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Probability density function and distribution function of distance from random point in polygon to the nearest corner in lattice
Methylamine crystals lattice vibrations IR spectra recordings, making spectral band assignments according to translational and librational motions
In this note we present two expressions relating the cohesive energy, E(sub coh), and the zero pressure isothermal bulk modulus, B(sub 0), of the alkali halides. Ag halides and TI halides, with the nearest neighbor distances, d(sub nn). First, we show that the product E(sub coh)d(sub 0) within families of halide crystals with common crystal structure is to a good approximation constant, with maximum rms deviation of plus or minus 2%. Secondly, we demonstrate that within families of halide crystals with a common cation and common crystal structure the product B(sub 0)d(sup 3.5)(sub nn) is a good approximation constant, with maximum rms deviation of plus or minus 1.36%.
Solubilized interstitial collagens will form a fibrillar, gel-like lattice when brought to physiologic conditions. In the presence of human dermal fibroblasts the collagen lattice will contract. The rate of contraction can be determined by computer-assisted planemetry. The mechanisms involved in contraction are as yet unknown. Using this system it was found that the rate of contraction was markedly decreased when collagen lacking telopeptides was substituted for native collagen. Histidinohydroxylysinonorleucine (HHL) is a major stable trifunctional collagen cross-link in mature skin that involves a carboxyl terminal, telopeptide site 16c, the sixteenth amino acid residue from the carboxy terminal of the telopeptide region of alpha 1 (I) in type I collagen. Little, if any, HHL was present in native, purified, reconstituted, soluble collagen fibrils from 1% acetic acid-extracted 2-year-old bovine skin. In contrast, HHL cross-links were present (0.22 moles of cross-link per mole of collagen) in lattices of the same collagen contracted by fibroblasts. However, rat tail tendon does not contain HHL cross-links, and collagen lattices made of rat tail tendon collagen are capable of contraction. This suggests that telopeptide sites, and not mature HHL cross-links per se, are essential for fibroblasts to contract collagen lattices. Beta-aminopropionitrile fumarate (BAPN), a potent lathyrogen that perturbs collagen cross-linking by inhibition of lysyl oxidase, also inhibited the rate of lattice cell contraction in lattices composed of native collagen. However, the concentrations of BAPN that were necessary to inhibit the contraction of collagen lattices also inhibited fibroblast growth suggestive of cellular toxicity. In accordance with other studies, we found no inhibition of the rate of lattice contraction when fibronectin-depleted serum was used. Electron microscopy of contracted gels revealed typical collagen fibers with a characteristic axial periodicity. The data provide evidence that collagen telopeptide sites play a role in collagen gel lattice contraction.
The spanwise vortex-lattice arrangement is mathematically established by lattice solutions of the slender wing which are shown to be analogous to the chordwise vortex-lattice thin wing solution. Solutions for any number N of panels wing theory lift and induced drag and thin wing theory lift and moment are predicted exactly. As N approaches infinity, the slender wing elliptic spanwise loading and thin wing cotangent chordwise loading are predicted, which proves there is mathematical convergence of the vortex-lattice method to the exact answer. Based on this planform spanwise lattice arrangement, an A-vortex-lattice spanwise system is developed for an arbitrary aspect ratio A. This A-lattice has the optimum characteristic of predicting lift accurately for any value of N.
Lattice block structures made up of high temperature superalloys could offer advantages in density, efficiency, and tailored properties. Three candidate lattice designs were screened through additive manufacturing of superalloy 718. Laser powder bed fusion was applied with 718 powder to produce lattice blocks for three-dimensional octahedral unit cell designs of three heights. As-designed versus additively manufactured 718 lattices were compared, including structural dimensions, densities, defects, grain structures, crystallographic texture, and surface residual stresses. Manufactured lattice I diverged the most from its design, with both corner and center nodes merged into continuous columns. Manufactured lattice III came closest to its design, though the actual nodes and struts were still larger in cross section than in the design. Only modest grain coarsening and negligible chemical segregation were observed for all three designs, suggesting uniform thermo-physical and mechanical properties are possible. Significant tensile residual stresses were present at the surface of the manufactured blocks, with highest stresses in the building direction. Current findings indicate this concept could produce useful superalloy structures for high-temperature-capable static components requiring tailored density, and thermo-physical/mechanical properties.
The three stable states of matter and the corresponding phase transitions were obtained with a single model. Patterned after Lennard-Jones and Devonshires's theory, a simple cubic lattice model containing two fcc sublattices (alpha and beta) is adopted. The interatomic potential is taken to be the Lennard-Jones (6-12) potential. Employing the cluster variation method, the Weiss and the pair approximations on the lattice gas failed to give the correct phase diagrams. Hybrid approximations were devised to describe the lattice term in the free energy. A lattice vibration term corresponding to a free volume correction is included semi-phenomenologically. The combinations of the lattice part and the free volume part yield the three states and the proper phase diagrams. To determine the coexistence regions, the equalities of the pressure and Gibbs free energy per molecule of the coexisting phases were utilized. The ordered branch of the free energy gives rise to the solid phase while the disordered branch yields the gas and liquid phases. It is observed that the triple point and the critical point quantities, the phase diagrams and the coexistence regions plotted are in good agreement with the experimental values and graphs for argon.
The application of the vortex lattice method to problems not usually dealt with by this technique is considered. It is shown that if the discrete vortex lattice is considered as an approximation to surface-distributed vorticity, then the concept of the generalized principal part of an integral yields a residual term to the vortex-induced velocity that renders the vortex lattice method valid for supersonic flow. Special schemes for simulating non-zero thickness lifting surfaces and fusiform bodies with vortex lattice elements are presented. Thickness effects of wing-like components are simulated by a double vortex lattice layer, and fusiform bodies are represented by a vortex grid arranged on a series of concentric cylindrical surfaces. Numerical considerations peculiar to the application of these techniques are briefly discussed.
The lattice-statics approach to the problem of providing a quantitative description of the microstructure near a defect is outlined. The method deals with an effectively infinite lattice and hence avoids interface problems; the lattice-statics method is not restricted by the range of the potential. The lattice-statics solution requires a harmonic solution for the lattice which is shown to be always obtainable by suitable Fourier transforms. Applications of the method to fracture and dislocations are discussed and areas that warrant further research are specified.
A method for computing the Lorentz tensor components in single crystals via rapidly convergent sums of Bessels functions is developed using the relationship between dipole-field sums and the tensor components. The Lorentz factors for simple, body-centered, and base-centered orthorhombic lattices are computed using this method, and the derivative Lorentz factors for simple orthorhombic lattices are also determined. Both the Lorentz factors and their derivatives are shown to be very sensitive to a lattice structure. The equivalent of the Clausius-Mossotti relation for general orthorhombic lattices is derived using the Lorentz-factor formalism, and the permanent molecular dipole moment is related to crystal polarization for the case of a ferroelectric of polarizable point dipoles. It is concluded that the polarization enhancement due to self-polarization familiar from classical theory may actually be a reduction in consequences of negative Lorentz factors in one or two lattice directions for noncubic crystals.
The formulation of the Dirac equation on a discrete lattice with half-integer spacing and periodic boundary conditions is investigated analytically. The importance of lattice formulations for problems in field theory and quantum mechanics is explained; the concept of half-integer Fourier representation is introduced; the discrete Dirac equation for the two-dimensional case is derived; dispersion relations for the four-dimensional case are developed; and the spinor formulation for the Dirac fields on the half-integer lattice and the discrete time variable for the four-dimensional time-dependent Dirac equation are obtained. It is argued that the half-integer lattice, because it takes the Dirac Lagrangian into account, is more than a mere relabeling of the integer lattice and may have fundamental physical meaning (e.g., for the statistics of fermions). It is noted that the present formulation does not lead to species doubling, except in the continuum limit.
A new lattice regularization of quantum general relativity based on Ashtekar's reformulation of Hamiltonian general relativity is presented. In this form, quantum states of the gravitational field are represented within the physical Hilbert space of a Kogut-Susskind lattice gauge theory. The gauge field of the theory is a complexified SU(2) connection which is the gravitational connection for left-handed spinor fields. The physical states of the gravitational field are those which are annihilated by additional constraints which correspond to the four constraints of general relativity. Lattice versions of these constraints are constructed. Those corresponding to the three-dimensional diffeomorphism generators move states associated with Wilson loops around on the lattice. The lattice Hamiltonian constraint has a simple form, and a correspondingly simple interpretation: it is an operator which cuts and joins Wilson loops at points of intersection.
A study is described that evaluates the accuracy of vortex-lattice methods when they are used to compute the loads induced on aircraft as they encounter lift-generated wakes. The evaluation is accomplished by use of measurements made in the 80- by 120-foot wind tunnel of the lift, rolling-moment, and downwash in the wake of three configurations of a model of a subsonic transport aircraft. The downwash measurements are used as input for a vortex-lattice code in order to compute the lift and rolling moment induced on wings that have a span of 0.186, 0.510, or 1.022 times the span of the wake-generating model. Comparison of the computed results with the measured lift and rolling moment distributions are used to determine the accuracy of the vortex-lattice code. It was found that the vortex-lattice method is very reliable as long as the span of the encountering of following wing is less than about 0.2 of the generator span. As the span of the following wing increases above 0.2, the vortex-lattice method continues to correctly predict the trends and nature of the induced loads, but it overpredicts the magnitude of the loads by increasing amounts. The increase in deviation of the computed from the measured loads with size of the following wing is attributed to the increase in distortion of the structure of the vortex wake as it approaches and passes the larger following wings.
Two techniques for extending the range of applicability of the basic vortex-lattice method are discussed. The first improves the computation of aerodynamic forces on thin, low-aspect-ratio wings of arbitrary planforms at subsonic Mach numbers by including the effects of leading-edge and tip vortex separation, characteristic of this type wing, through use of the well-known suction-analogy method of E. C. Polhamus. Comparisons with experimental data for a variety of planforms are presented. The second consists of the use of the vortex-lattice method to predict pressure distributions over thick multi-element wings (wings with leading- and trailing-edge devices). A method of laying out the lattice is described which gives accurate pressures on the top and part of the bottom surface of the wing. Limited comparisons between the result predicted by this method, the conventional lattice arrangement method, experimental data, and 2-D potential flow analysis techniques are presented.
If the discrete vortex lattice is considered as an approximation to the surface-distributed vorticity, then the concept of the generalized principal part of an integral yields a residual term to the vorticity-induced velocity field. The proper incorporation of this term to the velocity field generated by the discrete vortex lines renders the present vortex lattice method valid for supersonic flow. Special techniques for simulating nonzero thickness lifting surfaces and fusiform bodies with vortex lattice elements are included. Thickness effects of wing-like components are simulated by a double (biplanar) vortex lattice layer, and fusiform bodies are represented by a vortex grid arranged on a series of concentrical cylindrical surfaces. The analysis of sideslip effects by the subject method is described. Numerical considerations peculiar to the application of these techniques are also discussed. The method has been implemented in a digital computer code. A users manual is included along with a complete FORTRAN compilation, an executed case, and conversion programs for transforming input for the NASA wave drag program.
A simplified buckling analysis is presented for a family of periodic lattice structures such as those proposed for large space structures. A transcendental 6 x 6 matrix of eigenvalues is shown to be sufficient for modeling buckling behavior because member stiffnesses are based on an exact solution of the beam-column equation. Exact stiffnesses are derived for a curved member, thus allowing modeling of imperfect lattice structures. Comparisons of predictions of the lattice model with those available from shell and beam theory underscore the inaccuracies introduced by treating the lattice structure as a continuum. Sample calculations are provided for an isogrid cylinder and a three element double-laced truss.
High temperature X-ray diffraction techniques were used to determine the gamma-gamma prime lattice mismatch of three different nickel-base superalloys at temperatures between 18 and 1000 C. The measurements were performed on oriented single-crystal disks which had been aged to produce a semicoherent gamma-gamma prime structure. The thermal expansion of the lattice parameters of the gamma and gamma-prime phases was described by a second-order polynomial expression. The expansion of the gamma-prime phase was consistently smaller than that of the gamma phase, which caused the lattice mismatch to become more negative at higher temperatures. It was also shown that high values of lattice mismatch resulted in increased rates of directional gamma-prime coarsening during elevated temperature creep exposure.
The necessary phase diagram data for growing lattice-matched layers of PbSnSeTe on PbSe are presented. Solid compounds of Pb(1-x)Sn(x)Se(1-y)Te(y) lattice-matched to PbSe were grown from liquid melts consisting of (Pb/1-x/Sn/x/)(1-z)(Se/1-y/Te/y/)(z); phase equilibria data were determined together with liquidus data for values of x(liquid) from 0 to 40 percent and y(liquid) from 0 to 40 percent for temperatures between 450 and 540 C. It was found that relatively large amounts of Te must be added to the melt to achieve lattice matching because of its low segregation coefficient relative to Se. A significant lattice-pulling effect was discovered for the 5-percent Sn case, and a similar effect is expected for the 10- and 20-percent Sn cases.