Diffraction theory of nanotwin superlattices with low symmetry phase: Adaptive diffraction of imperfect nanotwin superlattices

Abstract

The diffraction behaviors of imperfect nanotwin superlattices were considered and the effects of random variations in nanotwin layer thicknesses and variant volume fractions on the diffraction peak intensity profiles investigated. Imperfect nanotwin superlattices exhibit a pronounced adaptive diffraction phenomenon when the Brillouin zone-dependent condition, , is met, where T is twin-related bilayer thickness, γ is twinning shear strain magnitude, s is twinning shear direction unit vector, and K is the reciprocal lattice vector of the constituent crystal denoting the fundamental reflection spots and Brillouin zones in reciprocal space. Nanotwin layer thickness variations produce little effect on the diffraction intensity distributions of adaptive Bragg reflection peaks, whereas variant volume fraction variations significantly reduce the height and broaden the width of adaptive peaks, but only affect their integrated intensities slightly and do not change their positions. In spite of imperfections, an extraordinary peak still appears at position k along twin peak splitting vector ΔK, as determined by a lever rule according to the average twin variant volume fraction : . Imperfect nanotwin superlattices exhibit a certain degree of diffuse reflection around the adaptive peaks with strong Lorentzian shape characteristics; adaptive peaks exhibit anisotropic broadening, i.e. peak is broader in the direction perpendicular to twin plane, and sharper parallel to the twin plane. The width of the adaptive peak does not provide direct information of nanodomain size, but the Brillouin zone-dependent diffraction behavior can be used to determine the nanodomain sizes.

Keywords:

• X-ray diffraction
• neutron diffraction
• nanotwin
• coherent scattering
• interference effect
• ferroelectric
• piezoelectric

Acknowledgements

NSF support under Grant No. DMR-0800048 is acknowledged. The calculations were performed on System X at Virginia Tech and Lonestar at Texas Advanced Computing Center.

Notes

Notes

1. For complete references to the monoclinic phases, see recent review 13.

2. For coherence length ∼1 µm and coherent scattering volume ∼1 µm³, a sample of volume 1 mm³ contains about 10⁹ coherent scattering volumes.