## 3 Magnetic Characterization Instruments
### Fluxgate Magnetometers
In the case of the single fluxgate arrangement, where the sensitive axis of the magnetometer has an orientation perpendicular to the magnetic field, the greatest coupling of a magnetic dipole to the fluxgate signal is achieved, in the case that the dipole is located at one of the rod-shaped core ends [42]. One drawback of this configuration is that only the nanoparticles close to the fluxgate sensor contribute significantly to the signal as a consequence of the \(1/\tau^{3}\) decay of the magnetic field of a magnetic dipole [42]. In the case of differential fluxgate arrangement, the sample is located with equal distances between the two magnetometers. The sensitive axes of each magnetometer are placed perpendicular to the magnetic field and consequently to the magnetic dipole axis. For this reason, in the case that the sample is absent, the magnetometer will register zero magnetic field. The total signal is the difference of the two signals. Consquently, the fields due to the environment are canceling each other, and the fields due to sample are constructivelly added, resulting in a total signal two times the signal that the single configuration provides. The topology of Figure 5b enables measurements in an unshielded environment, and the total signal is twice the signal of the single topology. Thus, the signal to noise ratio (SNR) is increased by a factor of \(\surd{2}\) compared to the values of the topology of Figure 5a [42]. Compared to a SQUID-based system, the system based on a differential fluxgate gradiometer is more economical and with lower weight, providing a simpler measurement setup. A main drawback of the second arrangement is the need for two complete electronic sets since there are two magnetometers with their respective output signals [42]. Table 2 summarizes all the aforementioned pro and cons for the two different fluxgate magnetometer setups. The most important studies that use the fluxgate magnetometer are summarized below. More specifically, in [19], two fluxgate magnetometers, in a gradiometer arrangement, have been used, for Fe\({}_{3}\)O\({}_{4}\)-based nanoparticle sample characterization, while the magnetization of samples is provided using Helmholtz coils. In [8] Fe\({}_{3}\)O\({}_{4}\) nanoparticles are also characterized using fluxgate magnetorelxometry, and the findings are compared with those provided by AC susceptibility, photon correlation, and transmission electron microscopy methods. In [26], a fluxgate magnetometer set up for cobalt-based nanoparticle characterization is proposed. The method is proven ideal for materials with high magnetic moment, while both the bed of the catalyst and peripherals, for temperature sensing or probes that monitor the reaction, can be successfully included in the measurement system. In [27], the behavior of magnetic nanoparticles in a rotating magnetic field is studied, using a setup based on fluxgate magnetometers. The configuration of the system is depicted in Figure 6. The investigation aims to detect the phase lag between the magnetization of the sample and the rotating magnetic field. This is also the novelty of the article, as the majority of the existing works on this topic focus on the magneto-mechanical coupling and not on the phase lag. In [42], the magnetorelxometry method has been applied for nanoparticle properties detection, while two magnetometers in differential fluxgate topology have been utilized. Moreover, Helmholtz coils are used instead of field coils, which can produce a homogeneous magnetic field in large samples. In [45], a fluxgate scanner that uses the magnetic relaxation imaging method is presented for the characterization of cell culture bag nanoparticles of immune cells. The schematic diagram of the setup is depicted in Figure 7. In [47], fluxgate magnetometers are used for nanoparticle characterization with the magnetorelxometry method and the results are compared with those of a SQUID instead of the fluxgate magnetometer. In [110], a fundamental mode orthogonal fluxgate magnetometer is