# Resonant Ultrasound Spectroscopy for Irregularly-Shaped Samples and its
Application to Uranium Ditelluride

Florian Theuss  Laboratory of Atomic and Solid State Physics, Cornell
University, Ithaca, NY 14853, USA  Gregorio de la Fuente Simarro  Laboratory
of Atomic and Solid State Physics, Cornell University, Ithaca, NY 14853, USA
Avi Shragai  Laboratory of Atomic and Solid State Physics, Cornell University,
Ithaca, NY 14853, USA  Gael Grissonnanche  Laboratory of Atomic and Solid
State Physics, Cornell University, Ithaca, NY 14853, USA  Ian M Hayes  Quantum
Materials Center, Department of Physics,University of Maryland, College Park,
Maryland 20742, USA  Shanta Saha  Quantum Materials Center, Department of
Physics,University of Maryland, College Park, Maryland 20742, USA  Tatsuya
Shishidou  Department of Physics, University of Wisconsin-Milwaukee,
Milwaukee, Wisconsin 53201, USA  Taishi Chen  The Institute for Solid State
Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Satoru
Nakatsuji  The Institute for Solid State Physics, The University of Tokyo,
Kashiwa, Chiba 277-8581, Japan  Department of Physics, The University of
Tokyo, Tokyo 113-0033, Japan  Institute for Quantum Matter and Department of
Physics and Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA
Trans-scale Quantum Science Institute, University of Tokyo, Tokyo 113-0033,
Japan  Sheng Ran  Department of Physics, Washington University in St. Louis,
St. Louis, MO 63130, USA  Michael Weinert  Department of Physics, University
of Wisconsin-Milwaukee, Milwaukee, Wisconsin 53201, USA  Nicholas P Butch
Quantum Materials Center, Department of Physics,University of Maryland,
College Park, Maryland 20742, USA  NIST Center for Neutron Research, National
Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg,
Maryland 20899, USA  Johnpierre Paglione  Quantum Materials Center, Department
of Physics,University of Maryland, College Park, Maryland 20742, USA  B. J.
Ramshaw<EMAIL_ADDRESS>Laboratory of Atomic and Solid State Physics,
Cornell University, Ithaca, NY 14853, USA

###### Abstract

Resonant ultrasound spectroscopy (RUS) is a powerful technique for measuring
the full elastic tensor of a given material in a single experiment.
Previously, this technique was limited to regularly-shaped samples such as
rectangular parallelepipeds, spheres, and cylinders [1]. We demonstrate a new
method for determining the elastic moduli of irregularly-shaped samples,
extending the applicability of RUS to a much larger set of materials. We apply
this new approach to the recently-discovered unconventional superconductor
UTe2 and provide its elastic tensor at both 300 and 4 kelvin.

##  I Introduction

UTe2 is a recently-discovered unconventional superconductor [2] with large
upper critical fields [3, 2] and an NMR Knight shift [4, 5, 6] that suggest
spin-triplet pairing. There are, however, many unsolved mysteries in UTe2,
including field-reentrant superconductivity [7, 8, 9, 10], time-reversal
symmetry breaking [11, 12], a phase transition between superconducting ground
states as a function of magnetic field [13], and the occurrence of two
superconducting transitions in certain samples [14, 12, 15, 16]. Underlying
many of these mysteries is the question of what is the symmetry of the
superconducting order parameter in UTe2.

Externally applied stress has proven to be a useful tuning parameter when
investigating these types of questions. For example, there are two unambiguous
superconducting phase transitions in UTe2 under hydrostatic pressure [17, 10],
and uniaxial pressure experiments [18] imply an insensitivity of the
superconducting order parameter to shear strain. However, while stress and
pressure are conceptually straightforward parameters to tune externally, the
more physically-relevant quantity—related to microscopic bond distances and
unit cell volumes—is strain. Stress, $\sigma$, and strain, $\varepsilon$, are
linearly related through the elastic tensor,
$\sigma=\boldsymbol{c}\varepsilon$, and converting from the experimentally-
determined stress to strain requires the full elastic tensor. For example, the
full elastic tensor was central in determining the quantitative relationship
between strain, the van Hove point, and superconductivity in Sr2RuO4 [19, 20],
as well as the relationship between anisotropic strains and superconductivity
in CeIrIn5 [21].

The full elastic tensor of a material can be determined with resonant
ultrasound spectroscopy (RUS). RUS measures the mechanical resonance spectrum
of a three dimensional solid. The resonance frequencies are determined by both
intrinsic sample properties—the density and elastic moduli—as well as by the
sample boundary conditions. If the elastic moduli, density, and sample
geometry are known, then the resonance frequencies are easily calculated
numerically—the “forward problem”—either using the method of Visscher  _et
al._ [1] or by finite elements [22, 23]. The inverse problem—obtaining the
elastic moduli from a measured resonance spectrum—is more challenging because
it requires multiple numerical evaluations of the forward problem and the
navigation of a complex parameter landscape with many local minima [24].

For simple geometries with easily-defined boundaries—typically rectangular
parallelepipeds, cylinders, or spheres—the method of Visscher  _et al._ [1]
can be combined with either Levenberg-Marquardt or heuristic (such as genetic
algorithm) fitting methods to solve the inverse problem and obtain the elastic
moduli from the resonance spectrum. For irregular samples, however, finite
elements has been traditionally used to solve the forward problem. The
difficulty with this approach is that finite elements is too computationally
intensive to use when solving the inverse problem. This has largely restricted
the applicability of RUS to materials where regularly-shaped samples can be
prepared.

In the case of UTe2, preparation of a rectangular parallelepiped is difficult
due to the brittle nature of the material (in addition to the potential
hazards associated with polishing uranium compounds). In this work, we take
advantage of the recent development by Shragai  _et al._ [25] that solves the
forward problem for irregularly-shaped samples in a way that is two orders of
magnitude faster than finite elements. This has allowed us to develop a
protocol for solving the inverse problem for irregularly-shaped samples. We
demonstrate this protocol on samples of SrTiO3 and Mn3Ge—compounds with known
elastic moduli—and then apply our new technique to obtain the full elastic
tensor of UTe2 at 300 K and at 4 K.

##  II Methods

Figure 1: CT-Scan Models. 3D models of the irregularly-shaped samples used for
RUS measurements based on CT-scans. From left to right: SrTiO3 sample B, Mn3Ge
sample B, UTe2.

Resonant ultrasound spectroscopy. We performed RUS measurements by placing a
sample in weak mechanical contact between two piezoelectric transducers,
providing nearly-free boundary conditions. One transducer is driven with an AC
voltage at fixed frequency, and the voltage generated on the other transducer
is measured using a lockin amplifier. By stepping the drive frequency from
roughly 100 kHz to 5 MHz, we obtain the first one hundred or so mechanical
resonances for a typical, mm-scale sample. More details of the experimental
setup and how to measure resonance spectra can be found in Ramshaw  _et al._
[24] and Balakirev  _et al._ [26]. Full lists of all experimental resonance
spectra used in this study are given in the supplement.

Fitting resonance spectra. Resonance spectra are calculated by minimizing the
linear elastic Lagrangian as a function of the displacement field
$\vec{u}\left(\vec{r},t\right)$,

$\mathcal{L}=\frac{1}{2}\int\left(\sum_{i}\rho\dot{u}_{i}^{2}(\vec{r})-\sum_{ijkl}c_{ijkl}\frac{\partial
u_{i}(\vec{r})}{\partial r_{j}}\frac{\partial u_{k}(\vec{r})}{\partial
r_{l}}\right)dV,$ (1)

where $\rho$ is the density, $c_{ijkl}$ is the elastic tensor, and the
integral is over the entire volume of the sample. The widely-adopted method of
Visscher  _et al._ [1] expands the displacements fields in a polynomial basis
and solves the volume integrals numerically. This method works well for
rectangular parallelepipeds, cylinders, and spheres [26], where the boundaries
are easily defined. For irregularly-shaped samples, however, this method is
insufficient and new methods for computing the resonance spectra must be used.

We implement two new resonance spectra calculation methods—two “forward”
solvers—into a genetic algorithm, extending the RUS fitting routine to
irregularly-shaped samples. Details of the genetic algorithm itself can be
found in Ramshaw  _et al._ [24] and in the supplement; here we focus on
incorporating and verifying the new forward solvers.

The first forward solver extends Visscher’s method to irregular samples by
converting the volume integrals to surface integrals [25]. We will refer to
fits performed with this forward solver as SMI (surface mesh integration
[25]).

The second forward solver is a commercially-available finite element solver:
Comsol. We implement a Comsol forward sovler to verify the newer SMI method
(note that the use of FEM to calculate the forward problem for RUS was
described earlier in Liu and Maynard [22]). We will refer to fits performed
using Comsol as FEM.

Both SMI and FEM-based fits are compared to fits using Visscher’s approach for
rectangular parallelepiped samples. Fits with Visscher’s approach are referred
to as RPR.

Sample digitization and alignment. Both the SMI and FEM methods require three-
dimensional digitizations of the samples. FEM uses the full, three-dimensional
tetrahedral mesh of the entire sample volume. SMI uses only the surface
triangles of the same mesh (this includes any “interior” surfaces around
voids). These digitizations were obtained with a Zeiss Xradia Versa XRM-520
X-ray nano-CT and are shown in Figure 1. The mesh size used for our fits
depends on the solver method, as well as on the sample shape and size. We use
a mesh with average linear dimension 10 $\mu$m for SMI, and 60 $\mu$m for FEM
(FEM uses a larger mesh because it is much more computationally intensive than
SMI). The samples are oriented to within 1 degree using Laue backreflection
diffractometry. The mesh is then aligned to the crystal axes by identifying
two flat faces on the sample which uniquely relate the surface mesh to the
orientation of the sample in the Laue apparatus. More details on sample
digitization can be found in the SI.

Measurement uncertainty. The uncertainties in our fits have two dominant
contributions. One contribution originates from small deviations from the
weak-coupling approximation: the resonance frequencies depending weakly on how
the sample is mounted in the apparatus. This is predominantly due to the
finite force exerted on the sample by the weight of the cantilever holding the
top ultrasonic transducer. By remounting the samples in different orientations
in our RUS apparatus, we find uncertainties in the elastic moduli of
approximately 2 GPa for the SrTiO3 and Mn3Ge samples, and 0.5 GPa for UTe2.
The second largest source of uncertainty is due to uncertainty in the
alignment between the crystal axes and the sample mesh. This uncertainty is
approximately 1 degree around each axis and results in uncertainties of
typically 0.03 GPa for all regularly-shaped samples. This uncertainty is
larger for all irregularly-shaped samples—on average 0.4 GPa, 2 GPa, 0.5 GPa
for the irregular SrTiO3, Mn3Ge, and UTe2 samples, respectively. A detailed
error analysis can be found in the SI.

Density functional theory calculations. We used density-functional theory to
produce estimates of the elastic moduli of UTe2. This involved total energy
calculations following the procedure of Ravindran et al. [27]. The full-
potential linearized augmented plane wave method [28] calculations employed
the generalized gradient approximation [29] for the exchange correlation, wave
function and potential energy cutoffs of 16 and 200 Ry, respectively, muffin-
tin sphere radii of 1.35 Å, and an $8\times 8\times 8$ $k$-point mesh. Spin-
orbit coupling was fully taken into account in the assumed nonmagnetic state.
The elastic tensor was extracted from fits of the total energy variations
around the experimental structure [30] to the energy-strain formula [27],
including linear terms. The resulting moduli are given in Table 3.

##  III Test Results

We test the implementation of two forward solvers—FEM and SMI—for fitting the
RUS spectra of irregularly-shaped samples using a genetic algorithm. We
compare these results to the moduli extracted for rectangular parallelepiped
samples that can be fit using the RPR forward solver, in addition to the FEM
and SMI methods. We find agreement between all methods and all sample
geometries to within our measurement uncertainties. It’s worth noting that
while both FEM and SMI are capable of fitting RUS spectra from irregularly-
shaped samples, the SMI method is two orders of magnitude faster than FEM,
taking under an hour to produce a fit while FEM takes several days or even
weeks.

SrTiO3. Our first test system for our new fitting method is SrTiO3, whose
elastic tensor consists of only three independent elements due to its cubic
crystal structure (point group $O_{h}$). RUS measurements and fits are
performed on two samples (Table 1): Sample A is was polished into a
rectangular parallelepiped with dimensions $\left(1.49\times 2.035\times
3.02\right)$ mm3, with edges oriented along the crystallographic axes. We can
perform fits using all three methods—RPR, SMI, and FEM—in this simple
geometry. Sample B was prepared with an irregular shape (see Figure 1 for a 3D
model based on a CT scan). Only SMI and FEM fits are possible for this
geometry.

The fit results are shown in Table 1. Comparing the fits within each sample,
we see that all methods yield identical results within uncertainties ($\pm 2$
GPa). This demonstrates that, given the same experimental data and sample
geometry, all three forward solvers used in a genetic algorithm yield the same
elastic moduli. This is consistent with previous demonstrations that the the
three methods of forward computation—RPR, SMI, and FEM—are consistent to
better than 1 part in 104 [25].

We observe differences of less than 2.5 % for all elastic moduli when
comparing the rectangular parallelepiped (sample A) and irregular (sample B)
samples. These difference are on the order of our estimated uncertainties due
to sample loading; alternatively, these deviations could be attributed to
systematic uncertainties in the RUS measurement process such as deviations
from a perfect rectangular parallelepiped for sample A, or sample
imperfections that were not captured by the CT scan such as small voids or
inclusions. Within our uncertainty, all of our fits are compatible with the
measurements of Bell and Rupprecht [31] and Poindexter and Giardini [32]
(bottom rows of Table 1). We therefore conclude that our new fitting methods
provide reliable elastic moduli, even for samples with irregular geometries.

Mn3Ge. Next, we test Mn3Ge, point group $D_{6h}$, which has 5 independent
elastic moduli. The results of the fits are given in Table 2. Similar to
SrTiO3, we show fit results for a rectangular parallelepiped sample (sample A)
with dimensions $\left(0.87\times 1.01\times 1.19\right)$ mm3 and corners
parallel to high-symmetry directions, as well as an irregularly-shaped sample
(sample B—see Figure 1 for a 3D model).

As with SrTiO3, all fit methods yield the same moduli for Mn3Ge for both
samples to within measurement uncertainty. Our results for sample A are also
in agreement with previously-published elastic moduli of Mn3Ge [33]. Comparing
samples A and B, the absolute difference in elastic moduli stays below 4 GPa.
This value is consistent with our results for SrTiO3 and is likely due to
similar systematic uncertainties described above.

Sample | Fit Method | $c_{11}$ | $c_{12}$ | $c_{44}$
---|---|---|---|---
SrTiO3 A | RPR | 322 | 104 | 125
FEM | 322 | 104 | 125
SMI | 322 | 104 | 125
SrTiO3 B | FEM | 317 | 103 | 122
SMI | 317 | 103 | 122
Bell et al. [31] |  | 317.3 | 102.3 | 123.4
Poindexter et al. [32] |  | 348.2 | 100.6 | 119.0
Table 1: Elastic moduli of SrTiO3. The elastic moduli in GPa for both SrTiO3 samples: the sample polished into the shape of a rectangular parallelepiped (Sample A), and the irregularly-shaped sample (Sample B). Uncertainties are approximately $\pm$2 GPa (due to the finite force applied to the sample by the RUS apparatus) and $\pm$0.4 GPa (from up to 1 degree misalignment between the crystal axes and the sample mesh). Literature values are provided for comparison.  Sample | Fit Method | $c_{11}$ | $c_{12}$ | $c_{13}$ | $c_{33}$ | $c_{44}$
---|---|---|---|---|---|---
Mn3Ge A | RPR | 130 | 44 | 13 | 202 | 48
FEM | 130 | 44 | 13 | 202 | 48
SMI | 130 | 44 | 13 | 202 | 48
Mn3Ge B | FEM | 127 | 40 | 14 | 203 | 49
SMI | 127 | 40 | 14 | 203 | 49
Table 2: Elastic moduli of Mn3Ge. The elastic moduli in GPa for both Mn3Ge
samples: a rectangular parallelepiped (sample A) and the irregularly-shaped
sample (sample B). Uncertainties are approximately $\pm$2 GPa (due to the
finite force applied to the sample by the RUS apparatus) and $\pm$2 GPa (from
up to 1 degree misalignment between the crystal axes and the sample mesh).

##  IV Application to UT$\rm{\bf e}_{2}$

Having demonstrated the successful extraction of elastic moduli from RUS
spectra of irregular samples, we now fit the elastic moduli for an
irregularly-shaped sample of UTe2. A 3D model of the sample is shown in Figure
1, and the moduli are given in Table 3.

|  $T$ (K) | $c_{11}$ | $c_{22}$ | $c_{33}$ | $c_{12}$ | $c_{13}$ | $c_{23}$ | $c_{44}$ | $c_{55}$ | $c_{66}$
---|---|---|---|---|---|---|---|---|---|---
RUS | 4 | 90.3 | 144.1 | 95.9 | 25.7 | 41.3 | 31.9 | 28.0 | 53.2 | 30.4
300 | 84.7 | 139.5 | 91.1 | 26.8 | 38.1 | 31.6 | 26.9 | 52.4 | 29.7
DFT |  | 96 | 136 | 90 | 28 | 46 | 26 | 28 | 57 | 31
Table 3: Elastic moduli of UTe2. The elastic moduli of the UTe2 sample shown
in Figure 1 at 300 K and at 4 K. Uncertainties are approximately $\pm$0.5 GPa
(due to the finite force applied to the sample by the RUS apparatus) and
$\pm$0.5 GPa (from a 1 degree misalignment between the crystal axes and the
sample mesh). Moduli obtained from DFT calculations, with atomic coordinates
optimized, are given on the bottom row. These values were used as rough guides
to constrain the parameter space of the genetic algorithm fits to the RUS
data.

UTe2 is orthorhombic (point group $D_{2h}$), with nine independent elastic
moduli. The population size—the number of initial guesses—required for a good
fit using a genetic algorithm scales roughly linearly with the number of free
parameters [34]. This means that the UTe2 fits require nearly twice the
population as compared to the previous Mn3Ge fits. Additionally, the samples
of SrTiO3 and Mn3Ge were produced by adding additional facets to what were
previously rectangular paraellelpiped samples, whereas our UTe2 sample is as-
grown. This results in smaller feature sizes on the UTe2 sample as compared to
the previous samples, requiring a finer mesh size for the 3D models (the full
meshes are shown in the supplement).

Both the dense mesh of our particular sample, and large number of moduli for
UTe2 in general, increase the fit time for the FEM method, which must re-
compute the entire spectrum at each stage of the fit. As noted in Shragai  _et
al._ [25], this makes FEM unsuitable for fitting under these circumstances, as
convergence would take upward of a month. The SMI method, on the other hand,
is largely unaffected by the increase in population size and mesh density
because the computationally-intensive step is performed only once, at the
start of the fit, and the results are stored for use in subsequent generations
of the genetic algorithm. We therefore only perform SMI fits to our UTe2
spectra. This approach is justified by the results of the previous section,
which demonstrated that fits using both FEM and SMI methods produce the same
elastic moduli.

The elastic moduli of UTe2, at both 300 K and 4 K are given in Table 3. For
comparison, we also report elastic moduli obtained from density-functional
theory calculations (bottom row in Table 3). These calculated values are in
remarkable agreement with the experimental results. Other experimentally-
relevant quantities, including the bulk modulus, the Young’s moduli, and the
Poisson’s ratios, are tabulated in the supplement.

##  V Conclusion

We implement two new methods for performing the forward calculation for
resonance spectra in a genetic algorithm. These methods—SMI and FEM—allow us
to fit the elastic moduli of irregularly-shaped samples using resonant
ultrasound spectroscopy. This is first demonstrated on materials with known
elastic moduli (SrTiO3 and Mn3Ge), demonstrating consistency between our new
methods and the older method of Visscher  _et al._ [1], which is only
applicable for rectangular parallelepipeds. We then apply our new method to
measure the full elastic tensor of the unconventional superconductor UTe2 at
300 K and at 4 K (Table 3).

We expect that the elastic moduli of UTe2 will be of use to researchers
studying the superconducting properties under both uniaxial strain and
hydrostatic pressure [15, 18, 35, 10, 17, 9]. We expect that the general
method that we have introduced—implementing the fast surface mesh integration
method into a genetic algorithm—will be broadly useful for measuring the
elastic moduli of samples that cannot be easily prepared into regular
geometric shapes.

##  VI Acknowledgments

This work was funded by the Office of Basic Energy Sciences of the United
States Department of Energy under award no. DE-SC0020143. Imaging data was
acquired through the Cornell Institute of Biotechnology’s Imaging Facility,
with NIH S10OD012287 for the ZEISS/Xradia Versa 520 X-ray Microscope (CT).
Laue backreflection diffraction measurements made use of the Cornell Center
for Materials Research Shared Facilities which are supported through the NSF
MRSEC program (DMR-1719875). Discussion of commercial software does not imply
endorsement by NIST.

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##  VII SUPPLEMENTARY INFORMATION

###  VII.1 Bulk Modulus, Young’s Moduli, and Poisson’s Ratios

In Table 4 we present the bulk modulus $B$, the Young’s moduli $E_{xx}$,
$E_{yy}$, $E_{zz}$, and the Poisson’s ratios $\nu_{xz}$, $\nu_{yz}$,
$\nu_{xy}$ for all measured samples and fits, calculated from the elastic
moduli shown in the main text. For cubic SrTiO3, $E_{xx}=E_{yy}=E_{zz}$ and
$\nu_{xz}=\nu_{yz}=\nu_{xy}$, whereas for hexagonal Mn3Ge, $E_{xx}=E_{yy}$ and
$\nu_{xz}=\nu_{yz}$. For orthorhombic UTe2, all moduli differ.

|  |  | Young’s Moduli (GPa) | Poisson’s Ratios
---|---|---|---|---
Sample |  $B$ (GPa) | $E_{xx}$ | $E_{yy}$ | $E_{zz}$ | $\nu_{xz}$ | $\nu_{yz}$ | $\nu_{xy}$
SrTiO3 | A | 177 | 271 | - | - | 0.24 | - | -
B | 174 | 266 | - | - | 0.25 | - | -
Mn3Ge | A | 66 | 115 | - | 200 | 0.04 | - | 0.33
B | 65 | 114 | - | 201 | 0.05 | - | 0.31
UTe2 | 300 K | 55 | 678 | 126 | 71 | 0.38 | 0.27 | 0.11
4 K | 57 | 71 | 131 | 74 | 0.40 | 0.26 | 0.09
Table 4: Bulk modulus and Young’s moduli

###  VII.2 Density Functional Theory

In Table 5 we compare the elastic moduli from density functional theory
obtained from calculations during which the atomic coordinates were frozen to
experimental values to calculations in which they were optimized. We also
compare to calculations presented in the supplement of [18].

Method | $c_{11}$ | $c_{22}$ | $c_{33}$ | $c_{12}$ | $c_{13}$ | $c_{23}$ | $c_{44}$ | $c_{55}$ | $c_{66}$
---|---|---|---|---|---|---|---|---|---
coordinates frozen | 100.2 | 140.0 | 99.3 | 28.7 | 56.4 | 27.1 | 33.8 | 69.4 | 36.1
coordinates optimized | 95.7 | 136.0 | 89.7 | 28.1 | 46.0 | 26.0 | 28.0 | 57.1 | 31.0
Girod et al. [18] | 97.0 | 140.7 | 101.3 | 40.9 | 48.6 | 46.7 | 19.6 | 57.3 | 27.1
Table 5: UTe2 DFT elastic tensor. Presented are the elastic moduli from
density functional theory calculations. The first two rows are our
calculations with atomic coordinates frozen to the experimental values and
optimized. The last row are values given in the SI of [18], where the atomic
coordinates were allowed to relax during minimization.

###  VII.3 Sample Digitization

Three-dimensional digitizations of the samples were obtained with a Zeiss
Xradia Versa XRM-520 X-ray nano-CT. The average mesh size varies between
samples and RUS solver methods (i.e. SMI vs. FEM). The meshes used for all FEM
fits are shown in Figure 2, and the average mesh size for all fits (FEM and
SMI) are given in Table 6. We used smaller meshes for irregularly-shaped
samples (compared to regularly shaped ones) as well as for fits using the SMI
method (compared to the FEM method). The latter is because a smaller mesh size
increases the time for a fit to converge significantly for the FEM method, but
leaves it almost unaffected for the SMI method [25].

Figure 2: CT-Scan Meshes. Shown are (from left to right) the digitizations for the SrTiO3 (samples A, B) and Mn3Ge (samples A, B) samples used for the RUS fits withe the FEM method. Meshes for fits using the SMI method are too dense for individual faces to be identifiable on the shown scale. Method | SrTiO3 | SrTiO3 | Mn3Ge | Mn3Ge | UTe2
---|---|---|---|---|---
| sample A | sample B | sample A | sample B |
FEM | 84 | 56 | 52 | 40 | -
SMI | 60 | 5 | 23 | 5 | 9
Table 6: Mesh Size. Average distance between two vertices (in $\mu$m) in the
meshes used in the RUS fits.

###  VII.4 Differential Evolution Algorithm

We use a differential evolution global minimizer to extract the elastic moduli
from the experimental resonance spectra [36, 37, 24]. A differential evolution
algorithm is initialized with its first generation by randomly generating $N$
sets of elastic moduli within given bounds. We call these sets parent sets,
$x_{i}^{parent}$, $i\in\left[1,2,...,N\right]$. Here every $x_{i}^{parent}$ is
a vector containing the independent elastic moduli of a given material. A
resonance spectrum is then calculated for all parent sets (see discussion
below on details on how resonances are calculated), and compared to the
experimental resonance frequencies. The best set of elastic moduli $x^{best}$
is characterized by the lowest $\chi^{2}$ value. The second generation of the
differential evolution algorithm is then created in three steps: First, a
mutated set of elastic moduli $x_{i}^{mut}$ is constructed for every parent
set $x_{i}^{parent}$ through

$x_{i}^{mut}=x^{best}+\epsilon\left(x_{j}^{parent}+x_{k}^{parent}\right).$ (2)

Here, $j,k\in\left[1,N\right]$ are random indices, and the mutation parameter
$\epsilon\in\left[0,2\right]$ ensures a controlled perturbation of $x^{best}$.
Secondly, trial sets $x_{i}^{trial}$ are created from each mutated parameter
set. For this procedure, each element $x_{i,j}^{trial}$—i.e. each elastic
modulus—of a given trial set is constructed as

$x_{i,j}^{trial}=\begin{cases}x_{i,j}^{mut}&\mathrm{if}~{}rand(j)\leq
p~{}\mathrm{or}~{}j=d\\\ x_{i,j}^{parent}&\mathrm{if}~{}rand(j)>p\end{cases},$
(3)

where $rand(j)\in\left[0,1\right]$ is a random number, $p\in\left[0,1\right]$
is the crossover probability, and $d$ is the number of elastic moduli. In the
last step, each original parent set $x_{i}^{parent}$ is compared to its
related trial set $x_{i}^{trial}$ and the set with smallest $\chi^{2}$ value
is chosen for the next generation. This procedure is repeated until the
standard deviation of the residuals of a given generation falls below a
predetermined tolerance level. The quality of this fit can be improved further
by performing a gradient descent fit, using the results of the differential
evolution algorithm as starting parameters.

###  VII.5 RUS Fit Uncertainties

The main sources of uncertainties in our fits are due to small deviations from
the weak-coupling approximation used in our analysis and from small
uncertainties in the alignment between the crystal axes and the sample mesh.

The former is caused by the weight of the cantilever holding the top
ultrasonic transducer and causes the measured resonance frequencies to weakly
depend on how the sample is mounted in the RUS apparatus. To estimate the
resulting uncertainty in elastic moduli, we arranged all irregularly-shaped
samples in three different ways between the transducers and re-performed RUS
measurements and fits. Figure 3 illustrates the different arrangements of
samples in the setup and the rows in Table 7 show the respective elastic
moduli (note that arrangement 2 is how the samples were mounted for the
results shown in the main paper). This analysis leads to an average
uncertainty of 1.7 GPa, 2.3 GPa, 0.5 GPa for SrTiO3 sample B, Mn3Ge sample B,
and the UTe2 sample at 300 K, respectively. The uncertainties are about a
factor of 4 larger for the SrTiO3 and Mn3Ge samples since they were measured
in a RUS apparatus with a cantilever weighing about 5 g, while the UTe2
spectra were recorded in an apparatus with a cantilever weighing about 0.5 g.

The latter contribution originates in an uncertainty of about 1 degree to
which we can align the crystal axes and the sample mesh using Laue
backreflection diffractometry. Refitting our RUS resonance spectra with a
sample mesh rotated by 1 degree results in average uncertainties of 0.4 GPa, 2
GPa, 0.5 GPa, for the elastic moduli of SrTiO3 sample B, Mn3Ge sample B, and
UTe2 at 300 K, respectively. Individual error bars for each elastic modulus
due to sample misalignment are given in Table 7.

Figure 3: RUS Sample Arrangements. Shown are different arrangements of the samples in the RUS apparatus. The golden discs represent the ultrasonic transducers (the bottom transducer is occluded by the sample in some images). The arrangement number relates to the correct row in Table 7. Sample | Arrangement | $c_{11}$ | $c_{22}$ | $c_{33}$ | $c_{12}$ | $c_{13}$ | $c_{23}$ | $c_{44}$ | $c_{55}$ | $c_{66}$
---|---|---|---|---|---|---|---|---|---|---
SrTiO3 (B) | 1 | 313.8 | - | - | 100.3 | - | - | 122.62 | - | -
2 | 316.7$\pm$0.5 | - | - | 102.9$\pm$0.7 | - | - | 121.95$\pm$0.05 | - | -
3 | 315.8 | - | - | 100.2 | - | - | 122.30 | - | -
Mn3Ge (B) | 1 | 126 | - | 200 | 38 | 12 | - | 48.7 | - | -
2 | 127$\pm$1 | - | 203$\pm$2 | 40$\pm$2 | 14$\pm$5 | - | 48.7$\pm$0.5 | - | -
3 | 129 | - | 196 | 42 | 18 | - | 49.6 | - | -
UTe2 (300 K) | 1 | 84.4 | 139.0 | 92.0 | 25.6 | 38.1 | 32.0 | 26.9 | 52.3 | 29.4
2 | 84.7$\pm$0.44 | 139.5$\pm$0.8 | 91.1$\pm$1.1 | 26.8$\pm$1.0 | 38.1$\pm$0.5 | 31.6$\pm$0.4 | 26.9$\pm$0.2 | 52.4$\pm$0.2 | 29.7$\pm$0.2
3 | 83.8 | 140.3 | 90.6 | 25.5 | 37.2 | 31.6 | 27.1 | 52.3 | 29.4
Table 7: RUS Uncertainty Analysis. Fit results for the different arrangements
of all irregularly-shaped samples shown in Figure 3 to estimate the
uncertainty introduced by the weight of the cantilever holding the top
ultrasonic transducer. For each sample and arrangement 2, we also show the
errors caused by a 1 degree misalignment between the crystal axes and the
sample mesh. All fits were performed with the SMI method.

###  VII.6 RUS Resonance Spectra and Fit Results

We show lists of experimental resonance frequencies $f_{exp}$ along with the
calculated frequencies $f_{calc}$ corresponding to the RUS fit results in the
main text. The lowest three resonances are excluded from all fits.

Table 8: SrTiO3 sample A (regular shape) |  | RPR | SMI | FEM
---|---|---|---|---
Index |  $f_{exp}$ (MHz) |  $f_{calc}$ (MHz) | diff (%) |  $f_{calc}$ (MHz) | diff (%) |  $f_{calc}$ (MHz) | diff (%)
1 | 0.71037 | 0.70491 | - | 0.70491 | - | 0.70489 | -
2 | 0.79197 | 0.79156 | - | 0.79156 | - | 0.79155 | -
3 | 0.89772 | 0.90185 | - | 0.90185 | - | 0.90184 | -
4 | 1.17353 | 1.17781 | 0.363 | 1.17781 | 0.363 | 1.17779 | 0.361
5 | 1.24570 | 1.24714 | 0.116 | 1.24714 | 0.116 | 1.24711 | 0.113
6 | 1.33132 | 1.32506 | 0.473 | 1.32506 | 0.473 | 1.32504 | 0.474
7 | 1.37709 | 1.38138 | 0.310 | 1.38138 | 0.310 | 1.38135 | 0.308
8 | 1.41735 | 1.40410 | 0.944 | 1.40410 | 0.944 | 1.40408 | 0.945
9 | 1.50363 | 1.49941 | 0.281 | 1.49941 | 0.281 | 1.49939 | 0.283
10 | 1.55724 | 1.55327 | 0.256 | 1.55327 | 0.256 | 1.55325 | 0.257
11 | 1.59727 | 1.58515 | 0.765 | 1.58515 | 0.765 | 1.58513 | 0.766
12 | 1.60045 | 1.58865 | 0.743 | 1.58865 | 0.743 | 1.58863 | 0.744
13 | 1.71219 | 1.71178 | 0.024 | 1.71178 | 0.024 | 1.71175 | 0.025
14 | 1.77642 | 1.77451 | 0.108 | 1.77451 | 0.108 | 1.77449 | 0.109
15 | 1.78574 | 1.79251 | 0.377 | 1.79251 | 0.377 | 1.79247 | 0.375
16 | 1.83710 | 1.84222 | 0.278 | 1.84222 | 0.277 | 1.84219 | 0.276
17 | 1.85498 | 1.85528 | 0.016 | 1.85528 | 0.016 | 1.85524 | 0.014
18 | 1.91594 | 1.92356 | 0.396 | 1.92356 | 0.396 | 1.92353 | 0.395
19 | 1.97811 | 1.97315 | 0.251 | 1.97315 | 0.251 | 1.97314 | 0.252
20 | 1.98353 | 1.98541 | 0.095 | 1.98541 | 0.095 | 1.98538 | 0.093
21 | 2.03547 | 2.04563 | 0.496 | 2.04563 | 0.496 | 2.04559 | 0.495
22 | 2.07281 | 2.07985 | 0.338 | 2.07985 | 0.338 | 2.07982 | 0.337
23 | 2.16306 | 2.17696 | 0.638 | 2.17696 | 0.639 | 2.17695 | 0.638
24 | 2.17570 | 2.17867 | 0.136 | 2.17867 | 0.136 | 2.17864 | 0.135
25 | 2.20327 | 2.21156 | 0.375 | 2.21156 | 0.375 | 2.21154 | 0.374
26 | 2.20830 | 2.21607 | 0.350 | 2.21607 | 0.350 | 2.21606 | 0.350
27 | 2.22903 | 2.22728 | 0.078 | 2.22728 | 0.078 | 2.22727 | 0.079
28 | 2.23226 | 2.23763 | 0.240 | 2.23763 | 0.240 | 2.23761 | 0.239
29 | 2.23351 | 2.23846 | 0.221 | 2.23846 | 0.221 | 2.23845 | 0.221
30 | 2.28139 | 2.27258 | 0.388 | 2.27258 | 0.388 | 2.27258 | 0.388
31 | 2.32126 | 2.31051 | 0.465 | 2.31051 | 0.465 | 2.31052 | 0.465
32 | 2.33100 | 2.32182 | 0.395 | 2.32182 | 0.395 | 2.32180 | 0.396
33 | 2.34719 | 2.34083 | 0.272 | 2.34083 | 0.272 | 2.34082 | 0.272
34 | 2.44146 | 2.44675 | 0.216 | 2.44675 | 0.216 | 2.44674 | 0.216
35 | 2.46013 | 2.45143 | 0.355 | 2.45142 | 0.355 | 2.45144 | 0.355
36 | 2.50210 | 2.50958 | 0.298 | 2.50958 | 0.298 | 2.50956 | 0.297
37 | 2.50662 | 2.50968 | 0.122 | 2.50968 | 0.122 | 2.50965 | 0.121
38 | 2.53756 | 2.53887 | 0.051 | 2.53887 | 0.051 | 2.53889 | 0.052
39 | 2.57650 | 2.57679 | 0.011 | 2.57679 | 0.011 | 2.57680 | 0.012
40 | 2.58500 | 2.59496 | 0.384 | 2.59496 | 0.384 | 2.59496 | 0.384
41 | 2.59850 | 2.59922 | 0.028 | 2.59922 | 0.028 | 2.59920 | 0.027
42 | 2.62865 | 2.63115 | 0.095 | 2.63116 | 0.095 | 2.63117 | 0.096
43 | 2.65167 | 2.65229 | 0.023 | 2.65229 | 0.023 | 2.65226 | 0.022
44 | 2.68309 | 2.68338 | 0.011 | 2.68338 | 0.011 | 2.68338 | 0.011
45 | 2.70518 | 2.70901 | 0.141 | 2.70901 | 0.141 | 2.70898 | 0.140
46 | 2.74782 | 2.74881 | 0.036 | 2.74881 | 0.036 | 2.74883 | 0.037
47 | 2.75176 | 2.75107 | 0.025 | 2.75107 | 0.025 | 2.75107 | 0.025
48 | 2.75392 | 2.76091 | 0.253 | 2.76091 | 0.253 | 2.76092 | 0.254
49 | 2.78789 | 2.79403 | 0.220 | 2.79403 | 0.220 | 2.79408 | 0.221
50 | 2.80071 | 2.79544 | 0.189 | 2.79545 | 0.188 | 2.79545 | 0.188
51 | 2.82197 | 2.82346 | 0.053 | 2.82346 | 0.053 | 2.82352 | 0.055
52 | 2.84439 | 2.85166 | 0.255 | 2.85167 | 0.255 | 2.85166 | 0.255
53 | 2.91635 | 2.91928 | 0.101 | 2.91928 | 0.101 | 2.91931 | 0.102
54 | 2.92322 | 2.92473 | 0.051 | 2.92473 | 0.051 | 2.92478 | 0.053
55 | 2.94885 | 2.92904 | 0.677 | 2.92904 | 0.677 | 2.92902 | 0.677
56 | 3.00333 | 2.99929 | 0.135 | 2.99929 | 0.135 | 2.99940 | 0.131
57 | 3.01642 | 3.01070 | 0.190 | 3.01070 | 0.190 | 3.01074 | 0.189
58 | 3.03353 | 3.03246 | 0.035 | 3.03246 | 0.036 | 3.03254 | 0.033
59 | 3.03869 | 3.04328 | 0.151 | 3.04328 | 0.151 | 3.04332 | 0.152
60 | 3.06889 | 3.06572 | 0.103 | 3.06572 | 0.103 | 3.06574 | 0.103
61 | 3.13815 | 3.14283 | 0.149 | 3.14283 | 0.149 | 3.14288 | 0.151
62 | 3.15099 | 3.15332 | 0.074 | 3.15332 | 0.074 | 3.15339 | 0.076
63 | 3.15918 | 3.16111 | 0.061 | 3.16111 | 0.061 | 3.16112 | 0.062
64 | 3.16664 | 3.16975 | 0.098 | 3.16975 | 0.098 | 3.16985 | 0.101
65 | 3.22486 | 3.22895 | 0.127 | 3.22895 | 0.127 | 3.22893 | 0.126
66 | 3.22600 | 3.23110 | 0.158 | 3.23110 | 0.158 | 3.23107 | 0.157
67 | 3.25102 | 3.24020 | 0.334 | 3.24020 | 0.334 | 3.24033 | 0.330
68 | 3.25333 | 3.24905 | 0.132 | 3.24905 | 0.132 | 3.24915 | 0.129
69 | 3.26656 | 3.25129 | 0.470 | 3.25129 | 0.470 | 3.25141 | 0.466
70 | 3.27348 | 3.28127 | 0.237 | 3.28127 | 0.237 | 3.28131 | 0.239
Table 9: SrTiO3 sample B (irregular shape) |  | SMI | FEM
---|---|---|---
Index |  $f_{exp}$ (MHz) |  $f_{calc}$ (MHz) | diff (%) |  $f_{calc}$ (MHz) | diff (%)
1 | 1.10869 | 1.09168 | - | 1.09156 | -
2 | 1.20480 | 1.19121 | - | 1.19112 | -
3 | 1.52331 | 1.50347 | - | 1.50341 | -
4 | 1.88059 | 1.86636 | 0.762 | 1.86620 | 0.771
5 | 1.97243 | 1.97144 | 0.050 | 1.97135 | 0.055
6 | 1.98450 | 1.98777 | 0.164 | 1.98756 | 0.154
7 | 2.07997 | 2.07118 | 0.424 | 2.07100 | 0.433
8 | 2.15054 | 2.14467 | 0.274 | 2.14453 | 0.280
9 | 2.23955 | 2.23207 | 0.335 | 2.23186 | 0.344
10 | 2.37823 | 2.36840 | 0.415 | 2.36833 | 0.418
11 | 2.43553 | 2.44027 | 0.194 | 2.44001 | 0.184
12 | 2.58487 | 2.57275 | 0.471 | 2.57258 | 0.478
13 | 2.72511 | 2.71927 | 0.215 | 2.71912 | 0.220
14 | 2.80098 | 2.79093 | 0.360 | 2.79078 | 0.366
15 | 2.83415 | 2.83210 | 0.072 | 2.83199 | 0.077
16 | 2.87280 | 2.86311 | 0.339 | 2.86304 | 0.341
17 | 2.92701 | 2.91494 | 0.414 | 2.91473 | 0.421
18 | 3.07796 | 3.07446 | 0.114 | 3.07432 | 0.118
19 | 3.13235 | 3.14863 | 0.517 | 3.14864 | 0.517
20 | 3.16734 | 3.16494 | 0.076 | 3.16486 | 0.079
21 | 3.18282 | 3.18195 | 0.027 | 3.18191 | 0.029
22 | 3.25382 | 3.24307 | 0.331 | 3.24298 | 0.334
23 | 3.29615 | 3.29027 | 0.179 | 3.29022 | 0.180
24 | 3.31731 | 3.33535 | 0.541 | 3.33524 | 0.538
25 | 3.48056 | 3.50474 | 0.690 | 3.50469 | 0.688
26 | 3.51006 | 3.52415 | 0.400 | 3.52412 | 0.399
27 | 3.65645 | 3.66182 | 0.147 | 3.66178 | 0.145
28 | 3.78318 | 3.79193 | 0.231 | 3.79191 | 0.230
29 | 3.84523 | 3.84207 | 0.082 | 3.84198 | 0.085
30 | 3.89894 | 3.90375 | 0.123 | 3.90368 | 0.122
31 | 3.98772 | 3.99729 | 0.239 | 3.99727 | 0.239
32 | 4.00064 | 4.00409 | 0.086 | 4.00407 | 0.086
33 | 4.06006 | 4.04985 | 0.252 | 4.04979 | 0.254
34 | 4.10005 | 4.10447 | 0.108 | 4.10444 | 0.107
35 | 4.15255 | 4.14028 | 0.296 | 4.14022 | 0.298
36 | 4.18979 | 4.20182 | 0.286 | 4.20174 | 0.284
37 | 4.23696 | 4.22814 | 0.209 | 4.22819 | 0.207
38 | 4.27449 | 4.28863 | 0.330 | 4.28857 | 0.328
39 | 4.30861 | 4.31115 | 0.059 | 4.31124 | 0.061
40 | 4.34986 | 4.35327 | 0.078 | 4.35335 | 0.080
41 | 4.39819 | 4.39097 | 0.165 | 4.39113 | 0.161
42 | 4.46403 | 4.46713 | 0.069 | 4.46719 | 0.071
43 | 4.48597 | 4.47481 | 0.249 | 4.47493 | 0.247
44 | 4.63845 | 4.62951 | 0.193 | 4.62957 | 0.192
45 | 4.68243 | 4.69013 | 0.164 | 4.69022 | 0.166
46 | 4.74359 | 4.74395 | 0.008 | 4.74417 | 0.012
47 | 4.75697 | 4.76987 | 0.270 | 4.76996 | 0.272
48 | 4.80223 | 4.79540 | 0.142 | 4.79552 | 0.140
49 | 4.82394 | 4.84266 | 0.387 | 4.84274 | 0.388
50 | 4.89389 | 4.90448 | 0.216 | 4.90445 | 0.215
51 | 4.93746 | 4.93653 | 0.019 | 4.93677 | 0.014
52 | 4.98372 | 4.98686 | 0.063 | 4.98700 | 0.066
53 | 5.00140 | 5.00252 | 0.022 | 5.00247 | 0.021
54 | 5.03260 | 5.03444 | 0.037 | 5.03457 | 0.039
55 | 5.06065 | 5.06660 | 0.117 | 5.06673 | 0.120
56 | 5.10892 | 5.10434 | 0.090 | 5.10460 | 0.085
57 | 5.12704 | 5.12970 | 0.052 | 5.12991 | 0.056
58 | 5.16940 | 5.18029 | 0.210 | 5.18061 | 0.216
59 | 5.20187 | 5.23093 | 0.556 | 5.23130 | 0.562
60 | 5.25259 | 5.25272 | 0.002 | 5.25278 | 0.004
61 | 5.27776 | 5.27884 | 0.020 | 5.27913 | 0.026
62 | 5.28863 | 5.29927 | 0.201 | 5.29969 | 0.209
63 | 5.33266 | 5.33158 | 0.020 | 5.33170 | 0.018
64 | 5.37012 | 5.35725 | 0.240 | 5.35743 | 0.237
65 | 5.40014 | 5.39589 | 0.079 | 5.39627 | 0.072
66 | 5.43224 | 5.43571 | 0.064 | 5.43599 | 0.069
67 | 5.45845 | 5.45699 | 0.027 | 5.45728 | 0.021
68 | 5.49232 | 5.49778 | 0.099 | 5.49816 | 0.106
69 | 5.54920 | 5.55358 | 0.079 | 5.55405 | 0.087
70 | 5.58061 | 5.58871 | 0.145 | 5.58914 | 0.153
Table 10: Mn3Ge sample A (regular shape) |  | RPR | SMI | FEM
---|---|---|---|---
Index |  $f_{exp}$ (MHz) |  $f_{calc}$ (MHz) | diff (%) |  $f_{calc}$ (MHz) | diff (%) |  $f_{calc}$ (MHz) | diff (%)
1 | 0.96469 | 0.95542 | - | 0.95544 | - | 0.95531 | -
2 | 1.31011 | 1.31762 | - | 1.31764 | - | 1.31751 | -
3 | 1.46748 | 1.46874 | - | 1.46876 | - | 1.46855 | -
4 | 1.50955 | 1.50952 | 0.002 | 1.50951 | 0.002 | 1.50947 | 0.005
5 | 1.56402 | 1.56137 | 0.169 | 1.56136 | 0.170 | 1.56127 | 0.176
6 | 1.57830 | 1.57952 | 0.077 | 1.57951 | 0.077 | 1.57945 | 0.073
7 | 1.58425 | 1.60049 | 1.015 | 1.60051 | 1.016 | 1.60029 | 1.002
8 | 1.60141 | 1.61340 | 0.743 | 1.61340 | 0.743 | 1.61327 | 0.735
9 | 1.60582 | 1.61949 | 0.845 | 1.61951 | 0.846 | 1.61938 | 0.838
10 | 1.78631 | 1.80175 | 0.857 | 1.80177 | 0.858 | 1.80162 | 0.850
11 | 1.82036 | 1.81183 | 0.471 | 1.81182 | 0.471 | 1.81169 | 0.478
12 | 1.87303 | 1.87322 | 0.010 | 1.87322 | 0.010 | 1.87313 | 0.005
13 | 1.94132 | 1.93235 | 0.464 | 1.93233 | 0.465 | 1.93222 | 0.471
14 | 1.95112 | 1.95456 | 0.176 | 1.95455 | 0.175 | 1.95451 | 0.173
15 | 2.03329 | 2.02996 | 0.164 | 2.02997 | 0.164 | 2.02986 | 0.169
16 | 2.09609 | 2.08355 | 0.602 | 2.08353 | 0.603 | 2.08339 | 0.610
17 | 2.10555 | 2.10872 | 0.150 | 2.10867 | 0.148 | 2.10854 | 0.142
18 | 2.14587 | 2.12376 | 1.041 | 2.12373 | 1.043 | 2.12363 | 1.047
19 | 2.16239 | 2.17045 | 0.371 | 2.17041 | 0.369 | 2.17050 | 0.374
20 | 2.16541 | 2.17570 | 0.473 | 2.17572 | 0.474 | 2.17548 | 0.463
21 | 2.25310 | 2.26434 | 0.496 | 2.26437 | 0.497 | 2.26423 | 0.491
22 | 2.32002 | 2.30809 | 0.517 | 2.30807 | 0.518 | 2.30797 | 0.522
23 | 2.37049 | 2.35061 | 0.846 | 2.35060 | 0.846 | 2.35052 | 0.849
24 | 2.38275 | 2.38340 | 0.027 | 2.38330 | 0.023 | 2.38334 | 0.024
25 | 2.43586 | 2.41898 | 0.698 | 2.41896 | 0.699 | 2.41890 | 0.701
26 | 2.46256 | 2.47420 | 0.471 | 2.47422 | 0.471 | 2.47413 | 0.468
27 | 2.47301 | 2.47498 | 0.080 | 2.47498 | 0.080 | 2.47483 | 0.073
28 | 2.52384 | 2.55083 | 1.058 | 2.55069 | 1.053 | 2.55063 | 1.051
29 | 2.55464 | 2.57203 | 0.676 | 2.57205 | 0.677 | 2.57190 | 0.671
30 | 2.57758 | 2.59117 | 0.524 | 2.59118 | 0.525 | 2.59105 | 0.520
31 | 2.61331 | 2.62284 | 0.363 | 2.62282 | 0.363 | 2.62273 | 0.359
32 | 2.65457 | 2.64939 | 0.196 | 2.64940 | 0.195 | 2.64940 | 0.195
33 | 2.71919 | 2.72259 | 0.125 | 2.72258 | 0.124 | 2.72248 | 0.121
34 | 2.75160 | 2.74615 | 0.198 | 2.74616 | 0.198 | 2.74602 | 0.203
35 | 2.78098 | 2.77733 | 0.131 | 2.77734 | 0.131 | 2.77722 | 0.135
36 | 2.84224 | 2.83905 | 0.112 | 2.83904 | 0.113 | 2.83904 | 0.113
37 | 2.86041 | 2.86306 | 0.092 | 2.86298 | 0.090 | 2.86289 | 0.086
38 | 2.87166 | 2.86844 | 0.112 | 2.86847 | 0.111 | 2.86843 | 0.113
39 | 2.91103 | 2.92610 | 0.515 | 2.92611 | 0.516 | 2.92617 | 0.518
40 | 2.93140 | 2.92769 | 0.127 | 2.92770 | 0.126 | 2.92756 | 0.131
41 | 2.93455 | 2.93191 | 0.090 | 2.93190 | 0.090 | 2.93189 | 0.091
42 | 2.94156 | 2.94237 | 0.027 | 2.94237 | 0.028 | 2.94228 | 0.024
43 | 3.04438 | 3.05124 | 0.225 | 3.05123 | 0.224 | 3.05109 | 0.220
44 | 3.10644 | 3.09616 | 0.332 | 3.09616 | 0.332 | 3.09622 | 0.330
45 | 3.12912 | 3.13303 | 0.125 | 3.13303 | 0.125 | 3.13297 | 0.123
46 | 3.19343 | 3.17795 | 0.487 | 3.17796 | 0.487 | 3.17801 | 0.485
47 | 3.21141 | 3.21040 | 0.031 | 3.21041 | 0.031 | 3.21032 | 0.034
48 | 3.28658 | 3.28210 | 0.136 | 3.28212 | 0.136 | 3.28219 | 0.134
49 | 3.30955 | 3.29828 | 0.342 | 3.29828 | 0.342 | 3.29851 | 0.335
50 | 3.31565 | 3.32261 | 0.209 | 3.32261 | 0.209 | 3.32258 | 0.209
51 | 3.35754 | 3.37386 | 0.484 | 3.37387 | 0.484 | 3.37382 | 0.482
52 | 3.38287 | 3.37789 | 0.147 | 3.37786 | 0.148 | 3.37795 | 0.146
53 | 3.44121 | 3.45795 | 0.484 | 3.45795 | 0.484 | 3.45796 | 0.484
54 | 3.47190 | 3.46067 | 0.324 | 3.46069 | 0.324 | 3.46107 | 0.313
55 | 3.49421 | 3.48695 | 0.208 | 3.48694 | 0.208 | 3.48696 | 0.208
56 | 3.49812 | 3.50059 | 0.071 | 3.50057 | 0.070 | 3.50061 | 0.071
57 | 3.49964 | 3.51232 | 0.361 | 3.51236 | 0.362 | 3.51229 | 0.360
58 | 3.54746 | 3.53238 | 0.427 | 3.53237 | 0.427 | 3.53259 | 0.421
59 | 3.57278 | 3.56487 | 0.222 | 3.56488 | 0.222 | 3.56526 | 0.211
60 | 3.58496 | 3.57191 | 0.365 | 3.57191 | 0.365 | 3.57200 | 0.363
61 | 3.58767 | 3.59720 | 0.265 | 3.59714 | 0.263 | 3.59709 | 0.262
62 | 3.60231 | 3.60308 | 0.022 | 3.60311 | 0.022 | 3.60304 | 0.020
63 | 3.60760 | 3.60521 | 0.066 | 3.60521 | 0.066 | 3.60540 | 0.061
64 | 3.61308 | 3.60655 | 0.181 | 3.60655 | 0.181 | 3.60663 | 0.179
65 | 3.61436 | 3.61431 | 0.001 | 3.61432 | 0.001 | 3.61455 | 0.005
66 | 3.61865 | 3.62068 | 0.056 | 3.62069 | 0.057 | 3.62084 | 0.060
67 | 3.64765 | 3.64158 | 0.167 | 3.64152 | 0.168 | 3.64153 | 0.168
68 | 3.67018 | 3.64580 | 0.669 | 3.64580 | 0.669 | 3.64602 | 0.663
69 | 3.71718 | 3.70179 | 0.416 | 3.70179 | 0.416 | 3.70203 | 0.409
70 | 3.72327 | 3.72386 | 0.016 | 3.72386 | 0.016 | 3.72412 | 0.023
71 | 3.78761 | 3.79633 | 0.230 | 3.79635 | 0.230 | 3.79659 | 0.237
72 | 3.81140 | 3.80554 | 0.154 | 3.80556 | 0.153 | 3.80579 | 0.147
73 | 3.84430 | 3.82293 | 0.559 | 3.82295 | 0.559 | 3.82310 | 0.554
74 | 3.86988 | 3.84886 | 0.546 | 3.84885 | 0.546 | 3.84915 | 0.539
75 | 3.91723 | 3.91340 | 0.098 | 3.91339 | 0.098 | 3.91352 | 0.095
76 | 3.94563 | 3.92345 | 0.565 | 3.92345 | 0.565 | 3.92395 | 0.552
77 | 3.97908 | 3.97784 | 0.031 | 3.97787 | 0.030 | 3.97802 | 0.027
78 | 3.99942 | 3.99986 | 0.011 | 3.99985 | 0.011 | 4.00031 | 0.022
79 | 4.01915 | 4.00900 | 0.253 | 4.00903 | 0.252 | 4.00943 | 0.242
80 | 4.03495 | 4.02355 | 0.284 | 4.02355 | 0.283 | 4.02443 | 0.262
81 | 4.04595 | 4.06045 | 0.357 | 4.06046 | 0.357 | 4.06074 | 0.364
82 | 4.05758 | 4.06567 | 0.199 | 4.06558 | 0.197 | 4.06568 | 0.199
83 | 4.06491 | 4.08637 | 0.525 | 4.08631 | 0.524 | 4.08631 | 0.524
84 | 4.0780 | 4.09008 | 0.295 | 4.09004 | 0.294 | 4.09015 | 0.297
Table 11: Mn3Ge sample B (irregular shape) |  | SMI | FEM
---|---|---|---
Index |  $f_{exp}$ (MHz) |  $f_{calc}$ (MHz) | diff (%) |  $f_{calc}$ (MHz) | diff (%)
1 | 0.77100 | 0.72927 | - | 0.72932 | -
2 | 1.08478 | 1.06421 | - | 1.06356 | -
3 | 1.22382 | 1.21029 | - | 1.21031 | -
4 | 1.33437 | 1.32618 | 0.618 | 1.32623 | 0.614
5 | 1.44953 | 1.43028 | 1.346 | 1.43030 | 1.344
6 | 1.60718 | 1.59563 | 0.723 | 1.59565 | 0.722
7 | 1.86078 | 1.84572 | 0.816 | 1.84544 | 0.831
8 | 1.88524 | 1.88848 | 0.172 | 1.88804 | 0.148
9 | 2.02059 | 2.01970 | 0.044 | 2.01929 | 0.064
10 | 2.06617 | 2.03724 | 1.420 | 2.03660 | 1.452
11 | 2.12700 | 2.11597 | 0.521 | 2.11593 | 0.523
12 | 2.30712 | 2.29890 | 0.358 | 2.29885 | 0.360
13 | 2.33951 | 2.33223 | 0.312 | 2.33215 | 0.316
14 | 2.44134 | 2.43977 | 0.064 | 2.43948 | 0.076
15 | 2.48376 | 2.48460 | 0.034 | 2.48477 | 0.040
16 | 2.66501 | 2.66902 | 0.150 | 2.66862 | 0.135
17 | 2.75780 | 2.74510 | 0.463 | 2.74453 | 0.484
18 | 2.78254 | 2.78677 | 0.152 | 2.78699 | 0.160
19 | 2.83647 | 2.82217 | 0.507 | 2.82196 | 0.514
20 | 2.97943 | 2.99342 | 0.467 | 2.99356 | 0.472
21 | 3.01159 | 3.01770 | 0.203 | 3.01823 | 0.220
22 | 3.07402 | 3.09174 | 0.573 | 3.09132 | 0.560
23 | 3.13148 | 3.13824 | 0.215 | 3.13768 | 0.198
24 | 3.16311 | 3.17632 | 0.416 | 3.17606 | 0.408
25 | 3.24362 | 3.25436 | 0.330 | 3.25461 | 0.338
26 | 3.25447 | 3.27058 | 0.492 | 3.27034 | 0.485
27 | 3.30929 | 3.30305 | 0.189 | 3.30309 | 0.188
28 | 3.40908 | 3.42759 | 0.540 | 3.42754 | 0.539
29 | 3.46136 | 3.46796 | 0.190 | 3.46799 | 0.191
30 | 3.53828 | 3.52248 | 0.449 | 3.52213 | 0.459
31 | 3.58063 | 3.59778 | 0.477 | 3.59745 | 0.468
32 | 3.64885 | 3.64490 | 0.108 | 3.64495 | 0.107
33 | 3.70451 | 3.68606 | 0.500 | 3.68572 | 0.510
34 | 3.73124 | 3.74476 | 0.361 | 3.74458 | 0.356
35 | 3.77514 | 3.78817 | 0.344 | 3.78828 | 0.347
36 | 3.81664 | 3.81972 | 0.081 | 3.82010 | 0.091
37 | 3.84659 | 3.86236 | 0.408 | 3.86303 | 0.426
38 | 3.89301 | 3.90315 | 0.260 | 3.90408 | 0.283
39 | 3.92147 | 3.92013 | 0.034 | 3.92011 | 0.035
40 | 4.00155 | 4.01985 | 0.455 | 4.01957 | 0.448
41 | 4.05232 | 4.04810 | 0.104 | 4.04792 | 0.109
42 | 4.07279 | 4.07673 | 0.096 | 4.07692 | 0.101
43 | 4.11185 | 4.10555 | 0.154 | 4.10515 | 0.163
44 | 4.16568 | 4.12779 | 0.918 | 4.12750 | 0.925
45 | 4.17169 | 4.15185 | 0.478 | 4.15181 | 0.479
46 | 4.23171 | 4.23428 | 0.061 | 4.23422 | 0.059
47 | 4.25632 | 4.26175 | 0.127 | 4.26195 | 0.132
48 | 4.28696 | 4.29318 | 0.145 | 4.29333 | 0.148
49 | 4.33501 | 4.33233 | 0.062 | 4.33252 | 0.057
50 | 4.39232 | 4.39776 | 0.124 | 4.39806 | 0.131
51 | 4.44617 | 4.44492 | 0.028 | 4.44480 | 0.031
52 | 4.45932 | 4.45990 | 0.013 | 4.45953 | 0.005
53 | 4.47269 | 4.48323 | 0.235 | 4.48383 | 0.248
54 | 4.50973 | 4.51226 | 0.056 | 4.51274 | 0.067
55 | 4.54993 | 4.53911 | 0.239 | 4.53896 | 0.242
56 | 4.57564 | 4.59124 | 0.340 | 4.59160 | 0.348
57 | 4.61763 | 4.59924 | 0.400 | 4.59942 | 0.396
58 | 4.64177 | 4.63850 | 0.070 | 4.63892 | 0.061
59 | 4.67340 | 4.67170 | 0.036 | 4.67168 | 0.037
60 | 4.67725 | 4.71140 | 0.725 | 4.71268 | 0.752
61 | 4.74328 | 4.72813 | 0.320 | 4.72823 | 0.318
62 | 4.75405 | 4.76937 | 0.321 | 4.76985 | 0.331
63 | 4.78548 | 4.79885 | 0.279 | 4.79964 | 0.295
64 | 4.83816 | 4.84783 | 0.199 | 4.84855 | 0.214
65 | 4.87454 | 4.89056 | 0.328 | 4.89094 | 0.335
66 | 4.89770 | 4.89955 | 0.038 | 4.89976 | 0.042
67 | 4.91976 | 4.92224 | 0.050 | 4.92262 | 0.058
68 | 4.94198 | 4.93395 | 0.163 | 4.93404 | 0.161
69 | 4.97544 | 4.95930 | 0.326 | 4.95855 | 0.341
70 | 5.00234 | 4.98595 | 0.329 | 4.98635 | 0.321
71 | 5.05521 | 5.05121 | 0.079 | 5.05153 | 0.073
72 | 5.06222 | 5.07826 | 0.316 | 5.07871 | 0.325
73 | 5.10734 | 5.10605 | 0.025 | 5.10572 | 0.032
74 | 5.14001 | 5.15663 | 0.322 | 5.15649 | 0.320
75 | 5.17375 | 5.18850 | 0.284 | 5.18928 | 0.299
76 | 5.20031 | 5.19720 | 0.060 | 5.19750 | 0.054
77 | 5.22839 | 5.23149 | 0.059 | 5.23154 | 0.060
78 | 5.24876 | 5.27701 | 0.535 | 5.27763 | 0.547
79 | 5.31483 | 5.33671 | 0.410 | 5.33550 | 0.387
80 | 5.35238 | 5.35144 | 0.018 | 5.35154 | 0.016
81 | 5.36938 | 5.36972 | 0.006 | 5.36912 | 0.005
82 | 5.37790 | 5.40947 | 0.584 | 5.40967 | 0.587
83 | 5.41390 | 5.42943 | 0.286 | 5.42989 | 0.295
84 | 5.43467 | 5.44317 | 0.156 | 5.44304 | 0.154
Table 12: UTe2 SMI fit results | 300 K | 4 K
---|---|---
Index |  $f_{exp}$ (MHz) |  $f_{calc}$ (MHz) | diff (%) |  $f_{exp}$ (MHz) |  $f_{calc}$ (MHz) | diff (%)
1 | 0.64645 | 0.64254 | - | 0.66344 | 0.65765 | -
2 | 0.80333 | 0.80092 | - | 0.81724 | 0.81425 | -
3 | 0.86443 | 0.85869 | - | 0.87702 | 0.87984 | -
4 | 1.09564 | 1.09407 | 0.143 | 1.11952 | 1.12080 | 0.114
5 | 1.18679 | 1.18992 | 0.263 | 1.21970 | 1.21818 | 0.125
6 | 1.32441 | 1.32471 | 0.023 | 1.34208 | 1.34680 | 0.350
7 | 1.34023 | 1.33357 | 0.499 | 1.36327 | 1.35737 | 0.435
8 | 1.49544 | 1.49019 | 0.353 | 1.51789 | 1.51792 | 0.002
9 | 1.56441 | 1.57149 | 0.450 | 1.59618 | 1.60073 | 0.284
10 | 1.64031 | 1.64096 | 0.040 | 1.67269 | 1.67216 | 0.032
11 | 1.67848 | 1.67996 | 0.088 | 1.71445 | 1.71359 | 0.050
12 | 1.71675 | 1.71182 | 0.288 | 1.75060 | 1.74551 | 0.291
13 | 1.77448 | 1.77473 | 0.014 | 1.80806 | 1.80856 | 0.028
14 | 1.83771 | 1.83767 | 0.002 | 1.87151 | 1.87196 | 0.024
15 | 1.86861 | 1.86685 | 0.094 | 1.90573 | 1.90183 | 0.205
16 | 1.93964 | 1.93352 | 0.317 | 1.97323 | 1.97067 | 0.130
17 | 2.02902 | 2.02240 | 0.327 | 2.06927 | 2.06282 | 0.313
18 | 2.06086 | 2.06388 | 0.146 | 2.09666 | 2.10371 | 0.335
19 | 2.13401 | 2.13646 | 0.115 | 2.17374 | 2.17693 | 0.147
20 | 2.18310 | 2.17778 | 0.244 | 2.23096 | 2.22600 | 0.223
21 | 2.21720 | 2.21601 | 0.054 | 2.26583 | 2.25830 | 0.334
22 | 2.25887 | 2.25321 | 0.251 | 2.30249 | 2.29602 | 0.282
23 | 2.27636 | 2.28024 | 0.170 | 2.32310 | 2.32435 | 0.054
24 | 2.33423 | 2.33367 | 0.024 | 2.37609 | 2.37215 | 0.166
25 | 2.35896 | 2.35370 | 0.223 | 2.40235 | 2.40166 | 0.029
26 | 2.37195 | 2.37131 | 0.027 | 2.41002 | 2.41040 | 0.016
27 | 2.40361 | 2.39962 | 0.166 | 2.46079 | 2.45405 | 0.275
28 | 2.45741 | 2.45189 | 0.225 | 2.50592 | 2.50395 | 0.079
29 | 2.47653 | 2.47529 | 0.050 | 2.51386 | 2.52139 | 0.299
30 | 2.49726 | 2.49516 | 0.085 | 2.55527 | 2.55033 | 0.194
31 | 2.53119 | 2.53407 | 0.113 | 2.58793 | 2.58510 | 0.109
32 | 2.56841 | 2.57293 | 0.176 | 2.61987 | 2.62518 | 0.202
33 | 2.58945 | 2.58640 | 0.118 | 2.64506 | 2.64006 | 0.189
34 | 2.67652 | 2.67401 | 0.094 | 2.72662 | 2.72716 | 0.020
35 | 2.69934 | 2.70106 | 0.064 | 2.75414 | 2.75458 | 0.016
36 | 2.70867 | 2.71641 | 0.285 | 2.76160 | 2.76697 | 0.194
37 | 2.75772 | 2.76163 | 0.142 | 2.81381 | 2.81674 | 0.104
38 | 2.80315 | 2.80522 | 0.073 | 2.85929 | 2.86408 | 0.167
39 | 2.83790 | 2.83401 | 0.137 | 2.88995 | 2.88518 | 0.165
40 | 2.84413 | 2.84397 | 0.006 | 2.89806 | 2.89733 | 0.025
41 | 2.85740 | 2.85678 | 0.022 | 2.91031 | 2.91158 | 0.044
42 | 2.89809 | 2.90021 | 0.073 | 2.95404 | 2.95353 | 0.017
43 | 2.91204 | 2.90687 | 0.178 | 2.96297 | 2.95926 | 0.125
44 | 2.93294 | 2.93629 | 0.114 | 2.98892 | 2.98868 | 0.008
45 | 2.97064 | 2.96887 | 0.060 | 3.03454 | 3.03881 | 0.140
46 | 2.98750 | 2.98376 | 0.125 | 3.04986 | 3.04773 | 0.070
47 | 3.04099 | 3.04359 | 0.086 | 3.10384 | 3.10431 | 0.015
48 | 3.05451 | 3.05513 | 0.020 | 3.11878 | 3.11603 | 0.088
49 | 3.07129 | 3.06617 | 0.167 | 3.12869 | 3.12381 | 0.156
50 | 3.11543 | 3.11631 | 0.028 | 3.17516 | 3.18070 | 0.174
51 | 3.14348 | 3.14681 | 0.106 | 3.21205 | 3.20841 | 0.113
52 | 3.16609 | 3.16333 | 0.087 | 3.23373 | 3.23001 | 0.115
53 | 3.19312 | 3.19392 | 0.025 | 3.25821 | 3.25521 | 0.092
54 | 3.21950 | 3.22107 | 0.049 | 3.28885 | 3.29187 | 0.092
55 | 3.26064 | 3.26213 | 0.046 | 3.32549 | 3.32873 | 0.097
56 | 3.27367 | 3.27710 | 0.105 | 3.33381 | 3.33748 | 0.110
57 | 3.28431 | 3.28793 | 0.110 | 3.35277 | 3.35414 | 0.041
58 | 3.33902 | 3.33560 | 0.103 | 3.39969 | 3.39764 | 0.060
59 | 3.37665 | 3.38210 | 0.161 | 3.44137 | 3.44067 | 0.020
60 | 3.38263 | 3.38841 | 0.171 | 3.44466 | 3.45563 | 0.317
61 | 3.40724 | 3.40264 | 0.135 | 3.47399 | 3.46879 | 0.150
62 | 3.42588 | 3.41984 | 0.176 | 3.48688 | 3.48075 | 0.176
63 | 3.46878 | 3.46450 | 0.123 | 3.52655 | 3.52462 | 0.055
64 | 3.48772 | 3.48306 | 0.134 | 3.55491 | 3.55194 | 0.084
65 | 3.49884 | 3.50575 | 0.197 | 3.56459 | 3.57159 | 0.196
66 | 3.51096 | 3.51080 | 0.005 | 3.58054 | 3.57953 | 0.028
67 | 3.54860 | 3.55098 | 0.067 | 3.61868 | 3.62048 | 0.050
68 | 3.56768 | 3.57417 | 0.182 | 3.64321 | 3.64728 | 0.112
69 | 3.57963 | 3.58351 | 0.108 | 3.65177 | 3.65702 | 0.144
70 | 3.59151 | 3.59154 | 0.001 | 3.66315 | 3.66157 | 0.043
71 | 3.63037 | 3.63146 | 0.030 | 3.70535 | 3.70318 | 0.059
72 | 3.66135 | 3.66185 | 0.014 | 3.73498 | 3.73249 | 0.067
73 | 3.67686 | 3.67367 | 0.087 | 3.75129 | 3.74617 | 0.137
74 | 3.70093 | 3.69729 | 0.098 | 3.77215 | 3.77270 | 0.015
75 | 3.72551 | 3.71979 | 0.154 | 3.79478 | 3.78954 | 0.138
76 | 3.74072 | 3.74038 | 0.009 | 3.81853 | 3.81297 | 0.146
77 | 3.75135 | 3.75430 | 0.079 | 3.82596 | 3.82515 | 0.021
78 | 3.76556 | 3.76595 | 0.010 | 3.83940 | 3.83786 | 0.040
79 | 3.77374 | 3.77640 | 0.071 | 3.84028 | 3.84954 | 0.240
80 | 3.81529 | 3.81717 | 0.049 | 3.89424 | 3.89363 | 0.016
81 | 3.84139 | 3.83861 | 0.072 | 3.91424 | 3.91245 | 0.046
82 | 3.85843 | 3.85711 | 0.034 | 3.93674 | 3.93303 | 0.094
83 | 3.87490 | 3.88271 | 0.201 | 3.94923 | 3.95735 | 0.205
84 | 3.89786 | 3.89296 | 0.126 | 3.97082 | 3.96312 | 0.194
85 | 3.90314 | 3.90305 | 0.002 | 3.98043 | 3.98563 | 0.130
86 | 3.91220 | 3.92250 | 0.263 | 3.98665 | 3.99710 | 0.261
87 | 3.94153 | 3.93803 | 0.089 | 4.01442 | 4.01238 | 0.051
88 | 3.99297 | 3.98031 | 0.318 | 4.06185 | 4.05116 | 0.264
89 | 3.99436 | 3.99491 | 0.014 | 4.07113 | 4.06908 | 0.050
90 | 4.00126 | 4.01160 | 0.258 | 4.07942 | 4.08544 | 0.147
91 | 4.03760 | 4.03141 | 0.154 | 4.11408 | 4.10575 | 0.203
92 | 4.07137 | 4.07176 | 0.010 | 4.15105 | 4.14854 | 0.061
93 | 4.08673 | 4.08572 | 0.025 | 4.16596 | 4.16925 | 0.079
94 | 4.11792 | 4.11397 | 0.096 | 4.19784 | 4.19441 | 0.082
95 | 4.12896 | 4.13582 | 0.166 | 4.20427 | 4.21294 | 0.206
96 | 4.14529 | 4.15212 | 0.164 | 4.22149 | 4.22953 | 0.190
97 | 4.15242 | 4.15367 | 0.030 | 4.23333 | 4.23415 | 0.019
98 | 4.17297 | 4.17074 | 0.053 | 4.24888 | 4.24690 | 0.047
99 | 4.18994 | 4.18888 | 0.025 | 4.27899 | 4.27740 | 0.037
100 | 4.20373 | 4.21285 | 0.216 | 4.28904 | 4.29176 | 0.064
101 | 4.22958 | 4.23006 | 0.011 | 4.30977 | 4.31112 | 0.031
102 | 4.24199 | 4.24296 | 0.023 | 4.32438 | 4.32504 | 0.015
103 | 4.25360 | 4.25075 | 0.067 | 4.33687 | 4.33486 | 0.046
104 | 4.28642 | 4.28345 | 0.069 | 4.36950 | 4.36567 | 0.088
105 | 4.30657 | 4.30624 | 0.008 | 4.39228 | 4.39207 | 0.005
106 | 4.31547 | 4.31995 | 0.104 | 4.40638 | 4.40694 | 0.013
107 | 4.34118 | 4.34193 | 0.017 | 4.42091 | 4.42370 | 0.063
108 | 4.34935 | 4.35598 | 0.152 | 4.43621 | 4.44172 | 0.124
109 | 4.36295 | 4.36176 | 0.027 | 4.44922 | 4.44494 | 0.096
110 | 4.37974 | 4.37984 | 0.002 | 4.46691 | 4.46202 | 0.110
111 | 4.39652 | 4.38614 | 0.237 | 4.47881 | 4.46618 | 0.283
112 | 4.40482 | 4.41073 | 0.134 | 4.48982 | 4.49513 | 0.118
113 | 4.42003 | 4.41888 | 0.026 | 4.49809 | 4.50730 | 0.204
114 | 4.42672 | 4.42968 | 0.067 | 4.51862 | 4.51611 | 0.055
115 | 4.44940 | 4.46193 | 0.281 | 4.53519 | 4.54067 | 0.121
116 | 4.46937 | 4.47430 | 0.110 | 4.55608 | 4.55307 | 0.066
117 | 4.48235 | 4.48380 | 0.032 | 4.56374 | 4.56785 | 0.090
118 | 4.50482 | 4.50569 | 0.019 | 4.59148 | 4.59609 | 0.100
119 | 4.50733 | 4.51327 | 0.132 | 4.59729 | 4.60078 | 0.076
120 | 4.52413 | 4.52782 | 0.081 | 4.60960 | 4.61562 | 0.131
121 | 4.53911 | 4.54276 | 0.080 | 4.62762 | 4.63458 | 0.150
122 | 4.56333 | 4.56602 | 0.059 | 4.64820 | 4.65923 | 0.237
123 | 4.57367 | 4.57286 | 0.018 | 4.66121 | 4.66642 | 0.112
124 | 4.60540 | 4.60303 | 0.051 | 4.69278 | 4.69228 | 0.011
125 | 4.61532 | 4.62032 | 0.108 | 4.70740 | 4.71043 | 0.064
126 | 4.62818 | 4.62519 | 0.065 | 4.71517 | 4.71609 | 0.020
127 | 4.63514 | 4.63026 | 0.105 | 4.72725 | 4.71991 | 0.155
128 | 4.64194 | 4.64199 | 0.001 | 4.73331 | 4.73754 | 0.089
129 | 4.66690 | 4.66430 | 0.056 | 4.75924 | 4.75618 | 0.064
130 | 4.66790 | 4.66597 | 0.041 | 4.76674 | 4.75845 | 0.174
131 | 4.69246 | 4.68730 | 0.110 | 4.78743 | 4.78038 | 0.147
132 | 4.70527 | 4.71080 | 0.118 | 4.80128 | 4.80542 | 0.086
133 | 4.72046 | 4.73008 | 0.203 | 4.81305 | 4.82061 | 0.157
134 | 4.75051 | 4.75395 | 0.072 | 4.84664 | 4.84559 | 0.022
135 | 4.75900 | 4.76169 | 0.057 | 4.85021 | 4.85330 | 0.064
136 | 4.76976 | 4.77384 | 0.085 | 4.86402 | 4.86321 | 0.017
137 | 4.77918 | 4.77950 | 0.007 | 4.87531 | 4.87473 | 0.012
138 | 4.79086 | 4.79493 | 0.085 | 4.88149 | 4.88582 | 0.089
139 | 4.80450 | 4.80244 | 0.043 | 4.89785 | 4.89637 | 0.030
140 | 4.81875 | 4.81484 | 0.081 | 4.91309 | 4.90863 | 0.091
141 |  |  |  | 4.93822 | 4.94346 | 0.106
142 |  |  |  | 4.95627 | 4.95918 | 0.059
143 |  |  |  | 4.96168 | 4.96342 | 0.035
144 |  |  |  | 4.98300 | 4.98143 | 0.032
145 |  |  |  | 4.98846 | 4.99420 | 0.115
146 |  |  |  | 4.99438 | 5.00516 | 0.215
147 |  |  |  | 5.01215 | 5.01596 | 0.076
148 |  |  |  | 5.02882 | 5.03051 | 0.034
149 |  |  |  | 5.04591 | 5.05275 | 0.135
150 |  |  |  | 5.06142 | 5.06270 | 0.025