Diffraction indexing under a cell change (${\mathbf{h}}_{\mathrm{new}}=P^T{\mathbf{h}}_{\mathrm{old}}$)

If new direct-cell vectors are columns of $P$ in old fractional coordinates, an unchanged reciprocal vector has the displayed new coordinates. Noninteger coordinates must not be rescaled when tracking the same diffraction order. Under the Bain path cell change, $P=\left(\begin{array}{ccc}1/2& -1/2& 0\\ 1/2& 1/2& 0\\ 0& 0& 1\end{array}\right)$. The old FCC $(100)$ reflection becomes $(1/2,-1/2,0)$ and remains absent; doubling that triple gives the different, allowed reflection that originated as FCC $(200)$. Orientation indices and order-resolved reciprocal vectors carry different information.