Predictive Formula for Electron Penetration Depth of Diverse Materials over Large Energy Ranges

Anne C. Starley, Utah State University
Gregory Wilson, Utah State University
Lisa Montierth Phillipps, Utah State University
JR Dennison, Utah State Univesity


An empirical model developed by the Materials Research Group that predicts the approximate electron penetration depth—or range—of some common materials has been extended to predict the range for a broad assortment of other materials. The electron range of a material is the maximum distance electrons can travel through a material, before losing all of their incident kinetic energy. The original model used the Continuous-Slow-Down-Approximation for energy deposition in a material to develop a composite analytical formula which estimated the range from <10 eV to>10 MeV with an uncertainty of <20% using a single fitting parameter, Nv, which describes the effective number of valence electrons. NV was empirically calculated for >200 materials which have tabulated range and inelastic mean free path data in the NIST ESTAR and IMFP databases. Correlations of NV with key material constants (e.g., density, atomic number, atomic weight, and band gap) were established for this large set of materials. Somewhat different correlations were found for different sub-classes of materials (e.g., solids/liquids/gases, conductors/semiconductors/insulators, elements/compounds/polymers/composites). Values of the average energy lost per inelastic collision were related to band gap and plasmon energies for solids and first excitation energies for liquids and gases. Simulations were performed to test the sensitivity of NV and the range to materials parameters; these suggest that reasonably accurate results were achievable with modest precision of the parameters. These correlations have led to methods using only basic material properties to predict Nv and the range for additional untested materials which have no supporting range data. Estimates for both simple compounds (e.g., BN and AlN), composites, and complex biological materials (e.g., brain tissue and cortical bone tissue) are presented, along with tests of the validity and accuracy of the predictive formula. These calculations are of great value for studies involving high energy electron bombardment, such as electron spectroscopy, spacecraft charging, or electron beam therapy. Efforts are underway to create a user tool available to the scientific community to estimate the range of an arbitrary material with modest accuracy over an extended width of incident electron energies.

*Supported through funding from NASA Goddard Space Flight Center and a USU URCO Fellowship.