Question

Exercises 23 and 24 give the threshold energies for which two particles of mass m can produce a given mass M in collidingbeam and stationary-target accelerator. Evaluate the two for a collision in which two protons become three protons and one antiproton. How much more energy is neededfor the stationary target?

Short answer

The energy needed for the colliding beam is$2m_p{c}^2$.

The threshold energy for the stationary target is about $6m_p{c}^2$.

Step-by-step solution

Given data

Let $m_p$be the initial mass of each proton.

Relation between energy and momentum

Firstly, choose to calculate this quantity after the collision in the reference frame where the final product is stationary.

It can be expressed as:

${\left(\frac{E_{\text{total}}}{c}\right)}^2-p_{\text{total}}^2=M^2{c}^2$

Step 3:Determine the energy needed for the colliding beam

Let$m_p$be the initial mass of each proton.

After the collision, the two protons become three protons and one antiproton.

So, the total mass of the four-particle after the collision is $M=4m_p$ .

For the case both protons are moving, the threshold energy for this reaction can be expressed as follows:

$\begin{array}{rcl}\left(M-2m_P\right)c^2& =& 4m_P{c}^2-2m_P{c}^2\\ & =& 2m_P{c}^2\\ & & \end{array}$

Therefore, the energy needed for the colliding beam is $2m_P{c}^2$.

Determine the threshold energy for the stationary target

For the case of the stationary target, the threshold energy for this reaction can be expressed, as shown below:

$\begin{array}{rcl}\left(\frac{M^2}{2m_P}-2m_P\right)c^2& =& \left(\frac{{\left(4m_P\right)}^2}{2m_P}-2m_P\right)c^2\\ & =& 6m_P{c}^2\\ & & \end{array}$

Thus, the threshold energy for the stationary target is about $6m_P{c}^2$.

Determine the extra energy required for the stationary target

The difference in threshold energy is, $6m_P{c}^2-2m_P{c}^2=4m_P{c}^2$.Therefore, $4m_P{c}^2$ more energy is needed for the stationary target.The energy needed for the colliding beam is $2m_p{c}^2$.The threshold energy for the stationary target is about$6m_p{c}^2$ . $4m_p{c}^2$more energy is needed for the stationary target.