Relativistic Bethe Research Articles

Calculations of electron inelastic mean free paths (IMFPs). XIV. Calculated IMFPs for LiF and Si3N4 and development of an improved predictive IMFP formula

We report inelastic mean free paths (IMFPs) of Si3N4 and LiF for electron energies from 50 eV to 200 keV that were calculated from their optical energy‐loss functions using the relativistic full Penn algorithm including the correction of the bandgap effect in insulators. Our calculated IMFPs, designated as optical IMFPs, could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter from 50 eV to 200 keV. The root mean square (RMS) deviations in these fits were less than 1% for Si3N4 and LiF. The IMFPs were also compared with the relativistic version of our predictive Tanuma–Powell–Penn (TPP‐2M) equation. We found that IMFPs calculated from the TPP‐2M equation are systematically larger than the optical IMFPs for both LiF and Si3N4. The RMS differences between IMFPs from the TPP‐2M equation and the optical IMFPs were 49.3% for LiF and 17.3% for Si3N4 for energies between 50 eV and 200 keV. These RMS differences are much larger than those for most of the inorganic compounds in our previous IMFP calculations where the average RMS difference was 10.7% for 42 inorganic compounds. We also report the development of an improved predictive IMFP formula, which we designate as the JTP equation. This formula is a refinement of the TPP‐2M equation and is based on the recent IMFP calculations for 100 materials including the present IMFPs for Si3N4 and LiF (41 elemental solids, 45 inorganic compounds, and 14 organic compounds) for 83 electron energies between 50 eV and 200 keV. Our predictive JTP equation gave satisfactory results in comparisons of optical IMFPs and IMFPs calculated from the JTP equation. The RMS difference between the 8300 optical IMFPs used for optimization and the IMFPs calculated from the JTP equation was 10.2%. This value is appreciably less than the RMS difference of 16.0% found in a similar comparison of the optical IMFPs and IMFPs from the TPP‐2M equation. Furthermore, IMFPs from the JTP equation were compared with measured IMFPs for energies between 50 eV and 200 keV for 16 elemental solids and 37 inorganic compounds. We found that the JTP equation gave satisfactory results that were comparable with previous comparisons of the optical IMFPs and measured IMFPs. We believe that the JTP equation will be applicable to a wider range of materials than the TPP‐2M equation.

Analytical solution of relativistic four quark bound systems

Since in recent years most of the heavy tetraquarks are discovered by Belle, LHCb, BESIII, etc., it motivated us to study these exotic hadrons. To the extent of our knowledge about the mass characteristics of heavy tetraquarks, we utilize the spinless relativistic Bethe–Salpeter equation by applying the Cornell potential in order to calculate the eigenvalue and mass of heavy tetraquarks. For this purpose, we present an ansatz solution to obtained Schrödinger-like equation to calculate the ground-state energy of diquarks and tetraquarks. Eventually, we compare our obtained predictions for heavy tetraquark masses with available experimental and theoretical data.

The partial waves of B^{*}_{2}(5747) and their contributions in strong decays

The relativistic Bethe–Salpeter method is adopted to study the OZI-allowed strong decays of the 2^+ state B^{*}_{2}(5747)^{0}, with emphasis on the relativistic corrections. We first study the partial waves in the wave functions used and find that there are P, D, and F partial waves in the B^{*}_{2}(5747)^{0} meson, with ratios of P/D/F=1:0.421:0.051. We also find that S/P/D=1:0.354:0.046 for B^*, and S/P=1:0.343 for the B meson. The large components of the D wave in B^{*}_{2}(5747)^{0} and the P wave in B^{(*)} mean that large relativistic effects exist in these states. Second, we calculate the strong decays, and the total decay width Gamma (B^{*}_{2}(5747)^{0})=25.9 MeV and the branching fraction Gamma (B^{*}_{2}(5747)^{0} rightarrow B^{*}pi ) / Gamma (B^{*}_{2}(5747)^{0} rightarrow Bpi )=0.96 obtained are consistent with experimental data. Third, we study the contributions of different partial waves in the initial and final wave functions, and find that the relativistic effects are about 15% and 11% for B^{*}_{2}(5747)^{0} rightarrow Bpi and B^{*}_{2}(5747)^{0} rightarrow B^{*}pi , respectively, which are much smaller than our expected effects, showing that the relativistic corrections cancel each other in these decays.

Relativistic Bethe logarithm for triplet states of helium-like ions

We report a calculation of relativistic corrections of order $m\alpha^7$ to the Bethe logarithm for the $2^3S$ and $2^3P$ states of helium-like ions. The calculation is required for improving the accuracy of theoretical energies of helium-like ions and for checking the evaluation of the $m\alpha^7$ effects in helium performed in [V. Patk\'o\v{s}, V. A. Yerokhin, K. Pachucki, Phys. Rev. A 103, 042809 (2021)], where a significant discrepancy with experimental results was found. The large-$Z$ limit of the relativistic Bethe logarithm is determined numerically, in excellent agreement with the analytical results obtained from the hydrogen theory.

Calculations of electron inelastic mean free paths. XIII. Data for 14 organic compounds and water over the 50 eV to 200 keV range with the relativistic full Penn algorithm

We have calculated inelastic mean free paths (IMFPs) for 14 organic compounds (26‐n‐paraffin, adenine, β‐carotene, diphenyl‐hexatriene, guanine, Kapton, polyacetylene, poly (butene‐1‐sulfone), polyethylene, polymethylmethacrylate, polystyrene, poly(2‐vinylpyridine), thymine, and uracil) and liquid water for electron energies from 50 eV to 200 keV with the relativistic full Penn algorithm including the correction of the bandgap effect in insulators. These calculations were made with energy‐loss functions (ELFs) obtained from measured optical constants and from calculated atomic scattering factors for X‐ray energies. Our calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter from 50 eV to 200 keV. The average root‐mean‐square (RMS) deviation in these fits was 0.17%. The IMFPs were also compared with a relativistic version of our predictive Tanuma–Powell–Penn (TPP‐2M) equation. The average RMS deviation in these comparisons was 7.2% for energies between 50 eV and 200 keV. This average RMS deviation is smaller than that found in a similar comparison for our group of 41 elemental solids (11.9%) and for our group of 42 inorganic compounds (10.7%) for the same energy range. We found generally satisfactory agreement between our calculated IMFPs and values from other calculations for energies between 200 eV and 10 keV. We also found reasonable agreement between our IMFPs for organic compounds and measured IMFPs for energies between 50 eV and 200 keV. Substantial progress for IMFP measurements for liquid water has been made in recent years through the invention of liquid water microjet photoelectron spectroscopy and droplet photoelectron imaging. We found that the IMFPs from these experiments and the associated analyses were larger than our IMFPs by factors between two and four for energies between about 30 eV and 1000 eV. The energy dependences of the measured IMFPs are, however, similar to that of our IMFPs in the same energy range. Since IMFPs calculated from the same algorithm for a number of inorganic compounds agree reasonably well with measured IMFPs for energies between 100 eV and 200 keV, the large differences between IMFPs for water from recent experiments and our results are surprising and need to be resolved with additional experiments.

Revealing the quasiparticle electronic and excitonic nature in cubic, tetragonal, and hexagonal phases of FAPbI3

The development of three-dimensional (3D) hybrid organic–inorganic perovskites has sparked much interest because of their rich light-harvesting capabilities in solar cells. However, the understanding of the electronic and optical properties, particularly the excitonic shifts upon structural phase transition with temperature in these materials, is not fully clear. Here, we report the accurate description of electronic and optical properties of mostly studied FAPbI3 across the cubic–tetragonal–hexagonal phases, using the relativistic GW method and Bethe–Salpeter Equation (BSE), including the spin–orbit coupling effects. Our GW calculations reveal that the bandgap values vary from 1.47 to 3.54 eV from the room temperature cubic phase to the low temperature hexagonal phase. Our optical analysis shows that excitonic peaks are blue-shifted, and exciton binding energies estimated by the model BSE approach increase from 74 to 567 meV going from the cubic to hexagonal phases. Our results may have important impacts on the practical uptake of hybrid perovskite based solar cells under different climatic conditions.

Mass spectra, wave functions and mixing effects of the (bcq) baryons

Mass spectra and wave functions of the J^P=frac{1}{2}^+ (bcq) baryons are calculated by the relativistic Bethe–Salpeter equation (BSE) with considering the mixing effects between the 1^+ and 0^+ (bc)-diquarks inside. Based on the diquark picture, the three-body problem of baryons is transformed into two two-body problems. The BSE and wave functions of the 0^+ diquark are given, and then solved numerically to obtain the effective mass spectra and form factors. Also we present the wave functions at zero point for the (bc)-diquark. Considering the obtained diquark form factors, the (bcq) baryons are then described by the BSE as the bound state of a diquark and a light quark, where the interaction kernel includes the inner transitions between the 0^+ and 1^+ diquarks. The general wave function of the frac{1}{2}^+ (bcq) baryons is constructed and solved to obtain the corresponding mass spectra. Especially, by using the obtained wave functions, the mixing effects between Xi _{bc}(Omega _{bc}) and Xi _{bc}'(Omega '_{bc}) in ground states are computed and determined to be small (sim !1%). The numerical results indicate that it is a good choice to take Xi _{bc} and Xi '_{bc} as the baryon states with the inside (bc)-diquarks occupying the definite spin.

Electronic stopping power of diamond for electrons and positrons

We assess a new data set of mass electronic stopping powers of diamond for electrons and positrons. To this end, an optical energy-loss function (OELF) for this material is generated from available experimental dielectric properties of the valence band and photoabsorption mass attenuation coefficients, supplemented with a Drude-type analytical expression to interpolate at intermediate energies. From this synthetic OELF, a mean excitation energy equal to (88.5 ± 2.0) eV is extracted and the density-effect correction is evaluated using Fano’s simplified method. These quantities then allow calculating the mass electronic stopping power of diamond for electrons and positrons by means of the relativistic Bethe formula. The present results for 1 keV–1 GeV electrons in diamond are 0.9%–3.3% smaller than those corresponding to graphite as tabulated in ICRU Report 90.

Calculations of electron inelastic mean free paths. XII. Data for 42 inorganic compounds over the 50 eV to 200 keV range with the full Penn algorithm.

We have calculated inelastic mean free paths (IMFPs) for 42 inorganic compounds (AgBr, AgCl, AgI, Al2O3, AlAs, AlN, AlSb, cubic BN, hexagonal BN, CdS, CdSe, CdTe, GaAs, GaN, GaP, GaSb, GaSe, InAs, InP, InSb, KBr, KCl, MgF2, MgO, NaCl, NbC0.712, NbC0.844, NbC0.93, PbS, PbSe, PbTe, SiC, SiO2, SnTe, TiC0.7, TiC0.95, VC0.76, VC0.86, Y3Al5O12, ZnS, ZnSe, and ZnTe) for electron energies from 50 eV to 200 keV. These calculations were made with energy-loss functions (ELFs) obtained from measured optical constants for 15 compounds while calculated ELFs were utilized for the other 27 compounds. Checks based on ELF sum rules showed that the calculated ELFs were superior to the measured ELFs that we had used previously. Our calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 200 keV. The average root-mean-square (RMS) deviation in these fits was 0.60 %. The IMFPs were also compared with a relativistic version of our predictive Tanuma-Powell-Penn (TPP-2M) equation. The average RMS deviation in these comparisons was 10.7 % for energies between 50 eV and 200 keV. This average RMS deviation is almost the same as that found in a similar comparison for a group of 41 elemental solids (11.9 %) although relatively large deviations were found for cubic BN (65.6 %) and hexagonal BN (34.3%). If these two compounds are excluded in the comparisons, the average RMS deviation becomes 8.7 %. We found generally satisfactory agreement between our calculated IMFPs and values from other calculations and from experiments.

Strong decays of the orbitally excited scalar D^{*}_{0} mesons

We calculate the two-body strong decays of the orbitally excited scalar mesons D_0^*(2400) and D_J^*(3000) by using the relativistic Bethe–Salpeter (BS) method. D_J^*(3000) was observed recently by the LHCb Collaboration, the quantum number of which has not been determined yet. In this paper, we assume that it is the 0^+(2P) state and obtain the transition amplitude by using the PCAC relation, low-energy theorem and effective Lagrangian method. For the 1P state, the total widths of D_0^*(2400)^{0} and D_0^*(2400)^+ are 226 and 246 MeV, respectively. With the assumption of 0^+(2P) state, the widths of D_J^*(3000)^0 and D_J^*(3000)^+ are both about 131 MeV, which is close to the present experimental data. Therefore, D_J^*(3000) is a strong candidate for the 2^3P_0 state.

On transfer matrices, Bethe ansatz and scale invariance

We explicitly calculate the transfer-matrix eigenvalues in the massless sector using the exact integrable S-matrix, for up to 5 particles. This enables us to conjecture the general pattern. We use the conjectured form of the eigenvalues to write down a set of massless Bethe ansatz equations. The same procedure applies to the relativistic as well as to the non-relativistic situation. In the relativistic case, the right and left modes decouple. We speculate that the relativistic massless Bethe ansatz we obtain in that case might capture the integrable structure of an underlying 2D critical theory. We finally take advantage of some remarkable simplifications to make progress in the massive case as well.

Total bremsstrahlung spectra of thick lead compounds produced by 90Sr beta emitter in photon energy region of 10–100 keV

The total bremsstrahlung spectra in the thick targets of lead acetate trihydrate (Pb(CH3COO)2·3H2O), lead nitrate Pb(NO3)2 and lead chloride (PbCl2) produced by 90Sr beta particles have been investigated in the photon energy region of 10–100keV. The experimental bremsstrahlung spectra have been compared with the theoretical models Elwert corrected (non relativistic) Bethe Heitler theory, modified Elwert factor (relativistic) Bethe Heitler theory for ordinary bremsstrahlung and modified Elwert factor (relativistic) Bethe Heitler theory which includes polarization bremsstrahlung in the stripped atom approximation. The experimental results show better agreement with theoretical model that includes polarization bremsstrahlung in stripped approximation in the photon energy region below 30keV. However, at higher photon energy region 30–100keV, the theoretical model which describes ordinary bremsstrahlung is more accurate to describe the experimental bremsstrahlung spectra. The experimental results show positive deviations from the entire theoretical models at higher energy end of the spectrum. The results indicate that polarization bremsstrahlung plays important role in the formation of total bremsstrahlung spectra in lead compounds produced by continuous beta particles at low photon energy region of 10–30keV.

Radiative rates and electron-impact excitation for then≤ 6 fine-structure levels in H-like ions with 13 ≤Z≤ 42

Context. Many observed emission lines from space missions are due to highly charged H-like ions. An analysis of the lines provides information on the temperature, density, and chemical composition of plasmas. A wide range of atomic parameters, such as energy levels, radiative rates, and excitation rate coefficients are needed to achieve this goal.Aims. In this paper we report on calculations for energy levels, radiative rates, collision strengths, and effective collision strengths for transitions among the 36 lowest levels of the n ≤ 6 configurations of highly charged H-like ions with 13 ≤ Z ≤ 42.Methods. The widely used Flexible Atomic Code (FAC) is adopted for the calculation. Energy levels and radiative rates are calculated within the relativistic configuration-interaction method. Employing relativistic distorted-wave approximation, direct excitation collision strengths are calculated at eleven scattered electron energies E ′ f = 0.001, 0.002, 0.005, 0.01, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0, and 2.5, where E ′ f is in units of Z 2 rydbergs. Collision strengths at higher energies are estimated by interpolation/extrapolation using relativistic Bethe form. Resonance contributions through the relevant He-like doubly excited n ′l ′n ′′l ′′ configurations with n ′ ≤ 7 and n ′′ ≤ 75 are explicitly taken into account using the independent-process isolated-resonance approximation. Radiation damping effects are taken into account.Results. We present the radiative rates, oscillator strengths, and line strengths for all electric dipole (E1), magnetic dipole (M1), electric quadrupole (E2), magnetic quadrupole (M2), electric octupole (E3), and magnetic octupole (M3) transitions. Assuming a Maxwellian electron velocity distribution, we report effective collision strengths over a wide temperature range between 2 × 103 × Z 2 and 2 × 106 × Z 2 K. We believe that the present results are the most extensive and definitive atomic dataset to date for highly charged H-like ions.

Studies of total bremsstrahlung in thick targets of Al, Ti, Sn and Pb for 90Sr beta particles in the photon energy region of 1–100keV

Abstract Total bremsstrahlung (BS) spectra in thick targets of Al, Ti, Sn and Pb produced by beta emitter 90Sr (End point energy=546 keV) are studied in the photon energy range of 1–100 keV. The experimentally measured BS spectra are compared with the theoretical spectral distributions calculated from Elwert corrected (non relativistic) Bethe–Heitler [EBH] theory, modified Elwert factor (relativistic) Bethe–Heitler [Fmod BH] theory for ordinary bremsstrahlung (OB) and the Avdonina and Pratt [Fmod BH+PB] theory, which include the contribution of polarization bremsstrahlung (PB) into OB. The present results are indicating the correctness of Fmod BH+PB theory in the low energy region, where PB dominates into the BS, but at the middle and higher photon energy region of the bremsstrahlung spectrum, the Fmod BH theory is more close to the experimental results. The description of the bremsstrahlung process in stripped atom (SA) approximation, which indicates the suppression of the bremsstrahlung at higher energy ends due to the production of PB in the low energy region, needs further considerations. Hence, the present measurements for BS for different target materials indicates that the considerations of the screening effects along with other secondary effects during the interaction of incident electrons with the target nuclei are important while describing the production of bremsstrahlung, particularly for the higher energy regions.

Calculations of electron inelastic mean free paths. X. Data for 41 elemental solids over the 50 eV to 200 keV range with the relativistic full Penn algorithm

We have calculated inelastic mean free paths (IMFPs) for 41 elemental solids (Li, Be, graphite, diamond, glassy C, Na, Mg, Al, Si, K, Sc, Ti, V, Cr, Fe, Co, Ni, Cu, Ge, Y, Nb, Mo, Ru, Rh, Pd, Ag, In, Sn, Cs, Gd, Tb, Dy, Hf, Ta, W, Re, Os, Ir, Pt, Au, and Bi) for electron energies from 50 eV to 200 keV. The IMFPs were calculated from measured energy loss functions for each solid with a relativistic version of the full Penn algorithm. The calculated IMFPs could be fitted to a modified form of the relativistic Bethe equation for inelastic scattering of electrons in matter for energies from 50 eV to 200 keV. The average root‐mean‐square (RMS) deviation in these fits was 0.68%. The IMFPs were also compared with IMFPs from a relativistic version of our predictive Tanuma, and Powell and Penn (TPP‐2M) equation that was developed from a modified form of the relativistic Bethe equation. In these comparisons, the average RMS deviation was 11.9% for energies between 50 eV and 200 keV. This RMS deviation is almost the same as that found previously in a similar comparison for the 50 eV to 30 keV range (12.3%). Relatively large RMS deviations were found for diamond, graphite, and cesium as in our previous comparisons. If these three elements were excluded in the comparisons, the average RMS deviation was 8.9% between 50 eV and 200 keV. The relativistic TPP‐2M equation can thus be used to estimate IMFPs in solid materials for energies between 50 eV and 200 keV. We found satisfactory agreement between our calculated IMFPs and those from recent calculations and from measurements at energies of 100 and 200 keV. Copyright © 2015 John Wiley & Sons, Ltd.

Studies of polarization bremsstrahlung and ordinary bremsstrahlung from 89Sr beta particles in metallic targets in the photon energy region of 1–100 keV

Abstract Studies of polarization bremsstrahlung (PB) and ordinary bremsstrahlung (OB) produced by the 89Sr beta emitter in Al, Ti, Sn and Pb targets were undertaken at photon energies of 1–100 keV. The experimental results are compared with the Elwert corrected (non-relativistic) Bethe–Heitler (EBH) theory and the modified Elwert factor (relativistic) Bethe–Heitler ( F mod BH) theory for OB and with the Avdonina and Pratt ( F mod BH + PB) theory for total bremsstrahlung (BS). These results are in agreement with the F mod BH + PB theory up to 13 keV, 16 keV, 22 keV and 28 keV energies for Al, Ti, Sn and Pb targets, respectively; F mod BH theory is more accurate at higher energies.

Semileptonic Production of Scalar and Tensor p-Wave Charmed Mesons

With the same constituents of quark and antiquark, the p-wave state has a higher mass than the corresponding s-wave state, but the relativistic corrections are much larger in the processes of which p-wave mesons are involved. Considering this, the relativistic Bethe—Salpeter method is applied to estimate the branching fractions of semileptonic decays B(s) → D*(sJ)lν (J = 0,2). Our predictions are Br[ → (2317)l−ν̄] = 0.50%, Br[ → (2573)l−ν̄] = 0.40%, Br[B− → (2400)l−ν̄] = 0.79%, Br[B− → (2460)l−ν̄] = 0.28%, Br[ → (2400)l−ν̄] = 0.58% and Br[ → (2460)l−ν̄] = 0.27%..

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.

Calculation of the relativistic Bethe logarithm in the two-center problem

We present a variational approach to evaluate relativistic corrections of order ${\ensuremath{\alpha}}^{2}$ to the Bethe logarithm for the ground electronic state of the Coulomb two-center problem. That allows us to estimate the radiative contribution at $m{\ensuremath{\alpha}}^{7}$ order in molecular-like three-body systems such as hydrogen molecular ions ${\text{H}}_{2}$${}^{+}$ and ${\text{HD}}^{+}$ or antiprotonic helium atoms. While we get ten significant digits for the nonrelativistic Bethe logarithm, calculation of the relativistic corrections is much more involved, especially for small values of bond length $R$. We were able to achieve a level of three to four significant digits starting from $R=0.2$ bohr, which will allow us to reach ${10}^{\ensuremath{-}10}$ relative uncertainty on transition frequencies.

Hybrid Vlasov–Fokker–Planck–Maxwell simulations of fast electron transport and the time dependance of K-shell excitation in a mid-Z metallic target

Using a hybrid Vlasov–Fokker–Planck–Maxwell code coupled to calculations using a modified relativistic binary-encounter Bethe model of ionizing collisions we study the time and space dependence of Kα photon generation by a fast electron beam injected into a solid density copper plasma target. The electron beam is chosen to be representative of that expected to be generated by a picosecond duration, ∼1019 W cm−2 intensity laser. Kα photons are produced as electrons reflux laterally across the target and are slowed and thermalized by collisions with, and Ohmic heating of, the background fluid over a ∼10 ps timescale, which dictates the timescale for Kα emission. The results show reasonable agreement with recent experimental results in terms of both the yield and time dependance. We show how lateral expansion of the electrons can be imaged in the Kα radiation.