Differential Bremsstrahlung Cross Section Research Articles

A model for the energy and angular distribution of x rays emitted from an x-ray tube. Part I. Bremsstrahlung production.

To develop an analytical model for bremsstrahlung production in a thick x-ray target (i.e., the x-ray tube anode) that takes into account the intrinsic bremsstrahlung angular distribution. X-ray spectrum models developed from theoretical principles have traditionally treated the angular distribution of the bremsstrahlung production as spherically uniform. This assumption stems from the rationale that electrons promptly attain a diffuse directional distribution in an x-ray target due to multiple scattering, thereby effectively masking the intrinsic bremsstrahlung angular distribution. In this work, a model that explicitly accounts for the angular distribution of the bremsstrahlung production is presented. The model combines Monte Carlo-calculated depth, energy, and angular distributions of electrons penetrating the x-ray target, and incorporates theoretical results for the differential bremsstrahlung cross section. The effects of using different simplified model assumptions for the electron penetration and the intrinsic bremsstrahlung angular distribution are analyzed for tungsten and molybdenum targets in the energy range 20-300keV. Typical assumptions of previous models are shown to introduce errors in calculated spectra. Particularly, it is shown that predictions of fluence and air kerma free-in-air can be overestimated by 15-30% (2-3% in aluminum half-value layer thickness) for clinically relevant beam qualities. The present model is able to reproduce comprehensive Monte Carlo calculations of the bremsstrahlung production generally to within 1%. The bremsstrahlung model developed in this work is an improvement over previous models in that the main features of the electron penetration and the resulting bremsstrahlung are considered in detail. The model can be used for more accurate predictions of the energy and angular distribution of x rays emitted from an x-ray tube.

The DSM theory for tip electron-atom bremsstrahlung at [formula omitted] MeV

The Dirac–Sommerfeld-Maue (DSM) theory for electron bremsstrahlung at the short-wavelength limit, based on a point-Coulomb description of the target nucleus, is tested against the Dirac partial-wave (DW) theory at collision energies up to 35 MeV. It is found that for the small photon angles, which contribute to the angle-integrated bremsstrahlung yield, the screening effects are at most in the percent region. Predictions are made for the nuclear charge dependence of the singly differential bremsstrahlung cross section at 500 MeV, where experiments are planned in the near future.

The [formula omitted] beam line of the São Paulo Microtron electron accelerator

The São Paulo Microtron 10−100keV electron beam-line described here was designed to allow precise measurements of the electron-atom interaction cross-sections using reproducible arrangements. The irradiation chamber has thirteen spectroscopy windows for x-ray detectors and a multi-target ladder. The photon background on the target was reduced by using a large and deep Faraday cup, lining the chamber's interior with plastic, and mounting the targets in carbon fiber frames. The current research plan includes measurements of: electron doubly differential bremsstrahlung cross sections; inner shell ionization cross sections; angular distributions of L X-rays, and the angular distribution of scattered electrons.

Bremsstrahlung production by colliding beams of protons and electrons

The differential cross section of bremsstrahlung emitted in the head-on collision of beams of energetic protons and electrons is calculated in the laboratory frame. The angular distribution of the photons is more isotropic than in the common bremsstrahlung production on stationary target atoms. The total cross section is determined by integration over the photon angles which are allowed kinematically.

Absolute differential bremsstrahlung cross sections for 0.4–2-keV electrons scattered by Ar, Kr, and Xe atoms

The dependences of absolute differential bremsstrahlung cross sections on 0.4--2-keV electron scattering by supersonic gas jets of Ar, Kr, and Xe versus electron energy as well as the atom number $Z$ were studied experimentally. The bremsstrahlung photons have been detected at an angle $97\ifmmode^\circ\else\textdegree\fi{}$ to the incident electron beam in the ultrasoft x-ray spectral region 70--200 eV. It is found that as the electron energy is increased from 0.4 to 2 keV, the absolute bremsstrahlung cross section for Xe increases throughout the whole electron energy range and the energy dependencies of cross sections for Ar and Kr displayed maxima at electron energies of $\ensuremath{\sim}0.7\text{ }\text{keV}$ (Ar) and $\ensuremath{\sim}1\text{ }\text{keV}$ (Kr). These experimental results cannot be explained in the framework of Bethe and Heitler and Sauter theory, based on the first Born approximation. The experimental absolute differential bremsstrahlung cross sections for Ar, Kr, and Xe are considerably larger then the tabulated values of ordinary bremsstrahlung. The obtained $Z$ dependence of bremsstrahlung cross section differs with calculated one, particularly for large atomic number $Z$.

Absolute differential bremsstrahlung cross section for the scattering of 0.6-keV electrons by xenon atoms

The spectral distribution of the absolute differential cross section for total bremsstrahlung, including ordinary and polarization, for the scattering of 0.6-keV electrons from free xenon atoms has been measured. The bremsstrahlung photons have been detected at an angle of 97° with respect to the electron motion direction. The investigations have been carried out in the ultrasoft x-ray spectral region 60–240 eV. The results are compared with the total cross sections for bremsstrahlung calculated in several approximations for the scattering of 5-and 25-keV electrons by Xe atoms. Both the value and the spectral distribution of the experimental absolute cross sections are most satisfactorily reproduced by the calculations in the distorted partial wave approximation.

Calculation of x‐ray spectra emerging from an x‐ray tube. Part II. X‐ray production and filtration in x‐ray targets

A new approach to the calculation of the x-ray spectrum emerging from an x-ray tube is proposed. Theoretical results for the bremsstrahlung cross section appearing in the literature are summarized. Four different treatments of electron penetration, based on the work presented in Part I, are then used to generate bremsstrahlung spectra. These spectra are compared to experimental data at 50, 80 and 100 kVp tube potentials. The most sophisticated treatment of electron penetration was required to obtain good agreement. With this treatment both the National Institute of Standards and Technology bremsstrahlung cross sections, based on accurate partial wave calculations, and the Bethe-Heitler cross section [H. A. Bethe and W. Heitler, Proc R. Soc. London, Ser. A. 146, 83-112 (1934)] corrected by a modified Elwert factor [G. Elwert, Ann. Phys. (Leipzig) 426, 178-208 (1939)], provided good agreement to measured data. An approximate treatment of the characteristic spectrum is suggested. The dependencies of the bremsstrahlung and characteristic outputs of an x-ray tube on tube potential are compared to experimentally derived data for 70-140 kVp potentials. Agreement is to within a few percent of the total output over the entire range. The spectral predictions of the semiempirical models of Birch and Marshall [R. Birch and M. Marshall, Phys. Med. Biol. 24, 505-513 (1979)] (IPEM Report 78) and of Tucker et al. [D. M. Tucker, G. T. Barnes, and D. P. Chakraborty, Med. Phys. 18, 211-218 (1991).] are also assessed. The predictions of Tucker et al. are very close to the model developed here. The predictions of IPEM Report 78 are similar, but consistently harder for the range of tube potentials examined (50-100 kV). Unlike the semiempirical models, the model proposed here requires the introduction of no empirical and unphysical parameters in the differential bremsstrahlung cross section, bar an overall normalization factor which is close to unity.

Electron–positron pair production and bremsstrahlung at intermediate energies in the field of heavy atoms

The Coulomb corrections (CC) to the processes of bremsstrahlung and pair production are investigated. The next-to-leading term in the high-energy asymptotics is found. This term becomes very essential in the region of intermediate energies. The influence of screening for CC is small for differential cross section, spectrum, and the total cross section of pair production. The same is true for the spectrum of bremsstrahlung, but not for the differential cross section, where the influence of screening can be very large. The corresponding screening corrections as well as the modification of the differential cross section of bremsstrahlung are found. A comparison of our results for the total cross section of pair production with the experimental data available is performed. This comparison has justified our analytical results and allowed to elaborate a simple ansatz for the next-to-leading correction. The influence of the electron beam shape on CC for bremsstrahlung is investigated. It turns out that the differential cross section is very sensitive to this shape.

Absolute doubly differential cross sections for electron bremsstrahlung from rare gas atoms at 28 and 50 keV.

Absolute doubly differential bremsstrahlung cross sections from Xe, Kr, Ar, and Ne have been measured for electron bombarding energies of 28 and 50 keV. Bremsstrahlung photons have been detected at 90 degrees to the incident electron beam at energies ranging from 5 keV up to the kinematic end point. The results are compared with predictions of ordinary bremsstrahlung and of total bremsstrahlung that include polarizational bremsstrahlung from the target atom calculated in the stripping approximation. All previous absolute cross sections have been from thin-film solid targets and have not shown any polarizational bremsstrahlung contribution. The present results, the first from free atoms, provide definitive evidence for the contribution of polarizational bremsstrahlung to the photon spectrum from electron bremsstrahlung.

Analysis of the differential cross sections for bremsstrahlung during collisions between two charged particles in a uniform electric field

Under the nonrelativistic Born approximation, differential cross sections are derived for elastic collisions between two point charged particles in an external constant uniform electric field and for bremsstrahlung during these collisions.An analysis of the cross sections obtained shows that, due to the interference of the wave functions of two colliding particles during their reflection from the potential barrier of an external electric field, the differential cross sections for elastic collisions and for unpolarized bremsstrahlung (i) are both oscillatory in character and (ii), instead of being linearly proportional to one another (as in the case without an externale electric field), are related in a more complicated manner.

Electron bremsstrahlung in collisions of 223 MeV/ u He-like uranium ions with gaseous targets

The bremsstrahlung process in the domain of strong Coulomb fields has been investigated for and Ar target electrons colliding with He-like uranium ions at . The differential cross sections for bremsstrahlung were measured at laboratory observation angles of , , and . Substantial discrepancies were found between the experimental cross sections and the first-order Born approximation calculations. The reported data provide a new testing ground for non-perturbative treatment of the coupling between radiation and matter in the presence of strong fields.

STUDY OF ELECTRON BREMSSTRAHLUNG IN STRONG COULOMB FIELDS AT THE ESR STORAGE RING

Electron bremsstrahlung has been investigated for collisions of 223.2 MeV/amu He-like uranium ions with N2 and Ar gaseous targets. The doubly differential cross-sections for bremsstrahlung are compared to the predictions of relativistic first order Born calculations (Bethe-Heitler formula with Elwert correction factor) and to the calculations based on the exact wavefunctions for electrons moving in the external point-Coulomb potential of the projectile. Whereas the “exact” IPA calculations give an improved agreement with experimental data, as compared to the Bethe-Heitler theory, in particular at the end-point region of the bremsstrahlung spectra, discrepancies still remain at lower photon energies.

Thin-target bremsstrahlung at 0° from 50 keV electrons

New results are reported for the relative doubly differential bremsstrahlung cross section for photons radiated at 0° for 50 keV electrons incident on Ag, TbF 3, Au and UF 4 targets. A “difference” method, particularly useful at 0°, is used and discussed. Problems and remedies unique to the 0° geometry are presented. A new method of fitting theory to data which includes both detector response and background contributions is also introduced. The measured photon energy dependence of the cross sections are found to agree with the theory within an error of about 3%.

Nonrelativistic calculation of an elementary process of electron bremsstrahlung from atoms.

We calculate the unpolarized triply differential cross section of bremsstrahlung from atoms with atomic number Z=13 and 79 for incident electrons of kinetic energy ${T}_{1}$=50 and 140 keV. It is an exact numerical nonrelativistic calculation in a partial-wave formulation. Our results show that the screening effect is important in the high-Z cases, as the behavior of angular distributions of recoiled electrons. When the Coulomb parameter is large, the nonrelativistic Born approximation is not good, as expected. Comparisons of our partial-wave results with experimental results of Faulk and Quarles [Phys. Rev. A 9, 732 (1974)] show large discrepancies.

Numerical calculation of classical bremsstrahlung.

We have carried out numerical calculations for the classical bremsstrahlung spectrum, angular distribution, and polarization resulting from electrons scattering in screened atomic potentials. The calculations utilize the classical theory of electromagnetic radiation from moving charges together with the classical mechanics of energy-loss-free orbits of charged particles in such potentials. The trajectories of incident electrons in the given central potential are calculated numerically, and the doubly differential bremsstrahlung cross section is obtained by taking the Fourier transform of the acceleration of the dipole moment of the projectile and integrating the radiation over the impact parameters of the beam. Results have been obtained for several neutral atoms for electron energies from 1 eV to 500 keV, using self-consistent Dirac-Slater atomic potentials. Comparisons with quantum-mechanical partial-wave results for the spectrum show that, as in the point Coulomb case, the classical method is generally good for screened potentials for low incident electron energies. The classical method therefore can be useful in low-energy situations for which the usual Elwert-Born form-factor method is not satisfactory. We have also obtained and examined the angular distributions of classical bremsstrahlung. Screening reduces the value of the asymmetry parameter which characterizes the angular distribution of dipole radiation. The difference of screened results from the point Coulomb results increases with decreasing incident electron energy. At very low energy the value of the asymmetry parameter oscillates with energy, and this also leads to oscillations in the degree of polarization.

Bremsstrahlung Cross-Section Formula Including a High-Order Coulomb Correction

A new expression of the differential cross section of bremsstrahlung in nuclear fields is given. Higher-order Coulomb correction terms are included, but the effect of the atomic electron screening, the radiative corrections, and the nucleus size are neglected. Validity of the formula obtained here is discussed. It is expected to give good accuracy in the energy region from a few MeV to about 50 MeV for high values of the atomic number of the target, as long as the screening effects are negligible. Some remarks concerning computation of the cross section are included.

Calculation of Bremsstrahlung Cross Sections with Sommerfeld-Maue Eigenfunctions

A formula for the differential cross section of bremsstrahlung is calculated with the aid of Sommerfeld-Maue eigenfunctions, i.e., under the assumption of a pure Coulomb field and low atomic numbers ($\ensuremath{\alpha}Z\ensuremath{\ll}1$). This expression is valid for all energies of both electrons and photons, and it is shown that the previously well-known formulas of Sommerfeld, Sauter, Bethe, Heitler, Scherzer, and Bethe and Maximon are contained as special cases. Hence this cross section is correct for high electron energies and, apart from a small spin correction, for nonrelativistic energies even if $\ensuremath{\alpha}Z\ensuremath{\ll}1$ is not satisfied. The formula has been programmed, and in addition total cross sections have been obtained by numerical integration over the angles of the emitted electrons and photons. Comparison with the Born approximation allows derivation of the Coulomb correction. Several cases of interest are compared with experiment, in particular the elementary bremsstrahlung process itself, for which experimental results are now available.

Nuclear Bremsstrahlung from p-p and p-d interactions near 140 MeV

Differential cross-sections for bremsstrahlung have been measured for p-p and p-d interactions near 140 MeV. The total cross-sections are, respectively, −0.01±0.06 μb and 8±1 μb for photon energies above 30 MeV and −0.06 ± 0.05 μb and 4.0 ± 0.3 μb for photon energies above 40 MeV. These results are compared with those of other experimenters and with theoretical predictions.

The influence of electronic screening on high-energy bremsstrahlung and pair production

The differential cross-sections for bremsstrahlung and for pair production, the cross-sections integrated over angles for bremsstrahlung and pair production, and the cross-section integrated over angles and energies for pair production are obtained as single integrals over the atomic form factor. It is shown that the shell structure of the atom does not give corresponding wiggles in the bremsstrahlung spectrum. It is shown that the shape of the bremsstrahlung spectrum is only slightly-dependent on the angle of emission. The total cross-section for pair production for 12 elements ranging from Li to U and for 41 energies ranging from 10 MeV to 100 GeV are computed numerically with the use of Hartree-Fock-Slater atomic form factors. These are compared with cross-sections computed from Thomas-Fermi-Moliere form factors. For 1 GeV the error in the Thomas-Fermi-Moliere cross-section is 2.2% in the case of Li, 0.51 % in the case of IT. These errors are of the same order of magnitude as the radiative correction, viz. of the order of 1%. The computed Hartree-Fock-Slater cross-sections agree fairly well with experiments. Radiative corrections have not been taken into account.

Form factor influence on processes of bremsstrahlung and pair production on protons

The differential cross-sections of the processes of electron bremsstrahlung and pair production by γ-quanta on protons in presence of the form factor were calculated. This calculation was made in the lowest order of perturbation theory. The formula of the differential bremsstrahlung cross-section was integrated over all angles, except the angle between the direction of the incident electron and that of the emitted photon. A comparison of the differential and integral bremsstrahlung cross-sections with the corresponding Bethe-Heitler formulae was made.