Quest for secondary $\mu $SR signals for Fe$_{\mathrm{3}}$O$_{\mathrm{4}}$ using MaxEnt: a Verwey phase transition study.

Most muon-spin rotation ($\mu $SR) time series for magnetite (Fe$_{\mathrm{3}}$O$_{\mathrm{4}})$ have been interpreted in terms of \textit{one} $\mu $SR frequency signal. [1] Its Fourier transform appears to confirm this internal magnetic field. Yet many time series show a beat pattern, strongly suggesting a second signal with a close-by frequency. We are searching for secondary signals in zero-field Fe$_{\mathrm{3}}$O$_{\mathrm{4}} \quad \mu $SR data using Maximum Entropy, a recently developed technique [2] more sensitive than curve fitting and/or Fourier transformation. There is also another dilemma namely: the upper signal found for Fe$_{\mathrm{3}}$O$_{\mathrm{4}}$ has a local magnetic field larger than the maximum allowable vectorial sum of external and internal contributions. However, the (non)occurrence of secondary signals may shed light on the nature of the Verwey phase transition and its precursors in the Fe$_{\mathrm{3}}$O$_{\mathrm{4}}$ Mott-Wigner glass [3] between T$_{\mathrm{v}}$ (123 K) and twice T$_{\mathrm{v}}$ (247 K). [4] Research supported by LANL-DOE, SETI-NASA, SJSU {\&} AFC. [1] C Boekema \textit{et al,} Hpf Interactions 31 (1986) 487; Phys Rev B31 (1985) 1233. [2] C Boekema and MC Browne, MaxEnt 2008, AIP Conf Proc {\#}1073 p260. [3] NF Mott, Metal-Insulator Transitions, Taylor {\&} Francis (1974); C Boekema \textit{et al,} Phys Rev B33 (1986) 210. [4] C Boekema \textit{et al,} Proc 11th Int M2S Conf (2015).