β-decay of Tritium

Tritium is an isotope of hydrogen with one proton and two neutrons. A hydrogen-like atom is formed with an electron bound to the tritium nucleus. The tritium nucleus undergoes β-decay, and the nucleus changes its charge state suddenly to +2 and becomes an isotope of helium. If the electron is initially in the ground state in the tritium atom, what is the probability that the electron remains in the ground state after the sudden –β-decay?

SOLUTION

We use the sudden approximation to calculate the probability that the electron remains in the ground state. One calculates the overlap integral I of the initial and final wave functions, and its square is the probability. The ground states in the initial and final states are called ѱ_i and ѱ_f, and a₀ is the Bohr radius:

(1)

(2)

(3)

(4)