Cauchy magnetic field component and magnitude distribution studied by the zero-field muon spin relaxation technique

Zero-field muon spin relaxation (ZF-$\mu $SR) data for dilute spin magnetic systems have been widely interpreted with what is called a Kubo-Toyabe form based on a Lorentzian distribution of local field components. We derive here the proper magnetic field \textit{magnitude} distribution using independent and uncorrelated \textit{component} distributions. Our result is then compared to the previously accepted formula for ZF-$\mu $SR. We discuss the origins of the magnetic field component and magnitude distributions. Further we found that after rescaling the magnetic field, the differences that are amenable to experimental examination are quite small, although the interpretations behind them are quite different.