Ionizing Deuterium

The hydrogen atom has an ionization energy of E_H = 13.5983 eV when an electron is bound to a proton. Calculate the ionization energy of deuterium: an electron bound to a deuteron. Give your answer as the difference between the binding energy of deuterium and hydrogen (δE = E_D – E_H). The deuteron has unit charge. The three masses are, in atomic mass units, m_e = 5.4858 × 10^-4, m_p = 1.00728, m_d = 2.01355.

SOLUTION

The ionization energy of hydrogen is just the binding energy of the electron which is given in terms of the reduced mass μ_ep of the electron–proton system. The same expression for deuterium contains the reduced mass μ_ed of the electron–deuteron system:

(1)

(2)

(3)

(4)

The difference is easily evaluated. The ratio m_e/m_p is a small number and can be used as an expansion parameter:

(5)

The ratio of masses gives 2.72 × 10^-4 and δE ≈ 3.700 meV.