Home
Search
Collections
Journals
About
Contact us
My IOPscience
Theory of electron impact ionization of atoms: The Born series
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2009 J. Phys.: Conf. Ser. 194 042040 (http://iopscience.iop.org/1742-6596/194/4/042040) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 181.215.99.35 This content was downloaded on 03/05/2017 at 08:20 Please note that terms and conditions apply.
You may also be interested in: Electronic conductance in mesoscopic systems: multichannel quantum scattering calculations I Tuvit, Y Avishai and Y B Band Inversion of scattering data to recover object detail where there is no deviation from the mean refractive index L Zapalowski and M A Fiddy Relativistic eikonal phase factors for electron capture B L Moiseiwitsch Multiphoton ionization by circularly and linearly polarized light: II F H M Faisal Ionization cross section for a strongly coupled partially ionized hydrogen plasma F B Baimbetov and Z A Kudyshev Application of the iterative method for calculating phase shifts S R Singh On the simplified second born approximation A R Holt A comparison of the second Born approximation using plane-wave and final-state Green functions A R Holt and B Santoso
XXVI International Conference on Photonic, Electronic and Atomic Collisions IOP Publishing Journal of Physics: Conference Series 194 (2009) 042040 doi:10.1088/1742-6596/194/4/042040
Theory of electron impact ionization of atoms: the Born series Yuri V. Popov∗, 1 , Vladimir L. Shablov† , and Konstantin A. Kouzakov§ ∗
†
Institute of Nuclear Physics, Moscow State University, Moscow 119991, Russia Obninsk State University for Nuclear Power Engineering, Obninsk, Kaluga Region 249040, Russia § Faculty of Physics, Moscow State University, Moscow 119991, Russia
Synopsis This contribution delivers a generalization of multichannel quantum scattering theory to (e, 2e) ionization of an atom. Since in this specific case the scattering amplitude given by a solution of the resolvent-like equation exhibits a Coulomb singularity on the energy shell, a rigorous definition of the physical amplitude is provided and a recipe for regularizing the divergent (e, 2e) Born series is formulated.
Theoretical treatments of ionization of atoms by fast electron impact are often carried out within the first Born approximation (FBA). FBA represents the lowest-order term of the Born series developed from the integral LippmannSchwinger equation for the T -matrix T (z), where z is the complex energy parameter. These Born series are known to diverge on the energy shell, i.e. when z → E + i0, due to a Coulomb singularity of the T -matrix. Therefore a perturbative Born treatment should be based on a theoretical approach which copes with the problem of divergence. Consider scattering from channel α to channel β in the system of charged particles. In a typical situation we have in the initial channel a charged particle with asymptotic momentum pα incident on a neutral complex of particles in the bound state φα . The physical amplitude for such transition can be presented in the form
with T (0) ≡ T FBA . It can be regularized following the procedure suggested in [2]: T (n) (z) = (n)
Here TR (z) are regularized Born terms and functions Rn (z) are singular on the energy shell if n ≥ 1, and R0 (z) = 1. Thus, the Born series (3) can be presented in the form of a product hpβ |Tβα (z)|pα i = R(z)TR (z),
R(z) =
(2)
where ηβ and Aβ are respectively the total Sommerfeld parameter and Dollard phase in channel β. In (2), the factor (z − E)−iηβ compensates for the Coulomb singularity of the LippmanSchwinger T -matrix on the energy shell. Let us formally develop the Born series for the Lippman-Schwinger T -matrix: hpβ |Tβα (z)|pα i =
∞ X
T (n) (z),
∞ X
(n)
TR (z).
n=0
[1] Shablov V L, Bilyk V A, and Popov Yu V 2002 Phys. Rev. A 65 042719. [2] Popov Yu 1981 J. Phys. B 14 2449. [3] Kouzakov K A et al 2007 J. Electron Spectrosc. Rel. Phenom. 161 35.
(3)
E-mail:
[email protected]
c 2009 IOP Publishing Ltd
TR (z) =
References
n=0 1
Rn (z),
It can be shown that R(z → E +i0) ∝ (z −E)iηβ , which ensures taking the on-shell limit in (2). Substitution of (4) into (2) provides a basis for the perturbative Born treatments of (e, 2e) processes. Note in this connection that while traditionally one developes the Born series in the potential between the incident electron and atom, in the case of (e, 2e) at high impact energy and large momentum transfer it is more natural to involve a final-state scattering potential, which is a sum of the pair potentials between the recoil ion, scattered and ejected electrons (see, for instance, [3]).
e− 2 πηβ +iAβ Γ(1 − iηβ )
(z − E)−iηβ hpβ |Tβα (z)|pα i,
∞ X n=0
1
z→E+i0
(4)
where the singular and regular factors are
where V α is the scattering potential in channel α and Ψ− β (pβ ) is the scattering state in channel β. The physical amplitude (1) is related to the Lippman-Schwinger T -matrix as follows [1]:
× lim
(m)
Rn−m (z)TR (z).
m=0
α tβα (pβ , pα ; E + i0) = hΨ− β (pβ )|V |φα , pα i, (1)
tβα (pβ , pα ; E + i0) =
n X
1
