Non-destructive method for measuring the Ca/P ratio in human bone

Hydroxyapatite (HA) [Ca$_{10}$(PO$_{4})_{6}$(OH)$_{2}$] is the main mineral constituent in human bone. It crystallizes in hexagonal and monoclinic phase, which are very similar in structure and properties. A critical measure for healthy bones is the Ca/P ratio which in turn affects the dielectric constant of the mineral constituent. The dielectric constant of HA varies between 5 and 20 depending on the Ca/P ratio in the sample [J. Mater. Sci.: Mater. Med. \textbf{21}, 399]. We suggest exploiting this large span in the dielectric constant in a non-destructive method to measure the Ca/P ratio in bone by optical spectroscopy. Using density functional theory we calculate the long-range corrected phonon dispersion. We find that only modes around 330 cm$^{-1}$ are strongly affected by the dielectric constant. The shifts in frequency can be up to 20 cm$^{-1}$ as you span the range of the dielectric constant. Thus, by measuring the optical shift and comparing with calibrated samples it is possible to draw conclusions on the Ca/P ratio in the mineral. Importantly, we find the same modes in both the monoclinic and hexagonal phases to be sensitive to changes in the dielectric constant.