The impact of a primary electron initiates a cascade of secondary electrons in solids, and these cascades play a significant role in the dynamics of ionization. Here we describe model calculations to follow the spatiotemporal evolution of secondary electron cascades in diamond. The band structure of the insulator has been explicitly incorporated into the calculations as it affects ionizations from the valence band. A Monte Carlo model was constructed to describe the path of electrons following the impact of a single electron of energy $E\sim 250\hspace{0.5em}\mathrm{eV}.$ This energy is similar to the energy of an Auger electron from carbon. Two limiting cases were considered: the case in which electrons transmit energy to the lattice, and the case where no such energy transfer is permitted. The results show the evolution of the secondary electron cascades in terms of the number of electrons liberated, the spatial distribution of these electrons, and the energy distribution among the electrons as a function of time. The predicted ionization rates $(\sim 5–13$ electrons in 100 fs) lie within the limits given by experiments and phenomenological models. Calculation of the local electron density and the corresponding Debye length shows that the latter is systematically larger than the radius of the electron cloud, and it increases exponentially with the radial size of the cascade. This means that the long-range Coulomb field is not shielded within this cloud, and the electron gas generated does not represent a plasma in a single impact cascade triggered by an electron of $E\sim 250\hspace{0.5em}\mathrm{eV}$ energy. This is important as it justifies the independent-electron approximation used in the model. At $1\hspace{0.5em}\mathrm{fs},$ the (average) spatial distribution of secondary electrons is anisotropic with the electron cloud elongated in the direction of the primary impact. The maximal radius of the cascade is about $50\hspace{0.5em}\mathrm{\AA }$ at this time. At $10\hspace{0.5em}\mathrm{fs}$ the cascade has a maximal radius of $\sim 70\hspace{0.5em}\mathrm{\AA },$ and is already dominated by low-energy electrons $(>\mathrm{}50\%,$ $E<\mathrm{}10\mathrm{}\hspace{0.5em}\mathrm{eV}).\mathrm{}$ These electrons do not contribute to ionization but exchange energies with the lattice. As the system cools, energy is distributed more equally, and the spatial distribution of the electron cloud becomes isotropic. At 90 fs, the maximal radius is about 150 Å. An analysis of the ionization fraction shows that the ionization level needed to create an Auger electron plasma in diamond will be reached with a dose of $\sim 2\times {10}^5$ impact x-ray photons per ${\mathrm{\AA }}^2$ if these photons arrive before the cascade electrons recombine. The Monte Carlo model described here could be adopted for the investigation of radiation damage in other insulators and has implications for planned experiments with intense femtosecond x-ray sources.

• Received 19 March 2002

DOI:https://doi.org/10.1103/PhysRevB.66.024116