Two-point Resolution in Partially Coherent Light

This paper is an extension of an earlier paper to cover the study of the effects of the annular source of illumination in ordinary microscopy on the resolution of general two-point targets. The limit of resolvable separation has been obtained, = (K _s + j)u/sin f, where K _s + 0·446 + 0·32 o gives the Sparrow limit when the illumination is such as to yield a degree of coherence, o, between the two particles at the limit of resolution. It is also seen that this resolution does not change significantly with the width of the annular source having a fixed mean radius.     

LIMITATIONS ON THE USE OF THE STRESS INTENSITY FACTOR, K, AS A FRACTURE PARAMETER IN THE FATIGUE PROPAGATION OF SHORT CRACKS

Abstract— It is well known that for very short cracks the stress intensity factor K is not a suitable parameter to estimate the stress level over the small but finite Stage II process zone activation region of size r_s near the crack tip, within which crack growth events take place. A critical appreciation of the reasons for the limitations on the applicability of ΔK as a fatigue crack propagation (FCP) parameter, when the crack length a is of the same order of magnitude or smaller than the size of the ‘fatigue-fracture activation region’, r_s is presented. As an alternative to ΔK the range Δσ_s of the cyclic normal stress at a point situated at the fixed distance s=r_s/2, ahead of the crack tip, inside the fatigue-fracture activation region, is proposed. It is observed that the limitation on the use of ΔK when the crack is short, is mathematical (and not physical) but this inconvenience is easily circumvented if the stress Δσ_s at the prescribed distance is used instead of ΔK since nowadays Δσ_s can be obtained numerically by using finite element methods (FEM). It follows that the parameter Δσ_s is not restricted by the mathematical limitations on ΔK and so it would seem that there is, a priori, no reason why the validity of the parameter Δσ_s cannot be extended to short cracks. It is shown that if the Paris law is expressed in terms of Δσ_s (πrr_s)^½ instead of ΔK the validity of the modified Paris law can be extended to short cracks. A coherent estimate of the value of the fatigue-fracture activation region r_s is derived in terms of the fatigue limit Δσ_FL obtained from S-N tests and of the threshold value ΔK_th obtained from tests on long cracks where both relate to Stage II crack growth that ends in failure, namely, r_s= (ΔK_th/Δσ_FL)²/π. An overall, threshold diagram is presented based on the simple criterion that, for sustained Stage II FCP, Δσ_s must be greater than Δσ_FL. The study is based on a simple continuum mechanics approach and its purpose is the investigation of the suitability of both ΔK and Δσ_s to characterise the crack driving force that activates complex fracture processes at the microstructure's scale. The investigation pertains to conditions that lead to the ultimate failure of the component at values of Δσ_s > Δσ_FL.     

Continuous cooling transformation behaviour and bainite formation kinetics of new bainitic steel

In order to avoid the appearance of soft particles composed of ferrite or pearlite in the actual production of new bainitic steel, the phase transformation behaviour and bainite formation kinetics were investigated by DIL805A dilatometer, optical microscopy, scanning electron microscopy and Vickers-durometer. The results show that the soft particles cannot appear when the cooling rate exceeds 0.025?K?s^?1, and this condition can be ensured by direct spray cooling in production. The local activation energy decreases with increasing transformed bainite volume fraction (f_b), and the average energy is about 136.7?kJ?mol^?1. The local Avrami exponent mainly lies between 0.5 and 3 in a wide f_b range, indicating that the dominating mechanism of bainite formation is two-dimensional and one-dimensional growth.     

High focal depth with a quasi-bifocus birefringent lens

The Strehl definition along the axis of a birefringent lens sandwiched between two polarizers is studied analytically. The optic axis of the birefringent lens made of a uniaxial crystal is perpendicular to the lens axis, and the system behaves like a bifocus lens for proper orientation of the polarizers. The Sparrow criterion is employed for designing an imaging system with high depth of focus. It is shown that, when the two foci are separated by the Sparrow limit of resolution, the focal depth is maximum and the intensity point-spread function remains almost identical within this limit. The resolution according to the Rayleigh criterion in this zone is more than that of an ideal lens.     

SHEAR LIPS ON FATIGUE FRACTURES OF ALUMINIUM ALLOY SHEETS SUBJECTED TO BIAXIAL CYCLIC LOADS AT VARIOUS R-RATIOS

An analysis is made of shear lip measurements as observed on fatigue crack surfaces of an aluminium alloy sheet material. It is shown that, for biaxial cyclic loads at stress ratios 1 and various R-ratios, shear lips were controlled by an equivalent intensity factor K_e=K₁F(λ,R). For crack growth in air the shear lip width t_s was approximately proportional to K²_e at t_s < t*_s= 2.5 × 10^?4 m and K⁴_E, at t_s > t*_s where t*_s is the critical value of the shear lip dimension. The initiation of shear lips and the orientation of crack growth for different parameters of the loading cycle are briefly considered in the discussion.     

Phase diagram for the helical texture in 3He-A

The stability region of the helical textures in³He-A in thev _s-H phase diagram is theoretically determined. Herev _sis the superflow velocity andH (v _s) is the external magnetic field. The pitch of the helixk ^–1 that minimizes the Gibbs free energy and the corresponding inclination angles ofl andd are determined numerically as functions ofv _sandH. The NMR frequencies (two longitudinal and four transverse frequencies) associated with the helical texture are obtained.Work supported by the National Science Foundation under Grant No. DMR 76-21032.     

Near-field microscopy by elastic light scattering from a tip

We describe ultraresolution microscopy far beyond the classical Abbe diffraction limit of one half wavelength (lambda/2), and also beyond the practical limit (ca. lambda/10) of aperture-based scanning near-field optical microscopy (SNOM). The 'apertureless' SNOM discussed here uses light scattering from a sharp tip (hence scattering-type or s-SNOM) and has no lambda-related resolution limit. Rather, its resolution is approximately equal to the radius a of the probing tip (for commercial tips, a < 20 nm) so that 10 nm is obtained in the visible (lambda/60). A resolution of lambda/500 has been obtained in the mid-infrared at lambda = 10 microm. The advantage of infrared, terahertz and even microwave illumination is that specific excitations can be exploited to yield specific contrast, e.g. the molecular vibration offering a spectroscopic fingerprint to identify chemical composition. S-SNOM can routinely acquire simultaneous amplitude and phase images to obtain information on refractive and absorptive properties. Plasmon- or phonon-resonant materials can be highlighted by their particularly high near-field signal level. Furthermore, s-SNOM can map the characteristic optical eigenfields of small, optically resonant particles. Lastly, we describe theoretical modelling that explains and predicts s-SNOM contrast on the basis of the local dielectric function.     

Microstructural development of Zn–40Al alloy during aging

Abstract

The microstructure development of a Zn–40 wt-%Al alloy during aging was studied using transmission electron microscopy and electron diffraction. The supersaturated solid solution α′_s phase was formed after solution quenching treatment, which exhibited a substructure with a structure factor contrast in some areas after aging for shorter times at ambient temperature. Electron diffraction results show that there are satellite reflections in the 〈110〉 _α′s direction. The α′_sphase decomposes during aging, forming an (α+η) lamellar structure by the discontinuous reaction or cellular decomposition and an equiaxed fine grained structure by the continuous reaction, which are α″_m and α″ or α′_m and α′ with twin relationships to each other. A very fine tweedlike structure was observed to spinodally decompose ahead of the cellular zone. The value of the modulation wavelength λ is about 70–85 nm. The bowing and migration of the grain boundary would occur in the evolution of the discontinuous precipitates. Precipitate free zones of the vacancy denuded type were found in the vicinity of the grain boundary and within grains. It has been observed that the formation of the metastable R phase is always associated with dislocations.     

Atomic Insights into Phase Evolution in Ternary Transition‐Metal Dichalcogenides Nanostructures

Phase engineering through chemical modification can significantly alter the properties of transition‐metal dichalcogenides, and allow the design of many novel electronic, photonic, and optoelectronics devices. The atomic‐scale mechanism underlying such phase engineering is still intensively investigated but elusive. Here, advanced electron microscopy, combined with density functional theory calculations, is used to understand the phase evolution (hexagonal 2H→monoclinic T′→orthorhombic T_d) in chemical vapor deposition grown Mo_{1− x} W _x Te₂ nanostructures. Atomic‐resolution imaging and electron diffraction indicate that Mo_{1− x} W _x Te₂ nanostructures have two phases: the pure monoclinic phase in low W‐concentrated (0 < x ≤ 10 at.%) samples, and the dual phase of the monoclinic and orthorhombic in high W‐concentrated (10 < x < 90 at.%) samples. Such phase coexistence exists with coherent interfaces, mediated by a newly uncovered orthorhombic phase T_d′. T_d′, preserves the centrosymmetry of T′ and provides the possible phase transition path for T′→T_d with low energy state. This work enriches the atomic‐scale understanding of phase evolution and coexistence in multinary compounds, and paves the way for device applications of new transition‐metal dichalcogenides phases and heterostructures.     

Characterizing Colloidal Structures of Pseudoternary Phase Diagrams Formed by Oil/Water/Amphiphile Systems

Two pseudoternary phase diagrams were constructed using ethyl oleate, water, and a surfactant blend containing poly (oxyethylene 20) sorbitan monooleate and sorbitan monolaurate with or without the cosurfactant 1-butanol. Two colloidal regions were identified in the cosurfactant-free phase diagram; a microemulsion (ME) and a region containing lamellar liquid crystals (LC). The addition of 1-butanol increased the area in which systems formed microemulsions and eliminated the formation of any liquid crystalline phases. Samples that form the colloidal regions of both systems were investigated by freeze-fracture transmission electron microscopy and by viscosity and conductivity measurements. The three techniques were compared and evaluated as characterisation tools for such colloidal systems and also to identify transitions between the colloidal systems formed. A droplet ME was present at a low water volume fraction (?_w) in both systems (?_w <0.15) as revealed by electron microscopy. At higher ?_w values, LC structures were observed in micrographs of samples taken from the cosurfactant-free system while the structure of samples from the cosurfactant-containing system was that of a bicontinuous ME. The viscosity of both systems increased with increasing ?_w to 0.15 and flow was Newtonian. However, formation of LC in the cosurfactant-free system resulted in a dramatic increase in viscosity that was dependent on ?_w and a change to pseudoplastic flow. In contrast, the viscosity of the bicontinuous ME was independent of ?_w. Three different methods were used to estimate the percolation threshold from the conductivity data for the cosurfactant-containing system. The use of nonlinear curve fitting was found to be most useful yielding a value close to 0.15 for the ?_w.     

Mechanism of the Vaterite-to-Calcite Phase Transition under Sonication

The vaterite-to-calcite phase transition in spheroidal vaterite particles produced by reacting aqueous Ca(NO₃)₂ and Na₂CO₃ solutions at 20 ± 1°C under 2.6-MHz sonication was studied by scanning electron microscopy and x-ray diffraction. The results are interpreted as evidence that the transformation of vaterite, a metastable phase under ordinary conditions, to calcite follows a “relay” recrystallization mechanism: vigorous ultrasonic stirring leads to energetic collisions between vaterite and calcite particles, resulting in a transition of the metastable phase to the stable one, followed by the formation of calcite aggregates consisting of 2 to 14 particles.     

Critical Velocities in Two-Component Superfluid Bose Gases

On the ground of the Landau criterion we study the behavior of critical velocities in a superfluid two-component Bose gas. It is found that under motion of the components with different velocities the velocity of each component should not be lower than a minimum phase velocity of elementary excitations (s ₋). The Landau criterion yields a relation between the critical velocities of the components (v _c1, v _c2). The velocity of one or even both components may exceed s ₋. The maximum value of the critical velocity of a given component can be reached when the other component does not move. The approach is generalized for a two-component condensate confined in a cylindrical harmonic potential.      

Hot deformation characterization of as-homogenized Al-Cu-Li X2A66 alloy through processing maps and microstructural evolution

Uniaxial compression tests were carried out on an Al-Cu-Li alloy at a temperature range of 300–500 °C and a strain rate range of 0.001–10 s⁻¹. Four representative instability maps based on Gegel, Alexander-Malas (A-M), Kumar-Prasad (K-P) and Murty-Rao (M-R) criteria were constructed. Through formula deduction and microstructural observation, it can be concluded that M-R criterion is more accurate than K-P criterion, and the first two criteria are better than Gegel and A-M criteria. From a power dissipation map and a M-R instability map, the optimized processing parameters are 480–500 °C/0.001–0.1 s⁻¹ and 420–480 °C/0.1-1 s⁻¹. The corresponding microstructural analysis shows that dynamic recovery and partial dynamic recrystallization are main dynamic softening mechanisms. Transmission electron microscopy observation indicated that a large number of primary coarse T₁ (Al₂CuLi) particles precipitated in the homogenized specimen. After deformation at 500 °C, most of the primary T₁ particles dissolved back into the matrix, and secondary fine T₁ particles precipitated at deformation-induced dislocations, high angle grain boundaries and other dispersed particles.     

Discontinuities in Amplitudes of Particles Micromotions Characterizing Phase Transitions in Liquid State

The behavior of amorphous solids below Vogel's (T _V ) or the glass transition (T _g ) temperature, as well as the solid-liquid transition, have been analyzed taking into consideration the anharmonicity of motion of microparticles forming the amorphous bodies. The T _g transition is explained within the logical association of this transition with the higher-temperature transitions, which can eventually involve the process of particle release into the gas phase through the process of a sudden vibrational amplitude growth. It follows from the mathematical solution of the anharmonicity problems that the pulses and the double amplitudes will always be present in aliquid matrix. The T _g temperature is considered as the boundary point for the liquid state at which the dynamical microcracks of a solid state matrix start to proceed. The processes at T _g are accompanied by appearance of new, highly agitated spots and the first microcracks (vacancies) filled up with the ‘semi-evaporated’ particles. In the mechanical sense, these vacancies form a new particle species characterized by quite different properties (different thermal expansion coefficient) as compared with the particles of the original matrix. It is assumed that a number of new mechanical units are growing up to the critical temperature when the original liquid frame, bonding the particles to lower amplitudes, is completely destroyed. The approach proposed does not contradict the traditional views reflected in the famous Adam-Gibbs-Di Marzio or WLF approaches, but allows a different approach to these theories. This revised version was published online in August 2006 with corrections to the Cover Date.     

Detection of a metastable magnetic phase in the La1.85Ba0.15CuO4−y system

We have made samples of La_1.85Ba_0.15CuO_4–y with varying heat treatments. We have measured their dc magnetization and electron paramagnetic resonance, and have examined them by electron microscopy and electron diffraction. For samples annealed only a few hours, cooling the material below 5 K causes a magnetic transition and a thermal hysteresis. These observations, taken with the EPR data, suggest a possible spin-glass or ferromagnetic phase. Annealing samples in an oxygen atmosphere appears to remove that magnetic phase. For the phase with the K₂NiF₄ structure, no superlattice spots are seen in electron diffraction at room temperature or at 77 K.     

A FRACTURE CRITERION FOR CRACKS UNDER MIXED-MODE LOADING

Abstract A fracture criterion is proposed, based on maximum energy release rates at the tips of short kinks when the main cracks are subjected to mixed mode loading. The criterion differs from existing energy based criteria in that the fracture toughness, g_c, is not independent of the stress mode prevailing in the region of the tip of the kink but is a function of the ratio of the mode II to mode I stress intensity factors at the tip of the kink, i.e., g_c is determined directionally by an elliptical region with major and minor axes equal to the fracture resistances of the material, K_Ir and K_IIr, for pure mode I and pure mode II, respectively. Points inside the elliptical region are considered safe. When K_IIr is equal to K_Ir the ellipse degenerates into a circle and the fracture criterion reverts to the existing familiar maximum energy release rate criterion based on a single value of the fracture toughness, irrespective of the active mode prevailing in the region at the tip of the kink. In this case, under pure shear (mode II) applied load, K_II, the angle of inclination of the fracture crack extension to the main crack, α, is in the region of ?76°, in general agreement with previous well established results. However, when the ratio r (=K_IIrK_Ir) is less than r′ (=0.82, approximately) a different pattern emerges and, in particular, under pure mode II load, the crack advance is co-planar with the main crack, i.e., in mode II. A lower transition value r″ (=0.582, approximately) was also detected under pure mode I applied load. Thus for values of r≥r″, the crack extension is in pure mode I and is co-planar with the main crack but when r < r″, the crack branches out at an angle (which can be positive or negative) in mixed modes I/II crack extension. Some implications of these results are discussed.     

Magnetic Phase Transitions in Cobalt Chromite Nanoparticles

Cobalt chromite (CoCr₂O₄), an insulating normal spinel compound, is a potential multiferroic material. We report that pure, nanosize CoCr₂O₄ particles are synthesized through a conventional coprecipiation technique by controlling the pH of the precipitation. Both single-crystal and polycrystalline samples develop long-range ferrimagnetic order below the Curie temperature, T _c (97 K), and a sharp phase transition at T _s??31 K, attributed to the onset of long-range spiral magnetic order. However, we observed a transition from paramagnetic to superparamagnetic phase at T _c. Further lowering the temperature below T _c (97 K), the superparamagnetic phase transforms to ferrimagnetic phase at blocking temperature, T _b, which is found to be between 50 and 60 K. This intermediate superparamagnetic phase in between paramagnetic and long-range ferrimagnetic phases is attributed to a nanosize effect.     

On the Diffraction Limit for Lensless Imaging

The diffraction limit for lensless imaging, defined as the sharpest possible point image obtainable with a pinhole aperture, is analyzed and compared to the corresponding limit for imaging with lenses by means of theoretical considerations and numerical computations using the Fresnel-Lommel diffraction theory for circular apertures. The numerical result (u = π) obtained for the best configuration parameter u which defines the optical setup is consistent with the quarter-wave criterion, and is the same as the value reported in a classical paper by Petzval but smaller than the value (u = 1.8π) found by Lord Rayleigh. The smallest discernible detail (pixel) in a composite image is defined by an expression found by Rayleigh on applying the half-wave criterion and is shown to be consistent with the Sparrow criterion of resolution. The numerical values of other measures of image size are reported and compared to equivalent parameters of the Fraunhofer-Airy profile that governs imaging with lenses.     

Synthesis of (K, Na)NbO3 particles by high temperature mixing method under hydrothermal conditions

In order to prepare the pure (K, Na)NbO₃(KNN) particles with higher crystallinity, the high temperature mixing method (HTMM) under hydrothermal conditions was carried out in this work. The obtained samples were characterized by X-ray diffraction (XRD), scanning electron microscopy (SEM), transmission electron microscopy (TEM), selected area electron diffraction (SAED), and high-resolution transmission electron microscopy (HRTEM). The results indicate that the KNN particles size decrease gradually with the increase of mineralization concentration in the starting solution. The ratio of K⁺/(K⁺ + Na⁺) in the starting solution has a great effect on the phase of the products, and several phases coexist in the product when the ratio of K⁺/(K⁺ + Na⁺) in the starting solution is 0.7.     

Phase transformation of the intermetallic compound Al4Ca

By measurement of physical properties and microscopic examination it was found that the phase Al₄Ca undergoes a martensitic transformation with M _s130° C. The low temperature structure is monoclinic, a=0.6158 nm, b=0.6175 nm, c=1.118 nm, =88.9°. The Dl₃ structure ascribed to Al₄Ca hitherto is valid only above M _s. Orientation relationships indicate that the transformation proceeds by a shear of 1.1° on (1 1 0) planes of the Dl₃ structure.