Problem 5 - An Introduction to Mechanics

Question

Threshold for nuclear reaction \(^{*}\) A nucleus of rest mass \(M_{1}\) moving at high speed with kinetic energy \(K_{1}\) collides with a nucleus of rest mass \(M_{2}\) at rest. A nuclear reaction occurs according to the scheme \(M_{1}+M_{2} \rightarrow M_{3}+M_{4}\) where \(M_{3}\) and \(M_{4}\) are the rest masses of the product nuclei. The rest masses are related by \(\left(M_{3}+M_{4}\right) c^{2}=\left(M_{1}+M_{2}\right) c^{2}+Q\). where \(Q>0 .\) Find the minimum value of \(K_{1}\) required to make the reaction occur, in terms of \(M_{1}, M_{2}\), and \(Q\).

Step-by-Step Solution

Answer

The minimum kinetic energy \(K_{1}\) required is \(M_{2}c^2 + Q\).

Step 1: Understand the Reaction Condition

For a nuclear reaction to occur, the total energy and total momentum must be conserved. We are given that the energy relation is \((M_{3} + M_{4})c^2 = (M_{1} + M_{2})c^2 + Q\), where \(Q\) is the energy release (or absorbed if negative).

Step 2: Write the Energy Conservation Equation

The initial total energy of the system is the sum of the rest energy of \(M_{1}\) and the kinetic energy \(K_{1}\). The kinetic energy \(K_{1}\) is part of the total initial energy: \[ E_{i} = M_{1}c^2 + M_{2}c^2 + K_{1} \].

Step 3: Express the Final Total Energy

The final total energy after the reaction accounts for the rest energies of the new nuclei and the energy \(Q\): \[ E_{f} = M_{3}c^2 + M_{4}c^2 = (M_{1}c^2 + M_{2}c^2 + Q) \].

Step 4: Relate Initial and Final Energies

Since momentum is conserved and \(M_{2}\) is at rest initially, we must provide enough kinetic energy \(K_{1}\) to ensure energy conservation including the factor that kinetic energy could be transformed into or from rest mass energy: \[ M_{1}c^2 + K_{1} = M_{1}c^2 + M_{2}c^2 + Q + P^{2}/(2M) \], where \(P\) is the total momentum and \(M\) the effective mass after the collision. We omit \(P\) simplifying to: \[ K_{1} = M_{2}c^2 + Q \].

Step 5: Formulate the Threshold Kinetic Energy Equation

For the reaction to occur, the kinetic energy must compensate for the rest energy deficity \(Q\) and the energy to bring the rest mass \(M_{2}\) into the reaction, which gives us the threshold energy equation \(K_{1} = - M_{2}c^2 + Q \). Ensure all other conditions on momentum conservation at the reaction threshold.

Key Concepts

Energy conservation in nuclear reactions

In the universe of nuclear reactions, energy conservation is a critical principle, ensuring that the total energy before and after a reaction remains constant. This principle indicates that energy cannot be created or destroyed, only transformed from one type to another. So, when two nuclei collide to form new products, the sum of their energies before collision equals the sum of the energies of the products.
This is expressed in the relation given in the exercise:

• The initial energy is the rest energy of the colliding nuclei plus any kinetic energy they have due to their motion.
• The final energy includes the rest energy of the newly formed nuclei plus any kinetic or excess energy, denoted as Q.

In our exercise, `E_{i}` for initial total energy is written as \[ E_{i} = M_{1}c^2 + M_{2}c^2 + K_{1} \]while the final energy after the reaction is \[ E_{f} = M_{3}c^2 + M_{4}c^2 = (M_{1}c^2 + M_{2}c^2 + Q) \].Therefore, energy conservation directs how much initial kinetic energy is essential for the reaction to conquer potential hurdles like binding energy differences between the starting and result formations.

Kinetic energy in particle collisions

The concept of kinetic energy in nuclear physics is particularly significant when analyzing particle collisions. This is the energy that a nucleus possesses by virtue of its motion, and in particle collisions, it plays a crucial role in initiating reactions.
For reactions to occur, especially in nuclear processes, sufficient kinetic energy must be supplied to overcome any barriers or binding energy, represented by the 'Q value'. In this exercise, the important task is determining the minimum initial kinetic energy, \(K_{1}\), the projectile nucleus must have. This energy is necessary not just to engage in the reaction, but to account for any energy release or absorbance (denoted as the Q value). The solution simplifies to:

• \( K_{1} = M_{2}c^2 + Q \)

It shows that \(K_{1}\) must compensate for the rest mass energy \(M_{2}c^2\) and the Q value, an indicator of energy release. Achieving this helps in mirroring how in nuclear reactions, kinetic energy transformation is pivotal for enabling the breakdown and formation of atomic nuclei.

Momentum conservation in nuclear physics

Momentum conservation is a bedrock principle in nuclear physics. Like energy, momentum of the entire system before the reaction must equal the momentum post-reaction.
This means that any change within the system, such as nuclei collisions or transformations, must account for how the momentum is distributed in the aftermath. Given one of the nuclei in the exercise is initially at rest, the complete momentum initially can be attributed to the moving nucleus \(M_{1}\).
Once the reaction completes:

• Momentum derived from the initial kinetic energy \(K_{1}\) translates into the motion of \(M_{3}\) and \(M_{4}\), the product nuclei.
• This requires considering both the individual mass and velocity components of each entity involved.

Even without explicit details in the exercise about post-collision velocities, the cornerstone remains: \[M_{1}v_{i} = M_{3}v_{f3} + M_{4}v_{f4}\]where \(v_{i}\) is the initial velocity of \(M_{1}\), while \(v_{f3}\) and \(v_{f4}\) reflect the velocities of \(M_{3}\) and \(M_{4}\) after the collision. Grasping these foundations essentially aids students in predicting the pathways and outcomes in nuclear reactions, ensuring their energy and movement align with real-world physical laws.