Stopping Power Theory Research Articles

Mean Excitation Energies of all ionization stages of all atoms with [formula omitted]

The presented work provides values of Mean Excitation Energy (MEE) for all atoms and their ions with atomic numbers 1≤Z≤86. To fill in the gaps in the available data, we propose an approximate atomic model for ions of high-Z elements, that uses a semi-empirical formula based on the Local Plasma Approximation (LPA). Despite the fact that the LPA, in its original form, poorly predicts MEE for high ionization states, a relatively simple modification utilizing a fit function can amend this shortcoming. We assess the importance of relativistic effects for the MEE for highly ionized atoms and compare the proposed formula to other approximations available for high-Z elements. We estimate the uncertainty of the presented data to be less than 40% in absolute value for the worst cases and less than 20% for most ions. This corresponds to an uncertainty of the order of few percent for the logarithm of MEE, which is the value of interest in the case of Bethe’s theory of stopping power.

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

3D Monte-Carlo model to study the transport of hot electrons in the context of inertial confinement fusion. Part I

We describe the development of a 3D Monte-Carlo model to study hot-electron transport in ionized or partially ionized targets, considering regimes typical of inertial confinement fusion. Electron collisions are modeled using a mixed simulation algorithm that considers both soft and hard scattering phenomena. Soft collisions are modeled according to multiple-scattering theories, i.e., considering the global effects of the scattering centers on the primary particle. Hard collisions are simulated by considering a two-body interaction between an electron and a plasma particle. Appropriate differential cross sections are adopted to correctly model scattering in ionized or partially ionized targets. In particular, an analytical form of the differential cross section that describes a collision between an electron and the nucleus of a partially ionized atom in a plasma is proposed. The loss of energy is treated according to the continuous slowing down approximation in a plasma stopping power theory. Validation against Geant4 is presented. The code will be implemented as a module in 3D hydrodynamic codes, providing a basis for the development of robust shock ignition schemes and allowing more precise interpretations of current experiments in planar or spherical geometries.

TÉCNICA ANALÍTICA PARA LA EVALUACIÓN DE LA CONTRIBUCIÓN A LA ENERGÍA MEDIA DE EXCITACIÓN DEL AGUA LÍQUIDA DEBIDO A LOS NIVELES DE EXCITACIÓN MOLECULARES

The mean excitation energy <I> is a fundamental quantity in radiation physics, concerning energy deposition in matter and absorbed dose analytical estimations for charged particles. The stopping of swift ions in different materials strongly depends on this parameter among others. This work intends to fill in part, an empty hole in the theory of stopping power: the need of analitically and theoretically assess the hIi-value for materials. The definition of the mean excitation energy using the dielectric response function is analytically integrable if the inelastic cross section parameters are known. Some dielectric models were studied, aimed at calculating the hIi-value for liquid water by theoretical means, reaching the conclusion that a decay of the order of ω −2 in frequency (energy) is needed as weak condition of the optical energy-loss function for the integrals to converge. Afterwards, the first four discrete excitation levels and the diffuse bands for water are treated in a fully analytical scheme, and further compared with numerical results, providing the contribution of these levels to hIi, with the aim of testing the proposed analytical model.

Analytical proposal for the assessment of the molecular excitation levels contribution to the mean excitation energy: Application to the water molecule

The mean excitation energy 〈I〉 is a fundamental quantity in radiation physics, concerning energy deposition in matter and absorbed dose analytical estimations for charged particles. The stopping of charged particles in different materials strongly depends on this parameter among others. This work intends to contribute with insights for some issues, usually treated not in detail in the theory of stopping power, as the analytic and theoretic assessment of 〈I〉-value for different relevant materials. A methodology is proposed and described aimed at providing the procedure for the 〈I〉 calculation framework, based on the definition of the mean excitation energy using the dielectric response function is analytically integrable if the inelastic cross section parameters are known. Some dielectric models were studied, aimed at calculating the 〈I〉-value for liquid water by theoretical means, reaching the conclusion that a decay at least of the order of ω−2 in frequency (energy) is needed as weak condition of the optical energy-loss function for the integrals to converge. Afterwards, the first four discrete excitation levels and the diffuse bands for water are treated in a fully analytical scheme, and further compared with numerical results, providing the contribution of these levels to 〈I〉, with the aim of testing the proposed analytical model.

On alpha-particle transport in inertial fusion

Analytical theory and models of inertial fusion implosion use parameterized microphysics models. In this paper, we consider the DT alpha-particle transport, and report new parameterizations of the range, heating efficacy, and energy partition using modern stopping-power theory. Our resulting heating efficacy is lower than previously published results, which reduces the temperature and pressure generated by a dynamic implosion hot-spot evolution model, and shifts the burning-plasma regime boundary slightly farther from current experimental results.

Simulations of the energy loss of ions at the stopping-power maximum in a laser-induced plasma

Simulations have been performed to study the energy loss of carbon ions in a hot, laser-generated plasma in the velocity region of the stopping-power maximum. In this parameter range, discrepancies of up to 30% exist between the various stopping theories and hardly any experimental data are available. The considered plasma, created by irradiating a thin carbon foil with two high-energy laser beams, is fully-ionized with a temperature of nearly 200 eV. To study the interaction at the maximum stopping power, Monte-Carlo calculations of the ion charge state in the plasma are carried out at a projectile energy of 0.5 MeV per nucleon. The predictions of various stopping-power theories are compared and experimental campaigns are planned for a first-time theory benchmarking in this low-velocity range.

Electronic stopping power data of heavy ions in polymeric foils in the ion energy domain of LSS theory

A continuous energy loss measurements of 63Cu, 28Si, 27Al, 24Mg, 19F, 16O and 12C ions over an energy range of (0.06–0.65)MeV/nucleon through thin polymeric foils (Mylar, Polypropylene and Formvar) were carried out by time of flight spectrometry. The deduced experimental stopping data have been used in order to assess our proposed semi empirical formula. The proposed approach based on the Firsov and Lindhard–Scharff stopping power models is provided for well describing-the electronic stopping power of heavy ions (3⩽Z<100) in various solids targets at low energy range. The ζe factor, which was approximated to be ∼Z11/6, involved in Lindhard, Scharff and Schiott (LSS) formula has been suitably modified in the light of the available experimental stopping power data. The calculated stopping power values after incorporating, effective charge Z1∗ of moving heavy ions with low velocities (v⩽v0Z12/3) and modified ζe in LSS formula, have been found to be in close agreement with measured values in various solids targets. A reason of energy loss measurements is to obtain data that help to assess our understanding of the stopping power theories. For this, the obtained results are compared with, LSS calculations, MSTAR and SRIM-2013 predictions code.

Energy partition and distribution of excited species in direction-sensitive detectors for WIMP searches

The Bragg-like curve for compounds is introduced for directional detection of galactic dark matter. The slow ion collisions are discussed in relation to direct dark matter searches. The Coulomb effect and the threshold effect in stopping power theory are examined. Ionization via molecular orbit (MO) is suggested for an additional contribution to the electronic stopping power at very slow energy.

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.

Collective deceleration: Toward a compact beam dump

With the increasing development of laser electron accelerators, electron energies beyond a GeV have been reached and higher values are expected in the near future. A conventional beam dump based on ionization or radiation loss mechanisms is cumbersome and costly, not to mention the radiological hazards. We revisit the stopping power theory of high-energy charged particles in matter and discuss the associated problem of beam dumping from the point of view of collective deceleration. The collective stopping length in an ionized gas can be several orders of magnitude shorter than that described by the Bethe-Bloch formulas and associated with multiple electromagnetic cascades in solids. At the same time, the tenuous density of the gas makes the radioactivation negligible. Such a compact beam dump without radioactivation works well for short and dense bunches, as they are typically generated from a laser wakefield accelerator. In addition, the nonuniform transverse wakefield can induce microbunching of the electron bunch by betatron oscillation. The microstructure could serve as a prebunched source for coherent radiation or feeding a free electron laser.

Differential and total ionization cross sections due to impact of swift heavy ions

In the present work, the Binary stopping power theory was used to calculate total ionization cross section and differential cross section for energy transfer from a swift heavy ion to the target electrons. The dependence on the projectile charge state is studied. The calculated values are compared with the experimental values collected from the literature.

Stopping power in a cylindrical system by the modified Bessel function’s method

On the basis of the general theory of stopping power for a system, where their electronic distributions are able to be treated as cylindrically symmetric, we derive the log-type formulae which are applied in the non-relativistic high-energy region. As well as the case of a free electron in a homogeneous system, we obtain analytically instructive solutions of stopping power in an inhomogeneous system of many-electrons, in which an effective impact parameter averaged over scattering angles is introduced. Numerical calculations are performed for fast protons into a Au〈1 0 0〉 string.

A dielectric response study of the electronic stopping power of liquid water for energetic protons and a new I-value for water

The electronic stopping power of liquid water for protons over the 50 keV to 10 MeV energy range is studied using an improved dielectric response model which is in good agreement with the best available experimental data. The mean excitation energy (I) of stopping power theory is calculated to be 77.8 eV. Shell corrections are accounted for in a self-consistent manner through analytic dispersion relations for the momentum dependence of the dielectric function. It is shown that widely used dispersion schemes based on the random-phase approximation (RPA) can result in sizeable errors due to the neglect of damping and local field effects that lead to a momentum broadening and shifting of the energy-loss function. Low-energy Born corrections for the Barkas, Bloch and charge-state effects practically cancel out down to 100 keV proton energies. Differences with ICRU Report 49 stopping power values and earlier calculations are found to be at the ∼20% level in the region of the stopping maximum. The present work overcomes the limitations of the Bethe formula below 1 MeV and improves the accuracy of previous calculations through a more consistent account of the dielectric response properties of liquid water.

The STEREO/PLASTIC response to solar wind ions (Flight measurements and models)

The Plasma and Supra-Thermal Ion Composition (PLASTIC) instrument is one of four experiment packages on board of the two identical STEREO spacecraft A and B, which were successfully launched from Cape Canaveral on 26 October 2006. During the two years of the nominal STEREO mission, PLASTIC is providing us with the plasma characteristics of protons, alpha particles, and heavy ions. PLASTIC will also provide key diagnostic measurements in the form of the mass and charge state composition of heavy ions. Three measurements (E/qk, time of flight, ESSD) from the pulse height raw data are used to characterize the solar wind ions from the solar wind sector, and part of the suprathermal particles from the wide-angle partition with respect to mass, atomic number and charge state. In this paper, we present a new method for flight data analysis based on simulations of the PLASTIC response to solar wind ions. We present the response of the entrance system / energy analyzer in an analytical form. Based on stopping power theory, we use an analytical expression for the energy loss of the ions when they pass through a thin carbon foil. This allows us to model analytically the response of the time of flight mass spectrometer to solar wind ions. Thus we present a new version of the analytical response of the solid state detectors to solar wind ions. Various important parameters needed for our models were derived, based on calibration data and on the first flight measurements obtained from STEREO-A. We used information from each measured event that is registered in full resolution in the Pulse Height Analysis words and we derived a new algorithm for the analysis of both existing and future data sets of a similar nature which was tested and works well. This algorithm allows us to obtain, for each measured event, the mass, atomic number and charge state in the correct physical units. Finally, an important criterion was developed for filtering our Fe raw flight data set from the pulse height data without discriminating charge states.

Theoretical solutions of stopping power in cylindrical many-electron systems

Using several kinds of mathematical formulae of the modified Bessel function, and taking into account the uncertainty principle at mathematically singular point for inelastic interactions between a projectile and electron, we develop the theory of stopping power for cylindrical many-electron systems. Addition theorem, the formula of Wronskian and integral formulae of the modified Bessel function perform important roles in solving complex integral equation of stopping power discussed here. For a cylindrical free electron gas, we derive the theoretical solution of stopping power analytically. We also show the equipartion rule of contributions due to surface plasmon mode to stopping power for random systems.

A computer program for scanning transmission ion microscopy simulation

With the installation of the Scanning Proton Microprobe system at Fudan University, we are in the process of developing a three-dimension reconstruction technique based on scanning transmission ion microscopy–computed tomography (STIM–CT). As the first step, a related computer program of STIM simulation has been established. This program is written in the Visual C++®, using the technique of OOP (Object Oriented Programming) and it is a standard multiple-document Windows® program. It can be run with all MS Windows® operating systems. The operating mode is the menu mode, using a multiple process technique. The stopping power theory is based on the Bethe–Bloch formula. In order to simplify the calculation, the improved cylindrical coordinate model was introduced in the program instead of a usual spherical or cylindrical coordinate model. The simulated results of a sample at several rotation angles are presented.

Electron stopping power and mean free path in organic compounds over the energy range of 20–10,000 eV

An empirical method to obtain optical energy loss functions is presented for a large number of organic compounds, for which optical data are not available, on the basis of structure feature analysis of the existed optical energy loss functions for certain organic compounds. The optical energy loss functions obtained by using this method are in good agreement with the experimental data. Based on the Penn's statistical model, a set of systematic expressions have been given for the calculation of the stopping powers and mean free paths of electrons penetrating into the organic compounds in the energy range of E⩽10 keV. Detailed comparison of the calculated data with other theoretical results is presented. The stopping powers and mean free paths for a group of important polymers, without available optical data, have been calculated. In the calculations, three different cases have been considered, i.e. exchange correction not being considered, Ashley exchange correction being involved, and Born–Ochkur exchange correction being included. The results indicate that for these compounds the calculated stopping powers agree well with those obtained by using Bethe–Bloch theory at high-energy limit E=10 keV, as expected for a stopping power theory that should be converged to Bethe–Bloch theory at high energies.

Bethe stopping power theory for heavy-element targets and relativistic projectiles

An approach is described by which the Bethe theory for the stopping power of heavy-element targets is extended to treat the projectile in a fully relativistic way. The target electrons are treated semirelativistically by use of a modified Foldy-Wouthuysen transformation, also known as the Douglas-Kroll transformation. This approach allows results to be obtained that are valid to all orders in v{sup 2}/c{sup 2}, with v being the initial speed of the projectile. The contributions from both the longitudinal and transverse parts of the differential cross section are included, yielding a full expression for the stopping power of these systems. Our results show that the relativistic nature of the electrons in heavy-element targets leads to significant corrections which can be greater than 3% of Bethe's original results.

Evidence for observed projectile- z dependence in values of target parameters extracted from stopping power measurements

One version of modified Bethe–Bloch stopping power theory features two parameters characteristic of the target material, the mean excitation energy and the Barkas-effect parameter, that can be evaluated through fits to accurate stopping power measurements. In several recent investigations, the values of these parameters thus determined have revealed a dependence on projectile atomic number. Most of those studies have featured target materials possessing aggregation effects, such as solid compounds and alloys. In the current investigation, the search for systematics in projectile- z dependence has been extended to encompass other target materials, including elemental targets. Most of the results are corroborative of the previously reported trends.