An empirical model 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 (CSDA) for energy deposition in a material to develop a composite analytical formula which estimated the range from10 MeV with an uncertainty of200 materials which have tabulated range and inelastic mean free path data in the NIST ESTAR and IMFP databases. Correlations of  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/semi-conductors/insulators, elements/compounds/polymers/ composites). A predictive formula was developed to accurately determine  for arbitrary materials

Dennison, JR

Phillipps, Lisa

Starley, Anne C

Wilson, Gregory

English

DigitalCommons@USU

Utah State University DigitalCommons@USU Posters Materials Physics 9-9-2016 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 Phillipps Utah State University JR Dennison Utah State Univesity Follow this and additional works at: https://digitalcommons.usu.edu/mp_post  Part of the Condensed Matter Physics Commons Recommended Citation Anne Starley, Gregory Wilson, Lisa Phillipps and JR Dennison, “Predictive Formula for Electron Penetration Depth of Diverse Materials over Large Energy Ranges,” USU Fall Undergraduate Research Orientation, Logan, UT September 9, 2016. This Conference Poster is brought to you for free and open access by the Materials Physics at DigitalCommons@USU. It has been accepted for inclusion in Posters by an authorized administrator of DigitalCommons@USU. For more information, please contact digitalcommons@usu.edu. I. Introduction to RangeThe range, commonly known as the penetration depth, describes the maximum distanceelectrons can travel through a material, given an initial incident energy, before they loseall of their kinetic energy and come to a rest.1,2 The primary energy loss mechanism whichcauses the electron to lose its kinetic energy is due to inelastic collisions within material.Due to the probabilistic nature of this mechanism, the Continuous Slow DownApproximation (CSDA) is often employed to simplify the problem where the stoppingpower is taken as a constant.Predictive Formula For Electron Range over a Large Span of EnergiesAnne Starley, Gregory Wilson, Lisa Phillipps, and JR DennisonUSU Materials Physics GroupUtah State University, Logan, UT 84332-4414III. Predictive Formula for NVA simple formula [Eq. (4)] was found to predict our single range parameter, Nv , as a function of only mean atomicnumber derived from the stoichiometric formula. This simple formula resulted from extensive analysis of muchmore complex predictive formula for NV involving sums of power law terms for density, effective atomic number andweight, and bandgap plus other properties including plasmon energy, conductivity, phase, and more. This general fitfor NV was evaluated using general least squares fit analysis methods to simultaneously determine the bestestimates for fitting parameters for each material property. Amazingly, the simple formula Eq. (4) involving only Zmwas the result.Fig. 3 and Table 1 show the results of the fits with Eq. (4) for A,B and C. The fitting parameters was then used tocalculate an estimate of Nv using the power law model. Plotting this estimate of Nv versus the true value of NvGoodness of fit metrics χ2Nv and R allows quantification of the quality of the fits.To refine Eq. (4), separate fits were made for materials subcategorized into grouping such as insulators,semiconductors, and conductors [see Figs. 3(b-d)] and solids, liquids, and gasses with the hope that thiscategorization might unearth additional information. Semiconductor show excellent agreement. Insulators showvery good agreement, with a slight downward concavity. Although conductors show good agreement, it is apparentthat an additional higher order correction for conductors needed is needed to account for electron overlap in the dand f orbitals of transition and rare earth/actinide elements.VI. Future Work We will create an online range prediction calculator that will be able to produce therange of a material with only input of the common material parameters mass densityρm, effective atomic number and weight NA and MA from the stoichiometric formula (orrelative amounts of elements in compounds and composites), and band gap Egap (orHOMO-LUMO gap). We propose to develop a better relativistic approximation for Eq. (1) to improverange predictions above mec2=0.5 MeV. We prose to improve NV predictions by adding addition Zm dependence in Eq. (4). Gas and liquid materials do not have band gaps, a necessary parameter in our rangemodel. The HOMO-LUMO energy gaps may be an appropriate surrogate. We will studythe HOMO-LUMO gap and its potential connection to the range.Fig. 3. The graphs examine NV found from fits to the NIST database data versus NV predicted using Eq. (4). Nominally, for exact agreement the slope of the fit would be 1 with an intercept of 0 and χ2Nvapproaching 0. 10% and 30% error lines to the slope are marked in dashed red and dashed purple lines, respectively. Values for the constants for predictive NV from Eq. (4) and the goodness of linear fitfor Fig. (3) graphs are found in Table 1.This idea is illustrated by aLichtenburg discharged treepictured in Fig. 1. This “tree”demonstrates a situationwhere accelerated highenergy electrons comes torest and deposit charge at agiven range in an insulatingmaterial.3 The side view ofthe Lichtenburg tree displaysthe melted plastic caused bythe energy of the depositedincident electrons at auniform penetration depth.Here the stored charge isdissipated through adischarge.1Fig. 1. Front (Left) and side (Right) views of a Lichtenbergdischarge tree. The white line (Right) indicates the narrowdistribution of deposited charge from a ~1 MeV electronbeam at R≈3 mm in a PMMA sample.Scan code to access the USU MaterialPhysics Group papers andpresentations.II. Original ModelThe previously developed model predicts the energy-dependent range, R(E), as a functionof incident electron energy for materials found in the NIST ESTAR database. In acontinuous composite analytic approximation to the range with a single fitting parameter,NV, spanning incident energies, E, from <10 eV to >10 MeV, the following functions (Eqs. 2,3, and 4) describe the energy-dependent range, R(E), in terms of NV and materialparameters mass density ρm, effective atomic number and weight NA and MA and band gapEgap.1Eq. (2). Mean energy lost percollision.Eq. (3). Plasmon energy.Manual fits to these rangeequations and optimumvalues of Nv were foundusing data for about 20 well-known elements andcompounds with range datafrom the ESTAR database.2Fig. 2 shows several approx-imate fits to the range data.Fig. 2. Comparison between several range approximations and the data from the ESTAR databasefor Au. The IMFP data for Au are also plotted along with the TPP-2M IMFP formula for λIMFP(E).V. Materials with Predicted RangesVII. References1. Wilson, G., & Dennison, J.R. (2010). Approximation of range in materials as a function of incident electron energy.2. Teancum Quist (with Greg Wilson and JR Dennison), “Compilation and Comparison of Electron Penetration Ranges as a Functionof Effective Number of Valence Electrons,” Utah State University, Logan, UT, April 2013.3. Starley, A., Phillipps, L., Wilson, G., Dennison, J.R., Predictive Formula For Electron Range Over A Large Span of Energies APSFour Corners Poster presented at APS Four Corners Meeting 2015,Arizona State University, Mesa, AZ, 2015.4. Meloy, R., private communications, 2016.All Materials InsulatorsConductorsSemiconductorsEq. (4). Predictive formula for effective number ofvalence electrons, where Zm the is mean atomic numberand A, B, and C are constants found through fits shownin Fig. 3.Solids Actinium 13.576 Adenine 2.408 Alumina (Aluminum Oxide) 4.538 Bakelite 2.047 A-150 Tissue-Equivalent Plastic 4.562Aluminium 5.195 Alanine 1.945 Barium Fluoride 7.125 C-552 Air-Equivalent Plastic 2.831 Adipose Tissue (ICRP) 1.554Americium 13.151 Cellulose Nitrate 31.722 Barium Sulfate 5.817 Cellulose Acetate Butyrate 2.059 Amber 1.462Antimony 10.59 Epoxy (EP) 1.805 Beryllium oxide 3.734 Cellulose Acetate, Cellophane 2.23 Anthracene 1.616Arsenic 8.404 Glutamine 2.064 Bismuth Germanium Oxide 7.025 Ethyl Cellulose 1.857 B-100 Bone-Equivalent Plastic 2.047Astatine 13.359 Guanine 2.581 Boron Carbide 2.7914 M3 Wax 1.452 Bone, Compact (ICRU) 2.129Barium 12.334 Stilbene 1.743 Boron Oxide 3.453 Mix D Wax 1.285 Bone, Cortical (ICRP) 2.391Berkelium 13.242 Sucrose 2.17 Cadmium Telluride 10.501 Nylon 11 (Rilsan) 1.63 Brain (ICRP) 2.73Beryllium 2.056 Cadmium Tungstate 6.4 Nylon 6 1.693 Calcium Carbonate 1.549Bismuth 13.063 Calcium Fluoride 5.317 Nylon 6/10 1.625 Eye Lens (ICRP) 1.658Boron 2.662 Calcium Oxide 5.555 Nylon 6/6 1.578 Lung (ICRP) 1.638Cadmium 9.712 Calcium Sulfate 4.804 Nylon, Du Pont ELVAmide 8062 1.622 MS20 Tissue Substitute 1.709Caesium 12.983 Calcium Tungstate 6.16 Paraffin Wax (26-n-paraffin) 1.17 Skin (ICRP) 1.642Calcium 7.062 Ceric Sulfate Dosimeter Solution 1.596 PC (Lexan) 2.067 Terphenyl 1.86Californium 14.302 Cesium Fluoride (Caesium Fluoride) 8.501 PE (polyethylene) 1.206 Tissue, Soft (ICRP) 1.563Carbon (Amorphous) 3.117 Ferric Oxide 5.339 PET (Mylar) 2.396 Tissue, Soft (ICRU four-component) 1.611Carbon (Diamond) 3.438 Ferroboride 5.03 PI (Kapton) 2.568Carbon (Graphite) 3.06 Ferrous Oxide 5.506 Plastic Scintillator (Vinyltoluene based) 1.592Cerium 11.451 Ferrous Sulfate Dosimeter Solution 1.585 PMMA (Lucite) 1.972Chromium 6.716 Gadolinium Oxysulfide 7.687 Polyacrylonitrile 1.995Cobalt 6.849 Gallium Arsenide 8.269 Polychlorostyrene 2.284Copper 7.125 Glass (Lead) 6.586 Polyoxymethylene 2.082Curium 13.296 Glass (Plate) 4.558 Polypropylene 1.184Dysprosium 11.717 Glass (Pyrex) 4.261 Polyvinyl Acetate (PVA) 2.002Erbium 11.762 Gypsum, Plaster of Paris 3.562 Polyvinyl Alcohol 1.876Europium 12.411 Lanthanum Oxybromide 8.214 Polyvinyl Butyral 1.558Francium 18.071 Lanthanum Oxysulfide 7.647 Polyvinyl Pyrrolidone 1.792Gadolinium 11.716 Lead Oxide 8.926 Polyvinylidene Chloride, Saran 3.629Germanium 8.328 Lithium Amide 1.5 Polyvinylidene Fluoride 2.753Gold 10.814 Lithium Carbonate (Lithium Salt) 3.105 PS (polystyrene) 1.74Hafnium 11.237 Lithium Fluoride 3.714 PTFE (Teflon) 3.755Holmium 11.729 Lithium Hydride 0.764 PVC (polyvinyl Chloride) 2.584Indium 10.142 Lithium Iodide 7.836 Rubber, Butyl 2.028Iodine 11.385 Lithium Oxide 2.657 Rubber, Natural 1.324Iridium 10.245 Lithium Tetraborate 3.251 Rubber, Neoprene 2.223Iron 6.881 Magnesium Carbonate 4.022 Valine 1.7Lanthanum 11.528 Magnesium Fluoride 4.813 Viton  Fluoroelastomer 3.388Lead 12.56 Magnesium Oxide 4.568Lithium 1.179 Magnesium Tetraborate 3.76Lutetium 11.874 Mercuric Iodide 11.744Magnesium 4.938 Plutonium Dioxide 7.553Manganese 6.816 Potassium Iodide 9.417Molybdenum 8.676 Potassium Oxide 5.768Neodymium 11.578 Silicon Glass 4.716Neptunium 11.814 Silver Bromide 9.22Nickel 6.971 Silver Chloride 8.186Niobium 8.858 Silver Iodide 10.518Osmium 10.145 Sodium Carbonate 4.035Palladium 8.776 Sodium Iodide 8.562Phosphorus (Black) 5.719 Sodium Monoxide 4.354Phosphorus (Red) 5.747 Sodium Nitrate 3.974Phosphorus (White) 5.756 Thallium Chloride 10.04Platinum 10.42 Titanium Dioxide 4.929Plutonium 12.101 Tungsten Hexafluoride 6.175Polonium 13.256 Uranium Dicarbide 7.233Potassium 6.848 Uranium Monocarbide 8.684Praseodymium 11.526 Uranium Oxide 7.481Promethium 11.576Protactinium 12.508Radium 15.23Rhenium 10.258Rhodium 8.606Rubidium 10.276Ruthenium 8.497Samarium 11.619Scandium 6.995Selenium (alpha) 8.886Selenium (grey) 8.785Selenium (vitreous) 8.912Silicon 5.493Silver 9.171Sodium 4.521Strontium 10.074Sulphur 6.044Tantalum 10.723Technetium 8.597Tellurium 10.874Terbium 11.703Thallium 12.4Thorium 13.28Thulium 11.785Tin (Grey) 10.291Tin (White) 10.291Titanium 6.88Tungsten 10.416Uranium 11.978Vanadium 6.749Ytterbium 12.618Yttrium 9.627Zinc 7.613Zurconium 9.5Liquids Gallium 8.051 1,2-Dichloroethane 2.922 Carbon Tetrachloride 5.793 Polytrifluorochloroethylene 4.22 Acetone 1.3971+B3:G96,2-Dichlorobenzene 2.922 Cesium Iodide (Caesium Iodide) 11.474 N,N-Dimethyl Formamide 1.546Aniline 1.828 Nitrobenzene 2.277Benzene 1.61 Pyridine 1.826Chlorobenzene 2.304Cyclohexane 1.122Dichlorodiethyl Ether 2.346Diethyl Ether 1.152Dimethyl Sulfoxide 1.962Naphthalene 1.869N-Heptane 1.001N-Hexane 0.844N-Pentane 0.946N-Propyl Alcohol 1.214Tetrachloroethylene 5.341Toluene 1.513Trichloroethylene 4.52Triethyl Phosphate 1.782Xylene 1.463Butane 0.397Chloroform 4.802Ethane 0.323Ethylene 0.705Gasses Carbon Dioxide 2.349 Photographic Emulsion 172.157 Acetylene 0.7Nitrous Oxide 2.381 Gel in Photographic Emulsion 1.856Silver Halides in Photographic Emulsion 7.602Tissue-Equivalent GAS (Methane based) 0.62Tissue-Equivalent GAS (Propane based) 0.637Elements Organic Compound Inorganic Compound Polymer CompositeIV. Accuracy of Range for Predicted NVMaterialEmpirical NVPredictedNVPercentErrorSiO2 4.716 4.646 1.5%Al2O3 4.538 4.652 2.5%Si 5.493 5.702 3.7%Al 5.195 5.468 5.1%Au 10.814 11.800 9% Sr 10.074 9.071 10%Table 2. Differences in the empirical and predicted NV for various materials.Fig. 4. (a) Strontium range calculated using both empirical and predicted NV values as compared to ESTARrange data. (b) Percent differences for both medium and high energy regimes for Sr range calculatedranges for both empirical and predicted NV values as compared to ESTAR range and inelastic mean freepath data. (c) Percent differences between ranges calculated with empirical and predicted NV values forSiO2, Al2O3, Si, Al, Au and Sr. (d) χ2 values for comparison of ESTAR range data to ranges calculated for boththe empirical and predicted NV values for all materials in the NIST material database (see Table 3). The χ2values for Si, Au and Sr with predicted NV are highlighted, illustrating their extreme variances from theempirical data.Fig. 5. (inset) Number ofelectrons deposited versusdepth of electronpenetration in Al for severalbeam energies usingGEANT4. The dotted linesindicate CSDA range aspredicted by Eq. (1). (Maingraph) Fraction of electronsdeposited normalized by themaximum deposition fractionas a function of penetrationdepth scaled by the CSDArange.We gratefully acknowledge contributions from othermembers of the Materials Physics Groups. This work wassupported by USU URCO fellowships and a NASA SpaceTechnology Research Fellowship.The fraction of electrons deposited as a function of penetration depth can be calculated withadvanced simulation programs such as GEANT4, as shown in the inset of Fig. 5. Fig. 5 shows thefraction of electrons deposited normalized by the maximum deposition fraction as a function ofpenetration depth scaled by the CSDA range. The dotted lines indicate CSDA range as predictedby Eq. (1). Note for energies of 10 keV and above, where GEANT4 simulations are valid, thescaled curves are the same with a deposition fraction ~0.2% at the CSDA range. This suggeststhat the single mean ranges for energy in our CSDA approximation can be broadened byconvolution with this universal curve to predict approximate internal charge deposition curvesfor a wide span of energies for any material using our composite formula for the range andpredictive formula for NV.𝑵𝑽(𝒁𝒎) = 𝑨𝒁𝒎𝑩 − 𝑪Eq. (1). Range for low, medium, and high energy regimes.Range values as a function of energy predicted withempirical NV values derived from fits to the NISTdatabase range and inelastic mean free path valueswere found to give good fits with differences typicallyless (often much less) than ±20% over full 10 eV<E<10MeV spans.1 Fig. 4(a) shows such a plot for one of theworst fit materials, Sr [see in Fig. 3(d) and Fig. 4(b) andTable 2].To better gauge the validity of the predictive formula forNV, Eq. (4), comparisons are made of ranges calculatedwith both empirical and predicted NV values (found inTable 3). Comparisons are shown in Figs. 4(b) and 4(c)for five different materials [including two materials withthe highest inconsistency between the two data sets--strontium (Sr) and gold (Au)]. Agreement between theranges calculated with predicted NV values and ESTARrange data are shown Fig. 4(d) for all 249 materials inTable 3.Table 1. Constants and Goodness of fit predictive NV fits.  Materials A B C χ2 Nv RAll 20.196 0.653 21.727 0.006 0.99Insulators 21.581 0.730 22.892 0.0032 0.99Conductors 22.898 0.604 24.982 0.0053 0.99Semiconductors 14.817 0.153 16.585 0.0055 0.99conductorssemi-conductorsinsulators Conductors

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

Montierth, Lisa D.

Utah State University DigitalCommons@USU Conference Proceedings Materials Physics 2016 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 D. Montierth Utah State University JR Dennison Utah State Univesity Follow this and additional works at: https://digitalcommons.usu.edu/mp_conf  Part of the Condensed Matter Physics Commons Recommended Citation Starley, Anne C.; Wilson, Gregory; Montierth, Lisa D.; and Dennison, JR, "Predictive Formula for Electron Penetration Depth of Diverse Materials over Large Energy Ranges" (2016). 14th Spacecraft Charging Technology Conference, ESA/ESTEC. Conference Proceedings. Paper 21. https://digitalcommons.usu.edu/mp_conf/21 This Conference Paper is brought to you for free and open access by the Materials Physics at DigitalCommons@USU. It has been accepted for inclusion in Conference Proceedings by an authorized administrator of DigitalCommons@USU. For more information, please contact digitalcommons@usu.edu. 14th Spacecraft Charging Technology Conference, ESA/ESTEC, Noordwijk, NL, 04-08 APRIL 2016                   1  PREDICTIVE FORMULA FOR ELECTRON PENETRATION DEPTH OF DIVERSE MATERIALS OVER LARGE ENERGY RANGES  Anne Starley, Gregory Wilson, Lisa Phillipps and JR Dennison  Utah State University Dept. of Physics, 4415 Old Main Hill, Logan, UT 84322-4415 USA, anne.starley@aggiemail.usu.edu   ABSTRACT An empirical model 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 (CSDA) 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 empirical fitting parameter, 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒, which is termed the effective number of valence electrons.    𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 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 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 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/semi-conductors/insulators, elements/compounds/polymers/ composites). A predictive formula was developed to accurately determine 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 for arbitrary materials.  1. INTRODUCTION The range or penetration depth, R, describes the maximum distance electrons can travel through a material, given an initial incident energy, before they lose all of their kinetic energy and come to a rest. It is a common way to parameterize electron interactions with materials.1,2   An approximate range expression was developed by merging well known semi-empirical models for the interaction of electrons with materials in different energy regimes by employing the continuous slow down approximation (CSDA); details are provided in [3].  Using the CSDA, a continuous, simple, composite, analytic formula—with the single empirical free parameter, 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒— approximates the range (10-9 m to 10-2 m) over an extended energy span (<10 eV to >10 MeV).  Agreement with available databases of electron interactions are within <20% for many conducting, semiconducting, and insulating materials.4    This paper describes efforts to develop a predictive formula for this single fitting parameter based solely on the stoichiometry of the material and its basic material properties. The validity of this process is examined by comparison of the range predicted with this formula to tabulated range data in the NIST databases.1,2  2. RANGE ACCURACY OF PREDICTED 𝑵𝑵𝑽𝑽𝒆𝒆𝒆𝒆𝒆𝒆 Range values as a function of energy were determined as empirical 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 values derived from fits to the NIST database range and inelastic mean free path values.  Original fits to ~20 materials3 have now been extended to almost all of the >200 diverse materials found in the NIST databases.1,2  These were found to give good fits with differences typically less (often much less) than ±20% over full 10 eV<E<10 MeV spans.3    3. PREDICTIVE FORMULA FOR 𝑵𝑵𝑽𝑽𝒆𝒆𝒆𝒆𝒆𝒆 A simple formula was found to predict the single range parameter, 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒:   𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒(𝑍𝑍𝑚𝑚) = 𝑁𝑁𝑜𝑜 𝑍𝑍𝑚𝑚𝑛𝑛0 + 𝑁𝑁𝑜𝑜𝑒𝑒𝑒𝑒𝑜𝑜𝑒𝑒𝑜𝑜  .            (1)  This formula was found through extensive analysis of much more complex predictive formula for 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒   involving products of power law terms for density, mean atomic number and weight, and bandgap plus other properties including plasmon energy, conductivity, phase, and more.5 This general fit for 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒  was evaluated using general least squares fit analysis methods to simultaneously determine the best estimates for fitting parameters for each material property.    Remarkably, this predictive formula for effective number of valence electrons was a function of only mean atomic number, Zm, which can be easily determined from the stoichiometric formula for compounds or from elemental fraction for composite materials.    4. RANGE ACCURACY OF PREDICTED 𝑵𝑵𝑽𝑽𝒆𝒆𝒆𝒆𝒆𝒆 Range values as a function of energy predicted with empirical 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 values derived from fits to the NIST database range and inelastic mean free path values were found to give good fits with differences typically less (often much less) than ±20% over full 10 eV<E<10 MeV spans.3   Fig. 1(a) shows the range of Sr calculated using both empirical 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒and predicted 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 values; Sr and Au are two of the materials with the worst agreement with measured data in the NIST databases.    This work was supported by a Utah State University Undergraduate Research and Creative Opportunities Grant and funding through NASA GSFC and the James Webb Space Telescope. 14th Spacecraft Charging Technology Conference, ESA/ESTEC, Noordwijk, NL, 04-08 APRIL 2016                   2  To better gauge the validity of the predictive formula for 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 as given by Eq. (1), comparisons of the range were made using the empirical 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 and predicted  𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 values with excellent results.  By plotting this prediction, 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒, versus the best values of 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 from direct fits to the NIST database, goodness of fit metrics χ𝑁𝑁𝑁𝑁2  and R allows quantification of the quality of the fits.  The fitting parameters were then used to calculate 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 using Eq. (1). Table 1 show the results of these comparisons, for all materials from the NIST database, and subdivided by conduction type.  Fig. 1(b) shows the percent differences between ranges calculated with empirical 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒and predicted 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 values for several representative materials.  5. CONCLUSIONS Simulations were performed to test the sensitivity of 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒and 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 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 and the range for additional untested materials which have no supporting range data.  These calculations are of great value for studies involving energetic 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 span of incident electron energies.   Future work will: • Create an online range prediction calculator that will be able to produce the range of a material with only input of the common material parameters. • Develop a better relativistic approximation to improve range predictions above mec2=0.5 MeV. • Imrove 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 predictions by adding additional Zm dependence and orbital corrections in Eq. (1). • Study the HOMO-LUMO gap, ELH, as a possible surrogate for liquids and gases of the band gap in solids. • Model the approximate internal charge deposition as the fraction of electrons deposited as a function of penetration depth scaled by the CSDA range determined by the predictive formula for 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒through convolution of a universal normalized deposition curve.   6. REFERENCES  1. National Institute of Standards and Technology, 2010, "ESTAR, Stopping Power and Range Tables for Electrons," (http://physics.nist.gov/PhysRefData/Star/Text/ESTAR.html). 2. National Institute of Standards and Technology, 2010, “NIST Electron Inelastic-Mean-Free-Path Database: Version 1.1,” (http://www.nist.gov/data/nist71.htm). 3. Wilson, G., & Dennison, J.R. (2010). Approximation of range in materials as a function of incident electron energy, IEEE Trans. on Plasma Sci., 40(2), 305-310. 4. T. Quist, B. Moore, G. Wilson and JR Dennison, Electron Penetration Ranges as a Function of Effective Number of Valence Electrons, Utah State University Student Showcase, Logan, UT, April 2013. 5. Starley, A., Phillipps, L., Wilson, G., Dennison, J.R., Predictive Formula for Electron Range over a Large Span of Energies, Am. Phys. Soc. Four Corners Meeting, Mesa, AZ, 2015.     Table 1. Constants and goodness of fit for predictive 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 model. Materials No no Noffset 𝝌𝝌𝑵𝑵𝑵𝑵𝟐𝟐  R All 21.223 0.112 22.854 0.0061 0.987 Insulators 22.712 0.101 24.114 0.0042 0.987 Conductors 24.219 0.107 26.690 0.0052 0.949 Semiconductors 14.817 0.153 16.585 0.0005 0.990  Figure 1. (a) Strontium range calculated using both empirical 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 and predicted 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 values as compared to ESTAR range data. (b) Percent differences between ranges calculated with empirical 𝑁𝑁𝑉𝑉𝑒𝑒𝑒𝑒𝑒𝑒 and predicted 𝑁𝑁𝑉𝑉𝑝𝑝𝑝𝑝𝑒𝑒 values, versus incident energy for Al, Au, Si, Sr, Al2O3 and SiO2. 

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

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