A phase field model of a crystalline material at the mesoscale is introduced to develop the necessary theoretical framework to study plastic flow due to dislocation motion. We first obtain the elastic stress from the phase field free energy and show that it obeys the stress strain relation of linear elasticity. Dislocations in a two dimensional hexagonal lattice are shown to be composite topological defects in the amplitude expansion of the phase field, with topological charges given by the Burgers vector. This allows us to introduce a formal relation between dislocation velocity and the evolution of the coarse grained envelopes of the phase field. Standard dissipative dynamics of the phase field crystal model is shown to determine the velocity of the dislocations. When the amplitude equation is valid, we derive the Peach-Koehler force on a dislocation, and compute the associated defect mobility. A numerical integration of the phase field crystal equations in two dimensions is used to compute the motion of a dislocation dipole, and good agreement is found with the theoretical predictions.

The resonant scattering and diffraction beamline P09 at PETRA III is designed for X-ray experiments requiring small beams, energy tunability, variable polarization and high photon flux. It is highly flexible in terms of beam size and offers full higher harmonic suppression. A state-of-the-art double phase-retarder set-up provides variable linear or circular polarization. A high-precision Psi-diffractometer and a heavy-load diffractometer in horizontal Psi-geometry allow the accommodation of a wide variety of sample environments. A 14 T cryo-magnet is available for scattering experiments in magnetic fields.

A consistent small-scale description of plasticity and dislocation motion in a crystalline solid is presented based on the phase field crystal description. By allowing for independent mass motion and lattice distortion, the crystal can maintain elastic equilibrium on the timescale of plastic motion. We show that the singular (incompatible) strains are determined by the phase field crystal density, while the smooth distortions are constrained to satisfy elastic equilibrium. A numerical implementation of the model is presented and used to study a benchmark problem: the motion of an edge dislocation dipole in a triangular lattice. The time dependence of the dipole separation agrees with continuum elasticity with no adjustable parameters.

We present a theoretical method for deriving the stress tensor and elastic response of ordered systems within a Ginzburg-Landau-type density field theory in the linear regime. This is based on spatially coarse graining the microscopic stress which is determined by the variation of a free energy with respect to mass displacements. We find simple expressions for the stress tensor for phase field crystal models for different crystal symmetries in two and three dimensions. Using tetradic product sums of reciprocal lattice vectors, we calculate elastic constants and show that they are directly related to the symmetries of the reciprocal lattices. We also show that except for bcc lattices there are regions of model parameters for which the elastic response is isotropic. The predicted elastic stress-strain curves are verified by numerical strain-controlled bulk and shear deformations. Since the method is independent of a reference state, it extends also to defected crystals. We exemplify this by considering an edge and screw dislocation in the simple cubic lattice.

We investigate numerically the statistics of quantized vortices in two-dimensional quantum turbulence using the Gross-Pitaevskii equation. We find that a universal −5/3 scaling law in the turbulent energy spectrum is intimately connected with the vortex statistics, such as number fluctuations and vortex velocity, which is also characterized by a similar scaling behavior. The −5/3 scaling law appearing in the power spectrum of vortex number fluctuations is consistent with the scenario of passive advection of isolated vortices by a turbulent superfluid velocity generated by like-signed vortex clusters. The velocity probability distribution of clustered vortices is also sensitive to spatial configurations, and exhibits a power-law tail distribution with a −5/3 exponent.

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Within the point-vortex model, we compute the probability distribution function of the velocity fluctuations induced by same-sign vortices scattered within a disk according to a fractal distribution of distances to the origin ∼r^{-α}. We show that the different random configurations of vortices induce velocity fluctuations that are broadly distributed and follow a power-law tail distribution P(V)∼V^{α-2} with a scaling exponent determined by the α exponent of the spatial distribution. We also show that the range of the power-law scaling regime in the velocity distribution is set by the mean density of vortices and the exponent α of the vortex density distribution.

Please note that terms and conditions apply.You may also be interested in: Double phase-plate setup for chromatic aberration compensation for resonant x-ray diffraction experiments T Inami, S Michimura and T Matsumura Complete polarization analysis in the 1keV to 2keV energy range using a high-precision polarimeter Hongchang Wang, Sarnjeet Dhesi, Peter Bencok et al. CVD diamond screens for photon beam imaging at PETRA III M Degenhardt, G Aprigliano, H Schulte-Schrepping et al. Characterization of an x-ray diamond phase plate by a polarization analyzer using multiple diffraction K Hirano, Y Ito, Y Shinohara et al.Abstract. Beamline P09 at PETRA III, DESY, is designed for general diffraction and resonant X-ray scattering experiments at low temperatures and high magnetic fields. The dependence of the X-ray cross-sections (Thomson, non-resonant magnetic, resonant exchange scattering, ATS) on the polarization state of the incident X-rays is an important property that one might want to capitalize on in a diffraction experiment. To that purpose, P09 is equipped with a double phase-retarder and diamond phase-plates making for the production of linearly and circularly polarized X-rays in the energy range between 3.5 and 8.5 keV as yet. Here we describe the double phase-retarder setup at P09, its principles of operation and its performances with respect to the generation of linearly polarized incident X-rays rotated by a variable angle η around the X-ray beam using two quarter-wave plates in series or a single half-wave plate.