Pozitron-Electron interaction(anihilation)

[pop_ianosd]

Homework Statement

A given pozitron with T kinetic energy is approaching an electron bound to a hard nucleus(in the K energy level, and the ionization energy for this level, E_k(>0), is given).
The energy of the two resulting photons is to be determined.

Homework Equations

Conservation of energy.

The Attempt at a Solution

So I wrote the conservation of energy as follows:
2m₀c² + T = 2ε + E_k ( ε is the energy of one photon )
Let me explain my way of thinking: After the anihilation, the electron has vanished, so the atom has become a ion(so it's energy has increased by E_k).

However, the proposers of the problem consider the conservation of energy as follows:
2m₀c² + T + E_k = 2ε;

So who is right?

 

[mfb]

If E_k is defined to be positive, it should be subtracted on the left side (or added on the right side).
Otherwise you can violate energy conservation: Take a free electron, let it get caught by a nucleus (emitting energy). Add a free positron, and get more than 2*511keV as energy => you gained the binding energy two times.