Formula For Cross Section Research Articles

Photodetachment of H − in electric and magnetic fields with arbitrary orientation

We present a derivation of the general quantum mechanical formula for the photodetachment cross section of H − in electric and magnetic fields with arbitrary orientation. The calculation displays the effects of the orientations of the fields upon the oscillatory structures on the cross section spectra, which are given a satisfactory physical interpretation with the help of closed-orbit theory.

Remark on the Kinematics of the One-Photon Production in Electron–Proton Scattering

We clarify some misleading mistakes in the literature about the phase space for the one-photon production in electron–proton scattering and present the correct overall phase space factor in the cross-section formula. Our conclusion is that the earlier diquark model predictions for the cross-sections of the one-photon production in electron–proton scattering are unreliable. Furthermore, we specify the phase space boundaries. In relation to the recent proposal to measure the off-diagonal quark distribution functions in the deeply virtual Compton scattering off the proton, we point out that the kinematics strongly disfavors such experiments.

Closed-orbit theory and the photodetachment cross section ofH−in parallel electric and magnetic fields

In this paper we obtain a simple analytic formula for the photodetachment cross section of H 2 in parallel electric and magnetic fields. The three-dimensional semiclassical approximation predicts oscillations in the spectrum and correlates these oscillations with closed classical orbits. The cylindrical symmetry of the Hamiltonian produces some interesting effects. In particular, at boundary energies the semiclassical approximation fails as a focused cusp approaches the origin. @S1050-2947~97!03306-4# PACS number~s!: 32.80.Gc I. INTRODUCTION Experimental measurements of the photodetachment cross section of H 2 in strong static electric fields were made by Bryant et al. @1#. For energies above the threshold energy the resulting cross section was found to be a smooth background upon which was superposed sinusoidal oscillations. Theoretical discussions of the measurements have been given by a number of authors @2,3#. The oscillations arise as an interference effect because the outgoing electron can move against the electric force and then return to the atom. It was shown in Ref. @3# that the cross section could be expressed in the following way: s~ E!5s 0~ E!1C~ E!sinF~ E!, where s 0 is the cross section in the absence of any external fields andC(E) and F(E) are called the recurrence strength and recurrence phase associated with the returning orbit. Theoretical calculations @4,5# show that analogous oscillations occur for photodetachment in crossed electric and magnetic fields, and that the oscillations are again associated with closed orbits. Quantum calculations have shown that very strong oscillations are present in parallel fields @6#, but the relationship to closed orbits was not made clear. Our original motivation in the present paper was to complete this subject by calculating the closed orbits and the resulting spectral oscillations for photodetachment of an electron from H 2 in parallel electric and magnetic fields. This is a system in which the recurrences are simple, strong, relatively easy to observe, and easy to understand. We found results which go far beyond this topic, and which are connected with many other problems of current interest. We know from study of nonlinear dynamics that even when long-time motion is chaotic, short-time motion remains relatively simple and predictable. Periodic orbits play a central role: in Poincare ´ ’s words, they offer ‘‘the only opening through which we might try to penetrate the fortress ~chaos! which has the reputation of being impregnable.’’ This classical statement holds also in quantum mechanics, wherein the periodic-orbit theory of Gutzwiller, Balian and Bloch, and Berry and Tabor @7# provides a general theoretical framework for studying quantum manifestations of classical chaos. Besides producing oscillations in absorption spectra @8#, periodic orbits produce scars in wave functions @9#, oscillations in the density of states @10#, and real-time recurrences that have been observed in many atoms and molecules @11#. @Similar phenomena are observed in microwaves in cavities @12~a!#, in microjunctions @12~b!#, and they are calculated to be consequences of certain models in nuclear physics @12~c!#.# Bifurcations of periodic orbits are of particular interest. A

Quantum manifestations of bifurcations of closed orbits in the photodetachment cross section ofH−in parallel fields

In the preceding paper, we showed that the semiclassical approximation diverges at a bifurcation, and that this divergence coincides with the passage of a focused cusp through the origin. Here we obtain a wave function in the vicinity of this cusp, and we use that wave function to eliminate the divergences in the photodetachment cross section. To describe the focused cusp, we first discuss the wave function of an ordinary two-dimensional (nonfocused) cusp. This wave function is known as a Pearcey function, and it has been studied extensively. Then we show how the formulas that lead to the Pearcey function have to be modified to describe a cylindrically focused cusp. The resulting wave function turns out to be given by an integral of Fresnel type containing within it a cylindrical Bessel function. This wave function is used to derive a formula for the photodetachment cross section near a bifurcation. That formula is a simple closed-form expression containing a Fresnel integral. Comparison with exact quantum calculations shows that this corrected-semiclassical formula is quite accurate.

Collisional frequency shift with a primary frequency standard

Collisional frequency shift can be a serious problem affecting the atomic fountain cesium standard. In this paper, I show that it can also be a problem for a traditional cesium standard employing an atomic beam that passes through two small, widely separated holes. The atomic wave packet is broadened perpendicular to the atomic beam. Atomic relative motion is parallel to the atomic beam and the wave packet is broadened transversally to relative motion. The collisional cross section is then much larger than that in a room temperature cell, as the collisional cross section is a value measured on a plane that is transverse to the relative motion. This cannot be derived from the previous formula for the collisional cross section, based on assumptions that the angular momentum of relative motion is deterministic. Note that the collisional cross section does not reach infinity even if the transverse velocity component is definitely zero.

Analyzing the photodetachment cross section ofH−in electric and magnetic fields with arbitrary orientation

We present the general quantum-mechanical formula for the photodetachment cross section of ${\mathrm{H}}^{\mathrm{\ensuremath{-}}}$ in the electric and magnetic fields with any orientation and the comparison of the quantum-mechanical cross sections to the cross sections calculated from the closed-orbit theory.

Optical theorem formulation of low-energy nuclear reactions

We present a detailed description of a recently proposed optical theorem formulation of nonresonant low-energy nuclear reactions between two charged nuclei. Based on the optical theorem formulation, we obtain an analytic formula for the reaction cross section \ensuremath{\sigma}(E) which exhibits explicitly the energy and charge dependences of \ensuremath{\sigma}(E). The formula may provide a physical understanding of the anomalous low-energy enhancement of \ensuremath{\sigma}(E) observed in sub-barrier heavy-ion fusions and also in light nuclei fusions relevant for primordial nucleosynthesis and stellar evolution. As examples of its application, the new formulation is used to analyze astrophysical S factors for ${}^{7}\mathrm{Li}(p,\ensuremath{\alpha}{)}^{4}\mathrm{He},$ ${}^{6}\mathrm{Li}(d,\ensuremath{\alpha}{)}^{4}\mathrm{He},$ and ${}^{6}\mathrm{Li}(p,\ensuremath{\alpha}{)}^{3}\mathrm{He}$ reactions.

Analytical parametrization for the shape of atomic ionization cross sections

The behavior of the ionization cross section of atoms is known classically in the limits of threshold energy and at high energies. These two limits are used to construct a simple analytical formula for the ionization cross section that depends upon two parameters: the magnitude of the maximum of the cross section and its position in energy. The parametrization has been tested for electron- and positron-impact ionization as well as for proton- and antiproton-impact ionization. It reproduces in all cases the shape of the cross section and offers a unified treatment for ionization by bare projectiles irrespectively of their charge and mass.

Numerical results for high-energy electron scattering cross sections from two-centre potentials

In this work we present the semiclassical formalism of relativistic electron scattering from two-centre potentials. The numerical results for the computation of cross sections have been developed and tested for the first time. Earlier results were limited up to the calculations of phase shifts only and thus had limitations for the application to practical cases. The expressions used for developing the programmes of general formulae for cross sections and the different parameters, i.e. separation constants , normalization constants , angular function , expansion coefficients and phase shifts have also been presented. The behaviour of the results of cross sections have been compared with those for the case of a one-centre potential in the limit of the intercentre distance approaching zero.

The influence of quasi-localized carrier states on the optical properties of size-quantizing structures

We considered the optical properties of size-quantizing semiconductor structures with quasi-localized electron states. In particular, the formulae we obtained allow us to determine the optical absorption due to electron transitions from the localized ground state to an excited quasi-localized (resonant) one. The calculations were carried out for objects of various dimensionalities, that is for a layered structure (quantum well) and a spherically symmetric irregularity (quantum dot), suggesting a certain model potential. The obtained results qualitatively correspond to the experimental observations, yielding an intermediate picture, when compared to the electronic transitions to an excited bound state and to a delocalized band state. Having derived a formula for the absorption cross-section of a single quantum well, we also obtained expressions for the corresponding corrections to the absorption coefficient, refractive index and dielectric constant of the multi-quantum well structure as functions of the frequency of incident light and the parameters of the quantum wells and barriers. The results may be useful in designing quantum well infrared photodetectors.

Analytical partitioning of total cross sections for electron scattering on noble gases

Empirical analytical formulae are proposed, aiming at the description of partial cross sections for electron scattering on noble gases. Selected measurements of total, excitation and ionization cross sections for these gases have been fitted by these analytical formulae in the 20-5000 eV energy range. The elastic cross sections, obtained as the difference, compare quite well with the elastic experimental data for all gases but for helium. The difference elastic cross sections were fitted by a cross section formula derived from electron scattering on a single or double Yukawa potential in the first Born approximation. Our analysis shows a good consistency between partial and total cross sections for the four heavier gases. For helium, we suggest the possibility of large errors in the high energy part of total, elastic and excitation cross sections. The analytical expressions given in this paper allow to reproduce the total and partial cross sections within the measurement errors in a large energy range from a few tens of eV, up to several keV.

On soft collisions of fast highly-charged ions with hydrogen and helium atoms

Ionization of hydrogen and helium atoms in small momentum transfer collisions with fast highly-charged ions is considered for the range of parameters , (Z and v are the ion charge and velocity, respectively, and is the characteristic velocity of the electron in the atom ground state). Analytical expressions have been derived for single and double differential cross sections of hydrogen ionization, accompanied by emission of slow electrons with the velocities . A generalization of the results obtained is proposed for helium single ionization. Soft collisions were shown to contribute mainly to hydrogen ionization and were found to determine practically the magnitude of the helium single ionization cross section. An asymmetry in the angular distribution of slow electrons is analysed. A formula for the helium single ionization cross section is proposed. A peculiarity of momenta transfer for soft collisions is briefly discussed.

A semi-empirical formula for total cross sections of electron scattering from diatomic molecules

A fitting formula based on the Born approximation is used to fit the total cross sections for electron scattering by diatomic molecules (CO, N 2, NO, O 2 and HCl) in the intermediate- and high-energy range. By analyzing the fitted parameters and the total cross sections, we found that the internuclear distance of the constituent atoms plays an important role in the e-diatomic molecule collision process. Thus a new semi-empirical formula has been obtained. There is no free parameter in the formula, and the dependence of the total cross sections on the internuclear distance has been reflected clearly. The total cross sections for electron scattering by CO, N 2, NO, O 2 and HCl have been calculated over an incident energy range of 10–4000 eV. The results agree well with other available experimental and calculation data.

Total reaction cross section calculations in proton-nucleus scattering.

We have developed a general semiempirical total reaction cross section formula for proton-nucleus reactions. It is applicable for all materials with nuclear charge Z greater than 5, and all proton kinetic energies from the lowest measured energies up to \ensuremath{\approxeq}20 GeV. While the regime of applicability is greatly enlarged, agreement between the calculated cross sections and the experimental data is comparable to or better than all earlier published results in their regime of applicability. \textcopyright{} 1996 The American Physical Society.

Hard pair production in large-angle Bhabha scattering

The cross section of hard pair production in large-angle Bhabha scattering is calculated in the leading and next-to-leading logarithmic approximations. Eight regions of the collinear kinematics, when the final particles imitate a process of the 2 → 2 type, and three semicollinear regions, when the final particles imitate a process of the 2 → 3 type, are considered. Analytical formulae for differential cross sections are presented.

Classical and Quantum Many-Body Description of Bremsstrahlung in Dense Matter: Landau–Pomeranchuk–Migdal Effect

Some considerations about the importance of coherence effects for bremsstrahlung processes in nonequilibrium dense matter (Landau–Pomeranchuk–Migdal effect) are presented. They are of particular relevance for the application to photon and di-lepton production from high energy nuclear collisions, to gluon radiation in QCD transport, or parton kinetics and to neutrino and axion radiation from supernova explosion and from hot neutron stars. The soft behavior of the bremsstrahlung from a source described by classical transport models is discussed and pocket correction formulas for the in-matter radiation cross sections are suggested in terms of standard transport coefficients. The radiation rates are also discussed within a nonequilibrium quantum field theory (Schwinger–Kadanoff–Baym–Keldysh) formulation. A classification of diagrams and corresponding resummation in physically meaningful terms is proposed, which considers the finite damping width of all source particles in matter. This way each diagram in this expansion is already free from infra-red divergences. Both, the correct quasi–particle and quasi-classical limits are recovered from this subset of graphs. Explicit results are given for dense matter in thermal equilibrium. The diagrammatic description may suggest a formulation of a transport theory that includes the propagation of off-shell particles in nonequilibrium dense matter

Dilepton bremsstrahlung from pion-pion scattering in a relativistic one boson exchange model.

We have made a detailed and quantitative study of dilepton production via bremsstrahlung of a virtual photon during pion-pion collisions. Most calculations of electromagnetic radiation from strong interaction processes rely on the soft photon approximation (SPA). The conditions underlying this approximation are generally violated when dilepton spectra are calculated in terms of their invariant mass, so that an approach going beyond the SPA becomes necessary. Superseding previous derivations, we derive an exact formula for the bremsstrahlung cross section. The resulting formulation is compared to various forms based on the SPA, the two-particle phase space approximation and R\"uckl's formula using a relativistic One Boson Exchange (OBE) model. Within the OBE approach, we show that approximations to the bremsstrahlung dilepton cross sections often differ greatly from the exact result; discrepancies become greater both with rising temperature and with invariant mass. Integrated dilepton production rates are overestimated by R\"uckl-based approximations by factors 1.5--8.0. The largest discrepancies occur for the reaction $\pi^+\pi^+ \to \pi^+\pi^+\ell^+\ell^-$, where such approximations overestimate the exact rate by factors ranging from 2 to 30 for invariant masses between 10 and 500 MeV. Our findings, combined with recent estimates of the Landau-Pomeranchuk effect, indicate that bremsstrahlung dileptons rates in ultrarelativistic heavy ion collisions should be even more suppressed than had been thought before.

Asymmetry of bremsstrahlung from polarized electrons

The triply differential cross section of unpolarized bremsstrahlung produced by polarized electrons in the Coulomb field of a nucleus is calculated with the aid of Sommerfeld-Maue wave functions. The photon asymmetry originating from transversely polarized electrons is calculated for various values of energies and angles, in particular for the parameters of recent experiments. As a function of the atomic number Z of the target nucleus, the asymmetry shows a distinct deviation from a linear dependence, in contrast to a previous calculation exact to lowest order in the Coulomb parameter Z/137. The cross-section formula is integrated numerically over the angles of the outgoing electrons to obtain the photon asymmetry observed in non-coincidence experiments.

Calculations of depth-dose distributions, cross sections and momentum loss

The ability to know with precision the depth-dose, dose average LET, fluence and energy distributions is of great importance in many research fields, including therapeutic and diagnostic medicine when using heavy ion beams, as well as in space research. We have therefore developed a model and a computer code for calculating these distributions when using high energy proton or heavy ion beams. In this model, we use semi-empirical total reaction and partial cross section formulas developed by us and a new prescription to take into account the energy and momentum loss of the secondary nuclei. In this paper, we will also present an empirical equation for the total inelastic 4He-p cross section, as well as the partial cross sections for the production of 3He, 3H, 2H, p and n.

Semiempirical formula for multiple ionization cross sections of atoms by electron impact

On the basis of experimental data on multiple-ionization cross sections σn of neutral atoms by electron impact a semiempirical formula for σn, n ⩾ 3, has been deduced where n is the number of ejected electrons. It has been shown that the cross section σn depends on three atomic parameters: the minimal ionization energy required to remove n electrons from the target, the total number of target electrons and the number of ejected electrons. The electron-impact energy dependence of the cross sections σn with different n is described by one universal function. As the number of ejected electrons n increases the cross section σn falls off approximately as σn ~ n−6.A comparison of σn calculated by the proposed semiempirical formula with available experimental data for the atomic targets from Ne to U and the number of the ejected electrons from n = 3 to n = 13 has been performed.For the first time, the formula suggested can be used for estimation of multiple-ionization cross sections σn, n ⩾ 3, from threshold up to high electron energies E ≈ 105 eV for an arbitrary atomic target.