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A conventional spin relaxation theory in zero and low external fields
(KuboToyabe theory [4,5,6]) begins with the assumption that
there are no fluctuations in the spin system. In this static situation, each muon
detects a local field () which causes Larmor precession of
the muon spin (Fig.15). The time evolution of the muon spin in the
static local field behaves as:
where is the angle between the static local field () and
the initial muon spin direction (), and
is the gyromagnetic ratio of the muon spin (=213.554 kHz/G).
Hereafter, the zaxis is defined to be the direction of the initial muon spin polarization.
Figure 15:
Larmor precession of a muon spin in a static local field .

When the local field has a distribution ,
the muon spin relaxation observed is
the ensemble average of the individual muon spin precession:
If the field distribution is isotropic, or the measurement is done on
a polycrystalline specimen, one can integrate the angular part:
The first term (1/3component) originates from the fraction of
the local field which is parallel to the initial muon spin
polarization. This term is an important signature of static
relaxation, because its existence doesn't depend on the
shape of the field distribution. The second term (2/3component)
is the Fourier transform of the field distribution, and hence, contains
all information about .
Figure 16:
Randomly oriented dense spin system. The local field is well approximated with
an isotropic Gaussian distribution.

In a randomly oriented frozen spin system, with a static moment at every
lattice point (Fig.16), the dipolar field at a muon site
is well approximated by an isotropic Gaussian distribution:
where is the width of the Gaussian distribution. The
corresponding muon spin relaxation is derived by performing
the integral:
This result, known as the `static Gaussian KuboToyabe
function' in zerofield, was first derived by R. Kubo and T. Toyabe in 1966
[4,5].
As shown later in this chapter, it is important to investigate the muon spin
relaxation in a `longitudinal field', which is the external magnetic
field applied parallel to the initial muon spin polarization
(; see Fig.15). The static Gaussian KuboToyabe function
in longitudinal fields is expressed as [6]:
In Fig.17, the static Gaussian KuboToyabe function
is shown for various longitudinal
fields. In zero magnetic field, the function first decays as a Gaussian and
then comes back up to the nonrelaxing 1/3component. With a
longitudinal field applied, the amplitude of the flat
component increases, because the fraction of the field which is
parallel to the initial muon spin polarization increases. In
static relaxation, the magnitude of the longitudinal field which
decouples the relaxation is comparable to the field distribution
width (). The distribution width can also
be extracted from the Gaussian decayrate at early times before the recovery.
Figure 17:
The muon spin relaxation in the randomly oriented frozen dense spin system:
static Gaussian KuboToyabe function .The circle in the figure represents the isotropic random field distribution,
which is centered at zero in zero external field; if the longitudinal field
() is applied, the random field is centered at .

Experimentally, the Gaussian KuboToyabe behavior of the muon spin
relaxation, as well as its longitudinal field dependence, is typically
observed in substances with nuclear magnetic moments, such as MnSi
[6] and copper [35].
Next: Effect of spin fluctuations
Up: 3 Spin relaxation theories
Previous: 3 Spin relaxation theories