Therefore, in this hv range, the ARPES final state incorporates two bands, one having the free-electron k z and another having k z +∆ k z shifted from the free-electron dispersion, with the ARPES intensity distribution between these bands defined by the energy-and kdependent photoemission matrix elements. In our case, we find ∆k z ≈ 1.0371 A -1 , equal to |Γ X ′ |. Conventionally, such final-state bands are called Umklapp bands, implying an Umklapp scattering process of the photoelectrons , although the corresponding momentum transfer is not necessarily an integer reciprocal-lattice vector. A rigorous description of such multi-band final states based on the one-step theory of photoemission, where they are treated as the time-reversed low-energy electron diffraction (LEED) states, can be found in Refs., and the references therein.