Packing Lattice Research Articles

Crystallographic Realization of the Mathematically Predicted Densest All‐Pentagon Packing Lattice by C5‐Symmetric “Sticky” Fluoropentamers

Crystallographic Realization of the Mathematically Predicted Densest All‐Pentagon Packing Lattice by <i>C</i>₅‐Symmetric “Sticky” Fluoropentamers

Surface structure determination of Pd on W(100) using X-ray photoelectron diffraction

Ferromagnetic ordering in non-ferromagnetic transition metals induced by structural effects such as lattice expansion and stacking faults have been predicted by several theoretical studies. Different surfaces such as Nb(100), W(100) and C(0001) have been suggested as potential templates which could drive a Pd ultrathin film to a stable ferromagnetic state, depending on growth packing, film thickness and lateral lattice parameter. In this work we present an X-ray photoelectron diffraction (XPD) study of sub-monolayer to a few layers of Pd grown on W(100) and its surface structure determination. XPD patterns of Pd 3d and W 4f peaks were measured using synchrotron radiation with photon energy of 1810eV provided by the SXS beamline at the Brazilian Synchrotron Light Laboratory. Our results indicate that the first few monolayers of Pd grow on W(100) with bcc packing and lateral lattice parameter dramatically changed in comparison with the Pd(fcc) bulk values. Based on these results we also present a discussion on the magnetic properties of this system derived from spin-polarized DFT calculation.

Form, symmetry and packing of biomacromolecules. I. Concepts and tutorial examples

The aim of this paper is to relate morphological properties of single biomacromolecules based on molecular enclosing forms indexed by an appropriate form lattice to the symmetry of the crystal where the molecules are periodically packed. Similar to the way in which the 'molécule intégrante' of Haüy permitted a molecular interpretation of the law of rational indices of crystal growth forms, alternative molecular enclosing forms, indexed by a so-called packing lattice, allow one to bridge the gap between form and crystal lattices. In this first part, selected tutorial examples illustrate the validity of the approach and the crystallographic compatibility between molecular and crystal structures. In particular, integral molecular lattices are shown to imply the observed axial ratios between crystal lattice parameters, leading sometimes to surprising results, like a cubic crystal lattice with a unit cell having a trigonal molecular filling with hexagonal enclosing form.

Three-dimensional Reconstruction of a Membrane-bending Complex: THE RC-LH1-PufX CORE DIMER OF RHODOBACTER SPHAEROIDES

A three-dimensional model of the dimeric reaction center-light harvesting I-PufX (RC-LH1-PufX) complex from Rhodobacter sphaeroides, calculated from electron microscope single particle analysis of negatively stained complexes, shows that the two halves of the dimer molecule incline toward each other on the periplasmic side, creating a remarkable V-shaped structure. The distribution of negative stain is consistent with loose packing of the LH1 ring near the 14th LH1 alpha/beta pair, which could facilitate the migration of quinone and quinol molecules across the LH1 boundary. The three-dimensional model encloses a space near the reaction center Q(B) site and the 14th LH1 alpha/beta pair, which is approximately 20 angstroms in diameter, sufficient to sequester a quinone pool. Helical arrays of dimers were used to construct a three-dimensional membrane model, which matches the packing lattice deduced from electron microscope analysis of the tubular dimer-only membranes found in mutants of Rba. sphaeroides lacking the LH2 complex. The intrinsic curvature of the dimer explains the shape and approximately 70-nm diameter of these membrane tubules, and at least partially accounts for the spherical membrane invaginations found in wild-type Rba. sphaeroides. A model of dimer aggregation and membrane curvature in these spherical membrane invaginations is presented.

Finite Lattice Packings and the Wulff‐Shape

This paper treats finite lattice packings Cn + K of n copies of some centrally symmetric convex body K in Ed for large n. Assume that Cn is a subset of a lattice Λ, and ϱ is at least the covering radius; namely, Λ + ϱK covers the space. The parametric density δ(Cn, ϱ) is defined by δ(Cn, ϱ) = n · V(K)/V(convCn + ϱK). It is shown that, if δ(Cn, ϱ) is minimal for n large, then the shape of conv Cn is approximately given by Wulff's condition, well-known from crystallography. Thus maximizing parametric density is equivalent to optimizing a certain Gibbs–Curie energy. It is also proved that, in case of lattice packings of K (allowing any packing lattice), for large n the optimal shape with respect to the parametric density is approximately a Wulff-shape associated to some densest packing lattice of K.

Multiuser Detection Techniques Using Maximum Likelihood Sphere Decoding in Multicarrier CDMA Systems

When performed using an exhaustive search, the maximum likelihood (ML) joint detection of all users in a multicarrier code-division multiple-access (MC-CDMA) system has a prohibitive complexity, growing exponentially with the number of users and the number of bits in each modulation symbol. In this paper, a novel ML multiuser detection algorithm is proposed, the complexity of which is a polynomial function of the number of users and is independent of the modulation size. The MC-CDMA system is modeled as a sphere packing lattice and a low-complexity optimum lattice decoder, the sphere decoder, is applied to jointly detect all users. Suboptimum simplifications, based on the orthogonal projection of the received signal on a facet of the lattice constellation, are also proposed to further decrease the complexity. Simulation results are shown with up to 64 users transmitting 16-quadrature-amplitude modulation symbols.

On Free Planes in Lattice Ball Packings

This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d‐dimensional packing lattice of balls one can find a four‐dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d.

Short Communication: The 1 :1 Co-Crystal of 2-Amino-5-nitro-1,3-thiazole with 4-Aminobenzoic Acid

The basic structural unit of the 1:1 co-crystal of 2-amino-5-nitro-1,3-thiazole with 4-aminobenzoic acid, [(2-antz)(4-aba)], comprises two constituent molecules associated by a hydrogen-bonded R2 2(8) graph set dimer through the carboxylic group across the N/N site of the thiazole [O–H···N, 2.614(3) Å; N–H···O, 2.991(3) Å]. Additional hydrogen-bonding associations from the three remaining amino hydrogen atoms construct a non-centrosymmetric packing lattice (space group: P21). The bulk ungraded powder exhibits a second harmonic signal of I/Iurea= 5.7. Manuscript received: 16 July 2001. Final version: 9 November 2001.

Large Finite Lattice Packings

A finite lattice packing of a centrally symmetric convex body K in \(\mathbb{R}\)d is a family C+K for a finite subset C of a packing lattice Λ of K. For ϱ>0 the density δ (C;K,ϱ) is defined by δ(C;K,ϱ) = card C·V(K)/V(conv C+ϱK). Assume that Cn is the optimal packing with given n=card C, n large. It was known that conv Cn is a segment if ϱ is less than the sausage radius ϱs (>0), and the inradius r(conv Cn) tends to infinity with n if ϱ is greater than the critical radius ϱc (≥ϱs). We prove that if ϱ>ϱc in \(\mathbb{R}\)d, then the shape of conv Cn is not too far from being a ball. In addition, if r(conv Cn) is bounded but the radius of the largest (d−2)-ball in Cn tends to infinity, then eventually Cn is contained in some k–plane and its shape is not too far from being a k-ball where either k=d−1 or k=d−2. This yields in \(\mathbb{R}\)3 that if ϱs<ϱ<ϱc, then conv Cn is eventually planar and its shape is not too far from being a disc. As an example, we show that ϱs=ϱc if K is a 3-ball, verifying the Strong Sausage Conjecture in this case. On the other hand, if K is the octahedron then ϱs<ϱc holds even for general (not only lattice) packings.

X-ray diffraction and structure of crystallins

The 3-dimensional organisation of crystallin polypeptides into globular proteins and their interactions into higher order structures are important factors governing optical functions related to refraction, accommodation and transparency. Single crystal X-ray diffraction studies have revealed the tertiary and quaternary structural organisation of β-, γ- and δ-crystallins. Regions of the lens with high refractive index contain high levels of monomeric y-crystallins while the accommodating, hydrated avian lens has largely replaced γ-crystallins with δ-crystallin. The βγ-crystallins form a superfamily of proteins of high symmetry and great diversity in which the basic building block is a 10 kD pseudo-symmetrical 2-Greek key domain. A γ-crystallin comprises two of these β-sheet domains, joined by a linker, and a short C-terminal extension. In β-crystallins the linker has an extended conformation resulting in dimer formation by a mechanism known as domain swapping. Crystallographic analysis of engineered single domains of γ-crystallins, analogous to the ancestral domain, has indicated the importance of the short C-terminal extension in directing domain pairing. γ-crystallins have numerous cysteine residues, some are conserved in the core of the protein molecule and some are variable on the protein surface. The structure of γB-crystallin has been determined at very high resolution using cryo-crystallography allowing the visualisation of the complete protein-protein and protein-water structure at the surface. β-crystallins are seen as tetramers in the crystal structures but their long sequence extensions are harder to visualise in the electron density of the hydrated crystal lattice structure. In one tight packing lattice of βB2 crystallin the N-terminal extension is seen to mediate protein-protein interactions between tetramers to form a 42 helix. The X-ray structure of the taxon-restricted avian δ-crystallin shows that the 50 kD subunit contains 22 helices that form three α-helical domains which dimerise followed by a dimer-dimer interaction to form a tetramer with a 20-helix bundle at the centre. Analysis of the spatial disposition of the sequence conserved regions showed the location of the active site cleft of the superfamily of enzymes related to δ-crystallin and argininosuccinate lyase. A different crystal structure of δ-crystallin solved under more physiological conditions revealed that tetramers assembled as higher order supramolecular helices and that the N-terminal extension may be involved. Combining the observations of higher order helical structures in both the oligomeric β-crystallin and δ-crystallin crystal lattices, we have proposed a highly speculative model for crystallin assembly in the lens fibre cells.

An indexed set of density bounds on lattice packings

It has long been known that the admissibility of a lattice Г with respect to a symmetric convex bodyB is equivalent to Г being a packing lattice for 1/2B. This fact is the basis of the interplay between the classical theory of the arithmetic minima of positive definite quadratic forms, on the one hand, and the dense lattice packing of spheres inR n , on the other.

Thermodynamics of Formation of Porous Polymeric Membrane by Phase Separation Method IV. Pore Formation by Contacting Secondary Particles: Computer Simulation Experiments

An attempt was made to investigate the process of forming porous polymeric membrane by computer simulation experiments based on Monte Carlo method. In this approach secondary particles (i.e., polymer-rich phase; referred to as polymer particles) and vacant particles (i.e., polymer-lean phase) are assumed to have the same radius S2 and the number ratio of those particles is predetermined by two-phase volume ratio R (≡V(1)/V(2); V(1) and V(2) are volumes of polymer-lean and -rich phases, respectively). These particles are arranged randomly on a hypothetical hexagonally closest packing lattice plane and all pores consisting of x vacant particles, which are referred to as “vacant particle pores,” are counted one by one to give probability of appearance of NP Vacant-particle pores in unit area of membrane, Q(NP), from which expected value of NP, {NP} are determined. Probability P(x) that a given pore contains x vacant particles and the average of number of vacant particles in one pore, x are also determined. In the range of R≤0.4, the theory on pore size distribution, proposed by Iijima, Matsuda, and Kamide in a previous paper, gives the same curves of P(x), NP, and x (accordingly, number-average pore radius of wet membrane, {rWet,1}) as those by the simulation, meaning that the theory holds its validity with respect to pore size distribution N(r) (r, radius of pore) and NP. Morphology and P(x) of the piled plane are shown to be practically the same as those of a hypothetical thin plane directly formed at an apparent phase volume ratio RA, evaluated from the number ratio of polymer particles and vacant particles on the surface of the piled plane by taking into consideration of overlapping. It is ascertained that computer experiments can produce membranes with morphology very similar to actual membranes and we can evaluate, by computer simulation, accurately the morphology and pore size distribution using RA, in place of R.

Thermodynamics of Formation of Porous Polymeric Membrane by Phase Separation Method III. Pore Formation by Contacting Secondary Particles: Theory and Its Comparison with Experiments

An attempt was made (1) to establish a theory of pore characteristics, including pore size distribution N(r) (r, radius of pore) for porous polymeric membranes, prepared by the phase separation method, using two-phase volume ratio R (≡V(1)/V(2); V(1) and V(2) are volumes of polymer-lean and -rich phases, respectively) and radius of secondary particle S2 and (2) to compare the N(r) calculated from R and S2 with that by an electron micrographic (EM) method. For this purpose, we assume that secondary particles (i.e., polymer-rich phase) and hypothetical vacant particles (i.e., polymer-lean phase) are placed randomly on a hexagonal closest packing lattice and that x vacant particles contact with each other to form a pore (referred to as vacant-particle pore). An expression of the probability P(x) that a given pore contains x vacant particles was derived. With consideration of an increase in pore size after drying, N(r) for vacant-particle pores, Nv(r) was derived, using R, S2 and pore density of vacant-particle pores NP (number/m2), and by translating x to pore radius r. The condition of determining NP from R and S2 Was established. N(r) for inter-polymer-particle pores, Ni(r) (i.e., crevasses of closest-packed secondary particles) was also calculated by using R and S2. The theory predicts that smaller pore size can be attained with smaller R and S2. Phase volume ratio R was found to be determined through use of a theoretical equation on porosity, using experimental porosity Pr(d4) determined from electron micrographs and approximate of degree of collapse of a membrane k′(=L0/Ld; L0, thickness of cast solution; Ld, that of dried membrane). Collapse of a hypothetical gel membrane during coagulation process explains well the findings that theoretical N(r) coincides fairly well with that by EM method only when apparent phase volume ratio RA is employed instead of R.

Electron diffraction studies of molecular ordering and orientation in phospholipid monolayer domains

The molecular order and orientation of phase separated domains in monolayers of DP(Me)PE and DP(Me)2PE were determined by electron diffraction. Dark and bright fluorescent domains at the air-water interface were observed by fluorescence microscopy. The monolayers were transferred to Formvar coated electron microscope grids for electron diffraction studies. The positions of domains on the marker grids were recorded in fluorescence micrographs, which were used as guide maps to locate these domains in the electron microscope. Selected area electron diffraction patterns were obtained from predetermined areas within and outside the dark domains. Sharp hexagonal diffraction patterns were recorded from dark domains, and diffuse diffraction rings from bright areas in between dark domains. The diffraction results indicated that the dark domains and bright areas were comprised of lipid molecules in solid and fluid states, respectively. The orientation of diffraction patterns from adjacent locations within a dark domains changed gradually, indicating a continuous bending of the molecular packing lattice vector within these domains. Orientation directors in U-shaped DP(Me)2PE domains followed the turn of the arm; no vortex nor branching was indicated by electron diffraction. Directors branching from the "stem" of highly invaginated DP(Me)PE domains usually occurred at twinning angles of n pi/3 from the stem director, which would minimize packing defects in the development of thinner branches. Electron diffraction from local areas of individual domains proved that dark fluorescent domains were solid ones, and that pseudo-long range order existed in these solid domains.

Refined crystal structure of an octanucleotide duplex with I.T. mismatched base pairs.

The structure of the synthetic deoxyoctamer d(GGIGCTCC) has been determined by single crystal X-ray diffraction techniques to a resolution of 1.7A. The sequence crystallises in space group P6(1), with unit cell dimensions a = b = 45.07, c = 45.49A. The refinement converged with a crystallographic residual R = 0.14 and the location of 81 solvent molecules. The octamer forms an A-DNA duplex with 6 Watson-Crick (G.C) base pairs and 2 inosine-thymine (I.T) pairs. Refinement of the structure shows it to be essentially isomorphous with that reported for d(GGGGCTCC) with the mispairs adopting a "wobble" conformation. Conformational parameters and base stacking interactions are compared to those for the native duplex d(GGGGCCCC) and other similar sequences. A rationale for the apparent increased crystal packing efficiency and lattice stability of the I.T octamer is given.

Effect of water on the molecular structure of a phosphatidylcholine hydrate: Raman spectroscopic analysis of the phosphate, carbonyl and carbon-hydrogen stretching mode regions of 1,2-dipalmitoylphosphatidylcholine dihydrate

The Raman spectra of 1,2-dipalmitoylphosphatidylcholine (DPPC) dihydrate crystals were examined in (a) the 1250 and 1080 cm −1 PO 2 − antisymmetric and symmetric stretching mode regions, (b) the 1730 cm −1 carbonyl stretching mode region and (c) the 2800–3100 cm −1 C-H stretching mode interval. No evidence was observed for hydrogen bonding effects involving the nonester phosphate oxygen atoms. Spectra of the DPPC dihydrate crystal in the 1250 cm −1 region are compared to spectra of both anhydrous and successively hydrated systems and to spectra of crystalline glycerylphosphorylcholine. The splitting pattern in the 1240 to 1280 cm −1 interval for the DPPC dihydrate was interpreted in terms of the different bonding arrangements assumed by the PO 2 − groups in the two molecule asymmetric unit of the eight molecule unit cell. An unusual triplet is observed in the carbonyl stretching mode region around 1730 cm −1. These features reflect differing bond environments, relative to the polar and hydrophobic regions of the bilayer, of the 1- and 2-chain carbonyl groups. Spectra for crystalline DPPC dihydrate in the 3000 cm −1 C-H stretching mode region suggest a hybrid chain packing lattice which is analogous to the packing arrangement determined for the isostructurally similar DMPC dihydrate species.

The liquid–solid transition of the hard sphere system from uniformity of the chemical potential

The search for solidlike singlet distribution functions in a system of hard spheres is undertaken. The analysis is based on the potential-distribution theory for nonuniform fluids. The ensuing requirement of constancy of the chemical potential throughout the system leads to a nonlinear integro-difference equation for the singlet density which is exact in one dimension. Bifurcations from its uniform solution, occurring before close packing and of an oscillatory nature, are found for both hard disks and hard spheres, but not for hard rods. The branching solution for hard spheres has the symmetry of a hexagonal close packing lattice, whereas that for hard disks is triangular. An investigation is also made of the stability of the fluid states. It is found that fluid instabilities and bifurcation points appear simultaneously.