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This section considers nuclear spin relaxation in a paramagnetic
localized spin system, with the interaction between the moments
described by the dipolar fields.
We assume the Hamiltonian of the spin system to be:
where, H_{0} is the external magnetic field applied in the
zdirection, is the spin of the ith localized electron
and D_{ij} represents the tensor for the dipolar interaction,
namely,
where, is the distance between
the ith and the jth electrons and is the unit vector parallel to the distance.
It is easily shown that the fluctuation rate () of the electron moments
doesn't depend on the external field; if one
writes the Hamiltonian (eq.68) in the rotating
reference frame (RRF) for the electrons, one can eliminate the external
field, while the dipolar interaction is expressed as:
where is the first term of the Hamiltonian
(eq.68) and the RRF representation
of the spin . This result indicates that the dipolar
interaction between the electrons (namely the associated fluctuation
rate ) doesn't change in RRF, because the interacting two spins
and have the same gyromagnetic ratio, and
conduct coherent precession in the external field H_{0}.
To the probe nuclear spin, the Larmor precession
of the electron moment in a reasonably large external field H_{0}
is so fast that its xy components becomes
invisible (Fig.64). In this limit, the probe
spin detects only the zcomponent of the electron moments, which
changes its length with the dipolar fluctuation rate (). In this
situation, the T_{1} relaxation rate of the probe spin may be reduced
by from that in the zerofield case, because the average
size of the zcomponent squared is 1/3 of the full spin:
. As shown below, the precise factor is 3/10.
Figure 64:
An electron moment and a muon spin in an external magnetic field
(H_{0}). If the Larmor precession of the electron is fast, muon
detects only the zcomponent of the electron spin S_{i}^{z}.

Generally, the T_{1} relaxation rate of a probe nuclear spin is
expressed as [34]:
where, is the Larmor precession frequency of the nuclear
spin (muon), (a=x,y) is the x,ycomponent of the
fluctuating local fields at the probe spin site and . The bracket denotes the ensemble
average. If the muonelectron interaction is the dipolar fields, the
local field at the muon site becomes:
where is the distance between the muon and the jth
electron and is the unit vector parallel to the distance.
To write , it is convenient to define
, where,
(a=x,y) is the x,ycomponent of the unit vector
. The local fields are expressed as:
Using the rotating reference frame (RRF) for the electrons, the electron
spin correlations <S^{a}_{j}(0)S^{b}_{k}(t)> () are expressed as:
where is the electron Larmor frequency, S^{a}_{j;R}(t)
() is the electron spins expressed in the RRF. Assuming a Markoffian
fluctuation processes, decay of the spin correlation is expressed as
where is the dipolar fluctuation rate, and . In
paramagnetic state, there is no correlation between the jth and the
kth spin, yielding the t=0 correlation functions as:
Using the above presented expressions, the field correlation at the probe
spin site becomes:
In the same manner, is calculated, yielding the T_{1}
relaxation rate as:
In an isotropic sample, the coefficients are replaced by the angular averages:
Introducing the dipolar field width () defined in eq.45,
the T_{1} relaxation yields:
Namely, if the Larmor precession of the electron moments are fast, the
local field width at the muon site is effectively reduced to
.
Next: A.3 Relaxation function in randomness
Up: A Appendix
Previous: A.1 Extended LiebShultzMattis Theorem