Nuclear Stopping Cross Section Research Articles

Calculation of the electronic, nuclear, rotational, and vibrational stopping cross sections for H atoms irradiation on H2, N2 and O2 gas targets at low collision energies

In this work, we present a study of hydrogen atoms interacting with hydrogen, nitrogen, and oxygen molecules in the gas phase in order to determine the projectile energy loss, the stopping cross section, and energy deposition with implication in hydrogen and proton dosimetry and radiotherapy. Our calculations are performed with a density-functional tight-binding (DFTB) as well as by an ab initio electron-nuclear dynamic (END). The H atoms have a low collision energy in the range of 1–100 eV for the DFTB and 1–1000 eV for the END. We decompose the projectile energy deposition into its electronic, nuclear, rotational, and vibrational energy target contributions. We find that for low collisions energy, DFTB and END are in fair agreement when comparing the nuclear and rotational contributions. However, we find discrepancies for the vibrational and electronic energy loss between both approaches. The largest discrepancy is found in the electronic energy loss as DFTB carries the electronic dynamics in the ground state. We confirm that the electronic stopping cross section is proportional to the projectile velocity as obtained by END showing a threshold effect and that it is underestimated by codes like SRIM. Our study shows that at low collision energies a quantum-classical molecular dynamics approach as DFTB can be used to determine the energy-loss process of a projectile penetrating a material target. Finally, we compare our results to available experimental data finding good agreement when extrapolated to high collision energies.

Highly accurate nuclear and electronic stopping cross sections derived using Monte Carlo simulations to reproduce measured range data

We have examined and confirmed the previously unexplored concept of using Monte Carlo calculations in combination with measured projected ranges of ions implanted in solids to derive a quantitative description of nuclear interaction and electronic stopping. The study involved 98 ranges of 11B in Si between 1 keV and 8 MeV, contained in 12 sets of 10 different groups. Systematic errors by up to ±8% were removed to establish a refined data base with 93 ranges featuring only statistical uncertainties (±1.8%). The Monte Carlo calculations could be set up to reproduce the refined ranges with a mean ratio 1.002 ± 1.7%. The input parameters required for this very high level of agreement are as follows. Nuclear interaction is best described by the Kr-C potential, but in obligatory combination with the Lindhard-Scharff (LS) screening length. Up to 300 keV, the electronic stopping cross section is proportional to the projectile velocity, Se = kSe,LS, with k = 1.46 ± 0.01. At higher energies, Se falls progressively short of kSe,LS. Around the Bragg peak, i.e., between 0.8 and 10 MeV, Se is modeled by an adjustable function serving to tailor the peak shape properly. Calculated and measured isotope effects for ranges of 10B and 11B in Si agree within the experimental uncertainty (±0.25%). The range-based Se,R(E) reported here predicts the scarce experimental data derived from the energy loss in projectile transmission through thin Si foils to within 2% or better. By contrast, Se(E) data of available stopping power tables exhibit deviations from Se,R(E) between −40% and +14%.

Misconceptions impairing the validity of the stopping power tables in the SRIM library and suggestions for doing better in the future

The popular SRIM library delivers data and simulation codes for describing the slowing down of energetic atoms in matter. This study explored the validity of the tables containing cross sections for electronic and nuclear stopping together with the respective range parameters. The electronic stopping cross sections Se were produced, much like in previous and subsequent attempts of other groups, by bringing together the limited number of available experimental results in the form of ratios, r(Z1,He,υ)=Se(Z1,Z2,υ)/Se(He,Z2,υ). Z1 and Z2 denote the atomic numbers of projectiles (velocity υ) and target atoms, respectively. Reference data for He impact are available in reasonable volume only for about 15% of all solid targets; missing information has to be derived by interpolation. The concept of data evaluation is based on the assumption that the Bethe–Bloch (BB) theory, developed for bare projectiles, can serve as a guide at all velocities. For the purpose in question, the theory features an indispensible property: Z1 and Z2 appear as straight pre-factors, not in coupled form, Se,BB(υ)∝Z12Z2L(Z2,υ), with L(Z2,υ) representing the stopping number. Therefore, Z2 and L(Z2,υ) cancel out when taking ratios for fixed Z1 so that data for arbitrary Z2 may be combined in one set of r-values to determine the best fit rfit(Z1,He,υ). The stopping cross sections of interest are then derived as Se(Z1,Z2,υ)=rfit(Z1,He,υ)Se(He,Z2,υ). At low energies these results were often refined, Se⇒Se,f, to account for experimental data that did not comply with the anticipated Z2-independent trend.Major findings and suggestions are as follows: (i) At high velocities, υ>4Z12/3υ0 (Bohr velocity υ0), the SRIM predictions (have to) rely primarily on BB theory. (ii) At intermediate velocities, i.e., around the Bragg peak, SRIM appears to produce reasonable results (ca. ±15%). (iii) Below the Bragg peak the tabulated data often deviate strongly and inconsistently from the predictions of Lindhard–Scharff (LS) theory; they also exhibit various forms of exotic velocity dependence. These deviations are primarily due to the fact that the range of validity of BB theory is artificially extended to velocities at which the ‘effective-charge’ concept is assumed to be applicable. Coupled Z1,2 scaling as in theories of LS or Firsov would be much more appropriate. Overall, the electronic stopping cross sections by SRIM are of unpredictable value and often strongly misleading below 1MeV/u. (iv) Another consequence of the tight link to the Z1,2 dependence of BB theory is that only 2×92 master sets of electronic stopping cross sections were required to generate all conceivable 89×92 tables from Se,f-ratios for elemental targets (the tables for H, He and Li projectiles are derived separately). The information contained in the SRIM library at large thus exhibits a highly redundant character. (v) The nuclear stopping cross sections Sn mirror the predictions of the universal potential due by Ziegler, Biersack and Littmark, which differ from alternative suggestions typically by less than 15%. With this uncertainty, range distributions may be calculated with the TRIM program of SRIM, but only at energies where Sn dominates so that uncertainties in Se play a minor role. (vi) As a side aspect, an example is presented illustrating the efforts required to identify incorrect experimental data, notably when respected authors are accountable. (vii) Other approaches to establish stopping power tables are shown to be subject to the same problems as SRIM. It is recommended to add a warning to all theses tables, informing users at which energies the data are likely to lack reliability. (viii) The currently unacceptable quality of Se,f-data below 1MeV/u could be improved significantly in the future if the user friendly TRIM(SRIM) code were modified to allow simulations with a free choice of nuclear and electronic stopping cross sections. Researchers would thus be enabled to identify optimum input parameters for reproducing measured range distributions at energies at which Sn and Se are often of similar magnitude so that their contribution to the total energy loss is difficult to quantify without simulations.

Mass scaling laws due to isotopic effects in the energy loss of He2+colliding with H, D, and T

Total, electronic, and nuclear energy loss as well as stopping cross sections are calculated for He${}^{2+}$ ions incident on atomic hydrogen, deuterium, and tritium at low to intermediate energies by means of an ab initio, nonadiabatic approach for solving the Schr\odinger equation incorporating coupled electron and nuclear dynamics. Comparison of this ab initio treatment with classical nuclear stopping cross-section formulations obtained from Coulomb and screened potential models, and an ab initio theory, allows us to deduce scaling laws that incorporate the different target masses (target isotopic effect). We provide a discussion on the range of validity of the use of classical trajectories. We verify that the nuclear energy loss is a universal function of the center-of-mass (c.m.) scattering angle. The scattering angle in the c.m. system shows a dependence on the projectile charge due to a strong isotopic effect on the charge-transfer cross section. In the case of a Coulombic interaction, that dependence disappears, and leads to a formulation based only on the c.m. collision energy for the nuclear stopping cross section. Due to the small electron capture cross section for He${}^{2+}$ on H, D, and T in the low-energy region, it is shown that the effective charge model that assumes neutralization of the projectile does not apply to the energy loss of He${}^{2+}$ ions in the calculation of stopping cross sections. Thus, for this system, there is no influence of the charge exchange isotope effect of the target in the nuclear stopping cross section of the projectile. In the electronic energy loss, such an isotopic effect is present through Stueckelberg oscillations, but its contribution is small. Furthermore, the electronic stopping cross section shows a threshold effect that results from the minimum momentum transfer necessary to produce an electronic excitation in the target. Finally, the nuclear stopping cross section has a universal behavior as a function of the c.m. collision energy, and care must be taken when comparing ionic systems stopping cross-section results to those of neutral systems as obtained by effective charge models.

Cross sections forH+and H atoms colliding with Li in the low-keV-energy region

State-to-state, summed charge transfer and stopping cross sections in collisions of protons and neutral hydrogen atoms with lithium atoms have been studied at collision energies ranging from $10\phantom{\rule{0.3em}{0ex}}\text{to}\phantom{\rule{0.3em}{0ex}}25\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$. Cross sections were calculated using electron-nuclear dynamics (END), which is a nonadiabatic, time-dependent, direct approach for the study of ion-atom-molecule interaction processes. Our results show good agreement when compared to available theoretical and experimental data. We find that the charge transfer cross section for protons shows a bump and a maximum as a function of the projectile energy, both of them as a result of the large probability for capture into the projectile $2p$ orbital. The bump corresponds to a projectile energy of approximately ${E}_{p}\ensuremath{\sim}0.7\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$, and results from the electron capture probability in the low impact parameter region of approximately $b\ensuremath{\sim}2.0\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ The maximum occurs at ${E}_{p}\ensuremath{\sim}5\phantom{\rule{0.3em}{0ex}}\mathrm{keV}$ as a result of the larger capture probability in the intermediate impact parameter region near $b\ensuremath{\sim}8\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ A similar behavior is found for the electron loss cross section for hydrogen projectiles. We find that the beam charge fraction, for all the energies considered, is nearly neutral. We also find that the largest contribution to the stopping cross section is for impact parameters around $b\ensuremath{\sim}5\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ Finally, we report the total, electronic, and nuclear stopping cross sections to be within a factor of 2 of the reported values in the SRIM 06 code (SRIM stands for stopping and range of ions in matter) and other available experimental data. The largest discrepancy is due to the charge transfer process as well as to a transient $\mathrm{Li}{\mathrm{H}}^{+}$ molecular ion formed in the low projectile energy region.

Theoretical investigation of energy deposition and electron capture cross-sections for helium ion impact on formaldehyde

The subject of the work presented here is related to damage caused by energetic, charged particle radiation such as electrons, protons, and alpha particles to prebiotic matter such as that found in interstellar space. The calculations are carried out using an all electron, all nuclei, scheme that explicitly treats the electron-nuclear coupling. We present results for the no-capture, as well as the single and double electron capture probabilities, as well as for the 1s and 2l (l=s,p) contributions to the electron capture cross-sections of 3He2+ projectiles on formaldehyde molecules. We find that the summed cross-section peaks at 10keV/amu, and has a plateau between 0.1 and 1keV/amu. We also present preliminary results for the nuclear, ro-vibrational and electronic stopping cross-section. We find a large contribution to the electronic stopping cross-section and a maximum shifted towards higher energies in the nuclear stopping cross-section, when compared to SRIM results. We interpret this to be a consequence of molecular bonding.

Stopping cross section in the low- to intermediate-energy range: Study of proton and hydrogen atom collisions with atomic N, O, and F

We report electron-nuclear dynamics calculations of energy loss, charge exchange, and nuclear, electronic, and total stopping cross sections for protons and hydrogen atoms incident on atomic nitrogen, oxygen, and fluorine targets for projectile energies from a few eV up to 25 keV. Particular emphasis is placed on the analysis of the impact parameter and charge-exchange dependence of the energy loss. It is found that at low energies the electronic stopping is not proportional to the projectile speed, but depends on the excitation spectrum of the target, manifested by having a threshold or a second maximum in the stopping cross section at low projectile energy. Agreement with other calculations and the available experimental data is good.

Study on the crystallization of nickel films prepared by the ion beam and vapour deposition method

The effect of ion irradiation on the crystallization of the nickel films prepared by the ion beam and vapour deposition (IVD) method was studied. The nickel films were prepared on the silicon (100) wafers by evaporation of nickel and simultaneous irradiation with inert gas ions, such as neon, argon, krypton and xenon. The energy of inert gas ion bombardment was changed in the range 0.5–10.0 keV. The crystallization states of the films were analyzed by X-ray diffraction. X-ray diffraction patterns indicated that the (111), (200) and (220) crystallinity depended strongly on both ion beam energy and ion species. We studied the effect of ion irradiation on the crystallization of nickel films in terms of stopping cross-section of the ions. It seems that in the case of ion irradiation which is dominated by the nuclear stopping cross-section, the crystallization of the 〈111〉 orientation is promoted strongly; and as the effect of the electronic stopping cross-section increases, the orientation of the crystallization is changed in the order of 〈100〉 and 〈110〉.

Projected ranges of energetic ions in solids

Projected range equations, including nuclear and electronic energy losses up to the third moment, are derived as a simple and accurate means to evaluate projected ranges of energetic ions in solids. With the proposed expressions for the nuclear stopping cross section and the third-moment of both nuclear and electronic energy losses, good agreement is observed between the calculated results and experimental data for the projected ranges of ions in both single- and multi-component target materials.

Stopping power of 100–600 keV F +, Ar +, As +, Br + and Xe + ions in silicon

The stopping power of 100–600 keV F +, Ar +, As +, Br + and Xe + ions in silicon have been obtained from measured range distribution obtained by NRA and RBS techniques. A deconvolution program was used in order to obtain true range distributions from the measured NRA excitation curves or RBS spectra. The total stopping powers were determined through fitting the projected ranges based on LSS transport theory to the experimentally determined projected ranges. After subtracting calculated nuclear stopping cross sections, the electronic stopping cross sections were obtained. These results indicate that the electronic stopping cross sections at low velocities may be described by the four-parameter formulae proposed previously and that deviation from the velocity-proportional electronic stopping is evident.

The power envelope correspondence between nuclear stopping and interatomic potential

The power envelope technique prescribed recently to deduce the interatomic potential from a known nuclear stopping cross section variation and vice-versa, comprises the construction of an envelope of piecemeal power law curves obtained using the power law formalism. The range of applicability of the technique is investigated by applying it to standard potentials and typical range-energy data of keV heavy ions.

Secondary ion emission from oxidized vanadium as a function of projectile energy in the range 10–90 MeV

The intensity of secondary ion emission from oxidized V bombarded by 10–90 MeV 79Br projectiles has been measured. The intensity of V + ion emission was proportional to the nuclear stopping cross section for Br in V indicating that these ions were produced as a consequence of a collision cascade. However, I(O +) and I(F +) were approximately constant across the projectile energy range demonstrating that those ions were produced by a different mechanism. The variation of I(VO +) with projectile energy was intermediate between that of I(V +) and I(O +). Possible mechanisms for the production of the O + and F + secondary ions are discussed.

A simple formula for low-energy sputtering yields

A simple formula for low-energy sputtering yields of elemental targets is presented. The formula follows directly from Sigmund's theory but includes a modified form of the functionα(Mt/Mp) and an accurate expression for the nuclear stopping cross section. Good agreement with experimental results is obtained for the projectiles Ne, Ar, Kr sputtering not too reactive surfaces at energies ≲1 keV. Further experiments are suggested as well as theoretical work concerning the meaning of the surface binding energy and the effect of the surface on the sputtering process in general.

A fresh look at nuclear stopping in repulsive power law potentials

Nuclear stopping cross sections in repulsive power law potentials have been computed as a function of the power. As the power exceeds unity, the cross section drops initially, reaches a minimum at the power value of 4.2 and rises gradually beyond. For very high powers, the cross section approaches an asymptotic ϵ/2 behaviour in contrast with the Lindhard-Nielsen-Scharff prediction of a monotonically decreasing cross section. An analytic expression has been fitted to the variation of the computed cross sections.

Theoretical energy straggling of 252Cf fission fragments in various absorbers

A new theoretical formula of energy straggling for heavy ions such as 252Cf fission fragments in various stopping media has been calculated in terms of incident heavy ion energy, differential scattering cross-section, electronic and nuclear stopping cross-section (data taken from Lindhard and co-workers [1,2]). However, no theory yet exists which is capable of predicting straggling behaviour of fission fragments. Comparison between our theoretical predictions and the previous work of Al-Bedri [3] and Sykes [4] has been made. Good agreement has been achieved at the beginning of the range of fission fragments but towards the end of the range our theoretical results were greater than the experimental results by about 0–20%. This discrepancy could be caused by the stopping cross-section, because the experimental stopping cross-section is greater than the theoretical data by about 0–30% [5–8].

Energy distributions of low energy H ions backscattered from graphite, stainless steel, and molybdenum

Energy distributions of H+ and H− backscattered from graphite, stainless steel and molybdenum were measured for low incident beam energies (150∼1500 eV). The energy distributions of H− resembled that of H+, while the intensity ratio of H− to H+ varied from material to material and with the incident proton energy. A peak in the energy distribution moved to the “cutoff” energy with decreasing incident energy, which can be attributed to an increase in nuclear stopping cross section and a decrease in electronic stopping cross section. The ratio of the peak energy to the incident energy is related to the reduced energy ɛ of incident beam independent of target materials in the measured region (0.03<ɛ< 3.3).

Theory of sputtering and comparison to experimental data

Two topics in sputtering are discussed. One is low energy sputtering and the other is non-linear sputtering. Based on the assumption that transient phenomena are dominant in the low-energy sputtering, a modified version of Sigmund's formula is derived. The same approach is successfully applied to analysis of the angular distribution of sputtered atoms. The formula obtained is shown to predict well the heart-shaped angular distribution for normal incidence In order to explain the non-linear effect of sputtering yields in heavy-ion bombardment on high Z material, a shock wave model is applied. The present approach has shown that the sputtering yield is nearly proportional to S n (E) 3 2 , not to S n( E) , where S n( E) is the nuclear stopping cross section. The non-linear Monte Carlo simulation is also performed in order to investigate the difference between linear cascade and non-linear cascade.

Recoil range distributions in multilayered targets

Range distributions of recoiled oxygen in silicon resulting from implantations of B, P, and As through films of SiO 2 on silicon are calculated by step-wise numerical integration of the Boltzmann transport equation. Using a nuclear stopping cross section proposed by Wilson, Haggmark and Biersack in place of the more standard Thomas-Fermi cross section, improved agreement with available experimental data is obtained. The recoil distributions are found to be characterized by exponential regions with spatial decay lengths proportional to the maximum energy transfer from incident to recoil atom γE o , this length being nearly independent of primary ion and film thickness. Total recoil flux calculations and a silicon recoil distribution resulting from a 400 keV Se implant through Si 3N 4 on GaAs are also found to agree well with experimental results. The device implications of these calculations are of considerable importance and will be discussed.

Ranges and standard deviations of implanted ions

This paper presents a theoretical analysis and simple expressions for the calculation of ranges and standard deviations of implanted ions in amorphous targets. Nuclear energy loss models are discussed to introduce an approximation formula for the nuclear stopping cross section appropriate for Thomas-Fermi, Lenz-Jensen and intermediate type potentials. The Firsov electronic stopping model has been used to show its successful application. Algebraical expressions for the total range of ions in monoatomic and biatomic targets as a function of ion energy result from the considerations presented. A quasi-elastic multiscattering model is suggested, which permits an easy estimate of projected ion ranges. The standard range deviation is obtained by determining the total average-square energy loss fluctuations. Finally a comparison of calculated and experimental results is made to show that calculations based on the Firsov electronic stopping model and the nuclear stopping cross section proposed here provide a better agreement with the experimental results than the wellknown Johnson-Gibbons LSS-calculations.

Parameter analysis of an electrostatic particle guide using radioactive recoil ions

A thorough parameter analysis of an electrostatic particle guide (EPG), constructed in the frame of a wider program for the study of nuclear stopping cross sections for slow heavy ions, has been performed. To obtain optimal performance for time-of-flight spectroscopy, the operational conditions of the system were studied in detail, using ∼ 100 keV alpha disintegration recoils. The parameters investigated were collector radial position, collector axial position, source radial position and wire voltage. Clear and sharp spectra with an energy resolving power of 1.5–3% fwhm, and transmission enhancement factors up to 300–400, were obtained.