Ag on Ge(111): 2D X-ray structure analysis of the (Wurzel)3 x (Wurzel)3 superstructure

We have studied the Ag/Ge(111)(Wurzel)3 x (Wurzel)3 superstructure by grazing-incidence X-ray diffraction. In our structural analysis we find striking similarities to the geometry of Au on Si(111). The Ag atoms form trimer clusters with an Ag-Ag distance of 2.94+-0.04°A with the centers of the trimers being located at the origins of the (Wurzel)3 x (Wurzel)3 lattice. The Ag layer is incomplete and at least one substrate layer is distorted

The Ge(001) (2 × 1) reconstruction: asymmetric dimers and multilayer relaxation observed by grazing incidence X-ray diffraction

Grazing incidence X-ray diffraction has been used to analyze in detail the atomic structure of the (2 × 1) reconstruction of the Ge(001) surface involving far reaching subsurface relaxations. Two kinds of disorder models, a statistical and a dynamical were taken into account for the data analysis, both indicating substantial disorder along the surface normal. This can only be correlated to asymmetric dimers. Considering a statistical disorder model assuming randomly oriented dimers the analysis of 13 symmetrically independent in-plane fractional order reflections and of four fractional order reciprocal lattice rods up to the maximum attainable momentum transfer qz = 3c* (c* = 1.77 × 10−1 Å−1) indicates the formation of asymmetric dimers characterized by R>D = 2.46(5) Å as compared to the bulk bonding length of R = 2.45 Å. The dimer height of Δ Z = 0.74(15) Å corresponds to a dimer buckling angle of 17(4)°. The data refinement using anisotropic thermal parameters leads to a bonding length of RD = 2.44(4) Å and to a large anisotropy of the root mean-square vibration amplitudes of the dimer atoms (u112) 1/2 = 0.25 Å, (u222)1/2 = 0.14 Å, (u332)1/2 = 0.50 Å). We have evidence for lateral and vertical disp tenth layer below the surface

Reduced basis isogeometric mortar approximations for eigenvalue problems in vibroacoustics

We simulate the vibration of a violin bridge in a multi-query context using reduced basis techniques. The mathematical model is based on an eigenvalue problem for the orthotropic linear elasticity equation. In addition to the nine material parameters, a geometrical thickness parameter is considered. This parameter enters as a 10th material parameter into the system by a mapping onto a parameter independent reference domain. The detailed simulation is carried out by isogeometric mortar methods. Weakly coupled patch-wise tensorial structured isogeometric elements are of special interest for complex geometries with piecewise smooth but curvilinear boundaries. To obtain locality in the detailed system, we use the saddle point approach and do not apply static condensation techniques. However within the reduced basis context, it is natural to eliminate the Lagrange multiplier and formulate a reduced eigenvalue problem for a symmetric positive definite matrix. The selection of the snapshots is controlled by a multi-query greedy strategy taking into account an error indicator allowing for multiple eigenvalues

Oval Domes: History, Geometry and Mechanics

An oval dome may be defined as a dome whose plan or profile (or both) has an oval form. The word Aoval@ comes from the latin Aovum@, egg. Then, an oval dome has an egg-shaped geometry. The first buildings with oval plans were built without a predetermined form, just trying to close an space in the most economical form. Eventually, the geometry was defined by using arcs of circle with common tangents in the points of change of curvature. Later the oval acquired a more regular form with two axis of symmetry. Therefore, an “oval” may be defined as an egg-shaped form, doubly symmetric, constructed with arcs of circle; an oval needs a minimum of four centres, but it is possible also to build polycentric ovals. The above definition corresponds with the origin and the use of oval forms in building and may be applied without problem until, say, the XVIIIth century. Since then, the teaching of conics in the elementary courses of geometry made the cultivated people to define the oval as an approximation to the ellipse, an “imperfect ellipse”: an oval was, then, a curve formed with arcs of circles which tries to approximate to the ellipse of the same axes. As we shall see, the ellipse has very rarely been used in building. Finally, in modern geometrical textbooks an oval is defined as a smooth closed convex curve, a more general definition which embraces the two previous, but which is of no particular use in the study of the employment of oval forms in building. The present paper contains the following parts: 1) an outline the origin and application of the oval in historical architecture; 2) a discussion of the spatial geometry of oval domes, i. e., the different methods employed to trace them; 3) a brief exposition of the mechanics of oval arches and domes; and 4) a final discussion of the role of Geometry in oval arch and dome design

Au/Si(111): Analysis of the (√3 × √3 )R30° and 6×6 structures by in-plane x-ray diffraction

In-plane, fractional-order diffraction-data sets from thin Au layers on Si(111) with (√3 × √3 )R30° and 6×6 structures were measured at the wiggler beamline W1 at the Hamburg synchrotron radiation laboratory. For the √3 × √3 structure the trimer model is confirmed with an Au-Au distance of 2.8 Å. In the √3 × √3 unit cell, two additional sites beside the Au trimer were found which can be identified with distorted substrate layers or additional partially occupied Au sites. The 6×6 structure is a sixfold twinned structure. The observed Patterson function clearly indicates the main features of the structural units. Each consists of three trimer clusters of Au atoms, forming a nearly equilateral triangle. The local structure of each trimer is either the original √3 × √3 structure or a twin structure where the Au trimers are rotated by 60°. Three of these structural units form the 6×6 unit cell. Model calculations with incoherent superposition of twin domains lead to Patterson maps very similar to the one observed