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One of the major goals of geophysical research is to understand deformation in the deep Earth. The COMPRES (Consortium for Materials Properties Research in Earth Sciences) workshop on `Rheology and Elasticity Studies at Ultra-High Pressures and Temperatures' was held on 21–23 October 2005 at the Advanced Photon Source, Argonne National Laboratory, organized by Haozhe Liu, Hans-Rudolf Wenk and Thomas S Duffy, and provided an opportunity to assemble more than 50 scientists from six countries. Experts in diamond anvil cell (DAC) design, large-volume high-pressure apparatus and data analysis defined the current state of ultra-high pressure deformation studies and explored initiatives to push the technological frontier. The DAC, when used in radial diffraction geometry, emerges as a powerful tool for investigation of plasticity and elasticity of materials at high pressures. More information regarding this workshop can be found at the website: http://www.hpcat.aps.anl.gov/Hliu/Workshop/Index1.htm. In this special issue of Journal of Physics: Condensed Matter, 17 manuscripts review the state-of-the-art and we hope they will stimulate researchers to participate in this field and take it forward to a new level.

A major incentive for high-pressure research has been the need of geophysicists to understand composition, physical properties and deformation in the deep Earth in order to interpret the macroscopically observed seismic anisotropy. In the mantle and core, materials deform largely in a ductile manner at low stresses and strain rates. From observational inferences and experiments at lower pressures, it is considered plausible that deformation occurs in the field of dislocation creep or diffusion creep and deformation mechanisms depend in a complex way on stress, strain rate, pressure, temperature, grain size and hydration state. With novel apparatus such as the rotational Drickamer press or deformation DIA (D-DIA) multianvil apparatus, large volumes (approximately 10 mm³) of materials can be deformed at pressure and temperature. Unfortunately these experiments do not currently extend to pressures of the lower mantle, which comprises most of the volume of the Earth. Thus deformation mechanisms of minerals such as perovskite (in the lower mantle), post-perovskite (in the anisotropic D" zone) and -iron (in the inner core) remain enigmatic. Here developments in the DAC offer new opportunities. At present, this is a novel, and in many ways still very primitive, method to deform minerals at high pressure, confined to room temperature and moderate strains. No doubt this will change in the near future as new technologies become implemented, for example laser heating, remote pressure control, especially fine control of strain rate during compression, decompression and cycling procedures for DAC radial diffraction studies.

The first paper, by Bassett, gives a perspective on the significance of stress in DAC experiments. An issue once considered by many a nuisance has become a gold mine when it comes to unravelling material properties at very high pressures. At high pressures many silicates and oxides become ductile, even at room temperature, and ductile deformation results in development of preferred orientation that can be used to infer deformation mechanisms as illustrated in the reviews by Wenk et al and Merkel. Mao et al investigate the strength of solidified argon and find it increases greatly and exceeds 2.7~GPa with applied pressure at 55 GPa. Singh et al investigate the dependence of strength on grain size by studying nanocrystalline gold, while Yoneda and Kubo use axial diffraction geometry to determine both mean pressure and deviatoric stress of gold. Miyagi et al illustrate the Rietveld method for quantitative texture analysis of CaSiO³ perovskite. Speziale et al map strain gradients in the DAC by investigating texture variations in copper to 25 GPa. Naturally, efficient and accurate image processing is a requirement for reduction and analysis of diffraction data (Hinrichsen et al). A complementary study by Conil and Kavner explores DAC heterogeneities with numerical methods. Using the D-DIA multianvil apparatus, the differential lattice strains in polycrystalline Fe₂SiO₄ and MgO are measured to investigate the evolution of the weighting factor for Reuss and Voigt bounds during plastic flow by Chen et al, while Weidner and Li study MgO and mixtures of MgO and spinel (MgAl₂O₄) for fabric and residual stresses development during a plastic deformation cycle. Mao and Mao describe an ingenious device for remote pressure control and demonstrate it with a sample of platinum to 230 GPa. Large strains can be obtained with a rotational DAC and this influences the pressure at which the α– transition occurs in iron (Ma et al). Chesnut et al introduce a new DAC adapted for radial diffraction geometry. While much of the current DAC work in radial geometry is done with monochromatic x-rays and angle dispersive 2D detectors, the radial x-ray diffraction program at X17C of NSLS at energy dispersive XRD geometry is reviewed by Hu et al. Meng et al introduce the double-sided laser heating technique at ID-B of HPCAT, and propose a rotational design for applying the laser heating technique to radial x-ray diffraction studies at simultaneous high pressure-temperature (PT) conditions. These contributions provide a sound introduction and overview of the current science for anyone interested in elasticity and plasticity of materials at ultrahigh pressure.

We are grateful to Dr Richard Palmer and Professor Marshall Stoneham for the opportunity to publish these carefully selected and reviewed proceedings in Journal of Physics: Condensed Matter where they reach a large audience. We acknowledge the authors for contributing exciting papers to this issue, to participants at the workshop for stimulating presentations and discussions, and to many reviewers whose suggestions improved the manuscripts. We are most grateful to COMPRES and its Director Dr Robert Liebermann for generous financial support of this workshop, and the support from local organizing committee Veronica O'Connor, Dave Mao, Guoyin Shen and other HPCAT staff. The goal of the project was not to provide final conclusions in a dynamic field but to stimulate progress.

Both synchrotron radiation and deviatoric stress were once considered to be nuisances. Now synchrotron radiation is one of the most important tools available to scientists of all disciplines and deviatoric stress is one of the most useful aspects of x-ray diffraction at extreme conditions. Samples in high-pressure devices are under true hydrostatic pressure only when surrounded by a fluid, thus limiting true hydrostatic pressure studies at ambient temperatures to pressures below about 11 GPa. Elevated temperature is able to extend this limit but has rarely been used for this purpose. Instead, noble gases have been used as pressure media as their solids are especially soft. Deviatoric stress and resultant anisotropic elastic strain in solid samples and solid media have led to many subtle errors in determinations of elastic properties and crystal structures, especially in the days before it was realized that they could be measured and were potentially a valuable source of information. In recent years, measuring anisotropic elastic strain by x-ray diffraction has provided new insights into materials strength, elastic properties, crystal structures, mechanisms of phase transitions, slip systems, lattice preferred orientation, and, of course, ways to make corrections when deviatoric stress is indeed a nuisance.

Diamond anvil cells may not only impose pressure upon a sample but also a compressive stress that produces elastic and plastic deformation of polycrystalline samples. The plastic deformation may result in texture development if the material deforms by slip or mechanical twinning, or if grains have a non-equiaxed shape. In radial diffraction geometry, texture is revealed by variation of intensity along Debye rings relative to the compression direction. Diffraction images (obtained by CCD or image plate) can be used to extract quantitative texture information. Currently the most elegant and powerful method is a modified Rietveld technique as implemented in the software package MAUD. From texture data one can evaluate the homogeneity of strain in a diamond anvil cell, the strain magnitude and deformation mechanisms, the latter by comparing observed texture patterns with results from polycrystal plasticity simulations. Some examples such as olivine, magnesiowuestite, MgSiO₃ perovskite and ε-iron are discussed.

This paper explores the applicability of x-ray diffraction measurements of stress to high pressure deformation experiments. We model measurements of elastic lattice strains in various geometries for both axial and rotational deformation apparatus. We then show that, for most cases, stresses can be inverted from the diffraction data. A comparison between the results of our models and actual experimental data also indicates that plastic deformation can have an influence that is not addressed properly in the elastic models of lattice strains and should therefore be treated with caution.

The elasticity and plasticity of materials at high pressure are of great importance for the fundamental insight they provide on bonding properties in dense matter and for applications ranging from geophysics to materials technology. We studied pressure-solidified argon with a boron–epoxy–beryllium composite gasket in a diamond anvil cell (DAC). Employing monochromatic synchrotron x-radiation and imaging plates in a radial diffraction geometry (Singh et al 1998 Phys. Rev. Lett. 80 2157; Mao et al 1998 Nature396 741), we observed low strength in solid argon below 20 GPa, but the strength increases drastically with applied pressure, such that at 55 GPa, the shear strength exceeded 2.7 GPa. The elastic anisotropy at 55 GPa was four times higher than the extrapolated value from 30 GPa. Extensive (111) slip develops under uniaxial compression, as manifested by the preferred crystallographic orientation of (220) in the compression direction. These macroscopic properties reflect basic changes in van der Waals bondings under ultrahigh pressures.

Two gold powder samples, one with average crystallite size of ≈30 nm (n-Au) and another with ≈120 nm (c-Au), were compressed under nonhydrostatic conditions in a diamond anvil cell to different pressures up to ≈60 GPa and the x-ray diffraction patterns recorded. The difference between the axial and radial stress components (a measure of the compressive strength) was estimated from the shifts of the diffraction lines. The maximum micro-stress in the crystallites (another measure of the compressive strength) and grain size (crystallite size) were obtained from analysis of the line-width data. The strengths obtained by the two methods agreed well and increased with increasing pressure. Over the entire pressure range, the strength of n-Au was found to be significantly higher than that of c-Au. The grain sizes of both n-Au and c-Au decreased under pressure. This decrease was much larger than expected from the compressibility effect and was found to be reversible. An equation derived from the dislocation theory that predicts the dependence of strength on the grain size and the shear modulus was used to interpret the strength data. The strength derived from the published grain size versus hardness data agreed well with the present results.

It is known that the {100} and {111} planes of cubic crystals subjected to uniaxial deviatoric stress conditions have strain responses that are free from the effect of lattice preferred orientation. By utilizing this special character, one can unambiguously and simultaneously determine the mean pressure and deviatoric stress from polycrystalline diffraction data of the cubic sample. Here we introduce a numerical tensor calculation method based on the generalized Hooke's law to simultaneously determine the hydrostatic component of the stress (mean pressure) and deviatoric stress in the sample. The feasibility of this method has been tested by examining the experimental data of the Au pressure marker enclosed in a diamond anvil cell using a pressure medium of methanol–ethanol mixture. The results demonstrated that the magnitude of the deviatoric stress is ∼0.07 GPa at the mean pressure of 10.5 GPa, which is consistent with previous results of Au strength under high pressure. Our results also showed that even a small deviatoric stress (∼0.07 GPa) could yield a ∼0.3 GPa mean pressure error at ∼10 GPa.

The Rietveld method is used to extract quantitative texture information from a single synchrotron diffraction image of a CaSiO₃ perovskite sample deformed in axial compression in a diamond anvil cell. The image used for analysis was taken in radial geometry at 49 GPa and room temperature. We obtain a preferred orientation of {100} lattice planes oriented perpendicular to the compression direction and this is compatible with slip.

The combination of the diamond-anvil cell, synchrotron x-ray diffraction in radial geometry and simultaneous Rietveld refinement with texture analysis allows us to quantitatively investigate the plastic deformation behaviour of materials at very high pressures. Our study of copper to 30 GPa shows in ideal experimental geometry a [110] fibre texture component, as is typical for axial compression of soft face centred cubic metals. Locally a plane strain texture develops which is energetically favoured (curling). A transition from compressional to plane strain/pure shear texture can be monitored by analysing individual images taken at different positions in the diamond cell.

Methods have been developed to facilitate the data analysis of multiple two-dimensional powder diffraction images. These include, among others, automatic detection and calibration of Debye–Scherrer ellipses using pattern recognition techniques, and signal filtering employing established statistical procedures like fractile statistics.

All algorithms are implemented in the freely available program package Powder3D developed for the evaluation and graphical presentation of large powder diffraction data sets.

As a case study, we report the pressure dependence of the crystal structure of iron antimony oxide FeSb₂O₄ (p≤21 GPa, T = 298 K) using high-resolution angle dispersive x-ray powder diffraction. FeSb₂O₄ shows two phase transitions in the measured pressure range. The crystal structures of all modifications consist of frameworks of Fe²⁺O₆ octahedra and irregular Sb³⁺O₄ polyhedra. At ambient conditions, FeSb₂O₄ crystallizes in space group P4₂/mbc (phase I). Between p = 3.2 GPa and 4.1 GPa it exhibits a displacive second order phase transition to a structure of space group P 2₁/c (phase II, a = 5.7792(4) Å, b = 8.3134(9) Å, c = 8.4545(11) Å, β = 91.879(10)°, at p = 4.2 GPa). A second phase transition occurs between p = 6.4 GPa and 7.4 GPa to a structure of space group P4₂/m (phase III, a = 7.8498(4) Å, c = 5.7452(5) Å, at p = 10.5 GPa). A nonlinear compression behaviour over the entire pressure range is observed, which can be described by three Vinet equations in the ranges from p = 0.52 GPa to p = 3.12 GPa, p = 4.2 GPa to p = 6.3 GPa and from p = 7.5 GPa to p = 19.8 GPa. The extrapolated bulk moduli of the high-pressure phases were determined to K₀ = 49(2) GPa for phase I, K₀ = 27(3) GPa for phase II and K₀ = 45(2) GPa for phase III. The crystal structures of all phases are refined against x-ray powder data measured at several pressures between p = 0.52 GPa, and 10.5 GPa.

We present isotropic, elastic–plastic finite element calculations detailing the pressure relationship between an inclusion and its surrounding matrix, subject to an externally imposed hydrostatic strain. In general, the inclusion and the matrix have different values of hydrostatic pressure, depending on their absolute and relative values of Young's modulus and Poisson's ratio. A series of finite element models was used to explore the parameter space of the elastic and plastic properties of an inclusion within a matrix. In all cases where there is insufficient relaxation of the nonhydrostatic stress, the material with the higher bulk modulus will also have a higher pressure, regardless of the shear moduli. The complete data set was subjected to a Pareto analysis to determine the main and secondary effects which influence the final result, expressed as the ratio of the pressure of the matrix to that of the inclusion. The four most important factors which determine the pressure ratio of an inclusion and matrix are the Young's modulus of the matrix, the interaction of the Young's modulus and the yield strength of the matrix material, the Young's modulus of the inclusion, and the interaction of the Young's modulus of the inclusion with the yield strength of the matrix material. The yield strength of the inclusion has a statistically insignificant effect on the results. This information provides guidelines for designing the most effective combinations of unknowns and material standards to minimize pressure errors in equation of state measurements.

Energy dispersive synchrotron x-ray diffraction is carried out to measure differential lattice strains in polycrystalline Fe₂SiO₄ (fayalite) and MgO samples using a multi-element solid state detector during high-pressure deformation. The theory of elastic modelling with Reuss (iso-stress) and Voigt (iso-strain) bounds is used to evaluate the aggregate stress and weight parameter, α (0≤α≤1), of the two bounds. Results under the elastic assumption quantitatively demonstrate that a highly stressed sample in high-pressure experiments reasonably approximates to an iso-stress state. However, when the sample is plastically deformed, the Reuss and Voigt bounds are no longer valid (α becomes beyond 1). Instead, if plastic slip systems of the sample are known (e.g. in the case of MgO), the aggregate property can be modelled using a visco-plastic self-consistent theory.

Stress analysis in polycrystalline materials reveals that stress can vary considerably among different subpopulations of grains. Samples of MgO and mixtures of MgO and spinel have been studied. After the onset of plastic flow, stronger grains or orientations will support more stress than the weaker grains. A grain to grain fabric develops that enables this stress partitioning. The stress partitioning and the resulting fabric can invalidate static measurements of elastic moduli. However, high temperature flow mechanisms reveal a more isotropic strength behaviour resulting in a more uniform variation of stress with orientation.

A Pt sample was compressed to ultrahigh pressures in a diamond anvil cell (DAC) using a motorized gearbox to change pressure remotely from outside the synchrotron x-ray hutch. In situ angle-dispersive x-ray diffraction (XRD) was used to determine pressure from known equations of state (EOS). The sample position was unperturbed during motor-driven pressure changes. By eliminating the need to realign the sample to the x-ray position after each pressure increment, 142 XRD patterns could be collected continuously over the course of three hours, and the maximum pressure of 230 GPa was reached before diamond failure ended the experiment. We demonstrate the advantages of this motor-driven assembly for smooth and efficient pressure change, and the possibility for fine pressure and temporal resolution.

The effect of shear strain on the iron α–ε phase transformation has been studied using a rotational diamond anvil cell (RDAC). The initial transition is observed to take place at the reduced pressure of 10.8 GPa under pressure and shear operation. Complete phase transformation was observed at 15.4 GPa. The rotation of an anvil causes limited pressure elevation and makes the pressure distribution symmetric in the sample chamber before the phase transition. However, it causes a significant pressure increase at the centre of the sample and brings about a large pressure gradient during the phase transformation. The resistance to the phase interface motion is enhanced due to strain hardening during the pressure and shear operations on iron and this further increases the transition pressure. The work of macroscopic shear stress and the work of the pressure and shear stress at the defect tips account for the pressure reduction of the iron phase transition.

We have designed a new diamond-anvil cell capable of radial x-ray diffraction to pressures of a few hundred GPa. The diffraction geometry allows access to multiple angles of Ψ, which is the angle between each reciprocal lattice vector g(hkl) and the compression axis of the cell. At the 'magic angle', Ψ≈54.7°, the effects of deviatoric stresses on the interplanar spacings, d(hkl), are significantly reduced. Because the systematic errors, which are different for each d(hkl), are significantly reduced, the crystal structures and the derived equations of state can be determined reliably. At other values of Ψ, the effects of deviatoric stresses on the diffraction pattern could eventually be used to determine elastic constants.

During the past decade, the radial x-ray diffraction method using a diamond anvil cell (DAC) has been developed at the X17C beamline of the National Synchrotron Light Source. The detailed experimental procedure used with energy dispersive x-ray diffraction is described. The advantages and limitations of using the energy dispersive method for DAC radial diffraction studies are also discussed. The results for FeO at 135 GPa and other radial diffraction experiments performed at X17C are discussed in this report.

An overview of a YLF:Nd laser heating system at the undulator x-ray diffraction station (16ID-B) of the high-pressure collaborative access team (HPCAT) of the Advanced Photon Source is presented. Based on the double-sided laser heating technique, the system is designed with considerable effort on the mechanical and optical stabilities, features for user-friendly operation, and the capability of accommodating diamond anvil cells of various heights up to 68 mm. This system has been used for x-ray diffraction studies of a wide range of materials to over 150 GPa and above 3000 K. Applying the laser heating technique to radial x-ray diffraction studies at simultaneous high-pressure and high-temperature (PT) conditions requires heating to be conducted at variable angles relative to the x-ray direction. A rotation laser heating design is discussed.