Question

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 $\eta °$ is $547.3\hspace{0.33em}MeV.$

Short answer

The minimum beam energy in a proton-proton collider to initiate the $p+p\to p+p+{\eta }^0$reaction is$E_m=1212\hspace{0.33em}MeV.$

Step-by-step solution

Definition of Law of conservation of energy

It states that total energy of a system is constant. It can change from one type of energy to another but never completely lost. System and enviroment interacts and exchange energy if the system allows.

Determine The minimum beam energy in a proton-proton collider to initiate the p+p→p+p+η0 reaction

The minimum available energy is:

$E_{\alpha }=m_p{c}^2+m_p{c}^2+m_{\eta }{c}^2$

Put known values, the rest energy of the proton is $938.3MeV$, and the rest energy of $\eta °$is $547.3MeV$

So,

$$\begin{gathered} E_{\alpha }=938.3\hspace{0.33em}MeV+938\hspace{0.33em}MeV+547.3\hspace{0.33em}MeV \\ {E}_{\alpha }=2423.9\hspace{0.33em}MeV \end{gathered}$$

The total energy equals half the available energy in colliding beams

So

$$\begin{gathered} E_m=E_{\alpha }/2 \\ {E}_m=(2423.9\hspace{0.33em}MeV)/2 \\ {E}_m=1211.95\hspace{0.33em}MeV \\ {E}_m=1212\hspace{0.33em}MeV \end{gathered}$$

Hence, the minimum beam energy in a proton-proton collider to initiate the

$p+p\to p+p+{\eta }^0$ reaction is$E_m=1212\hspace{0.33em}MeV.$