A.3 Relaxation function in Néel state with randomness

 In the true Néel state, a muon spin exhibits a Larmor precession. In a randomly frozen spin system, it exhibits the Gaussian Kubo-Toyabe function. In the following, an intermediate situation between these cases is considered.

If a spin system is frozen almost randomly, but having a non-zero sublattice magnetization (Fig.65), the local field at muon site will be a Gaussian distribution around a static field:

where, $\rho_{G}({\mbox{\boldmath$H$\unboldmath}})$ is the isotropic Gaussian distribution induced by the randomly frozen spin component and ${\mbox{\boldmath$H$\unboldmath}_0}$ is a well defined static field from the Néel ordered component.

  

Figure 65: A Néel order with randomness. Each spin bears a sublattice magnetization, but the remaining spin component is randomly frozen.

Muon spin relaxation in this field distribution is expressed as:

where the integral about ${\mbox{\boldmath$H$\unboldmath}_0}$ is taken over the sample geometry. In a polycrystalline sample (, isotropic) , the integral about ${\mbox{\boldmath$H$\unboldmath}_0}$ is analytically performed, yielding the muon relaxation as:

Here, the integral over ${\mbox{\boldmath$H$\unboldmath}}$ is expressed in polar coordinate. Since the Gaussian distribution is isotropic, the angular part is easily integrated; it gives the 1/3 and 2/3 components. The integral of the radial part is also performed analytically:

In the limit, this function becomes the Gaussian Kubo-Toyabe function (eq.18), and in the large H₀ limit, it exhibits a damped oscillation with a frequency associated with H₀. In Fig.66, the function obtained is drawn for various H₀. A crossover from the Gaussian Kubo-Toyabe behavior to a damped oscillation is shown.

  

Figure 66: Behavior of the Gaussian Kubo-Toyabe function with a static local field H₀. The dashed lines are the envelope for the damped oscillation in the large H₀ limit.