Single-crystal Elastic Moduli Research Articles

Elastic properties of iron-bearing wadsleyite to 17.7 GPa: Implications for mantle mineral models

The sound velocities and single-crystal elastic moduli of iron-bearing wadsleyite with [Fe]/[Fe+Mg] molar ratio of 0.075 have been measured by Brillouin scattering experiments at high pressures up to 17.7GPa. Pressure derivatives for the adiabatic bulk modulus (KS0) and shear modulus (μ0) are 4.1(1) and 1.45(4), respectively. A comparison of our results with previous Brillouin scattering results on the Mg end-member wadsleyite shows that incorporating 7.5mol% iron in wadsleyite at high-pressure conditions decreases the shear moduli by ∼4–5%, but does not have a discernable effect on the bulk modulus. The effects of iron on the elastic moduli of wadsleyite at ambient pressure persist to high-pressure conditions at a relatively constant level. Using our results on iron-bearing wadsleyite at high pressure, we conclude that less olivine than in the pyrolite model of mantle composition provides a satisfactory explanation for 410km seismic discontinuity at the top of the transition zone.

Single-crystal elasticity of diopside to 14 GPa by Brillouin scattering

The single-crystal elastic moduli (Cij) of diopside have been measured up to 14GPa using Brillouin spectroscopy, from which the aggregate compressional and shear velocities, adiabatic bulk modulus, shear modulus, and their pressure derivatives were obtained. A least-squares fit of the velocity–pressure data to third-order finite strain equation yields KS′=4.8(2), G′=1.7(1) with ρ0=3.264(6)g/cm3, Ks=114.6(7)GPa and G=72.7(4)GPa. The current study provides the first high-pressure experimental values for the individual Cij’s of diopside and extends the range of direct measurements on the shear modulus to higher pressure. From the single-crystal moduli, the acoustic anisotropy of diopside is calculated and found to be higher than that of other major mantle minerals such as olivine, orthopyroxene, or garnet. The high anisotropy of diopsidic pyroxene is solely responsible for the observed acoustic anisotropy of eclogitic rocks. In mantle rocks containing both olivine and clinopyroxene, such as lherzolites, the diopside component could either enhance or diminish the bulk acoustic anisotropy, depending on the relationship of preferred orientation of these phases.

Elasticity of franklinite and trends for transition-metal oxide spinels

The pressure dependence of single-crystal elastic moduli of a natural Mn-rich franklinite, (Mn 0.40 Fe 2+ 0.16 Zn 0.37 Mg 0.03 )(Fe 3+ 1.94 Al 0.08 )O 4 , has been determined by GHz-ultrasonic interferometry in a diamond-anvil cell to 9.8 GPa. The room-pressure elastic constants of franklinite are C 11 = 244(3) GPa, C 12 = 142(4) GPa, and C 44 = 77(2) GPa. Linear pressure derivatives of C 11 and C 12 are 4.3(3) and 3.8(3), respectively, whereas the C 44 modulus exhibits softening, fitted in the P ≤ 10 GPa pressure range to C 44 = 77(2) + 0.29(2) P – 0.018(2) P 2 GPa. The average of Hashin-Shtrikman bounds on the adiabatic bulk modulus ( K S0 ) of franklinite is 175(3) GPa, with pressure derivative K S ′ = 4.3(3), and the shear modulus G 0 = 66(2) GPa with G ′ = 0.09(3). The isothermal compressibility of franklinite was determined from a separate high-pressure, single-crystal X-ray diffraction experiment to 7.8 GPa, yielding K T0 = 173.5(7) GPa fitted with a fixed pressure derivative of K T ′ = 4. When K ′ is fixed to the ultrasonic value of 4.3, we obtain K T0 = 172.2(7) GPa. In contrast to iron-free gahnite (ZnAl 2 O 4 ), franklinite exhibits pressure-induced mode softening of C 44 similar to magnetite (Fe 3 O 4 ). Between end-member compositions ZnFe 2 O 4 (franklinite) and MnFe 2 O 4 (jacobsite), the bulk modulus decreases linearly with increasing %Mn, however we observe non-linear behavior in other elastic moduli, especially C 44 , which displays a pronounced negative anomaly for the mid-range Mn composition. Applying Birch’s law to AB 2 O 4 -type spinels reveals that oxide spinels containing transition metals on both A and B sites follow a distinct trend from other spinels.

X-ray diffraction from stishovite under nonhydrostatic compression to 70 GPa: Strength and elasticity across the tetragonal → orthorhombic transition

The tetragonal phase of silica (stishovite) was synthesized at high pressure and temperature in a laser-heated diamond anvil cell. Nonhydrostatic pressure condition was produced by pressurizing the sample without any pressure transmitting medium. The tetragonal→orthorhombic transition could be detected from the X-ray diffraction patterns at ∼40GPa. In contrast, the orthorhombic phase has been reported to occur only above ∼60GPa in an earlier experiments under hydrostatic pressure. However, the transition pressures derived from the square of the symmetry-breaking strain versus pressure data in the two cases differ only marginally, the values being 44(8)GPa and 49(2)GPa under nonhydrostatic and hydrostatic compressions, respectively. We combine the d-spacings measured under nonhydrostatic and hydrostatic compressions to derive a parameter Q(hkl) that contains the information on differential stress t (a measure of compressive strength) and single-crystal elasticity. The compressive strengths derived from the average value of Q(hkl) and line-width analysis agree well. It increases from ∼4GPa at 20 GPa to ∼8GPa at 40GPa and decreases as the transition pressure is approached. In the orthorhombic phase, t increases with pressure monotonically. The mean crystallite size of the sample decreases from ∼5000Å to ∼1000Å as the pressure is increased from 20GPa to 45GPa and remains nearly unchanged between 45GPa and 70GPa. The single-crystal elastic moduli derived from the X-ray diffraction data indicate that (C11-C12) decreases rapidly as the transition pressure is approached. Line-width analysis of the diffraction lines suggests that near-hydrostatic pressure condition is achieved by laser annealing of the compressed sample.

Sound velocities of hydrous ringwoodite to 16 GPa and 673 K

To understand the effect of hydration on the sound velocities of major mantle minerals and to constrain the mantle's H2O budget, we have measured the single-crystal elastic moduli of hydrous ringwoodite, (Mg1.633Fe2+0.231Fe3+0.026)Si1.00H0.179O4 with 1.1wt.% H2O using Brillouin scattering combined with X-ray diffraction in an externally-heated diamond anvil cell up to 16GPa and 673K. Up to 12GPa at 300K, the presence of 1.1wt.% H2O lowers the elastic moduli of ringwoodite by 5–9%, but does not affect the pressure derivatives of the elastic moduli compared to anhydrous ringwoodite. The reduction caused by hydration is significantly enhanced when temperatures are elevated at high pressures. At 12GPa, increasing temperature by ΔT=100K leads to a 1.3–2.4% reduction in the elastic moduli (C11, C12, and C14). Comparing our results with seismic observations, we have evaluated the potential H2O content in the lower part of the transition zone. Our results indicate that the observed seismic velocity anomalies and related depth depression of the 660-km discontinuity could be attributed to thermal variations together with the presence of ~0.1wt.% H2O.

Texture and elastic strains in hcp-iron plastically deformed up to 17.5 GPa and 600 K: experiment and model

Internal elastic strains and textures are measured using monochromatic x-ray diffraction in ϵ-Fe plastically deformed up to 17.5 GPa and 600 K in the deformation-DIA. We observe the development of a strong 0 0 0 1 compression texture along with a strongly non-linear behavior of 0 0 0 2 lattice strains. We then use an elastoplastic self-consistent polycrystal model to simulate the macroscopic flow curves, internal strain and texture development within the sample. Input parameters are single-crystal elastic moduli, critical resolved shear stresses, and hardening behavior of the slip and twinning mechanisms. The model is found to reproduce the experiment data with basal slip as the easiest and most active deformation mechanism. Other active mechanisms are tensile twinning, prismatic and pyramidal 〈c + a〉 slip. Tensile twinning is most active at lower temperatures (e.g. 400 K) and a higher strain rate. In most cases, the twinning activity occurs early in the deformation and later saturates. It is also responsible for the non-linear behavior of 0 0 0 2 lattice strains. Later in the deformation, the plastic activity of ϵ-Fe is controlled by basal, prismatic and pyramidal 〈c + a〉 slip.

Evolution of the crystal stress distributions in face-centered cubic polycrystals subjected to cyclic loading

Due to the heterogeneous nature of polycrystalline metals, the stress distribution at the crystal level is influenced by the complex interplay of factors such as the orientations of the crystal lattices, the elastic and plastic mechanical properties, the interactions between neighboring crystals, and the type of loading conditions. In this paper, we investigate the evolution of the crystal scale stress and elastic strain distributions under cyclic loading. More specifically, we examine the orientation dependent lattice strains in face-centered cubic polycrystals under fully reversed, strain-controlled cyclic loading using a cyclic hardening model implemented within a crystal-based finite element formulation. The directional strength-to-stiffness ratio and the single crystal yield surface (SCYS) topology are used to provide a quantitative explanation of the observed hysteresis behavior when the lattice strains are plotted as a function of the macroscopic stress. The directional strength-to-stiffness ratio influences the progression of yielding in the elastic–plastic transition regime, leading to a nonlinear response in the elastic–plastic transition regime and causing hysteresis behavior in the macroscopic stress versus lattice strain curves. The lattice strains after the elastic–plastic transition regime are influenced by the single crystal elastic moduli and the facet/vertex attributes of the SCYS. We found that the same trends hold in the absence of slip system hardening and in the presence of a strong rolling texture.

Far-field high-energy diffraction microscopy: a tool for intergranular orientation and strain analysis

The far-field high-energy diffraction microscopy technique is presented in the context of high-energy synchrotron x-ray diffraction. For each grain in an illuminated polycrystalline volume, the volume-averaged lattice orientations, lattice strain tensors, and centre-of-mass (COM) coordinates may be determined to a high degree of precision: better than 0.05°, 1 × 10−4, and 0.1 pixel, respectively. Because the full lattice strain tensors are available, corresponding mean stress tensors may be calculated unambiguously using single-crystal elastic moduli. A novel formulation for orientation indexing and cell refinement is introduced and demonstrated using two examples: first, sequential indexing and lattice refinement of a single-crystal ruby standard with known COM coordinates; and second, indexing and refinement of simulated diffraction data from an aggregate of 819 individual grains using several sample rotation ranges and including the influence of experimental uncertainties. The speed of acquisition and penetration depth achievable with high-energy (that is, >50 keV) x-rays make this technique ideal for studies of strain/stress evolution in situ, as well as for residual stress analysis.

Strength and elasticity of niobium under high pressure

High purity polycrystalline niobium contained in boron-epoxy gasket was compressed in a diamond anvil cell (DAC). The pressure was increased in steps of ∼3 GPa and the diffraction patterns recorded at each pressure with the incident x-ray beam perpendicular to the load axis of the DAC (radial diffraction). The maximum pressure reached was 37.6 GPa. The compressive strength (differential stress) derived from the radial diffraction data is 0.44(1) GPa at 2.1 GPa and shows a shallow maximum at ∼5 GPa, and then decreases to 0.35(5) at 12 GPa. At higher pressures, strength increases nearly linearly and the extrapolated value at 40 GPa is 0.94(6) GPa. At any pressure, the single-crystal elastic moduli derived from the diffraction data can be made to match well those obtained from the extrapolation of the elasticity data at ambient pressure by adjusting the weight parameter α that appears in the lattice strain theory. The parameter α is found to decrease from 2.00(8) at 2.1 GPa to 1.35(4) at 37.6 GPa.

Elasticity of cubic boron nitride under ambient conditions

As a superhard material with properties similar to diamond including chemical inertness, cubic boron nitride (cBN) is an excellent candidate as a pressure calibration standard for high-temperature high-pressure research using its pressure-volume-temperature (P-V-T) equation of state. However, the elastic properties of cBN at ambient conditions reported in the literature vary by up to 8%, which can likely be attributed in part to variability in the cBN composition and defect structure and measurement uncertainties. We have measured the single-crystal elastic moduli of high-purity cBN with high precision by Brillouin scattering measurements, making an effort to minimize experimental uncertainties. We obtain values of C11 = 798.4 ± 1.7 GPa, C44 = 469.0 ± 1.0 GPa, and C12 = 172.4 ± 1.1 GPa, from which the isotropic aggregate bulk modulus Ks = 381.1 ± 1.3 GPa and shear modulus G = 398.8 ± 1.2 GPa (the Hill average) were calculated. Our results improve the precision and reduce the uncertainties in the elastic moduli of high-purity cBN as a reference for future high P-T pressure scales.

The elastic properties of diopside, CaMgSi2O6

The 13 single-crystal elastic moduli of two samples of diopsidic pyroxenes close to the ideal composition (CaMgSi2O6) from different locations have been measured at ambient pressure and room temperature by Brillouin spectroscopy. We obtain (in GPa, ±1σ uncertainty): C11 = 229.0(4), C22 = 179.0(4), C33 = 242.5(4), C44 = 78.9(3), C55 = 68.1(2), C66 = 78.2(3), C12 = 78.0(7), C13 = 69.8(6), C23 = 58.0(7), C15 = 9.9(3), C25 = 6.1(5), C35 = 40.9(3), C46 = 6.6(2), and C11 = 226.1(9), C22 = 179.5(8), C33 = 239.2(9), C44 = 78.1(6), C55 = 69.2(4), C66 = 76.4(8), C12 = 77.4(10), C13 = 70.2(13), C23 = 56.7(16), C15 = 9.9(7), C25 = 5.9(12), C35 = 41.0(7), C46 = 6.8(4) for the two diopside samples. There are no major differences in the elastic tensor of the nearly pure diopside composition compared with the second sample containing slightly higher Al and Fe contents. Polycrystalline averaging of the Cij for the aggregate bulk and shear moduli yield Ks = 114.6(7) GPa, G = 72.7(4) GPa and Ks = 113.7(8) GPa, G = 72.2(5) GPa for the two diopside samples. The shear moduli reported here are 8% larger than those determined from previous measurements on a similar natural sample. We confirm the existence of systematic correlations between the composition of the M2 site and most of the Cij values. No unusual compositional dependence of Cij values or the shear modulus near end-member diopside composition is observed.

Elasticity of glaucophane, seismic velocities and anisotropy of the subducted oceanic crust

Upon subduction, the oceanic crust transforms to blueschists and eclogites, with seismic properties that gradually become similar to those of the surrounding mantle. In order to evaluate the anisotropy of blueschists and glaucophane-bearing eclogites, the elastic constants of glaucophane single-crystal plates from the Sesia–Lanzo Zone (Aosta Valley, Western Alps) were measured using Brillouin spectroscopy at ambient conditions. The mean P- and S-wave velocities are 7.8 and 4.6 km s −1 respectively, and the anisotropy is high (38.1% (AV P) and 27.3% (AV S)). Glaucophane develops strong LPO, characterized by the [001]-axes concentrated sub-parallel to the lineation, and the {110} poles concentrated sub-perpendicular to the foliation in both blueschist and eclogite rocks. The measured LPO is in good agreement with viscoplastic self-consistent numerical models. Seismic properties of glaucophane-bearing blueschist and eclogite are calculated by combining measured LPO and the single-crystal elastic moduli of glaucophane with the other main mineral constituents of the rock: mostly epidote for blueschist, and omphacite and garnet for eclogite. Blueschists present stronger anisotropies (AV P = 16.1% and AV S = 10.3%) than eclogites (AV P = 2.9% and AV S = 1.7%). The shear-wave splitting and resulting delay times for a 7-km thick layer of eclogite or blueschist are low for the eclogite (< 0.03 s), but significant for blueschist (0.16 s). Application to the subducted oceanic crust yields a decrease of velocity contrast with the surrounding mantle and of anisotropy at depth depending on the temperature of the slab. The details of the velocity variations in subducted oceanic crust are difficult to reconcile with the blueschist–eclogite transition as probed by exhumed metamorphic rocks and may require additional phases such as lawsonite or chemical variations such as oxidation state.

A method for measuring single-crystal elastic moduli using high-energy X-ray diffraction and a crystal-based finite element model

This paper presents a method – based on high-energy synchrotron X-ray diffraction data and a crystal-based finite element simulation formulation – for understanding grain scale deformation behavior within a polycrystalline aggregate. We illustrate this method by using it to determine the single-crystal elastic moduli of β21s, a body-centered cubic titanium alloy. We employed a polycrystalline sample. Using in situ loading and high-energy X-rays at the Advanced Photon Source beamline 1-ID-C, we measured components of the lattice strain tensor from four individual grains embedded within a polycrystalline specimen. We implemented an optimization routine that minimized the difference between the experiment and simulation lattice strains. Sensitivity coefficients needed in the optimization routine are generated numerically using the finite element model. The elastic moduli that we computed for the β21s are C11=110GPa, C12=74GPa and C44=89GPa. The resulting Zener anisotropic ratio is A=5.

Elastic relaxations associated with the – transition in LaAlO3: I. Single crystal elastic moduli at room temperature

Values for all six independent components of the elastic modulus tensor of LaAlO3 perovskite are reported. These were determined by means of Brillouin scatteringmeasurements of acoustic velocities in single crystal plates cut parallel to (110) and(100), as defined with respect to the cubic parent structure, and by pure-modelongitudinal and transverse sound velocity measurements along [100], [110] and[111] directions using GHz pulse-echo ultrasonics. The crystals contained intimateintergrowths of twins arising from the transition at higher temperature but, in combination with a careful analysis of twin orientationrelationships, the two sets of data have allowed a unique solution to be obtained for individual twincomponents. The new data set represents an important contribution to the characterization ofLaAlO3 single crystals which are widely used as the substrate for a plethora of different thin filmswith technological applications.

Analysis of nonhydrostatic high-pressure diffraction data (cubic system): Assessment of various assumptions in the theory

The mathematical formulation commonly used to analyze the high-pressure diffraction data from the sample under nonhydrostatic compression is based on three assumptions: A1—a weighted harmonic mean of the diffraction shear moduli under Reuss and Voigt limits with a weight parameter α that lies between 0.5 and 1 describes adequately the diffraction shear modulus; A2—a stress tensor with only the diagonal terms describes the stress state at the center of the sample under nonhydrostatic compression; and A3—the lattice-strain equations derived using only the linear elasticity theory are adequate to derive strength and elastic moduli from the diffraction data. To examine A1 we derive compressive strength, diffraction shear moduli, and single-crystal elastic moduli from the experimental high-pressure x-ray diffraction data on bcc Fe, Au, Mo, and FeO. These data contain plastic deformation effects. The diffraction shear modulus in the limit of small deformation (elastic) is computed using rigorous formulae derived by Kröner [Z. Phys. 151, 504 (1958)] and de Wit [J. Appl. Crystallogr. 30, 510 (1997)]. The elastic moduli are derived from the computed shear moduli assuming the validity of A1. The results show that A1 with α≅0.5 is valid for small deformation in all four cases. The analysis of the experimental data suggests that A1 is valid with α&amp;lt;1 for solids with x&amp;gt;1 where x=2C44/(C11−C12); for solids with x&amp;lt;1, the validity of A1 requires α&amp;gt;1. At least for solids of the cubic system, the effect of plastic deformation appears to be fully contained in a single parameter α. In practice, deviations from A2 of varying magnitudes occur mainly because of the difficulty in avoiding diffraction from regions of stress gradient in the sample. A discussion of A3 is presented.

Temperature Dependence of Single-Crystal Elastic Constants of Flux-Grown α-GaPO4

The lattice parameter change with respect to temperature (T) has been measured using high-temperature powder X-ray diffraction techniques for high-temperature flux-grown GaPO(4) single crystals with the alpha-quartz structure. The lattice and the volume linear thermal expansion coefficients in the temperature range 303-1173 K were computed from the X-ray data. The percentage linear thermal expansions along the a and c axes at 1173 K are 1.5 and 0.51, respectively. The temperature dependence of the mass density rho of flux-grown GaPO(4) single crystals was evaluated using the volume thermal expansion coefficient alpha(V)(T) = 3.291 x 10(-5) - 2.786 x 10(-8) [T] + 4.598 x 10(-11)[T](2). Single-crystal high-resolution Brillouin spectroscopy measurements have been carried out at ambient pressure from 303 to 1123 K to determine the elastic constants C(IJ) of high-temperature flux-grown GaPO(4) material. The single-crystal elastic moduli were calculated using the sound velocities via the measured Brillouin frequency shifts Deltanu(B). These are, to our knowledge, the highest temperatures at which single-crystal elastic constants of alpha-GaPO(4) have been measured. Most of the room-temperature elastic constant values measured on flux-grown GaPO(4) material are higher than the ones found for hydrothermally grown GaPO(4) single crystals. The fourth-order temperature coefficients of both the Brillouin frequency shifts T(nuB)((n)) and the single-crystal elastic moduli T(C(IJ))((n)) were obtained. The first-order temperature coefficients of the C(IJ) are in excellent agreement with previous reports on low-temperature hydrothermally grown alpha-GaPO(4) single crystals, while small discrepancies in the higher-order temperature coefficients are observed. This is explained in terms of the OH content in the GaPO(4) network, which is an important parameter in the crystal thermal behavior.

Single-crystal elastic constants of the zeolite analcime measured by inelastic X-ray scattering

The single-crystal elastic moduli of the zeolite analcime, | Na 16 + ( H 2 O ) 16 | [ Al 16 Si 32 O 96 ] (structure type ANA), have been determined by inelastic X-ray scattering (IXS). The room temperature values (GPa) are C 11 = 103.5 ± 5, C 44 = 30.3 ± 2, C 12 = 21.7 ± 2. The results demonstrate the applicability of this technique for measuring elastic constants of open three-dimensional structures that include extra-framework species.

Modeling analysis of the influence of plasticity on high pressure deformation of hcp-Co

Previously measured in situ x-ray diffraction is used to assess the development of internal elastic strains within grains of a sample of polycrystalline cobalt plastically deformed up to a pressure of 42.6 GPa. An elastoplastic self-consistent polycrystal model is used to simulate the macroscopic flow curves and internal strain development within the sample. Input parameters are single-crystal elastic moduli and their pressure dependence, critical resolved shear stresses, and hardening behavior of the slip and twinning mechanisms which are active in Co crystals. At 42 GPa, the differential stress in hcp-Co is $1.9\ifmmode\pm\else\textpm\fi{}0.1\text{ }\text{GPa}$. The comparison between experimental and predicted data leads us to conclude that: (a) plastic relaxation plays a primary role in controlling the evolution and ordering of the lattice strains; (b) the plastic behavior of hcp-Co deforming under high pressure is controlled by basal and prismatic slip of $⟨a⟩$ dislocations, and either pyramidal slip of $⟨c+a⟩$ dislocations, or compressive twinning, or both. Basal slip is by far the easiest and most active deformation mechanism. Elastoplastic self-consistent models are shown to overcome the limitations of models based on continuum elasticity theory for the interpretation of x-ray diffraction data measured on stressed samples. They should be used for the interpretation of these experiments.

Multiphase improved Steinberg–Guinan model for vanadium

Vanadium has been observed recently to transform from the body-centered cubic (bcc) crystal structure to a rhombohedral structure at high pressure (∼0.69 Mbar) [Y. Ding et al., Phys. Rev. Lett. 98, 085502 (2007)]. Recent theoretical work predicts a transformation to a second rhombohedral phase at 1.2 Mbar before transforming back to the bcc structure at 2.8 Mbar at absolute zero temperature [B. Lee et al., Phys. Rev. B 75, 180101(R) (2007)]. Here we develop an analytic model for the shear modulus in these phases based on ab initio calculations of the single-crystal elastic moduli and a finite element based homogenization technique. The form of the shear modulus is suited to application in strength models such as in the Steinberg–Guinan form and other analogous continuum-level models.

Single-crystal elasticity of iron-bearing majorite to 26 GPa: Implications for seismic velocity structure of the mantle transition zone

Single-crystal elastic moduli of synthetic iron-bearing majoritic garnet were measured using Brillouin spectroscopy up to a pressure of 26 GPa. We found that the iron-bearing majoritic garnet (Mg 0.79Fe 0.08Al 0.30Si 0.84O 3) remains elastically isotropic throughout the pressure range of this study. The shear and adiabatic bulk moduli and their pressure derivatives at zero pressure were determined to be µ 0 = 88.7(7) GPa, K S0 = 166(2) GPa, (∂ µ / ∂ P) 0 = 1.28(8), (∂ K / ∂ P) 0 = 4.2(2). These values are almost indistinguishable, within the experimental uncertainties, from those obtained in previous studies of pure MgSiO 3 majorite and other solid solutions of the majorite–pyrope join. Our direct acoustic velocity measurements of iron-bearing majorite indicate that the incorporation of iron into majoritic garnet does not drastically affect its elastic moduli and their pressure derivatives. The measured pressure derivatives of iron-bearing majorite are about 3.6 times lower than required to account for the high seismic velocity gradients observed in the mantle transition zone.