Hexagonal Elements Research Articles

A Coarse-Mesh Method for Multidimensional, Mixed-Lattice Diffusion Calculations

A coarse-mesh finite difference method has been developed for multidimensional, mixed-lattice reactor diffusion calculations, both statics and kinetics, in hexagonal geometry. Results obtained with the coarse-mesh (CM) method have been compared with a conventional mesh-centered finite difference method and with experiment. The results of this comparison indicate that the accuracy of the CM method for highly heterogeneous (mixed) lattices using one point per hexagonal mesh element (''hex'') is about the same as the conventional method with six points per hex. Furthermore, the computing costs (i.e., central processor unit time and core storage requirements) of the CM method with one point per hex are about the same as the conventional method with one point per hex.

An economical construction of large astronomical lightweight mirrors made of glass

The building of large flux collectors for infrared astronomy necessitates the construction of mirrors less precise than those of conventional telescopes (the optical quality is restricted to a value which takes into consideration, in particular, the dimensions of the detector that is used and of the collector itself). We are trying to make mirrors with new designs, light and less costly. With this aim, a prototype of hexagonal element, 98 cm across, of a primary mosaic of a projected collector of large diameter, was built, tested and polished. We give the description of the mirror and details of the making. The weight reduction of the mirror to 70 per cent and the optical quality obtained will allow the effective utilisation of this new mirror.

A hexagonal finite element with one degree‐of‐freedom per hexagon

A hexagonal finite element with one degree‐of‐freedom per hexagon

The design of 2300 MW(e) twin high temperature gas-cooled reactors for the philadelphia electric company

The basic design features of a 2300 MW(e) twin high temperature gas-cooled reactor (HTGR) power plant are described. The reactor core consists of vertical columns of hexagonal graphite fuel-moderator elements and graphite reflector blocks which are grouped into a cylindrical array and supported by a graphite core support structure. Reactivity control is accomplished by means of 146 control rods. The distribution of helium coolant flow through the core is controlled by variable orifice valves. Each of the six primary coolant loops is equipped with a helium circulator. The main steam/water section of each steam generator consists of a single helical tube bundle arranged in an annulus around the center duct. A core auxiliary cooling system is provided to furnish an independent means of removing reactor afterheat. The inherent safety characteristics and the design safety features of the large HTGR are discussed. Station arrangement, steam cycle and twin turbine generators, plant performance and control, containment and fuel handling, and environmental controls, are described.

Finite difference techniques for variable grids

Partial differential equation problems arising in contemporary engineering problems are often solved numerically using finite difference techniques. The majority of the attention has been devoted to the rectangular grid formulation. The complications of applying rectangular grids on curved boundaries, however, has motivated active study of finite difference methods using non-rectangular grids since about 1953. The majority of the effort on variable grid techniques appears to have been directed toward the solution of self adjoint systems using an energy formulation. A hexagonal mesh element is generally used in these approaches. Another approach which has received minimal attention is finite difference evaluation on an arbitrary grid using, e.g. two dimensional Taylor expansion. The truncation error using this approach can be rigorously shown to converge to zero with increasing nodal density. There appear to be two pitfalls in the practical application of this approach, namely; (1) how to select a set of neighboring nodes for a given node to use for the finite difference evaluation and (2) how to efficiently handle the vast number of difference coefficients that can result from a given grid. Fortunately neither of these problems is insurmountable. Preliminary studies indicate that for many shell problems this variable grid technique will yield improved efficiency as well as a simple method for handling curved boundaries and varying stress patterns.

Kristallstruktur und Doppelbrechung fehlgeordneter ZnS Einkristalle

By comparing the crystal structure (investigated by means of X-ray diffraction) and the birefringence of disordered ZnS crystals it is shown that the birefringence Δn is proportional to the hexagonality σ, i.e. Δn = σ . ΔnWurt. The hexagonality σ is the part of hexagonal packing elements, ΔnWurt the birefringence of Wurtzite.

Stress Concentrations from Single-Filament Failures in Composite Materials

The solution of the two-dimensional, elastic, multiple-filament-failure stress concentration problem led to the treatment of three-dimensional, elastic failure models and a two-dimensional, plastic failure model where an ideally plastic behavior of the matrix material adjacent to a broken filament was assumed. Another plastic behavior is proposed wherein the bond between the broken filament and the adjacent matrix material fails completely after reaching a prescribed stress level. This failure formulation is applied to five- and seven-element-width models as well as to the infinite element case. Both the bond failure and matrix yield models are then extended to the three-dimensional cases with both square and hexagonal element configurations.

FINE STRUCTURE OF NUCLEAR CRYSTALS IN LEAVES OF TWO DIANTHUS SPECIES

Examination of leaf cells of Dianthus barbatus with the election microscope revealed the presence of crystals in the nuclei, but not in the cytoplasm or in other organelles. They were at first thought to be viral crystalline inclusions, but no evidence could be adduced to support this possibility. The crystals were found in the nuclei of cells from all parts of the leaf, including parenchyma cells of the vascular elements. They were usually 0.5–2 μ long, and occasionally up to 5 μ. In longitudinal section they appeared as electron‐dense striations, about 10–15 mμ wide, alternating with electron‐transparent bands, 6–8 mμ wide. In cross section they had a honeycomb appearance, resulting from closely packed hexagonal elements, with electron‐transparent centers. Similar crystals were found in nuclei of D. chinensis leaf cells, but not in other members of the Caryophyllaceae, including two varieties of D. caryophyllus, D. deltoides, D. arenarius, Silene acaulis, Lychnis chalcedonica, Saponaria calibrica, and Stellaria media. This appears be the first fine structural report of a lattice inclusion in nuclei of cells of healthy higher plants.

Fine Structure of Nuclear Crystals in Leaves of Two Dianthus Species

Examination of leaf cells of Dianthus barbatus with the election microscope revealed the presence of crystals in the nuclei, but not in the cytoplasm or in other organelles. They were at first thought to be viral crystalline inclusions, but no evidence could be adduced to support this possibility. The crystals were found in the nuclei of cells from all parts of the leaf, including parenchyma cells of the vascular elements. They were usually 0.5–2 μ long, and occasionally up to 5 μ. In longitudinal section they appeared as electron-dense striations, about 10–15 mμ wide, alternating with electron-transparent bands, 6–8 mμ wide. In cross section they had a honeycomb appearance, resulting from closely packed hexagonal elements, with electron-transparent centers. Similar crystals were found in nuclei of D. chinensis leaf cells, but not in other members of the Caryophyllaceae, including two varieties of D. caryophyllus, D. deltoides, D. arenarius, Silene acaulis, Lychnis chalcedonica, Saponaria calibrica, and Stellaria media. This appears be the first fine structural report of a lattice inclusion in nuclei of cells of healthy higher plants.

High-resolution monochromatization of neutrons and X-rays by multiple Bragg reflection

A simple scheme for obtaining highly monochromatic, reproducible beams of slow neutrons or X-rays at fixed wavelengths by means of crystal diffraction is described. The method is based on the well known phenomenon of `umweganregung', in which a `forbidden' Bragg reflection is simulated under conditions of multiple reflection, i.e. when three or more reciprocal lattice points lie on an Ewald sphere. It is shown that there are orientations at which the Bragg angle (and wavelength) of a simulated reflection has local maxima. With only coarse collimation in the neighborhood of these extrema, perfect crystals can give wavelength resolutions of Δλ/λ ≃ 10−5. Mosaic crystals would give less resolution but greater intensity. Since the various simulations of a `forbidden' reflection may interfere with each other, it is necessary to carry out systematic computer calculations to check for interference after the extrema have been located. The scheme is applicable to a large set of crystals, including several diamond-structure and hexagonal close-packed elements and compounds such as quartz and calcite. Detailed numerical results show that germanium can reflect about 80 potentially useful wavelengths in the range 1.2 to 5 Å (neutron energy 0.003 to 0.05 eV). A number of other interesting properties of such beams are pointed out. Potential uses in neutron research and several problems are discussed briefly.

Anisotropic Thermal Expansion of Single Crystals of Thallium, Yttrium, Beryllium, and Zinc at Low Temperatures

The anisotropic thermal expansion of the hexagonal elements, thallium, yttrium, beryllium, and zinc, has been measured. The temperature range for the thallium, yttrium, and zinc measurements was 4.2−273.2°K, while for beryllium the range was 77−273.2°K. Dimensional changes were detected by an interferometric technique, and the movement of the interference fringes was detected with a photomultiplier tube. From the measured values of ΔL/L0 the coefficients of thermal expansion and the c/a ratios of the metals were calculated as a function of temperature. A review of previous measurements of the thermal expansion of these metals is also included.

Studies on 4CaOAl2.O3.13H2O and the Related Natural Mineral Hydrocalumite

Single‐crystal X‐ray and electron‐diffraction studies show the existence in one polymorph of 4CaO.Al2O3. 13H2O of a hexagonal structural element with α= 5.74 a.u., c= 7.92 a. u. and atomic contents Ca2(OH)7‐ 3H2O. These structural elements are stacked in a complex way and there are probably two or more poly‐types as in SiC or ZnS. Hydrocalumite is closely related to 4CaO.A12O3.13H2O, from which it is derived by substitution of CO32‐for 20H‐+ 3H2O once in every eight structural elements; similar substitutions explain the existence of compounds of the types 3CaO Al2O3.CaY2‐ xH2O and 3CaO Al2O3 Ca Y xH2O. On dehydration, 4CaO.Al2O3.13H2O first loses molecular water and undergoes stacking changes and shrinkage along c. At 150° to 250°C., Ca(OH)2 and 4CaO.3Al2O3.3H2O are formed and, by 1000°C., CaO and 12CaO.7Al2O8. The dehydration of hydrocalumite follows a similar course, but no 4CaO.3Al2O3.3H2O is formed.

The Lindemann and Grüneisen Laws

The Lindemann assumption of direct contact of neighboring atoms at fusion is replaced by the criterion that melting occurs when the root-mean-square amplitude of thermal vibration reaches a critical fraction $\ensuremath{\rho}$, presumed the same for all isotropic monatomic solids, of the distance of separation of nearest-neighbor atoms. The Debye-Waller theory of the temperature dependence of the intensity of Bragg reflection of x-rays is used, without further assumptions, to derive a generalized Lindemann law. In contrast to the Lindemann form, all physical quantities involved in this formulation are evaluated at the fusion point, and departure of the average energy of an atomic oscillation from the equipartition value is taken into account by the quantization factor of the Debye-Waller theory. If the Gr\uneisen constant ${\ensuremath{\gamma}}_{m}$ of the solid at fusion is evaluated by its definition from the Debye frequency of the solid, use of the generalized Lindemann law and Clapeyron's equation permits one to express ${\ensuremath{\gamma}}_{m}$ in terms of the bulk modulus of the solid at melting and the latent heat and volume change of fusion. By means of Gr\uneisen's law applied to the solid at fusion, ${\ensuremath{\gamma}}_{m}$ can be expressed likewise in terms of the corresponding bulk modulus, thermal expansion, volume, and heat capacity at constant volume; the two evaluations of ${\ensuremath{\gamma}}_{m}$ connect the Lindemann and Gr\uneisen laws. These relations permit one to evaluate the slope and curvature of a fusion curve as functions of ${\ensuremath{\gamma}}_{m}$, and thus to express in terms of ${\ensuremath{\gamma}}_{m}$ the conditions that a fusion curve be normal in the sense of Bridgman. Experimental fusion data on 13 cubic metals are used to evaluate the constant of proportionality (inversely proportional to $\ensuremath{\rho}$) in the Lindemann frequency; the values are reasonably constant. The corresponding values of $\ensuremath{\rho}$ for Al and Cu show reasonable agreement with values deduced from x-ray intensity measurements. The average $\ensuremath{\rho}$ seems significantly below values estimated by Gr\uneisen. Good agreement with the cubic metals is found for hexagonal close-packed elements, but not for elements with more complex lattices, in general. The two evaluations of the Gr\uneisen constant ${\ensuremath{\gamma}}_{m}$ of the solid at fusion are shown to be in good agreement, experimentally, for 14 elements checked, but agreement fails for the elements Ga, Bi, and Sb, which show abnormal fusion curves.