Small Quartz Tuning Forks as Potential Magnetometers at Room Temperature. Peter Lunts*, Daniel M. Pajerowski, Eric L. Danielson

Transcription

1 Small Quartz Tuning Forks as Potential Magnetometers at Room Temperature Peter Lunts*, Daniel M. Pajerowski, Eric L. Danielson Department of Physics, University of Florida, Gainesville, FL July 25, 28 Abstract Small quartz tuning forks with resonance at 32 khz are tested to observe their capability as magnetometers of nanoparticles at room temperature. The forks are characterized and compared to each other both in and outside of their initial packaging. They are sprinkled with magnetic particles and placed in a magnetic field. The full quantitative analysis is not completed, but a shift in the height of the magnitude of the signal is observed for a fork immersed at different distances into a magnet. This shift is approximately.6 mv, which is about 3 times the usual deviation for the setup. The conclusion is that the tuning fork method of a magnetometer is promising, but more analysis needs to be done. *Department of Physics, Indiana University, Bloomington, IN

2 Introduction In view of the increased use of nanotechnology, a growing need exists for more precise measurements of the properties of the materials employed in these novel devices. Physicists use various technologies for these measurements, but cheap, accessible, and relatively painless new techniques are always favored. A possible way to measure the magnetic properties is to use a technique that takes advantage of the high sensitivity of small quartz tuning forks [1]. These tiny ( mm in length and about 1 mm in diameter) devices used in wrist watches are very inexpensive (averaging on the order of $. - $.2) and can be ordered in large quantities. They can be piezoelectrically excited for long periods of time (they run in watches for years), which allows for a long, low energycost measurement (as opposed to a SQUID Magnetometer, for example, which uses liquid helium an expensive fluid). In our attempt to exploit this technique we used only a limited set of tools. We used only one type of fork. This invariance would mean that the resonant frequency and the geometry and size of all the forks used are the same (to within the manufacturing variance). If the reader wished to further research this subject, we would recommend trying forks with different resonant frequencies and, in particular, different sizes for the possibility of greater sensitivity. Also, we used only one setup, in which all of the instruments stayed the same, along with most of their settings (see Measurements and Fig. 1(b) for details). Our analysis is mainly qualitative, not quantitative. Although we show results in the form of graphs, we provide very few numbers. This incomplete analysis is due to lack of time, and we encourage the reader to carry out the numerical analysis. 2

3 The Fork and the Setup The quartz tuning fork is a simple harmonic oscillator (SHO) and so, when it is excited by a sinusoidal wave, it fits the equation of the driven SHO: 2 ' t mx ' + γ x' + mω x = F cos( ω ), (1) whereγ is the damping coefficient, ω is the resonant frequency, and the right hand side is the driving function [2]. The solution to this equation, according to [3], has an amplitude of / 2 / m) /( γ ω / + ( ω ) ) R = ( F m ω (2) and a phase (relative to driving function) of 2 2 θ = arctan( ωγ / m /( ω ω )). (3) These are the polar coordinates of the solution. One can also give the X and Y components [4], also called the real and imaginary or the in-phase and out-of-phase components: X = ( F / m)( ω ω ) /( γ ω / m + ( ω ω 2 2 ) ), (4) and Y = ( F / m)( γω / m) /( γ ω / m + ( ω ω 2 2 ) ). (5) The original resonance frequency of the system, when it is unforced and undamped, isω. But in our case the frequency of resonance becomes, according to [3], ω / 1 = ( ω γ / 2m ) 2, (6) so the higher the damping coefficient γ and/or the larger the mass m of the fork the smaller ω 1. Also, with increase inγ there is a decrease in R, as can be seen in Eq. 1. When the forks are originally purchased, they are in a low-pressure can, which means 3

4 that γ is relatively low. When the fork is taken out of the can and operated in air, γ increases slightly. When grease and magnetic particles are added onto the fork and it is immersed into a magnetic field (see Measurements ), an even greater increase occurs in γ because the force on the magnetic particles by the field produces a drag. We show in Measurements that this increase in γ is what we observed. The setup uses a lock-in amplifier and a function generator (see Fig. 1). The latter produces AC current in the form of a sine wave, which excites the fork by the piezoelectric effect, and the resulting current, produced by the mechanical vibrations of the fork, travels to the input of the lock-in amplifier. The input frequency, at which the mechanical vibrations produced have the greatest amplitude, is at resonance. So, the function generator sweeps through a range of frequencies, while each out-coming wave is measured by the lock-in. The lock-in measures the X and Y components of the signal, which can then be converted into the magnitude and phase of the signal by the relations described earlier. This sweep will then have produced 4 graphs: X and Y vs. frequency, and R and theta vs. frequency (see Fig. 2). As stated before, the frequency at which the amplitude R peaks is at resonance, but the same is true for the Y channel. Either pair of graphs characterizes the oscillation of the fork completely, so we reserve the right to provide either one or the other for the forks tested here. 4

5 Function Generator Fork A Lock-in Amplifier REF REF IEEE Control USB Computer (a) (b) Figure 1: (a) Schematic of the setup. (b) The function generator on top of the lock-in amplifier. The program Labview, which takes the data, is brought up on the screen. The following settings were used for the measurements. Sensitivity: or 2 mv. Dynamic Resolution: Normal. No offset for either channel. Pre- Time Constant: ms. Post- Time Constant: None. 5

6 (a) 15 X channel Y channel 25 X channel (mv) Y channel (mv) (b) magnitude of signal (mv) magnitude phase phase (radians) Hz Figure 2: (a) X and Y channels vs. frequency; (b) magnitude of signal and phase vs. frequency. Fork #11 in its can; time between data points is 2.5 s; 13 Hz / 4 pts; resonance is at Hz (±.2 Hz); connecting lines added as visual aid. The outlier point of the phase in between and is what s called wrapping of the phase and occurs because the lock-in amplifier at that point can t decide whether it wants to label the phase π /2 or π /2. 6

7 Measurements The first problem we encountered was the length of time over which data were taken. Once the function generator switches to a new frequency, the lock-in amplifier needs a certain amount of time to lock in to the new signal being fed to it (usually about 3-5 cycles of wave). So, if the time with which we ask the function generator to change frequencies is too small, then the data that we collect will be invalid. For our scans with the fork taken out of the can, a time of 25 ms was sufficient. However, when the forks are still in the can, their Q is extremely high [5] (on the order of 4 4 time sweep of 25 ms is not enough, since around resonance the fork has such strong vibrations that more time is needed to find the exact peak of oscillations (see Fig. 3). ), and a X channel (mv) Y channel (mv) Figure 3: Fork #8 in its can; X and Y channels vs. frequency; time between data points is 25 ms, 3 Hz / 25 pts; too high of a Q gives this shape; connecting lines as visual aid. Moving the time sweep to 2.5 s gives the results of Fig. 2, and going to 3 s and 6 s gives the results of Fig. 4. These graphs are not identical, but the resonance for them all is 7

8 15 3s 6s 3s 6s X channel (mv) Y channel (mv) Figure 4: Fork #11 in its can; X and Y channels vs. frequency; time between data points is 3 s and 6 s; resonance frequencies are at Hz ±.1 Hz for both. within ±.1Hz, where the error is determined by approximately half the distance between neighboring data points. For this reason we consider these scans close enough for our purposes, and so we have shown that data taken at 2.5 s time intervals between points are good data for the fork inside its can. Next we tested the equivalence of the different forks. We ran three forks under all the same conditions and found that their resonance frequencies varied by up to.82 Hz (see Fig. 5), which is almost what their specification sheets predict[6]. The next stage was to study the forks outside of their containers. The forks had to be extracted in such a way so as to not damage any part of the fork significantly. The extraction is done with a pair of dikes, and the resulting fork is bare except that its leads still go through and are imbedded in a glass disk (see Fig. 6). This arrangement reduces the risk of the leads breaking off of the fork. 8

9 (a) 25 magnitude of signal (mv) #11 #12 # (b) 1.5 phase (radians) #11 #12 # Figure 5: (a) Magnitude of signal vs. frequency; (b) phase vs. frequency. Forks #11, 12, 13 in cans; 4 pts / 13 Hz; resonance frequencies at Hz, Hz, Hz (±.2 Hz), respectively. Low phase points around are explained in Figure 2 caption. 9

10 Figure 6: A fork in its can; the dikes used to remove the cans; an open fork. For the fork in air, a shift of the resonance to lower frequency is observed (see Fig. 7). Also, an interesting feature that appears for the forks in air is that the X channel, although it retains roughly the same shape, no longer asymptotes at zero. This change causes the amplitude and phase of the signal to have a distorted look (see Fig. 7(b)). We ran three different forks with their cans removed to see how much variation they had (see Fig. 8). Two of them were very close to each other, while the third had a frequency shift of about 2.2 Hz from the others and its amplitude peaked about.3 mv lower. This variance would imply that when the forks are opened there is a possibility of minor damage and the forks should not be treated as identical after they are extracted from their cans. However, it is also possible that this fork (#9) was a outlier and if more

11 forks were tested, one would find that, for the most part, they do not differ as much as these three. We encourage the reader to explore this question further. (a) X channel (mv) Y channel (mv) (b) magnitude of signal (mv) phase (radians) Figure 7: (a) X and Y channels vs. frequency; (b) magnitude of signal and phase vs. frequency. Fork #8 in air with nothing on it; 1 Hz / 2 pts; resonance at Hz (±.3 Hz); connecting lines added for visual aid. See the caption of Fig. 2 for a discussion of the phase outliers. 11

12 (a) 1.4 #8 #9 # magnitude of signal (mv) (b) #8 #9 # phase (radians) Figure 8: (a) Magnitude of signal vs. frequency; (b) phase vs. frequency. Forks #8,9, in air with nothing on them; 2 pts / Hz, resonances at Hz, HZ, Hz (±.3 Hz), respectively. Fork #9 exhibits a clear difference in magnitude of signal and phase from the other two forks. 12

13 Finally, we singled out one fork ( 3 rd opened ) and did runs with it through the magnet. The setup for the magnet is shown in detail in Fig. 9. Frequency scans were Figure 9: Setup of fork going into magnet. Exact magnet strength is unknown, but it is estimated that in the middle of the magnet the field is of the order of.5 T. taken at mm intervals, and the fork was moved very slowly from one point to the next. To compare these measurements, we had to know the relative strength of the force acting on the particles at each point where data were taken. We know that this force is proportional to B, or B B, where B is the strength of the magnetic field. We could not determine for sure which on of these it is due to lack of time. So all that we had to do was find the relationship between B, B B and d the reading on the actuator. The formulas for these relationships are too long to be shown here and can be derived easily with the help of a textbook on electromagnetism [7]. We present the graph of the relations in Fig.. 13

14 B gradb BgradB B grad(b), BgradB d (mm) Figure : B, B, and B B (relative scales) vs. reading on actuator (d). The point of the fork is taken to be its tip. The blue lines represent the edges of the magnet. The values at which the greatest change is seen for the measurements of the forks are d = 2 mm and d = 3 mm. Next, we started to add various objects to the fork. First, we put grease onto the end of the prongs. We ran the fork through the magnet to see if the magnetic field had any effect on the oscillations. The results are displayed in Fig. 11. The shifted resonance peak is due to the additional mass added on to the fork, as discussed previously. Fig. 11 shows an increase of the amplitude of the peak, but there is no consistent order to the increase, not in terms of the strength of the magnetic field and not in terms of the progression with which the fork is inserted into the magnet. Also, the resonance is roughly in the same spot, Hz ±.3 Hz. These two observations lead to the conclusion that the magnetic field has negligible effect on the fork when it is tainted with grease. 14

15 (a') 1.2 distance (mm) magnitude of signal (mv) magnitude of signal (mv) (a'') 1.12 magnitude of signal (mv) distance (mm) Figure 11(a): Magnitude of signal vs. frequency. (a ) Graphs of all scans; (a ) close up of peaks. 3rd opened fork; with grease, no particles; run through magnet; peak for all scans at Hz ±.3 Hz. In the left graph of (a ) the scans are displaced by an even amount. In the other two graphs they are all plotted one on top of the other. In (a ) one can see that the height of the peaks varies by up to.2 mv. For our purposes we consider these scans equivalent. 15

16 (b') distance (mm) phase (radians) 1..5 phase (radians) (b'') phase (radians) distance (mm) Figure 11(b): Phase vs. frequency. (b ) Graphs of all scans; (b ) close up of peaks. For the rest of the information see Figure 11(a). We do not have an explanation for the strange behavior of the scan at distance d = 3 mm. A reasonable assumption would be that the lock-in amplifier was having technical difficulties for an unknown reason. Next, we put magnetic particles of iron oxide onto the prongs of the tuning fork where the grease was. Because of the additional mass and the presence of a magnetic field, we expected to see another jump downward in the resonance frequency. We ran the fork in and out of the magnet, obtaining the data displayed in Fig. 12. An obvious shift in 16

17 (a') magnitude of signal (mv) magnitude of signal (mv) distance (mm) start from top (a'').87 magnitude of signal (mv) distance (mm) Figure 12(a): Magnitude of signal vs. frequency. (a ) Graphs of all scans; (a ) close up of peaks. 3rd opened fork; with grease and particles; run through magnet. Plotting assignment same as for Figure11. In (a ) one can see that there are two groups : one consisting of higher peaks and one of lower peaks. The one with lower peaks are the scans at d = 2 mm and d = 3mm. The difference in height between the two groups is about.6 mv, which is three times the difference in height for the forks without the particles (Fig. 11). 17

18 (b') 1.4 phase (radians) phase (radians) distance (mm) start from top (b'') phase (radians) distance (mm) start from top Figure 12(b): Phase vs. frequency. (b ) Graphs of all scans; (b ) close up of peaks. For the rest of the information see Figure 12(a). the height of the magnitude of the signal (R) is seen, but no notable ordered change in resonance frequency can be seen. The resonance frequency ranges from Hz ±.3 Hz to Hz ±.3 Hz. The last two scans give the largest resonance frequencies; we 18

19 hypothesize that this result is due to a hysteresis effect, but one would need to run the fork in and out again to confirm this. The scans in which the peak magnitudes were greatest are those made at d = 2 mm and d = 3 mm, at which B B was most positive ( B has no outstanding values at both those points compared to the others). However, this result is interesting as we assumed that the force would be proportional to the magnitude of B B and would be oblivious to its sign. We have no explanation for this occurrence and encourage the reader to follow up on it. We want to point out that the difference between the heights of the peaks in the first test with the magnet, where we greased the fork but had not yet put particles on it, was approximately.2 mv. As stated previously, the conclusion for this test was that the scans at different distances of insertion into the magnet were equivalent enough for our purposes. Fig. 12, however, shows the difference in the heights of the peaks between the two formed groups is approximately.6 mv, which is three times the difference before and so does not fall under our category of equivalence. This change in height is one of the expected results, and although we didn t observe the change in resonance, the result we found proves the susceptibility of the forks with particles to the magnetic field of the magnet. Conclusion The forks prove to be sensitive enough instruments for use as magnetometers for measuring the magnetization of small particles. However, the particles used in this experiment, iron oxide, have a relatively high magnetization. The shift we observed indisputably exists, but it is not large in scale. Amplitude peaks for particles with smaller magnetization would be much harder to detect. We suggest that those who would want to 19

20 see such changes repeat this experiment with different variables, notably those mentioned earlier (smaller tuning forks, different setup). Acknowledgements I would like to thank the Physics REU program at UF for selecting me for this project and giving me an opportunity to work in a professional lab setting. I would like to thank Dr. Mark Meisel for supervising this project. I would also like to thank Prof. Kevin Ingersent and Ms. Kristin Nichola for taking students to various places and worrying about our problems. This work was partially supported by the NSF via the UF Physics REU program and DMR 714 (MWM). References 1. J.-M. Friedt and E. Carry, Am. J. Phys. 75, 418 (27). 2. R. Blaauwgeers, M. Blazkova, M. Človečko, V. B. Eltsov, R. de Graaf, J. Hosio, M. Krusius, D. Schmoranzer, W. Schoepe, L. Skrbek, P. Skyba, R. E. Solntsev and D. E. Zmeev, J. Low Temp. Phys. 146, 539 (27). 3. G.R. Fowles, Analytical Mechanics, 4 th ed. (CBS College Publishing, 1986), pp Information on lock-in amplifier and how it works taken from 5. R. Blaauwgeers, M. Blazkova, M. Človečko, V. B. Eltsov, R. de Graaf, J. Hosio, M. Krusius, D. Schmoranzer, W. Schoepe, L. Skrbek, P. Skyba, R. E. Solntsev and D. E. Zmeev, J. Low Temp. Phys. 146, 54 (27). 6. Tuning forks purchased from Digi-Key ( and produced by Epson Toyocom Corporation. Specifications sheet found at 7. D.J. Griffiths, Introduction to Electrodynamics, 3 rd ed. (Prentice Hall, New Jersey, 1981), pp. 22 and