Resonant Ultrasound Spectroscopy
and the Elastic Properties of Several Selected Materials
D. Litwiller
Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011
July 31, 2000
This paper provides an introductory explanation of the technology, procedures and applications
pertinent to resonant ultrasound spectroscopy (RUS). Also included, is RUS analysis of
quasicrystalline and new superhard materials.
I. INTRODUCTION
II. ELASTIC CONSTANTS
As early as the 19th century, and throughout the
industrial revolution, the information provided by the
sound of an object’s resonance was utilized by a wide
range of people in a variety of occupations, if only on a
very superficial level. The resonance spectra of an object
can provide useful information about its structural
integrity. For instance, the harmonic resonance of a
cracked bell sounds distinctly different from one whose
structural integrity is preserved. This same idea was
widely employed throughout the early history of industry,
as a quality check for manufactured parts as the steel
smelting process was developed and refined. In the same
way the timbre of a musical instrument is dependent on its
structure, the resonance spectra of any solid state material
is similarly contingent upon the properties of its physical
composition.
Resonant ultrasound spectroscopy (RUS) can be used
to quantify the elastic properties of a wide range of solid
state materials by measuring their resonance spectra.
Elastic moduli are considered to be a thermodynamic bulk
property, just like specific heat, thermal expansion and
magnetization. These four properties are all related to
each other via basic thermodynamic laws. Until recently,
within the last decade or so, it has been relatively difficult
to make a comprehensive measurement of such elastic
properties. However, the commercialization of RUS has
made it relatively cheap and easy to determine a material’s
complete elastic tensor, in addition to its bulk modulus.
This technology has immediate physics-oriented
applications, because the information it provides allows a
more complete analysis of pressure-related condensed
matter experiments. Specifically, it allows one to know
how pressure will affect the other three bulk properties, for
instance, how a material’s magnetization will change as a
function of hydrostatic pressure. An accurate knowledge
of material’s elastic tensor also allows the analysis of a
number of non-bulk properties. Furthermore, the
technology utilized by the RUS instrument is used as a
non-destructive testing (NDT) device in a variety of
engineering applications. These topics will be further
discussed later in this paper.
How an object responds to a “small” surface stress, is
given by the generalized form of the well-known Hooke’s
law, such that
σ ij = cε ij
(2.1)
where σ is the applied stress, c is an elastic constant
specific to the material, and ε is the resulting strain. In this
generalized form, the subscripts indicate that stress and
strain are both tensor quantities, specifying any directional
σ33
3
1
2
σ31
σ32
σ13
σ23
σ11
P
σ21 σ12
σ22
dependence, where i refers to the direction of applied
stress, and j to the direction of the resulting strain. For a
three-dimensional object, an off-axis surface stress
produces a total of nine stress components (3
compressional, 6 shear), and consequently, nine
corresponding equations (see diagram). These nine
equations can be generalized into a 9x9 matrix, which,
because of certain symmetries/redundancies, can be
reduced to a smaller but less intuitive 6x6 matrix through
the subscript translation described below. The original
subscripts are altered so that each individual number refers
back to a single set of one of the nine pairs described
earlier. The translation is as follows:
11 → 1, 22 → 2, 33 → 3
23 (or 32) → 4, 13 (or 31) → 5, 12 (or 21) → 6
1
monitor the transducers are very sophisticated, they are
extremely easy to use. The setup consists of a main board
and a preamplifier (see Figure 3b). The main board, which
contains the frequency synthesizer that drives the first
transducer, is also responsible for receiving the preamp
signal and communicating with the PC used to control the
experiment. The preamp is a smaller device, which
receives and subsequently relays the sample’s response
back to the main board (see Figure 3c).1
The new “simplified” matrix below represents a crystal
with twenty-one independent elastic constants. (Notice the
symmetry around the matrix diagonal.)
 σ 1   c11
  
 σ 2   c12
σ   c
 3  =  13
 σ 4   c14
  
 σ 5   c15
σ   c
 6   16
c12
c22
c23
c24
c25
c26
c13
c23
c33
c34
c35
c36
c14
c24
c34
c44
c45
c46
c15
c25
c35
c45
c55
c56
c16  ε1 
 
c26  ε 2 
c36  ε 3  (2.2)
 
c46  ε 4 
 
c56  ε 5 
c66  ε 6 
Sample Geometry
Because it is much easier to model a sample of regular
geometry than one of arbitrary shape, the mathematical
tools used to derive a given material’s elastic constants
rely on samples that exhibit symmetry within traditional
coordinate systems (i.e. Cartesian, cylindrical and
spherical). The DRS program has provisions for both
rectangular parallelepiped (RP) and cylindrical sample
geometry. In order to achieve high-accuracy results
(±0.1%), it is necessary that the geometrical errors
associated with the sample be equally as small. This
means our samples must have polished faces, sharp edges
and corners, in addition to being homogeneous, and free of
cracks (as well as grain boundaries for single crystals).1
The most common
sample shape used in our
measurements was the
rectangular parallelepiped.
To prepare a material for
testing, it is first cut with a
diamond
saw
to
an
approximate size of several
millimeters per side.
In
case of a single crystal, the
cuts are made parallel and
perpendicular
to
the
direction of growth in order
to preserve the lattice
orientation. This proves to
Figure 3a
Some of these constants are specific to certain types of
elastic behavior, such as c11 and c33 (compressional
moduli), as well as c44 and c66 (shear moduli). For
crystals exhibiting a high symmetry, such as cubic or
hexagonal, the number of independent elastic moduli is
decreased significantly (to 3 and 6, respectively), and
many of the matrices’ off-diagonal elements become zero,
due to isotropy within certain planes of the crystal lattice.
In fact, a completely isotropic solid only has two elastic
constants, c11 and c44.
Another important elastic property, one that is the
main interest of this paper, is the bulk modulus cB, which
is defined as
cB =
σ
∆V
(2.3)
where σ is an applied hydrostatic stress (acting from all
sides) and ∆V is the object’s fractional volume change due
to this stress.
A material’s bulk modulus can be
determined from a its elastic tensor, and is widely used to
characterize general elastic response.
III. RUS BASICS
Resonant ultrasound spectroscopy (RUS) exploits the
aforementioned characteristics by measuring a sample’s
inherent resonance spectrum and fitting it to a theoretical
model, in order to derive the material’s elastic constants.
Apparatus
For the RUS measurements described in this paper,
we used the Dynamic Resonance Systems (DRS), Inc.
Modulus I. The actual heart of this system is a simple
piezo-electric transducer stage used to excite the sample’s
resonances over a wide range of frequencies (see Figure
3a). There are two transducers involved in this process.
The first transducer drives the sample, while the second is
used to measure the sample’s response; they are connected
to the instrument’s electronics via a pair of coaxial cables.
While the two pieces of electronics used to control and
Figure 3b
2
Transducer
condition. However, for a satisfactory fitting routine to
take place, test samples must be mounted and scanned so
that the upper transducer exerts the least possible amount
of pressure that still allows sufficient contact with the
sample. Transducer pressure that is too high tends to
dampen the vibrations, shifting the frequencies of the
resonance spectrum, and decreasing correlation between
theory and experiment. Although it is not obvious, where
the corners contact the transducers is also an important
issue. In order to detect all of the RP’s modes of
oscillation, the sample must be mounted near the
transducer edges. Some resonances produce displacements
in the horizontal direction, making detection very difficult
with a device that is mounted within the horizontal plane.
Fortunately, the transducer faces do not distort evenly, and
as a result, there is some bulging at the edges, which can
be exploited to detect lateral motion as long as care is
taken to place the sample correctly.1
Preamplifier
Amplifier
Sample RP
Mixer
A/D Converter
DualSynth.
Transducer
To Computer
Figure 3c
be an important step when it comes time to actually fit the
sample’s measured resonance spectrum to a theoretical
model. After cutting the sample into a rough rectangular
shape, its faces are hand-polished with a number of
different sandpaper grades (down to 3µm). To ensure that
the edges are not rounded in the process, the sample is
wax-mounted on the end of a small aluminum cylinder,
which slides inside a stainless steel ring to prevent the
apparatus from rocking back and forth. Furthermore, the
polishing is carried out in a figure-eight pattern on a
precision ground granite block to provide a flat surface for
the paper, thereby minimizing any other imperfections that
might degrade the sample surface. The combination of
these two precautions results in a rectangular
parallelepiped sample with exceptionally smooth surfaces
and sharp edges.
As it turns out, polycrystalline samples tend to polish
fairly easily because there is no distinct macroscopic
lattice where accidental cleaving might occur. However,
experience reveals that special care should be exercised
when preparing single crystals and brittle samples. Not
only are they difficult to cut without chipping, but they can
be somewhat difficult to polish as well, especially with
lower grade papers which tend to spawn cracks and cause
chipping along the sample’s edges.
Finally, after being checked and rechecked for
significant cracks and chips, the sample is cleaned in an
ultrasonic acetone bath to remove residual wax and dust
collected during the polishing process. Now the material
is ready for testing.
Scanning
A desktop Pentium III PC is used to run the scans,
collect and analyze the data; it communicates with the
main board through the serial COM1 port connection. The
DRS program (version 1.14.1R) is compatible with
Windows and allows the user to control scan settings such
as the frequency range, number of data points as well as a
short list of more technical parameters. When the scan is
completed, the data is displayed as a plot of sample
response versus frequency. Typical scans run anywhere
from 100 kHz to 3 MHz, while the sample response is
usually limited to eight volts to prevent peak distortion.
An extensive list of tools is available for controlling the
display and manipulating data.
Analysis
The measured peaks in the resonance spectrum
represent the sample’s normal modes of resonant
oscillation; they are electronically marked and recorded.
After this tagging procedure, the fitting routine can be
executed. The actual fitting process is extremely
complicated and involves a sophisticated Lagrangian
minimization procedure, i.e. minimizing the difference
between the kinetic and potential energies over the entire
sample volume. During this procedure, the computer
generates a theoretical list of resonances based on the
given sample parameters. It then compares its list with the
experimental input and tries to match experiment and
simulation by adjusting the elastic constants used to
calculate the theoretical spectrum. The computer repeats
this process until the RMS error between the two lists has
been minimized. It then displays the two lists along with
the associated error and the elastic contants of the
theoretical list. If the fitting error is not too high (typically
0.1-0.2%) one can assume that the elastic contants
generated by the computer accurately represent those of
the test material.1
Sample Mounting
The actual transducers themselves are a mere
millimeter in diameter, but they are mounted behind a thin
Kapton film, which provides structural support and
damping (citation). Transducer contact with the sample
only occurs through a tiny hole in the Kapton film. The
requirements for a proper sample mounting are quite
simple. First, the RP is first mounted on two of its corners
between the transducers. During the fitting process, the
computer’s algorithm calculates a resonance spectrum for
a free body, that is, a body that is free of external damping.
A diagonal mount is the easiest way to achieve this
3
shifted, however, no obvious improvement was observed
during the refitting process. It is believed that internal
dissipation of the vibrational energy in those soft samples
may be responsible for the poor performance of their
resonance spectra.
On the other hand, single crystals such as the silicon
standard, seem to be easier to test and produce more
reliable results. In this case, however, the 0.6% fitting error
may be a little deceiving. There is an available option with
the fitting routine that allows the program to “float” or
adjust the sample dimensions to better fit the experimental
spectra. This option is useful when the fitting error is
already low, because the program does not need to make
significant adjustments. For this study, sample dimensions
were typically determined with a pair of high-accuracy
digital calipers (±1 µm). Therefore, allowing computer to
adjust the measurements more than a few micrometers is
somewhat unreasonable.1 Because of this oversight, the
error associated with the single crystal silicon may be an
inaccurate depiction of its goodness of fit. Of the
standards, silicon proved to be the most reproducible, so as
to why the sample may not have fit better in the first place,
there is only one possible explanation. It is likely that
there may be some systematic error associated with the
unusual way the silicon was cut (along the [110] axis).
Without x-ray diffraction, such a cut would be difficult to
prepare at best, and an error or a degree or more could
easily throw off the fitting routine.1
In order to test experimental reproducibility, several
spectra were taken under different conditions on the same
sample. The observed peak shifting from trial to trial was
observed to be in the range of 100-200 Hz and was
randomly distributed, independently of frequency, i.e. for a
typical RUS trial, the associated relative error decreases as
a function of frequency. This is one of the reasons it is
important to include a substantial list of peaks when trying
to fit the spectra; as the scan frequency increases, the
effective fitting error decreases in a similar fashion,
meaning that larger discrepancies have lessened effects. In
addition, it was also observed that precision increased for
IV. STANDARD SAMPLES
In order to gain an idea of the RUS instrument’s
capabilities, in terms of both accuracy and reproducibility,
several common materials with well-known bulk moduli
were analyzed. The tests also served as a way to learn
sample preparation and the computerized fitting routine.
Over the course of several weeks, a total of 6 standard
samples were prepared and analyzed, including
polycrystals of the following materials: aluminum, copper,
tool steel and stainless steel (see Appendix A). Tests were
also run on a single crystal of ultra-pure silicon, cut along
the [110] direction of its crystal lattice. Rectangular
parallelepipeds of each material were prepared and tested.
Results and Discussion
Generally, a good scan quality was obtained for the
measured samples. In some cases, however, the fitting
procedure did not produce satisfactory results (see Table
4.1). ‘Soft’ materials like copper and aluminum generally
perform less well in terms of their fitting errors,2 as was
the case for our experiments (aluminum usually tested
above 1%). As was noted before, the fitting algorithm has
been idealized to deal with non-dissipative (hard)
materials, which a valid approximation in most cases.
Another possible source of error for the copper standard #1
may have been its relatively large mass (about three times
that of the other standard samples), which added
substantial inertia to the system and may have increased
sample/transducer pressure.
In an effort to investigate the nature of some
inconsistencies with respect to the spectra of copper and
aluminum, the samples were annealed at two-thirds their
melting temperature for eight hours each. The heat
treatment effectively increases the mobility of the metals’
atoms, which helps heal out imperfections in the crystal
lattice, thereby decreasing internal stress. Subsequently,
the samples were rescanned and fitted. As a result of the
annealment, the resonance spectrum appeared to have
Table 4.1: Comparison of experimental elastic moduli with literature values.
Experimental
Material
Aluminum
Copper #1
Copper #2
Silicon
Stainless Steel
Tool Steel
Fit
Isotropic
Isotropic
Isotropic
Cubic 3:6
Isotropic
Isotropic
c11
1.1228
2.7063
5.0461
1.6359
2.7555
2.8013
c44
0.2629
0.4732
0.4623
0.7844
0.7698
0.8442
cB
0.772
2.075
1.43
0.958
1.729
1.676
Literature
Fitting Error (%) cB Error (%)
1.319
~ 0.76
1.58
2.3136
~ 1.4
48.2
0.7379
~ 1.4
2.14
0.6202
~ 0.98
2.24
0.5301
~ 1.7
1.71
0.1205
~ 1.7
1.41
12
2
Elastic moduli are in units of 10 dyn/cm .
4
samples that were harder and smaller. The number of
missed peaks from scan to scan was always higher for the
softer materials (i.e. copper and aluminum) which seems to
be consistent with internal damping.2
Accuracy of the standard’s fits seems to be in good
agreement with the literature values for each respective
material.
With the exception of the silicon, the
discrepancies (on the order of a few percent) may be
mostly attributed to inherent variation in the purity of
materials.
In addition to the systematic error caused by sample
imperfections and the discrepancies that arise from the
idealistic modeling of a physical system (in this case about
0.1%), there is one other important source of systematic
error to be reckoned with. Even though typical peak
shifting is small from scan to scan, peak amplitude can
change dramatically, depending on alterations to the
sample mount and transducer pressure, which can be a
source of systematic error during the fitting procedure,
especially if peaks are accidentally overlooked. As was
discussed earlier, too much pressure can result in peak
shifting and sample dampening. Furthermore, an RP’s
rotational orientation around its vertical axis seems to play
a role as well. This is related to the fact that transverse
modes of oscillation that are also tangential to the
transducer’s circumference cannot be detected. Therefore,
a small sample rotation can have a significant effect on its
measured resonance spectrum and also the quality of its fit.
Ultimately, a given scan’s quality is best determined
by its resulting fitting error. With the present state of this
technology, there is an effort to achieve a fitting error of
less than 0.2% RMS, in other words, accuracy to within
one or two parts per thousand; this is considered
acceptable.1 However, values around 1% also tend to
provide fairly reliable elastic constants, although one
should treat the numbers with caution and use them
conservatively. Errors exceeding 1% were sometimes seen
to provide a good ballpark values, although there is no
obvious way of determining to exactly what extent they are
accurate. Fitting error greater than 1% is usually indicative
of some source of systematic error, whether it be samplerelated or peak-related. Peaks that are accidentally missed
during the marking process produce very significant fitting
errors (typically above 5%), rendering the generated elastic
moduli totally useless. This is another reason why scan
quality is of such importance.
Having discussed experimental procedures and the
reliability of the RUS system, I would like to discuss
measurements of elastic properties of several new
materials, such as quasicrystalline samples, and a family of
new superhard materials recently synthesized at Ames
Laboratory.
V. QUASICRYSTALS
Since their discovery in 1984, quasicrystals have
provided a rich field of study for a number of physicists
and mathematicians. What distinguishes a quasicrystal
from a traditional crystalline solid is its “quasiperiodic”
lattice structure, a property that was once believed to be
fundamentally impossible in three-dimensional space.
Unlike a regular crystalline lattice which consists of selfrepeating fundamental units (unit cells), a quasicrystal
does not exhibit long-range translational or rotational
symmetry. In order for a material to be classified as a
periodic crystal, it must exhibit both local symmetry and
long-range order. For quasicrystalline materials, however,
one observes local symmetry without the existence of
traditional long-range order.3 In the following section, we
will determine whether certain elastic properties of these
materials can be emulated by assuming certain isotropic
characteristics.
Decagonal Al-Ni-Co
In its lattice plane, Al-Ni-Co exhibits a decagonal
structure, that is, a ten-fold local symmetry. In this case,
the lattice is comprised of stacks of interlocking decagons
and pentagons. The material’s macroscopic crystal
morphology within the staggered planes is pentagonal (see
Figure 5.1).
A flux-grown single crystal was obtained from Ian
Fisher (Ames Lab). Single-crystalline Al-Ni-Co is a
tough, scratch resistant material with a shiny mirror-like
surface.
The sample was rough-cut and polished
preserving its original orientation of growth. The polishing
process proved to be somewhat time-consuming due to the
sample’s hardness, however, the sample also resisted
chipping, surprisingly, which made it fairly easy to polish
to near rectangular perfection. Its final dimensions were
2.122x1.910x1.760 mm3.
Scan quality for the Al-Ni-Co was remarkably good,
and background noise was extremely low for most of the
scanned region. It revealed a resonance spectrum with
Figure 5.1: Al-Ni-Co Quasicrystal.
a) decagonal
atomic
structure
5
b) pentagonal
macroscopic
structure
c) hexagonal
approximation
Table 5.1: Quasicrystals
Independent Elastic Moduli
Material [Batch No.] Lattice
Fit
cB Fitting Error (%) c11
c33
c13
c12
c44
c66
Al-Ni-Co [AP 428] Decagonal Hexagonal 1.159
0.2889
2.2911 2.2382 0.6274 0.5486 0.6676 0.8704
Y-Mg-Zn [AP 426] Icosahedral Isotropic 0.742
0.993
1.3012
0.4195
12
2
Elastic moduli are in units of 10 dyn/cm .
Y-Mg-Zn, notably its low-temperature thermal
conductivity. Thus far, studies have been conducted on the
material’s
electronic
transport
(conductivity),
magnetoresistance, as well as specific heat and optical
properties.5 To date, there have been no reported
measurements on the elastic properties of this particular
quasicrystal. In Table 5.1 are the first results on the elastic
constants of this material.
A flux-grown sample of Y-Mg-Zn provided by Ian
Fisher was cut to size and polished to the final dimensions
of 1.687x1.608x1.268 cm3. In this case we did not attempt
to preserve the macroscopic morphology of the grown
sample shape, because electron and x-ray diffraction
measurements show that Y-Mg-Zn is nearly isotropic at
the microscopic level. In this case, the quasicrystal proved
to be difficult to polish due to its softness, which will be
reflected later in its relatively low bulk modulus.
Furthermore, the flux growth method can leave tiny
gaps/bubbles in the crystal lattice. We generally try to
avoid these by polishing such inhomogeneities out of the
sample faces.
Although the sample proved to be testable, it should
be noted that scans for the Y-Mg-Zn were of much lower
lower than for the Al-Ni-Co, possibly due to sample
softness or geometrical imperfections. The combination of
enhanced background noise and broadened (low-Q) peaks
complicated proper identification. Nevertheless, we were
able to fit the 53 measured peaks to an isotropic
simulation, with a fitting error of slightly less than 1%
RMS, resulting in a bulk modulus of approximately 74
GPa. In elastic terms, icosahedral Y-Mg-Zn seems thus to
be well described by an isotropic model.
sharp, well defined (high-Q) peaks, owing to the fact that
the sample was an ideal size, in addition to being hard and
close to geometrical perfection. Several fitting approaches
using different hypothetical lattice approaches were used.
A hexagonal approximation turned out to fit the
experimental data extremely well, suggesting isotropy
within the decagonal plane, anisotropy perpendicular to it,
as was expected. Results of the computerized fitting
routine are shown in Table 5.1. Our findings are in
reasonable agreement with those reported earlier by
Chernikov et al.4 However, in addition to our fitting error,
which may have originated from minute sample defects or
slight lattice misalignment, variations of our elastic moduli
typically lie within 5% of the published values (Table 5.2).
It is also worth noting that the crystal used by Chernikov et
al. was grown using a slightly different approach, and
there are undoubtedly small variations between the two
sample batches, as quasicrystals are difficult to prepare.
Table 5.2: Al-Ni-Co, correlation with literature.
12
2
Elastic moduli are in units of 10 dyn/cm .
Elastic Moduli
c11
c33
c13
c12
c44
c66
Experimental Published
2.2911
2.343
2.2382
2.3221
0.6274
0.6662
0.5486
0.5736
0.6676
0.7019
0.8704
0.8845
average:
% error
2.22
3.61
5.82
4.36
4.89
1.59
3.75
Figure 5.2: Y-Mg-Zn Quasicrystal.
Icosahedral Y-Mg-Zn
We also studied a second quasicrystal of Y-Mg-Zn,
which exhibits local icosahedral symmetries, i.e. an
arrangement of twenty equilateral triangles (see Figure
5.2). Surprigingly, the crystal assumes a dodecahedral
shape at the macroscopic level, an Archimedean polytope
comprised of 12 pentagons.
A recent paper by Gianno et al. reports a general
overview of the material’s properties of single crystalline
a) icosahedral
atomic
structure
6
b) dodecahedral
macroscopic
structure
appear on the RUS scan as a single peak. Fortunately, the
computer-generated theoretical list of resonances can
provide some guidance for instances like these. Fitting
results improved after degenerate resonances were added
to the list of experimental peaks. Interestingly, in the case
of these highly symmetric wafers, the more symmetrical
the sample, the more difficult its spectrum is to mark and
fit. Again, this is due to the significant number of
degenerate resonance peaks, making their identification
more difficult without the guidance of the theoretical
simulation.
Table 6.1 shows, for the first time, experimental
results on the elastic properties of these materials.
Obviously, for the sake of sample consistency, there is
much work to be done in terms of refining the material’s
preparation process. In fact, the reason the different
batches are so inconsistent with each other may be related
to the fact that they are polycrystals and probably exhibit
different grain sizes, and therefore different elastic
properties. In the future, RUS analysis will be performed
on a number of additional samples, including higher
quality polycrystals and single-crystalline versions of a
variety of dopings. Thus the numbers reported here are
only the beginning of what may become an extensive look
at this particular family of superhard materials.
VI. NEW SUPERHARD MATERIALS
In October of 1999, Ames Lab researchers discovered
what is believed to be one of the hardest materials known.
The material consists of a base alloy of aluminum,
magnesium and boron (AlMgB14), the hardness of which
can be optimized by proper doping. In this particular case,
the material was silicon-doped, however, efforts have been
made to increase the material’s hardness even further by
experimenting with alternative dopings.
The materials were provided by Bruce Cook of Ames
Laboratory. They included of two different batches of the
base compound, AlMgB14, as well as two TiB2-doped
versions. Because of the material’s unusually high
hardness, attempts to use a diamond saw proved to be
inefficient in terms of time. An attempt to fashion an RP
by means of spark erosion (Ames Lab machine shop) was
problematic because of the materials low conductivity.
However, one RP was prepared by first scoring the surface
and then breaking the material; it was subsequently
polished and tested. The other three samples were tested in
their original cylindrical form, specifically, the thin
circular wafers in which they were first pressed and
sintered. The samples were typically 0.2 cm thick with a
1.5 cm diameter, which is about the maximum size that the
transducer stage will allow. In fact, two of the wafers were
so large that they were too wide for the spring-mounted
transducer spread, which had to be stretched to
accommodate their size. Obviously, for a circular wafer no
sharp corner exists, meaning that the rounded edge is too
flat to fit down into the depression in the Kapton film. As
a result, vibrational contact between the transducer and
sample was limited to communication via the Kapton film
and a thin layer of dried epoxy. Also, the force required to
hold the sample in this position, with enough pressure to
establish good sample/transducer contact via the edges of
the Kapton foil, should have proven to be problematic for
the accuracy of the resonance spectra obtained.
Nevertheless, the experiment was commenced and
although the scans were noisier than previous experiences,
the samples did indeed show sharp resonances. A
cylindrical fitting option was used during the analytical
portion of this experiment. A further complication that
arose while testing these wafers, was the existence of
degenerate peaks, pairs of overlapping resonances that
SUMMARY AND OUTLOOK
The measurement of a material’s elastic tensor and
bulk modulus can have extensive applications within the
realm of physics. Because the elasticity of a material is
such a basic property, it is related to a number of other
non-bulk properties as well, properties such as electron
transport (resistivity). As a result, the knowledge of a
material’s elastic tensor can be utilized by both theorists
and experimentalists in order to gain a better overall
understanding of a material’s general physical properties.
Crystal elasticity can even be related to more unique
properties, such as superconductivity. The ability for a
pair of electrons to travel in a tandem arrangement that is
in phase with a material’s thermal lattice vibration is what
is known as cooper pairing, and gives rise to
superconductivity. The potential for cooper pair
propagation can be dependent on lattice stiffness.
Therefore, a material’s elastic tensor may provide a certain
Table 6.1: Superhard Materials
Material
AlMgB14 (RP)
AlMgB14
AlMgB14 +TiB 2
AlMgB14 +TiB 2
Batch No. Sample Quality
3B
Good*
AL-03
Excellent
SH-12
Poor**
4B
Fair*
Fit
Isotropic
Isotropic
Isotropic
Isotropic
cB Fitting Error (%)
1.372
0.6845
1.949
0.1247
1.8339
1.9526
1.7519
0.5723
c11
3.1831
4.2343
4.0084
3.8153
12
c44
1.3580
1.7140
1.6309
1.5436
2
Elastic moduli are in units of 10 dyn/cm .
*chipped edges
**probable microcracking
7
amount of insight when it comes to studying
superconductivity and searching for new and improved
superconductors. In addition to basic research, the
technology of resonant ultrasound spectroscopy is also
widely used in industry as a means for non-distructive
materials and device testing, giving rise to a new level of
quality control and product reliability. Future applications
for Ames Lab include expanded RUS capabilities that
include temperature-dependence (300-1K). This will
allow a more complete look at a material’s elastic
properties as they are affected by temperature, allowing
experimentalists to detect solid-state phase transitions and
lattice softening effects.
Acknowledgements
Special thanks would like to be extended to Robert
Modler, Marzia Rozati, Ian Fisher and Bruce Cook, in
addition to the National Science Foundation, Iowa State
University and Ames National Laboratory.
References
1
A. Migliori and J. L. Sarrao, Resonant Ultrasound
Spectroscopy (Wiley, New York, 1997).
2
F. Willis, DRS Inc., private communication.
3
M. Senechal, Quasicrystals and Geometry (Cambridge
Univ. Press, Cambridge, 1995).
4
M.A. Chernikov, H.R. Ott, A. Bianchi, A. Migliori, and
T.W. Darling, Phys. Rev. Lett. 80, 321 (1998).
5
K. Gianno, A.V. Sologubenko, M.A. Chernikov, and
H.R. Ott, Phys. Rev. B 62, 292 (2000).
8
Appendix A
Purity, Mass and Dimensions of RUS Test Samples
Dimensions (cm)
Material
Purity
Mass (g)
x
y
z
Shop Grade
0.7628
0.6682
0.6578
0.6455
Copper #1
HOFC*
2.5099
0.6693
0.6561
0.6425
Copper #2
HOFC*
0.6975
0.6087
0.3991
0.3221
Standard RPs: Aluminum
Silicon
99.999%
0.2415
0.8206
0.3763
0.3368
Stainless Steel
Shop Grade
0.4267
0.4704
0.3788
0.3042
Tool Steel
Shop Grade
0.4059
0.464
0.3724
0.3051
*High-purity Oxygen-Free Copper
Quasicrystalline RPs: Y-Mg-Zn (AP 426)
-
0.01734
0.16873
0.16076
0.1268
Al-Ni-Co (AP 428)
-
0.02936
0.2126
.1910
.1760
-
0.0621
0.597
0.222
0.186
Diameter (cm)
Thickness (cm)
Superhard RP: AlMgB14 (3B)
Superhard Wafers: AlMgB14 (AL-03)
AlMgB14 +TiB 2 (SH-12)
AlMgB14 +TiB 2 (4B)
-
0.4424
1.040
0.183
-
0.5798
1.295
0.149
-
0.7835
1.273
0.226
9
Appendix B
Density, Hardness, Bulk Modulus and Shear Modulus
of Selected Hard Materials
Density
Material
C (diamond)
BN (cubic)
C3N4 (cubic)
AlMgB14+TiB2 (4B)
AlMgB14+TiB2 (SH-12)
AlMgB14 (AL-03)
AlMgB14 (3B)
TiB2
TiC
WC
SiC
AlB12
Al2O3
Si3N4
3
(g/cm )
3.52
3.48
2.724*
2.954*
2.846*
2.519*
4.5
4.93
15.72
3.22
2.58
3.98
3.19
Hardness
Bulk Modulus
Shear Modulus
(Gpa)
70
~ 48
~47
~ 43
~ 43
~ 34
~ 34
~ 32
~ 29
~ 27
~ 26
~ 26
~ 22
~ 19
(Gpa)
443
400
496
175*
183*
195*
~ 137*
244
241
421
226
246
249
(Gpa)
535
409
332
154*
163*
171*
~ 136*
263
188
196
162
123
*Values obtained during this study (please note the errors in Table 6.1)
10
2
One gigapascal is equal to 1x10 dyn/cm .
Other values borrowed from table published
by Cook et al. (Scripta mater. 42 (2000) 597-602)
10