Muon-spin-relaxation studies of magnetic order and dynamics of the $n=2\mathrm{}$ Ruddlesden-Popper phases ${\mathrm{Sr}}_{2}R{\mathrm{Mn}}_2{\mathrm{O}}_7$ $(R=\mathrm{Pr},$ Nd, Sm, Eu, Gd, Tb, Dy, and Ho)

Zero field muon spin relaxation (μSR) has been used to study the magnetic properties of $n=2\mathrm{}$ Ruddlesden-Popper phases ${\mathrm{Sr}}_{2}R{\mathrm{Mn}}_2{\mathrm{O}}_7,$ where $R=\mathrm{Pr},$ Nd, Sm, Eu, Gd, Tb, Dy, and Ho. The results show that the size of the lanthanide ion is crucial in determining the magnetic state and dynamics of the system. Because muons are implanted throughout the bulk of the sample, impurity phases contribute only according to their volume fraction. Hence in the case of biphasic samples the data are dominated by the majority phase. Although none of our samples has a ferromagnetic ground state, colossal magnetoresistance (CMR) is observed over a wide temperature range, $4\mathrm{K}<T<\mathrm{}150\mathrm{}\mathrm{K},$ for both the Pr and Nd compounds. The μSR results show that the magnetic transition in both these samples is broad. Ordered, but fluctuating, regions form at $\sim 150\mathrm{K},$ the reported onset of CMR, with the fluctuation rates gradually decreasing with temperature. Even at 5 K, fluctuations are still observed. The ferromagnetic double exchange between Mn ions becomes weaker as the size of the lanthanide ion decreases. ${\mathrm{Sr}}_2{\mathrm{SmMn}}_2{\mathrm{O}}_7$ shows weak clustering at a much reduced temperature of 30 K whereas ${\mathrm{Sr}}_2{\mathrm{EuMn}}_2{\mathrm{O}}_7$ shows spin-glass-like behavior. For all lanthanide ions smaller than Eu no long range magnetic ordering of the spins is observed and the observed relaxation rates follow an activated dependence. The technique allows us to extract the effective activation energy associated with the magnetic fluctuations of the lanthanide moments in samples with $R=\mathrm{Sm},$ Eu, Gd, Tb, Dy, and Ho. CMR is only observed where μSR measurements show a broad magnetic transition associated with fluctuations. We therefore believe that these fluctuating ordered regions are responsible for the extended temperature regime in which CMR has been observed in these nonferromagnetic $n=2\mathrm{}$ Ruddlesden-Popper phases.