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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).

      1.1.8.2 The Heating of Magnetic Nanoparticles in an Alternating Magnetic Field

      Due to the reduced dimension at nanoscale of magnetic materials, another very important aspect from a practical point of view is the fact that in an alternating harmonic magnetic field, the nanoparticles heat up (Pankhurst et al. 2003) due to the superparamagnetic relaxation processes that take place in nanoparticles up to 20–25 nm (in the case of soft magnetic materials). This effect is used in magnetic hyperthermia (MHT) as an alternative method for tumor therapy, a matter of great interest in current research.

      The power dissipated in such a process is given by the following relationship (Rosensweig 2002):

      (1.36)

      where H and f are the amplitude and frequency of the harmonic alternative magnetic field, tau the magnetic relaxation time, and h₀ is the static magnetic susceptibility. For the usual magnetic fields used in magnetic hyperthermia (until several tens of kA m⁻¹), which generally exceed the linear range of the magnetization variation (Figure 1.16a), the magnetic susceptibility χ₀ will no longer be given by the initial susceptibility (χ_i), but by the following:

(1.37)

because the Langevin variation (Langevin 1905) of the magnetization with the magnetic field must be taken into account (Jacobs and Bean 1963). In Eq. (1.37), ξ is the Langevin parameter, which in the case of nanoparticles is

      (1.38)

      In the case of a monodisperse nanoparticles system, under adiabatic conditions, the specific absorption rate (SAR) or specific loss power (SLP) will be

      (1.39)

      where ρ is the density of the nanoparticle material. Thus, the heating rate (∆T/∆t) of the biological tissue is

      (1.40)

      In the given formula, c is the specific heat of the environment.

These are very important observables used in magnetic hyperthermia in order to quantitatively establish its effectiveness in the thermal destruction of tumors. Figure 1.20 shows such a variation of SLP calculated in the case of γ‐Fe₂O₃ monodispers nanoparticles system for 10 kA m⁻¹ magnetic field (H) and frequency (f) in the range (100–500) kHz (Caizer 2010), from which it results that a maximum power can be obtained only at a certain size of the nanoparticle diameter. This is a very important aspect for the practical application of magnetic hyperthermia. Thus, it results that the size of the nanoparticles is very important in magnetic hyperthermia, the diameter of the nanoparticles being a critical parameter. Outside the critical diameter, the dissipating power decreases rapidly to zero, which shows the ineffectiveness of the method at values greater or less than the critical diameter.

Figure 1.20 SLP for γ‐Fe₂O₃ nanoparticles.

      Source: Caizer (2010). West University Publishing.

      In the case of real magnetic nanoparticle systems, the polydispersity of the nanoparticles must also be taken into account (taking into account in this case a suitable distribution function) (Bacri et al. 1986; O’Grady and Bradbury 1994; Baker et al. 2006), their magnetic packing fraction (Caizer 2003a) as well as the dipolar magnetic interactions between them (Caizer 2008) or even the nonlinearity of magnetization at high amplitudes of the magnetic field (Déjardin et al. 2020). These and other aspects of magnetic hyperthermia are presented in detail in the other chapters of this book.

1.2 Magnetic Nanoparticles as a New Tool for Biomedical Applications

      1.2.1 Magnetic Nanoparticles for Diagnosis and Detection of Diseases

Magnetic NP's detection and diagnosis strategies have recently received considerable attention. Magnetic nanoparticles combined with other diagnostic systems offer exclusive advantages over conventional methods. Specifically, deep tissue imaging magnetic nanomaterials are used because a biological sample does not exhibit magnetic background so it has become a point‐of‐care diagnostics as well. The magnetic NPs are widely used in many applications (Figure 1.21). Magnetic Resonance Imaging (MRI) is a noninvasive imaging modality that has been widely used in clinical diagnosis (Sun

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