  The temperature dependence of the zero-field splitting (ZFS) between the
$|m_{s}=0\rangle$ and $|m_{s}=\pm 1\rangle$ levels of the nitrogen-vacancy (NV)
center's electronic ground-state spin triplet can be used as a robust nanoscale
thermometer in a broad range of environments. However, despite numerous
measurements of this dependence in different temperature ranges, to our
knowledge no analytical expression has been put forward that captures the
scaling of the ZFS of the NV center across all relevant temperatures. Here we
present a simple, analytical, and physically motivated expression for the
temperature dependence of the NV center's ZFS that matches all experimental
observations, in which the ZFS shifts in proportion to the occupation numbers
of two representative phonon modes. In contrast to prior models our expression
does not diverge outside the regions of fitting. We show that our model
quantitatively matches experimental measurements of the ZFS from 15 to 500 K in
single NV centers in ultra-pure bulk diamond, and we compare our model and
measurements to prior models and experimental data.
