  Pump-and-probe photoelectron spectroscopy (PES) with femtosecond pulsed
sources opens new perspectives in the investigation of the ultrafast dynamics
of physical and chemical processes at the surfaces and interfaces of solids.
Nevertheless, for very intense photon pulses a large number of photoelectrons
are simultaneously emitted and their mutual Coulomb repulsion is sufficiently
strong to significantly modify their trajectory and kinetic energy. This
phenomenon, referred as space-charge effect, determines a broadening and shift
in energy for the typical PES structures and a dramatic loss of energy
resolution. In this article we examine the effects of space charge in PES with
a particular focus on time-resolved hard X-ray (~10 keV) experiments. The
trajectory of the electrons photoemitted from pure Cu in a hard X-ray PES
experiment has been reproduced through $N$-body simulations and the broadening
of the photoemission core-level peaks has been monitored as a function of
various parameters (photons per pulse, linear dimension of the photon spot,
photon energy). The energy broadening results directly proportional to the
number $N$ of electrons emitted per pulse (mainly represented by secondary
electrons) and inversely proportional to the linear dimension $a$ of the photon
spot on the sample surface, in agreement with the literature data about
ultraviolet and soft X-ray experiments. The evolution in time of the energy
broadening during the flight of the photoelectrons is also studied. Despite its
detrimental consequences on the energy spectra, we found that space charge has
negligible effects on the momentum distribution of photoelectrons and a
momentum broadening is not expected to affect angle-resolved experiments.
Strategy to reduce the energy broadening and the feasibility of hard X-ray PES
experiments at the new free-electron laser facilities are discussed.
