High-pressure melt curve of shock-compressed tin measured using pyrometry and reflectance techniques

We have developed a new technique to measure the melt curve of a shocked metal sample and have used it to measure the high-pressure solid-liquid phase boundary of tin from 10 to 30 GPa and 1000 to 1800 K. Tin was shock compressed by plate impact using a single-stage powder gun, and we made accurate, time-resolved radiance, reflectance, and velocimetry measurements at the interface of the tin sample and a lithium fluoride window. From these measurements, we determined temperature and pressure at the interface vs time. We then converted these data to temperature vs pressure curves and plotted them on the tin phase diagram. The tin sample was initially shocked into the high-pressure solid γ phase, and a subsequent release wave originating from the back of the impactor lowered the pressure at the interface along a constant entropy path (release isentrope). When the release isentrope reaches the solid-liquid phase boundary, melt begins and the isentrope follows the phase boundary to low pressure. The onset of melt is identified by a significant change in the slope of the temperature-pressure release isentrope. Following the onset of melt, we obtain a continuous and highly accurate melt curve measurement. The technique allows a measurement along the melt curve with a single radiance and reflectance experiment. The measured temperature data are compared to the published equation of state calculations. Our data agree well with some but not all of the published melt curve calculations, demonstrating that this technique has sufficient accuracy to assess the validity of a given equation of state model

Suitability of Magnesium Oxide as a Visar Window

Impedance matching of a velocity interferometer for any reflector (VISAR) window to a material under study helps simplify a shock experiment by effectively allowing one to measure an in situ particle velocity. The shock impedance of magnesium oxide (MgO) falls roughly midway between those of sapphire and LiF, two of the most frequently used VISAR window materials. A series of symmetric impact experiments was performed to characterize the suitability of single crystal, (100) oriented magnesium oxide as a VISAR window material. These experiments yielded good results and show the viability of MgO as a VISAR window up to 23 GPa. Results were used to determine window correction factors and, subsequently, to estimate the pressure induced change in index of refraction. In many of the shots in this work we exceeded the Hugoniot elastic limit (HEL) of MgO, and both elastic and plastic waves are evident in the velocity profiles. The presence of both waves within the VISAR window complicates the typical VISAR window correction analysis. Preliminary analysis of the elastic and plastic contributions to the window correction is presented

Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions

We introduce an iterative method for computing the first eigenpair $(\lambda_{p},e_{p})$ for the $p$-Laplacian operator with homogeneous Dirichlet data as the limit of $(\mu_{q,}u_{q})$ as $q\rightarrow p^{-}$, where $u_{q}$ is the positive solution of the sublinear Lane-Emden equation $-\Delta_{p}u_{q}=\mu_{q}u_{q}^{q-1}$ with same boundary data. The method is shown to work for any smooth, bounded domain. Solutions to the Lane-Emden problem are obtained through inverse iteration of a super-solution which is derived from the solution to the torsional creep problem. Convergence of $u_{q}$ to $e_{p}$ is in the $C^{1}$-norm and the rate of convergence of $\mu_{q}$ to $\lambda_{p}$ is at least $O(p-q)$. Numerical evidence is presented.Comment: Section 5 was rewritten. Jed Brown was added as autho

High-pressure melt curve of shock-compressed tin measured using pyrometry and reflectance techniques

We have developed a new technique to measure the melt curve of a shocked metal sample and have used it to measure the high-pressure solid-liquid phase boundary of tin from 10 to 30 GPa and 1000 to 1800 K. Tin was shock compressed by plate impact using a single-stage powder gun, and we made accurate, time-resolved radiance, reflectance, and velocimetry measurements at the interface of the tin sample and a lithium fluoride window. From these measurements, we determined temperature and pressure at the interface vs time. We then converted these data to temperature vs pressure curves and plotted them on the tin phase diagram. The tin sample was initially shocked into the high-pressure solid γ phase, and a subsequent release wave originating from the back of the impactor lowered the pressure at the interface along a constant entropy path (release isentrope). When the release isentrope reaches the solid-liquid phase boundary, melt begins and the isentrope follows the phase boundary to low pressure. The onset of melt is identified by a significant change in the slope of the temperature-pressure release isentrope. Following the onset of melt, we obtain a continuous and highly accurate melt curve measurement. The technique allows a measurement along the melt curve with a single radiance and reflectance experiment. The measured temperature data are compared to the published equation of state calculations. Our data agree well with some but not all of the published melt curve calculations, demonstrating that this technique has sufficient accuracy to assess the validity of a given equation of state model

Spallation reactions. A successful interplay between modeling and applications

The spallation reactions are a type of nuclear reaction which occur in space by interaction of the cosmic rays with interstellar bodies. The first spallation reactions induced with an accelerator took place in 1947 at the Berkeley cyclotron (University of California) with 200 MeV deuterons and 400 MeV alpha beams. They highlighted the multiple emission of neutrons and charged particles and the production of a large number of residual nuclei far different from the target nuclei. The same year R. Serber describes the reaction in two steps: a first and fast one with high-energy particle emission leading to an excited remnant nucleus, and a second one, much slower, the de-excitation of the remnant. In 2010 IAEA organized a worskhop to present the results of the most widely used spallation codes within a benchmark of spallation models. If one of the goals was to understand the deficiencies, if any, in each code, one remarkable outcome points out the overall high-quality level of some models and so the great improvements achieved since Serber. Particle transport codes can then rely on such spallation models to treat the reactions between a light particle and an atomic nucleus with energies spanning from few tens of MeV up to some GeV. An overview of the spallation reactions modeling is presented in order to point out the incomparable contribution of models based on basic physics to numerous applications where such reactions occur. Validations or benchmarks, which are necessary steps in the improvement process, are also addressed, as well as the potential future domains of development. Spallation reactions modeling is a representative case of continuous studies aiming at understanding a reaction mechanism and which end up in a powerful tool.Comment: 59 pages, 54 figures, Revie

Free-Surface Optical Scattering as an Indicator of the Shock-Induced Solid-Liquid Phase Transition in Tin

When highly polished metal surfaces melt upon release after shock loading, they exhibit features that suggest significant surface changes accompany the phase transition. The reflection of light from such surfaces changes from specular (pre-shock) to diffuse upon melting. A familiar manifestation of this phenomenon is the loss of signal light in VISAR measurements, which occurs at pressures high enough to melt the free surface. Unlike many other potential material phase-sensitive diagnostics (e.g., reflectometry, conductivity) that show relatively small (1%–10%) changes, the specularity of reflection provides a more sensitive and definitive indication of the solid-liquid phase transition. Data will be presented that support the hypothesis that specularity changes indicate melt in a way that can be measured easily and unambiguously

A posteriori error estimates for the virtual element method

An a posteriori error analysis for the virtual element method (VEM) applied to general elliptic problems is presented. The resulting error estimator is of residual-type and applies on very general polygonal/polyhedral meshes. The estimator is fully computable as it relies only on quantities available from the VEM solution, namely its degrees of freedom and element-wise polynomial projection. Upper and lower bounds of the error estimator with respect to the VEM approximation error are proven. The error estimator is used to drive adaptive mesh refinement in a number of test problems. Mesh adaptation is particularly simple to implement since elements with consecutive co-planar edges/faces are allowed and, therefore, locally adapted meshes do not require any local mesh post-processing

Organizational innovation in the multinational enterprise: internalization theory and business history

This article engages in a methodological experiment by using historical evidence to challenge a common misperception about internalization theory. The theory has often been criticized for maintaining that it assumes a hierarchically organized MNE based on knowledge flowing from the home country. This is not an accurate description of how global firms operate in recent decades, but this article shows it has never been true historically. Using longitudinal data on individual firms from the nineteenth century onwards, it reveals evidence of how entrepreneurs and firms with multinational activity faced with market imperfections changed the design of their headquarters and their organizational structures