The Hadrons — Question 4

Question 4

Charge exchange changes hadron identity while preserving total baryon number

Consider $\pi^-+p\to\pi^0+n$. Take rest energies $m_{\pi^-}c^2=139.57$, $m_pc^2=938.27$, $m_{\pi^0}c^2=134.98$ and $m_nc^2=939.57$ MeV. Pions have B=0 and nucleons B=1.

1. Check electric charge and baryon number, then calculate Q.

2. If initial center-of-mass kinetic energy is 2.00 MeV, find final total kinetic energy and explain what conservation alone does not determine.

Question 4 - Solution

Charge exchange changes hadron identity while preserving total baryon number

(a) Method and calculation

$$\boxed{Q_{\mathrm{charge}}/e:\ -1+1=0+0,\quad B:\ 0+1=0+1}.$$$$\boxed{Q=(139.57+938.27)-(134.98+939.57)=3.29\,\mathrm{MeV}}.$$

(b) Interpretation and further calculation

$$\boxed{K_f=K_i+Q=2.00+3.29=5.29\,\mathrm{MeV}}.$$$$K_f=K_{\pi^0}+K_n,\quad\mathbf p_{\pi^0}=-\mathbf p_n\quad\text{in the center-of-mass frame}.$$

Physical check

Conservation establishes allowed kinematics but not a reaction cross section. The positive Q permits energy release rather than a rest-energy threshold.