Chapter 32, Problem 54GP

To determine

To find: The minimum pion kinetic energy needed to produce this reaction.

Answer to Problem 54GP

  $767.8\text{MeV}$

Explanation of Solution

Given data:

The reaction ${\pi }^{-}+\text{p}\to {\Lambda }^0+{\text{K}}^0$

Calculation:

The $Q$ -value is the mass energy of the reactants minus the mass energy of the products. $\begin{array}{c}Q=m_{{\pi }^{-}}{c}^2+m_{\text{p}}{c}^2-\left(m_{{\Lambda }^0}{c}^2+m_{{\text{K}}^0}{c}^2\right)\\ =139.6\text{MeV}+938.3\text{MeV}-(1115.7\text{MeV}+497.6\text{MeV})\\ =-535.4\text{MeV}\end{array}$

The velocity of both products is same, one can assume sum of mass of product as one mass. $M=m_{{\Lambda }^0}+m_{{\text{K}}^0}=1613.3\text{MeV}/{\text{c}}^2$ .

From the conservation of energy

  $E_{{\pi }^{-}}+m_{\text{p}}{c}^2=E_M\mathrm{...}(1)$

From the conservation of momentum

  $\begin{array}{l}{p}_{{\pi }^{-}}=p_{\text{M}}\\ {\left(p_{{\pi }^{-}}c\right)}^2={\left(p_{\text{M}}c\right)}^2\end{array}$

Using the relativistic energy relationship that $E^2=p^2{c}^2+m^2{c}^4$

  $E_{{\pi }^{-}}^2-m_{{\pi }^{-}}^2{c}^4=E_M^2-M^2{c}^4$

Substituting the $E_M$ from conservation of energy

  $\begin{array}{l}{E}_{{\pi }^{-}}^2-m_{{\pi }^{-}}^2{c}^4={\left(E_{{\pi }^{-}}+m_{\text{p}}{c}^2\right)}^2-M^2{c}^4\\ E_{{\pi }^{-}}^2-m_{{\pi }^{-}}^2{c}^4=E_{{\pi }^{-}}^2+2E_{{\pi }^{-}}{m}_{\text{p}}{c}^2+m_{\text{p}}^2{c}^4-M^2{c}^4\\ E_{{\pi }^{-}}=\frac{M^2{c}^4-m_{{\pi }^{-}}^2{c}^4-m_{\text{p}}^2{c}^4}{2m_{\text{p}}{c}^2}\\ \because E_{{\pi }^{-}}={\text{KE}}_{{\pi }^{-}}+m_{{\pi }^{-}}{c}^2\Rightarrow {\text{KE}}_{{\pi }^{-}}=E_{{\pi }^{-}}-m_{{\pi }^{-}}{c}^2\end{array}$

The minimum pion kinetic energy needed to produce this reaction.

  $\Rightarrow {\text{KE}}_{{\pi }^{-}}=\frac{M^2{c}^4-m_{{\pi }^{-}}^2{c}^4-m_{\text{p}}^2{c}^4}{2m_{\text{p}}{c}^2}-m_{{\pi }^{-}}{c}^2$

  $\begin{array}{l}=\frac{{(1613.3\text{MeV})}^2-{(139.6\text{MeV})}^2-{(938.3\text{MeV})}^2}{2(938.3\text{MeV})}-(139.6\text{MeV})\\ =767.8\text{MeV}\end{array}$

Conclusion:

The minimum pion kinetic energy needed to produce this reaction is $767.8\text{MeV}$