It is advantageous in automatic computers to employ methods of integration which do not require preceding function values to be known. From a general theory given by Kutta, one such process is chosen giving fourth-order accuracy and requiring the minimum number of storage registers. It is developed into a form which gives the highest attainable accuracy and can be carried out by comparatively few instructions. The errors are studied and a simple example is given.
We present a cross-sectional scanning-tunneling microscopy investigation of the shape, size, and composition of InAs quantum dots in a GaAs matrix, grown by molecular beam epitaxy at low growth rate. From the dimensional analysis we conclude that the investigated quantum dots have an average height of 5 nm, a square base of 18 nm oriented along ͓010͔ and ͓100͔ and the shape of a truncated pyramid. From outward relaxation and lattice constant profiles we conclude that the dots consist of an InGaAs alloy and that the indium concentration increases linearly in the growth direction. Our results justify the predictions obtained from previous photocurrent measurements on similar structures and the used theoretical model.
Analytical models for strained heteroepitaxial quantum dot systems have invariably assumed that the dots have a low-aspect ratio (small slopes) and that the elastic properties of the dot and the substrate are identical. In this paper, a three-dimensional analytical model for the energetics of an array of axisymmetric quantum dots is developed from physical principles. This is valid for high-aspect ratio dots (such as GeSi and InGaAs) and allows the dot and substrate to have different elastic properties. It is shown that these features are very important in determining the strain energy of both isolated dots and arrays of interacting dots. Both the elastic relaxation energy (per unit volume) of a single dot and the elastic interaction energy (per unit volume) between multiple dots are found to be greatest for tall, steep dots and for dots which are stiffer than the substrate. The equilibrium of two-facet dots is investigated and shape transition phase diagrams for small slope monoelastic theory, GeSi and InGaAs are compared. Different features of the bimodal dot size distributions in these systems are explained.
Saucer-shaped sills are common in sedimentary basins worldwide. The saucer shape relates to asymmetric sill-tip stress distributions during intrusion caused by bending of the overburden. Most saucer-shaped sill models are constructed using a magma-analogue excess source pressure (Po) to drive host-rock failure, but without tectonic stress. Here we present axisymmetric finite-element simulations of radially propagating sills for a range of tectonic stress (σr) conditions, from horizontal tension (σr < 0) to horizontal compression (0 < σr). Response to σr falls into four regimes, based on sill geometry and failure mode of the host rock. The regimes are considered in terms of the ratio of tectonic stress versus magma source pressure R = σr/Po: (I) initially seeded sills transition to a dike during horizontal extension (R < 0); (II) with R increasing from 0 towards 1 (compressive σr), sill base length increases and sill incline decreases; (III) where 1 < R < 2, sill base length relatively decreases and sill incline increases; and (IV) where R > 2, sills grow as inclined sheets. Sills in regimes I–III grow dominantly by tensile failure of the host rock, whereas sills in regime IV grow by shear failure of the host rock. Varying σr achieves a range of sill geometries that match natural sill profiles. Tectonic stress therefore represents a primary control on saucer-shaped sill geometry and emplacement mechanism.
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