AI 中文总结
研究晶体学中四元数的应用问题，通过推广四元数乘积法则，引入交叉张量，使其能直接在晶体基中使用，给出仅取决于度量张量的乘积公式，并探讨了在电子背散射衍射中的应用。
AI 中文摘要
单位四元数在科学中广泛用于编码旋转，因其在计算两个旋转的合成时比矩阵乘积更高效，比罗德里格斯乘积更稳定。但通常形式的四元数基于笛卡尔基，无法直接用于晶体学。解决方法通常是借助结构张量在晶体基和附着于晶体的笛卡尔基之间来回变换坐标。本文表明，通过推广四元数乘积法则，四元数可直接在晶体基中使用。为此引入了一个称为交叉张量的矩阵，它能在晶体基中计算叉积，类似于度量张量用于标量积。还表明交叉张量与度量张量的逆成比例，进而给出了晶体学四元数乘积的公式，该公式仅取决于度量张量。最后讨论了晶体学四元数在电子背散射衍射中的应用。
英文摘要
Unit quaternions are widely used in science to encode rotations because the quaternion product is more efficient than matrix product and more stable than Rodrigues product to calculate the composition of two rotations. However, quaternions in their usual form refers to a Cartesian basis; they cannot be used in crystallography as they are. The usual way to solve this issue to apply back-and-forth coordinate changes from the crystal basis to a Cartesian basis attached to the crystal with the help of the structure tensor. Here, we show that actually quaternions can be used directly in the crystal basis by generalizing the quaternion product law. In that aim, we introduced a matrix that we called cross tensor. It allows the calculation of the cross product in the crystal basis, a bit like the metric tensor allows it for the scalar product. We also show the cross tensor is proportional to the inverse of the metric tensor. The formula of crystallographic quaternion product is then given; it depends uniquely on the metric tensor. The application of the crystallographic quaternions to Electron Back Scatter Diffraction is discussed.
Comments15 pages, 1 figures, 25 equations, 2 appendices
