High-Resolution Phase Velocity Measurements in an
Acoustic Cavity
Eduardo M. Calleja
University of Florida
Shanshan Qi
Barnard College, Columbia University
July 29, 2003
Abstract
A method for making high-resolution phase velocity measurements has been developed.
A cryostat probe and spectrometer have been specifically designed and constructed for
this experiment. An experimental cell containing a variable path length acoustic cavity
has been constructed and assembled. The cell was attached to the probe and cooled down
to liquid Nitrogen temperatures with 4He gas inside. Data for the first two resonance
frequencies and for the displacement of the bimorph were acquired. Unfortunately,
certain complications did not allow for further data collection.
2
Introduction
Sound propagation in superfluids has been studied extensively. For instance, the
determination of collective modes in 3He has been obtained through measurements of the
sound propagation of the material [1]. The Path Length Modulation (PLM) method is one
of the most useful acoustical techniques, which has been applied to make high resolution
phase velocity measurements [1]. In order to achieve this method, it is necessary to build
an acoustical cell that has an adjustable path length.
The assembly of this acoustic cell usually has the form of a piezo electric bimorph
attached with a reflector and an electrode on each side. A transducer is placed parallel to
one side of the reflector; another electrode is positioned on the side of the electrode that is
connected to the bimorph, hence forming a parallel plate capacitor. The displacement of
the bimorph can be determined with a high resolution by measuring the capacitance
between the two electrodes. The transducer is able to convert mechanical energy to
electrical energy, allowing us to measure signals in the cavity using the spectrometer. At
the cavity resonance, the length of the acoustical cavity (Z) is Z = N (λ / 2), where N is
the number of nodes, and λ is the wavelength of the acoustic wave.
In order to make a high-resolution phase velocity measurement, we must
determine λ accurately. The phase velocity (vφ) is defined as vφ = ω / k, where ω is the
angular frequency, and k is the wave number. The frequency is normally a controlled
variable, because one can input it from the signal generator. Hence, our goal is to
measure λ with a very high resolution, so that the sound velocity can be computed.
Previous works incorporating the PLM method include the experiments that were
done by Grimsrud and Werntz, who used a hearing aid transducer with a frequency
3
response at liquid Helium temperature of 1000Hz to 3500Hz. By applying a method
similar to the one discussed in this paper, the researchers were able to obtain sound
velocity measurements of 3He and 4He gas with an uncertainty of 0.07% [2]. More
recently, Hamot et al. [3] have published a paper explaining a similar method but
working in the MHz frequency range that allowed a resolution of 10 -4 to 10-5 in the phase
velocity measurement.
Equipments
The Experimental Cell
A piezo electric bimorph (61942 PZT -5A) was ordered, and a hole was drilled in
its center in order to mount a macor shaft piece to hold the reflector on one side and the
electrode on the other side of the bimorph. The center piece was glued onto the bimorph
by using Stycast 1266, and the reflector and one electrode were also glued onto the center
piece on each side of the bimorph. The bimorph was then mounted on to a macor piece
that contains a top and a bottom part, as shown in Fig.1.
Three equally spaced holes were drilled on both parts of the macor piece near its
edge. On the top segment, each hole was loaded with a short spring attached with a
sapphine ball, which supports the bimorph but also allows the bimorph to move freely
relative to the transducer at the same time. Three sapphine balls were directly glued in the
holes on the bottom side of the macor piece, which were in a fixed position relative to the
transducer. The advantage of using this method to support the bimorph is that it allows
maximum displacement of its center relative to the transducer with its edges free.
4
FIG. 1
Macor piece with bimorph; the reflector and electrode are not shown.
(Designed by Jose Cancino.)
The transducer was then mounted and glued on to the bottom part of macor piece
by using Stycast 1266. At the same time, the second electrode was also glued on to the
top part of macor piece using Stycast 1266. Along with the parallel electrode that was
glued onto the bimorph, these two electrodes together create a capacitor whose value can
be measured using a capacitance bridge with an accuracy of 10-6; hence we have a very
accurate measurement of the displacement of the bimorph. Additionally, all wires on the
transducer (X-out quartz transducer with 10 MHz fundamental resonance frequency) and
electrodes were attached by using silver epoxy. Finally, both of the top and the bottom
parts were screwed together. The macor piece was then mounted in the experimental cell
as shown in Fig. 2. The cell was sealed using an indium o - ring. In addition, a small
capillary tube, called the fill line, was soft soldered on the top of the cell and allows us to
load a fluid sample to be tested.
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FIG. 2
Schematic of the experimental cell assembly with bimorph and Macor piece.
(Designed by Jose Cancino.)
The Cryostat
A cryostat was designed and constructed specifically for this acoustic experiment.
The cryostat is shown in Fig.3. The probe has an Inner Vacuum Chamber (IVC) to
provide a layer between the cryogenic liquid (liquid N2 or liquid 4He) and the
experimental cell. An electrical box was constructed as well to house all the electrical
wirings, such as the wires that connected to the transducer and the capacitance bridge.
Flexible coaxial cables are run from the room temperature to the IVC for the electrical
wirings. The probe is inserted into a low temperature dewar shown in Fig. 4. As can be
seen from Fig. 4, the bath space filled with liquid He4 is enclosed by a nitrogen jacket,
and the liquid nitrogen is protected from room temperature by a vacuum jacket. The
detailed design for the cryostat probe is provided in Appendix A.
6
Top Flange
Radiation Shield
Pumping Lines
Vacuum Can
FIG. 3
Cryostat probe. (Designed by Aaron Gray.)
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FIG. 4
Nitrogen/He4 Dewar. (Industrial design.)
The Spectrometer
The spectrometer includes a SR844 Lock-In amplifier and a quadrature hybrid as
its key components (see Fig. 5). The heart of the spectrometer lies in the
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FIG. 5
Spectrometer. (Designed by Jose Cancino.)
quadrature hybrid, which has four arms. Two of its arms serve to send a signal with the
appropriate attenuation and phase shift, in order to excite the transducer and balance the
9
detecting arm. When the electrical impedance of the transducer changes (due to the
changes of waves in the cavity), the quadrature hybrid detects the difference in the
reflection of the waves in the fourth arm and send the imbalanced signal to the lock in
amplifier. Since the signal is very small, an rf amplifier is used to amplify the signal and
then the signal is sent to the lock-in amplifier. In this experiment, we used an Agilent
8648A signal generator to generate signals between 10 -60 MHz. For a mapping of the
spectrometer board, refer to Appendix B.
Methods
Our first priority in this experiment was to check that all the equipments were
working properly. Hence, the data for the spectrometer were first taken at room
temperature with the probe inserted in the Dewar. The data for the first two resonance
points are given in Fig. 6.
First Resonance at Room Temperature
(IN Dewar)
X Out
Y Out
Second Resonance at Room Temperature
(IN Dewar)
0.036
0.002
0.035
0.034
0.0000
0.000
0.0150
-0.002
0.0148
-0.006
0.030
-0.0010
0.0146
-0.0015
0.0144
0.029
Y Out (Vrms)
-0.004
0.031
Y Out (Vrms)
0.032
X Out (Vrms)
-0.0005
0.033
X Out (Vrms)
X Out
Y Out
0.0152
-0.008
0.028
-0.010
-0.0020
0.0142
0.027
10300000103500001040000010450000105000001055000010600000
31320000 31380000 31440000 31500000 31560000 31620000
Frequency (Hz)
Frequency (Hz)
FIG. 6
Test data for 10 MHz quartz transducer at room temperature.
The first resonance was found at 10.449MHz, and the second resonance was found at
31.494 MHz. Finding these frequencies gave us a standing wave with wavelength
10
λ/2, when we start to move the bimorph. Thus, the cavity length would be
Z=N(λ / 2), where Z is known to be about 149 micro meters, and N is the number of
nodes and can be determined by using the spectrometer. Hence, when the bimorph is
moved, the position can be determined by the AH 2500A high precision capacitance
bridge (1 ppm resolution). Since we know the bimorph displacement, then we can find
the new cavity length. To check whether the bimorph moves, we swept a DC bias voltage
to the bimorph. Data were then taken to verify that the bimorph was indeed moving up
and down. The data for the displacement of the bimorph are shown in Fig. 7. The applied
voltage on the bimorph varied from -60 to 60 volts, and the bimorph moved at a rate of
about 0.30 micrometers per volt at room temperature. A stray capacitance of about 5 pF
was observed at room temperature. There was also a considerable hysterisis observed, but
that is expected to reduce substantially at lower temperatures.
Bimorph Moving DOWN
(Vacumn)
Trial 1
Trial 2
Trial 3
Trial 4
23
22
Bimorph Moving UP
(in Vacumn)
16.5
Trial 1
Trial 2
Trial 3
Trial 4
16.0
21
15.0
Capacitiance (pF)
Capacitance (pF)
15.5
20
19
18
17
14.5
14.0
13.5
13.0
16
12.5
15
12.0
-60
-50
-40
-30
-20
-10
11.5
0
0
Voltage (V)
10
20
30
40
Voltage (V)
FIG. 7
Test data for the bimorph moving up and down with an
applied voltage of -60 to 60 Volts.
50
60
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Using the fact that the diameter of the electrodes was 9.55 mm and that the
distance between the plates (d) was 49 micrometers, we could calculate the elemental
capacitance to be about 12 pF, using
C = ε0 A / d, where A is the area and ε0 is the permittivity constant. Nonetheless, this
formula does not consider the edge effects. So, using the formula derived by Sloggett,
Barton, and Spencer [4], we found an elemental capacitance of about 10 pF. During the
experiment at room temperature, a capacitance of about 15 pF was observed; this was due
to the hystatresis present at room temperature.
When all the electronics were carefully checked, it was then necessary to
determine the resonance of the transducer at liquid nitrogen temperatures (T=77K) with
4
He gas. Since the sound velocity in 4He gas at 77K is very well known, we are be able to
calibrate the wavelength by measuring the interference patterns. Once this is complete,
we can cool down to 4He temperatures (T=4.2K) and make high resolution measurements
of the phase velocity of liquid 4He, by again finding the resonance frequency and moving
the bimorph.
Results and Analysis
The probe was cooled down to liquid nitrogen temperature (T=77K), and
frequency sweeps were made in order to determine the resonance frequency, while the
bimorph was moved. The results are given in Fig. 8. The first resonance was found to be
10.462 MHz, and the second resonance was found to be 31.533MHz. We noticed that
both resonant frequencies shifted as expected. The data are shown in Fig. 9.
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Second Resonance of He 4 at T=77 K
0.0160
0.0005
0.0158
0.0000
X Out
Y Out
First Resonance of He 4 at T=77 K
X Out
Y Out
0.036
0.002
0.034
0.0152
-0.0020
0.0150
Y Out (Vrms)
X Out (Vrms)
-0.0015
X Out (Vrms)
0.032
-0.0010
0.0154
-0.0025
0.0148
-0.002
-0.004
0.030
-0.006
0.028
-0.0030
0.0146
-0.008
0.026
-0.0035
31300000
31400000
31500000
31600000
Y Out (Vrms)
0.000
-0.0005
0.0156
-0.010
10320000 10380000 10440000 10500000 10560000 10620000
31700000
Frequency (Hz)
Frequency (Hz)
FIG. 8
First and second resonance frequencies for He4 at T=77 K.
Bimorph Moving Up in Nitrogen Temp
Trial(1)
Trial(2)
Trial(3)
Trial(4)
10.2
10.8
10.0
10.6
9.8
Capacitance (pF)
Capacitance (pF)
Trial(1)
Trial(2)
Trial(3)
Trial(4)
Bimorph Moving Down in Nitrogen Temp
11.0
9.6
9.4
10.4
10.2
10.0
9.2
9.8
9.0
-10
0
10
20
30
40
50
60
70
80
-80
-70
-60
-50
-40
-30
-20
-10
0
10
Voltage (V)
Voltage (V)
FIG. 9
Data for the bimorph moving up and down at T=77K.
Once the probe was cooled down, the stray capacitance was negligible, as the elemental
capacitance was measured to be about 10 pF in agreement with the predicted value. The
bimorph displacement was found to be 0.05 micrometers per volt, which again was
expected.
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Due to certain complications with the Dewar and the probe involving certain
types of leaks, we were unable to proceed to calibrate the wavelength. Hence, our
experiment has come to a halt while we attempt to fix these leaks. Currently, the biggest
problem is that we are unable to determine whether or not the line that transports our
sample and pressurizes the cell is clogged. Our immediate goal is to solve this problem
without having to take the probe apart, and, so far, little progress has been made.
Conclusion
In conclusion, the construction of the experimental cell (including the bimorph),
cryostat probe, and spectrometer allow us to make the measurement of the first two
resonances and find the bimorph displacement at T=77K. Once the current obstacles are
overcome, it will take a short period of time to finish this experiment.
Acknowledgments
First and for most, we would like to show our sincere appreciation to Dr.
Yoonseok Lee for being a terrific mentor and allowing us to work and learn in his lab.
We would also like to thank Jose Cancino, Aaron Gray, Hyunchang Choi, and Pradeep
Bhupathi, whose dedication and help made this project possible. We would also like to
show our appreciation to the Physics’ department’s machine shop for building all of our
probes and gadgets.
Finally, we would like to thank the University of Florida and the
National Science Foundation for providing us this wonderful physics research
opportunity and funding this Research Experience for Undergraduate program.
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References
1. Helium Three, edited by W. P. Halperin and L. P. Pitaevskii, (Elsevier, 1990).
2. D.T. Grimsrud and J.H. Werntz, Phys. Rev. 157, 181-190 (1967).
3. P.J. Hamot, H.H. Hensley, and W.P. Halperin, J. Low. Temp. Phys. 77, 429447 (1989).
4. G.J. Slogget, N.G. Barton, and S.J. Spencer, J. Phys. A : Math. Gen 19, 27252736 (1986).
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Appendix – A
Cryostat Probe Specifics
16
17
All designs by Aaron Grey
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Appendix – B
Spectrometer Board Specifics
Designed by Jose Cancino