Rectangular Parallelepiped Resonance Research Articles

Temperature variation of elastic constants of quartz across the α - β transition

The α − β transition of quartz was successfully observed with using a single sample by means of the rectangular parallelepiped resonance (RPR) method. An oriented rectangular parallelepiped of α-quartz single crystal was prepared and the resonant frequencies of 30–11 vibrational modes were measured from room temperature to 700°C. The softening of quartz crystal was observed as the significant reduction of resonant frequencies near the α–β transition. The present study is the first application of the RPR method to the study of phase transition. The complete set of elastic constants of α- and β-quartz were determined as a function of temperature by the least-squares inversion of the measured frequency data obtained by a single run. This is a merit yielded by the RPR method. It is shown near the α − β transition in both α- and β-quartz that the elastic parameters decrease proportionally to |T−T 0|−n , where T is temperature and T 0 is the transition temperature, 573.0°C for α-quartz and 574.3°C for β-quartz. It was also seen that linear incompressibilities K 1 = (C 11 +C 12 +C 13)/3 and K 3 = (C 33 +2C 13)/3 decrease rapidly toward the transition, whereas, shear moduli C 44, C S1 = (C 11 +C 33 -2C 13)/4 and C S3 = (C 11 -C 12)/2 = C 66 decrease only slightly. The shear modulus C S3 = C 66 increased slightly in α-quartz. The elastic properties of isotropic aggregate of quartz were calculated, and it is shown that the longitudinal wave velocity significantly decreases at the α − β transition, whereas, the shear wave velocity decreases only slightly.

Elastic constants of face-centered cubic and L12 Ni-Si alloys: Composition and temperature dependence

The adiabatic elastic stiffness constants C ij of Ni-Si single-crystal solid-solution alloys of two slightly different compositions, 10.78 and 11.17 at. pct Si, were measured over the temperature range from 20 °C to 900 °C using the rectangular parallelepiped resonance method. The isotropic elastic constants of the polycrystalline ordered intermetallic compound Ni3Si containing 23 at. pct Si were also measured over this temperature range. Values of the C ij for Ni3Si were estimated from the data on the polycrystalline alloy, as well as from published data in the literature on isomorphous ternary ordered intermetallic compounds containing different amounts of Si. Using measured values and previously published data, the stiffness constants of Ni3Ti were estimated; these are the only available data on this alloy. The estimated single-crystal elastic constants of Ni3Si, as well as the experimentally measured bulk modulus, are considerably smaller than published values calculated from first-principles methods. The same is true for the C ij of Ni3Ti, but the discrepancies are smaller.

Premartensitic attenuation in Ti–30Ni–20Cu shape memory alloy

Rectangular parallelepiped resonance (RPR) method has been mostly used to measure elastic constants of crystals. Here we show that this method can also yield useful information about the attenuation of samples. We measured the change in the height of resonance peak with temperature for a Ti–30Ni–20Cu shape memory alloy prior to its transformation temperature. From this measurement we found that attenuation increases with approaching transformation temperature. This is consistent with the result obtained from pulse-echo method and strongly suggests there exists premartensitic attenuation well above M s temperature. We further show that the premartensitic attenuation increase just corresponds to a softening in zone boundary TA2 phonon, as well as an increase in diffuse scattering intensity at Brillouin zone boundary. All these results are consistent with the existence of hetero-phase fluctuation prior to martensitic transformation.

A comparative study of elastic constants of Ti–Ni-based alloys prior to martensitic transformation

Single crystal elastic constants of Ti–Ni alloys without (quenched) and with (aged) Ti 3Ni 4 precipitates were measured systematically by rectangular parallelepiped resonance method as a function of composition and temperature, and compared with Ti–Ni–Cu and Ti–Ni–Fe alloys, in an attempt to answer some long-standing questions as to the origin of the unique monoclinic B19′ martensite, and why composition and thermomechanical treatment greatly changes the path of martensitic transformation. The results showed that softening in c 44, in additional to c′, is a common feature for all Ti–Ni binary (both quenched or aged) and ternary alloys. This general feature just corresponds to the fact that all these alloys ultimately transform into B19′, suggesting that softening in c 44 is responsible for the unique B19′ martensite, which found no analogy in other β phase alloys. We also found an interesting correspondence between the temperature dependence of anisotropy factor and transformation path. Prior to B2–B19′ transformation anisotropy shows a decrease with lowering temperature; prior to B2–B19 an anisotropy increase, while prior to B2–R transformation a constant anisotropy. We further showed that three possible martensite candidates (R, B19, B19′) are rooted in anomalies in specific phonon modes and elastic softening. We showed that the multi-stage transformations are restricted by a general rule: multi-stage transformation occurs in the sequence of increasing transformation strain. With this rule we can explain all known transformation paths by considering the effect of alloying addition and fine precipitates/dislocation network on relative stability of different martensites. We further predict that there may exist a new transformation path in Ti–Ni-based alloys: B2–R–B19–B19′.

Evaluation of elastic strain energy associated with the formation of hydride precipitates in LaNi 5

The elastic energy associated with coherent precipitation of hydrides with various shapes in LaNi 5 has been calculated by the Eshelby method with the single-crystal elastic constants experimentally determined by a rectangular parallelepiped resonance method. The calculated elastic energy amounts to about 70% of the change in chemical energy that occurs upon hydride formation. This implies the difficulty for coherent hydrides to nucleate in the perfect lattice of LaNi 5. This is in contrast to the case of titanium or zirconium, in which the associated elastic energies are only about 15% of the chemical energy change. Hydride precipitation in LaNi 5 is thus suggested to occur preferentially at free surfaces and lattice defects such as dislocations and grain boundaries.

Temperature and composition dependence of the elastic constants of Ni3Al

The stiffness constants, c ij , of monocrystalline Ni3Al of three different compositions, 23.2, 24.0, and 25.0 at. pct Al, were measured over the temperature range from 300 to 1100 K using the rectangular parallelepiped resonance (RPR) method. The bulk modulus, as well as the shear modulus, Young’s modulus, and Poisson’s ratio for randomly oriented polycrystalline stoichiometric Ni3Al, were derived from the stiffness constants. The data indicate that c 44 is essentially independent of composition, decreasing slightly with increasing temperature for all three alloys. The values of c 11 and c 12, however, decrease with increasing aluminum content, the difference being small at room temperature but becoming larger at higher temperatures. We find that c 11 and c 12 are not as sensitive to aluminum concentration as is implied by previous results. A comparison of different shear moduli of Ni3Al and the saturated Ni-Al solid solution in equilibrium with it indicates that the ordered phase is generally elastically stiffer than the solid solution over the range of temperatures at which coarsening of the Ni3Al precipitate has been heavily investigated.

Elastic Constants of Ti-48 at%Ni-2 at%Fe Single Crystal Prior to B2→R Transformation

Elastic constants of a Ti-48 at%Ni-2 at%Fe single crystal prior to B2→R transformation was measured by using the rectangular parallelepiped resonance (RPR) method. Both shear modulus c' and c 44 were found to soften with decreasing temperature towards the transformation temperature R,. It was also found that in the vicinity of the transformation temperature, the anisotropy factor A of this alloy shows little temperature dependence, which is in contrast to the decrease for as-quenched binary Ti-50 at%Ni alloy (B2→ B19'), and the increase for Ti-30 at%Ni-20 at%Cu alloy (B2→ B19), respectively reported. Therefore, this phenomenon appears to be closely related to B2→ R transformation.

Elastic constants of Ti50Ni30Cu20alloy prior to martensitic transformation

Abstract Elastic constants of Ti50Ni30Cu20 alloy prior to the martensitic transformation B2 → B19 was measured for the first time by using the rectangular parallelepiped resonance method. The softening in both shear modulus ć and c44 was found with approaching transformation temperature. As a result of soft c 44, the anisotropy factor shows a low value of 2.8 in the vicinity of the transformation temperature. c 11, c 12, and bulk modulus were found to show little temperature dependence in the vicinity of the transformation temperature. Compared with the binary Ti50Ni50 alloy, Ti50Ni30Cu20 alloy exhibits a 40% higher and increasing anisotropy with approaching transformation temperature. Based on the experimental results, we explained why Ti50Ni30Cu20 transforms into B19 martensite, while Ti50Ni50 transforms into a strange monoclinic B19 martensite.

Temperature dependence of single-crystal elastic constants of Mo(Si,Al)2

The temperature dependence of single-crystal elastic constants of Mo(Si,Al) 2 with the hexagonal C40 structure has been measured by the rectangular parallelepiped resonance method over the temperature range from room temperature to 1373 K. The silicide has the largest value of Poisson's ratio among transition-metal silicides with the C40 structure. This is interpreted in terms of the reduced directionality in atomic bonding due to the partial substitution of Al for Si atoms. The expression of the elastic constants on the basis of simple interatomic interactions cannot reproduce the elastic anisotropy of crystal with the C40 structure. The disagreement is considered to be a consequence of the occurrence of internal displacements during elastic deformation in crystals with the non-centrosymmetry.

Single-crystal elastic constants of MoSi2 with the C11b structure

Single-crystal elastic constants of MoSi2 have been measured by the rectangular parallelepiped resonance (RPR) method over the temperature range 4–1373 K. The values at room temperature agree with those reported by Nakamura et al. [1], and the values extrapolated to 0 K are in good agreement with theoretical calculates by Alouani et al. [2]. Values of cij except for c12 and c13 decrease with increasing temperature while those of c12 and c13 are virtually temperature independent. Elastic property of the compound is discussed in terms of the strength of atomic bonds by adopting the approach introduced by Musgrave and Pople [3]. The elastic constants are expressed with three parameters corresponding to first-nearest-neighbor central, first-nearest-neighbor non-central and second-nearest-neighbor non-central interactions. The extent of contribution of the directional nature in atomic bonding to the elastic properties of MoSi2 is discussed in the light of the temperature variation of the three parameters we introduced.

Elastic constants of Al-based icosahedral quasicrystals

The elastic constants of several Al-based icosahedral alloys were measured over a temperature range from 4K to 1073 K by the rectangular parallelepiped resonance method. The values of the bulk moduli of these quasicrystals are similar to those of conventional cubic crystals having similar melting temperatures, while the values of Young's and shear moduli are larger in comparison with conventional aluminium alloys. The latter feature is considered to be due to the strong directional bonding in the quasicrystals.

Elastic constants and their temperature dependence for the intermetallic compound Ti3Al

Abstract Single-crystal elastic contents of Ti3Al have been measured from 3.3 to 290K by a rectangular parallelepiped resonance method. The elastic anisotropy of Ti3Al is similar to, but slightly larger than, that of pure Ti. The values of the measured elastic constants are compared with those calculated for pure Ti stressed to such an extent that it has the same lattice constants as Ti3Al. The difference is interpreted as indicating the existence of directional bonding in the compound.

New pressure measurements of elasticity using the rectangular parallelepiped resonance method.

The rectangular parallelepiped resonance (RPR) method has proven to be an effective means to measure the adiabatic elastic moduli, Cij, at very high temperatures. The extension of the RPR technique to elevated pressure is an attractive proposition because bonding problems associated with coupling between transducers and specimens are essentially eliminated. However, RPR data have not yet been reported at elevated pressure because of (1) the small frequency shifts observed over the accessible pressure range, and (2) complications in the data reduction scheme by which primary measurements of a resonance spectrum are interpreted in terms of the specimen Cij’s. The experimental and analysis problems encountered when pressure dependences of modal frequencies are measured will be discussed, and ways in which these problems can be overcome will be presented. A new apparatus in which RPR measurements at elevated pressure have been made will be described. Initial RPR pressure data on several materials will be presented and compared to results obtained when using thermodynamic considerations that relate the temperature dependences of the resonant frequencies to their pressure dependences.

The determination of physical properties at high pressure using a high‐temperature database from resonant ultrasound spectroscopy (RUS) measurements.

Using the RUS technique called rectangular parallelepiped resonance, a large database on elastic constants and associated thermoelastic parameters extending from 300 K up to as high as 1800 K (P=0) has been established at UCLA. Temperature derivatives of the Cij’s have been determined with considerable precision. It has been shown how the pressure derivatives of elastic constants (∂Cij/∂P)T can be approximated from (∂Cij/∂T)P. Agreement with experiment is quite good in some cases. Extending this method to high pressure (≳3 GPa) requires evaluation of the volume dependence of the parameter, (∂P/∂T)V=αKT, where α is thermal expansivity and KT is the isothermal bulk modulus. Again, this is done from temperature Cij data, but it requires one datum on pressure, K0′=(∂KT/∂P)T (P=0). From the data, the temperature at which αKT becomes independent of volume was predicted. The theory agrees well with experiment for all solids tested (NaCl, MgO, Al2O3, Mg2SiO4, CaO).

Temperature variation of elasticity of α-quartz by the rectangular parallelepiped resonance method.

The resonant ultrasound spectroscopy of a specimen of α-quartz single crystal was carried out up to 558 °C, just below the α–β transition (T0=573 °C), and all of the independent elastic constants were determined simultaneously by the rectangular parallelepiped resonance method. The results show that bulk moduli decrease largely toward T0, while the shear moduli show only slight decreases. The constant, C14, characteristic of trigonal crystals, decreases to just below T0, but it seems not to vanish, but to remain a finite value, even at T0. It is noted that the isotropic aggregate of α-quartz is a curious and scarce material, in the sense that its bulk modulus is smaller than the shear modulus, and Poisson’s ratio is very small or even negative at temperatures higher than 450 °C. The effect of piezoelectricity of α-quartz on the theoretical frequencies was investigated in order to know whether the ordinary nonpiezoelectric theory gives considerably inaccurate resonance frequencies. It was shown that the nonpiezoelectric theory gives a frequency lower than the frequency given by piezoelectric theory, but the difference is 0.5% at most.

Single-crystal elastic constants of intermetallic compounds

The elastic constants of single crystals of several intermetallic compounds have been measured in our laboratory by the rectangular parallelepiped resonance method. This paper first describes the experimental procedure and the method of data analysis of the resonance spectrum to derive values of the elastic constants, and then gives the numerical results for materials measured so far: Ni-base L12 compounds Ni3X (X = Mn, Fe, Al, Ga, Ge and Si), TiAl, Ti3Al, CoTi, Co3Ti and CoSi2. The elastic constants of various intermetallic compounds reported by other investigators are also given in tabulated form.

Elastic and viscoelastic properties of α iron at high temperatures

Recent experiments done at low driving frequencies suggest that a large degree of dispersion exists in the measured value of the shear modulus, μ, of α iron at high temperature. Discrepancies between values for μ from ultrasonic measurements and those from low‐frequency torsional measurements have been interpreted in terms of viscoelastic relaxation. However, the ultrasonic data are not in agreement with one another, and the degree of dispersion is not accurately known. We present new high‐temperature data for the elastic moduli of single‐crystal iron (α phase). The elastic moduli were measured using the rectangular parallelepiped resonance method (0.27–0.59 MHz) from room temperature to 925 K. Our data show that the difference in μ at high temperature between ultrasonic‐based measurements and low‐frequency (1 Hz) torsional measurements is only 14 GPa, rather than 29 GPa, as inferred from previous analyses. Thus the possible effects of viscoelastic relaxation are reduced but not eliminated. We find no dispersion in measurements for μ of α iron when considering frequencies ranging from 0.27 to 70 MHz and discuss the possibility that significant viscoelastic effects on measurements of μ at high temperature are limited to frequencies below 3 Hz.

Softening of the elasticity tensor of Na 2[Fe(CN) 5NO] · 2H 2O upon generation of metastable electronic states by irradiation with light

The elastic behaviour of Na 2[Fe(CN) 5NO] · 2H 2O in the ground state and after population of the two metastable states (SI,SII) was investigated between 293 and 88 K by the aid of the Rectangular Parallelepiped Resonance (RPR) method. For SI, a slight decrease of all elastic constants was found, whereas SII showed a strong attenuation and broadening of the acoustic resonances. The values for the elastic constants and their temperature variation in the ground state are given.

Temperature Variation of Elastic Properties of .ALPHA.-Quartz up to the .ALPHA.-.BETA. Transition.

The temperature variations of normal mode frequencies of a specimen of α-quartz single-crystal have been measured nearly up to the α-β transition (T0 573°C), and all of the independent elastic constants were determined simultaneously by the rectangular parallelepiped resonance method. The results demonstrate that bulk moduli decrease largely toward T0, while shear moduli show only a slight decrease. The trigonal constant C14 decreases just below T0, but seems to remain a finite value even at T0. It is noted that the isotropic aggregate of α-quartz is a curious and scarce material in the sense that its bulk modulus is smaller than the shear modulus, and Poisson's ratio is very small or even negative at temperatures higher than 450°C. In the measured temperature variations of mode frequencies, a characteristic feature was observed when two branches of Ag modes came close, suggesting mode coupling. Except for this coupling between Ag modes themselves, no definite evidence was observed which was indicative of coupling between modes of different symmetry. So it can be said the present sample is uniform in its property enough to suppress the coupling between modes of different symmetry.

Elastic constants of some intermetallic compounds as determined by the rectangular parallelepiped resonance method

The three elastic stiffness constants of various Ni-base intermetallic compounds Ni 3X with L1 2 structure have been determined by the rectangular parallelepiped resonance method. Two types of specimen chamber were constructed for low temperature (down to 90 K) and for high temperature (up to 1200 K) measurements. In order to reduce the amount of numerical calculation needed to derive the values of elastic constants from the resonance spectrum, we have developed a systematic and efficient method of analysis, which is valid for crystals of cubic symmetry.