Born Approximation Results Research Articles

Cross Section for Pair Production in Al, Cu, Cd, and Pb at 2.615-MeVγ-Ray Energy

Cross sections for pair production by 2.615-MeV $\ensuremath{\gamma}$ rays incident on Al, Cu, Cd, and Pb have been found by subtracting tabulated values of the photoelectric cross section from measured values of total cross sections. The latter were determined to be 129.8, 137.0, 147.6, and 181.9 mb per electron, respectively, for the above elements. The statistical uncertainty is \ifmmode\pm\else\textpm\fi{}0.2%. Subtraction of photoelectric cross sections gives 0.0642 (\ifmmode\pm\else\textpm\fi{}5%), 0.343 (\ifmmode\pm\else\textpm\fi{}2.3%), 0.991 (\ifmmode\pm\else\textpm\fi{}1.4%), and 3.55 (\ifmmode\pm\else\textpm\fi{}0.8%) b per atom for pair production. The present result in Pb is 38.9% higher than the Bethe-Heitler Born approximation result and is about 14% higher than Jaeger-Hulme non-Born theoretical results but only 6% higher than the recent, exact non-Born calculation (neglecting screening) by \O{}verb\o{}, Mork, and Olsen. Differences between our values for Al, Cu, and Cd and the calculated values of these authors are even smaller, namely, 0.5% for Al and Cd, and 3% for Cu. It is pointed out that pair-production cross sections can be found from Born-approximation values by multiplication by $a+bZ$ for $0<Z<50$, with $a=0.97\ifmmode\pm\else\textpm\fi{}0.01$ and $b=4\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, and by a factor $1+\ensuremath{\alpha}{Z}^{2}$ for $Z\ensuremath{\ge}50$ with $\ensuremath{\alpha}=(5.8\ifmmode\pm\else\textpm\fi{}0.1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$.

Effect of nucleon continua on inelastic electron-scattering form factors

It is shown that the diffraction zeroes in the Born approximation result for the differential electron scattering cross section are replaced by finite minima if the continuum structure of the excited nuclear states is treated properly.

The role of volume dependence in the resistivity of a liquid metal

The experimental evidence suggesting that the Born approximation result for the temperature dependence of the resistivity of a liquid metal is inadequate is examined and compared with a calculation which reproduces the correct behaviour.

Excitation of Molecular Rotation by Slow Electrons

The cross sections for the rotational excitation of ${\mathrm{H}}_{2}$ and ${\mathrm{N}}_{2}$ by slow electrons are evaluated in a distorted-wave approximation with the use of a simple model potential. The changes from the previous Born-approximation results are quite pronounced, particularly for ${\mathrm{H}}_{2}$.

Partial-Wave Analysis of the Inelastic Scattering of Electrons by Nuclei. I. Results for Quadrupole Excitations

>The problem of nuclear Coulomb excitation by inelastic scattering is approached by using a distorted partial-wave description for the initial and final states of the electron and a multipole expansion of the interaction with the nucleus which is treated in the semiclassical approximation. The method is facilitated by use of a closed analytic form for the contribution to the scattering of partial waves of sufficiently high angular momenta to be insensitive to the shape (but not the charge) of the target nucleus. The contribution of partial waves which do penetrate the nuclear charge distribution is found by numerical evaluation of the various matrix elements using electron wave functions obrained by numerical solution of the radial part of the Dirac equation in a central field. The particular case of electric quadrupole excitation assuming a unlform charge distribution is calculated and compared with the usual Born approximation treatment of inelastic electron scattering. The results of the calculations for 200-Mev electrons differ from the Born approximation result in two ways: for scattering angles less than 90 deg the Coulomb field causes an apparent, and significant, change in the radius of the charge distribution; for scattering angles greater than 90 deg the cross sectionmore » differs qualitatively from the Born approximation result. (auth)« less

On the high energy behavior in field theory

The high energy behavior of cross sections and vertices is of great interest in dispersion theory and bears heavily on the consistency problem of field theory. Using the methcd of Lehmann et al., it is shown that the pi yields N + N + pi vertex cannot go to the Born approximation result in the limit of infinite invariant momentum transfer. These conclusions are validated by those of Aramaki, and Kallen for a meson propagator and an effective coupling constant. (B.O.G.)

Circular Polarization of Internal Bremsstrahlung

A calculation is made of the circular polarization of the internal bramsstrahlung in allowed $\ensuremath{\beta}$-decay with the most general $\ensuremath{\beta}$-interaction. The polarization is found to rise from zero at the low-energy end of the spectrum to a maximum value at the high-energy end. This maximum value depends critically on the relative magnitudes of the coupling constants in the $\ensuremath{\beta}$-interaction. With the two-component neutrino theory and only the scalar and tensor interactions present, the polarization is complete at the high-energy end of the spectrum. Plots have been made, with this $\ensuremath{\beta}$-coupling, of the Born approximation result for the circular polarization as a function of photon energy for ${\mathrm{S}}^{35}$ and ${\mathrm{P}}^{32}$. The effects of the nuclear Coulomb field on the $\ensuremath{\gamma}$-ray spectrum and polarization are discussed.

Detection of Electron Polarization by Double Scattering

12:000356A calculation is made of the circular polarization of the internal bremsstrahlung is allowed beta decay with the most general beta interaction. The polarization is found to rise from zero at the low-energy end of the spectrum to a maximum value at the high-energy end. This maximum value depends critically on the relative magnitudes of the coupling constants in the beta interaction. With the two-component neutrino theory and only the scalar and tensor interactions present, the polarization is complete at the high- energy end of the spectrum. Plots have been made, with this beta coupling, of the Born approximation result for the circular polarization as a function of photon energy for S/sup 35/ and P/sup 32/. The effects of the nuclear Coulomb field on the gamma -ray spectrum and polarization are discussed. (auth)

Application of Variational Methods to Intermediate and High-Energy Scattering

The application of Schwinger's variational principle to scattering by a Yukawa potential is studied. A detailed comparison of the Born approximation, variational, and exact results is given in the energy region from 20 to 150 Mev for a potential strength and range appropriate to the $^{3}S$ neutron-proton interaction. Comparison with the second Born approximation is also made. The utility of the variational principle is discussed and consideration is given to the high-energy limit. It is judged that for most potentials, the variational formulation without elaborate trial functions will be better than the Born approximation.The relation between the variational principles for the total scattering amplitude and for the phase shifts is analyzed. A new formulation intermediate between these two is presented as a compromise between the simplicity of the former and the accuracy of the latter. With the potential studied, use of this formulation leads to considerably better results for the total amplitude than does the original principle.

Higher-Order Potential Effects in the Radiative Correction to Scattering of Slow Electrons

The radiative correction to the scattering of low-energy electrons by an external field has been calculated in first Born approximation in the external field. For strong fields or slow electrons this is not a valid approximation. With the assumption that only low-energy virtual photons contribute significantly to the effect, the radiative correction to the scattering of slow electrons is calculated to all orders of the external field. There is no change in the fractional radiative correction from the Born approximation result.

Radiative Processes in the Presence of Heavy Nuclei

Calculations of the cross sections for bremsstrahlung and pair production are carried out without the use of the Born approximation, but by reducing the calculation of these cross sections to the calculation of the exact elastic scattering of electrons and positrons, the last quantities having been tabulated to some extent. Large deviations from the Born approximation results are obtained in the angular distributions of these radiative processes, but arguments are presented that the Born approximation should yield the integrated cross sections correctly at sufficiently high energies. The failure of the Born approximations for relativistic equations is explained in a purely classical way.