Dense Sphere Packings Research Articles

Hyperfine fields and local environment in iro-boron metallic glasses

The concentration dependence of the average hyperfine field and the hyperfine field distribution in Fe100-xBx metallic glasses (0<or=x<or=25) is explained in terms of a local environment model based on dense radiation packing of hard spheres' model structures. It is assumed that the hyperfine field at iron sites without boron neighbours is sensitive to local compression or dilation in analogy to crystalline close-packed iron. In this way even the strong decrease of the average hyperfine field at low boron content can be explained.

Effect of the structure difference between a crystal and its liquid on the interfacial roughening

The effect of structure difference between a crystal and its liquid on the interfacial roughening has been investigated. Jackson's theory on the interfacial roughening is modified by taking the mismatch between the two phases and the dangling bonds at the interface into consideration. The mismatch yields a tendency of decreasing the degree of the interfacial roughness. This tendency is strong, if the ratio of the latent heat of vaporization to that of fusion, δ h v/δ h f, or the anisotropy of the crystal structure is l When a dense random packing of hard spheres is assumed for the liquid structure, crystals of P 4, Sn, In, Hg, Zn, Cd and Al growing from their own liquids have intrinsically smooth surfaces, in contrast with the conclusion drawn from the Jackson theory. The present theoretical result for P 4 is consistent with the experimental observations of growing into a faceted form.

Cavities in dense random packings

The structure of dense random packings of hard spheres is analyzed in terms of the polyhedral cavities between spheres. Two different self-consistent schemes of polyhedron classification and identification are presented: polyhedra with all triangular faces, called deltahedra; and local arrangements allowing both triangular and four-edged faces. Two models were analyzed according to these schemes: the mechanical model of Finney Proc. R. Soc. 319A, 479 (1970), and the computer-generated model of Bennett J. appl. Phys. 43, 2727 (1972). In addition, the number and sizes of the sites for small interstitial spheres were calculated for both models, without regard to the polyhedral classification of the cavities. The structural units described and the results may be applicable to the structure of liquid metals and metallic glasses.

Dense lattice packings of spheres in Euclidean spaces of dimension n⩽16 *

We present a number of lattice packings of equal spheres in ℝn for n⩽16 For n⩽15, these packings have the same density as the densest known lattice packings. For n=16, the packing described here is denser than the known ones.

Crystal electric field and local symmetry in rare-earth amorphous alloys

Amorphous alloys containing small concentrations of rare earths exhibit very different magnetic properties for Pr (strong singlet ground state effects), Tm (smaller singlet effects), and Dy or Er (small deviations from the free ion behavior). All the experimental results for different rare earths in a given host can be accounted for by a unique distribution of nonuniaxial quadratic crystal fields. The distribution of local symmetries is not consistent with random dense packing of hard spheres and suggests well defined local angular correlations.

Limiting Densities for Dense Random Packing of Spheres

Theoretical packing densities are derived for dense random packing of hard spheres for single-valued and for multivalued particle diameters. The optimum particle-size distribution for maximum density is calculated. The results are compared with packing densities for close-packed arrangements. The results may be useful in estimating inhomogeneities in density of green-powder compacts. They may also be useful in understanding the role of particle rearrangement during sintering.

Computer simulation of isotropic, homogeneous, dense random packing of equal spheres

A computer program generated 3000 interpenetrating spheres in a cubic container (with periodic boundary conditions which ensured complete continuity at all boundaries) and used a relaxation technique to reduce overlaps. When the average overlap fell below (exceeded) an empirical value, sphere radii were increased (decreased). This simulated vibration caused the nominal solids fraction to increase slowly to 0.6381. Further iterations with a very gradual decrease in sphere radii produced an overlap-free packing with a density of 0.6366. This packing, which is isotropic and homogeneous, is characterized by very few contacts and many non-touching neighbours. The radial distribution function is similar to that obtained experimentally. In particular, the first peak is split into two maxima.

On the density of transition metal-metalloid glasses

Dependence of the density of transition metal-metalloid glasses on the nature and concentration of the metalloid element has been examined by comparing published density measurements for glasses with values derived from the lattice parameters of a wide range of crystalline alloys. An appealing similarity is revealed between the effects of concentration and radius ratio on the packing fractions of atoms in each phase. A first-order description of the density of alloy glasses is given in terms of the number of metalloid atoms occupying cavities in the metal sub-lattice and the dilatation which such occupancy entails. A linear relationship allows the density to be predicted with reasonable accuracy from the concentration and metalloid radius ratio. This relationship highlights similarities in structure which appear to exist between crystalline and amorphous phases: in particular, it seems that the metalloid is preferentially situated in trigonal prismatic cavities and there is no evidence for any other occupied polyhedron in these glasses. Packing of metal atoms, and therefore of trigonal prisms,is surprisingly efficient and approaches that of face-centred cubic close-packing. The properties of this model are compared with two earlier descriptions based on dense random packing of hard spheres and both are found to give less satisfactory explanations of experimental measurements.

Formation and stability of nickel-zirconium glasses

Glasses may be formed in two composition ranges: 33–42 and 60–76 at.% Zr. The crystallisation temperatures ( T x), activation energies ( Q x), crystallisation morphologies, densities, Young's moduli and peak in the X-ray diffraction pattern ( Q p) are reported as a function of composition. T x, Q x and Q p do not interpolate smoothly between the two composition ranges. Maximum thermal stability occurs at 36 at.% Zr, well away from the value predicted by the free electron approach of Nagel and Tauc. The glasses are denser than would be expected from a simple dense random packing of hard spheres. Some implications for the structure of all-metal glasses are discussed.

Interstice correlation functions; a new, sensitive characterisation of non-crystalline packed structures

A dense random packing of soft spheres is developed as a model of an “ideal” single component metallic glass. This structure can be considered as an assembly of distorted octahedra and tetrahedra, as can simple close-packed crystal structures. The more complex polyhedra (Bernal holes) previously identified in hard sphere models are largely squeezed out as a result of the additional free volume conferred by the soft potential. The local correlations of these basic tetrahedral and octahedral building blocks are discussed as possible characterisations of non-crystalline packed structures, together with their organisation around the component atoms. These characterisations of local correlations seem more sensitive and more reliably interpreted than the Voronoi polyhedra, particularly in detecting local icosahedral structures. Extensions of the descriptive techniques developed include examinations of weighted interstice connectivity, using data on the sizes of the interconnecting largely triangular “necks” as weighting factors. Departures from this idealised framework of linked tetrahedra and octahedra can be considered as defects, or distortions from the idealised structure, and the relative populations of local groupings provide a possible route to configurational entropy calculations.

A three dimensionally periodic, eleven coordinated, dense packing of symmetry equivalent spheres

Symmetry equivalent equal size spheres can be packed in an array filling 71.87% of space. Each sphere in this packing has eleven neighbors at unit distance. The density of this tetragonal close packing is second only to closest packing with twelve coordination. Anions in numerous compounds, including some which are solid electrolytes, arrange themselves in this pattern. It occurs among others in rutiles, β-BeO, Li 4GeO 4 and CaSO 4.

Structure simulation of amorphous monatomic and binary systems

A simplified approach to computer simulation of amorphous structures by the dense random packing of hard spheres (DRPHS) has been developed and applied to amorphous Tb and TbFe 2. The computer algorithms devised are rapid and result in uniform, isotropic packing for both the monatomic as well as binary systems; in the latter case, small-atom drift toward the center of the cluster is reduced to the uncertainty in the random number generator. Energy relaxation has been performed using the first term in Keating's expression for the elastic energy; relaxation causes the peaks in the pair correlation functions to broaden, but their relative intensities (peak areas) remain essentially the same. In all cases investigated, the packing fractions are 2% (or less) smaller than those of Finney's ball bearing measurements, and the calculated densities are 13–23% lower than their crystalline counterparts. The pair correlation functions of the monatomic and binary models have been compared with X-ray results on CoP and neutron results on TbFe 2 and found to be in good agreement.

A note on the leech lattice as a code for the gaussian channel

The Leech lattice A is a very dense packing of spheres in 24-dimensional Euclidean space, discovered by Leech (1967). Its automorphism group was determined by Conway (1969), and its usefulness as a source of codes for the Gaussian channel was studied by Blake (1971). The present note contains some comments on and corrections to the latter paper. Both Leech (1967) and Conway (1969) use essentially the same definition of A. This can be stated succinctly using the binary Golay code g24 of length 24 and minimum Hamming distance 8: A consists of those points in real 24dimensional space R 24 with coordinates

Anomalous behavior of amorphous Fe-B alloys

The density of Fe-B amorphous alloys is constant at 7.5 g/cm 3 for 13 ≤ at.% B 20 at.%. Hardness increases with B content up to 20% and less steeply for B > 20 at.%. A model based on the dense random packing of spheres of two sizes with densities d_{1}p and d_{2}p is presented to account for the anomalous behaviour in density of Fe-B alloys.

EXAFS STUDIES OF GLASSY METALLIC ALLOYS

The near neighbor correlations have been obtained from EXAFS measurements at Stanford Synchrotron Radiation Laboratory for a series of metal-metal glassy alloys. In particular, the structures of Mo50Ni50 alloy in amorphous and crystalline states are presented. The structure of the a-MoNi may be described by a more or less dense random packing of binary spheres in contrast to the c-MoNi which is described by a distorted bcc structure.

Triplet correlation in metallic glasses a field-ion microscopy study

Abstract The triplet correlation of atomic positions in iron-boron based metallic glass has been studied by field-ion microscopy. The distribution function for angles between directions to the near neighbours of atoms exhibits a fairly distinct peak at 60°, a broad peak between 90° and 125°, and a shoulder around 150°. These observations are in general agreement with the predictions of a model for metallic glass based on dense random packing of hard spheres.

The structure of some refractory transition metal-metalloid glasses

The radial distribution functions of four refractory alloys of composition (W 0.5Ru 0.5) 0.8 M 0.2 where M = some metalloid from the group B, P, Al 0.5B 0.5, Si 0.5B 0.5 are presented along with densities, atomic volumes of metalloids and coordination numbers. The atomic distributions functions of these materials have been found to exhibit characteristics typical of many other amorphous metal-metalloid alloys. A comparison was made between the experimentally determined distribution functions for the WRu alloys and the Bernal Finney model calculations of dense random packings of hard spheres using the average WRu Gold-schmidt radii as the hard sphere radius. The agreement was found to be quite good in nearly all places, the most notable exception being in the vicinity of the second maxima, where the relative heights of the double peaks is found to be different from the DRPHS for the cases of M = B, P and B 0.5Si 0.5. In all cases for the WRu alloys the reduced radial distribution functions suggest a local short range order favoring common-base tetrahedra configurations as opposed to coplanar tetrahedra bases predominant in the DRPHS.

ChemInform Abstract: THE ROLE OF STRUCTURE IN THE MAGNETIC PROPERTIES OF AMORPHOUS ALLOYS

Abstract The crystal structure and symmetry of crystalline magnetic materials determine the nature of their magnetic anisotropy including the direction of the easy axis of magnetization. In amorphous metallic alloys, particularly those containing non S-state rare earth ions, the coupling to the immediate environment gives rise to a local magnetic anisotropy (RMA) which reflects the real space structure of such systems. This article is an attempt to review The origin and consequences of these local couplings in systems which lack long range periodic order. The paper begins with a description of the structure of amorphous metals and its similarity to the dense random packing of hard spheres (DRPHS), a mode ideally suited for detailed computer simulation. The experimental data are presented with a view to assessing the effects of RMA on the magnetic properties of amorphous rare earth alloys. The main emphasis of the text is a detailed examination of the various theoretical models for RMA which have been introduced either in model calculations or in conjuction with computer generated clusters. Magnetization, Mossbauer and specific heat calculations in the molecular field approximation are presented and related to data for both ferromagnetic rare earth-noble metal and ferromagnetic rare earth-transition metal alloys. It is shown that this approximation works well at high magnetic fields and low temperatures and also provides a good description of the hysteresis effects. On the other hand, the remanent state of these alloys often appears to a be a spin glass-like state which cannot be described by the molecular field equations but can be found in Monte Carlo simulations based on the RMA. Finally, the spin excitations of systems with RMA are analyzed within a random phase approximation as well as by a Monte Carlo calculation. The article concludes with an evaluation of the progress to data and an outline of related problems yet outstanding.

Short-range order in Fe-B metallic glass alloys

Measured hyperfine fields at iron sites in Fe-B glass alloys are compared to calculations based on two models. The first model is based on the concept of dense random packing of spheres; the second emphasizes the short-range order found in crystalline materials with similar compositions. The crystalline short-range order is found to give a better fit and suggests specific improvements in the concepts used to understand the atomic structure of glasses.

The polyhedron and cavity analyses of a structural model of amorphous iron

Local atomic arrangements were investigated as a model of amorphous iron in terms of Voronoi polyhedra for a dense random packing of hard spheres with and without relaxation. The structure of liquid iron was also simulated in a computer. The amorphous structure was microscopically described as a mosaic of crystalline and noncrystalline polyhedra. The authors have also searched for possible cavities involved in the amorphous structure and found all the Bernal holes. A chain of independent tetragonal dodecahedron (TDs) and trigonal prisms capped with three half-octahedra (TPs), which contact each other on common edges or common faces, could be constructed by trial and error, preferentially retaining the face-contacting holes. If the dislocation model is accepted and the TDs and TPs are regarded as the core structure, the dislocation density is estimated to be 3.4*1014/cm2.