Critical Number Of Atoms Research Articles

Atomic soliton reservoir: How to increase the critical atom number in negative-scattering-length Bose-Einstein gases

We study the stabilization of a large negative scattering length Bose-Einstein condensate by using a multisoliton fringe system. Each fringe behaves like a quasiparticle and the whole cloud forms a stable system when the trapping in one or two directions is switched off. The resulting system can be used as a coherent source of atomic solitons. A discussion on how to generate the soliton fringe system experimentally is made.

Impact of spontaneous spatial symmetry breaking on the critical atom number for two-component Bose-Einstein condensates

The consequences of spatial symmetry breaking in two-component Bose-Einstein condensates are explored for the ${}^{85}{\mathrm{R}\mathrm{b}+}^{87}\mathrm{Rb}$ system. It is shown that the mean-field critical number of ${}^{85}\mathrm{Rb}$ atoms depends strongly on whether the single-particle wave function is symmetry preserving or symmetry breaking. This system thus provides a uniquely straightforward experimental means of observing spatial symmetry breaking.

Critical number of atoms for the magnetically trapped Bose-Einstein condensate with negative s-wave scattering length

We predict the collapse of the Bose-Einstein condensate in a magnetic trap where the atoms have a negative s-wave scattering length. To describe the time evolution of the condensate, we employ the Gross-Pitaevskii equation. Generalizing the Zakharov theory, we prove that the singularity of wavefunction emerges in a finite time even when the total energy of the system is positive. In addition, we present an improved formula for a critical number of atoms above which the collapse of the condensate occurs. This number for 7Li atoms can be the same as the one in the recent experiment.

Effective potentials for dilute Bose-Einstein condensates

We present a theoretical formulation of trapped, dilute Bose-Einstein condensates (BEC's) at zero temperature based on ordinary Schr\"odinger quantum mechanics. By a judicious choice of coordinates and of a variational trial wave function we reduce the many-atom problem to a linear Schr\"odinger equation that is easier to handle and interpret than the usual nonlinear Schr\"odinger equation of BEC theory. Ordinary quantum mechanics then reproduces, semiquantitatively, many of the main features of zero-temperature BEC, including the critical number of atoms in a condensate with negative scattering length. The procedure is similar in results, but completely different in spirit, to recent variational approaches to solving the nonlinear Schr\"odinger equation. Moreover, the present procedure represents a step in a systematic alternative method for computing quantitatively accurate wave functions for trapped bosonic atoms.

Bose-Einstein condensation in an atomic gas with attractive interactions

The behavior of a trapped atomic gas undergoing a Bose-Einstein condensation is strongly influenced by the attractive interaction between particles. The role played by such an interaction is analyzed by means of a variation method. The critical number of atoms for a metastable state to exist is determined. The aspect ratio of condensed bosons in an anisotropic trap, the loss rate of atoms, and the excitation spectrum are studied. The vortex state is discussed.

Bound electron states in clusters of inert atoms in a magnetic field

An external magnetic field is shown to stabilize negatively charged clusters of inert atoms. In an external magnetic field the critical number of atoms in a cluster, necessary to bound an excess electron to a neutral cluster, is less than that in the absence of a field. General conditions for creation of electron bound states in a cluster in a magnetic field are considered. PACS: 36.40. +d; 31.20.Ej; 33.80.-b We witness an increased interest in the study of charged clusters of inert gases [1 ‐ 3]. The critical number of atoms at which an electron can be trapped by a cluster were found in [2] for ( Ar) n , (Kr) n , (Xe) n , clusters. These charged clusters are difficult to observe near the critical size because the electron wave function is highly diffuse and the bound state energy is close to zero. The formation of deeper electron levels requires clusters with the number of atoms by an order of magnitude greater than their critical values. In this work electron states are examined in the clusters of inert gases placed in a magnetic field. In the case of an external magnetic field the electron level becomes deeper, thus, the charged cluster is stabilized and the critical number of atoms in a cluster necessary to bound an electron is less than that without a magnetic field. According to the simple continuum model [2], the potential energy of an electron in a cluster is defined by the relation

Dislocation structure formation due to ion implantation

A mechanism of dislocation structure formation and its dynamics is presented. Coalescence of the interstitial atoms leads to the formation of dislocation loops. The formed loops grow due to precipitation of the interstitial atoms. Interaction between loops results in their motion deep into material, where loops form a developed dislocation structure. 1. There are many experimental evidence of an influence of ion bombardment on the material structure and properties in the region disposed much deeper than the ion penetration depth. One of this long range effects is the dislocation structure formation at a depth up to about 100 microns.'-4 A number of mechanisms explaining that phenomenon has been suggested. However, a general theory of dislocation structure formation has not been found yet. A mechanism of dislocation structure formation and its dynamics in the implanted specimen is considered in the present paper. It is suggested that the coalescence of the interstitial atoms emerging due to ion bombardment results in the creation of dislocation loops. Then the loops move deep into the specimen because of interaction with each other. Collisions between the loops result in pinning and formation of immobile dislocation structure. 2. Under irradiation a lot of interstitial atoms and vacancies appear in the material. They are created in a narrow region at the ion penetration depth and then diffuse deep into the sample. The diffusion coefficient of interstitial atoms, D, is much greater than that of vacancies, D,. Therefore, at the depth z > (D\ty5 the concentration of interstitial atoms, c, exceeds that of vacancies, c,. This is the condition for the preferable formation of interstitial dislocation loops and their growth. The loops are formed by means of fluctuation. The critical number of atoms in the loop is equal to7

Numerical simulation of cluster growth during the transient nucleation period in melt-spinning

Classical nucleation theory proposes that first order phase transformations start with the formation of small, unstable clusters (embryos) of the new phase. Eventually, embryos grow to a critical size, beyond which they are stable, i.e. they become nuclei. The nucleation temperature T N and time t N are where and when a nucleus reaches a critical size, i.e. contains a critical number of atoms. A steady state rate of formation of nuclei per unit volume of the untransformed phase is reached after a “transient” or “induction” time. Various theoretical modellings of solidification calculate T N using analytical expressions for the steady state nucleatioin kinetics. In rapid quenching processes, such as the one that occurs in melt-spinning, the “transient” period cannot be ignored or approximated. A numerical simulation method is proposed for the reactions involved in embryo cluster formation and growth, at the atomic level. The simulation is adapted to the particular flow characteristics of the liquid in melt-spinning. It starts at the liquidus temperature, then goes through the “transient” period for nucleation, and ends when the critical number of critical nuclei is formed, at the nucleation temperature T N, and the amount of undercooling is deduced. The simulation enables the calculation of the nucleation temperature for each alloy considered. Values for the nucleation parameters appear to depend primarily on the number of atoms available, for embryo clusters to grow, on the liquid surface in isothermal contact with the substrate. Results for Al ribbons are compared with the analysis of melt-spinning by steady state nucleation kinetics.

Relationships between phase stability and void swelling in Fe-Cr-Ni alloys during irradiation

Austenitic Fe-Cr-Ni steels are potential candidate alloys for structural materials in both fast breeder and magnetic fusion reactors. However, void swelling and phase instability during irradiation have been major problems limiting the properties and useful lifetimes of these materials and thus have been the subject of intensive investigations. Cavity nucleation in steels subjected to displacement damage is strongly influenced by the interactions between helium atoms and precipitates which are formed during irradiation. Several important mechanisms regarding gas atom-precipitate interactions and principles for the design of radiation-resistant alloys are critically examined. Central concepts derived from theory, including the critical radius and critical number of gas atoms, cavity-precipitate interactions, and relative sink strengths for point defects, are applied to the interpretation of experimental data.

Molecular geometry and chain entanglement: parameters for the tube model

AbstractThe tube diameter in the reptation model is the distance between a given chain segment and its nearest segment in adjacent chains. This dimention is thus related to the cross‐sectional area of polymer chains and the nearest approach among chains, without effects of thermal fluctuation and steric repulsion. Prior calculated tube diameters are much larger, about 5 times, than the actual chain cross‐sectional areas. This is ascribed to the local freedom required for mutual rearrangement among neighboring chain segments. This tube diameter concept seems to us to infer a relationship to the corresponding entanglement spacing. Indeed, we report here that the critical molecular weight, Mc, for the onset of entanglements is found to be Mc = 28 A/(〈R2〉0/M), where A is the chain cross‐sectional area and 〈R2〉0 the mean‐square end‐to‐end distance of a freely jointed chain of molecular weight M. The new, computed relationship between the critical number of backbone atoms for entanglement and the chain cross‐sectional area of polymers, Nc = A0,44, is concordant with the cross‐sectional area of polymer chains being the parameter controlling the critical entanglement number of backbone atoms of flexible polymers.

Pre-amorphization damage in ion-implanted silicon

Ion implantation in silicon with doses below the amorphization threshold can lead to the formation of dislocations after high-temperature annealing. We have studied this for implants of 0.1–1 MeV B, Si, P, Ga, As, In, and Sb ions after annealing at 900°C using cross-sectional transmission electron microscopy. Pre-amorphization damage, also called category I dislocations, is observed if the total number of silicon atoms displaced by the implant exceeds a critical value before reaching the threshold dose for amorphization. These dislocations are of interstitial type and result from agglomeration of mobile silicon interstitials. The critical number of displaced Si atoms required for pre-amorphization damage formation increases with the mass of the implanted species and was determined by Rutherford backscattering spectrometry and channeling analysis to range from 1.5 × 10 16/cm 2 for B ions to (1.5-2) × 10 17/cm 2 for Sb ions. This increase with mass is attributed to an increasing collision cascade density resulting in a lower fraction of the measured damage being in the form of mobile Si interstitials needed for dislocation formation. In contrast to keV implants, category I defects are observed for high-mass species at MeV energies because the critical number of mobile interstitial silicon atoms is reached prior to the amorphization threshold. The critical number can be used to manipulate secondary defect formation. First, introducing a second damage profile can influence where the secondary defects form. Results are presented for MeV B or As implants in combination with low-energy Si irradiations. Depending on the separate implant parameters the position where secondary defects form can be influenced. Second, a comparison between channeling and random implants of B or P ions in Si(100) wafers shows that higher doses can be reached without formation of secondary defects by channeling implants than by random implants due to the lower amount of damage produced by a channeled ion. In either case, secondary defect formation is observed after high-temperature (900°C) annealing only if the total number of displaced Si atoms exceeds a critical value of ∼ 1.5 × 10 16/cm 2 and ∼ 5 × 10 16/cm 2 for the B and P implants, respectively. Third, higher total doses can be introduced without forming secondary defects by repetitive subthreshold implants each followed by an anneal to remove the implant damage. While a single 6 × 10 13 In/cm 2 implant results in a high density of dislocation loops after annealing, we demonstrate that instead using four separate 1.5 × 10 13 In/cm 2 implants each followed by an anneal leads to the formation of only a few partial dislocations. Pre-amorphization damage formation and annihilation is shown to influence transient tail diffusion of B. This has been investigated as a function of B implant condition, dose, energy, time, temperature, and as a function of further Si or Ge implants. Significantly longer transient tail diffusion is observed for B implants along [100] than for random implants reflecting the differences between the random and channeling implants in the position of the damage distribution relative to the B profile. A second implant with 1 MeV 29Si ions below the amorphization threshold can significantly reduce B tail diffusion if the damage for the Si dose is high enough to form pre-amorphization damage during the anneal. Lower Si doses do not influence B diffusion. Annealing of extended defects results in anomalous diffusion as well. These results demonstrate that Si interstitials cause the enhanced B diffusion. Transient B tail diffusion is completely prevented only if the B-implanted silicon is amorphized so that the B profile is completely incorporated in the amorphized region.

Theoretical basis for unified analysis of experimental data and design of swelling-resistant alloys

Essential aspects of the theory of radiation-induced swelling are reviewed. In particular, concepts central to the understanding of experimental data and the control of swelling by alloy design are discussed. The knowledge that a critical number of gas atoms is required in a cavity before point defect-driven swelling can begin is a most important contribution of theory. The mathematical expressions that have been derived for calculating this critical quantity are given in terms of materials parameters and irradiation conditions. After swelling begins, its magnitude as a function of dose is governed strongly by the relative sink strengths for point defects of dislocations and cavities, expressed in terms of an index of the microstructure, Q. High swelling and low swelling microstructures can be categorized into four types based on this index. Wide ranges of experimental swelling results covering ferritic/martensitic and austenitic alloys, neutron and ion irradiations and a variety of compositions and irradiation conditions are analyzed and found to be explained consistently within this framework. Based on the understanding gained, approaches to alloy design for swelling resistance are recommended.

Collective oscillations in Rydberg atomic systems

The macroscopic oscillatory regime obtained for a Rydberg atomic system in high- Q cavities is discussed from a phase transition point of view. The path integral method is used to demonstrate the equilibrium phase transition. It is shown that the exhibition of the phase transition is equivalent to the occurrence of an oscillatory regime in the temporal evolution of the system, expressed by a necessary critical number of atoms in the cavity.

Mechanisms of swelling suppression in cold-worked phosphorous-modified Fe-Ni-Cr alloys

Abstract Three cold-worked phosphorous-modified Fe-15Ni-13Cr-base alloys were irradiated with simultaneous Fe- and He-ion beams to doses of about lOOdpa at a helium implantation rate of 20at.p.p.m./dpa. The three alloys correspond to compositions that led to increasing densities of a very fine phosphide precipitate. In the case of the alloy with the highest precipitate density, the swelling was suppressed to the greatest extent. The results are interpreted in terms of the theory of cavity swelling, which provides the central concept of a critical radius and critical number of gas atoms required for cavity swelling. The large interfacial area of the needle-shaped phosphide precipitates led to profuse bubble nucleation at the interface. The high density of bubbles slowed the accumulation of gas atoms in individual bubbles and thus delayed the onset of point-defect-driven cavity swelling. After interpreting the present data in terms of these concepts, more general features of cavity swelling reduction by an ...

Unified theoretical analysis of experimental swelling data for irradiated austenitic and ferritic/martensitic alloys

Central concepts in the theory of swelling based on defect reactions are reviewed. The critical radius and critical number of gas atoms, which determine the initiation of swelling, and the ratio of the dislocation and cavity sink strengths, which dictates the swelling rate, are demonstrated to be of great utility in the understanding and control of swelling. Over the past two decades, a large data base has been accumulated covering austenitic and ferritic/martensitic alloys, the leading candidate materials for both fusion and fast fission reactor applications. This collection of data naturally serves as the largest source of information on which to develop and test mechanistic understanding of swelling. Over wide ranges in materials and irradiation parameters, including composition, temperature, damage rate, and helium generation rate, we find that apparently divergent swelling behaviors can be explained in a unified manner within the present theoretical framework. Principles for microstructures that insure swelling resistance, together with results from the necessary confirmatory experiments, are described.

Effect of residual and injected oxygen on swelling in irradiated Fe-Ni-Cr alloys

Abstract Mechanisms by which oxygen affects swelling during Fe-ion irradiation are investigated in a series of Fe-15Ni-13Cr-base alloys containing minor additions of titanium, silicon and carbon. The irradiations are carried out at 948 and 1000 K to doses up to 100 dpa. Effects of both residual oxygen content and accelerator-injected oxygen are studied. It is found that pre-injected oxygen promotes cavity swelling and, in some cases, leads to a fine distribution of stable bubbles for oxygen injection levels as high as 1000 at.p.p.m. The results are interpreted in terms of the chemical activity of oxygen in forming precipitates with alloying elements, and in terms of radiation dissolution of oxygen leading to oxygen-pressurized bubbles. Accounting for the interaction with alloying elements, we conclude that oxygen exhibits a direct cavity-pressurization effect on swelling similar to that previously reported for helium. The results are analyzed in terms of the theory of the critical radius and critical number of gas atoms, together with Gibbs free energies or precipitate formation.

Unified theoretical analysis of experimental swelling data for irradiated austenitic and ferritic/martensitic alloys

Central concepts in the theory of swelling based on defect reactions are reviewed. The critical radius and critical number of gas atoms, which determine the initiation of swelling, and the ratio of the dislocation and cavity sink strengths, which dictates the swelling rate, are demonstrated to be of great utility in the understanding and control of swelling. Over the past two decades, a large data base has been accumulated covering austenitic and ferritic/martensitic alloys, the leading candidate materials for both fusion and fast fission reactor applications. This collection of data naturally serves as the largest source of information on which to develop and test mechanistic understanding of swelling. Over wide ranges in materials and irradiation parameters, including composition, temperature, damage rate, and helium generation rate, we find that apparently divergent swelling behaviors can be explained in a unified manner within the present theoretical framework. Principles for microstructures that insure swelling resistance, together with results from the necessary confirmatory experiments, are described.

Comparison of retention mechanisms of homologous series and triglycerides in non-aqueous reversed-phase liquid chromatography

The retention behaviour of saturated homogeneous triglycerides was compared to that of single chain homologous compounds in reversed-phase liquid chromatography on silica-based octadecyl bonded phases with non-aqueous binary eluents containing chloroform. For both series, in all eluents and at all investigated temperatures, a discontinuity (1–6% slope change) in log k′ vs. the carbon atom number, n c, of each solute hydrocarbonaceous chain was observed for a given critical number of carbon atoms, n c, crit ≈ 12–13. Similar discontinuities were observed in Δ H° and Δ S° vs. n c for the same n c, crit value. These and other phenomena (existence of two convergence points for log k′ vs. n c plotted at different eluent compositions; existence of two convergence temperatures in log k′ vs. 1/T plotted for several members of a given solute series) reflect the mechanism of penetration of the solute aliphatic chains into the bonded layer. The eluotropic strength of acetonitrile—chloroform mobile phases was determined and seen to have a larger rate of variation with the solvent composition than that of some other non-aqueous binary eluents. The ratio of the slope of plots of log k′ vs n c curves for triglycerides and single chain homologous series is not a whole number but has a value lying between 2 and 3. This explains why triglyceride retentions cannot be predicted from retention data for the fatty acid methyl esters. More importantly, this indicates that the triglycerides may interact with the stationary phase using various conformations, one, two or three chains penetrating into the bonded layer. These conformations, in dynamic equilibrium with each other, contribute differently to the retention. This offers, in principle, the possibility to separate unsaturated triglycerides having the same number of carbon atoms and of double bonds but differing in the distribution of the unsaturations along the chains, if the double bonds located in a non-penetrating chain can selectively interact with a mobile phase component.

ChemInform Abstract: C‐H···O Interactions and the Adoption of 4 Å Short‐Axis Crystal Structures by Oxygenated Aromatic Compounds

Oxygen atoms, pendant as substituents or occurring within the ring systems of planar aromatic molecules, have a pronounced tendency to direct crystallisation patterns of such compounds to the β-structure, characterised by a 4 A short axis. They seem to perform this function by stabilising a critical number of hydrogen atoms, covalently bonded to carbon, through short and directionally specific intermolecular C–H ⋯ O hydrogen bonds. Consequently, the number of ‘free’ hydrogen atoms which contribute to crystal stabilisation through non-β steering C ⋯ H interactions is reduced. Both these factors result in the formation of C–H ⋯ O stabilised two-dimensional entities such as sheets and ribbons. Such entities may be stacked at 4 A translational separation to generate the entire structure. These concepts are illustrated for some methylenedioxy and alkoxy aromatic systems, quinones, and heterocycles. However, both intra- and inter-sheet C–H ⋯ O interactions may sometimes be present and the unusual crystal structure of 7-acetoxycoumarin (5) shows how a significantly non-planar molecule may still adopt the β-structure if it is particularly well suited for the formation of C–H ⋯ O bonds. Yet 4-acetoxycinnamic acid (6), the crystal structure of which was determined in this work and which has almost the same C:H:O ratio as (5), adopts a non-β structure because the number of oxygen atoms available for C–H ⋯ O bond formation is greatly reduced. The crucial role of the number of such ‘available’ oxygen atoms and ‘H-bonded’ and ‘free’(C–)H atoms vis-a-vis the carbon content is exemplified by the β-steering behaviour of oxygen in some large fused-ring quinones and heterocycles. These trends may also be extended to nitrogen and sulphur heterocycles.

C–H ⋯ O interactions and the adoption of 4 Å short-axis crystal structures by oxygenated aromatic compounds

Oxygen atoms, pendant as substituents or occurring within the ring systems of planar aromatic molecules, have a pronounced tendency to direct crystallisation patterns of such compounds to the β-structure, characterised by a 4 A short axis. They seem to perform this function by stabilising a critical number of hydrogen atoms, covalently bonded to carbon, through short and directionally specific intermolecular C–H ⋯ O hydrogen bonds. Consequently, the number of ‘free’ hydrogen atoms which contribute to crystal stabilisation through non-β steering C ⋯ H interactions is reduced. Both these factors result in the formation of C–H ⋯ O stabilised two-dimensional entities such as sheets and ribbons. Such entities may be stacked at 4 A translational separation to generate the entire structure. These concepts are illustrated for some methylenedioxy and alkoxy aromatic systems, quinones, and heterocycles. However, both intra- and inter-sheet C–H ⋯ O interactions may sometimes be present and the unusual crystal structure of 7-acetoxycoumarin (5) shows how a significantly non-planar molecule may still adopt the β-structure if it is particularly well suited for the formation of C–H ⋯ O bonds. Yet 4-acetoxycinnamic acid (6), the crystal structure of which was determined in this work and which has almost the same C:H:O ratio as (5), adopts a non-β structure because the number of oxygen atoms available for C–H ⋯ O bond formation is greatly reduced. The crucial role of the number of such ‘available’ oxygen atoms and ‘H-bonded’ and ‘free’(C–)H atoms vis-a-vis the carbon content is exemplified by the β-steering behaviour of oxygen in some large fused-ring quinones and heterocycles. These trends may also be extended to nitrogen and sulphur heterocycles.