Showing papers on "Cluster (physics) published in 1973"
TL;DR: Sibson gives an O(n 2) algorithm for single-linkage clustering, and proves that this algorithm achieves the theoretically optimal lower time bound for obtaining a single- linkage dendrogram.
Abstract: Main point Sibson gives an O(n 2) algorithm for single-linkage clustering, and proves that this algorithm achieves the theoretically optimal lower time bound for obtaining a single-linkage dendrogram. This improves upon the naive O(n 3) implementation of single linkage clustering. A single linkage dendrogram is a tree, where each level of the tree corresponds to a different threshold dissimilarity measure h. The nodes of a dataset are grouped into \" equivalence classes \" c(h) at each level of the dendrogram, where two classes C i and C j are merged if there is a pair of \" OTU's \" (vertices) v i ∈ C i and v j ∈ C j such that the dissimilarity measure between v i and v j is less than h, or D(v i , v j) < h. For example, consider a set of 10 vertices v 1 ,. .. , v 10 for which the dissimilarity matrix D is given below, with D ij equal to the dissimilarity between v i and v j. Suppose we take four cutoff dissimilarity measures h 1 , h 2 , h 3 , h 4 and produce the dendrogram according to these thresholds. An example illustrating how the 10 vertices are grouped into equivalence classes at each level is shown in Figure 1. Since no dissimilarity is at or below 1, each vertex or \" OTU \" is its own equivalence class at the level corresponding to h 1 = 1. At the next level, however, we see that some classes have been merged together because several dissimilarity measures are below h 2 = 2. We can see that c(h 2) consists of 6 equivalence classes, c(h 3) has 3 equivalence classes, and c(h 4 = 4) aggregates all the vertices into one equivalence class. In single linkage clustering, the number of levels in the tree is determined by the nearest-neighbor criterion – at each level, at least one new merge is made between two clusters, and the merge is made for clusters C i and C j if the minimal distance between vertices v i ∈ C i and v j ∈ C j is the smallest such distance across all the clusters. In other words, the nearest neighbors between clusters C j and C i are found, and if these neighbors are closer than all the other nearest-neighbor pairs, then C i and C …
1,208 citations
TL;DR: In this article, the nucleation and growth of crystals on a substrate are discussed in terms of rate equations for the atom cluster concentrations as a function of time and simple approximations allow this general set of equations to be reduced to three coupled equations.
Abstract: The nucleation and growth of crystals on a substrate are discussed in terms of rate equations for the atom cluster concentrations as a function of time. Simple approximations allow this general set of equations to be reduced to three coupled equations. Many physical processes can be incorporated into these rate equations, including coalescence of clusters, and cluster mobility. The problem of increasing correlation between single atoms and stable clusters as growth proceeds is studied. It is shown that the problem can be solved self-consistently using an auxiliary diffusion equation and that approximations may be obtained which give upper and lower bounds for the cluster growth rates. These diffusion equations also give expressions which enable the cluster-cluster correlations and cluster size distributions to be discussed. With these approximations, expressions are derived for observable quantities and the expressions are compared with one experimental example. In this case, that of gold on alka...
542 citations
TL;DR: In this article, a formal physical cluster theory for an imperfect gas, valid for an arbitrary definition of a ''physical cluster'' is described and the role of the definition of the physical cluster is stressed.
Abstract: A formal physical cluster theory for an imperfect gas, valid for an arbitrary definition of a ``physical cluster,'' is described The role of the definition of the physical cluster is stressed For a particular definition of the physical cluster, which may be appropriate in nucleation theory, the Helmholtz free energy of 13‐, 43‐, 60‐, 70‐, 80‐, 87‐, and 100‐atom argon clusters is calculated in the classical limit for temperatures ranging from absolute zero to 100°K using Monte Carlo techniques It is found that a cluster's free energy is almost independent of its definition provided that the definition is reasonable and the temperature is sufficiently low The results are compared with the predictions using the harmonic approximation
378 citations
194 citations
TL;DR: In this article, the Gibbs free energy of a cluster of 15, 30, 45, 60, 80, and 100 argon atoms at temperatures of up to 75°K is compared with those obtained from the microcrystal model.
Abstract: Molecular dynamics calculations have been performed on clusters of 15, 30, 45, 60, 80, and 100 argon atoms at temperatures of up to 75°K. Values of the independent‐cluster thermodynamic functions are presented and compared with those obtained from the microcrystal model. This comparison indicates good agreement for values of the Gibbs free energy of formation. The transition from solidlike to fluidlike diffusion in the cluster occurs gradually; no semblance of a phase transition is noted. The radial variation of density in the clusters have maxima and minima reminiscent of those in the radial distribution function for bulk liquids. The temperature dependence of that function indicates that clusters expand quite rapidly as the temperature is increased. The radial distribution of potential energy indicates that there is no region inside the clusters where the environment resembles that in the bulk phases; the properties of the clusters are dominated by the ``surface region'' in which nearly all the atoms exist.
168 citations
126 citations
TL;DR: In this paper, an electron diffraction study of extremely small clusters of argon atoms formed by homogeneous nucleation in an expansion nozzle system is presented, in which the clusters yielded good quality Debye-Scherrer patterns under conditions essentially free from all substrate interactions.
Abstract: We present an electron diffraction study of extremely small clusters of argon atoms formed by homogeneous nucleation in an expansion nozzle system. The clusters, in general comprising less than 1000 atoms, yielded good quality Debye‐Scherrer patterns under conditions essentially free from all substrate interactions. The high precision of the diffraction patterns, even at large angles, makes it possible to assign an effective fcc structure to the particles. However, an anomalously large 111 peak is observed consistently and may be interpreted as resulting from vestiges of noncrystalline structure exhibited by the aggregates in the early stages of their growth.
118 citations
105 citations
TL;DR: In this paper, the authors analyzed the X-ray data for the extended sources in Coma, Perseus, and Virgo in terms of an isothermal gas-sphere model emitting thermal-bremsstrahlung radiation.
Abstract: Analysis of the X-ray data for the extended sources in Coma, Perseus, and Virgo in terms of an isothermal gas-sphere model emitting thermal-bremsstrahlung radiation. It is found that the gas distribution is somewhat less centrally condensed than the galaxies, and that the gas mass is generally a small fraction of the binding mass of each cluster. In the case of the Coma cluster, approximately one-eighth of the binding mass (with an uncertainty of about 50%) may be in the form of hot gas at 100,000,000 K if H sub zero = 50 km/sec/Mpc.
90 citations
TL;DR: In this paper, the dependence of cluster ion intensities on the bombarding gases He+, Ar+ and Xe+ (energies: 4 to 12 keV, current densities: 100 mA/cm2) was investigated.
Abstract: This paper reports previously unpublished results which were obtained in 1966. We systematically investigated the dependence of cluster ion intensities on the bombarding gases He+, Ar+ and Xe+ (energies: 4 to 12 keV, current densities: 100 mA/cm2). Frequently, the observed structures in the relative cluster intensities were quite puzzling, e.g. for A1 and Si. Attempts to correlate these structures to crystal configurations failed, nor did any pattern develop from simple valency considerations alone. Initial ion energy distribution measurements from 0 to 1200 eV showed significant differences for atomic ions and cluster ions. This effect is used to reduce interference problems caused by cluster ion peaks in SIMS applied to trace analysis of solids. The results are discussed and compared with those of other investigators, also including cluster formation by vaporization and sparking. Extending known theoretical considerations may possibly afford a general understanding of the intensity structure. T...
TL;DR: Extended Huckel and complete neglect of differential overlap (CNDO) molecular orbital calculations were performed for clusters of metallic atoms to determine possible catalytic properties as discussed by the authors, which indicated the presence of d-band holes in Ni and Pd clusters with a greater amount in Ni.
TL;DR: In this article, a high energy electron beam of 39.5 keV was used to obtain Debye-Scherrer diffraction patterns from carbon dioxide and water clusters and the largest size clusters produced are solid and have the structure of the bulk material which is simple cubic for CO2 and diamond cubic for H2O.
Abstract: A nozzle molecular beam is operated so as to optimize the molecular clustering in the free jet expansion. Continuous beams of carbon dioxide and water are sampled with an ionization gauge detector. The increased beam intensity under some operating conditions can only be attributed to clustering. A high energy electron beam of 39.5 keV is used to obtain Debye‐Scherrer diffraction patterns from carbon dioxide and water clusters. The largest size clusters produced are solid and have the structure of the bulk material which is simple cubic for CO2 and diamond cubic for H2O. The average diameters for CO2 and H2O are 52±5 and 54±5 A, respectively, corresponding to 1600 and 2600 molecules per cluster. The CO2 data are in agreement with other beam results and, to the authors' knowledge, these are the first data published on the structure of water clusters formed from the vapor phase via homogeneous nucleation. Based on the diffraction data the structure of clusters down to average sizes in the range of 300–450 molecules per cluster can be treated as bulk phase.
TL;DR: In this article, a special version of time-of-flight mass spectrometry was used to study hydrogen and nitrogen clusters originating from condensing supersonic nozzle flows, and the molecular weights of the clusters to be investigated range from 103 to more than 106 and are determined from the change of the cluster speed which is effected by a longitudinal electric field after ionization of clusters by electron impact.
Abstract: Hydrogen and nitrogen clusters originating from condensing supersonic nozzle flows are studied by a special version of time‐of‐flight mass spectrometry. The molecular weights of the clusters to be investigated range from 103 to more than 106 and are determined from the change of the cluster speed which is effected by a longitudinal electric field after ionization of the clusters by electron impact. Clusters with up to four positive charges can be distinguished, and the probability for multiple ionization as a function of the electron energy is given for hydrogen clusters containing 6.5 × 104 molecules and for nitrogen clusters containing 1.2 × 104 molecules. Formation of singly charged negative cluster ions of both gases is observed for the first time and becomes more probable with increasing cluster size. While neither fragmentation nor evaporation as a result of the ionization process is observed, beams containing two fractions of masses prior to ionization are found at certain conditions of the cluster beam generation. Except for such cases, the distribution of sizes of singly charged clusters can be derived if the measured narrow speed distribution of the cluster beam is assumed to be valid also for each separate cluster size.
TL;DR: In this paper, the authors describe an integration method which is used to calculate the orbits of individual members subject to their mutual attractions and a simple regularization procedure is outlined, which can be used to study collision orbits as well as perturbed two-body motion with increased speed and accuracy.
TL;DR: In this article, two difference equations are proposed, that could be considered as most general forms of the Thomson-Gibbs formula relating the equilibrium vapor pressure of small phases with their dimension.
Abstract: Following Mayer's and Frenkel's statistical treatment of condensation, two difference equations are proposed, that could be considered as most general forms of the Thomson‐Gibbs formula relating the equilibrium vapor pressure of small phases with their dimension. The first equation (18) assumes the knowledge of the complete ``cluster integral'' for particles of each size. The second one (19) takes into account the internal partition function of the most stable configuration only. Both equations, being discontinuous, give supersaturation ranges in which particles could be considered as critical nuclei in the condensation process. The computer calculation of the cluster integrals for (2‐ to 14‐atomic) particles of a monatomic substance with central forces, using the Lennard‐Jones 6–12 potential, shows a negligible difference between the numerical values of the supersaturations obtained by Eqs. (18) and (19). On the other hand, from the comparison of these data with the Thomson‐Gibbs formula for the same model, it follows that the latter is a fairly good approximation even for particles of only a few atoms. A discussion of the most stable structure of small particles and of the smoothing effect of their entropy terms on the vapor pressure curve, enables us to affirm that such particles have a pronounced liquid character even at low temperatures.
TL;DR: In this paper, an atomistic calculation of the free energies of small clusters having 3 to 87 atoms has been carried out and the most probable configurations can be described by a simple analytical expression and differ from the predictions of the classical capillarity model.
Abstract: An atomistic calculation of the free energies of small clusters having 3 to 87 atoms has been carried out. The free energies of the most probable configurations can be described by a simple analytical expression and differ from the predictions of the classical capillarity model. We also examine the importance of considering all possible configurations of a cluster when computing thermodynamic properties. Nucleation rates are calculated from the exact thermodynamic properties and compared with the predictions of classical nucleation theory. Exact rates differ from the classical rates by factors from 10–2 to 107. The correction depends strongly on both supersaturation and temperature.
TL;DR: In this article, the abundance of different clusters is found to rise to a flat maximum, at optimum source conditions for a specific cluster size, reaching values of about 10% of the H2+ signal without a strong tendency to decrease with increasing mass.
TL;DR: The thermodynamic properties of small clusters of atoms and molecules are of interest for nucleation theory as discussed by the authors, and several researchers have examined in detail the properties of these small clusters, including the phase diagram.
Abstract: The thermodynamic properties of small clusters of atoms and molecules are of interest for nucleation theory1–7. Recently several investigators have examined in detail the properties of small clusters8–19. A particularly important problem for small clusters is to determine the phase diagram.
TL;DR: In this paper, the anisotropy of positively charged clusters sputtered with 150 keV Ar+ ions from a tungsten single crystal has been investigated with a mass and energy spectrometer.
Abstract: The anisotropy of positively charged clusters sputtered with 150 keV Ar+ ions from a tungsten single crystal has been investigated with a mass and energy spectrometer. Not only the monomer W+ but also the clusters W+ 2, W+ 3 and W+ 4 (so far investigated) show maxima of emission along close packed directions. The maximum cluster yield along a direction normalized to the yield along a random direction increases with increasing cluster size and cluster energy, although the absolute cluster yield decreases. An interpretation of the experimental results on the basis of a simple statistical model of cluster formation kinetics is presented which explains the yield profiles of the anisotropic contribution fairly well.
TL;DR: In this article, the model of the center as deduced from the experiments yields a highly covalent almost planar cluster involving the impurity cation and four Cl- neighbors.
TL;DR: In this article, essential features of the band structure of a covalent crystal, as silicon, emerge in the level scheme of a cluster calculation utilizing the SCF Xα scattered wave method.
TL;DR: In this article, a general exact law of mass action which relates concentrations and activity coefficients of physical clusters to concentration-independent equilibrium constants is derived, and further expansions are obtained by the methods of the Mayer ionic solution theory.
Abstract: The configurations of point defects in a solid can be uniquely classified in terms of interacting physical clusters. Two point defects are counted as in the same physical cluster if their separation is less than or equal to some characteristic distance. The methods of Mayer cluster theory are used to derive a general formal exact law of mass action which relates concentrations and activity coefficients of physical clusters to concentration‐independent equilibrium constants. The activity coefficients and distribution functions for physical clusters are obtained as series expansions in powers of physical cluster concentrations. From these results further expansions, individual terms of which converge even for ionic crystals, are obtained by the methods of the Mayer ionic solution theory. The relationship of these results to the Frenkel‐Band theory of association in imperfect gases and to the Lidiard‐Teltow theory of defect interactions in ionic crystals is briefly indicated and possible applications of the formalism suggested. A detailed application is given in the accompanying paper.
TL;DR: In this paper, the electrical resistivity of spherical clusters of vacancies (voids) in the three simple metals sodium, magnesium and aluminium has been calculated by two different methods, one based on phase shift scattering and the other on pseudopotentials.
Abstract: The electrical resistivity of spherical clusters of vacancies (voids) in the three simple metals sodium, magnesium and aluminium has been calculated by two different methods, one based on phase shift scattering and the other on pseudopotentials. In the phase shift method the clusters were represented by a spherical potential well, whose radius was adjusted to give a volume equal to that of the vacancies in the cluster and whose depth was determined so as to satisfy the Friedel sum rule for the phase shifts. In the other method, local screened pseudopotentials, situated at the vacant lattice sites, were used and the cross section of the cluster was determined from perturbation theory. The absolute values of the resistivity found by the two methods were in close agreement for magnesium and aluminium, though there was a substantial discrepancy in the case of sodium. Similar calculations for spherical interstitial clusters were also performed, although such clusters have not been observed experimentally. The results have the same general features as for the voids.
TL;DR: In this article, a scanning electron microscopy study of aluminum deoxidation products in molten pure iron was conducted, and the results obtained were summarized as follows: well developed dendritic alumina inclusions were mainly observed in unstirred iron bath, and arms tended to be cut off from the stems.
Abstract: Aluminum-deoxidation products in molten pure iron were investigated by a scanning electron microscopy. The results obtained are summarized as follows. (1) Well developed dendritic alumina inclusions were mainly observed in unstirred iron bath, and arms of those dendritic alumina tended to be cut off from the stems. (2) D eoxidation productsfrom stirred bath mainly consisted of spherical alumina particles. (3) The neck growth of alumina inclusions observed was succesifully explained by the introduction of the sintering theory of solid particles.