Antiproton impact ionization of hydrogen atom: Differential cross sections computed by Coulomb wave function discrete variable representation method

. Our aim is using the Coulomb wave function discrete variable representation method (CWDVR) for the calculation of collision problem in first time. Nonrelativistic collision of antiproton with hydrogen atom is described by solving the time-dependent Schrodinger equation numerically. Two collision amplitudes are used for calculation of the differential cross sections, one of them corresponds to impact parameter of the projectile while other one is determined by projectile momentum transfer and found by Fourier transform of the first one. The ionization amplitude calculated by projecting of the wave function onto continuum wave function of the ejected electron. The differential cross sections calculated depending on projectile impact energy, scattering angle and electron ejection energy and angles, which is a result that can be measured experimentally. Our results are in good agreement with the relativistic calculation results.


Introduction
Antiproton impact ionization of hydrogen atom is an important benchmark test of theoretical method for charged particle atom collision. The perturbative calculations of triply differential cross section (TDCS) for ionization in antiproton-hydrogen collision have been performed in refs. Jones and. Madison [1], Voitkiv and Ullrich [2].
Doubly differential cross section's had been studied by several non-perturbative methods. Igarashi et al [3] developed an approach using one-center close-coupling (CC) calculation with large basis set. Tong et al [4], calculated ionization total cross section antiproton impact ionization of hydrogen atom using generalized pseudospectral method (GPSM). McGovern et al. [5,6] developed a method for extracting the TDCS from an impact-parameter treatment of the collision within a coupled pseudostate (CP) formalism. Abdurakhmanov et al. worked out the convergent-close-coupling (QM-CCC) [7] and wave-packet convergent-close-coupling (WP-CCC) [8] approaches to studies in ion-atom collisions. Ciappina et al. [9] applied the time-dependent closecoupling (TDCC) technique to investigate the role of the nucleus-nucleus interaction in the TDCS. Recently Bondarev et al [10] developed new relativistic method based on the Dirac equation for calculating TDCS's for ionization of hydrogen atom by antiproton impact. One simple and accurate non perturbative method named as Coulumb wave function discrete variable representation (CWDVR ) method was developed by Dunseath et al [11] and Peng and Starace successfully applied it to laser atom interaction [12].
In this paper we introduce implementation of CWDVR method to antiproton-hydrogen atom collision problem. Atomic units are used throughout this paper unless otherwise specified.

Theory
Hydrogen-antiproton collision process is expressed by time-dependent Schrödinger equation (TDSE).
Antiproton impact ionization of hydrogen atom 3 The time propagation of the wave function can be performed using second-order splitoperator method. [4,13] In spherical coordinate system the wave function can be written as where . , spherical harmonics, , r, t time-dependent radial function. Atomic Hamiltonian for the angular momentum is expressed as follows As well known, the hydrogen atom Hamiltionian has infinite number of discrete and continious spectrum, that are the main difficulty to use them for numerical calculations. To avoid this difficulty, we used CWDVR [11]. In the CWDVR method regular Coulomb wave function , is used to construct the pseudospectral basis functions. Parameters Z and k are control radial grid points distribution, which are roots of the . On this radial grid following eigen problem is solved Here is pseudospectral base corresponding to the quantum number and spectral number . N is number of radial grid points. Now we expand the radial function in pseudospectral base.
To perform the time propagation (3) we used the power of the Wolfram Mathematica software.
Antiproton impact ionization of hydrogen atom 5 Transition amplitudes in terms of the transverse (perpendicular to ⃗) component ⃗ of the projectile momentum transfer ⃗ rather than the impact parameter ⃗ is obtained by a two-dimensional Fourier transform.
where is the additional phase due to the projectile and target interaction [10] • • ln • .
Integrating the TDCS over corresponding variables, one can obtain various doubly differential cross sections (DDCS). Electron ejection energy and angular distribution DDCS is obtained integrating of the TDCS by the scattering angle Ω which is equivalent to the integral obtained by the impact parameter integration ⃗ ⃗ .
Integrating over the ejection angle and momentum transfer angle we get the DDCS ∬ Ω .
By integrating the DDCS over corresponding variables, one can obtain various singly differential cross sections (SDCS). Electron angular distribution is expressed by the SDCS: .
Another SDCS give the electron ejection energy distribution: 6 Zorigt Gombosuren et.al

Details of calculation
Coulomb wave function parameters Z and k are chosen 120 and 2 respectively which give 600 radial nodes up to rmax = 793.3. Maximum electron angular momentum number lmax = 5. The projectile z-component lies between -80 to 560 with the step Δ 0.32. 225 different values of impact parameters are chosen in an interval from 0.001 to 100. The polar angle of the ejected electron runs relative to direction of the momentum transfer.

Fully Differential Cross Sections
It is important to mention that our results are in good agreement with the the Dirac equation calculation results of relativistic-CC [10]. It is seen that the binary peak lower than of QM-CCC [7] results. However when the ejection energy became smaller than 10eV, this discrepancy became smaller (See [14]).

DDCS
First of all we interested in DDCS for electron energy and angular disterbutions,which is shown in Fig. 6. In our observation present results are similar with the results of Bondarev et al [15] for surface shape and value for all the region and also similar with the results of Abdurakhmanov et al [7] except the region where ejection energy is lower than 0.1 eV. We extend the DDCS up to 600eV ejection energy, and observed the shift of the maximum to zero ejection angle for about 350eV (Fig. 7.(a),(b). We also calculated the DDCS at lower projectile incident energy of 30keV, ejection energy of 5eV. In Fig. 8.  Small discrepancy observed for the 30keV incident energy. For higher incident energy of 500keV our DDCS coincides whith the results of McGovern et al [5] (Fig. 9.).  DDCS dependence on the transferred momentum at different ejection energies shown in Fig.10, again our CWDVR results are in good agreement with the relativistic calculations of Bondarev et al [10]. In this case one can see that, the maximum of the DDCS shifts to the higher value of the transferred momentum,due to the momentum conservation law. 10 Zorigt Gombosuren et.al

Conclusion
Ionization differential cross sections of antiproton impact hydrogen atom is calculated with CWDVR method by directly solving the TDSE. Present results of triply, doubly and singly cross sections have good agreement with some of the nonperturbative method results such as the relativistic-CC Bondarev et al [10].
From the analysis of the DDCS (which depends on electron ejection energy and angle) we conclude that the maximum of the DDCS shifts from the direction of antiproton incident at low ejection energy to the opposite direction at high ejection energy. This is the effect due to the post collision interaction between the projectile and ejected electron. Also we observed the shift of the maximum of the DDCS to the higher value of the transferred momentum with the increase of the electron ejection energy. We explain this shift as the effect of the momentum conservation law.