# Intermediate and Advanced Labs PHY3802L/PHY4822L

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1 Intermediate and Advanced Labs PHY3802L/PHY4822L Torsional Oscillator and Torque Magnetometry Lab manual and related literature

2 The torsional oscillator and torque magnetometry 1. Purpose Study the torsional oscillator as an example of harmonic oscillator. As an application, use it as torque magnetometer to find the magnetic moment per Nd 2 Fe 14 B formula. 2. Apparatus Torsional Oscillator apparatus from TeachSpin, oscilloscope, multimeter, DC power supply and low frequency source generator. 3. Description of experiment The apparatus allows to apply various types of torques, such as gravitational or magnetic, static, periodic or chaotic, to rotate a mass suspended on a wire with a known torsion constant. In your work, you will apply only magnetic torques by driving an electric current in a pair of Helmholtz coils. The wire passes through the middle of the split coils and has a stack of 4 permanent magnets (Nd 2 Fe 14 B) attached to the wire. The magnetic field of the coils will induce a torque on the magnets (and thus to the wire), generating a torsional deformation. Fig.1 Sketch showing how the field of the split coils generates a torque onto the magnetic moment of the permanent magnets, which are attached to the torsional oscillator (not shown). In the background part of the lab report, give the entire theoretical discussion presented here as well as the error propagation study. Damped harmonic oscillator In the first part of the experiment, you will apply a static magnetic torque and study the dependence between torque and angular displacement due to torsion. When you will suddenly turn off the torque, you will observe oscillations back-and-forth, similar to the motion of an harmonic oscillator. The torsional oscillator has an angular degree of freedom, let s call it θ, which follows the equation of motion of an harmonic oscillator [1]: " + " + " = 0 where is mass, = is the angular acceleration, 0 is a friction constant, = and > 0 is a torsion (spring) constant defining the restoring force. With the notations: damping constant = /(2) and resonance frequency = / the equation of motion becomes + 2" + = 0 with solutions of the form ". After substitution of and solving for, one gets ± = ± 1

3 and = + with ± constants depending on the initial conditions. The square root in imposes three situations: A. Overdamped oscillator >, ± ℝ Write and explain which of the -values describes the exponential decay of back to equilibrium. B. Critical damping = Write. C. Underdamped oscillator >, ± ℂ In this case, ± = ± " with = and = " cos " + sin = " cos " where is the amplitude of the exponentially damped oscillations and is the initial phase of the motion. It is essential to note that the resonance frequency of the torsional oscillator is decreased by damping effects. If, the Taylor expansion of gives = 1 2. As you will find out in your work, when the eddy currents brakes are fully retracted, 2 is of the order of Only in this case, one can consider and you will find and the so-called quality factor of the oscillator = 2. Driven harmonic oscillator In the second part of the experiment, you will apply a small periodic magnetic torque, using an alternating current sin rather than a dc current as in the previous experiment. The equation of motion becomes: + 2" + = sin where and are the drive frequency and its amplitude imposed by the external sinusoidal current generator. The solution in this case shows a transient regime exponentially damped in a time scale 1 followed by a steady-state regime with a solution: = where = "# and tan =. This means that the oscillator will rotate back-and-forth with the same frequency as the external drive. Two essential observations are to be made here. First, while is fixed by the drive, the amplitude of the oscillations strongly depends on the relationship between and : it is very small when " and maximum at the resonance condition,"# = 2 (you can demonstrate this by solving " = 0). Note the difference between,"# and. Only at resonance, the transfer of energy between the driving force and the oscillator is optimal, thus generating a periodical torsion with the largest amplitude. Second observation: there is a phase shift between the drive and oscillator. In this second part of the experiment, you will verify quantitatively the shape of the function and identify the resonance frequency. You will also check qualitatively the variation of 2

6 Part 1 Read carefully [2] to get acquainted with the following reasoning. To get to the correlation between applied torque and torsion angle, let's start with the static equilibrium condition: "#\$%&'( = "#\$%"& or in other words " cos = ". Since the field is directly proportional to the current imposed by the DC power supply, one can write = ". Discuss why = Nm/rad and = mt/a. Assume that these values have no uncertainty. Justify why a plot / cos vs is more appropriate to our study, and not vs. Give the value of θ0. Show the data in a table with columns, "#,, cos, "# where "# is the uncertainty of / cos and is evaluated following the error propagation theory you gave in the background section. Execute the plot / cos vs with the appropriate error bars (like in Fig. 2) and perform a linear interpolation to obtain the slope and its uncertainty. Calculate and its uncertainty. Fig.2 Data (dots) and fit (line) of / cos vs. To calculate the Q-factor, show the screen capture of the decaying oscillations in the case where the brakes were fully retracted. Explain how you got the values for and ; give them with your estimated uncertainties. The Q-factor is defined as = and has the meaning of a ratio "energy stored/energy dissipated by cycle times 2π". In the case studied here, you will substitute with. Solve for from the equation = 1 2 and show that the Q-factor can be expressed as = ". Calculate and its uncertainty, following the error propagation theory. Part 2 Show the data in a table with columns,, where = is the drive frequency. Plot it similarly to the example in Fig.3, with error bars (your plot will have more data points). The red fit is optional, and you can use a data analysis software for such a purpose. Otherwise, use your data to estimate and the oscillator bandwidth ". The bandwidth is defined as the frequency separation between points where the amplitude drops by a factor of 1 2 (and not by 1/2 explain why). The fit would give you a better resolution of and = " 2. 5 Fig. 3 Data (dots) and fit (line) of vs.

7 In Fig. 3, the amplitude shows a quality factor a little bit reduced by the effect of the eddy currents brake, to about ~ The Q-factor is given by =. Calculate Q and its " uncertainty. Show the 3 screen shots and discuss the phase shift between the two channels. An example is shown in Fig. 4 for =0.83, (resonance) and 0.91 Hz. Your frequencies, especially the resonance value, may be different. Use the vertical cursors to show minima (or maxima) on different channels, to be able to calculate the phase shift. Knowing that the two signals are sine waves with the same frequency, calculate the phase shift = for all three situations. What is the range over which should vary when going from 0 to infinity, through the resonance (see the equation of tan )? Within this range, where is at resonance located? Is your data qualitatively in agreement with the theoretical expectations? Can you imagine a reason that could create an additional constant shift between the coil current (or B) and the signal monitoring the angular position? Fig. 4 Example of screen shots using the USB port (data is exported as a table as well). The vertical cursors show successive minima on different channels to show the phase difference. Note the values of for =0.83, and 0.91 Hz, starting from top left, right, bottom. 6

8 Part 3 Use the total magnetic moment ± calculated in Part 1, to find the density of magnetic moment, as explained in [2]. Assume that the volume of the 4 magnets is known precisely (no uncertainty). Finding is already an example of torque magnetometry and in the following you will use crystallographic information to estimate the magnetic moment per formula of Nd 2 Fe 14 B. As described in [3], this crystal has a tetragonal unit cell (shown in Fig. 5) with dimensions =8.78 Å and =12.21 Å. The unit cells contains = 4 formula, that is 17x4=68 atoms per unit cell. Since you know the density of magnetic moment, find the total magnetic moment per formula and its uncertainty. Compare your value with the value measured in [3] at 293K using neutron diffraction. Use Table II in [3] and detail how you calculate their total moment per formula. 6. Additional questions Fig. 5 Unit cell of Nd 2 Fe 14 B, from [4]. 6.1 Both and " are quantities which are specific properties of the oscillator in its environment, independent of any drive. Actually, in the case of freely decaying oscillations in Part 1 (that is, no drive ) what is the meaning of a finite bandwidth; it is the effect of what phenomena? 6.2 Optionally, as a follow-up to 6.1, discuss also the meaning of a curve looking like the one in Fig. 3 but in the case of no external drive. More precisely, how can a decaying oscillation by related to such a curve? 7. References [1] J. R. Taylor, Classical Mechanics, chapter 5, University Science Books, USA (2005). [2] Torsional Oscillator - Instructor Guide by SpinTeach, sections , available online (PDF). [3] J.F. Herbst, J.J. Croat, W.B. Yelon, "Structural and magnetic properties of Nd 2 Fe 14 B", Journal of Applied Physics 57, 4086 (1985). [4] J.F. Herbst, J.J. Croat, F.E. Pinkerton, "Relationships Between Crystal Structure and Magnetic Properties in Nd 2 Fe 14 B", Physical Review B 29, 4176(R) (1984). 7

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