Particle size and its distribution deduced from magnetic properties of magnetite nanoparticles by the modified Langevin equation

The magnetic hysteresis curve can provide information about magnetic properties, particle size, and its distribution. For superparamagnetic materials, the characteristics of the hysteresis curve are very typical, that is, Hc is equal to zero with a certain Ms value related to particle size in the order of nanometers. Here, we report the analysis of the magnetization hysteresis curve by applying a modified Langevin equation with and without a log-normal distribution for nearly superparamagnetic Fe3O4 and Fe3O4 encapsulated by SiO2 (Fe3O4–SiO2). To study the reliability of the modified Langevin equation, we compared the particle size values obtained from fittings analysis of the modified Langevin equation with those obtained from the TEM measurement. It was found that particle size from TEM measurement for Fe3O4 and Fe3O4–SiO2 is 11 and 14 nm, respectively. These values had a very high match with the particle size obtained from fitting analysis of the modified Langevin equation with or without a log-normal distribution, which is 12.6 and 12.9 nm for Fe3O4 and 13.8 and 14.5 nm for Fe3O4–SiO2, respectively. The fitting results of the modified Langevin equation with a log-normal distribution showed a value closer to the measurement results. By using the modified Langevin equation with a log-normal distribution, it was also obtained that the probability density function for Fe3O4 was 20.69% and that for Fe3O4–SiO2 was 0.55%. It was concluded that the modified Langevin equation with and without log-normal distribution was able to be used to obtain the particle size and its distribution for nearly superparamagnetic samples.