EPR on free atoms

First task:

When a free La atom in an externally applied magnetic field absorbs a microwave photon, it undergoes a transition between quantized energy levels. This will cause the orientation of the magnetic moment of the electron cloud to change relative to the external field.

Second task:

Without hyperfine interaction, the energy of a magnetic sublevel is determined only by the Zeeman effect:

E = m_J g_J mu_B B_0.

The energy of the absorbed photon is the difference between two adjacent levels:

Delta E = | g_J mu_B B_0 | (accounting for selection rule).

With g_J = 0.8, B_0 = 2T and mu_B = 5.788 * 10^-5 eV / T, we find

Delta E = 9.26 * 10^-5 eV

In absence of hyperfine interaction, all allowed transitions require the same photon energy.

With hyperfine interaction, the effective magnetic field seen by the electrons becomes B_eff = B_0 + m_I B_hf. The transition energy is now

Delta E = | g_J mu_B (B_0 + m_I B_hf) |.

Since I = 1/2, the nuclear magnetic quantum number m_I can be +1/2 or -1/2. So when the hyperfine interaction is turned on, the single resonance line splits into two distinct lines:

Delta E = 0.8 * mu_B * 7 T = 32.41 * 10^-5 eV for I = +1/2,

Delta E = 0.8 * mu_B * 3 T = 13.89 * 10^-5 eV for I = -1/2.