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Below are some important aspects regarding the magnetic relaxation in nanoparticles and the determination of the threshold volume (V_th) from Eq. (1.29) in dynamic conditions (Caizer 2004a).

However, in dynamic conditions, the measurement time (t_m) (observation duration of the process) relative to the relaxation time is important. Thus, depending on the ratio in which the two times are found there may be the following cases:

1 The case τN ≪ tm, which corresponds to the superparamagnetic state; in this case, the height of the energy barrier is very low, and after the application of a field (or its removal), the magnetization quickly reaches the thermodynamic equilibrium.

2 The case τN ≫ tm, which corresponds to the stable state; in this situation, the height of the energy barrier is very high, and the probability of the magnetic moment passing over the barrier is very low and, therefore, the magnetization does not change over time tm.

3 The case τN ≈ tm (when the two times are of the same order of magnitude), which corresponds to the intermediate state; when the magnetization, on the one hand, does not reach immediate thermodynamic equilibrium, and on the other hand, does not remain in the state of balance (stable) for a long time; this is the case of magnetic relaxation (under dynamic conditions).

Under dynamic conditions, the threshold volume of the nanoparticle can be determined from the condition:

(1.33) (1.33)

Thus, imposing this equality in Eq. (1.29), the threshold volume (magnetic volume [V_mp]) result is

(1.34)

When the relaxation process takes place in an alternating magnetic field (harmonic) with small amplitude, the period of the alternative field (T_H) is considered as the measurement time (t_m = T_H). Under these conditions, the blocking temperature, according to relation (1.29), will be

(1.35)

However, when an alternative (harmonic) high amplitude magnetic field is applied on the nanoparticles system, the measurement time (t_m), threshold volume (V_th), and blocking temperature (T_B) of magnetic moments change. A study on this aspect is presented in Ref. (Caizer 2005b).