Figure 4: The application of distortion symmetry to a distortion of β-BaB₂O₄.

The mostly rigid rotation of the B₃O₆ rings leads to two variants of β-BaB₂O₄ with R3c symmetry group, the λ=−1 variant (inset in orange in a and the λ=+1 variant (inset in cyan in a), transforming through a transition state at λ=0 (inset in magenta in a) with a symmetry of . The symmetry of this path, , requires that the energy profile in a is symmetric. (b) The superimposed images of β-BaB₂O₄ along the distortion pathway; their colour varies from orange, through magenta to cyan as λ varies from −1 through 0 to +1. (c) The optical second harmonic generation tensor coefficients along this pathway calculated by Cammarata and Rondinelli³⁷ (red, green and blue circles) and a polynomial fit (red, green and blue lines) using only the coefficients that are consistent with point-group symmetry. Distortion symmetry predicts that these coefficients will be odd functions of the distortion parameter, λ, and zero when λ=0.