B. Schönfeld

The asymptotic leading term of anisotropic small-angle scattering intensities. II. Non-convex particles

For anisotropic particulate samples with scattering contrast (delta n)(2), the leading asymptotic term of the scattering intensity, along a direction q (q/q) of reciprocal space, is [4 pi(2)(delta n)(2)/q(4)]sigmaj [1/kappaG.j(+/-q)]. Here, kappaG.j(+/-q)denotes the Gaussian curvature value at the points (labelled by j) of the interphase surface where the normal is either parallel or antiparallel to q. If the Gaussian curvature vanishes at, say, the jth of these points, the corresponding contribution takes the form Cj/qalpha with 2< or = alphaj<4, Cj and alphaj being determined by the local behaviour of the surface. However, the intensity detected by a counter pixel, with opening solid angle deltaomega(q0) along (mean) direction q0, asymptotically still behaves as 4pi(2) (delta n)(2)(deltaomega(q0))/q(4), where S(deltaomega(q0)) is the area of that part of the interface that has its normals inside deltaomega(q0).

Investigation of short-range order in Ni–10 at.% Al single crystals by diffuse X-ray scattering

The diffuse scattering from single crystals of nominally Ni-10 at.% Al, quenched after heat treatment at 973 K, has been measured with Mo Kα radiation. The data were analysed for the contributions due to short-range order and static atomic displacements. Consistency of the structural parameters as determined by three different methods, the separation methods of Georgopoulos-Cohen and Boric-Sparks and the least-squares method of Williams, is demonstrated for the first time. The variation of the first few short-range-order parameters αlmn with the coordination shell Imn strongly resembles that of the L12 superstructure. Employing these parameters to model a short-range-ordered computer crystal with 13 104 atoms, a preference for configurations of the L12 superstructure type is found, though no Ni3Al embryos are observed. The values of αlmn rapidly reach those for a random solid solution.

Small-angle neutron scattering of precipitates in Ni-rich NiTi alloys—I. Metastable states in poly- and single crystals

Abstract Two successive metastable regimes of decomposition are revealed in Niue5f8(10.8–11.9) at.% Ti single crystals and polycrystals by small-angle neutron scattering (SANS) after various aging steps at 853 K. During the first two hours of aging, the Ti concentration in Ti-rich regions grows linearly with time, while their size remains unchanged. After a few hours, an intermediate plateau in the integrated SANS intensity is reached and indicates an intermediate metastable state. This γ″ state, fully developed after about 12 h, corresponds to coherent non-spherical precipitates containing (18.4 ± 2.5) at.% Ti. After 50–100 h, a further increase in Ti concentration in the precipitates sets in, while they grow and tend to form clusters. After 400–1000 h, a second metastable state, γ′, with (22 ± 2) at.% Ti in the precipitates, is reached. The growth rate in this range is higher than for classical coarsening, and very pronounced spatial correlations of precipitates appear along cubic directions.

Phase separation in Ni-rich Ni–Ti: the metastable states

Abstract The microstructure of Ni-11.3 at.% Ti single crystals was investigated after quenching from the solid solution at 1440 K and after ageing at 873 K for various periods of time by using transmission electron microscopy and X-ray scattering near fundamental reflections and superstructure reflections of the ordered precipitates. The two-stage decomposition sequence with the initial formation of metastable γ″ precipitates followed by metastable γ′ precipitates as found by small-angle neutron scattering is confirmed. The long-range order parameter of these precipitates is always close to the maximum value achievable at a given composition. The presence of the first metastable state (γ″) is related to a rearrangement of the precipitates on a macro-lattice in order to minimize the elastic energy due to coherency strains.

Small-Angle Neutron Scattering of Precipitates in Ni-Rich Ni–Ti Alloys. II. Methods for Analyzing Anisotropic Scattering Data

Methods for analyzing small-angle neutron scattering data for single crystals with anisometric precipitates and anisotropic arrangement are described. A method for projecting the measured scattering intensity in a (110) plane and obtaining the anisotropic correlation function is described. The method uses an indirect Fourier transformation with Hermitian functions as the basis. The correlation function gives model-independent information on the precipitate form and arrangement. A modified paracrystal cluster model is also introduced and size polydispersity is included in a local monodisperse approximation. The size distribution is obtained by least-squares fitting. The correlation function and the cluster model give consistent results for the morphology of precipitates in Ni-rich Ni–Ti single crystals.

Shape of precipitates in Ni-Al-Mo single crystals

A method is presented to determine the shape of the average precipitate from anisotropic small-angle scattering patterns from single crystalline alloys during decomposition. The form factor squared of an average precipitate is described by an analytical expression with a single parameter, the morphology parameter , to account for changes from spherical to cuboidal shape. The method is applied to microstructural states of Ni-12 at.% Al-2 at.% Mo and Ni10 at.% Al-5 at.% Mo after aging treatments of up to 10 h at 970 or 1070 K. The value of increases with aging time and smaller Mo content, in good agreement with the average shape of precipitates obtained from transmission electron microscopy.

Guinier-preston I zones in Al-1.75 at.% Cu single crystals

Abstract Two Al-1.75 at.% Cu single crystals aged together for 60 min at 353 K were investigated by elastic diffuse neutron scattering. A large range of the scattering vector from 0.175 to 6.9 reciprocal lattice units along 〈h00〉 and the choice of isotopically pure 63Cu and 65Cu for contrast variation allowed the variation of the extracted results for short-range order scattering due to the finite series expansion of the static atomic displacement scattering to be followed systematically. Short-range order scattering was best separated if the scattering within the first Brillouin zone was included in the evaluation. Only then the short-range order scattering is found to be nearly independent of the scattering vector which indicates a considerably lower amount of multilayer Guinier-Preston I zones than found by other authors from three-dimensional diffuse X-ray scattering measurements.

Lattice misfit in CMSX4-like nickel-base superalloys and its temperature dependence

Abstract The temperature dependence of the γ – γ′ lattice misfit (δ) between precipitates and matrix in three modified CMSX4-type nickel-base superalloys was investigated. The effect of the so-called standard heat treatment and of the addition of elements like Nb, Ti and Re on δ was also monitored. The misfit was measured by in situ high-resolution X-ray diffraction (up to 1375 K), and the microstructure was investigated by means of optical and transmission electron microscopy. It was found that with the addition of Nb the microstructure becomes more sensitive to thermal treatments, while the presence of Ti strongly influences the lattice misfit δ. Re also influences the value of δ, but in combination with other elements. A standard CMSX4 alloy shows relatively less influence of the thermal treatments on the misfit values and the microstructural properties. For all samples investigated, the misfit is negative at all temperatures.

In-situ neutron scattering studies of order and decomposition in Ni-rich Ni–Ti

Abstract Diffuse wide-angle neutron scattering experiments have been performed for two states of thermal equilibrium within the γ solid solution, 58Ni–5.8 at.% Ti at 777 K and 58Ni–9.6 at.% Ti at 1103 K. The analysis of the Clapp configurations yields the basic cluster of the L12 structure as the most important local configuration. Neither the static nor the dynamic atomic displacements indicate any precursor of the stable hexagonal phase (D024 structure) containing 25 at.% Ti. Metastable states with precipitates showing non-stoichiometric L12 structure evolve during the decomposition of supersaturated Ni–11.3 at.% Ti polycrystals as seen by in-situ small-angle neutron scattering. From the integrated small-angle scattering intensity, the Ti content of the cuboidally shaped precipitates in the metastable γ′′ state is found to be about 17 at.% between 870 and 950 K.

Small-angle neutron scattering investigation of the early stages of decomposition in Ni-rich Ni-Ti

In-situ small-angle neutron scattering was performed on polycrystals and single crystals of Ni-10 to 12 at.% Ti at temperatures between 900 and 1240 K. A plateau in the integrated intensity as a function of aging time for polycrystalline Ni-11.3 at.% Ti at 900 K indicates the presence of a precursor state preceding the formation of the metastable state, with a Ti concentration of 16.5(1.0) at.% in the particles. In the scattering patterns of a Ni-12.0 at.% Ti single crystal aged at 1200 and 1240 K, strong scattering intensities along related to the coherent / states, and sharp streaks along related to plates of the stable phase appear successively. At 1240 K, the presence of the L12-related intensities along is unexpected, as this temperature lies above the metastable miscibility gap suggested in the literature.