← Particle Physics

# Energy and Momentum in Particle Reactions

Saturday, April 9, 2022

## Particle Collisions

In a particle collision, a high-speed particle is often collided with particles at rest. Since the energies involved are so great, relativistic formulas must be used.

In the generic reaction

$m_1+m_2\rightarrow m_3+m_4+m_5+...$

each particle has some mass $m$. The first particle (the one not at rest) has some energy $E_1$, kinetic energy $K_1=E_1-m_1c^2$, and momentum $cp_1=\sqrt{E_1^2-m_1^2c^4}$ in the frame of reference of the laboratory. We define the Q value as such:

$Q=\left(m_i-m_f\right)c^2=\left[m_1+m_2-\left(m_3+m_4+m_5+...\right)\right]c^2$

If $Q$ is positive, some rest energy is converted into kinetic energy. If $Q$ is negative, some kinetic energy has been stored as rest energy in the products.

## Threshold Energy

If the $Q$ value is negative, there must be some threshold kinetic energy $K_\text{th}$ larger than the magnitude of $Q$ needed to initiate the reaction. Additionally, $K_\text{th}$ must be sufficiently large to conserve linear momentum in the products.

If the kinetic energy is above the threshold energy, the resulting particles will have some nonzero transverse momentum which sums to zero in order to conserve linear momentum (for example, momentum in the $y$ direction that adds to zero).

### Efficiency

The ratio of rest energy produced in the reaction to the amount of kinetic energy supplied to the first particle measures the efficiency of the reaction. The higher the rest energies of the products, the less efficient the reaction. To combat this fact, reactions can be done in the center-of-mass frame, called the CM frame. In this case, the linear momentum of the reacting particles is zero, so no energy is wasted in linear momentum of the products.