In order to estimate the numbers of unpaired spins, we analyzed the
paramagnetic region of the susceptibilities with the Curie-Weiss law
, a model which assumes that all the
doping effect is a creation of local moments. The results are shown in
Fig.47.

In the vacancy doped systems, the Mg concentration (*y*) dependence of
the Curie constant (*C*) is consistent with the heuristic
`singlet-triplet' model, which assumes that *two* Mg^{2+} ions
effectively create one * S=1* spin [104]. In the
charge doped systems, one Ca ion seems to create =1 spin, or =1/2 spins. Since susceptibility has no
information about the local structure of the unpaired spins, it is not
possible to distinguish these two situations. For simplicity, we assume hereafter
that =1 effective spins are created for one Ca ion, and =1 spins for one Mg ion.

It is known that the paramagnetic moments of the nominally pure
Y_{2}BaNiO_{5} are created by excess oxygen, which works as a hole-dopant
[105]. The Curie constant of the nominally pure specimen
corresponds to the native charge concentration of at.%,
which has been estimated by extending the Ca concentration
dependence of the Curie constant (Fig.47a) to
the negative *x* side.

In order to estimated the Lorentzian field width (*a*) generated from the
unpaired spins, we utilized
the procedure developed for analysis of the dilute spin-glass alloys
[7]:

- (1)
- obtain the hypothetical Gaussian field width () for
the situation in which all the spin sites are filled up with randomly
oriented moments. In zero external field, this Gaussian width
() is expressed as [6]:

*S*is the size of the spins at each site, () is the muon(electron) gyromagnetic ratio. (, where*g*is the*g*-factor and is the electron Bohr magneton.) - (2)
- using eq.26, obtain the Lorentzian width (
*a*) from the hypothetical Gaussian width.

Since we do not know the muon site in Y_{2}BaNiO_{5}, we assumed locations
shown in Fig.48. These sites assumed are all
Å away from a oxygen ion, where a muon usually resides in
oxides [113,114,115,116,117].
We numerically performed the dipolar sum (eq.45), assuming
that the *g*-factor is 2 for the doping induced *S*=1 spins. The results
() are shown in Fig.48.

If an external field is applied to a paramagnetic spin system, the
electron moments undergo Larmor precession. Since the gyromagnetic
ratio of muon and electron differ by two orders of magnitude, the
precessing electron spin component becomes invisible to a muon.
Namely, only the secular part of the dipolar field contributes to muon
spin relaxation in longitudinal field measurements. For paramagnetic
*T _{1}* relaxation, the secular contribution is 3/10 of the zero-field

Using the Gaussian width , the estimated
unpaired spin concentration (*y* is
the Mg concentration) and eq.26, the
Lorentzian field widths (*a*) were estimated as shown in Table
5. Although the ambiguity from the muon
site is large, the experimental data is within the range of the
estimated magnitude.

If the fluctuations of the unpaired spins are governed by their dipolar
interactions, the fluctuation rate () should scale with the
Lorentzian width (*a*). The ratio will be the order of the
ratio of the electron/muon gyromagnetic factors (), where the
factor comes from the fact that an electron moment
detects the entire spin of other electrons, and the factor corrects the units (*a* in rad/sec and in Hz). The
fluctuation rates estimated as above are shown in Table
5. The experimental results again are in
the same order of magnitudes with these estimates.

The above estimates of the field width (*a*) and the fluctuation rate
() in the nominally pure and Mg-doped systems supports the scenario
in which the muon detects the dipolar fields of the unpaired spins in these
systems.