  $\textit{Ab initio}$ calculations were employed to investigate the effect of
oxygen concentration dependent Ti vacancies formation on the core electron
binding energy shifts in cubic titanium oxynitride
(Ti$_{1-\delta}$O$_x$N$_{1-x}$). It was shown, that the presence of a Ti
vacancy reduces the 1s core electron binding energy of the first N neighbors by
$\sim$0.6 eV and that this effect is additive with respect to the number of
vacancies. Hence it is predicted that the Ti vacancy concentration can be
revealed from the intensity of the shifted components in the N 1s core spectra
region. This notion was critically appraised by fitting the N 1s region
obtained via X-ray photoelectron spectroscopy (XPS) measurements of
Ti$_{1-\delta}$O$_x$N$_{1-x}$ thin films deposited by high power pulsed
magnetron sputtering. A model to quantify the Ti vacancy concentration based on
the intensity ratio between the N 1s signal components, corresponding to N
atoms with locally different Ti vacancy concentration, was developed. Herein a
random vacancy distribution was assumed and the influence of surface oxidation
from atmospheric exposure after deposition was considered. The so estimated
vacancy concentrations are consistent with a model calculating the vacancy
concentration based on the O concentrations determined by elastic recoil
detection analysis and text book oxidation states and hence electroneutrality.
Thus, we have unequivocally established that the formation and population of Ti
vacancies in cubic Ti$_{1-\delta}$O$_x$N$_{1-x}$ thin films can be quantified
by XPS measurements from N 1s core electron binding energy shifts.
