A New Look at Energy — Question 9

Question 9

A two-body radiative decay must allocate energy to recoil

A parent of invariant mass M at rest decays into a daughter of mass m and one photon. Use photon momentum $p_\gamma=E_\gamma/c$ and daughter energy $E_d^2=p_d^2c^2+m^2c^4$. Assume $M>m$.

1. Derive the photon energy including daughter recoil.

2. For $M=3m$, find both final energies and the daughter speed.

Question 9 - Solution

A two-body radiative decay must allocate energy to recoil

(a) Method and calculation

$$\mathbf p_d=-\mathbf p_\gamma,\qquad Mc^2=E_\gamma+\sqrt{E_\gamma^2+m^2c^4}.$$$$(Mc^2-E_\gamma)^2=E_\gamma^2+m^2c^4\quad\Rightarrow\quad\boxed{E_\gamma=\frac{M^2-m^2}{2M}c^2}.$$

(b) Interpretation and further calculation

$$M=3m:\quad\boxed{E_\gamma=\tfrac43mc^2},\qquad\boxed{E_d=\tfrac53mc^2}.$$$$\boxed{v_d/c=p_dc/E_d=E_\gamma/E_d=4/5}.$$

Physical check

Assigning all the rest-energy difference to the photon neglects the kinetic energy required by momentum conservation.