, how much was there initially?

4. A radioactive sample decays to 20% of its original amount in 5 years. What is its half-life?

Activity 4.16 Flashcard Matching Challenge

Objective: To reinforce your understanding of nuclear physics terms by matching terms with their definitions using provided flashcards.

Materials Needed

1. Flashcards

• Set A: Flashcards with key terms

• Set B: Flashcards with corresponding definitions

2. Pen or Pencil

Set A

Figure 4.18: Flashcards with key terms on nuclear physics Set B

Figure 4.19: Flashcards with some definitions on nuclear physics What to do

1. Take a few minutes to read through both sets of flashcards. Familiarise yourself with the terms in Set A and their corresponding definitions in Set B.

2. To test your memory, close your eyes or turn away from the learner material to avoid looking at the definitions. This will help you recall the definitions based on the terms.

3. Start with the first term in Set A. Think about its meaning and try to recall which definition in Set B matches it.

4. Write down your answer

5. Proceed through all the terms in Set A, matching each one to its definition in Set B. Take your time and think carefully about each match.

6. Once you have matched all the terms, go back through the flashcards to check your answers. Look at each term and its corresponding definition to see if they align correctly.

7. For any incorrect matches, review the definitions again to understand why they do not match. Take notes on any challenging terms.

Activity 4.17 Online quiz Visit the website linked below to practice applying your understanding of half life: click here

Figure 4.20: QR link to multiple choice quiz