temperature. This length of time can be anywhere from a few nanoseconds to years or much longer. In particular, it can be seen that the Néel relaxation time is an exponential function of the grain volume, which explains why the flipping probability becomes rapidly negligible for bulk materials or large nanoparticles. Blocking temperature Let us imagine that the magnetization of a single superparamagnetic nanoparticle is measured and let us define as the measurement time. If , the nanoparticle magnetization will flip several times during the measurement, then the measured magnetization will average to zero. If , the magnetization will not flip during the measurement, so the measured magnetization will be what the instantaneous magnetization was at the beginning of the measurement. In the former case, the nanoparticle will appear to be in the superparamagnetic state whereas in the latter case it will appear to be “blocked” in its initial state. The state of the nanoparticle (superparamagnetic or blocked) depends on the measurement time. A transition between superparamagnetism and blocked state occurs when . In several experiments, the measurement time is kept constant but the temperature is varied, so the transition between superparamagnetism and blocked state is seen as a function of the temperature. The temperature for which is called the blocking temperature: For typical laboratory measurements, the value of the logarithm in the previous equation is in the order of 20–25. Equivalently, blocking temperature is the temperature below which a material shows slow relaxation of magnetization. Effect of a magnetic field When an external magnetic field H 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: 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 (TB < T ≲ KV/(10 kB)), then the magnetization of the assembly is . If all the particles are identical and the temperature is high enough (T ≳ KV/kB), then, irrespective of the orientations of the easy axes: In the above equations: n is the density of nanoparticles in the sample is the magnetic permeability of vacuum is