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A versatile Hall magnetometer has been developed, manufactured, calibrated, and turned operational for measurements of the magnetic properties of bulk materials and magnetic micro- and nanoparticles. The magnetometer was constructed from the combination of various equipments, which was usually available in most laboratories, such as a Hall effect sensor, an electromagnet, a current source, and a linear actuator. The achieved sensitivity to the magnetic moment was approximately 10
^{-8} Am
^{2}. The results were compared to measurements performed with commercial vibrating-sample magnetometers and superconductor quantum interference devices (SQUID) and showed errors of around 1.7% and a standard deviation of 1.2% in relation to measures themselves. The constructed Hall magnetometer records a magnetic hysteresis loop of up to 1.2 T at room temperature. This magnetometer is cost-effective, versatile, and suitable for research.

During the last decade, scientific research on the diverse magnetic properties of bulk materials, particles, microparticles, and nanoparticles has been adequately supported by various new and sophisticated characterization techniques. Properties like the Curie temperature, saturation magnetization, coercivity, magnetic anisotropy, and polarization rotation are some of the most important parameters for magnetic materials that guarantee their applications [

Among the sensing technologies used, the one based on the Hall effect has played an important role owing to its ability to measure dc fields (i.e., constant in time), large frequency bandwidth, non-hysteretic behavior, operation at high magnetic fields, reliability, and low cost. Traditionally, Hall magnetometry has been performed without the use of moving parts [

In this paper, we present a different approach to measuring magnetization curves: a Hall magnetometer built from various equipment that is usually available in most laboratories, such as a Hall effect sensor, a small electromagnet, a current source, a voltmeter, and a linear actuator. One advantage of this magnetometer is its easy assembly and customization. The nucleus of the magnetometer consists of a small acrylic camera, which works as an electric device for the electromagnet poles, and it is built with a path to guide the sample. We used a low- cost GaAs Hall effect sensor as a magnetic sensor. A model that considered the geometry of the sample was developed in order to increase the precision of the obtained magnetic moment. The magnetometer resolution was limited by the employed Hall sensor to 10^{−8} Am^{2} and recorded the magnetic hysteresis loop up to 1 T at room temperature. The device was calibrated independently and its performance was compared to that of commercial VSM (LakeShore, model 7400) and SQUID (Quantum Design, MPMS-XL) magnetometers, showing errors smaller than 1.7% in the magnetization obtained for various samples. All the equipments involved in the operation of magnetometers were controlled using LabVIEW^{®}.

In VSM magnetometers, the sample vibrates at a specific position, in contrast with this type of magnetometer. The Hall magnetometer must move the sample during the measurement; for this purpose, it requires a track from the sample to move in constant form, and the reading is simultaneously acquired by the Hall effect sensor. The configuration of the magnetometer is illustrated schematically in Figures 1(a)-(d).

Thus, the first restriction is to ensure the homogeneity of the magnetic field applied by the electromagnet (GMW 3470). In order to achieve at least 0.1% homogeneity, the employed electromagnet (with poles of 40 mm in diameter) has a distance of 5.3 mm between the poles to allow the displacement of the sample, which is approximately mm from the center of the poles along its diameter (see

A Hall effect sensor (model HG-166A) with dimensions 2.5 mm × 1.5 mm × 0.7 mm is located on the holder with its plane perpendicular to the shaft fastening. According to the direction z shown in

The Hall sensor has a magnetic field resolution of 600 nT and an active area of approximately 1.26 10^{−7} m^{2} [

To optimize the Hall sensor for readings of the magnetic field induced in the sample, we must determine the feeding current sensor. We conducted a study of the behavior of the signal noise of the sensor for various values of applied current (

Tests were performed in the presence of magnetic fields. The results of one of these tests (Hall sensor measurement in the presence of a magnetic field of 335 mT), displayed in

A current of 5 mA was then applied to the Hall sensor using a programmable current source (Keithley, model 6220) for the reading of the detected magnetic field. This source has a precision of 0.05% and a resolution of 100 nA in the range from 2 mA to 10 mA [

Therefore, the reading system consisted of a current source and a voltmeter that were integrated in a system called Delta Mode and communicated with each other using an RS-232 connection and a trigger signal.

The current source applied a square wave of pre-defined amplitude (±5 mA) to the sensor with a frequency of 100 Hz and provided a nanovoltmeter with the timing information of the signal through the trigger. After reading the signals from several consecutive cycles, the nanovoltmeter calculated the mean and communicated the

result to the current source through the RS-232 connection, which sent this information to the computer via GPIB.

Although the VSM magnetometer can also be used for absolute measurements, in practice, it is typically implemented for calibrations through the comparison with the magnetization patterns measured for a nickel sphere in a high field. The SQUID magnetometer usually needs to be calibrated by the final user [

In the constructed Hall magnetometer, we used 99% pure nickel spheres with a 3 mm diameter in order to perform the calibration independently from the other magnetometers. The magnetization results were compared to values obtained using the same spheres in the two commercial magnetometers. In this case, we used a magnetic dipole model placed at the center of the sphere, which coincided with the center of the sample holder. Before obtaining the value of magnetization, we used the spatial dependency of the magnetic dipole model to determine the distance between the center of the sample holder and the center of the Hall sensor. According to

The proposed model for the calibration of the nickel sphere was a magnetic dipole located at the origin of the system of coordinates, with a magnetic moment m_{x} pointing to the positive direction of the x axis. The center of the sensor was at the position (x, y, z) = xi, yj, zk.

The equation that represents the magnetic field produced by the point magnetic dipole is:

Because the magnetic moment was oriented in the direction of the x axis: m = m_{x} i. Simplifying Equation (1), the equation used for the construction of this model is:

The magnetic flux from the z component of the magnetic field of a dipole through an area π² is given by [

where x, y, z represent the coordinates of the sensor in relation to the dipole. The dipole is located exactly in the region between the poles of the electromagnet, thus defining the sphere to be analyzed.

This sphere is displaced in the y direction (negative), the greatest intensity of the magnetic field occurs when the sample passes by the sensor, out of its center. According to Equation (2), the unknown values are x, y, z, and m_{x}. The value of y_{max} can be determined through the region of maximum intensity of the magnetic field read by the sensor.

We can substitute the integral calculation in Equation (3) using the reciprocity principle, which estimates the

where _{r}, which can be written in terms of complete elliptic integrals of the first and second kind (K and E, respectively) [

where _{s} represents the radius of the fictitious coil,

Because only m_{x} differs from zero, we can calculate B_{x} as:

The magnetic flux of the x-component of the magnetic field of a dipole through an area is given by:

The calibration process of the Hall magnetometer begins by obtaining the distance from the center of the sample holder to the center of the Hall sensor. These values are subsequently used to determine the desired magnetic moment. The values of x_{0} and z_{0} used in the dipole model (Equation (6)) represent the horizontal and vertical distance, respectively, from the center of the sensor to the center of the sample.

To determine x_{0} and z_{0}, we used a sample holder with two cylindrical cavities, where we two nickel spheres of equal mass (126 × 10^{−6} kg) were placed. These were positioned at a distance of 22.5 mm in relation to the other (minimum distance required to avoid influencing each other’s induced magnetic fields, see _{0} and z_{0}, in relation to the center of the sensor.

Since the examined magnetization value is independent from the spatial dependence of the field induced in the spheres, we can normalize the measured magnetic field and only consider the relative distances between the spheres and the sensor. If we measure precisely the values of dz_{1}, dz_{2,} and dx_{2}, we have a system of two equa-

tions, corresponding to the two measures, and two variables, the distances x_{0} and z_{0}.

Analyzing this sample holder in an optical microscope we can determine the values of dz_{1} and dx_{2} (Figures 4(a)-(d)). Once the relative distances are obtained, we denormalize the measures and obtain the desired magnetic moment. In this study, we repeated this procedure many times and obtained the average values x_{0} = 1.63 mm and z_{0} = 3.02 mm. It must be noted that these values were obtained considering the flux through the area of the sensor.

In the literature, the saturation magnetization of nickel occurs approximately at 0.5 T and has the value 55.18 Am^{2}/kg at room temperature [

For the maximum field, 1 T, the Hall magnetometer obtained a magnetization of 55.38 Am^{2}/kg. For the same field, the VSM device measured 54.94 Am^{2}/kg and the SQUID magnetometer 54.42 Am^{2}/kg.

The difference in the magnetization curves was 0.8% and 1.7% in relation to the values obtained by the VSM and the SQUID magnetometers, respectively.

It is worth noting that our result was obtained independently from the measurements with the other magnetometers. Regarding the precision of our magnetometer, after repeated many times measurements performed during several months, we noticed that the standard deviation was 1.2%.

The automation of the Hall magnetometer was achieved using a LABVIEW program (

After the sample holder calibration, we removed the nickel sphere and placed the magnetic material in the cylindrical cavity located at the end of the sample holder.

We can observe in

According to Equation (7), the B_{z} component for a cylinder of length L is:

After establishing the geometric parameters, we need to determine the magnetic moment m of Equation (8), which adjusts the model to the experimental measurements. Using the data acquisition program developed with LabVIEW^{®} (

generates an induced field curve versus distance traveled, where each curve was optimized by the cylindrical model. By adjusting each of these curves, we can obtain the magnetic moment of the sample for each applied magnetic field.

As an example of the application of the cylindrical model, ^{−5} Am^{2}.

Nanoparticles are important tools used in medicine, both in diagnosis and for the treatment of various diseases. Their sizes can be controlled, varying from tens up to hundreds of nanometers, making their dimensions smaller or comparable to those of cells, bacteria, or viruses.

In

magnetization values obtained with the Hall magnetometer for fields above 1 T, we traced a curve M x 1/H. Extrapolating to 1/H equals zero, we estimated the saturation magnetization of cobalt ferrite to be 6 Am^{2}/kg. By analyzing the data, we found a remanent magnetization of approximately 0.29 Am^{2}/kg and a coercivity near 5.5 mT. The lowest magnetic moment detected in the cobalt ferrite was 3.74 × 10^{−6} Am^{2}, which is the lowest measured value to date. The same sample was analyzed with an atomic force microscope from Puc-Rio. In the upper corner of

We manufactured magnetic nanoparticles of magnetite using the co-precipitation method. This method is quick and versatile and allows the fabrication of a large variety of magnetic nanoparticles with a great number of advantages, such as short time of reaction, particles with small agglomeration, low cost, and large quantities. The technique allows us to control the size and distribution of the obtained nanoparticles by varying parameters like the pH [

Samples of these nanoparticles were prepared from magnetite coated with silica using the Stöber method. The Stöber method is a coating process that involves hydrolysis and condensation of metal alkoxides and inorganic salts [

According to the results shown in ^{2}/kg and a coercivity of 0.6 mT.

We designed, manufactured, and calibrated a magnetometer using autonomous and low-cost instruments available in laboratories and a low-cost GaAs Hall effect sensor. The magnetometer was calibrated independently using two nickel spheres of 99% purity. Its performance was compared to that of commercial VSM and SQUID magnetometers, exhibiting magnetization errors smaller than 1.7%. Owing to the proximity of the Hall sensor to the sample, which had a cylindrical geometry, we used a model of a current cylinder to obtain its magnetic moment. The magnetometer proved to be effective for the characterization of magnetic nanoparticles, obtaining complete magnetization curves and reaching sensitivity of the order of 10^{−8} Am^{2}, depending on the employed sensor. It was also demonstrated to measure the magnetostriction and magneto-crystalline anisotropy of magnetic materials, not only particles but also in the form of bulk. This configuration is cost-effective but versa-

tile and will be adequate for research and educational purposes.

This work was financed in part by the Brazilian agencies CNPq and FAPERJ. We would like to thank Professors R. L. Sommer from CBPF, N. Massalani, and M. A. Novak from UFRJ for the measurements with the VSM and SQUID magnetometers. We would like to thank Professor Antonio Carlos Bruno for lengthy discussions on the Hall magnetometer.

Jefferson F. D. F.Araujo,11,Joao M. B.Pereira, (2015) A Practical and Automated Hall Magnetometer for Characterization of Magnetic Materials. Modern Instrumentation,04,43-53. doi: 10.4236/mi.2015.44005