Distribution of Fields from Randomly Placed Dipoles: Free-Precession Signal Decay as Result of Magnetic Grains

Free-precession signals were observed from fluids in samples containing randomly distributed ferromagnetic grains. The local free-precession phase shift was calculated by computing volumes of space for various ranges of perturbing field strength near individual grains. The frequency of occurrence of a given phase shift caused by individual grains is inversely proportional to the square of the phase shift, this distribution being a limiting case of the Cauchy form. The resultant distribution of phase shifts from effects of many grains is then still of the Cauchy form. This leads to an exponential signal decay, with the rate independent of diffusion. If $M$ is the algebraic sum of the individual dipole moments of the individual magnetic grains per unit volume, and $\gamma$ the magnetogryic ratio, $\frac{1}{T_2}=\left(\frac{8{\pi }^2}{9\sqrt{3}}\right)M\gamma$ if all grains are magnetized parallel to the precession field; $\frac{1}{T_2}=\left(\frac{4\pi }{3}\right)M\gamma$ if perpendicular. Within 10%, $\frac{1}{T_2}=4.6M\gamma$ for any random or systematic orientation of the grains. Measurements on water containing magnetite powder stabilized by carboxymethylcellulose and on glycerine containing magnetite powder, as well as on sands containing magnetite powder and saturated with water or glycerine, verified the exponential decay, independence of decay rate on diffusion or viscosity, and the above numerical value of decay rate (with small geometrical correction applied to results for the sand system).