Plane Of Reciprocal Space Research Articles

A method for the recovery of local order coefficients from two-dimensional intensity data for binary f.c.c. alloys

A method is derived which permits the Cowley–Warren order parameters to be recovered from diffuse scattering measurements made in one plane in reciprocal space for systems with only short-range correlations. The procedure is demonstrated first by application to some at-temperature measurements on Cu3Au and then by considering some simulated diffraction profiles, with and without controlled errors. From the simulated scattering without error, an empirical rule is developed which allows one to decide if a given profile is amenable to this recovery method. By simulation of data with varying constant-background contributions, it is shown that such an error can be recognized and corrected.

X-ray scattering from graphite crystals containing interstitial basal-plane loops

The diffraction effects arising from the presence of interstitial basal-plane loops in a graphite crystal are analyzed using a high-speed computer. The atomic displacements were computed using expressions for the displacement field of a prismatic dislocation loop in an anisotropic hexagonal crystal given by Ohr. The loops are assumed to be circular clusters of atoms in the $C$ position of the normal $\mathrm{ABAB}\ensuremath{\cdots}$ sequence of graphite planes. The resulting defect is a finite extrinsic fault and its associated strain field. The effects of loop size and concentration on the lattice parameters, Bragg intensities, and diffuse scattering are given. Graphs are presented which allow loop size and concentration to be determined from measurements of lattice parameters and Bragg intensities. Isodiffusion contour maps of the diffuse scattering in the ($HH\ifmmode\cdot\else\textperiodcentered\fi{}L$) and ($HO\ifmmode\cdot\else\textperiodcentered\fi{}L$) planes of reciprocal space are presented. The diffuse scattering around reciprocal-lattice points for which $H\ensuremath{-}K=\mathrm{mod}(3)$ is different from those for which $H\ensuremath{-}K\ensuremath{\ne}\mathrm{mod}(3)$. The former depends on the symmetry of the loop strain field and is concentrated around the reciprocal-lattice point, while the latter reflects the disruption of the stacking sequence and is characterized by streaks connecting reciprocal-lattice points. The size of the loops can be determined approximately by the full width at half-maximum of the streak connecting the (10 \ifmmode\cdot\else\textperiodcentered\fi{} 0) and (10 \ifmmode\cdot\else\textperiodcentered\fi{} 1) reflections. These predicted effects are compared with diffraction effects seen in neutron-irradiated graphite.

Fermi Surface of Magnesium. IV. Magnetoresistance Study of Open Trajectories

Measurements of the magnetoresistance of high-purity single crystals of magnesium were made in order to determine the magnetic field directions for which open trajectories exist on the magnesium Fermi surface. Nine distinct series of open trajectories with net directions parallel to the basal plane of reciprocal space were observed. These trajectories have basic periodicities extending up to nine Brillouin zones and result from magnetic breakdown between the monster and cigar. The magnetic field directions for which these trajectories exist are shown to be in excellent agreement with the predictions of the Fermi-surface model proposed by Kimball, Stark, and Mueller. Measurements for the open trajectory parallel to the hexagonal axis in reciprocal space are also reported; these show a clear indication of the onset of magnetic breakdown across the spin-orbit gap separating the monster and the cap.

Electron Microscopic Studies of Some Lunar Minerals

Ion-thinned samples of rock 12052 have been examined in the electron microscope at 100 kV. This rock contains numerous phenocrysts of pyroxene, Ca1 – P (Mg, Fe)1 + P Si2O6, and a few rounded olivine crystals (Mg, Fe)2 SiO4 set in a finer-grained groundmass of pyroxene, plagioclase (Na, Ca) (Al, Si)4O8, ilmenite, FeTiO3 and other minor minerals (Champness et al., 1971). The cores of the pyroxene phenocrysts consist of a calcium-poor pyroxene, pigeonite, with an epitaxially grown rim of a calcium-rich pyroxene, augite. Both the pigeonite and augite are zoned, and indication of non-equilibrium crystallisation (Bence et al., 1970).The outer parts of the pigeonite cores of the phenocrysts show very finescale composition modulations on (001) (major) and (100) (minor) (Figure 1). This texture is interpreted as exsolution by a spinodal mechanism (Cahn, 1968).The augite rim of the phenocryst exhibits a coarse exsolution of pigeonite on (001) on which is superimposed fine-scale composition modulations on (001) and (100) (Figure 2).The two-stage exsolution process indicates that the cooling rate of the pyroxene increased suddenly, possibly when the magma was extruded on the Moon's surface.Antiphase domains produced during the polymorphic transition from space group C2/c to P21/c have been found in all primary and precipitated pigeonites (Figure 3). This confirms Morimoto and Tokonami's (1969) suggestion that all clino-pyroxenes have the C2/c space group at high temperatures.Cristobalite, a high-temperature polymorph of silica, occurs as very small grains in the groundmass. It contains numerous microtwins on {100} planes produced during the β→α polymorphic transition which occurs at 268°C (Figure 4a). Diffraction patterns show extensive diffuse streaks in various directions (Figure 4b) which correspond to intersections of planes in reciprocal space with the Ewald Sphere. These diffuse planes originate from the cooperative thermal vibration of Si-O chains parallel to {100} and {111}.

X-Ray Scattering from hcp Crystals Containing Interstitial Basal-Plane Loops

The diffraction effects arising from the presence of interstitial basal-plane loops in an ideal hcp crystal are analyzed using a high-speed computer. The atomic displacements were computed assuming that the crystal responds to the loop as an isotropic elastic medium with a Poisson's ratio of 0.3 into which a "penny-shaped" inclusion is inserted. The loops are assumed to be circular clusters of atoms in the $C$ position of the normal $\mathrm{ABAB}\ensuremath{\cdots}$ sequence of hcp planes. The resulting defect is a finite extrinsic fault and its associated strain field. The effects of loop size and concentration on the lattice parameters, Bragg intensities, and diffuse scattering are given. Graphs are presented which allow the determination of loop size and concentration from the measurement of lattice parameters and Bragg intensities. Isodiffusion contour maps of the diffuse scattering in the ($\mathrm{HH}\ifmmode\cdot\else\textperiodcentered\fi{}L$) and ($\mathrm{HO}\ifmmode\cdot\else\textperiodcentered\fi{}L$) planes of reciprocal space are presented. The diffuse scattering around reciprocal-lattice points for which $H\ensuremath{-}K=\mathrm{modulo} 3$ is quite different from that around reciprocal-lattice points for which $H\ensuremath{-}K\ensuremath{\ne}\mathrm{modulo} 3$. The former depends strongly on the symmetry of the loop strain field and is concentrated around the reciprocal-lattice point, while the latter reflects more strongly the disruption of the stacking sequence, having characteristically S-shaped streaks connecting the reciprocal-lattice points. The size of the loops can be determined approximately from the full width at half-maximum of the streak connecting the (10 \ifmmode\cdot\else\textperiodcentered\fi{} 0) and (10 \ifmmode\cdot\else\textperiodcentered\fi{} 1) reflections. The diffraction effects produced by basal-plane loops are discussed in connection with those seen in neutron-irradiated BeO. All of the predicted effects are seen in neutron-irradiated BeO, including the duplex nature of the reflections with $L\ensuremath{\ne}0$ and the crosses around reciprocal-lattice points for which $H\ensuremath{-}K=\mathrm{modulo} 3$ and $L=0$.

Electron diffraction effects in silicon carbide. I. Pure polytypes. II. whiskers

Pure samples of the three most common hexagonal SiC polytypes have been examined in transmission electron diffraction. The diffraction patterns from the most densely populated planes in reciprocal space, the (h0l)* planes, differ considerably from the calculated patterns. The difference can be attributed to ubiquitous multiple diffraction. Calculated intensity distributions for 100 kV electrons are given for 4H, 6H, and 15R SiC. SiC whiskers have also been examined, and the unusual diffraction effects seen can, in general, be explained as a combination of (a) extreme disorder, (b) distorted lattices and (c) multiple diffraction.

X-Ray study of short-range order in nickel alloys containing 10.7 and 20.0 at. % Molybdenum

The diffuse intensity in the h 1, h 2, 0 plane of reciprocal space was measured for Ni-rich alloys containing 20.0 and 10.7 at. % Mo after varied thermal history. A very unusual distribution of diffuse intensity indicating considerable short-range order was observed for samples of both alloys which had been quenched from 1000°C. After correcting for atomic size effects, two-dimensional short-range order parameters defined in terms of the usual three-dimensional parameters by A lm= ∑ n α lmn were determined. Comparison of these observed A lm with values A lm calculated from models of the short-range structures of the alloys led to the conclusion that small regions existed within which the atomic arrangement was similar to that in long-range ordered Ni 4Mo. Reheating the Ni-10.7 at. % Mo alloy near 479°C after quenching from 1000°C resulted in a sharpening of the diffuse intensity distribution corresponding to an increase in the degree of order. Short heat treatments of the Ni-20.0 at. % Mo alloy near 650°C (below the critical temperature, 868°C) caused the short-range order peaks observed in samples quenched from above the critical temperature to split into peaks located near the positions of the superlattice reflections for long-range ordered Ni 4Mo. This is interpreted to mean that reasonably well defined domains of order exist at this stage of heat treatment. The critical size of the domains necessary to cause the observed splitting was estimated to be of the order of 10 a 0.

A construction giving the projection of the point h00 on to the 0kl plane in reciprocal space, for non-orthogonal axes

A construction giving the projection of the point h00 on to the 0kl plane in reciprocal space, for non-orthogonal axes

Approximate Elastic Spectrum of Acoustic Waves in AgCl from X-Ray Scattering

The temperature diffuse scattering of x-rays from a large single crystal of AgCl has been measured at room temperature using a Geiger counter spectrometer and crystal monochromated CuK-α radiation. Measurements were made throughout the hk0 plane in reciprocal space, and in certain other regions. The data are analyzed on the basis of the theory developed by Laval and James, in which temperature vibrations are expressed in terms of traveling elastic waves. An assumed set of vibrational frequencies is used for the optical branch. Dispersion curves for waves in the acoustical branch traveling in the [100], [110], and [111] directions are obtained. The velocities at infinite wavelength are shown to agree fairly well with those given by the static elastic constants. An approximate frequency spectrum is then calculated, and suggestions are made as to how the effects of the optical and acoustical branches may be separated.