Local Plasma Approximation 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.

L- and M-subshell ionization cross-sections of heavy atoms by electron and proton impact

Inner-shell ionization has many applications related to material analysis. However, full theoretical calculations in heavy multielectronic atoms are scarce. In this contribution, we assess proton and electron impact Li and Mi-subshell ionization cross-sections of heavy targets (72≤Z≤83) using two theoretical models. These calculations are based on the Distorted Plane Wave Born Approximation for electron impact and the shellwise local plasma approximation for the proton impact, combined with relativistic calculations of these deep subshells wave functions and binding energies. We analyze the L-shell ionization of W, Au, and Bi and the M-shell of Hf, Ta, W, Re, Os, Pt, Au, Pb, and Bi. Present values are compared with available experimental data from the literature, with disparate agreement. The convergence of proton and electron impact cross-sections at high-impact velocities is also discussed.

Stopping force of protons in Pt, Mo, and Cr foils at energies between 325 keV and 900 keV using the forward scattering method

In this work, the stopping force of protons in Pt, Mo, and Cr foils was measured at energies between 325 and 900 keV using the forward scattering method. Obtained results were compared with predictions from the semi-empirical code SRIM, the purely theoretical code DPass, and shellwise local plasma approximation with Levine-Mermin dielectric function (SLPA-LM). Both theoretical predictions underestimate experimental data for Pt at the energies measured. However, the theoretical calculation results for Mo, and Cr displayed good agreement with experimental data over the measured energy range. The obtained results were also compared with data from the literature, which displayed experimental discrepancies, especially for Pt and Cr stopping while a very good agreement between all experimental data was observed for Mo data.

Stopping-power calculations and the Levine-Mermin dielectric function for inner shells

The contribution of bound electrons to the electronic stopping power is modeled by using the shellwise local plasma approximation (SLPA). An improvement to this formalism is proposed by including the Levine-Mermin dielectric function (SLPA-LM). This approach considers a more realistic description of the bound electron response by facing the problem of the damping in the collective excitations. We introduce a local damping that depends on the density of electrons of each shell, keeping the full-theoretical characteristic of the SLPA. We implement the present model to obtain the stopping power for metal targets of period-6 elements of the periodic table (Hf, Ta, W, Os, and Pt) and we analyze the importance of the SLPA-LM description of heavy multielectronic targets. Furthermore, we investigated metal targets of group VI of the periodic table (Cr, Mo, and W) to inspect the dependence of the SLPA-LM with the number of subshells.

Understanding the mechanisms of L-shell x-ray emission from Os atoms bombarded by 4–6 MeV/u fluorine ion

The L-subshell ionization mechanism is studied in an ultra-thin Os target bombarded by 4–6 MeV/u fluorine ions. Multiple ionization effects are considered through the change of fluorescence and Coster-Kronig yields while determining L-subshell ionization cross sections from L x-ray production cross sections. The present experimental values are compared with various theoretical approximations: (i) the relativistic semi-classical approximation (RSCA), (ii) the shellwise local plasma approximation (SLPA), and (iii) the ECUSAR theory. We also take into account the vacancy sharing among the subshells by the coupled-states model (CSM) and the electron capture (EC) by a standard formalism. We find that the ECUSAR-CSM-EC describes the measured excitation function curves the best. However, the theoretical calculations are still about a factor of two smaller than the measured values even though the recent fluorescence and Coster-Kronig yields are considered. Hence, a re-evaluation of these parameters is a challenge for the theoretical works. Whatsoever, this work leads to demonstrate that in the present energy range the heavy-ion induced inner-shell ionization of the heavy atoms can be understood by combining the direct Coulomb ionization, the electron capture, and the vacancy sharing among subshells, together with optimizing the atomic parameters. Optimization of the atomic parameters shows that our experimental results agree with theoretical vacancy production theories if the L1 fluorescence yield is nearly doubled. Such a optimization is validated by the proton induced L-shell ionization data of uranium atoms.

Stopping power of plasma free and bound electrons using dielectric formalism

In this work, a method is proposed that accounts for the stopping power of free and bound electrons of a partially ionized plasma. To formulate this method, the well-known dielectric formalism is used, combining a dielectric function of quantum plasmas, which includes temperature, for free electrons, and the shellwise local plasma approximation (SLPA) for bound electrons. Since few experimental data are available for plasmas, our method calculations at T=0 for solids are compared with SRIM code. The agreement with experimental data is excellent. Also, a cold gas target is considered at different ionization states; results show the expected behavior, but with a little overestimation of the stopping peak and underestimation in the low energy region. Finally our results for plasmas at diverse conditions of temperature are compared with other methods from the literature that contemplate bound electrons explicitly: Mehlhorn, Zimmerman and Unified Wave Packet models. Differences observed between our method, and Mehlhorn and Zimmerman models can be explained by the limited range of application of these old models. In contrast better agreement is obtained when our method is compared with Unified Wave Packet model. These comparisons also serve to prove the validity of our method in different elements.

Stopping power of hydrogen in hafnium and the importance of relativistic 4f electrons

The stopping power of protons through Hf foil has been studied both experimentally and theoretically. The measurements were performed at the Laboratory of Accelerators and X-Ray Diffraction in Lisbon by using the transmission method on self-supporting stopping material. The overall uncertainty of around $5%$ was established over the protons energy range (0.6--2.5) MeV. The theoretical developments involved fully relativistic atomic structure calculations for Hf, which required the solution of the Dirac equation. The shell-wise local plasma approximation was used to describe the energy transferred to the bound $1s\text{\ensuremath{-}}4f$ electrons, and the outer four electrons were considered as a free electron gas. We found the relativistic description of the $4f$ shell and the screening between $4f$ and $5p$ electrons to be decisive around the stopping maximum. Present theoretical and experimental results are in very good agreement in the energy region of the new measurements. However, our theoretical stopping cross sections show substantial differences with the most used semiempirical models (SRIM2013 and ICRU-49) at intermediate to low energies. Our calculations suggest the stopping maximum to be higher and shifted to lower energies than these previous predictions. Future measurements around the maximum and below would be necessary for a better understanding of the stopping power of hafnium.

Experimental and theoretical L-shell ionization cross sections of heavy atoms by impact of Si ions

We present a theoretical and experimental study of the subshell resolved L-shell ionization of relativistic targets such as 73Ta, 78Pt, 90Th, and 92U. The measurements of x-ray production cross sections by (84–140 MeV) Si+q ions (q = 8; 12), were held at the Inter-University Accelerator Centre of New Delhi. Multiple-hole fluorescence and Coster-Kronig yields were used to obtain the Li(i = 1–3) ionization cross sections from the measured x-ray production cross sections of Lℓ, Lα, and Lβ, Lη, and Lγ lines. The experimental results are compared with ab initio theoretical calculations by means of the shell-wise local plasma approximation (SLPA). This model uses the quantum dielectric formalism to obtain the total ionization cross sections from an initial ground state. The wave functions and binding energies of the different targets were obtained by solving the fully relativistic Dirac equation using the HULLAC code package. These calculations are based on first order perturbation theory with a central field, including Breit interaction and quantum electrodynamics corrections. The present SLPA ionization cross sections of the L-shell are found to be independent of the charge state of the Si ions. The experimental observations display also quite similar character if the correct mean projectile charge state inside the target is used for including the multiple ionization effect during ion-solid collisions. A general good agreement between the experimental measurements and full theoretical calculations supports the reliability of present results. The comparison also includes the well-known ECPSSR and ECUSAR semi empirical approximations. We noted that the ECUSAR results agree well with the SLPA, while the ECPSSR cross sections are rather low.

Low- and intermediate-energy stopping power of protons and antiprotons in solid targets

In this work we propose a non-perturbative approximation to the electronic stopping power based on the central screened potential of a projectile moving in a free electron gas, by Nagy and Apagyi. We used this model to evaluate the energy loss of protons and antiprotons in ten solid targets: Cr, C, Ni, Be, Ti, Si, Al, Ge, Pb, Li and Rb. They were chosen as canonicals because they have reliable values of the Seitz radio, rS=1.48-5.31 a.u., which cover most of the possible metallic solids. Present low velocity results agree well with the experimental data for both proton and antiproton impact. Our formalism describes the binary collision of the projectile and one electron of the free electron gas. It does not include the collective or plasmon excitations, which are important in the intermediate to high velocity regime. The distinguishing feature of this contribution is that by using the present model for low to intermediate energies (below the appearance of plasmon excitations), the Lindhard dielectric formalism (including plasmons) for intermediate to high energies, and the shellwise local plasma approximation to account for the inner shell contribution, then a good full-theoretical description of the available experimental data is obtained in an extensive energy range, covering the low, intermediate and high energy region.

Lindhard's polarization parameter and atomic sum rules in the local plasma approximation: a case for excited states

ABSTRACTIn this work, we analyze the effects of Lindhard polarization parameter, χ, on the sum rule, , within the local plasma approximation (LPA) as well as on the logarithmic sum rule , in both cases for the system in an initial excited state. We show results for a hydrogenic atom with nuclear charge Z for the spherically symmetric (l=0) states with n=1, 2 and 3 with p=−1,0,1, and 2. Our calculations are complemented with comparisons to the exact results for this system for the ground state. As a natural extension of our treatment, we report analytical results for helium-like systems in terms of a screened charge for the ground state. Our study shows that by increasing χ, the sum rule for p<0 decreases while for p>0 it increases, and the value p=0 provides the normalization/closure relation which remains fixed to the number of electrons for the same initial state. When p is fixed, the value of increases as the initial excited state increases for p<0 and decreases for p>0. Thus, when using a specific value of χ to adjust to a particular sum rule value, the sum rule becomes unbalanced for other values of p. Our results show that the dipole oscillator strength distribution, within the LPA, provides good results for the sum rules for excited states.

Stopping and straggling of H and He in ZnO

We present experimental and theoretical values for the energy loss of H and He ions in Zinc oxide, in mean value (stopping per unit path length) and mean square value (energy loss straggling). The measurements were carried out using the Rutherford Backscattering technique for (300–2000) keV H ions and (300–5000) keV He ions. Present experimental data are the first set of stopping and straggling values in this oxide. The theoretical research was encouraged considering the molecular description of ZnO as crystal solid using the density functional theory. The energy loss calculations for H and He ions with different charge states were performed with the shelwise local plasma approximation (SLPA). The molecular versus the Bragg-rule description is also discussed. The equilibrium charge state of He inside ZnO is analyzed based on the present stopping measurements, and a semiempirical charge state distribution is proposed. Present experimental and theoretical values show good agreement for both the stopping and the straggling. We also compare our data with the SRIM2013 and with CasP5.2 values.

Accurate evaluation of pressure effects on the electronic stopping cross section and mean excitation energy of atomic hydrogen beyond the Bethe approximation

Atomic hydrogen is used as a fundamental reference target system to explore pressure effects on the electronic stopping cross section, Se, of swift bare ions such as protons and α-particles. This is achieved by considering the hydrogen atom under pressure as a padded spherically-confined quantum system. Within this scheme, Se is calculated rigorously in the first Born approximation taking into account the full target excitation spectrum and momentum transfer distribution for different confinement conditions (pressures) and fixed projectile charge states. Pressure effects on the target mean excitation energy, I, are also formally calculated and compared with corresponding accurate calculations based on the Local Plasma Approximation (LPA). Even though atomic hydrogen is the simplest target system, its accurate treatment to account for the role of pressure in the stopping dynamics is found to provide useful means to understand the behavior of more complex systems under similar conditions. It is found that: (i) the region of projectile velocities for which the Bethe approximation remains valid is shifted towards higher values as pressure increases; (ii) shell corrections are enhanced relative to the free-atom case as pressure increases, and (iii) the LPA seems to underestimate I as pressure is increased. The results of this work for atomic hydrogen may serve as accurate benchmark reference values for studies of pressure effects on Se and I using different methodologies.

Stopping power of Zn for heavy ions

We present stopping power measurements of Zn for C and O ions and compare them with a theoretical description given by the Transport Cross Section - Extended Friedel Sum Rule (TCS-EFSR) for the valence electrons, and two different models for the inner-shells: the Shellwise Local Plasma Approximation (SLPA) and the CasP approach. The SLPA, which successfully applies to projectiles from H to B, is slightly high for C ions and clearly overestimates the data for O ions. On the other hand, total stopping results using the CasP description for the inner-shells show good agreement with the data for C and O ions, and also with the SRIM predictions.

L and M-shell ionization of very heavy targets

We present a theoretical study on the inner-shell ionization of Au, Pb, and Bi. A collective response model, the shellwise local plasma approximation (SLPA), is employed, which works within the dielectric formalism. The wave functions and binding energies were obtained in fully-relativistic way by solving numerically the Dirac equation. The theoretical results describe nicely the experimental data for M-shell ionization of Au and Bi above 2MeV/amu, and for L-shell ionization of Au and Pb above 10 MeV/amu. The SLPA tends to underestimate the data for energies below the range of validity of the model and approach to them for higher energies.

Experimental and theoretical study of the energy loss of C and O in Zn

We present a combined experimental-theoretical study of the energy loss of C and O ions in Zn in the energy range 50--1000 keV/amu. This contribution has a double purpose, experimental and theoretical. On the experimental side, we present stopping power measurements that fill a gap in the literature for these projectile-target combinations and cover an extended energy range, including the stopping maximum. On the theoretical side, we make a quantitative test on the applicability of various theoretical approaches to calculate the energy loss of heavy swift ions in solids. The description is performed using different models for valence and inner-shell electrons: a nonperturbative scattering calculation based on the transport cross section formalism to describe the Zn valence electron contribution, and two different models for the inner-shell contribution: the shellwise local plasma approximation (SLPA) and the convolution approximation for swift particles (CasP). The experimental results indicate that C is the limit for the applicability of the SLPA approach, which previously was successfully applied to projectiles from H to B. We find that this model clearly overestimates the stopping data for O ions. The origin of these discrepancies is related to the perturbative approximation involved in the SLPA. This shortcoming has been solved by using the nonperturbative CasP results to describe the inner-shell contribution, which yields a very good agreement with the experiments for both C and O ions.

A collective model for inner shell ionization of very heavy targets

We present a theoretical study of the ionization of inner shells, such as L and M shells of Au, Pb and Bi or K-shell of Sb. The ionization is described using a collective response model, the shellwise local plasma approximation (SLPA). This model deals with each sub-shell of target electrons as an inhomogeneous electron gas with an ionization threshold. No parameters are included, just the theoretical wave functions of the ground state and the binding energies. The validity range is that of the perturbative approximation, high energies and asymmetric projectile target relation. In the case of Sb, known Hartree–Fock wave functions and energies are used. Instead, for Au, Pb and Bi, they were obtained in fully relativistic way by solving numerically the Dirac equation. The SLPA describes well the experimental data for K-shell ionization of Sb by O and F positive ions. However, it underestimates the data for C or lighter ions. Good results are obtained for M-shell ionization of Au and Bi above 2 MeV/amu, and for L-shell ionization of Au and Pb above 10 MeV/amu. For L- and M-shells, the SLPA tends to underestimate the data for energies below the range of validity of the model but approaches the experimental data for higher energies.

Ionization of He, Ne, Ar, Kr, and Xe by impact ofHe+ions

In this article we report ionization cross sections of neutral He, Ne, Ar, Kr, and Xe by impact of ${\mathrm{He}}^{+}$ ions with energies ranging from 25 to $1000$ keV/amu. A general theory to deal with dressed projectiles is developed in the context of the Continuum-Distorted-Wave Eikonal-Initial-State approximation. The strategy is based on the use of an effective charge defined in terms of the momentum transfer extracted from the first Born approximation and extensively used in the local plasma approximation to deal with ions in solids. The difference between the actual potential of the projectile and the Coulombic effective one used in the distorted wave functions is treated in first perturbative order. Our results show that the proposed approach gives a very good account of available experiments.

Energy loss of protons in W using fully relativistic calculations and mean excitation energies of W, Au, Pb, and Bi

We present a theoretical study on the energy loss of protons in wolframium by calculating target fully relativistic wave functions and binding energies. The HULLAC code is employed to obtain numerical solutions of the Dirac equation. We use the shellwise local plasma approximation (SLPA) to evaluate the different moments of the energy loss. The partial contribution of each subshell of target electrons is calculated separately, including the screening among the electrons of the same binding energy. We pay special attention to the role of the outer $4f$ shell and the screening between electrons of near subshells (i.e., the $5p$ and $4f$ electrons). Results for stopping and straggling cross sections are compared to the experimental data available. Our calculations describe rather well the stopping measurements around the maximum and for very high energies, but overestimate the data for impact energies around 1MeV. We find that the SLPA results tend clearly to Bethe limit, but show a systematic overestimation in the energy region of 1--2 MeV. This overestimation may indicate the presence of other mechanisms included neither in the SLPA nor in Bethe formulations. We also present results for the stopping number of W, Au, Pb, and Bi which follow quite well the Lindhard scaling. A theoretical mean excitation energy $I(W)=710\text{ }\text{eV}$ is obtained, in good agreement with the suggested value of $727\ifmmode\pm\else\textpm\fi{}30\text{ }\text{eV}$. Theoretical mean excitation energies for Au, Pb, and Bi are also presented, which are in good agreement with the experimental ones.

Experimental and theoretical study of the energy loss of Be and B ions in Zn

Energy-loss measurements and theoretical calculations for Be and B ions in Zn are presented. The experimental ion energies range from 40 keV/u to 1 MeV/u, which includes the energy-loss maximum and covers a lack of experimental data for these systems from intermediate to high energies. The measurements were performed using the Rutherford backscattering technique. The ab initio calculations are based on the extended Friedel sum rule--transport cross-section method for the valence electrons and the Shellwise local plasma approximation for the bound electrons. A comparison of these calculations to the present experimental data for Be and B and previous values for H, He, and Li ions on the same target is included. This confirms the applicability of the employed theoretical framework also for ions of intermediate atomic number.

Energy loss of protons in Au, Pb, and Bi using relativistic wave functions

We present a theoretical study on proton energy loss in solid targets of atomic number greater than 54. Fully relativistic wave functions and binding energies are obtained by solving numerically the Dirac equation. Ab initio calculations are developed for the first stopping and second straggling moments of the energy transferred from the ion to the target electrons. The shellwise local plasma approximation is employed for the inner shells, and the Mermin dielectric function is employed for the valence electrons. The dielectric response of each subshell is calculated separately, including in this way the screening among the electrons of the same binding energy. Results for stopping and straggling cross sections of protons in Au, Pb, and Bi are compared with the available experimental data. The theoretical stopping results are very good in the case of Au, reproducing the experimental data in an extensive energy region 10 keV‐100 MeV. For Pb and Bi, the stopping results agree with the measurements for energies above 300 keV, for which the inner shells play a major role. However, we found some difficulties around the stopping maximum. For the energy-loss straggling, we obtained reasonably good agreement with the experiments for the three targets studied.