Mossbauer Effect

An atom in its ground state has mass m. It is initially at rest in an excited state of excitation energy Δε. It then makes a transition to the ground state by emitting one photon. Find the frequency of the photon, taking into account the relativistic recoil of the atom. Express your answer also in terms of the mass M of the excited atom. Discuss this result for the case of a crystalline lattice (Mossbauer effect).

SOLUTION

Write the energy and momentum conservation equations:

(1)

(2)

where p is the momentum of the atom after emitting the photon, p_ph = hω/c, the momentum of the photon, and ω is the photon’s frequency. Substituting p = -hω/c from (1) into (2) and rewriting it in the form we can find, after squaring both sides,

(3)

Now taking into account that ∆ε + mc² = Mc², we can rewrite (3) as

(4)

which is smaller by the amount of (∆ε)²/2Mc²h than it would have been without relativistic effects. In the case of a crystalline lattice (Mossbauer effect), the atoms are strongly coupled to the lattice and have an effective mass M₀ >> M. From equation (S.2.11.4) we can see that in this case the atom practically does not absorb energy, which all goes into the energy of the photon, and therefore there is no frequency shift due to this effect.