Enabling resonant ultrasound spectroscopy in high magnetic fields

Elastic constants are second derivatives of the free energy with respect to strain which describe the resistance of a material to elastic deformation. They are indispensable in characterizing mechanical performance^1–3 and sensitive to atomic structure and bonding⁴ in all types of materials.⁵ As the thermodynamic susceptibility of strain, elastic constants also reflect strain couplings to structural, electronic, and magnetic degrees of freedom.^6–8 This makes elastic constants ideal to understand changes in materials as a function of external fields (temperature, magnetic field, etc.) and symmetry-breaking at phase transitions.^6–8 

There are numerous static, quasi-static, and dynamic methods to measure elastic constants.⁹ Among these, resonant ultrasound spectroscopy (RUS)¹⁰ is a particularly powerful technique for the non-destructive examination of elastic properties because the mechanical resonances of a sample with a known geometry and mass provide access to the complete set of elastic constants, as well as symmetry-resolved ultrasonic attenuation,^9,11 with high accuracy and precision.^12–14 The simultaneous determination of the entire elastic constant tensor makes RUS well-suited for studying elastic anisotropy (e.g., texture in metals¹⁵) and the level of precision enables detailed studies of phase transitions with relatively small elastic signatures [e.g., superconducting and skyrmion phase transitions correspond to relative changes in elastic constants ∼10⁻⁵–10⁻³ (Refs. 16–18)]. Together, these aspects have made RUS an important tool for studying elastic properties with implications in both fundamental and applied science.

Significant progress has been made since the inception of RUS,^19–22 especially in the areas of computational efficiency²³ and instrumentation.^24,25 Recent advances in generalizing RUS to complex geometries through finite-element²⁶ or surface-mesh²⁷ approaches also offer promising paths toward efficient elastic constant determination on irregularly shaped samples. Central to all these approaches is that information about elastic moduli is encoded into mechanical resonant frequencies, with the exact relationship between the two depending on geometry, materials properties (density and internal symmetry), and mechanical boundary conditions. Free boundary conditions are most often employed in RUS inversion because they yield well-defined solutions and make the inversion computationally efficient.^12,13,28

While free boundary conditions are most commonly used to quantitatively extract elastic moduli from measured resonant frequencies, RUS experimental implementations only approximate free boundary conditions. The most prevalent RUS methods require transducers to contact the sample and the resulting contact forces cause deviations from free boundary conditions which shift resonant frequencies from their free resonance value.²⁹ In well-designed experiments, these forces are sufficiently small and applied at an appropriate location on the sample such that the resonant frequency shifts are below the uncertainty of the inversion process.¹³ Contact effects are a negligible perturbation to the free resonance condition in these scenarios, but computational simplicity comes at the expense of sample stability: samples are typically positioned in unstable configurations to minimize contact with transducers. Therefore, RUS is ill-suited for experiments in “extreme” environments where sample stability is paramount, such as high magnetic fields. RUS approaches which utilize non-contact ultrasonic excitation/detection³⁰ offer closer approximations to the free resonance condition than transducer-based measurements, but are only compatible with electrically conductive materials,^31,32 necessitate coating samples with thin films,³³ or face similar sample stability challenges as transducer-based RUS owing to the need to support samples.³⁴ Additionally, while sample orientation is often not considered in conventional RUS, orientation becomes important if an external field (magnetic,^17,18,35,36 electric,³⁷ etc.) must be applied along a specific direction in a sample. While there have been some efforts towards applying RUS in magnetic fields,^17,18,35,36 they involved immense work to maintain sample positioning, are incompatible with samples with appreciable magnetic moments, and/or are limited to low magnetic fields. Together, sample movement and orientation represent two significant challenges which limit practical applications of RUS in extreme environments.

The central result of this paper is that the use of adhesives between samples and ultrasonic transducers, under specific conditions, introduces a perturbation to the free resonance condition akin to the effect of contact in conventional RUS. The resonant frequency spectrum obtained in this manner is similar to conventional spectra measured under free boundary conditions with a shift in the resonant frequencies to higher values. These effects can be captured by a simple model and effectively minimized, such that elastic constants are determined when a sample is adhered to transducers to within the same uncertainty of conventional RUS, but with the benefits of enhanced sample stability and controlled sample orientation imbued by the adhesive. Moreover, efficient elastic constant analysis is maintained because established RUS-inversion frameworks are compatible with this approach. After showing that we recover the accuracy of conventional RUS, we demonstrate these adaptions enable RUS measurements of changes in elastic constants and ultrasonic attenuation in magnetic fields up to 14 T with a comparable sensitivity to phase transitions. This advancement makes RUS amenable to high magnetic fields, and other extreme environments, with improved reproducibility, sample stability, and control over sample orientation.

RUS measurements were performed on a custom-built system using a similar experimental approach described in Ref. 24 and summarized in Fig. 1. A sample shaped into a suitable geometry for RUS (rectangular parallelepiped or cylinder) was placed between two ultrasonic transducers. The transducers consisted of a metallic enclosure filled with an epoxy, a piezoelectric active element (lead zirconate titanate for ambient measurements and lithium niobate for low temperature measurements), and an alumina or sapphire hemisphere (to electrically insulate the piezoelectric from conductive samples, minimize wear on the piezoelectric, and guarantee point contact with the sample³⁸). For conventional RUS, care was taken to ensure the sample was lightly held between the transducers and the sample was positioned to maximize the number of measurable resonances. Then, a field-programmable gate array-based system generated a single-frequency excitation in one transducer and measured the response to this excitation frequency with the other transducer.²⁴ Sweeping the excitation frequency allowed for the identification of the resonant frequencies of the sample from modified Lorentzian fits to the frequency spectrum. For the geometries used in this work, elastic constants were then obtained from measured resonant frequency spectra via an iterative, Lagrangian minimization approach.²³ Temperature-dependent measurements were performed on a custom-built piston probe^17,18 and a custom-built cantilever probe compatible with a Physical Property Measurement System (Quantum Design).

The data reported here were obtained on four samples. Data at ambient conditions were collected on 1100 aluminum alloy³⁹ and high-density tungsten alloy (90% W, 7% Ni, 3% Fe) samples.^15,40 Temperature-dependent measurements on a single crystal SrTiO₃ sample ( Appendix C) and temperature- and magnetic field-dependent measurements on a pure gadolinium polycrystal were used to examine the sensitivity to well-known phase transitions.^41,42 These four samples were selected because they span a range of geometries, masses, stiffnesses, and elastic-anisotropies. The approach has also been successful in additional polycrystalline and single crystal systems which will be the subject of future work. The adhesive used here was a commercially available cyanoacrylate-based adhesive which could be removed from the samples with acetone. Other fast curing adhesives were also tested, yielding comparable results to those presented in this work. The amount of adhesive used for optimized adhesion was ≲0.5 mg for samples with masses and dimensions spanning 0.1–2 g and 3–13.5 mm, respectively.

First, we consider the effects of coupling the sample and transducers via an adhesive by measuring the resonance spectra of an 1100 aluminum alloy sample without an adhesive, with a small amount (⁠ $≈$2 mg) of adhesive, and with a large amount (⁠ $≈$20 mg) of adhesive (Fig. 2). When the sample is adhered to the transducers, all the resonant frequencies of the sample are shifted to higher frequencies and broadened, and the amplitude of the resonances is increased. Figure 2 demonstrates that these effects become more pronounced as the coupling between the sample and transducer is increased by using more adhesive. As discussed later, the increased amplitude may have important applications in non-linear RUS measurements or for poorly resonating materials like polymers or geological materials.^43,44 However, for the purposes of elastic constant determination, frequency shifts and broadening are deleterious: shifts in the resonant frequency will affect elastic constant determination from the inversion process, and excessively broadened peaks can be difficult to identify leading to missing resonant frequencies.⁴⁵ For example, the individual resonances in the frequency range of 280–320 kHz become difficult to distinguish when an excessive amount of adhesive is used [top-most spectrum in Fig. 2(a)]. In such cases, the elastic constants will not be able to be fit from the resonance spectrum, or they can be fit but will be erroneous.

In addition, it may be important to consider the location at which the adhesive is applied to the sample. In conventional RUS, the point of contact between the sample and the transducers can affect elastic constant determination because certain resonances may not have measurable amplitudes where the transducer contacts the sample or may be strongly affected by the transducer weight. Given the complex shapes of the resonant modes, the ideal location to apply the adhesive may differ for each sample. In principle, the resonant mode shapes can be measured or simulated (e.g., Refs. 26, 47) for a particular experiment. In practice, adhering the sample close to its corners appears to be sufficient. This works particularly well in our experiments because the hemispheres on the transducers inherently minimize the contact area of the adhesive.

Figure 3 demonstrates that when only a small, optimized amount of adhesive is used (⁠ $≲$0.5 mg), sharp resonances from the sample are present which exhibit small frequency shifts and increased width. The resonances measured from the adhered sample form circles when plotted in the complex plane indicating they possess Lorentzian shapes, just like those measured with conventional RUS [Figs. 3(b) and 3(d)]. To obtain the same amplitudes in resonance spectra with and without the adhesive, the excitation voltage used with the adhesive was reduced by a factor of ∼3 because of the enhanced coupling between the transducers and the sample. A consequence of this stronger coupling is that direct transmission between the transducers is possible and spurious resonances from the system may be measured. Care should be taken to minimize these effects by reducing the excitation voltage.

To quantify the changes in the resonance spectra stemming from the adhesive, we compare the resonant frequencies obtained on a high-density tungsten alloy sample with and without an adhesive (data in Fig. 3 and Tables II–IV in  Appendix B). Figure 4 demonstrates that the change in the frequencies caused by the adhesive is well-described by Eq. (1), i.e., the dominant effect of the adhesive is a shift in the resonant frequencies of the sample to higher values. The slope of this line is 1 within the uncertainty of the fit and the intercept (which is a measure of the size of the adhesive effect) is small compared to $f 0 2$⁠. Moreover, fixing the slope to 1 in Fig. 4(a) has a minimal impact on the fit quality or the elastic constant determination described below. A direct consequence of Eq. (1) is that that higher resonant frequencies possess smaller relative changes from the adhesive. At sufficiently high frequencies (resonance number ∼15 in this case), these variations are within the usual reproducibility of conventional RUS as determined by changing transducer positioning on the sample. This suggests that it may be possible to perform quantitative RUS measurements of elastic constants without the corrections described below if only higher frequency resonances can be used. Last, we find no systematic relation between the frequency shifts and the composition or symmetry of the resonance. This supports using the simple model in Eq. (1), which does not include mode-dependent corrections.

Having established the frequency shift stemming from the use of an adhesive, we now demonstrate that elastic constants determined with an adhesive are within the uncertainty of elastic constants determined from conventional RUS after adhesive effects have been accounted for. Table I and Fig. 5 provide comparisons of the isotropic elastic constants obtained from measurements on a high-density tungsten alloy sample with and without an adhesive. Elastic constants were determined via Visscher's method²³ with the same number of resonances used for the inversions. Measured frequencies and fit results are provided in Tables II–IV in  Appendix B.

We first note that for the un-adhered case, the elastic constants are well characterized with either 20 or 52 resonances. However, as shown in Fig. 5 and Table I, when elastic constants are determined from the first 20 as-measured frequencies with an adhered sample, the elastic constants differ substantially from the elastic constants obtained with conventional RUS. The situation is improved by increasing the number of frequencies included in the inversion to the first 52 as-measured frequencies, but the resulting elastic constants are still clearly outside the uncertainty of the elastic constants found with conventional RUS. This improvement in accuracy from increasing the number of resonant frequencies included in the inversion arises because the adhesive-induced shift in the resonant frequencies becomes relatively smaller at higher frequencies (Fig. 4).

In contrast, when the frequencies measured with the adhesive are corrected using the fit parameters found from applying Eq. (1), quantitative agreement is obtained with the elastic constants determined by conventional RUS (Fig. 5). More specifically, the correction process requires finding the fit parameters $A$ and $B$ according to $f ′ 2=A f 0 2+B$⁠, where $f′$ are the as-measured resonant frequencies with an adhesive applied to the sample and $f 0$ are the free resonant frequencies of the sample measured with conventional RUS. The agreement between the elastic constants found with this method and those determined from conventional RUS holds when either the first 20 or 52 resonant frequencies are included in the inversion. Note, the small differences in c₁₁ between inversions using the first 20 and 52 frequencies is the result of better c₁₁ representation in the larger number of frequencies, whereas c₄₄ has sufficient representation in every resonance (see  Appendix B).

Having demonstrated elastic constants can be determined to within the uncertainty of conventional RUS when an adhesive is used, we now turn to using our modified RUS approach for temperature- and magnetic field-dependent studies. The application of RUS in these contexts is highly desirable because elastic constants in the vicinity of phase transitions are renormalized by strain-order parameter and electron-phonon couplings.^6–8 Such deviations from typical elastic constant behavior⁴⁸ can be used to understand the nature of phase transitions and place constraints on order parameter symmetries.^6,16,49 As previously discussed, the principal effect of adhering a sample to a transducer is a constant shift in the resonant frequencies to higher frequencies [Eq. (1)]. Consequently, changes in resonant frequencies (and elastic constants), which are the important quantity in temperature- and magnetic field-dependent studies,⁴⁶ should be largely unaffected by the adhesive. We experimentally confirm this is the case using a single crystal of SrTiO₃ ( Appendix C) and polycrystalline Gd. Gd was selected to test our approach because it has well-known magnetic phase transitions as a function of temperature and applied magnetic field, including a paramagnetic-ferromagnetic transition and a spin-reorientation transition at zero applied magnetic field (e.g., see Ref. 41, and references therein). These phase transitions are an ideal test because they involve large changes in magnetization, causing the sample to experience substantial torque⁵⁰ which tends to knock samples out of transducers in conventional RUS setups.³⁵ Indeed, every attempt to apply a magnetic field to Gd samples in conventional RUS without an adhesive led to the sample flying away at low and irreproducible fields (⁠ $μ$₀H  $∼$ 0.1 T).

Figure 6 shows the temperature-dependent behavior of a resonant frequency of a polycrystalline Gd cylinder at fixed magnetic fields under free resonance conditions and with an adhesive used to affix the sample to the transducers. Focusing first on the zero applied magnetic field data, these measurements demonstrate that both the ferromagnetic and spin-reorientation transitions are clearly resolved in the adhered sample with transition temperatures that agree with the literature.⁴¹ Furthermore, the temperature dependence at zero applied magnetic field is essentially unchanged by the adhesive: maxima and minima occur at the same temperatures and the data show similar concavity [Fig. 6(b)]. The acoustic quality factors, while reduced, are still sufficient to observe changes in ultrasonic attenuation at phase transitions. See  Appendix D for additional discussion of the effects of the adhesive on ultrasonic attenuation. Note, the good agreement between the temperature dependences measured with and without adhesive in Fig. 6(b) indicates the adhesive is not substantially changing the relative amounts of different elastic constants in each normal mode and provides additional evidence that the dominant effect of the adhesive in these measurements is a rigid shift in the resonant frequencies. There also do not appear to be any thermal-strain effects associated with the adhesive which alter the temperature dependence of the sample's resonant frequency spectra, likely because the amount of adhesive is minimal. That being said, in general it is important to characterize the temperature-dependent properties of the adhesive for a particular sample/application over the measurement conditions. By maintaining the assumptions which led to Eq. (1), i.e., $f ′ 2− f 0 2≈0$⁠, contributions to the temperature-dependent elastic response from the adhesive should be minimized (Fig. 6). Additionally, the adhesive may improve thermal conductivity to the sample which can benefit very low temperature RUS applications which commonly face sample thermalization difficulties.⁵¹ 

Measuring the temperature-dependent behavior of a resonant frequency at a fixed applied magnetic field of 2 T on the adhered samples shows both the ferromagnetic and spin-reorientation phase transitions are suppressed, in agreement with the literature.⁴¹ Comparable measurements with conventional RUS at 2 T could not be performed because the sample flew away at substantially smaller applied magnetic fields (<0.1 T). To further demonstrate our adhesive-enabled RUS capabilities at high magnetic fields, we measured the same resonant frequency shown in Fig. 6 as a function of magnetic field up to 14 T at a fixed temperature of 260 K (Fig. 7). This temperature places the sample below the ferromagnetic transition and above the spin-reorientation transition, a region of phase space where it is known that Gd undergoes a phase transition at $μ$₀H  $≈$ 0.4 T.⁴¹ As shown in Fig. 7, we can perform high signal-to-noise RUS measurements up to 14 T and identify signatures of this phase transition⁴¹ in both resonant frequencies and ultrasonic attenuation. Therefore, this new RUS approach provides the same sensitivity to phase transitions at zero field compared to conventional RUS and enables reproducible RUS measurements in large magnetic fields.

RUS is an ideal technique to study elasticity and determine phase diagrams under extreme environments because of its high precision and accuracy, and its ability to simultaneously determine the entire elastic constant tensor from a single measurement. The latter benefit is especially significant because obtaining comparable datasets with alternative approaches (e.g., pulse-echo ultrasound techniques⁶) requires combining multiple samples and measurements, introducing the possibility of large systematic uncertainties. We have shown that the primary challenge in utilizing RUS in extreme environments, namely, sample stability, is readily addressed by adhering a sample to the transducers in an otherwise conventional RUS experiment. The adhesive improves sample stability and orientation control, allowing for reproducible RUS in extreme environments such as high magnetic fields. The presence of the adhesive does not reduce sensitivity to phase transitions, as demonstrated by the fact that phase transitions are readily observed in both elastic constants and ultrasonic attenuation. Modelling the adhesive effects even allows one to obtain elastic constants in excellent quantitative agreement with conventional RUS. Furthermore, we anticipate our findings may also be beneficial to other RUS approaches such as non-contact methods.³⁰ 

Beyond the opportunities to perform RUS measurements in new environments and regions of phase space, the use of the adhesive may improve RUS investigations under historically challenging conditions and on difficult samples. As shown in Figs. 2 and 3, the excitation voltage used in RUS experiments when an adhesive is present must be substantially reduced to avoid saturating detection electronics. This enhanced sample-transducer coupling can assist in the study of polymeric and geological materials which are often weak resonators^43,44 while maintaining pseudo-free resonance conditions. The ability to tune sample-transducer coupling may also find applications in high-pressure RUS experiments, where reduced resonant frequency amplitudes and increased ultrasonic attenuation owing to the pressure medium are major obstacles.⁵² Last, the adhesive acts as a thermal link to the sample which can improve thermalization and facilitate ultralow temperature RUS measurements.⁵¹ 

This approach also provides an opportunity to investigate the properties of the adhesive materials themselves (e.g., to understand the efficacy of transducer couplants⁵³). As shown in Fig. 8, changes in the resonant frequencies of a sample as a function of time reflect the dynamics of the curing process. The adhesive stiffens as it cures, shifting the resonant frequencies of the sample to higher values, increasing the resonant quality factor, and enhancing the coupling to the transducers. By pairing time-dependent RUS measurements on adhered samples with controlled external conditions (gas composition, humidity levels, light exposure, etc.), intrinsic and dynamic polymeric properties can be studied. For the purposes of these measurements, the data in Fig. 8 also demonstrate the importance of waiting for the adhesive to completely cure before beginning measurements. If the adhesive is not fully cured, changes in the adhesive may be erroneously attributed to the sample.

Finally, we provide a typical workflow for performing RUS measurements in high magnetic fields (or other extreme environments) where the sample has been adhered to the transducers (Fig. 9). Naturally, details are sample and adhesive dependent, but we have found the following steps are a useful guideline.

1. Measure a resonant frequency spectrum with conventional RUS at ambient conditions. Ensure a high-quality inversion can be obtained with results which are not especially sensitive to transducer placement.

2. Apply a sample-compatible adhesive and measure changes in the resonance spectrum. Wait until there are no significant changes in the resonance spectrum. Ideally, measure the changes as a function of time to check the adhesive has fully cured, at least the first time a particular adhesive is used.

3. Once the adhesive has cured, measure a resonance spectrum with the adhesive at the same conditions as (1).

4. Check that frequency shifts before and after the application of the adhesive are small (∼1%) and correct the resonant frequencies measured with the adhesive according to the Eq. (1). Confirm inversion with corrected frequencies gives reasonable elastic constants. If not, re-adhere sample.

5. Perform RUS measurements. Analyze relative changes in elastic constants from relative changes in frequency⁴⁶ or changes in absolute values with corrected frequencies. Remove adhesive if desired.

In this work, we demonstrated how to perform quantitative elastic constant measurements and phase boundary determination under extreme conditions, such as high magnetic fields, using a variation of resonant ultrasound spectroscopy in which the sample is adhered to ultrasonic transducers. We outlined the experimental conditions in which the adhesive can be treated as a perturbation to the free resonance condition and show that, in this regime, the adhesive acts to shift the resonant frequencies by a constant value. These effects can be accounted for by a simple model such that elastic constants can be efficiently obtained with the same accuracy and sensitivity to phase transitions as conventional RUS, but with substantial improvements in sample stability and controlled sample orientation because of the adhesive. This development facilitates the application of RUS to new, extreme environments including reproducible measurements of elastic constants and ultrasonic attenuation in magnetic fields up to 14 T.

This work was supported by the LANL LDRD Program and performed at the National High Magnetic Field Laboratory (which is supported by the National Science Foundation Cooperative Agreement No. DMR-2128556 and the State of Florida). We thank Tannor T. J. Munroe and Fedor Balakirev for assistance with sample preparation and instrumentation.

The authors have no conflicts of interest to disclose.

Data used in this work are provided in  Appendix B. Additional data available upon request.

The tungsten alloy sample, shaped into a rectangular parallelepiped with dimensions of 4.024(5) mm × 5.036(5) mm × 6.038(5) mm, was used for quantitative analyses. Its mass was 2.0582(5) g yielding a density of 16.82(3) g/cm³, which is comparable to the literature.⁴⁰ The tungsten alloy sample exhibited isotropic elasticity based upon an elastic constant determination with the first 52 resonant frequencies (see Tables II–IV). Initial conditions were varied to confirm the reported sets of elastic constants correspond to a global minimum solution. Errors reported correspond to 2% change in the $χ 2$ value.¹³ Explanations for the (k, i) mode naming convention are provided in Ref. 13.

We also applied this method to a SrTiO₃ single crystal sample to show the adhered RUS approach works for anisotropic single crystals. The SrTiO₃ single crystal sample was cut into a cylindrical shape [diameter of 2.984(5) mm and thickness of 3.002(5) mm] with the [100] crystallographic direction along the long axis of the cylinder. The mass of the sample was 0.1057(5) g [density of 5.04(4) g/cm³]. Table V provides the elastic constants determined using conventional RUS prior to the application of the adhesive, which are in good agreement with literature values (Ref. 13 and references therein). After correction according to Eq. (1), the elastic constants determined from RUS measurements where the sample has been adhered to the transducers are within the uncertainties of the conventional RUS measurements.

Next, we examine the behavior of a resonant frequency of the SrTiO₃ single crystal near the structural phase transition at ∼105 K.⁴² For this example, we have purposefully used an excessive amount of adhesive and did not let it fully cure before cooling down the sample. Even with too much adhesive, Fig. 11 demonstrates the softening prior to the onset of the cubic-tetragonal phase transition in SrTiO₃ single crystals⁴² is apparent. The maximum in the resonant frequency also occurs at the same temperature with and without the adhesive. Though some details are affected when an excessive amount of uncured adhesive is used [e.g., curvature in Fig. 11(b)], such un-ideal circumstances still allow for the study of phase transitions. This may be important for very small samples or those which are extraordinarily unstable in magnetic fields.

Changes in the resonant frequency widths in adhered samples were investigated through measurements on a high-density tungsten alloy sample with and without adhering the sample to the ultrasonic transducers. The widths of the resonant frequencies increased in the presence of the adhesive with the extent of the increase ranging from factors of 2–15. Compared to the change in the frequencies (at most ∼1%), this change in width was large and not perturbative. There does not appear to be a strong correlation between the widths of a given resonance mode before and after the application of the adhesive. However, as shown in Fig. 12, the resonance width increase from the adhesive does seem to decrease with resonant frequency. Our simple model does not capture these effects on widths, but the frequency-dependence of the width increase in the presence of the adhesive suggests a possible connection to the spatial extent of the resonance mode vibrational pattern. Depending on the sample, such width increases may preclude quantitative analyses of internal friction, but they do not impact elastic constant determination for strongly resonating samples because resonant frequencies only weakly depend on the resonant quality factor.¹²