Fast broadband cluster spin-glass dynamics in PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$

PbFe$_{1/2}$Nb$_{1/2}$O$_{3}$ (PFN) is a relaxor ferroelectric (T$_{c}$ $\sim$ 400 K) consisting of disordered magnetic Fe$^{3+}$ (S=${5\over2}$, L$\approx$0) ions resulting in a low temperature ``cluster glass" phase (W. Kleemann $\textit{et al.}$ Phys. Rev. Lett. ${\bf{105}}$, 257202 (2010)). We apply neutron scattering to investigate the dynamic magnetism of this phase in a large single crystal which displays a low temperature spin glass transition (T$_{g} \sim$ 15 K), but no observable spatially long-range antiferromagnetic order. The static response in the cluster glass phase (sampled on the timescale set by our resolution) is found to be characterized by an average magnetic spin direction that lacks any preferred direction. The dynamics that drive this phase are defined by a magnetic correlation length that gradually increases with decreasing temperature. However, below $\sim$ 50 K the spatial correlations gradually becoming more short range indicative of increasing disorder on cooling, thereby unravelling magnetism, until the low temperature glass phase sets in at T$_{g}$ $\sim$ 15 K. Neutron spectroscopy is used to characterize the spin fluctuations in the cluster glass phase and are found to be defined by a broadband of frequencies on the scale of $\sim$ THz, termed here ``fast" fluctuations. The frequency bandwidth driving the magnetic fluctuations mimics the correlation length and decreases until $\sim$ 50 K, and then increases again until the glass transition. Through investigating the low-energy acoustic phonons we find evidence of multiple distinct structural regions which form the basis of the clusters, generating a significant amount of local disorder. We suggest that random molecular fields originating from conflicting interactions between clusters is important for the destruction of magnetic order and the eventual formation of the cluster glass in PFN.