Measuring magnetic susceptibility using a balance and LEGO

Jessica Stromer, Tina Curtins, Oskar Leibnitz, Ambra Zaccarelli and Elena Kropf
D-MATL, ETH Zürich

Theory

Methods

Results

Conclusion

Motivation

The goal of the experiment was to construct a device in order to measure magnetic susceptibility and to study the magnetic behaviour when cooling a substance down with liquid nitrogen. Therefore, a laboratory balance and LEGO building blocks were used and its limitation was studied.

Magnetic Susceptibility

Diamagnetic materials repell external magnetic currents, as shown in Fig. 1 (left). Diamagnetism is caused by paired electrons in the orbitals, whereas paramagnetic materials have disordered and unpaired electrons. When an external magnetic field is applied, the unpaired electrons align and are attracted to the magnetic field as seen in Fig. 1 on the right. Magnetic susceptibility measures the response of a material to an external magnetic field.

Fig. 1: Influence of diamagnetism (left) and paramagnetism (right) on an external magnetic field [1].

Magnetic susceptibility \(χ < 0\) is a value that describes the response of a material to an external magnetic field. The substance is diamagnetic if If \(χ < 0\), and paramagnetic if \(χ > 0\).
If the resultant force acts vertically and the sample is connected to a balance, there is a change of weight introduced by the magnetic field. Mathematically, this can be expressed by using the following equation

where g is the gravitational acceleration, \(∆m\) the mass change, χ_M the molar magnetic susceptibility, μ₀ the vacuum permeability, n_M the number of moles, H₁ and H₂ the magnetic field at the bottom resp. at the top of the sample and \(l\) the height from the magnet surface to the bottom of the sample.
If the measurement conditions, e.g the height of the sample, are kept equal during all experiments it is possible to reduce the equation to only two variables, resulting in that the susceptibility is proportional to the change of weight divided by the number of moles of the substance. The value \(β\) contains all the parameters that are constant and can be calculated once the magnetic field and the height are determined [1].

Neodymium Magnet

For the experiment a neodymium magnet was used. Fig. 2 shows the adhesive force diagram of the magnet. The neodymium magnet used has a diameter of 20 mm, a height of 10 mm and a magnetic field of \(1.29 T\) to \(1.32 T\) directly on the surface of the magnet [2]. When looking at the diagram, it makes sense to choose a height l that is above \(2 mm\). If it would be below, minor changes in l would lead to bigger error measurements. At the same time, the height should not be to big, as the measurement of the change in weight ∆m will be very small and lead to bigger errors.

Figure 2: Adhesiv force diagram of the neodymium magnet [2].

Construction

The neodymium magnet was placed on the top of the LEGO construction, leaving about \(2 mm\) to the glass ceiling, where the sample was placed for measurement. When the magnet was attracted to the sample, the balance measured a lower weights. In the case of repulsion, more weight was measured. The final setup is shown in Fig. 3.

Figure 3: Constructed magnetic balance (left) and schematic structure of the magnetic balance (right).

Calibration Results

As seen on the graph, six different substances were used to calibrate the magnetic susceptibility leading to a value of β=-990.8⋅10^-6 cm³/g. Using this calibrated value, we receive a magnetic susceptibility for FeSO₄⋅7 H₂O and MnSO₄ of 12559⋅10^-6 cm³/mol and 24823⋅10^-6 cm³/mol respectively. The literature values are 11200⋅10^-6 cm³/mol and 13660⋅10^-6 cm³/mol respectively. After a month, a sample of FeCl₂⋅4 H₂O with the same weight was newly prepared, and the resulting measurements matched with the previous measurement of that substance.

Figure 4: Calibration of the magnetic balance with six different materials.

Conclusion of the calibrated measurements

The measurement was not very stable, ”flickering” or changing into a direction for several seconds after putting the sample on. It was noticable that movement around the balance would influence the result. A marble table in a non-ventilated room would probably lead to more stable results . The preparation of the sample is also error-prone in that it is possible to have voids in the volume and the final height varied. This would influence the magnetic field H₂, and therefore the resulting weight difference. The re-measurement of FeCl₂⋅4 H₂O confirmed that the LEGO construction seemed to be stable enough to not alter certain parameters like the distance from the neodym magnet to the sample. So as long as the preparation of the sample would be consistent, the calibration data would still be valid for measurement. While FeSO₄⋅7 H₂O was measured within the 95% CI, the measurement of MnSO₄ didn't give a value close to the literature value. The most probable explanation would be a measurement error.

Figure 5: Comparison of the susceptibility values measured using the calibrated magnetic balance with the literature values.

Ferro- & Paramagnetism

A paramagnet reaches maximum magnetisation at absolute zero. Above zero, thermal motion prevents the spins from being fully aligned with the magnetic field. Paramagnetism can also occur above the Curie point. The Curie or Neel point, describes a materialspecific temperature at which a phase transition takes place. The phase transition causes a change from ferromagnetic to paramagnetic properties [3].

Figure 6: The difference between ferromagnetic and paramagnetic behaviour [3].

Cooling MnSO₄

A sample of MnSO₄ was cooled in liquid nitrogen, which has a temperature of approx. \(-200 °C\). After cooling, the sample was placed on the magnetic balance as quickly as possible and the left there until it reached room temperature. Unfortunately, the temperature could not be measured correctly, as there was rime on the surface, which made the infrared temperature measurement impossible.

Figure 7: Cooling with liquide nitrogen [4].

Change In Magnetism Results

The magnetic susceptibility changes over time because the sample is quickly heating up. First, the magnet is attracted more strongly by the sample, which reduces the measured weight. After \(100 s\), the slope decreases and the attraction changes less.

Figure 8: The Y-axis displays the weight difference and the X-axis displays the time. The magnet is attracted by the Sample, wich reduces the mesured weight.

Temperature Dependence

In the plot "Temperature chande and magnetic behaviour" a phase change from ferromagnetic to paramagnetic can be observed between \(20 s\) and \(100 s\). After \(100 s\), it is visible that the paramagnetic properties decrease due to the increasing entropy, which increases due to the temperature.