Magnetic order and spin dynamics in the helical magnetic system Fe$_3$PO$_4$O$_3$

The 3$d$-electronic spin dynamics and the magnetic order in Fe$_3$PO$_4$O$_3$ were investigated by muon spin rotation and relaxation ($\mu$SR) and $^{57}$Fe M\"ossbauer spectroscopy. Zero-field (ZF)-$\mu$SR and $^{57}$Fe M\"ossbauer studies confirm static long range magnetic ordering below $T_{\mathrm{N}}$ $\approx$ 164\,K. Both transverse-field (TF) and ZF-$\mu$SR results evidence 100\% magnetic volume fraction in the ordered state. The ZF-$\mu$SR time spectra can be best described by a Bessel function, which is consistent with the helical magnetic structure proposed by neutron scattering experiments. The M\"ossbauer spectra are described in detail by considering the specific angular distribution of the local hyperfine field $B_{\mathrm{hyp}}$ with respect to the local electric field gradient. The $\mu$SR spin-lattice relaxation rate exhibits two peaks: One at the magnetic ordering temperature related to critical magnetic fluctuations and another peak at 35\,K signaling the presence of a secondary low energy scale in Fe$_3$PO$_4$O$_3$.