Magnetic moment measurements Davíð Örn Þorsteinsson (1), Guðjón Henning Hilmarsson (2) and Saga Huld Helgadóttir (3) 1) dth21@hi.is 2) ghh9@hi.is and 3) shh14@hi.is Abstract: A uniform magnetic field makes a magnetic dipole experience a net magnetic torque. By measuring precessional motion of a rotating body and period of an oscillating pendulum in a varying uniform magnetic field, the magnetic dipole can be obtained. Using a superconducting magnet, with VSM, measurements of an unknown sample are performed. The magnetic moment for different orientations of the magnetic field on the sample is measured and from that the sample s composition is determined. Introduction Magnetic moment Because a magnetic dipole in a uniform magnetic field experiences a net magnetic torque, the dipole s magnetic moment can be determined. This knowledge is exploited in two experiments. In both experiments, the objective is to measure the dipole moment of a magnet inside a cue ball, but with different methods. In the first one, the torque is used to cause a rotating cue ball to precess. Then, in the second one, the torque is made to act as a restoring torque on a physical pendulum, i.e. a cue ball. The uniform magnetic field is acquired by putting a current through wires of Helmholtz coils. By knowing relations between precessional angular velocity, Ω p, period of oscillation, T, and magnetic field strength gives one the ability to calculate the magnetic moment by measuring Ω p and T in varying uniform magnetic fields. Theory The central principle of this experiment is that when the magnetic moment is displaced some angle from the direction of the magnetic field, the magnetic dipole experiences a torque that causes a change in the ball s angular momentum in the direction of the torque. The ball is displaced from the vertical position and spun, its spin-axis is the axis that runs through the handle of the ball. This creates a large spin angular momentum. The spin axis will remain in a fixed position until the uniform magnetic field is turned on. When it is turned on the magnetic dipole will experience a torque which will cause a change in angular momentum in the direction of the torque. But because the ball already has a large spin angular momentum it will change the rotational axis of the ball. The differential equation for the motion of the ball is µ B = L t (1) The relation between Ω p, the precessional angular velocity, and µ, the magnetic moment, is Ω p = µ L B (2) The precessional frequency is the dependent variable. It can be determined, in radians/second, by measuring the time needed for the handle of the ball to precess through 2π radians and then dividing that time by 2π. The magnetic field is the independent variable. The magnitude of the angular momentum L is a constant that can be measured using a strobe light. The handle of the ball has a white dot on its top. As the ball spins the strobe light reflects off of this white dot. When the strobe light is flashing at the same frequency at which the white dot is spinning, the dot will appear stationary. Then from the displayed strobe frequency and the measurement of the moment of inertia, the spin angular momentum of the ball at that time can be calculated. The graph of Ω p as a function of B will give a straight line if L is held constant throughout the experiment. From the slope of this line, µ can be determined. The magnetic field, B, is generated by introducing current through Helmholtz coils. While spinning the cue ball to the correct angular velocity, a field gradient is turned on at the center of the coils to keep the ball from starting its precessional motion. When ready, the field gradient is then turned off which gives a uniform magnetic field and the ball s precession starts and its period can be measured. Fig. 1: Cue ball with spin angular momentum about its handle s axis This experiment involves dynamics principles. From classical mechanics it is known that a net torque on an object causes a change in that object s angular momentum. For the system in this
experiment, if the cue ball is placed in the air bearing with a uniform magnetic field present, and if the intrinsic dipole moment of the ball is displaced some angle away from the direction of the magnetic field, the ball will experience a net torque and will change its angular momentum. It is important to note the direction of the magnetic moment relative to the magnetic field. If the magnetic moment in the ball is displaced by an angle θ from the axis of the coils, in the direction of the field, it experiences a restoring torque that acts against the angular displacement of µ. So the differential equation that describes the motion of the ball with moment of inertia I is µ B = I 2 Θ 2 t (3) where θ is the angular displacement from the direction of B. The minus sign indicates that the torque is restoring in nature. For small angle displacements sinθ θ and in scalar form µbθ = I 2 Θ 2 t (4) The solulution of this equation is Θ(t) = Acosωt, where ω and A are constants. Plugging the value of Θ into eq. 3 gives µbacosωt = IAω 2 cosωt (5) For this to be true for all times t, the following relation needs to be fulfilled, ω 2 = µ I B (6) where ω is the angular frequency of oscillation. If T is the period of oscillation T = 2π ω then the final equation for small angles is : T 2 = 4π2 I µb (7) (8) I can be approximated as the moment of inertia of a uniform solid sphere I = 2 5 MR2 (9) where M is the mass of the ball and R is the ball s radius. B is the independent variable and T can be measured using a stopwatch. From the graph of T 2 as a function of 1/B it is possible to get a straight line whose slope includes the magnetic moment, µ. Fig. 2: Vertical angular displacement of cue ball Experimental First the constants are measured, the radius, the mass and the spin angular momentum of the ball. A constant value of the spin angular momentum of the ball is accomplished by fixing the frequency of the strobe light. The strobe light is set to frequency 5 Hz because in the range of frequency between 4.5 and 6 Hz the rotational frequency does not change significantly during the time it takes the ball to precess through one period. The ball is spun and adjusted so its handle is bathed in the strobe light. The white dot on the handle is observed and as the ball slows down, the white dot s rotation slows down until it appears stationary under the strobe light. At that moment the field gradient is turned off and a magnetic field is introduced to the system. This results in a precessional movement of the ball and its period is measured. Measurements are carried out from 1 A to 4 A with 0.5 A intervals. First the moment of inertia of the cue ball is determined using its mass and its radius. The mass can be determined using a scale and the radius by using the calipers. In this experiment, the field gradient and strobe light should be off, the air on and the field s direction up. Because in this experiment the magnetic torque is the only torque involved, the experiment can be performed at low currents, which results in a small magnetic field, B. The cue ball is placed on the air bearing and the current set at 1 A. Then handle of the ball is given a small angular displacement from the vertical. The ball is released from rest and starts to oscillate. The amount of time it takes the ball to complete 20 full cycles of motion is measured with a stopwatch. This measured time divided by 20 will be period of oscillation for the ball at that particular magnetic field. This is repeated for currents up to 3 A with 0.5 A intervals. The small angular displacement of the ball s han-
Fig. 3: a) Magnetic torque apparatus, b) Helmholtz coils, c) cue ball, d) air bearing, e) strobe light dle, from vertical, is 15 20. Instruments used to conduct these experiment were: Magnet, power supply, air bearing, Helmholtz coils, strobe light, cue ball, stopwatch, calipers and scale. The strobe light was only used in the precessional motion part of the experiment. The schematic setup can be seen in fig. (3). Calculations The radius of the ball is measured as R = 2.650 ± 0.005 cm and the mass M = 140 ± 2 g. The moment of inertia of the ball is assumed to be that of a solid sphere. It is calculated, with eq. (9), as I = (3.933 ± 0.002) 10 5 kg m 2. The angular velocity, ω = 31.4 rad/s is calculated from the frequency, f, with the well known relation The angular momentum ω = 2πf (10) L = Iω (11) is then L = (12.3 ± 0.1) 10 4 kg m 2 /s. From the precessional period, t, the precessional frequency, Ω p is calculated with Ω p = 2π t (12) Fig. 4: Ω p as a function of B same as mentioned above. The slope of the best fit line is then 4π 2 I/µ, and since I is known, calculating the magnetic moment is easy. The slope is h h = 0.004 ± 0.001 so the magnetic moment is µ = 0.4 ± 0.1 J T Conclusion The value of the dipole s magnetic moment acquired in these two experiments is essentially the same. They overlap each others uncertainty which is a very satisfying result. The magnetic moment value given by the manufacturer is 0.4 J/T so these results are more than acceptable [2]. The precessional period could have been measured more precisely by videotaping the cue ball s motion and determining the period from the video. It might also have been better to spin the ball more slowly to make it easier to see if the ball s rotation was syncronized with the strobe light. The pendulum period could also have been videotaped for more precise measurements. With better equipment the vertical angular displacement could have Then, Ω p is plotted as a function of the magnetic field, B. The magnetic field is related to the applied curent by the value 0.00137 T/A. From eq. (2) it shows that the slope of the best fit line is µ/l. Knowing the value of L, one can then calculate the magnetic moment, µ. From fig. (4), the slope of the best fit line is h p = 280 ± 70, which gives, by multiplying with L, the magnetic moment µ = 0.34 ± 0.08 J T By measuring the oscillation period of the pendulum, T, and plotting T 2 as a function of 1/B, it is possible to determine the magnetic moment from eq. (8). This is shown in fig. (5). The relationship between the magnetic field and current is the Fig. 5: T 2 as a function of 1/B
been performed with more accuracy. VSM Measurements Using a Vibrating sample magnetometer(vsm) it is possible to identify an unknown metal sample. A VSM is an instrument that measures magnetic properties of a sample, which is placed in a uniform magnetic field and vibrated. The sample is a very thin sheet of metal and its magnetic moment is measured by vibrations along the sample s easy, hard and perpendicular axis. By plotting the magnetic moment as a function of the magnetic field strength for all the axes it is possible to determine the sample s magnetic saturation and from that the metal can be identified. A simple schematic can be seen in fig. 6. The magnetic sheet has a triaxial anisotropy and therefore has an easy axis, hard axis and a perpendicular axis. The easy axis is the direction which the magnetization wants to head in order to minimize the energy and is parallel to the direction in which the sample was grown. The hard axis is the direction of maximum energy and is perpendicular to the sample s growth direction. The perpendicular axis is then perpendicular to both the hard and easy axes. The sample is a circular sheet. Its thickness is 30 nm and its diameter is 5 ± 0.5 mm. Volume of the sample is then Table 1: Magnetic saturation from different axes Axis M s Perpendicular axis 580 ± 150 emu/cm 3 Hard axis 520 ± 130 emu/cm 3 Easy axis 480 ± 100 emu/cm 3 Table 2: List of sample s potential identity and magnetic saturation Metal M s Fe 1600 1700 emu/cm 3 Co 1200 1400 emu/cm 3 Ni 500 600 emu/cm 3 Ni 80 Fe 20 700 800 emu/cm 3 2. From the information gathered it is clear that the sample s identity is Nickel. M s values found from all the axes intersect each others uncertainty and are all within the boundaries of the given value of M s for Ni. V = (6 ± 1) 10 7 cm 3 The magnetic saturation, M s, can be determined from the hysteresis graphs of the easy and hard axes, but a different method is nescessary for the perpendicular axis. Best fit lines are drawn through the measurements and their intersections determine the saturation. From the graphs, figs. 7 9, the magnetic saturation is calculated and the results are listed in table 1. A list of potential identity and the magnetic saturation of the given sample was handed out and can be seen in table Fig. 7: Easy axis Fig. 6: VSM schematic [3] Fig. 8: Hard axis
Fig. 9: Perpendicular axis References [1] teachspin.com, Magnetic force balance, Magnetic torque [2] http://teachspin.com/brochures/magnetic%- 20Torque%20.pdf [3] http://www.el.utwente.nl/tdm/istg/research/- vsm/thema39.gif [4] http://en.wikipedia.org/wiki/magnetic_anisotropy