Quantum Geometry: Generalized version of an en- ergy approach in scattering theory and its appli- cation to electron-collisional excitation of multi- charged ions

Within quantum geometry it is presented an Generalized version of an energy approach in scattering theory and its application to electroncollisional excitation of multicharged ions. The reestimated numerical data for electron-collisional excitation cross-sections are presented for barium.


Introduction
This paper goes on our investigations of the energy, spectral and geometric features of electron-collisional parameters of atomic (ionic) systems. The main attention is devoted to quantitative studying the eigen energy values and eigen function of electrons in the scattering processses. The dierent algorythms of an energy amplitude approach have been presented in [1] [12]. Let us re remind that at present time a great progress can be noted in development of a quantum geometry and quantum dynamics, that is mainly provided due to the carrying out more correct and eective mathematical methods of solving eigen function and eigen values tasks for multi-body complex quantum sustems in relativistic approximation and new algorythms of accounting for the complex exchangecorrelation eects. Nevertheless in many calculations there is a serious problem of the gauge invariance,connected with using non-optimized one-electron representation. In fact it means uncorrect accounting for the complex exchangecorrelation eects (such as polarization and screening eects, a continuum pressure etc.). In this paper, which goes on our studying [4][12], we present an generalized version of an energy approach in scattering theory and its application to calculation of cross-sections for some complex atomic systems.It is based on the relativistic many-body perturbation theory (PT) with the Dirac-Kohn-Sham zeroth approximation and more correct numerical accounting for the complex polarization, screening eects and continuum pressure. As application of generalized version of energy approach, the numerical data for electron-collisional excitation cross-sections are presented for some complex atomic ions.

Formal energy approach in scattering theory
We start from the formal energy approach presented in ref. [1]. The new original moment of our scheme is in using more corrected in comparison with [3], [10] gauge invariant procedure for generating the atomic functions basis's (optimized basis's) The lather includes solution of the whole dierential equations systems for Dirac-like bi-spinor equations [10]. More exactly, we use the Dirac-Kohn-Sham zeroth appraoximation in a formally exact relativistic perturbation theory. As an exchange potential we use the known-Kohn-Sham one in the consistent relativistic form. Besides, the correct Gunnarsson-Lundqvist correlation potential in a relativistic form is also used (look details regarding the used potentials, for example, in ref. [13].
Besides using this eective one-quasiparticle representation in relativistic perturbation theory the rest computational scheme remains the same [4] [10].
Follwoing to [4], as an example, we consider the collisional de-excitation of the Ne-like ion: Here Φ o is the state of the ion with closed shells (ground state of the Ne-like ion); J i is the total angular moment of the initial target state; indices iv, ie are related to the initial states of vacancy and electron; indices ε in and ε sc are the incident and scattered energies, respectively to the incident and scattered electrons.
Further it is convenient to use the second quantization representation. In particular, the initial state of the system atom plus free electron can be written Here C Ji,Mi mie,miv is the Clebsh-Gordan coecient.
The justication of the energy approach in the scattering problem is in details described in ref. [2]. For the state (1) the scattered part of energy shift Im ∆E appears rst in the second order of the atomic perturbation theory (fourth order of the QED perturbation theory) in the form of integral over the scattered electron energy ε sc [2]: with Im∆E = πG(ε iv , ε ie , ε in , ε sc ).
Here G is a denite squired combination of the two-electron matrix elements of the interelectron interaction. The value σ = −2Im∆E (5) represents the collisional cross-section if the incident electron eigen-function is normalized by the unit ow condition and the scattered electron eigen-function is normalized by the energy δ function.
The collisional de-excitation cross section is dened as follows [2]: Here B IK ie,iv is a real matrix of eigen-vectors coecients, which is obtained after diagonalization of the secular energy matrix. The amplitude like combination in the above expression has the following form: Here values Q Qul λ and Q Br λ are dened by the standard Coulomb and Breit expressions [2]. For the collisional excitations from the ground state (inverse process) one must consider a + in Φ o as the initial state and as a nal state. The cross-section is as follows: The dierent normalization conditions are used for the incident and for the

Some examples and conclusions
We applied our approach to estimate of the electron collisional excitation crosssections, strengths and rate coecients for electron-collisional excitation for Neand Ar-like ions. To test our theory we compare our calculations on collisional cross-sections for Ne-like iron with known calculations [2], [12]. Table 1 compares the experimental results with our calculations and with those of other theoretical works [2], [5], [7], [11].
It should be noted that the experimental information about the electroncollisional cross-sections for high-charged Ne-like ions is very scarce and is extracted from indirect observations. In any case implementation of such new elements as indicated above, allows to meet more ne agreement between theoretical relativistic energy-approach data and empirical results. An analysis of theoretical results accuray shows that a gauge invariance principle fullling in collisional processes, more exact account of the complex exchange-correlation eects can lead to increasing accuracy of computed collisional parameters.