Superparamagnetism - Effect of A Magnetic Field

Effect of A Magnetic Field

When an external magnetic field is applied to an assembly of superparamagnetic nanoparticles, their magnetic moments tend to align along the applied field, leading to a net magnetization. The magnetization curve of the assembly, i.e. the magnetization as a function of the applied field, is a reversible S-shaped increasing function. This function is quite complicated but for some simple cases:

1. If all the particles are identical (same energy barrier and same magnetic moment), their easy axes are all oriented parallel to the applied field and the temperature is low enough (T_B < T ≲ KV/(10 k_B)), then the magnetization of the assembly is
   .
2. If all the particles are identical and the temperature is high enough (T ≳ KV/k_B), then, irrespective of the orientations of the easy axes:
   

In the above equations:

• n in the density of nanoparticles in the sample
• is the magnetic permeability of vacuum
• is the magnetic moment of a nanoparticle
• is the Langevin function

The initial slope of the function is the magnetic susceptibility of the sample :

in the first case

in the second case.

The later 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.