The assumption that electrons in deep core states do not participate in the bonding of solids, and thus do not show band-like dispersion, is at the base of our understanding of the solid state. It is also of essential practical significance for many experiments, such as x-ray photoelectron spectroscopy. This technique derives its power from the fact that the precise value of the core level binding energy depends on the chemical environment of the emitting atoms, but it is tacitly assumed that it has a single, well-defined energy, i.e. it does not show any dispersion. Violations of this assumption have only been found for small molecules in the gas phase such as C$_{2}$H$_{2}$ or N$_{2}$ with much stronger bonding and shorter bonding distances than present in solids [1, 2]. In this paper we report the observation of a considerable band-like dispersion of the C 1 s core level in graphene. The dispersion is observed as an emission-angle dependent binding energy modulation and it is shown that under appropriate conditions only the bonding or anti-bonding states can be observed. 

The binding energy modulations of the C 1 s core state are illustrated in Fig. 1(a)-(e). 

Each panel shows a group of spectra taken at a fixed polar emission angle θ as a function of azimuthal emission angle φ . Clear shifts of the peak position are observed. Fig. 1(f) shows a comparison of the spectrum taken at normal emission and one taken at θ = 25 ◦ together with the result of a peak fit to these two spectra. Details of the fitting procedure are described in the supporting material. 

The binding energy variation obtained from the peak fitting is given in Fig. 1(g) with the markers corresponding to the spectra in (a)-(e). Strong changes are evident with the largest difference of binding energies spanning ≈ 60 meV. The variation is consistent with the point symmetry of the graphene lattice. Fig. 1(h) shows the intensity variations of the peaks, displayed as the modulation function (see supplementary material). Strong inten- sity modulations are observed, caused by photoelectron diffraction in the final state. The variations follow the point symmetry of the graphene lattice, but they do not appear to be correlated with the binding energy in (g), with the main structures being in phase for some polar emission angles and out of phase for others. 

While the data of Fig. 1 serve to illustrate the nature of the effect and the fitting procedure, they are insufficient to pin down the physical origin of the modulation. Fig. 2 therefore shows a much more extensive data set measured over many polar and azimuthal angles and at different photon energies. Again, a good fit to all the spectra in the data set was obtained for a single C 1 s component using always the same line shape. Note that this 

