the magnetic moment of a nanoparticle is the Langevin function The initial slope of the function is the magnetic susceptibility of the sample : The latter susceptibility is also valid for all temperatures if the easy axes of the nanoparticles are randomly oriented. It can be seen from these equations that large nanoparticles have a larger µ and so a larger susceptibility. This explains why superparamagnetic nanoparticles have a much larger susceptibility than standard paramagnets: they behave exactly as a paramagnet with a huge magnetic moment. Time dependence of the magnetization There is no time-dependence of the magnetization when the nanoparticles are either completely blocked () or completely superparamagnetic (). There is, however, a narrow window around where the measurement time and the relaxation time have comparable magnitude. In this case, a frequency-dependence of the susceptibility can be observed. For a randomly oriented sample, the complex susceptibility is: where is the frequency of the applied field is the susceptibility in the superparamagnetic state is the susceptibility in the blocked state is the relaxation time of the assembly From this frequency-dependent susceptibility, the time-dependence of the magnetization for low-fields can be derived: Measurements A superparamagnetic system can be measured with AC susceptibility measurements, where an applied magnetic field varies in time, and the magnetic response of the system is measured. A superparamagnetic system will show a characteristic frequency dependence: When the frequency is much higher than 1/τN, there will be a different magnetic response than when the frequency is much lower than 1/τN, since in the latter case, but not the former, the ferromagnetic clusters will have time to respond to the field by flipping their magnetization. The precise dependence can be calculated from the Néel–Arrhenius equation, assuming that the neighboring clusters behave independently of one another (if clusters interact, their behavior becomes more complicated). It is also possible to perform magneto-optical AC susceptibility measurements with magneto-optically active superparamagnetic materials such as iron oxide nanoparticles in the visible wavelength range. Effect on hard drives Superparamagnetism sets a limit on the storage density of hard disk drives due to the minimum size of particles that can be used. This limit on areal-density is known as the superparamagnetic limit. Older hard disk technology uses longitudinal recording. It has an estimated limit of 100 to 200 Gbit/in2. Current hard disk technology uses perpendicular recording. drives with densities of approximately 1 Tbit/in2 are available commercially.