An electron with a total energy of 30.030.0 GeV collides with a stationary positron. What is the available energy?

Ch 44: Particle Physics and Cosmology

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 44: Particle Physics and Cosmology

Problem 8a

Chapter 43, Problem 8a

An electron with a total energy of $30.0$ GeV collides with a stationary positron. What is the available energy?

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Step 1: Understand the concept of available energy in particle collisions. The available energy is the total energy that can be converted into mass or other forms of energy in the center-of-mass frame of the system. For collisions involving a stationary particle, the available energy depends on the total energy of the moving particle and the rest energy of both particles.

Step 2: Recall the rest energy formula for a particle, which is given by $ E_{rest} = m c^2 $, where $ m $ is the mass of the particle and $ c $ is the speed of light. For an electron and positron, their rest energy is identical because they have the same mass.

Step 3: Use the formula for available energy in the center-of-mass frame for a collision between a moving particle and a stationary particle: $ E_{available} = \sqrt{2 E_{moving} E_{rest} + (E_{rest})^2} $. Here, $ E_{moving} $ is the total energy of the moving electron, and $ E_{rest} $ is the rest energy of the positron.

Step 4: Substitute the given values into the formula. The total energy of the moving electron is $ E_{moving} = 30.0 \, \text{GeV} $, and the rest energy of the positron is $ E_{rest} = 0.511 \, \text{MeV} $, which needs to be converted to GeV ($ 1 \text{GeV} = 1000 \text{MeV} $).

Step 5: Perform the calculation step-by-step to find $ E_{available} $. First, convert $ E_{rest} $ to GeV, then compute $ 2 E_{moving} E_{rest} $, add $ (E_{rest})^2 $, and finally take the square root to determine the available energy.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Total Energy in Particle Physics

Total energy in particle physics includes both the rest mass energy and the kinetic energy of a particle. For an electron, this is calculated using Einstein's equation E=mc², where m is the rest mass. In high-energy collisions, the kinetic energy becomes significant, and the total energy is the sum of the rest mass energy and the kinetic energy.

Center of Mass Energy

The available energy in a collision is often referred to as the center of mass energy, which is the energy available for particle interactions. It is calculated by considering the energies of both colliding particles in their center of mass frame. For a stationary positron, the available energy is the total energy of the moving electron plus the rest mass energy of the positron.

Rest Mass Energy

Rest mass energy is the energy equivalent of a particle's mass when it is at rest, given by E=mc². For an electron and positron, their rest mass energies contribute to the total energy available in a collision. In this scenario, the rest mass energy of the positron must be added to the total energy of the electron to determine the total available energy for the interaction.

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