Question

A nuclear reactor contains \(\alpha\) - and \(\beta\) particles. In every second each \(\alpha\) -particle splits into three \(\beta\) -particles, and each \(\beta\) -particle splits into an \(\alpha\) -particle and two \(\beta\) -particles. If there is a single \(\alpha\) -particle in the reactor at time \(t=0,\) how many \(\alpha\) -particles are there at \(t=20\) seconds? [Hint: Let \(x_{k}\) and \(y_{k}\) denote the number of \(\alpha\) - and \(\beta\) -particles at time \(t=k\) seconds. Find \(x_{k+1}\) and \(y_{k+1}\) in terms of \(x_{k}\) and \(y_{k}\).]

Short answer

There are 21 \( \alpha \)-particles at \( t=20 \) seconds.

Step-by-step solution

Understanding the Initial Conditions

At time \( t=0 \), there is 1 \( \alpha \)-particle and 0 \( \beta \)-particles. Thus, the initial conditions are \( x_0 = 1 \) and \( y_0 = 0 \).

Recurrence Relation Setup for Alpha Particles

Each \( \beta \)-particle splits into one \( \alpha \)-particle and two \( \beta \)-particles. Therefore, the number of \( \alpha \)-particles at \( t=k+1 \) is equal to the number of \( \alpha \)-particles at \( t=k \) plus the number of \( \beta \)-particles at \( t=k \): \[ x_{k+1} = x_k + y_k \].

Recurrence Relation Setup for Beta Particles

Each \( \alpha \)-particle splits into three \( \beta \)-particles, and each \( \beta \)-particle contributes to two more \( \beta \)-particles (besides forming one \( \alpha \)-particle). Therefore, the number of \( \beta \)-particles at \( t=k+1 \) is three times the number of \( \alpha \)-particles plus two times the number of \( \beta \)-particles at \( t=k \): \[ y_{k+1} = 3x_k + 2y_k \].

Calculating Particle Numbers Iteratively

Apply these recurrence relations from \( t=0 \) to \( t=20 \) to compute the number of \( \alpha \)-particles. Start with \( x_0 = 1 \), \( y_0 = 0 \). Use the relations: \( x_{k+1} = x_k + y_k \) and \( y_{k+1} = 3x_k + 2y_k \) to find subsequent terms until \( t=20 \).

Performing the Iterations

For \( t=1 \) to \( t=20 \), update \( x_k \,\) and \( y_k \,\) using the recurrence relations iteratively. Continue iterating to find the number of \( \alpha \)-particles at each second up to \( t=20 \).

Extract the Result for Alpha Particles

After computing iteratively, determine the value of \( x_{20} \). This represents the number of \( \alpha \)-particles at \( t=20 \) seconds.

Key concepts

Understanding Alpha Particles

Alpha particles, also represented as \( \alpha \)-particles, are one type of particle emitted during nuclear reactions. They consist of two protons and two neutrons, making them the nucleus of a helium atom. This composition grants alpha particles a relatively large mass compared to other fundamental particles in nuclear reactions, such as beta particles.
Alpha particles have a +2 charge, making them positively charged. In nuclear reactions, they typically originate from heavier nuclei undergoing a process called alpha decay. During this decay, an unstable nucleus emits an alpha particle, reducing the atomic number by two and the mass number by four.
Alpha particles have limited penetration power and are generally stopped by a few centimeters of air or a piece of paper. Although they cannot penetrate deeply, they are highly ionizing, meaning they can easily remove electrons from atoms they interact with.
In nuclear reactors, alpha particles can play a transformative role. For instance, in our original exercise, alpha particles convert into beta particles. Understanding this conversion is key to tracking the changes in particle numbers over time.

The Role of Beta Particles

Beta particles are another type of particle commonly involved in nuclear reactions. These particles are smaller than alpha particles and can be either electrons or positrons. Beta particles are represented as \( \beta \)-particles and can have a negative or positive charge, depending on whether they are electrons or positrons.
Much like alpha particles, beta particles are frequently emitted during radioactive decay. There are two types of beta decay: beta-minus (\( \beta^- \)) and beta-plus (\( \beta^+ \)). In beta-minus decay, a neutron transforms into a proton, emitting an electron and an antineutrino. In beta-plus decay, a proton transforms into a neutron, emitting a positron and a neutrino.

• Beta-minus particles are negatively charged and are essentially high-energy electrons.
• Beta-plus particles, or positrons, have a positive charge equivalent to an electron's negative charge.

In the content of our exercise, beta particles play a crucial role in the feedback loop that creates both alpha and beta particles. Each alpha particle generates three beta particles, which in turn, transform into a combination of alpha and new beta particles.

Exploring Nuclear Reactions

Nuclear reactions are processes that alter the nucleus of an atom. They involve the rearrangement of protons and neutrons and often result in the emission or absorption of particles like alpha and beta particles.
A significant aspect of nuclear reactions is the exchange of energy that occurs. This energy exchange is frequently manifested as kinetic energy in the emitted particles or as radiation.
Nuclear reactions can occur naturally, such as the fusion reactions within the sun, or artificially, within a controlled setting like a nuclear reactor or particle accelerator. In reactors, reactions can include fission, where heavy nuclei split into smaller fragments, or fusion, where light nuclei combine to form heavier ones.
In the given exercise, we are dealing with a hypothetical nuclear reaction within a reactor where

• Alpha particles convert into beta particles.
• Beta particles transform back into a combination of alpha and additional beta particles.

Such transformation processes drive the dynamic evolution of particle numbers as described by recurrence relations, showcasing the complexity and constant change inherent in nuclear reactions.