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 32611-8440
July 25, 2008
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 0.06 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 47405
1
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 (≤ 10 mm in length and about 1 mm in diameter)
devices used in wrist watches are very inexpensive (averaging on the order of $0.10 $0.20) 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
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:
mx' '+γx'+ mω 0 x = F0 cos(ωt ) ,
2
(1)
where γ is the damping coefficient, ω 0 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
R = ( F0 / m) /(γ 2ω 2 / m 2 + (ω 0 − ω 2 ) 2 )1 / 2
2
(2)
and a phase (relative to driving function) of
θ = arctan(ωγ / m /(ω 0 2 − ω 2 )) .
(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 = ( F0 / m)(ω 0 − ω 2 ) /(γ 2ω 2 / m 2 + (ω 2 − ω 0 ) 2 ) ,
2
2
(4)
Y = ( F0 / m)(γω / m) /(γ 2ω 2 / m 2 + (ω 2 − ω 0 ) 2 ) .
(5)
and
2
The original resonance frequency of the system, when it is unforced and undamped, is ω 0 .
But in our case the frequency of resonance becomes, according to [3],
ω1 = (ω 0 2 − γ 2 / 2m 2 )1 / 2 ,
(6)
so the higher the damping coefficient γ and/or the larger the mass m of the fork the
smaller ω1 . Also, with increase in γ 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
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
Fork
Function
Generator
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: 10 or 20 mV. Dynamic
Resolution: Normal. No offset for either channel. Pre- Time Constant: 10 ms. Post- Time
Constant: None.
5
(a) 15
X channel
10
25
Y channel
20
15
0
10
-5
5
-10
-15
Y channel (mV)
X channel (mV)
5
32763
32764
32765
32766
32767
32768
32769
0
32770
frequency (Hz)
(b)
1.5
25
magnitude
1.0
0.5
15
0.0
10
-0.5
phase (radians)
magnitude of signal (mV)
phase
20
-1.0
5
-1.5
0
32763
32764
32765
32766
32767
32768
32769
32770
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 / 400 pts;
resonance is at 32766.28 Hz (± 0.02 Hz); connecting lines added as visual aid. The
“outlier” point of the phase in between 32763 and 32764 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
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 250 ms was sufficient. However, when
the forks are still in the can, their Q is extremely high [5] (on the order of 4 × 10 4 ), and a
time sweep of 250 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).
15
15
10
5
0
-5
Y channel (mV)
X channel (mV)
10
5
0
-10
-5
-15
32763
32764
32765
32766
32767
32768
32769
32770
frequency (Hz)
Figure 3: Fork #8 in its can; X and Y channels vs. frequency; time between data points is
250 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 30 s and 60 s
gives the results of Fig. 4. These graphs are not identical, but the resonance for them all is
7
15
10
30s
60s
30s
60s
25
20
15
0
10
-5
5
-10
-15
32765.6
Y channel (mV)
X channel (mV)
5
0
32765.8
32766.0
32766.2
32766.4
32766.6
32766.8
frequency (Hz)
Figure 4: Fork #11 in its can; X and Y channels vs. frequency; time between data points
is 30 s and 60 s; resonance frequencies are at 32766.27 Hz ± 0.01 Hz for both.
within 32766.27 ±0.01Hz, 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 0.82 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
magnitude of signal (mV)
(a)25
#11
#12
#13
20
15
10
5
0
32763
32764
32765
32766
32767
32768
32769
32770
frequency (Hz)
(b)
1.5
#11
#12
#13
phase (radians)
1.0
0.5
0.0
-0.5
-1.0
-1.5
32764
32766
32768
32770
frequency (Hz)
Figure 5: (a) Magnitude of signal vs. frequency; (b) phase vs. frequency. Forks #11, 12,
13 in cans; 400 pts / 13 Hz; resonance frequencies at 32766.28 Hz, 32766.67 Hz,
32765.85 Hz (± 0.02 Hz), respectively. Low phase points around 32764 are explained in
Figure 2 caption.
9
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 0.3 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
10
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.
0.9
1.5
0.6
1.2
0.3
0.9
0.0
0.6
-0.3
0.3
-0.6
0.0
32740
32745
32750
32755
32760
32765
32770
Y channel (mV)
X channel (mV)
(a)
32775
frequency (Hz)
(b)
1.6
1.5
1.4
0.5
1.0
0.0
0.8
0.6
-0.5
phase (radians)
magnitude of signal (mV)
1.0
1.2
0.4
-1.0
0.2
-1.5
0.0
32740
32750
32760
32770
32780
frequency (Hz)
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 32759.8 Hz (± 0.3
Hz); connecting lines added for visual aid. See the caption of Fig. 2 for a discussion of
the phase outliers.
11
(a)
#8
#9
#10
magnitude of signal (mV)
1.4
1.2
1.0
0.8
32752
32754
32756
32758
32760
32762
32764
frequency (Hz)
(b) 1.5
#8
#9
#10
phase (radians)
1.0
0.5
0.0
-0.5
-1.0
32750
32760
32770
32780
32790
frequency (Hz)
Figure 8: (a) Magnitude of signal vs. frequency; (b) phase vs. frequency. Forks #8,9,10
in air with nothing on them; 2 pts / Hz, resonances at 32760.0 Hz, 32757.8 HZ, 32760.0
Hz (± 0.3 Hz), respectively. Fork #9 exhibits a clear difference in magnitude of signal
and phase from the other two forks.
12
Finally, we singled out one fork (“3rd 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 0.5 T.
taken at 10 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.
10.
13
1.6
B
gradB
BgradB
1.4
0.08
0.06
0.04
1.0
0.02
0.8
0.00
0.6
-0.02
grad(B), BgradB
B
1.2
-0.04
0.4
-0.06
0.2
-0.08
0.0
0
10
20
30
40
50
d (mm)
Figure 10: 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 = 20
mm and d = 30 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, 32325.8 Hz ± 0.3 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
distance
(mm)
0
10
20
30
40
50
(a')
1.2
1.0
magnitude of signal (mV)
magnitude of signal (mV)
1.2
1.0
0.8
0.6
0.4
0.2
32200
0.8
0.6
0.4
0.2
32250
32300
32350
32400
32450
32200
32250
32300
32350
32400
32450
frequency (Hz)
frequency (Hz)
(a'')
1.12
distance
(mm)
0
10
20
30
40
50
magnitude of signal (mV)
1.11
1.10
1.09
1.08
1.07
32325.0
32325.5
32326.0
32326.5
32327.0
frequency (Hz)
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 32325.8 Hz ± 0.3 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 0.02 mV. For our purposes we
consider these scans equivalent.
15
(b')
distance
(mm)
0
10
20
30
40
50
2.0
1.5
1.0
phase (radians)
phase (radians)
1.5
0.5
0.0
1.0
0.5
0.0
32220
32250
32280
32310
32340
32370
32400
32430
32220 32250 32280 32310 32340 32370 32400 32430
frequency (Hz)
frequency (Hz)
(b'')
0.02
distance
(mm)
0
10
20
30
40
50
phase (radians)
0.00
-0.02
-0.04
-0.06
32329
32330
32331
32332
32333
frequency (Hz)
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 = 30 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
(a')
0.9
0.8
magnitude of signal (mV)
0.9
magnitude of signal (mV)
0.8
0.7
0.7
0.6
0.5
0.4
0.6
0.3
0.5
3150031550316003165031700317503180031850319003195032000
frequency (Hz)
0.4
distance
(mm)
start
from top
0
10
20
30
40
50
40
30
20
10
0
0.3
31500
31600
31700
31800
31900
32000
frequency (Hz)
(a'')
distance
(mm)
0
10
20
30
40
50
40
30
20
10
0
magnitude of signal (mV)
0.87
0.84
0.81
0.78
31744
31745
31746
31747
31748
31749
31750
31751
31752
frequency (Hz)
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 = 20 mm and d = 30mm. The difference in height between the two groups is
about 0.06 mV, which is three times the difference in height for the forks without the
particles (Fig. 11).
17
(b')
1.4
phase (radians)
1.2
1.0
0.8
0.6
0.4
0.2
31500
31550
31600
31650
31700
31750
31800
31850
31900
31950
frequency (Hz)
1.6
phase (radians)
1.4
1.2
1.0
0.8
0.6
0.4
32000
distance
(mm)
start
from top
0
10
20
30
40
50
40
30
20
10
0
0.2
31500
31600
31700
31800
31900
32000
frequency (Hz)
(b'') 0.6
distance
(mm)
start
from top
0
10
20
30
40
50
40
30
20
10
0
phase (radians)
0.5
0.4
0.3
31750
31752
31754
31756
31758
31760
frequency (Hz)
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 31747.6 Hz ± 0.3
Hz to 31749.3 Hz ± 0.3 Hz. The last two scans give the largest resonance frequencies; we
18
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 = 20 mm and d = 30 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 0.02 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 0.06 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
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–0701400 (MWM).
References
1. J.-M. Friedt and E. Carry, Am. J. Phys. 75, 418 (2007).
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 (2007).
3. G.R. Fowles, Analytical Mechanics, 4th ed. (CBS College Publishing, 1986), pp.
70-75.
4. Information on lock-in amplifier and how it works taken from
http://www.cpm.uncc.edu/programs/lia.pdf
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, 540 (2007).
6. Tuning forks purchased from Digi-Key (www.digi-key.com) and produced by
Epson Toyocom Corporation. Specifications sheet found at
http://www.eea.epson.com/portal/pls/portal/docs/1/745499.PDF
7. D.J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, New
Jersey, 1981), pp. 220 and 263.
20