Relaxation Phenomena

There are many contributions to the hyperfine field at the nucleus as seen in Equation 2.17 but the major contributor for transition metals such as $^{57}$Fe, when in zero applied field, is $H_{S}$. This arises from the polarising effect of unpaired electron spins with the direction of the field being related to that of the electron spins. However, this direction is not invariant and can flip after a period of time. This is the relaxation phenomenon. The effects upon the Mössbauer lineshape depend upon the relative time scales of measurement and the relaxation mechanism, there being three time scales to consider: the lifetime of the Mössbauer event, the Larmor precession time and the relaxation time.

The lifetime of the Mössbauer event, $\tau_{m}$, which is also the limiting time scale of the measurement technique, is determined by the Heisenberg uncertainty relationship as shown in Equation 2.3. For $^{57}$Fe this is of the order of $10^{-7}\ensuremath{\unskip\,\mathrm{s}}$.

The second time scale to consider is the minimum time required for the nucleus to detect the hyperfine field. This is usually assumed to be equal to the Larmor precession time, $\tau_{l}$, which can be considered as the time taken for a nuclear spin state, $I$, to split into $(2I+1)$ substates under the influence of a hyperfine field. $\tau_{l}$ is proportional to the magnitude of the hyperfine field (and hence related to the nuclear energy levels as in Equation 2.16) with the following relation

$$\tau_{l} = \frac{2\pi\hbar}{g\mu_{n}B}$$ (2.26)

where $g$ is the gyromagnetic constant and $\mu_{n}$ is the nuclear Bohr magneton. In iron oxides the hyperfine field is $\sim 400 \rightarrow 500\ensuremath{\unskip\,\mathrm{kG}}$ giving $\tau_{l}$ of the order of $10^{-8}\ensuremath{\unskip\,\mathrm{s}}$. This means that $\tau_{m} \gg \tau_{l}$ and hence the hyperfine fields are detectable by the technique.

The final time scale is the relaxation time, $\tau_{r}$, associated with the time dependent fluctuations of the electron spin. For the hyperfine field to be observed it must remain constant at the nucleus for at least one Larmor precession period.

There are three regimes which are important when considering the effect of relaxation on the Mössbauer lineshape:

1. If $\tau_{r} \gg \tau_{l}$ then the hyperfine field is static during a single Larmor precession period. The spectral lines are narrow and Lorentzian in shape.
2. If $\tau_{r} \ll \tau_{l}$ then the nucleus experiences a time averaged hyperfine field. The magnitude is less than the value obtained for a static field as the interaction will have changed many times during a single precession period and tends to zero as $\tau_{r}$ decreases. Narrow Lorentzian lines are still observed.
3. If $\tau_{r} \approx \tau_{l}$ then resonance between the relaxation and the precession occurs leading to complex spectra and broadened lineshapes. As $\tau_{l}$ is proportional to the energy difference between the spectral lines $\tau_{l}$ for the outer lines will be less than for the inner lines, causing the inner lines of a sextet to broaden and disappear before the outer ones.[6]

The two main mechanisms involved in the spin relaxation are Spin-Spin and Spin-Lattice relaxation.