Low-angle grain boundaries (LAGBs) are ubiquitous in natural and man-made materials and profoundly affect many of their mechanical, chemical, and electrical properties. The properties of LAGBs are understood in terms of their constituent dislocations that accommodate the small misorientations between grains. Discrete dislocations result in a heterogeneous local structure along the boundary. In this article, we report the lattice rotation across a LAGB in olivine (Mg1.8Fe0.2SiO4) measured at the nanometer scale by using quantitative high-resolution transmission electron microscopy. The analysis reveals a grain boundary that is corrugated. Elastic calculations show that this waviness is independent of the host material and thus a general feature of LAGBs. Based on our observations and analysis, we provide equations for the boundary position, local curvature, and the lattice rotation field for any LAGB. These results provide the basis for a reexamination of grain-boundary properties in materials such as high-temperature superconductors, nanocrystalline materials, and naturally deformed minerals.dislocations ͉ geometric phase analysis ͉ high-resolution transmission electron microscopy ͉ olivine ͉ interface T he properties of materials are influenced by the presence of grain boundaries. Low-angle grain boundaries (LAGBs) consist of arrays of discrete dislocations that separate two crystals of slightly different orientation (1). The dislocation model for LAGBs was first proposed by Taylor (2) 70 years ago and further developed by Read and Shockley (3) to predict dislocation density from macroscopic boundary geometry. Read and Shockley used their model to calculate boundary energy based on the summed elastic energy of the individual dislocations. The dislocation model remains the basis for understanding LAGB structures and interpreting their properties.In this article, we reexamine the structure of LAGBs by using a combination of high-resolution transmission electron microscopy (HRTEM) and geometric phase analysis, a recently developed technique that is sensitive to small displacements of lattice fringes in HRTEM images (4). Previous analyses of LAGBs by HRTEM relied on visual inspection of images (5). Phase analysis has been used to accurately measure displacements around a [100] dislocation in olivine to Ͻ0.09 Å (6) and a [110] dislocation in Si to 0.03 Å (7). The technique also has been used to examine the distribution of dislocations along heterophase interfaces and LAGBs in semiconducting materials (8, 9). Here we employ the phase technique to analyze LAGBs in naturally deformed olivine, a widespread rock-forming mineral that is an important constituent of Earth's upper mantle and crust.
Materials and MethodsThe olivine sample (Mg 1.8 Fe 0.2 SiO 4 ; space group Pbnm) is from a mantle xenolith collected in San Carlos, AZ. Grain boundaries in ion-milled samples were imaged with a JEOL 4000EX transmission electron microscope. We observed arrays parallel to (100) consisting primarily of [100] dislocations and mixed ar...