CHAOTIC SCATTERING AND INVERSE SCATTERING.
C.Jung (Cuernavaca, Mexico), O.Merlo, T.H.Seligman (Cuernavaca, Mexico and Mexico City) and D.Trautmann
In the last two decades, the interest in nonlinear systems has increased enormously. The investigations done in very different fields of science have lead to the understanding that nonlinear behaviour and chaotic dynamics are the rule rather than the exception. Recently, open systems, i.e. chaotic scattering, have attracted particular attention. Chaotic scattering is hereby the general behavior of a physical system that evolves from a state that was stationary in the far past to another state that will be stationary in the far future (B.Eckhard, Physica 33 (1998), 89). The benefits from the study of chaotic scattering systems is therefore not restricted to a limited area in physical research but can produce results that help to understand a wide range of physical processes (L.Benet et al., Celest.Mech. and Dyn.Astr. 66 (1997), 203; C.Jung et al., J.Phys. A25 (1992), 3929; R.Guantes et al., Phys.Rev. E56 (1997), 378); E.Doron at al., Phys.Rev.Lett. 65(1990), 3072; J.Dorfman et al., Phys.Rev. E (1995), 28).
The goal of every scattering experiment is to reveal as much information as possible of the hidden scattering region. The only informations available are the asymptotically stable states in the far past and in the far future of the scattering event and how they are transformed from one into the other. The quest is to make statements on the scattering process based on these restricted informations. Although it seems to be impossible to conclude from scattering data on the specific form of the interaction, there are a number of other meaningful properties. A very important ingredient in the transient chaotic dynamics of chaotic scattering are the unstable periodic orbits, since they strongly influence the phase flow. Therefore it is very useful to have methods that give some information about the properties of these periodic orbits. Such a method, that finds periods, stability exponents and symmetries of unstable periodic orbits only from scattering data has been developed by us recently. The compiled results are given in T.Bütikofer, C.Jung and T.H.Seligman (Phys.Lett. A265 (2000), 76) and in T.Büti-kofer (PhD-thesis (2000), unpublished). The method has already been successfully applied to scattering off a magnetic dipole, to the three-disk billiard and to periodically kicked systems, but unfortunately this method can only be applied to scattering systems with two degrees of freedom.
A new method to get the symbolic dynamics and the branching tree was tested successfully with a kicked system which was analysed in great detail by B.Rückerl and C.Jung (J.Phys. A27 (1994), 55). Now the method will be tested in a more realistic scattering system and the results will soon be published.
CHAOTIC SCATTERING IN HIGHER DIMENSIONS.
L.Benet (Cuernavaca, Mexico), J.Broch, C.Jung (Cuernavaca, Mexico), O. Merlo, T.H.Selig-man (Cuernavaca, Mexico and Mexico City) and D.Trautmann
Although there exists very extended work on two-dimensional chaotic scattering (C.Lipp and C.Jung, J.Phys. A28 (1995), 6887; W.Breymann et al., Phys.Rev. E50 (1994), 1994) only little is known about higher dimensional scattering (K.M.Atkins and J.M.Hutson, J. Chem. Phys. 103 (1995), 9218; Z.Kovacs and L.Wiesenfeld., Chao-dyn. 98/10006; Wiggins Phys.Rev.Lett. 86 (2001), 5478). Since in higher dimensions all parts of the phase space are connected there is a qualitative difference to the two-dimensional case.
Up until now we have focused our attention on model systems as well as on more realistic systems intended to describe general features of chaotic scattering systems in classical mechanics. On one hand we are working on billiard-like models, the so called hard disk scattering. The interest comes from the fact that those models exhibit already all the interesting features of chaotic scattering but they are still reasonable to be handled by computational numerics. Hereby we concentrate on systems with an explicit time-dependence, i.e. systems in which the kinetic energy is not a constant of motion.
As a special model L.Benet et al. have investigated the dynamics of rotating targets, e.g. they have studied the case where the target rotates with constant angular velocity , which implies the existence of a constant of motion (the Jacobi integral) in two different systems, and also the case, where the rotating target consists of two rotating stars of given masses attracting by means of the gravitational force (N.Meyer et al., J.Phys. A28 (1995), 2529; L.Benet et al., loc. cit. and Celest.Mech. and Dyn. Astr. 73 (1998), 167).
On the other hand we are concentrating on general 4D scattering systems with one open channel. There we would like to study the invariant sets and the symbolic dynamics for this high- dimensional scattering system.
We now did already a lot of numerical simulations on rotating discs moving on keplerian orbits, a stadium orbit and on a elliptic orbit with constant angular velocity. We found a great stability of the structure of the scattering against perturbation of the movement of the disc. The numerical results indicate that the general structure of the scattering seems to be the same for all three systems even for very high eccentricity. These results will now be published.
CORRELATION EFFECTS IN ATOMIC FEW-BODY SYSTEMS.
T.A.Heim and A.R.P.Rau (Baton Rouge, Louisiana)
We study correlations in doubly excited states. Our pair-Rydberg description of doubly excited states has now matured into a consistent framework providing a unified treatment of both the wave functions and levels, as well as the dynamic transitions between such states. This approach represents a qualitative shift in the treatment of doubly excited states. The key ingredient in our derivation is a complete one-to-one mapping of the doubly excited states onto a six-dimensional Coulomb system. Specifically, the rich set of degeneracies of the six-dimensional Coulomb spectrum holds the key to understanding the structure and interrelation of correlated multiply excited states in atoms. In a detailed study of photo double excitation of He (T.A.Heim and A.R.P.Rau, Physics Essays 13 (2002), 358) we explained why earlier attempts at treating this process in a hyperspherical framework failed. Only by stressing the pair-features of the two electrons, that is, without ever resorting to single-particle aspects of the electrons, we obtained the correct magnitude for the cross sections for double excitation.
With this solid foundation of our model, we have started to apply the six-dimensional generalizations of hydrogenlike continuum wave functions in studies of processes involving ionization and breakup of doubly excited atoms. Numerically stable and very efficient routines for the determination of continuum (regular and irregular) Coulomb functions have already been extended to the case of six dimensions of space, requiring among other changes a replacement of the conventional integer orbital angular momentum quantum number l by half-integral numbers. Special care must be taken in exploiting the recurrence relations between Coulomb wave functions, because these relations read qualitatively different for half-integral generalized angular momenta.
Having implemented and thoroughly tested these preliminary modifications, we are currently applying our model in the determination of the angular and energy distribution of electrons in the so-called (e,2e) experiments, such as e+H -> H+ +e+e (the classic "Wannier" process, with a rather slow incoming electron and two slow outgoing electrons), or e+He -> He+(n)+e+e (with a fast incoming electron and one fast and one slow outcoming electrons with simultaneous high excitation of the remaining ion).
Besides the comparison with experimental results, these calculations afford comparing our hyperspherical wave functions with other (approximate) model wave functions for the continuum states of three charged particles, such as the Brauner-Briggs-Klar (BBK) wave functions or those investigated by Berakdar and others. Further applications of our model pertain to (g,2e)-processes. The calculations will then be compared with existing experimental data.
A.Aste, G.Baur (FZ-Jülich and Basel), K.Hencken, Y.Kharlov (IHEP-Protvino), S.Sadovsky (IHEP-Protvino) and D.Trautmann
We have continued our studies of electromagnetic processes due to the strong fields in peripheral relativistic heavy ion collisions. These processes are of interest for both photon-photon and photon-hadron physics at the heavy ion colliders RHIC and LHC. In the preparation of the experimental program at the CMS detector at the LHC, we have contributed to their Heavy Ion Chapter, where the possibilities of such peripheral collisions are discussed (G.Baur et al., CMS-Note 2000/060 to appear in EPJ C (2002)). We have also written a review of this field for Physics Reports (G. Baur, K. Hencken et al., Phys.Rep. 364 (2002), 359). This report is written in collaboration with S. Sadovsky and Y. Kharlov (both IHEP-Protvino and CERN), two experimentalists from the ALICE detector at the LHC at CERN. Apart from the discussion of possible physics processes that can be studied in these collisions in both photon-photon and photon-hadron collisions, an overview of existing experiments, as well as, possible trigger for such collisions are adressed.
In the CERES experiment at CERN one looks for electron-positron pairs coming from the central collisions. It is expected that these pairs can provide informations about the formation of the quark-gluon plasma as well as changes in the properties of the hadrons in the dense and hot nuclear matter in these collisions. As electron-positron pairs will also be produced from the electromagnetic interactions, they are a possible background. We have therefore calculated this process for small impact parameters and incorporated the experimental conditions on transverse momentum, invariant mass and rapidity used in the experiment. We also give predictions for RHIC conditions. We find electromagnetic pairs to be only a small background (K.Hencken et al., Phys.Rev. C61 (2000), 027901).
In connection with higher order effects there has been an ongoing controversy about the size of the so-called Coulomb corrections. In a series of paper these higher order effects have been derived in a high-energy formalism to all orders (B.Segev and J.Wells, Phys.Rev. A57 (1998), 1849; A.Baltz and L.McLerran, Phys.Rev.,C58 (1998), 1679; U.Eichmann et al., Phys.Rev. A59 (1999), 1223). In the meantime the validity of this high-energy limit used in these articles has been questioned, especially since these results are in contradiction to the predictions of the Bethe-Maximon-Davis theory (D.Ivanov et al., Phys.Lett. B454 (1999), 155; R.Lee and A.Milstein, Phys.Rev. A61 (2000), 032103; U.Eichmann et al, Phys.Rev. A61 (2000), 062710). We have contributed to this question by an explicit calculation of the impact parameter dependent probability using the high-energy approach of the first group of authors (K.Hencken et al., Phys.Rev. C59 (1999), 841). As the effects of the Bethe-Maximon-Davis theory are mainly due to large impact parameters, which dominate in the single-pair production cross section, there is still the possibility that these calculations are valid for small impact parameters and therefore for multiple pair production. We have started to investigate this question within our numerical program, where we are able to study the difference of the impact parameter dependent probabilities for the full calculation, as well as, within the so-called sudden limit.
In order to provide a different approach to these questions of higher order corrections we have also looked into the possibility to describe them within second order Magnus theory. We keep all lowest order terms in the Magnus theory, which corresponds to including all Coulomb-scattering terms in the sudden limit. The second order Magnus theory is necessary, as pair production (and annihilation) are only present here and not in the first order. As these second order terms are supposed to be small, we keep only a single pair production term (The effects of keeping more pair production and annihilation terms but without higher order Coulomb-scattering terms have already been studied by us (K.Hencken et al., Phys.Rev. A51 (1995), 998) and were found to be small). We have already derived an expression for this process, which looks very promising in terms of the included diagrams. Work on the explicit evaluation of this expression is currently under way.
In a recent publication (A.J.Baltz, et al., Nucl. Phys. A695 (2001), 395) it was claimed that the discrepancy between the Bethe-Maximon results and the high-energy approximation could be attributed to the fact, that not the single-pair cross section but the average-multiplicity cross section is calculated in the high-energy approximation, where retarded boundary conditions are used. Within the external field approximation we have shown that such a claim cannot hold as especially for the asymmetric collisions both cross sections do agree with each other, whereas the discrepancy between the two approaches remains. In a paper by A. Aste et al. (Eur.Phys.J C23 (2002), 545) we clarified in a very straightforward way the link between the single-particle S-matrix elements obtained from the solution of the Dirac equation and the multiplicity (i.e. total pair production rate), as well as, the single pair production rate, which has to be calculated in a fully second quantized theory. It turned out that the naive use of the pair production matrix element from the Dirac equation (e.g., wave mechanics) leads indeed to the multiplicity, whereas the single pair production rate is a quantity much more difficult to calculate. This fact can also be understood by a careful analysis of the perturbative aspects of the Dirac theory. For the Dirac equation, which can be solved in the idealized case where the colliding ions have a velocity infinitely close to the speed of light, the relevant object for calculations is the retarded propagator, but in a full quantum theory the more fundamental object becomes the Feynman propagator. Additionally, there is a vacuum-to-vacuum transition amplitude in the external field problem which is absent in wave mechanics. After the clear analysis of the external field problem, we have to conclude that the problem of the absence of Coulomb corrections seems to lie in the way the high energy limit is made, and that more detailed calculations of Coulomb corrections for arbitrarily charged ions are still missing. These calculations are subject to our future considerations.
One of the dominant loss processes in relativistic heavy ion colliders is the so-called 'bound-free pair production' process, where an electron-positron pair is created with the electron produced into the bound state of one of the ions. As this changes the charge state of the ion, it is lost from the beam; therefore a precise knowledge of the cross section is necessary in order to pre-estimate the accessible beam luminosity at the LHC. We have made exact calculations within the DWBA (distorted wave Born approximation). Transitions not only into the 1s-, 2s- and 3s-states were calculated but also into the 2p-states. For the ns-states we find the 1/n3 scaling to be rather accurately fulfilled. For the p-states we find that the 2p1/2-state already contributed more than the 3s-state for heavy ions, due to its strong s-wave character in the Dirac equation. The total cross section we calculate deviates in some cases by a factor of two from the ones used, e.g., in the ALICE Technical Notes. A recent experiment at the CERN-SPS has also measured this process at a g???factor of 168. We have compared our results for this case with them. Apart from an uncertainty of the capture process into higher states, our calculations are in good agreement. Since this process is of importance for the heavy ion run with Pb-ions at LHC, we have also made a comparison of all calculations of this process in the literature. In general, a good agreement between the different approaches was found, even though most of them were either made at low energies and extrapolated, or where done within some approximation scheme (H.Meier et al., Phys.Rev. A63 (2001), 032713).
On March 8-9 (2002), a workshop on 'Ultraperipheral Collisions at Relativistic Heavy Ion Collisions' took part within the Heavy Ion Forum at CERN. We have been involved in the planning and organization of this workshop. It was the intention of the workshop to gauge the interest of the community in the physics of ultraperipheral collisions in connection with both the RHIC and the LHC. The program and transparencies of the different talks can be found at the URL: http://phys-merger.physik.unibas.ch/upc/. One outcome of the workshop is the formation of a working group that is currently looking at the possibility to write a document, in which the physics topics of interest are pointed out for the experiments at both RHIC and LHC.
BREAKUP OF HALO NUCLEI FROM NUCLEAR REACTIONS AND COULOMB EXCITATION.
P.Banerjee (VECC-Calcutta, India), G.Baur (FZ-Jülich and Basel), H.Esbensen (ANL, USA), M.Fujiwara (Osaka, Japan), K.Hencken, H.Rebel (FZ-Karlsruhe), K.Schwarz (FZ-Karlsruhe and GSI-Darmstadt), C.Samanta (Saha Institute-Calcutta, India), and R.Shyam (Saha Institute-Calcutta, India).
For the proton-halo nucleus B8 we have made a systematic study how good the nuclear structure can be predicted by looking at a combination of both nuclear break and Coulomb excitation reactions. We found that, in agreement with others, there is an indication that the size of the proton halo is wider by about 20% compared to what has been used up to now. With this adjusted model good agreement between the nuclear breakup and the El strength is found. The magnitude of the E2 transition is still to large in the calculations compared to experiments. We attribute this to the still missing interference of nuclear and Coulomb interaction in the calculations. Therefore the simultaneous description of both within the Glauber theory is currently under way (H.Esbensen and K.Hencken, Phys.Rev. C61 (2000), 0654606).
As a new application of our reaction formalism we have made calculations of the scattering of Li6 on different targets at an energy of 100MeV/nucleon. Both elastic scattering as well as breakup reactions to alpha-particles or deuterons were studied. These calculations were made in collaboration with a group from the Forschungszentrum Karisruhe for an experiment done in Osaka (Japan). The analysis of the elastic scattering cross section had already shown that lowest order calculations, neglecting the internal structure of Li6 are not sufficient. A phenomenological way to incorporate these internal excitation is through a dynamic polarisation potential. Alternatively calculations using the CDCC (continuum discretized coupled channel) calculations were already made (K.Schwarz et al., EPJ A7 (2000), 367). Here our treatment within Glauber theory, which includes all orders even though in a simplified way, was found to give rather good agreement with the data. Different interaction-potentials were used in order to assess the uncertainty coming from these potentials. Where available phenomenological optical potential for the alpha- and deuteron-scattering on the targets were used, they were found to give a good agreement in general. A density dependent finite range double folding potential based on DDM3Y was found to be equally good (when the imaginary part of the potential was adjusted), whereas models based on zero-range folding approaches were found to overestimate the data at large angles. These models have the advantage of providing both real and imaginary part without adjustable parameters. On the other hand the influence of different potentials was found to be only insignificant for the breakup reactions to alpha or deuteron. Whereas the shape of the energy distribution of the fragments was found to be in good agreement with the data, the overall size was larger than the data. Apart from the C-target an almost constant factor was found between calculations and measurements. This can partly be attributed to the simplifying model used for Li6, where other channels like He3 or t are not included.(K.Schwarz et al., submitted to Phys.Rev. C).
In Coulomb excitation processes with neutron halo nuclei, e.g., with C19 or Be11, one open question is the importance of higher order Coulomb interactions. This so-called 'Coulomb postacceleration effect' was found to be present and very strong in low energy reactions. These effects would spoil the simple interpretation of the cross section being directly related to the photon absorption cross section. More recently the presence of this postacceleration effect was studied at higher energies (S.Typel and G.Baur, Phys.Rev. C64 (2001), 024601). In order to investigate this question in more detail, a calculation of these effects was done within a simplified model (zero range approximation of the neutron halo wave function and a pure Coulomb interaction), which allows for an analytic treatment. Again the importance of these Coulomb corrections was found to be small at high energies and in the forward direction - even for highly charged targets, that is, for strong Coulomb fields. The analytic expression could be related to the Born result in the semiclassical approximation by doing a transformation of the argument of the hypergeometric function and expanding this function for small arguments (G.Baur, K.Hencken and D.Trautmann, Proc. of the ENAM 2001-Conference, to be published).
We have studied the effect of higher-order Coulomb effects also in a numerical way, looking especially at their importance as a function of beam energy. Finite range corrections were used in addition. In agreement with the analytical results above it was found that whereas postacceleration is important as low energies, it is a minor effect at high energies (P.Banerjee et al., to appear in Phys.Rev. C (2002))
Finally we have been invited to write a review article for 'Progress in Nuclear and Particle Physics' on the use of Coulomb excitation for nuclear structure and astrophysics, which we are currently working on.
COULOMB EXCITATION AND BREAKUP OF p +p-?ATOMS AT HIGH ENERGIES.
G.Baur (FZ-Jülich and Basel), T.Heim, K.Hencken, M.Schumann, D.Trautmann
The experiment DIRAC currently being carried out at CERN requires very precise calculations of electromagnetic excitation and ionization cross sections for pionium (hydrogenlike atoms consisting of a p + and a p??) interacting with target atoms. The required accuracy of 1% for these input data implies that all aspects of the process under investigation need to be analyzed in great detail. In our initial study (Z.Halabuka, T.Heim, K.Hencken, D.Trautmann and R.Viollier, Nucl.Phys. B554 (1999), 86), we implemented a fully quantum mechanical treatment of the transitions of the pionium system, however using a simplified description for the target system. The sequel of this investigation then refined the results by including additional effects on the target side: In addition to the target-elastic (coherent) scattering process considered in our previous investigation, we have derived the formalism to treat incoherent (target-inelastic) processes as well. We also demonstrated that simplifying models attempting to express the incoherent scattering functions directly in terms of the atomic elastic formfactors are not accurate enough. Instead we calculated accurate scattering functions directly from numerically determined wave functions for all occupied electronic orbitals within the framework of Dirac-Hartree-Fock theory. Our results for the scattering functions are in good agreement with tabulated values, the minor deviations being explained by our using of fully relativistic orbitals as opposed to the non-relativistic calculations in the tables published in 1975. Because we determine the electronic orbitals explicitly, we can also calculate the atomic contributions to the incoherent scattering cross section shell by shell. This investigation allowed us to obtain information about the most relevant momentum transfer in the interaction between pionium and normal target atoms. Since the two systems differ by roughly a factor 137 in scale (energy / momentum / size), it is not clear from the outset which one of the two systems will set the stage for the dominant contribution to the cross section. In fact, we identified the most relevant range of momentum transfer to be the region between atomic and pionium scale, with a complex interplay of both composite scatterers. For the target atoms, our study shows that the cross section is dominated by the outer shells with rather loosely bound electrons. Our analysis of the differential cross section as a function of photon momentum transfer then demonstrates that the contributions from quasi-free or from conduction band electrons need not be considered separately beyond the level is already included in our treatment. Based on these findings we conclude that (a) the average excitation energy for the atom is negligible and may safely be set to zero in the closure approximation, at variance with the situation for the pionium; (b) solid state effects of the target foil are included adequately in our description; and (c) magnetic terms or contributions from the transverse current are negligible on the atomic side. These results have been published in a comprehensive paper (T.Heim, K.Hencken, D.Trautmann and G.Baur, J.Phys. B33 (2000), 3583). Since we calculate the electronic orbitals for the incoherent scattering functions, we now also use these numerical wave functions to determine the elastic formfactors, although these are well approximated by the analytical parameterizations used in our previous study.
As a further refinement of the description of the interaction between pionium and target atoms, we have extended our computer program to include the magnetic terms explicitly. As expected, these contributions amount to less than 1% for pionium in its ground states, vanishing rapidly for pionium excited states. We also carefully analysed the reduction from the Klein-Gordon equation for pionium and its relativistic coupling with the target atom to a non-relativistic description for pionium in its rest frame. We identified an additional type of Feynman diagrams linear in the external fields which can easily be calculated with our existing form factors. The resulting modifications of the cross section are found to be of the order of 10-10 as are the contributions from the terms quadratic in the fields. At this point, the calculation of the interaction between pionium and the target atoms, accurate to far better than 1%, is completed in first order of the interaction. (T.Heim, K.Hencken, D.Trautmann and G.Baur, J.Phys. B34 (2001), 3763). Using this framework, we have tabulated extensively the full spectroscopy of pionium with transitions between all states up to n=10, including ionization probabilities and total cross sections, for all target materials of interest to the experiment. These tables have been transferred to the group at CERN where they are needed as an input in the analysis of the experiment.
The distribution of pions from the breakup of pionium can be obtained from a straightforward extension of our existing computer program for the first order calculation. These calculations help to reduce the background in the analysis of the experimental data. In fact, we found, that the angular distribution, when integrated over the momentum, shows the two pions to be preferentially emitted perpendicular to the beam. This distribution is in contrast to the isotropic distribution one expects for the free pions (T.Heim, K.Hencken, M.Schumann, D.Trautmann and G.Baur, Proc. of HadAtom01, hep-ph/0112293, 13).
Our current efforts concentrate on calculating the higher order contributions to the breakup and excitation of pionium. This is handled within the framework of the Glauber approximation. The systematic study of transition probabilities between arbitrary bound initial and final states of the pionium, determined in the Glauber approximation and thus including multi-photon exchange, presents a major computational challenge. Not surprisingly, there are no such calculations available from other groups. All previously existing calculations pertained to total cross sections determined using the closure approximation. As these calculations already demonstrated the importance of higher order contributions, amounting to up to 15 or 20% for the heavier targets, the multi-photon exchange must really be included explicitly in all transition probability calculations. Our systematic tabulation of these transition probabilities for all initial and final states up to reasonably large principal quantum number n is completed for the target materials used in the experiment DIRAC. The main idea behind this calculation is to evaluate the difference between first order and full Glauber approximation which converges relatively quickly. The first order term can be evaluated separately, in the same way as the calculations for the Born approximation, by a one-dimensional integral. We have checked the validity of the sudden approximation, i.e., neglecting the energy difference between final and initial state of the pionium, and found that this approximation to be valid as the inclusion of the energy difference into the calculations let to differences of the order of 0.1% or less (M.Schumann, T.Heim, K.Hencken, D.Trautmann and G.Baur, J.Phys. B, to be published, and in Proc. of HadAtom01, hep-ph/0112293, 14).
As the importance of the higher order contributions turned out to be so large, we need to make sure the approximations used are sufficiently accurate. Since the Glauber approximation includes all orders of the multi-photon exchange, we cannot simply calculate a 'next-higher' order. Therefore, a coupled channel calculation is in preparation to confirm the results obtained in the Glauber approximation. This type of calculation is very computational intensive as a large number of transitions need to be included. We are using our Beowulf cluster to speed up the calculation using parallel programming techniques.
RADIATIVE AND COULOMB CORRECTIONS IN (e, e'p) - SCATTE-RING.
A.Aste, G.Baur (FZ-Jülich and Basel), K.Hencken, D.Rohe, I.Sick, P.Stagnioli and D.Traut-mann
The standard radiative correction procedure used for the last decade considers only (e,e')-inclusive scattering processes and it is desirable to have a more rigorous theory applicable to exclusive measurements. The possibility of emitting additional photons, both real and virtual, has to be included in the PWIA model describing the (e,e'p)-scattering where only the exchange of a single virtual photon between the electron and the struck proton is taken into account.
The emission of real photons (bremsstrahlung) causes a discrepancy between the measured particle momenta and the actual momenta at the scattering vertex. To reconstruct the missing momenta from the measured quantities in exclusive (e,e'p)-scattering, the experimental data have to be corrected for these effects. One approach including radiative corrections on the hadronic part can be found in N.C.R.Makins (Ph.D. thesis, MIT (1994), unpublished; and R.Ent et al., Phys.Rev. C64 (2001), 054610), where the soft-photon limit and the peaking approximation are employed. It is currently used in the experimental data analysis in a (e,e'p)-scattering experiment at the Jefferson Lab by the group of I.Sick (A.Honegger et al., TJNAF-proposal, (1998)). This approach by Makins is in part based on the older results of Y.Tsai (L.W.Mo and Y.Tsai Rev.Mod.Phys. 41 (1969), 205; Y.Tsai, Phys.Rev. 120 (1960), 269 and Y.Tsai, Phys.Rev.122 (1961), 1898). Especially the peaking approximation has to be improved since it seems to break down especially in those cases where a photon is emitted by the proton. A general QED framework of radiative corrections in scattering involving any number of leptonic and/or hadronic particles was given in 1991 by de S.Calan et al. (Nucl.Phys. B384 (1991), 47). The results of this work seem to be unknown to most people working in the field since it is cited only twice in the last ten years. We have looked at a deduction of the results of Makins from this more general framework in order to compare the two results as they are not completely identical but differ in the treatment of the hadronic contributions.
Another question is the importance of Coulomb corrections especially in the case of quasielastic scattering on heavier nuclei. Whereas rigorous calculations solving the radial Dirac equation in a DWBA approach exist (Y.Jin et al., Phys.Rev. C47 (1994), 2024; K.S.Kim et al., nucl-th/0103032), these calculations are very involved for the high energies one currently uses and in addition they are difficult to interpret in an intuitive way. We are currently looking at the Coulomb corrections in an eikonal-DWBA approach, where one replaces the wave function of the incoming and outgoing electron by the eikonal approximated wave function. We have derived the general form of this cross section from a field theoretic point of view, making use of the eikonal form as found, e.g., in (M.Levy and J.Sucher, Phys.Rev. 186 (1969), 1656). In the static limit, where it is assumed that the heavy nucleus stays at rest after the scattering process, the cross section is proportional to a convolution of the matrix element of the hard (quasielastic) process with the eikonal phase. Since we intend to be able to treat quite general types of hard processes, we want to do the necessary integrations numerically, in contrast to, e.g., some recent work (B.Z..Kopeliovich et al., Eur.Phys.J. A11 (2001), 345), where the meson production on a heavy ion is studied in a similar fashion, but using only an analytically solvable form for the hard meson production process. There it was also found that the use of analytically solvable models is an oversimplification which is unsufficient to predict quantitatively the influence of Coulomb corrections in a satisfactory way. The three dimensional Fourier transformation of the eikonal phase, which would have to be performed numerically by a Fast Fourier Transform (FFT), was found to be intractable due to the singular structure of the object and the in principle unlimited integration range in coordinate space. Therefore we decided to perform a three dimensional Fourier transformation of the hard process from momentum to coordinate space, where direct access to the analytic structure of the eikonal phase is guaranteed for a general class of charge distributions.
The fact that the occurring integrals are highly oscillatory and do exist only in a distributional sense makes it necessary to implement the numerical integrations in a careful way. We therefore developed a special integration routine which will serve for this purpose.
IONIZATION AND EXCITATION OF INNER-SHELL ELECTRONS IN RELATIVISTIC AND NON-RELATIVISTIC ION-ATOM COLLISIONS.
We are permanently asked by different experimental groups for calculating impact-parameter and energy dependent K-, L-, M- and N- shell ionization probabilities for the collision of light to medium heavy projectiles with various medium and heavy target atoms. Our theoretical cross sections are calculated using our efficient computer code based on the relativistic semiclassical model and using hydrogenic wave functions for the bound and continuum electronic states according to the Slater screening recipe in the separated atom-, as well as, in the united atom limit or using exact Dirac-Hartree-Fock wave functions. In this code we describe also the correct relativistic collision dynamics by using the exact Liénard-Wichert potential for the field of the moving projectile in complete analogy to our work on the Coulomb breakup of pionium (see above).
The corresponding experiments are performed, e.g., in Warsaw and Kielce (Poland - M.Jaskola, M.Pajek et al.), Erlangen (M.Kretschmer et al.) or at GSI-Darmstadt (L.Tribedi et al.). For the ionization of inner shells we always get quite good agreement between experiment and theory especially if we use relativistic Dirac-Hartree-Fock wave functions for the active electron. A further improvement has been achieved quite recently for the ionization from higher shells by introducing the coupling between the different magnetic sub-shells. In close collaboration with M. Pajek (Kielce, Poland) and T.Mukoyama et al. (Osaka, Japan), we have developed an approximate scheme to include these couplings. Hereby we calculate the ionization cross section in a full coupled channel approach but by using restricted physical parameters (straight-line path for the projectile, restricted energy in the continuum, restricted number of partial waves, etc.). Comparing these calculations with the corresponding first-order perturbation theory leads to a correction function, which then can be included in our first-order semiclassical model. The results of these calculations are very promising: the agreement between the new experiments and our ab initio calculations is excellent. These results will now be published (M.Pajek et al., Phys.Rev. A (2002)).
As a further result of this new approximate calculation scheme, we have found new insights in the correct treatment of the changing binding energy during the collision. We will now develop a simple approximate model to include this so-called binding-energy effect in our first-order code in an easy way.