Calculate the minimum beam energy in a proton-proton collider to initiate the $p+p\to p+p+{\eta }^0$ reaction. The rest energy of the ${\eta }^0$ is $547.3$ MeV (see Table $44.3$).

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to calculate the minimum beam energy required for the proton-proton collision to produce the η^0 particle. This involves using the principles of energy conservation and relativistic kinematics.
2. Step 2: Identify the key quantities. The rest energy of the η^0 particle is given as 547.3 MeV. The rest energy of a proton is approximately 938.3 MeV. In the center-of-mass frame, the total energy must be sufficient to account for the rest energies of the two protons and the η^0 particle.
3. Step 3: Write the energy conservation equation. In the center-of-mass frame, the total energy before the collision is the sum of the kinetic energy and rest energy of the two protons. After the collision, the total energy is the sum of the rest energies of the two protons and the η^0 particle. The minimum energy occurs when the η^0 is produced at rest. The equation is: $E_{\mathrm{total}}=2E_{\mathrm{proton}}=2E_{\mathrm{rest}}+E_{\eta }$
4. Step 4: Relate the beam energy to the center-of-mass energy. In a symmetric collider, the beam energy of each proton contributes equally to the center-of-mass energy. Therefore, the beam energy per proton is half of the total energy required in the center-of-mass frame. Use the equation: $E_{\mathrm{beam}}=E_{\mathrm{total}}2$
5. Step 5: Substitute the values into the equations. The rest energy of each proton is 938.3 MeV, and the rest energy of the η^0 particle is 547.3 MeV. Calculate the total energy required in the center-of-mass frame and then divide by 2 to find the minimum beam energy for each proton.

Key Concepts

Conservation of Energy

In particle physics, the conservation of energy principle states that the total energy in a closed system remains constant. When calculating the minimum beam energy required for a reaction, one must account for the rest mass energies of the particles involved. The energy must be sufficient to create new particles, which means the initial kinetic energy of the colliding protons must equal the total rest mass energy of the final products.

Threshold Energy

Threshold energy is the minimum energy required for a specific reaction to occur. In the context of particle collisions, it is the energy needed to produce the rest mass of the resulting particles. For the reaction p + p → p + p + η^0, the threshold energy can be calculated by summing the rest mass energies of the initial and final particles, ensuring that the kinetic energy of the colliding protons meets this requirement.

Center of Mass Frame

The center of mass frame is a reference frame in which the total momentum of the system is zero. In collider experiments, calculations are often simplified by analyzing the reaction in this frame. The minimum energy required for a reaction can be more easily determined in the center of mass frame, as it allows for a direct comparison of the rest mass energies of the particles before and after the collision, facilitating the calculation of the necessary beam energy.