Zeeman

For the discussion of the principles of EPR we start with the simple case of a two-level system for a paramagnetic centre with an electron spin S = 1/2. In the absence of any magnetic field the magnetic moment, associated with the electron spin, is randomly oriented and the two energy levels are degenerate. The application of an external magnetic field B₀ results in a splitting of the two energy levels as the electron spin S can only be oriented parallel or anti-parallel to the magnetic field vector. The quantization of the energy levels is due to the quantum-mechanical nature of the electron spin. The potential energy of this system is derived from the classical expression for the energy of a magnetic dipole in a magnetic field and is described by the spin Hamilton operator, expressed in frequency units:

$$ \mathcal{H}={g}_e\frac{\beta_e}{h}\mathbf{S B}_0 \qquad (1) $$

where g_e = 2.0023, the dimensionless g-value of the free electron, β_e denotes the Bohr Magneton (J·T^-1) and h the Planck constant (J·s); B₀ is given in Tesla (T). The energies for the two spin states are then given by the eigenvalues of H in Eq. (1) and characterized by the spin quantum numbers m_S = ± 1/2.

$$ E(m_s)=g_e\frac{\beta_e}{h}B_0 {m}_s \qquad (2) $$

The splitting between the two energy states is called electron Zeeman interaction (EZI) and is proportional to the magnitude of B₀, as illustrated in Figure 1. The energy difference between the two Zeeman states is given by ΔE = E(m_S = +1/2) - E(m_S = -1/2) = g_eβ_eB₀/h (in Hz).

Fig. 1: Illustration of the Zeeman splitting for a S = 1/2 system with one unpaired electron in an external magnetic field B₀. For a given irradiation frequency ν a transition between the Zeeman states occurs if the resonance condition is fulfilled (see inset).

The most simple EPR experiment one could imagine involves the application of an electromagnetic radiation field of variable frequency ν to this system. If the energy of the radiation field matches the energy gap ΔE, transitions between the two spin states can be induced, i.e. the spin can be flipped from one orientation to the other. In this case the resonance condition is fulfilled:

$$ \Delta {E}=\nu = {g}_e \frac{\beta_e}{h}{B}_0 \qquad (3) $$

For experimental reasons, however, the microwave frequency is usually held constant in a CW EPR experiment and the magnetic field is swept linearly.