Representation of Nye’s Lattice Curvature Tensor by Log Angles

Abstract

The log angles of a rotation matrix are three independent elements of the logarithm of the rotation matrix. Nye’s lattice curvature tensor κ_ij is discussed by using the log angles. For the change in a crystal orientation ΔR with the change in a position Δx_i, it is shown that the elements of κ_ij are written as κ_ij=Δω_i/Δx_j using the log angles Δω_i of ΔR. The log angles for the crystal rotation given by the axis/angle pair are also discussed.

Fig. 2

The change in a crystal orientation as much as ΔR with the change in a position from （x₁, x₂, x₃） to （x₁＋Δx₁, x₂＋Δx₂, x₃＋Δx₃）. δR is a small-angle rotation which satisfies ΔR≈（δR）^N where N is a sufficiently large positive integer. The relationship ΔR≈（δR）^N means that the N times continuous rotations of δR with an interval of δx＝（Δx₁/N, Δx₂/N, Δx₃/N） is equivalent to ΔR. The angles δϕ_i are the small rotation angles of δR around the x_i axis.