Stopping Power Formula Research Articles

Dose increment in ultrashort particle irradiations via coherent stopping power

We present a relativistic self-consistent theory of the coherent stopping power (CSP) of ultradense charged particle beams propagating in a dense matter. CSP corresponds to a collective inelastic collision which adds to the ordinary stopping power of individual particles. Unlike the latter, which depends only on the particles' energy, CSP depends upon many more parameters such as the total charge of the ensemble and its charge density and shape. This paves the way for a broad variety of novel methods to tailor particle absorption and penetration in a dense matter. CSP losses can be explained both by collective excitations of single or multiple molecules and by the emission of coherent Cherenkov radiation. Here we demonstrate that the former mechanism is more relevant than the latter. We find that the coherent energy absorption of subpicosecond particle bunches in water occurs exciting a broad Debye process in the GHz range and an intermolecular stretching vibration mode in the THz region. We generalize the Bethe-Bloch stopping power formula to coherent effects including self-forces, dielectric screening, and absorption by the dense matter. For the sake of a self-consistent dynamical theory including phase space evolution, we have also generalized the Fermi-Eyges theory of particle diffusion to the presence of forces. Nonlinear dynamics is demonstrated, inducing nonlinear dose release, beam self-focusing, and self-enhancement of coherent losses. Given the advent of ultraintense particle sources and their use for biomedical and other relevant applications, our results may be of paramount importance for contemporary and future developments of science and technology. Published by the American Physical Society 2024

Precise measurement of the Bragg curve for 800 MeV/u 238U ions stopping in polyethylene and its implications for calculation of heavy ion ranges

Stopping power predictions in radiation transport codes are based on the Bethe-Bloch formula and different corrections. For very heavy ions at relativistic energies the available experimental data are scarce and therefore verification of stopping power predictions is only possible to a limited extent.In this work, a full experimental Bragg curve for 800 MeV/u 238U ions stopping in polyethylene is presented. The measurements were conducted at the experimental area Cave A at GSI Helmholtzzentrum für Schwerionenforschung in Darmstadt, Germany. The 800 MeV/u 238U beam was provided by the SIS18 heavy ion synchrotron. The Bragg curve was measured with a setup consisting of a binary range shifter and two large area parallel plate ionization chambers. Complementary Monte Carlo simulations using the FLUKA code were performed and compared with the experimental Bragg curve. The mean ionization potential of polyethylene was fine-tuned to match the measured primary ion range with FLUKA simulations. The impact of the Bloch and Mott corrections to the stopping power calculation were studied by switching them off intentionally in separate simulations. A detailed description of the implementation of the stopping power formulae and the Mott correction in FLUKA is provided.Supplementary data for this article is available online.

Bethe stopping-power formula and its corrections

The classical and quantum theories leading to the asymptotic Bethe formula of the stopping power of matter for charged particles heavier than the electron are briefly reviewed. Models and approximations for the practical calculation of various corrections that extend the validity range of the formula are described. The asymptotic formula and the associated shell correction were determined previously from an extensive database of atomic generalized oscillator strengths, calculated for an independent-electron model with the Dirac-Hartree-Fock-Slater (DHFS) self-consistent potential, with due account for relativistic departures from the Bethe sum rule. The nonrelativistic Bloch correction is extended to the relativistic domain by means of the Lindhard-S\o{}rensen formulation, and an accurate parametrization for point projectiles with small charges is proposed. The density-effect correction and the Barkas correction are obtained from a semiempirical model of the optical oscillator strength (OOS), built from the calculated DHFS contributions of inner electron subshells plus the OOS of outer-shell electrons represented by an analytical expression, which is determined by the composition, mass density, and empirical mean excitation energy, or $I$ value of the material. Inclusion of the shell, density-effect, Lindhard-S\o{}rensen, and Barkas corrections into the asymptotic formula leads to the corrected Bethe formula. A general strategy is proposed to determine the stopping power in terms of only the $I$ value of the material. It is shown that, with the empirical $I$ values recommended in Report 37 of the International Commission on Radiation Units and Measurements, the stopping powers calculated numerically from the corrected formula are in close agreement with available measurements of the stopping power of elemental materials for protons and alpha particles with energies higher than 0.75 and 5 MeV, respectively.

A primary proton integral depth dose calculation model corrected with straight scattering track approximation

PurposeProton therapy requires a highly accurate dose calculation model. The Bortfeld analytical formula (BAF) is a classical depth-dose calculation model that exhibits a dose deviation compared to the Monte Carlo simulation. This study focuses on the lateral scattering effect on depth dose deviation between model calculations and Monte Carlo simulations and provides a modified model to correct. MethodsOnly the dose contributed by the primary particles was studied in this work. The Bethe stopping power formula was applied to reduce the deviation in the flat region. The proton track lateral scattering is considered to reduce the deviation in the Bragg peak region. Lateral scattering tracks were approximated as straight lines and assumed to have a Gaussian slope distribution. The tracks were projected to the central axis to correct the depth-dose curve. The calculated dose was compared with a Geant4 simulation. Four proton beams with energies of 50 MeV, 100 MeV, 150 MeV, and 200 MeV were calculated and verified. The dose deviation between the model calculation and Monte Carlo simulation was quantified under a dose and depth tolerance of 2%/1 mm with gamma analysis. ResultsThe straight scattering track approximation significantly reduced the primary dose deviation between the model calculation and the Geant4 simulation. The gamma values of all verified cases are below 1, which means that the deviation is acceptable under a given tolerance of 2%/1 mm. ConclusionsThis study proposes a modification of the primary depth-dose calculation model involving scattering-track correction. Good dose agreement between the calculation and Geant4 simulation was obtained. The modified model shows good accuracy in primary depth dose calculation and will be helpful for clinical application together with further secondary dose models, which are still under investigation.

Development and validation of proton track-structure model applicable to arbitrary materials

A novel transport algorithm performing proton track-structure calculations in arbitrary materials was developed. Unlike conventional algorithms, which are based on the dielectric function of the target material, our algorithm uses a total stopping power formula and single-differential cross sections of secondary electron production. The former was used to simulate energy dissipation of incident protons and the latter was used to consider secondary electron production. In this algorithm, the incident proton was transmitted freely in matter until the proton produced a secondary electron. The corresponding ionising energy loss was calculated as the sum of the ionisation energy and the kinetic energy of the secondary electron whereas the non-ionising energy loss was obtained by subtracting the ionising energy loss from the total stopping power. The most remarkable attribute of this model is its applicability to arbitrary materials, i.e. the model utilises the total stopping power and the single-differential cross sections for secondary electron production rather than the material-specific dielectric functions. Benchmarking of the stopping range, radial dose distribution, secondary electron energy spectra in liquid water, and lineal energy in tissue-equivalent gas, against the experimental data taken from literature agreed well. This indicated the accuracy of the present model even for materials other than liquid water. Regarding microscopic energy deposition, this model will be a robust tool for analysing the irradiation effects of cells, semiconductors and detectors.

Dose measurement in the TEM and STEM

Practical aspects of dosimetry are considered, including the measurement of electron-beam current and current density. Complications that arise in the case of a focused probe or a STEM image are discussed and solutions proposed. Advantages of expressing the radiation dose in Grays are listed and a simple formula given for converting electron fluence to Gray units, based on a near constancy of the stopping power per atomic electron. Comparisons with stopping-power calculations and EELS measurements suggest that this formula is accurate to within 5%. Based on the stopping power formula, a new way of measuring the local mass-thickness of light-element specimens is proposed. The average energy loss per inelastic collision is shown to be higher than previous expectations.

Bohr's stopping-power formula derived for a classical free-electron gas

Bohr's centenary stopping-power formula is rederived for a free electron gas (FEG) system within the framework of nonrelativistic classical mechanics. A simple and more concise expression for the stopping power of charged particles in FEG is demonstrated on classical grounds. Using semiclassical arguments and the Euler-Maclaurin well-known mathematical formula, Bloch's correction that links Bethe's quantum theory to Bohr's classical model is also recovered. The proposed semiclassical stopping-power formula contains the main physical ingredients for a general stopping formula applicable for different systems and energies and facilitates computational calculations.

Calculation of stopping power and range for heavy ions at intermediate energies by using Tietz screening function

In this study, a stopping power formula and range calculation procedure for incoming heavy ions were presented. For this purpose, simple analytical expressions are obtained for effective atomic number in the modified Lindhard Scharff Schiøt formula by using Tietz screening function. Also, quasi-molecule contribution to stopping power, Sei, due to the modified Lindhard Scharff Schiøt formula, were found for a wide energy region. Stopping powers for a few target elements were calculated and compared on the basis of Öztürk et al., Montenegro et al., Wang and He, Gümüş and Köksal models. Comparisons have also been made with selected experiments. The calculated results are in good agreement with experimental data for various ions and targets to within 5–10%. In addition, range values were predicted by using these formulae and were compared with SRIM results.

Proton-induced secondary electron emission from elemental solids over the energy domain 1 keV–1000 MeV

We propose a new model to calculate the proton-induced secondary electron yield (SEY) from elemental solids. The model combines the stopping power formula generated by Lindhard dielectric response function and a modified version of Bethe formula for electron stopping. The former works well at low energies while the latter at high energies. The present work further modifies the Bethe formula to adapt it to the proton stopping to complement the stopping formula generated by the dielectric theory. The combination of the stopping power formulae of the dielectric and Bethe theories are connected with the SEY formula of Batrakin et al. [Sov. Phys. JETP 62(3) (1985) 633–634] to frame the present new model. The proposed model is applied to calculate the SEY values of 18 elements with atomic numbers Zt = 3–83 over a broad energy domain 1 keV–1000 MeV. The present results show a reasonable agreement with the available experimental data and other calculations. Universal curves of the normalized SEY values against the reduced energy is drawn to verify the validity of our calculations.

Measurements of 252Cf fission product energy loss through thin silicon nitride and carbon foils, and comparison with SRIM-2013 and MCNP6.2 simulations

Gas ionization detectors are very widely used for radiation detection and measurement, but for external sources an entrance window is used which can reduce particle energy. For heavy ions this energy loss can be significant enough to affect measurements and very thin windows, such as those composed of silicon nitride (SiN), may be utilized to minimize this effect. For fission spectroscopy, carbon conversion foils are also common in measurements. In the current work, energy losses were measured for 252Cf spontaneous fission products passing through thin foils of carbon and of silicon nitride. The foils ranged in from 22.5 to 131.0 µg/cm2 for C and from 56.4 to 402.2 μg/cm2 for SiN. For comparison, simulations were performed with the TRIM program in SRIM-2013 and with MCNP6.2. To understand calculation differences, effective charge from partial ionization was predicted by several methods and compared with results directly using the Bethe stopping power formula.

A Theoretical Study of Stopping Power and Range For Low Energy (<3.0mev) Protons In Aluminium, Germanium, Lead, Gold and Copper Solid Materials

A new empirical relation was obtained by modifying an empirical relation deduced by Chaubey (1977) based on Bohr’s classical mechanics by using least squared fitting method for stopping powers from 0.20MeV to 2.90MeV protons in Aluminium (Al), Germanium (Ge), Lead (Pb), Gold (Au) and Copper (Cu) solid target materials and the results compared with some available experimental values and earlier investigations as well as PSTAR and SRIM (2013) results. The proton range relation was obtained by directly integrating the stopping power formula and the values of the ranges for the elements are calculated and compared with PSTAR and Janni (1982) values. The calculated stopping powers and range values were in excellent agreement with the experimental values of Bichsel since the percentage uncertainty was within 10% and the theoretical values of Janni (1982) and, the PSTAR and SRIM-2013 codes generated values had the percentage difference approximately within 10%. The cross section was also calculated and the results discussed. The practical applications of the stopping power, range and cross section values of the selected materials are discussed.

CSDA range, stopping power and mean penetration depth energy relationships in some hydrocarbons and biologic materials for 10 eV to 100 MeV with the modified Rohrlich–Carlson model

In this study, for some hydrocarbons and biological compounds, stopping power formula are presented, being valid for low and intermediate electron energies. In addition, calculation of the continuous slowing down approximation range (CSDA range) from the stopping power is also made. Calculation of the CSDA range for some hydrocarbons: C2H6 (ethane), C4H10 (butane), C6H14 (hexane) C8H18 (octane), C5H5N5 (adenine) and C5H5N5O (guanine) have been introduced for incident electrons in the energy range 30 eV to 1 MeV. The range of electrons has been calculated within the continuous slowing down approximation (CSDA) using modified Rohrlich and Carlson formula of stopping power. Besides, we have calculated the mean penetration depths using a spherical geometric model developed by Bentabet (Vacuum 86:1855–1859, 35). The results have been compared with the other theoretical results, Monte Carlo code such as PENELOPE predictions and semi-empirical results. The calculated results of CSDA ranges for electrons in the energy range from 20 eV to 100 MeV are found to be in good agreement to within 10% with available date.

Alternative treatment for the energy-transfer and transport cross section in dressed electron-ion binary collisions

A formula for determining the electronic stopping power and the transport cross section in electron-ion binary collisions is derived from the induced density for spherically symmetric potentials using the partial-wave expansion. In contrast to the previous one found in many textbooks, the present formula converges to the Bethe and Bloch stopping-power formulas at high ion velocities and agrees rather well with experimental stopping-power data, as shown here for Al, C, and ${\mathrm{H}}_{2}\mathrm{O}$ targets. It can be employed in plasma physics and particularly in any application that requires electronic stopping-power values of quasifree electrons with high accuracy.

Modified Bethe formula for low-energy electron stopping power without fitting parameters.

We propose a modified Bethe formula for low-energy electron stopping power without fitting parameters for a wide range of elements and compounds. This formula maintains the generality of the Bethe formula and gives reasonable agreement in comparing the predicted stopping powers for 15 elements and 6 compounds with the experimental data and those calculated within dielectric theory including the exchange effect. Use of the stopping power obtained from this formula for hydrogen silsesquioxane in Monte Carlo simulation gives the energy deposition distribution in consistent with the experimental data.

Extrapolation of the Bethe equation for electron stopping powers to intermediate and low electron energies by empirical simulation of target effective mean excitation energy and atomic number

A series of simple stopping power (SP) formulas, modified from the relativistic Bethe equation, is presented that is based on the concepts of target effective atomic number and mean excitation energy (MEE). The analytical model function is constructed to approximate experimental or calculated SPs at low electron energies and tend asymptotically to the relativistic Bethe function at high energies. The energy dependencies of our effective values, in contrast with theoretical approaches, are defined empirically by parametrization with tuning parameters. A least-squares fitting routine based on the Levenberg–Marquardt algorithm was developed. We utilize the material parameters and numerical calculations of SPs from optical data using the full Penn-algorithm. Our formula is thought to be applicable for energies above 60eV. Our simulations of SPs for 41 elemental solids are found to be in good agreement with published numerical results. The flexibility of a general empirical formula is shown. Shortened formulas were developed that are applicable for particular energy ranges, and effective MEEs are proposed that differ from previously recommended values. The presented formulas may be used for analytical calculation of SPs over a broad projectile energy region.

Formula for the total stopping power from 2 keV to 10 keV for a metal

On the basis of the range-energy relationship, the relationships (L 2–10) among the total stopping power from 2 keV to 10 keV for a metal (S 2–10), the energy exponent (n 2–10), the primary energy at the surface (W p0), and the parameter (A 2–10) were deduced. In addition, the relationships (L 10–30) among the total stopping power from 10 keV to 30 keV for a metal (S 10–30), the energy exponent (n 10–30), W p0 and the parameter (A 10–30) were obtained. According to some relationships between the parameters of the secondary electron yield from 2 keV to 10 keV for a metal (δ 2–10), the composition of the secondary electron yield from 10 keV to 30 keV for a metal (δ 10–30), L 2–10, and L 10–30, the universal formula for expressing S 2–10 as a function of S 10–30, δ 2–10, δ 10–30, the backscattered coefficient (η) from 2 keV to 10 keV, η from 10 keV to 30 keV and W p0 was deduced. The S 2–10 calculated from this universal formula and the S 2–10 measured experimentally were compared, and we conclude that the formula presented in this paper is universal for S 2–10.

Stopping power for particle therapy: The generic library libdEdx and clinically relevant stopping-power ratios for light ions

Purpose: Stopping-power data enter at a number of different places in particle therapy and their uncertainties have a direct impact on the accuracy of the therapy, e.g., in treatment planning. Furthermore, for clinical quality assurance, the particle beam stopping-power ratios (STPR) have to be known accurately for dosimetry.Methods: An open-source computer library called libdEdx (library for energy loss per unit path length, dE/dx, calculations) is developed, providing stopping-power data from data tables and computer programs as well as a stopping-power formula comprising a large list of target materials. Calculations of STPR in the case of spread-out Bragg-peaks (SOBP) are performed with the Monte Carlo transportation code SHIELD-HIT (SHIELD-Heavy Ion Transport) using different ions relevant for particle therapy.Results: For SOBP the water-to-air STPR depends on the residual range and is qualitatively very similar for different ions; however, small quantitative differences exist between the considered ion species.Conclusions: libdEdx allows for a convenient and efficient treatment of stopping powers in numerical applications. It can be applied to estimate the dependence on the accuracy of the stopping power and to provide data for an extended number of target materials. The STPR for SOBP for different ions are found to be qualitatively the same which may allow for an analytical description valid for all ions.

Measurement of absolute detection efficiencies of a microchannel plate using the charge transfer reaction

A new method for determining the absolute detection efficiencies of a microchannel plate (MCP) detector is presented. This method uses a charge transfer reaction and can be applied to determine not only the detection efficiencies for ions but also those for neutral atoms. The measurements were carried out for Ar+ and neutral rare gas atoms (Ne, Ar, Kr and Xe) in the 0.5–4.8 keV energy range. The detection efficiencies for all species increase with increasing impact energy and approach the open area ratio of the MCP used (about 50%). The measured detection efficiencies were found to scale with the Lindhard, Scharff and Schiott formula for electronic stopping power at keV energies.

Monte Carlo calculations of low-energy electron dose-point-kernels in water using different stopping power approximations

The spatial distribution of absorbed energy in irradiated matter can be conveniently described by dose-point-kernel (DPK) distributions that characterize the average energy deposition around single charged-particle tracks during their slowing down process. In the present work, electron DPKs in liquid water in the energy range from 100eV to 10keV are presented based on Monte Carlo simulation of electron transport in the continuous-slowing-down-approximation (csda). Elastic collisions are individually simulated using the screened Rutherford formula, whereas the energy loss from inelastic interactions is determined from stopping power (SP) theory. Along with the standard Bethe SP formula we examine different empirical expressions of general-use which are meant to improve the performance of the Bethe formula at low electron energies. Comparison is also made with a recent Bethe-type parametric expression obtained from a dielectric optical data model of liquid water. Our findings indicate that for electron energies below ∼1keV the discrepancies between the DPKs calculated by the general-purpose SP formulae become apparent. Moreover, the results obtained by the empirical expressions compare rather poorly with those from the dielectric model over the entire energy range examined.

On the clinical spatial resolution achievable with protons and heavier charged particle radiotherapy beams

The ‘sub-millimetre precision’ often claimed to be achievable in protons and light ion beam therapy is analysed using the Monte Carlo code SHIELD-HIT for a broad range of energies. Based on the range of possible values and uncertainties of the mean excitation energy of water and human tissues, as well as of the composition of organs and tissues, it is concluded that precision statements deserve careful reconsideration for treatment planning purposes. It is found that the range of I-values of water stated in ICRU reports 37, 49 and 73 (1984, 1993 and 2005) for the collision stopping power formulae, namely 67 eV, 75 eV and 80 eV, yields a spread of the depth of the Bragg peak of protons and heavier charged particles (carbon ions) of up to 5 or 6 mm, which is also found to be energy dependent due to other energy loss competing interaction mechanisms. The spread is similar in protons and in carbon ions having analogous practical range. Although accurate depth–dose distribution measurements in water can be used at the time of developing empirical dose calculation models, the energy dependence of the spread causes a substantial constraint. In the case of in vivo human tissues, where distribution measurements are not feasible, the problem poses a major limitation. In addition to the spread due to the currently accepted uncertainties of their I-values, a spread of the depth of the Bragg peak due to the varying compositions of soft tissues is also demonstrated, even for cases which could be considered practically identical in clinical practice. For these, the spreads found were similar to those of water or even larger, providing support to international recommendations advising that body-tissue compositions should not be given the standing of physical constants. The results show that it would be necessary to increase the margins of a clinical target volume, even in the case of a water phantom, due to an ‘intrinsic basic physics uncertainty’, adding to those margins usually considered in normal clinical practice due to anatomical or therapeutic strategy reasons. Individualized patient determination of tissue composition along the complete beam path, rather than CT Hounsfield numbers alone, would also probably be required even to reach ‘sub-centimetre precision’.