Behavior of Quartz Tuning Forks in Liquid Helium-4
Matt Jachowski, A.J. Kumar, Naveen Sinha, Aaron Ligon, John Rutherford, and Ed Fei
TA: Charis Quay
Our group studied the behavior of quartz tuning forks in liquid helium-4 (LHe). Previous work by
other groups [1, 2] has been done to study piezoelectric crystals in low temperature environments;
however, ours is the first extensive study to measure its behavior in separate environments. We
found stark changes in tuning fork behavior that indicated a superfluid transition at 2.17 ± 0.04 K
and strong correlation to proposed models that could lead to applications in density
measurements. Our work opens the possibility to use quartz tuning forks as an inexpensive means
for liquid property measurements.
Introduction
Over the past several decades, a variety of methods have been developed for studying the
physical properties of liquid helium. The two primary techniques utilize vibrating wires and
torsional oscillators [3, 4]. These techniques measure the oscillations of mechanical objects
immersed in the liquid helium. However, these techniques suffer from the need for an external
magnet to excite vibrations, complicating the setup and preventing experiments in high magnetic
field. We explored the possibility of using quartz tuning fork crystals as a simple sensor to study
the properties of LHe. Such crystals are used in digital watches, making them inexpensive and
readily available. As these crystals are piezoelectric, relatively simple electronics can be used to
both excite vibrations in the crystal and to measure the response. Consequently, this approach
offers advantages over the other techniques.
Experimental Setup
In order to extract a meaningful relation between the LHe properties and the quartz tuning
fork behavior, we sought to isolate effects due purely to the LHe from other effects that could
potentially affect tuning fork behavior. Two of the primary concerns going into the experiment
were whether the tuning forks would experience mechanical damping as a result of thermal
contraction and whether the physical act of removing the tuning forks from their sealed factory
cans would cause a significant change to the intrinsic properties of the tuning fork resonator. To
isolate the effects of the liquid from these two effects, we decided to follow the behavior of
tuning forks in three environments as we cooled.
We constructed two experimental stages on our probe to mount tuning forks in three
different environments. One stage was mounted inside a brass vacuum can on the end of the stick
while the other stage was mounted on the outside end of the vacuum can cover, an expanded
view of which is shown in Figure 1.
Figure 1: Exploded view of setup
The three different tuning fork environments studied were: (a) open tuning forks (i.e. removed
from factory vacuum seal) in LHe, (b) sealed tuning forks in LHe, and (c) open tuning forks in
our home made vacuum can in LHe. To accomplish this, we inserted the can into a power drill
collar to gently squeeze the ends of the can near the leads until the can broke. Forks of type a and
b were mounted on the outside of the vacuum can on a single stage.
The outer stage was fairly easy to construct since all that was needed was to secure the
forks in place and provide good electrical contact to the manganin wire leads used to connect the
forks to the top of the stick. This stage was constructed from a DIP socket as shown in Figure 2,
allowing for modularity and ease of tuning fork replacement.
Figure 2: View of the bottom of the stick with type a. and b. forks mounted on the DIP socket.
The inner stage provided more challenging design considerations. The vacuum was
pulled from the inside of the dipping stick which connected to the top of the can. All inside wires
ran through this stick as well. Unlike the outer stage, which is protected by several baffles, the
middle of the stick left a straight shot of room temperature black body radiation to the mounted
tuning forks, potentially making cooling below 4 K difficult. Thus, it was necessary to establish
a thermal contact for the tuning forks with the vacuum can walls while maintaining electrical
insulation. This was accomplished by mounting the inner tuning forks on a G10 disk upside
down in the can, to shield them from radiation (Figure 3).
Figure 3: View of the inside stage. Note this is from an older version of the experiment with
smaller tuning forks and soldered leads rather than sockets.
The forks were epoxied onto the G10 so that one side had the leads sticking out and the
forks were exposed on the other side. On the fork side, a square piece of copper ribbon was
attached with GE varnish and silver paint to provide thermal contact with the forks (Figure 4).
Figure 4: View of fork surface with thermal contacts.
This ribbon rested on the brass mount in the can, cooling the forks to the wall
temperature. A silicon diode thermometer (Lakeshore DT-470) was attached with GE varnish to
the side of the disk and given a similar thermal contact to verify that the inner tuning forks were
in fact being cooled. A four point measurement was used with the Si diode. Sockets soldered
onto the end of manganin wires were then used to plug into the tuning forks. The can was sealed
with an indium wire ring.
The final design described above was the result of modifications over several cool-downs.
Originally, both stages were G10 disks, and wires were soldered directly to the tuning fork leads.
The final design provided the maximum simplicity and most versatility, as it was very easy to
swap tuning forks in and out when they broke.
Electronics
The tuning fork can be characterized as a simple LRC circuit in parallel with a capacitor
Cn[1], as shown in Figure 5. The value of Cn is the sum of the inherent pad capacitance of the
tuning fork and the capacitance in the leads. In our experiment, Cn is a non-negligible value (on
the order of 100pF); thus, to obtain our resonance signal, we must cancel the capacitance. Since
Cn varies for each individual tuning fork circuit, we constructed a digital variable capacitor from
an array of fixed-value capacitors to consistently reproduce capacitances. Our circuit uses a
transformer to apply a 180phase shift to the excitation signal so the voltage on the adjustable
capacitor is opposite that applied to the tuning fork. By varying the adjustable capacitor to equal
Cn, the current from the two capacitors cancel, and we are left with the desired resonance.
Figure 5: Tuning fork circuit diagram
There are three possible methods of exciting the tuning fork to measure the
quality factor and the resonant frequency. It can be excited with a single pulse, a frequency
sweep, or white noise. To our knowledge, our group is the first to use white noise to excite the
tuning forks at LHe temperatures. We performed a Fourier transform on the resulting output of
the tuning fork using a spectrum analyzer (Agilent 89410A).
We built two boxes to house all of the electronics necessary for the experiment. The first
box contained a pre-amplifier composed of an LF 411 op-amp configured as a non-inverting
amplifier and powered by an external, adjustable power supply. The second box contained the
capacitance circuit described above along with a dual switching device that allowed us to address
each tuning fork individually (Figure 6).
The tuning fork was excited with white noise from the spectrum analyzer, and the
response was taken from the output of the capacitance circuit box and fed to the pre-amplifier
before being measured on the spectrum analyzer.
Figure 6: Electronics configuration
The data form the spectrum analyzer was collected with a custom LabView program. A
screen shot of the user interface is shown in Figure Gratuitous.
Figure Gratuitous: A classy graphical user interface.
Results
The frequency response of each fork was obtained at temperatures ranging from 4.2 K to
1.3 K. Figure 7 shows a typical resonance curve for a tuning fork open to liquid helium. Theory
predicts that the tuning fork resonance curves should be Lorentzian [2]. However, we empirically
determined the Voigt profile to be a better fit, implying that some other Gaussian process is
affecting the resonance.
32.00
4
3
2
1
31.7
31.8
31.9
32.0
32.1
32.2
Resonant Frequency (kHz)
Magnitude (arbitrary)
5
31.98
31.96
31.94
31.92
1.0
1.5
Frequency (kHz)
2.5
3.0
3.5
4.0
4.5
Temperature (K)
Figure 7: A typical resonance curve for a tuning
fork open to LHe. The data is fit with a Voigt
profile (red).
Figure 8: Resonant frequency vs.
temperature. Notice the dramatic change in
behavior of the curve around the lambda
temperature (2.17 K).
5000
30
25
4000
Quality Factor
Resonance Width (Hz)
2.0
20
15
10
3000
2000
5
1000
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature (K)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature (K)
Figure 9: Resonance width vs. temperature. The
width tends to decrease with temperature.
Figure 10: Quality factor vs. temperature.
The quality factor increases rapidly as
temperature decreases beyond the lambda
temperature (2.17 K).
Though the Voigt profile was found to be a better fit to the data overall, fitting to
asymmetric data yielded resonant frequencies that were systematically too high. Asymmetric
peaks resulted from incomplete cancellation of Cn. The Lorentzian fit, though systematically too
short on each peak to justifiably describe the data, fit more robustly to the centers of asymmetric
data. We thus extracted the resonant frequency from the Lorentzian fit. The Voigt profile was
still used to obtain the resonance width, and the quality factor was calculated from the
combination of these two results.
Figures 8-10 display the behavior of the resonant frequency, resonance width, and quality
factor, respectively, of a tuning fork open to liquid helium as the temperature is varied. Most
striking is the behavior of the resonant frequency with respect to temperature, which shows a
dramatic change in slope as the temperature passes through the superfluid transition around 2.17
K. The resonance width is observed to decrease steadily with temperature. The quality factor is
observed to increase gradually as temperature decreases, and as the temperature goes through the
lambda point, increase rapidly.
To ensure that the observed effects were only due to the changing properties of the liquid
helium, we analyzed the data from tuning forks in the two other environments, as described in the
Experimental Setup section. The sealed tuning forks in liquid show negligible change in resonant
frequency and width. The open tuning forks in the vacuum chamber underwent similar, but
smaller, changes in these resonance characteristics. Examination of our experimental setup
suggested a liquid helium leak in out vacuum chamber, which would explain this behavior. A
further discussion of this issue is included in the appendix. Taken together, the results from the
three different tuning fork environments imply that the changes in the tuning fork resonance were
entirely caused by the liquid helium properties.
According to our data, the superfluid transition occurs at 2.17 ± 0.04 K. The error in this
value comes almost entirely from the coarse resolution with which the temperature was scanned.
Without any change to the experimental setup, a more precise value of the transition could be
obtained by stepping more finely through the temperatures around the lambda point.
Discussion
Despite their widespread use, quartz tuning forks are surprisingly difficult to model
accurately, evidenced by the lack of literature describing theories for a tuning fork in viscous
media. Several models do exist, however. The seminal work of Kanazawa and Gordon [5] on
quartz resonators gives the relation
f  
(1)
where η is the viscosity of the liquid and ρ is the density. This model applies to a very simple
class of quartz resonators, known as thickness shear mode (TSM) resonators, which are defined
as slabs of resonating crystal with one face in contact with liquid. The relationship (1) has been
experimentally verified in solvents of glucose and alcohol by [5, 6], but to our knowledge not in
LHe. Matsiev and colleagues [6] assert that the TSM model is appropriate for crystal tuning
forks. We find, though, that for a tuning fork in LHe this relationship fails. Figure 11 clearly
demonstrates there is no correlation. Values of η and ρ were taken from [7] to create the plot of
the Kanazawa model and were scaled by a linear constant – which is otherwise difficult to
extricate – to put both sets of data in the same regime.
A more accurate model has been derived by Zhang [8] for tuning forks exclusively.
Under the approximation that one tine is fixed in space, the tuning fork acts like a rectangular
beam, and one can derive with suitable approximations:
1

K
 f  2 
 0   1
 f 

(2)
where ρ is the density of LHe, K is a constant which depends on the intrinsic properties of the
tuning fork on the order of unity, f0 = 32,768 Hz is the resonant frequency in vacuum, and f is the
resonant frequency in the liquid. Figure 12 shows the accepted value for density from [7]
alongside our data, which has been substituted into equation (2). Again, the experimental values
have been scaled by an appropriate linear constant.
E xperim ental A greem ent w ith Zhang Theory
E xp erim en tal A g reem en t w ith K an azaw a T h eo ry
0.150
900
Zhang M odel
Derived Experim ental
Values
0.145
850
(Pa*s)
f (Hz)
800
750
700
Kanazawa M odel
Derived Experim ental
Values
650
0.140
0.135
0.130
600
550
0.125
500
1.0
1.5
2.0
2.5
3.0
3.5
4.0
Temperature (K)
Figure 11: Kawanaza Theory: No agreement.
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature (K)
Figure 12: Zhang Theory: Reasonable agreement after
adjustment of linear parameter.
The two trends in Figure 12 show remarkable similarities, especially around the lambda
point. The experimental values, however, broaden at higher temperatures and slope downward
more sharply at low temperatures, compared to the model. Our current understanding of how the
rectangular beam model could apply to the oscillation of both tines is too limited to predict
whether these broadening and constricting trends would shift either way.
Conclusion
We observed the resonance behavior of quartz tuning fork crystals in LHe. The resonant
properties display clear trends with temperature: the resonant frequency shows a marked change
in slope around the superfluid transition, the resonance width decreases with temperature, and the
quality factor increases rapidly as the temperature decreases below the lambda point. We found
suggestive agreement between our results and the theoretical model proposed by Zhang [9]. This
raises the possibility of using tuning forks as an inexpensive and simple means to measure
density in LHe and other liquids.
This study opens up many possibilities for further work. First, a more thorough study of
the tuning fork behavior around the superfluid transition could provide a more precise value for
the lambda temperature. Additionally, a more complete theoretical model of the tuning fork
could provide accurate and quantitative determination of liquid properties.
Appendix:
Discussion of data from tuning forks in vacuum
35
Resonance Width (Hz)
Resonant Frequency (kHz)
32.70
32.68
32.66
32.64
32.62
32.60
1.0
1.5
2.0
2.5
3.0
3.5
4.0
30
25
20
15
10
5
0
1.0
4.5
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature (K)
Temperature (K)
Figure A.1: Resonant frequency vs. temperature.
Figure A.2: Resonance width vs.
Note capacitance was changed around 2.3 K to keep
symmetric peak shape.
temperature. Notice the dramatic change in
behavior of the curve around the lambda
temperature (2.17 K).
12000
Quality Factor
10000
8000
6000
4000
2000
0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature (K)
Figure A.3: Quality factor vs. temperature data from tuning fork in home made vacuum can.
Figures A.1-A.3: Results from open tuning fork in homemade vacuum can.
Figures A.1-A.6 give the results from the open fork in our home made vacuum chamber
and the factory sealed tuning fork. As previously mentioned, the factory sealed can shows no
statistically significant changes in resonance curve properties as temperature decreases,
suggesting that thermal damping of the fork resonance is minimal. This result then allows us to
attribute the change in resonance character experienced by the open fork in LHe to the LHe itself
rather than the tuning fork.
The data from the open fork in the vacuum can should ideally give the same results as the
factory sealed cans so that we could rule out the physical trauma of opening the forks as a
relevant factor effecting resonance behavior. However, we found some evidence that our can
may have had a cold leak that was letting helium in.
Tuning Fork 1 in Sealed Can: Resonant Frequency
32697.0
Tuning Fork 1 in Sealed Can: Resonance W idth
1.00
32696.8
Resonant Frequency [Hz]
Resonance W idth [Hz]
32696.6
0.75
0.50
0.25
32696.4
32696.2
32696.0
32695.8
32695.6
32695.4
32695.2
32695.0
0.00
1.0
1.5
2.0
2.5
3.0
3.5
4.0
1.0
4.5
1.5
Temperature [K]
2.0
2.5
3.0
3.5
4.0
4.5
Temperature [K]
Figure A.4: Resonant frequency vs. temperature.
Figure A.5: Resonance width vs.
temperature.
Tuning Fork 1 in Sealed Can: Quality Factor
60000
Quality Factor
50000
40000
30000
20000
10000
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature [K]
Figure A.6: Quality factor vs. temperature data from tuning fork in factory sealed can.
Figures A.4-A.6: Results from open tuning fork in homemade vacuum can.
As we warmed up after the third cooldown, the pump to the inner chamber was turned off
and the thermocouple gauge for the inner chamber shot up from below 50 milliTorr to about 300
milliTorr and then continued to rise slowly, passing 2 Torr and off the scale of the inner gauge.
The pressure never built up enough to blow open the relief chamber, which is why we had paid
less attention to the gauge on the previous two cooldowns. It was only after we analyzed the data
from the second run that we realized something might be wrong with our chamber, since it
displayed the uncharacteristic behavior seen in Figure A.1. If a leak had occurred letting He gas
and then superfluid He in, the quality factor would decrease until the forks were submerged in
LHe at which point the curve would then follow that of the open tuning forks in LHe. Indeed,
this qualitatively matches with our experimental data and explains the hybrid shape of fork in our
vacuum can (Figures A.7 and A.8).
Open Fork in LHe
Open Fork in Home made Can
5 point AA smoothing
5 point AA smoothing
12000
50000
40000
Quality Factor
Quality Factor
10000
8000
6000
open tuning fork in LHe
sealed tuning fork in LHe
open tuning fork in vacuum
5 point AA smoothing
5 point AA smoothing
5 point AA smoothing
30000
20000
4000
10000
2000
0
0
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature (K)
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5
Temperature (K)
Figure A.7: Comparison of quality factor for
Figure A.8: Comparison of same forks
open tuning fork in Lhe and open fork in our
homemade can with suspected leak.
including sealed factory can tuning fork data.
Figures A.7-A.8: The fork in the homemade can has a Q value that approaches the fork in
LHe and at the lambda point follows behavior that is qualitatively similar to the immersed
fork. The higher value could be explained by the fork not being completely immersed and
thus not being exposed to the full damping force of the LHe.
Without the vacuum can data to show that removing the fork’s factory can is a nondamaging process, we resort to a slightly more qualitative analysis. At room temperature, we
monitored forks as we pulled a vacuum on both our vacuum chamber and the helium dewar.
These peaks sharpened and frequencies shifted up as the chambers were evacuated. Analysis of
these resonance curves compared to those of the factory sealed forks at room temperature (298
K) show that there is not a significant systematic difference to the resonance curve properties that
can be associated with the forks being open or sealed (Figure A.9).
Q
6000
5000
Qfactor
4000
3000
2000
1000
0
0
1
2
3
4
5
6
7
8
9
Tuning Fork
Figure A.9: Excepting fork 5, which was believed faulty for other reasons, the seven other
forks show similar quality factors. Forks 4, 7, and 8 are open forks in the home made
vacuum can averaging a Q of 4230 ± 460. Forks 1 and 2 were sealed and had an average
Q of 3970 ± 310. Forks 3 and 6 were open in the He dewar vacuum and had an average Q
of 3280 ± 240. Hence, all the forks were all very close (excepting fork 5).
If removing the factory can has any effect, we would expect it to be a single shift, most likely
making Q lower for all circumstances. We do not expect there to be any thermally dependent
change and thus seeing no significant effect at room temperature implies that removing the can
on the fork in the LHe did not affect its resonance peaks or behavior.
Despite the likely presence of a leak in our vacuum chamber, we were able to extract
enough data that allowed us to rule out physical effects of removing the factory cans. Added to
the results showing no significant thermal damping effects, we can say with certainty that the
changing resonance characteristics observed on our tuning fork in LHe were a result of the
properties of the LHe alone.
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