Taking into account the slow paramagnetic relaxation caused during the
fast fluctuation time, the muon spin relaxation due to this unconventional
field fluctuations may be expressed as :
Using the same hypothetical relaxation function
(eq.44), we have analyzed the spectra from other
Ca-doped systems (x=4.5, 14.9 and 30.5%). The fraction parameter
(f) and the instantaneous Gaussian width () are shown in
Fig.46. It was found that the Gaussian width
() is almost independent of the Ca concentrations (x), while
the fraction parameter (f) reflects the concentrations: a smaller Ca
concentration results in a smaller fraction parameter (f),
indicating more unconventional fluctuations (note that regular
Markoffian spin fluctuations are described with f=1). The indifference
of the Gaussian width () to the charge concentrations suggests
that the muon spin relaxation mechanism is common to all the
charge doped systems, but how frequently the relaxation is caused
() is determined by the charge concentration.
One possible description of the relaxation mechanism is that the doped hole, which takes a localized state with hopping , occasionally comes close to the muon site and induces muon spin relaxation. When the charge is far away, the muon relaxation should be small and dynamic, because the majority of the spins on the chain may stay in the non-magnetic ground state.
Phenomenologically, the magnetic behavior of the charge doped Haldane material (Y2-xCax)BaNiO5 is very similar to that of the Kagomé-lattice compound (SrCrzGa12-zO19), a geometrically frustrated antiferromagnet of Cr moments (S=3/2). The susceptibility measurements of the Kagomé-lattice system [107,108,109] have revealed the existence of a small portion of unpaired spins (% of the total Cr ions for the z=8 sample), which exhibit a spin-glass-like history dependence below K. The unpaired moments observed in the susceptibilities are probably caused by Ga substitutions to the Cr sites, which are inevitable in this series of Kagomé compounds . The ZF-SR spectrum of the z=8 Kagomé material approaches a Gaussian shape as , while LF-SR measurements at 100 mK suggest fast field fluctuations [95,106,110]. Neutron scattering measurements of the z=7.1 Kagomé compound  have also suggested persistent dynamics below the cusp temperature; a large fraction () of the scattering intensity was found to remain in the inelastic channel at .
Theoretically, the S=1/2 Kagomé-lattice system may have a ground state composed of many singlet pairs , as expressed by the resonating valence bond (RVB) state. In this situation, the unpaired spins created by non-magnetic ion doping still have the ability to migrate spatially, because the surrounding spins have a large number of combinations for their singlet pairings. In the charge doped Haldane system, the doped charge may also move, with the surrounding spins in the singlet state. Considering these similarities, the persistent dynamics below the spin-glass like cusp temperature, as well as the hardly decoupled Gaussian relaxation of SR spectra may be common signatures for migrating unpaired spins in singlet ground state materials.