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In the true Néel state, a muon spin exhibits a Larmor precession. In a
randomly frozen spin system, it exhibits the Gaussian KuboToyabe function.
In the following, an intermediate situation between these cases is considered.
If a spin system is frozen almost randomly, but having a nonzero
sublattice magnetization (Fig.65), the local
field at muon site will be a Gaussian distribution around a static field:
where, is the isotropic Gaussian distribution induced by
the randomly frozen spin component and 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 is taken over the sample
geometry. In a polycrystalline sample (, isotropic) ,
the integral about is analytically performed, yielding the
muon relaxation as:
Here, the integral over 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
KuboToyabe function (eq.18), and in the large H_{0}
limit, it exhibits a damped oscillation with a frequency associated
with H_{0}. In Fig.66, the function obtained
is drawn for various H_{0}. A crossover from
the Gaussian KuboToyabe behavior to a damped oscillation is shown.
Figure 66:
Behavior of the Gaussian KuboToyabe function with a static local
field H_{0}. The dashed lines are the envelope for the damped
oscillation in the large H_{0} limit.

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