QED Theory of Elastic Electron Scattering on Hydrogen-Like Ions Involving Formation and Decay of Autoionizing States

PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

QED theory of elastic electron scattering on hydrogen-like ions involving formation and decay of autoionizing states

K. N. Lyashchenko ,1,2 D. M. Vasileva,1 O. Yu. Andreev ,1,3,* and A. B. Voitkiv4 1Department of Physics, St. Petersburg State University, 7/9 Universitetskaya nab., St. Petersburg, 199034, Russia 2Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China 3Center for Advanced Studies, Peter the Great St. Petersburg Polytechnic University, 195251 St. Petersburg, Russia 4Institute for Theoretical Physics I, Heinrich-Heine-University of Düsseldorf, Düsseldorf 40225, Germany

(Received 10 July 2019; published 28 January 2020)

We develop ab initio relativistic QED theory for elastic electron scattering on hydrogen-like highly charged ions for impact energies where, in addition to direct (Coulomb) scattering, the process can also proceed via formation and consequent Auger decay of autoionizing states of the corresponding helium-like ions. Even so, the primary goal of the theory is to treat electron scattering on highly charged ions, a comparison with experiment shows that it can also be applied for relatively light ions covering thus a very broad range of the scattering systems. Using the theory we performed calculations for elastic electron scattering on B4+,Ca19+,Fe25+,Kr35+, and Xe53+. The theory was also generalized for collisions of hydrogen-like highly charged ions with atoms considering the latter as a source of (quasi)free electrons.

DOI: 10.1103/PhysRevResearch.2.013087

I. INTRODUCTION Resonant scattering was extensively studied for nonrelativistic electrons incident on light ions [for instance, Atomic systems with two or more electrons possess au- − + ∗∗ + − e + He (1s) → He → He (1s) + e ]. This scattering toionizing states which can be highly visible in many pro- becomes especially important at large scattering angles (as cesses studied by atomic physics, e.g., photo and impact viewed in the rest frame of the ion) where the contribution ionization, dielectronic recombination, and photon or electron of the potential Coulomb scattering is minimal. Therefore, scattering. In particular, when an electron is incident on an experimental investigations (which often replace electron-ion ion, for certain (resonant) energies of the incident electron an collisions by ion-atom collisions in which atomic electrons autoionizing state can be formed. This state can then decay are regarded as quasifree) were mainly focused on such angles either via spontaneous radiative decay or via Auger decay (see, e.g., [1–6]). There exist also a large number of calcula- due to the electron-electron interaction. In the former case tions for resonant electron-ion scattering in which a nonrela- dielectronic recombination takes place, whereas in the latter tivistic electron interacts with a light ion (see, e.g., [5–8]). resonant electron scattering occurs. Depending on whether the In sharp contrast, the studies on resonant scattering of an initial and ﬁnal ionic states coincide or not, the energy of the electron on a highly charged ion, in which relativistic and scattered electron can be equal to the energy of the incident QED effects can become of importance, are almost absent electron (elastic resonant scattering) or differ from it (inelastic with no experimental data and merely two theoretical papers resonant scattering). [9,10]. In [9] scattering of an electron on a hydrogen-like An incident electron can also scatter on an ion without uranium ion was considered, however, only the resonant excitation of the internal degrees of freedom of the ion. In part of the scattering was calculated, whereas the potential such a case, scattering proceeds via the Coulomb force acting Coulomb part as well as the interference between them were between the electron and the (partially screened) nucleus of not taken into account. In [10] resonant electron scattering on the ion. This scattering channel is nonresonant and elastic and helium-like ions with atomic numbers 60 Z 92 proceed- its amplitude should be added coherently to the amplitude ing via the formation of autoionizing states [(1s2p / )nl] / for the elastic resonant scattering. As a result, there appears 1 2 1 2 and [(1s2s)nl] / with the principal quantum numbers 5 interference between these two channels which has to be taken 1 2 n 7 is presented. The goal of [10] was to investigate the into account in a proper treatment of elastic scattering. possibility of experimental study of the parity nonconserva- tion effects in this process. We note that for the heavy ions the potential Coulomb part is much larger than the resonant part *[email protected] (particularly, this concerns the contribution of highly excited states), so that the contribution of the resonant part including Published by the American Physical Society under the terms of the its interference with the Coulomb part is negligible. Creative Commons Attribution 4.0 International license. Further In case of scattering on highly charged ions, the electrons distribution of this work must maintain attribution to the author(s) are subjected to a very strong ﬁeld generated by the ionic nu- and the published article’s title, journal citation, and DOI. cleus. As a result, the account of relativistic and QED effects

2643-1564/2020/2(1)/013087(16) 013087-1 Published by the American Physical Society K. N. LYASHCHENKO et al. PHYSICAL REVIEW RESEARCH 2, 013087 (2020) may become of great importance for a proper description of m and parity defined by the orbital momentum l, and φ jl are the scattering process. the phases determined by the external field [see Eq. (B1)in In this paper we consider elastic electron scattering on Appendix B]. These wave functions are normalized according hydrogen-like highly charged ions for impact energies where to the presence of autoionizing states of the corresponding + ψ ψ = δ ε − ε δ δ δ , helium-like ions can actively influence this process. To this dr ε jlm (r) ε jlm(r) ( ) j j l l m m (5) end, we shall develop ab initio relativistic QED theory of this process, which enables one to address both its resonant and where δ denotes either the delta function or the Kronecker Coulomb parts in a unified and self-consistent way. Relativis- symbol, respectively. The spherical bispinor reads as [13] tic units are used throughout unless otherwise is stated. ν = ls ν χ , jlm( ) Cjm(ml ms )Ylml ( ) ms (6) II. GENERAL THEORY ml ms ls = | We consider elastic electron scattering on a hydrogen-like where Cjm(ml ms ) lml sms jm are the Clebsch-Gordan ν χ ion which is initially in its ground state, coefficients, Ylml ( ) are spherical harmonics, and ms , e− + X (Z−1)+(1s) → e− + X (Z−1)+(1s), (1) p ,μi p ,μ f 1 0 i f χ+ / = ,χ− / = (7) 1 2 0 1 2 1 where Z is the atomic number of the ion X. If the energy of the initial state of the electron system, which consists of the ν − are spinors. Further, in (3) there are also spinors vμ( ), which incident electron ep,μ with an asymptotic momentum p and are defined according to polarization μ and the 1s electron bound in the ion, is close 1 to the energy of a doubly excited (autoionizing) state of the νσ μ ν = μ μ ν , 2 v ( ) v ( ) (8) corresponding helium-like ion, the resonant scattering channel − (Z−1)+ (Z−2)+ (vμ (ν), vμ(ν)) = δμμ, (9) e ,μ + X (1s) → X (d) pi i − (Z−1)+ = χ , → e ,μ + X (1s), (2) vμ(ez ) μ (10) pf f where d is a doubly excited state, becomes of importance. where σ is the Pauli vector, ez is the unit vector along the z ψ (±) Here, the scattering proceeds via the formation of a doubly axis. Then, the wave functions pμ (r) are normalized as excited state (d) and its subsequent Auger decay. This channel is driven by the interelectron interaction. (±)+ (±) 3 dr ψ μ (r)ψ μ (r) = (2π ) δ(p − p)δμμ One can expect that in case of highly charged ions the main p p channel of the electron-ion scattering process (1) is Coulomb 3 (2π ) scattering, which is nonresonant. Therefore, in order to in- = δ(ε − ε)δ(cos θ − cos θ ) εp vestigate the resonant structure experimentally, the electron ◦ scattering to very large angles (180 ), for which the contri- × δ(ϕ − ϕ)δμμ. (11) bution of the main channel is minimal, should be considered [5]. Accordingly, one needs to calculate the differential cross For the process of elastic electron scattering (1), the section. contribution of its Coulomb part to the amplitude is very We shall consider the scattering process (1) using the important. The Coulomb scattering amplitude is usually Furry picture [11], in which the action of a strong external calculated by studying the asymptotics of the electron wave field (for example, the field of the ionic nucleus) on the functions in the Coulomb field [12]. electrons is taken into account from the very beginning. The resonant part of the scattering process (2) is due to The electrons interact with each other via the interaction with the formation and decay of autoionizing states of the corre- the quantized electromagnetic and electron-positron fields, sponding helium-like ion. In this paper, for the description which is accounted for by using perturbation theory. of autoionizing states within the QED theory the line-profile The ingoing (+) and outgoing (−) wave functions of an approach (LPA) will be employed [14], where the QED electron in an external central electric field with a asymptotic perturbation theory is used. In particular, it is necessary to momentum (p) and polarization (μ) can be presented as [12] take into account various corrections such as the interelectron interaction corrections and the relativistic corrections to the ± + ± φ ψ ( ) = ν ν i jl l ψ , scattering amplitude. pμ (r) N jlm( )vμ( )e i ε jlm(r) (3) jlm In order to make the consideration of both Coulomb and resonant parts of the scattering amplitude self-consistent, we where shall apply the formal theory of scattering for the Coulomb / (2π )3 2 potential considering it by using perturbation theory. N = √ (4) pε is the normalization factor, ν = p/|p| is the unit vector A. Coulomb scattering amplitude defining the angular dependence of the momentum p.The For simplicity, in this section we limit ourselves to the ψ wave functions ε jlm(r) describe electrons with the energy consideration of a one-electron system. Within the scattering ε = 1 + p2, the total angular momentum j, its projection theory developed in [15,16], the in (+) and out (−)states

013087-2 QED THEORY OF ELASTIC ELECTRON SCATTERING ON … PHYSICAL REVIEW RESEARCH 2, 013087 (2020) satisfy the Lippmann-Schwinger equation A 1 ψ (±) = φ + ˆ ψ (±) , r pμ (r) pμ(r) V pμ (r) (12) k , e ε − Hˆ0 ± i0 where Vˆ represents the scattering potential. Here, the function n = a φpμ(r) describes a free electron and can be obtained from Eq. (3) by setting Z = 0: r k , e ipr φpμ(r) = uμ(p)e , (13) √ A 1 ε + 1vμ(ν) uμ(p) = √ √ . (14) 2ε ε − 1(σν)vμ(ν) FIG. 1. Feynman graph describing the elastic photon scattering on an atomic electron. The double solid line denotes the electron in Following [15] we introduce the R-matrix elements according the external potential V F (the Furry picture). The wavy lines with to the arrows describe the emission and absorption of a photon with (+) R = φ μ Vˆ ψ μ momentum k and polarization e. if pf f pi i

1 (+) = φ μ Vˆ φ μ + φ μ Vˆ Vˆ ψ μ . time variables the matrix element can be written as [12] pf f pi i pf f pi i εi − Hˆ0 + i0 SQED = (−2πi)δ(ε − ε )U QED, (23) (15) if f i if The S-matrix elements are related to the R matrix as follows: where U QED is the amplitude. In QED of strong ﬁelds the if S-matrix elements and, correspondingly, the amplitude Uif are Sif = φ μ Sˆ φ μ = δif − 2πiδ(εi − ε f )Rif. (16) pf f pi i evaluated with the use of perturbation theory [12]. For the Since the S-matrix elements can be also written as description of highly charged ions within QED theory, spe-

(−) (+) cial methods are employed. Most of QED calculations were Sif = ψ μ ψ μ , (17) pf f pi i performed using the adiabatic S-matrix approach [18–20], we obtain that ( f = i) the two-time Green’s function method [21], the covariant-

(−) (+) evolution-operator method [22], and the LPA [14]. In this ψ μ ψ μ = (−2πi)δ(ε f − εi ) Rif. (18) pf f pi i paper we employ the LPA. =−α / We shall now establish the relationship between the ampli- In the case of Coulomb scattering, Vˆ Z r, the states Coul (±) tude Rif defined by Eq. (15) and the corresponding ampli- ψ μ can be conveniently evaluated using the expansions (3), pi i tudes derived within the LPA. Let us evaluate the scattering which enables us to obtain the following expression: amplitude corresponding to the interaction with an external − ext Coul 2 ( 1)p ∗ field V in the one-electron case. Within the LPA the initial R = N (vμ (ν )) M μ (θ,ϕ) (19) if (2π )2 f f m m i and final states (the reference states) are described as reso- m nances in the process of scattering. As an auxiliary process, it (a detailed derivation of RCoul is presented in Appendix A). is normally most convenient to take elastic photon scattering. Following [17] we introduced the matrix M: For the properties of the reference states to be independent of −iϕ the details of the scattering process, the resonance approxima- M+ / ,+ / (θ,ϕ) = f (θ ), M+ / ,− / (θ,ϕ) = g(θ )e , 1 2 1 2 1 2 1 2 tion is employed [14]. In this approximation, the line profile iϕ M−1/2,+1/2(θ,ϕ) =−g(θ )e , M−1/2,−1/2(θ,ϕ) = f (θ ), is interpolated by the Lorentz contour, the position of the resonance and its width define the energy and width of the (20) corresponding state. where f (θ ) and g(θ ) are the relativistic scattering amplitudes We consider the process of elastic photon scattering on 1 one-electron ion initially being in its ground state. The Feyn- f (θ ) = |κ|(e2iφκ − 1)P (cos θ ), (21) man graphs corresponding to this process in the zeroth order 2πi l jl of the perturbation theory are depicted in Fig. 1. The corre- 1 φ φ sponding S-matrix element reads as [12,14] g θ = e2i κ=−l−1 − e2i κ=l P1 θ . ( ) π ( ) l (cos ) (22) 2 i μ μ l QED = − 2 4 4 ψ¯ γ u , γ d ψ Sif ( ie) d xud xd u(xu ) S(xu xd ) d (xd )

B. Coulomb scattering amplitude within the LPA × A∗(k ,λ )(x )A(k,λ)(x ). (24) μu u μd d In the QED theory the scattering matrix (S) can be rep- We assume that ψu = ψd describes the 1s electron. The resented as the sum of the normal ordered products of ﬁeld , operators corresponding to different processes of particle scat- electron propagator S(xu xd ) can be written as tering. Each normal ordered product and, consequently, any i − ω − ψn(ru )ψ¯n(rd ) , = ω i 1 (tu td ) , scattering process can be represented graphically according to S(xu xd ) d 1e (25) 2π ω1 − εn(1 − i0) the Feynman rules. In the corresponding matrix elements, the n integration over xμ = (t, r) 4-vectors is performed. If in the where the sum runs over the entire Dirac spectrum. Further, process the energy is conserved, after the integration over the A(k,λ)(x ) and A∗(k ,λ )(x ) refer to the absorbed and emitted μd d μu u

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A Within the LPA the reference states (the initial and ﬁnal states) are deﬁned via resonances in the process of elastic r k, e photon scattering. Accordingly, we are interested in (we note that εu = εd ) ext × V ω + εu ≈ ε f , (32) r k , e ω + εd ≈ εi. (33)

A In the resonance approximation, the reference states are described in such a way that their properties do not depend on a FIG. 2. Feynman graph corresponding to the insertion describing ext specific scattering process. Accordingly, retaining only terms the interaction with the external field V . The dotted line with a with n = f , n = i, in the resonance approximation we get cross denotes the interaction with the field V ext. The notations are the 1 2 same as in Fig. 1. ∗ ,λ A (k ) SQED(1)res = (−2πi)δ(ω + ε − ω − ε ) e2 uf if u d ω + ε − ε photons, respectively, kμ = (ω, k) is the 4-vector of photon ( u f ) ,λ momentum, λ describes the photon polarization. By inserting A(k ) × ext id . (25) into Eq. (24) and integrating over the time and frequency Vfi (34) (ω + εd − εi ) variables tu, td , and ωn we obtain ,λ A∗(k ,λ )A(k ) The second order of the perturbation theory with respect to SQED = (−2πi)δ(ω + ε − ω − ε ) e2 un nd , the interaction with the external field reads as if u d ω + ε − ε n d n QED(2) = − π δ ω + ε − ω − ε 2 (26) Sif ( 2 i) ( u d ) e ∗(k,λ ) where one-electron matrix elements A 1 × un1 V ext ω + ε − ε n1n2 ω + ε − ε ,λ μ ,λ u n1 d n2 (k ) = 3 ψ¯ γ (k ) ψ , n1n2n3 And d r n(r) Aμ (r) d (r) (27) (k,λ) An d ∗ ,λ μ ∗ ,λ × ext 3 . (k ) = 3 ψ¯ γ (k ) ψ Vn n (35) Aun d r u(r) Aμ (r) n(r) (28) 2 3 ω + ε − ε d n3 were introduced. In the resonance approximation we keep only the term with In the first order of the perturbation theory we consider n2 = n3 = i: ext ext one interaction with an external field Aμ (x) = (V (r), 0). ∗ ,λ The Feynman graph describing this interaction is presented in A (k ) SQED(2)res = (−2πi)δ(ω + ε − ω − ε ) e2 uf Fig. 2. The corresponding S-matrix element reads as if u d ω + ε − ε ( u f ) QED(1) 3 4 4 4 μu ∗(k ,λ ) (k,λ) S = (−ie) d xud xd d x ψ¯u(xu )γ Aμ (xu ) 1 A if u × ext ext id . Vfi Vii (36) (ω + εd − εi ) (ω + εd − εi ) μ ext (k,λ) μd × S(xu, x)γ Aμ (x)S(x, xd )Aμ (xd )γ ψd (xd ). d The third order of perturbation theory in the resonance (29) approximation reads as The integration over the time and frequency variables leads to QED(3)res = − π δ ω + ε − ω − ε Sif ( 2 i) ( u d ) ∗(k,λ ) A SQED(1) = (−2πi)δ(ω + ε − ω − ε ) e2 un1 ∗(k ,λ ) if u d Auf 1 ω + εu − εn × 2 ext n1n2 1 e Vfi (ω + εu − ε f ) (ω + εd − εi ) ,λ n1n2 A(k ) ext n2d ,λ ×V , (30) 1 A(k ) n1n2 ω + ε − ε × ext ext id . d n2 Vii Vii (37) (ω + εd − εi ) (ω + εd − εi ) where In the resonance approximation the series of the perturbation V ext = e d3r ψ+(r)V ext (r)ψ (r). (31) theory composes a geometric progression n1n2 n1 n2

∞ ∞ ∗(k,λ ) ,λ A 1 l A(k ) QED(l)res = − π δ ω + ε − ω − ε 2 uf ext ext id Sif ( 2 i) ( u d ) e Vfi Vii (ω + εu − ε f ) (ω + εd − εi ) (ω + εd − εi ) l=0 l=0 ∗(k,λ ) ,λ A A(k ) = (−2πi)δ(ω + ε − ω − ε ) e2 uf V ext id . (38) u d ω + ε − ε fi ω + ε − ε − ext ( u f ) d i Vii

013087-4 QED THEORY OF ELASTIC ELECTRON SCATTERING ON … PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

Thus, the summation of the infinite series resulted in a shift of a b the position of the resonance, which represents a correction to the energy of the reference state. This procedure was first applied in [23] for the derivation of the natural line shape within the QED theory, where the correction due to the electron self-energy was considered. We also note that r r the corrections to the photon scattering amplitude for a few electron ions were discussed in [14]. Within the LPA we introduce the amplitude ∞ = (l), a b Uif Uif (39) l=1 FIG. 3. Feynman graph corresponding to the first-order in- where terelectron interaction in two-electron ion (one-photon exchange (1) = ext, graph). Uif Vfi (40) 1 U (2) = V ext V ext, (41) We note that the amplitude (47), in which the exact state if fn (ω + ε − ε ) ni n d n is given by (48), essentially coincides with the amplitude 1 1 U (3) = V ext V ext V ext. (42) (15), where the exact state is given by (12). Therefore, we if fn1 n1n2 n2i ω + εd − εn ω + εd − εn conclude that the amplitude Uif obtained within the LPA, n1n2 1 2 a QED approach, coincides with the amplitude Rif, which In the resonance approximation we can set ω + εd = εi.The follows from (relativistic) quantum mechanics and which, in amplitude (39) describes the transition (i → f ) caused by the particular, can be evaluated using Eq. (18). One should add external field V ext. that in the framework of the LPA also QED corrections, such Introducing the operator Vˆ , which is defined by its matrix as electron self-energy, vacuum polarization, photon exchange elements as corrections, can be taken into account using the procedure ext described above. Thus, in general, the operator Vˆ includes Vn n = V , (43) 1 2 n1n2 also the corresponding QED corrections. the amplitude can be rewritten as 1 = (0) ˆ (0) + (0) ˆ ˆ (0) C. Implementation of the LPA for the description Uif f V i f V V i εi − Hˆ0 + i0 of elastic resonant scattering 1 1 The scattering amplitude corresponding to the interelectron + (0) ˆ ˆ ˆ (0) f V V V i interaction is given by Feynman graphs depicted in Figs. 3 εi − Hˆ0 + i0 εi − Hˆ0 + i0 +··· and 4. The graph in Fig. 3 represents the one-photon exchange correction, the graphs in Figs. 4(a) and 4(b) refer to the ∞ l 1 two-photon exchange corrections; the three- and more-photon = (0) Vˆ Vˆ (0) . (44) f ε − ˆ + i exchange corrections could also be considered. One should = i H0 i0 l 0 note, however, that in the process under consideration all (0) The one-electron wave functions n describe electrons non- these graphs need a special treatment because they contain interacting with the external field V ext : divergences. n1n2 ˆ (0) = ε (0). H0 n n n (45) Here, Hˆ0 is the Dirac Hamiltonian a b a b F Hˆ0 = αpˆ + βm + V (r), (46) r r r r where α and β are the Dirac matrices and the choice of the F potential V defines the Furry picture employed. n1 n2 n1 n2 The amplitude Uif in Eq. (44) can be also written as

= (0) ˆ | , r r r r Uif f V i (47) where a b a b 1 = (0) + ˆ . (a) (b) i i V i (48) εi − Hˆ0 + i0 FIG. 4. Feynman graphs corresponding to the second-order in- The function i is a solution of the following equation: terelectron interaction in two-electron ion (two-photon exchange (Hˆ0 + Vˆ )i = εii. (49) graphs).

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For the description of processes, which involve electrons a b a b interacting with the field of a highly charged nucleus, the Z − 1 s Furry picture is normally used, in which the interaction of × the electrons with the potential of the nucleus V F =−αZ/r Z − 1 is fully taken into account from the onset, whereas the inter- × s Z − 1 electron interaction is considered using a QED perturbation × s theory. However, in the case of elastic scattering such an approach leads to divergent results arising from the application of the perturbation theory to a long-range Coulomb interaction a b a b between bound and free electrons. (a) (b) In order to modify the “standard” approach, we note that the incident and scattered electron most of the time is mov- FIG. 5. Feynman graphs representing the first (a) and second ing in the Coulomb field of a hydrogen-like ion with the (b) terms in the right side of Eq. (15) for two-electron ion. The single net charge Z − 1. Therefore, we shall avoid the use of the solid line denotes free electron, the double line denotes electron perturbation theory for the interaction of the free electron in the field of potential V F =−αZ/r. The dashed line with the with the bound electron by employing the Furry picture in cross designates interaction with external field Vcont =−α(Z − 1)/r. which the free electron is supposed to move in the field of The double striped line describes electron in the Furry picture with F =−α − / the nuclear charge Z − 1. Then, the interaction of the free potential Vcont (Z 1) r. electron with the bound electron together with its interaction with the “remaining” charge of the nucleus [Z − (Z − 1) = 1] where r12 =|r1 − r2| and is already a short-range interaction since it corresponds to the δμ δμ c 10 20 interaction with an electrically neutral system [the charge of Iμ μ = , (51) 1 2 r the bound electron (e) smeared out over the size of its bound 12 ir δμ μ ∂ ∂ 1 − e 12 state and the “remaining” charge of the nucleus (−e)]. Now, t 1 2 ir12 Iμ μ =− e + μ μ 1 2 ∂ 1 ∂ 2 2 the divergences do not arise and this short-range interaction r12 x1 x2 r12 can be accounted for by using the perturbation theory. × 1 − δμ 1 − δμ . (52) Thus, our consideration involves the following points. (i) 10 20 The wave functions of the incident and scattered electron While the exchange contribution [Fig. 6(c)] does not contain are obtained by solving the Dirac equation with the potential any divergence, the direct contribution [Fig. 6(a)] does. How- −α(Z − 1)/r. (ii) The wave functions of all other electrons ever, the contribution to the amplitude given by Fig. 6(b) is (bound and virtual) are derived from the Dirac equation with also divergent and it turns out that the divergences in Figs. 6(a) the potential −αZ/r. (iii) The interaction of the continuum and 6(b) cancel each other. electrons with the “remaining” charge of the nucleus is calcu- Let us consider this point more in detail. First of all we note lated with the use of perturbation theory. (iv) The interelectron that the problem with Fig. 6(a) arises only due to the Coulomb interaction is considered as the interaction with the quantized part of the transition matrix element. We now assume that electromagnetic and electron-positron fields within the QED d1 = u1 = ep,μ [the incident and scattered electron having perturbation theory. (v) The divergences arising from the long- the same energy (ε)] and d2 = u2 = 1s (the 1s electron) and range Coulomb interaction of the continuum electron with the consider only the above-mentioned part Ic of the transition remaining charge of the nucleus and with the bound electron matrix element can be regularized and cancel each other. 1 (Ic) = α dr dr ψ+(r )ψ+(r ) ψ (r )ψ (r ). The amplitude of the Coulomb scattering given by Eq. (19) u1u2d1d2 1 2 u1 1 u2 2 d1 1 d2 2 formally corresponds to taking into account the Feynman r12 graphs depicted in Fig. 5, where graphs (a) and (b) represent (53) the first and second terms in the right side of Eq. (15). For the derivation of the amplitude for the resonant scattering channel a b a b a b [see (2)] the Feynman graphs depicted in Fig. 6 have to be taken into account. Figures 6(a) and 6(c) present the direct and 1 exchange graphs, respectively, of the one-photon exchange. r r × The double line describes an electron in the Furry picture with the potential V F =−αZ/r, the double striped line describes continuum electrons in the Furry picture with the potential F =−α − / Vcont (Z 1) r. Figure 6(b) represents the interaction of a b a b a b = F − F = the incident electron with the potential Vcont V Vcont (a) (b) (c) −α/r (i.e., with the potential of the remaining charge of the nucleus). FIG. 6. The direct (a) and exchange (c) Feynman graphs repre- The contribution of Fig. 6(a) [or Fig. 6(c)]isgivenby[14] senting the one-photon exchange. The double line describes electron in the Furry picture with potential V F =−αZ/r, the double striped μ1 μ2 F = α ψ¯ ψ¯ γ γ μ μ ||, line describes electron in the Furry picture with potential V = (I)u1u2d1d2 dr1dr2 u1 (r1) u2 (r2 ) 1 2 I 1 2 ( r12 ) cont −α(Z − 1)/r. The graph (b) represents interaction of the incident ×ψ ψ , =−α/ d1 (r1) d2 (r2 ) (50) electron with the potential Vcont r.

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Using the expansion [13] [see Eq. (3)]. The divergence appears only for the case of identical partial waves for the incident and scattered electrons, ∞ K 1 = r< θ , which we shall now consider. The wave function of 1s electron K+1 PK (cos ) (54) r r> r 12 K=0 decreases exponentially, accordingly, the integration over 2 is convergent. Asymptotically (when r →∞), the upper and = , = , where r< min(r1 r2 ), r> max(r1 r2 ) and retaining only lower components of the incident electron wave function are = > the term with K 0 (the terms with K 0havenodiver- given by gences) we can write gε (r) ∼ C cos(pr + ξ log 2pr + δ ), (56) 1 jlm g jl c = α ψ+ ψ+ ψ ψ . I0 dr1dr2 u (r1) u (r2 ) d1 (r1) d2 (r2 ) u1u2d1d2 1 2 r> fε jlm(r) ∼ Cf sin(pr + ξ log 2pr + δ jl ), (57) (55) where ξ = αZε/p is the Sommerfeld parameter, δ jl are the The wave function of an electron with a given momentum phase shifts, Cg and Cf are constants [12]. Accordingly, the and polarization can be decomposed into the complete set of integration over r1 contains a divergent part at the upper limit partial waves with a certain energy and angular momentum (r1 →∞):

1 2 2 2 2 dr1 [|Cg| cos (pr1 + ξ log 2pr1 + δ jl ) +|Cf | sin (pr1 + ξ log 2pr1 + δ jl )]. (58) r1

After a regularization of the integral over r1, the divergent part of Eq. (55) can be singled out and, as will be shown, it will be exactly canceled by the divergence contained in the contribution to the amplitude due to the interaction of the continuum electron with the potential Vcont. It is convenient to split the integral Ic into two parts: 0 1 1 1 c = α − ψ+ ψ ψ+ ψ + α ψ+ ψ ψ+ ψ I0 dr1dr2 u (r1) d1 (r1) u (r2 ) d2 (r2 ) dr1dr2 u (r1) d1 (r1) u (r2 ) d2 (r2 ) u1u2d1d2 r> r 1 2 r 1 2 1 1 1 1 1 = α dr dr − ψ+(r )ψ (r ) ψ+ (r )ψ (r ) + α dr ψ+(r )ψ (r ) δ . (59) 2 1 u1 1 d1 1 u2 2 d2 2 1 u1 1 d1 1 u2d2 |r1|<|r2| r2 r1 r1

Here, we used the fact that the wave function of the 1s electron is normalized to unity. The integral containing the term (1/r2 − 1/r1) is obviously convergent. The integral with 1/r1 is divergent but is exactly canceled out by the divergent part of the contribution to the amplitude due to the interaction Vcont of the incident electron with the remaining charge, which reads as −1 (V ) = α dr ψ+(r ) ψ (r ) δ . (60) cont u1u2d1d2 1 u1 1 d1 1 u2d2 r1 The consideration given in the previous paragraphs is based on the decomposition Z = (Z − 1) + 1. Its ﬁrst term corresponds to the effective charge of the nucleus (as seen asymptotically by the incident and scattered electron) screened by the bound electron which essentially is regarded as a pointlike charge (−1) placed at the origin. Using the Furry picture, the Coulomb interaction between the continuum electron and such a pointlike electron is taken into account to all orders. At the same time, the difference between this interaction and the Coulomb interaction between the continuum electron and the bound electron in the 1s state is considered in the ﬁrst order. We note that this difference is of short range corresponding to the interaction with an c electrically neutral system, which is considered using the perturbation theory. Indeed, the sum of I and Vcont 0

c 1 1 + + I + Vcont = α dr2 dr1 − ψ (r1)ψd (r1) ψ (r2 )ψd (r2 ) (61) 0 u1u2d1d2 u1 1 u2 2 |r1|<|r2| r> r1 is finite and represents a correction due to interaction with the direct (nonresonant) elastic scattering. Let us now briefly a short-range potential. Within our approach, Fig. 6(a) must discuss the description of the resonant channel of the scat- be always considered together with Fig. 6(b). For light ions tering process which becomes relevant when the sum of the the accuracy of our approach may be improved if we replace energies of the incident and the bound electrons is close to the the interaction with a pointlike charge by the interaction with energy of a doubly excited (autoionizing) state. In such a case, the charge density given by the 1s electron wave function and the contribution of two- and more-photon exchange between adjust the Furry picture accordingly. The influence of this electrons in low-lying states becomes of importance. More- replacement on the results is discussed in the next section over, the doubly excited states are normally quasidegenerate, (Figs. 11 and 12). hence, a perturbation theory for quasidegenerate states has to The Feynman graph depicted in Fig. 6(c) represents the be used. For this purpose, the LPA [14] is employed. exchange graph of the one-photon exchange, it is finite and For application of the quasidegenerate perturbation the- does not require a special treatment. Up to now, we considered ory within the LPA we introduce the set of two-electron

013087-7 K. N. LYASHCHENKO et al. PHYSICAL REVIEW RESEARCH 2, 013087 (2020) configurations (g)inthe j- j coupling scheme which includes is represented by Feynman graphs depicted in Figs. 3 and 4, all two-electron configurations composed by a certain set of respectively. electrons (for example, 1s, 2s, 2p, 3s, 3p, 3d electrons) The total amplitude of the process (1), including its reso- (0) (r , r ) nant part (1), is given by JM j1 j2l1l2n1n2 1 2 j j = Coul + Auger, = 1 2 , ψ ,ψ , Uif Uif U (67) N CJM (m1 m2 ) det n1 j1l1m1 (r1) n2 j2l2m2 (r2 ) if m1m2 where the Coulomb and Auger contributions are given by (62) Eqs. (19) and (66), respectively. The numerical calculation of Coul where j and m are the total angular momentum and its the Coulomb amplitude Uif is discussed in Appendix B. projection, l is the orbital angular momentum defining the The transition probability is expressed via the amplitude parity, n denotes the principal quantum number or energy (for according to [12] the continuum electrons), J and M are the total angular mo- d3 p mentum of the two-electron configuration and its projection, 2 f dwif = 2π|Uif| δ(Ei − E f ) , (68) N is the normalizing constant. (2π )3 V In the LPA a matrix is introduced, which is determined where E , E are the energies of the initial and final states of by the one- and two-photon exchange, electron self-energy i f the system and pf is the momentum of the scattered electron. and vacuum polarization matrix elements and other QED cor- The cross section is defined as rections [14] and which can be derived order by order within the QED perturbation theory. The matrix V = V (0) + V is dwif dσif = , (69) considered as a block matrix j (0) V V V + V11 V12 = /ε V = 11 12 = 11 . (63) where j pi i is the flux of the incident electrons having an V21 V22 (0) + ε V21 V22 V22 energy i and a momentum pi. Accordingly, the double and single differential cross sections for elastic electron scattering The matrix V is defined on the set g, which contains con- 11 read as figurations mixing with the reference state ng ∈ g.The matrix V (0) is a diagonal matrix composed of the sum of dσ ε p ε if (ε ,θ ) = 2π|U |2δ(ε − ε ) i f f , (70) the one-electron Dirac energies. The matrix V is not a f f if f i 3 11 dε f d f pi (2π ) diagonal matrix, but it contains a small parameter of the QED σ ε ε d if 2 i p f f perturbation theory. The matrix V11 is a finite-dimensional (ε = ε ,θ ) = 2π|U | , (71) f i f if π 3 matrix and can be diagonalized numerically: d f pi (2 ) diag = t , t = . where ε f and f are the energy and solid angle (with polar V11 B V11B B B I (64) angle θ f ) of the scattered electron, respectively. The matrix B defines a transformation to the basis set in which the matrix V11 is a diagonal matrix. Then, the standard perturbation theory can be applied for the diagonalization of III. RESULTS AND DISCUSSION the infinite-dimensional matrix V . The eigenvectors of V read In this section we discuss results of applications of our as [14] theory to electron scattering on hydrogen-like ions ranging from boron to uranium. The calculated cross sections will be (0) Blgng (0) n = Bk n + [V21]kl +··· , g g g kg g (0) − (0) k given in the rest frame of the ion and for those impact energies ∈ ∈/ , ∈ Eng E kg g k g lg g k where only LL autoionizing states participate in the scattering (65) process. Since the Coulomb scattering is especially strong in forward angles, we restrict ourselves to the consideration of ≡ where ng (JM j1 j2l1l2n1n2 ) is a complete set of quantum electron scattering to backward angles for which the Coulomb numbers describing the reference state, indices k, lg denote contribution is minimal. the two-electron configurations: the index lg runs over all con- In experiments on electron-ion scattering free electrons are figurations of the set g; the index k runs over all configurations often replaced by electrons bound in light atomic (molecular) not included in the set g (this implies the integration over (0) targets which serve as a source of (quasi)free electrons. There- the positive- and negative-energy continuum). Here, Ek is fore, in what follows we consider collisions of hydrogen-like (0) the energy of the two-electron configuration k given by ions not only with free electrons, but also with molecular the sum of the one-electron Dirac energies. hydrogen. The amplitude of the scattering process is given as a matrix In the bottom panels of Figs. 7–10 the single differential element of the operator (Vˆ ) cross section for scattering of free electrons is presented as a 19+ Auger (0) function of the scattered electron kinetic energy for Ca (1s), U = Vˆ | . (66) + + + if fin ini Fe25 (1s), Kr35 (1s), and Xe53 (1s). In these figures we The bra vector corresponds to the wave function describing observe a smooth background, caused by Coulomb scattering, noninteracting electrons, Eq. (62), the ket vector is given which is superimposed by maxima and minima arising due by the eigenvector, Eq. (65). The operator Vˆ is derived to the resonant scattering as well as interference between the within the QED perturbation theory order by order [14,24]. resonant and the Coulomb parts of the scattering process. In the first and second orders of the perturbation theory it The figures show that the differential cross section strongly

013087-8 QED THEORY OF ELASTIC ELECTRON SCATTERING ON … PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

19+ FIG. 7. Double-differential cross section for the elastic electron scattering in collisions of 5.178 MeV/uCa (1s) with H2 target (top) and single differential cross section for the elastic electron scattering in collisions of Ca19+(1s) with free electrons (bottom). depends on the charge of the ionic nucleus (Z) both qualita- 1/Z. Further, the Schrödinger equation predicts that, provided tively and quantitatively. the total width of an autoionizing state is determined mainly The total scattering amplitude can be written as a sum by the electron-electron interaction, the amplitude for res- of the two parts: the nonresonant (Coulomb) part and the onant scattering scales as 1/Z as well. Therefore, for not resonant part. Accordingly, the cross section can be split too heavy ions one could expect that all the contributions into three terms corresponding to the Coulomb scattering, the to the cross section, the Coulomb and the resonant parts resonant scattering, and their interference. When the charge and the interference term, scale with Z roughly as 1/Z2. Z of the ionic nucleus varies, the amplitude for Coulomb Indeed, our calculations for ions ranging between Z ∼ 5 and scattering in the vicinity of resonances effectively scales as Z ∼ 20 are in qualitative agreement with this scaling (for

FIG. 8. The same as in Fig. 7 but for 8.694 MeV/uFe25+(1s). The solid black lines represent the results of the exact QED calculation, the dashed red line corresponds to the calculation with disregard of the Breit part of the interelectron interaction and the real part of the radiative corrections.

013087-9 K. N. LYASHCHENKO et al. PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

FIG. 9. The same as in Fig. 8 but for 16.731 MeV/uKr35+(1s). illustration see Figs. 7 and 11). For heavier ions the amplitude In order to investigate the inﬂuence of the Breit interac- for the resonant scattering begins to decrease with Z faster tion and the radiative corrections (self-energy and vacuum than 1/Z. This is mainly caused by a rapid growth of the polarization), we performed calculations where these correc- radiative contribution to the total widths which in heavy ions tions were omitted. The red dotted lines in the bottom panels outperforms the Auger decay. Accordingly, with increasing of Figs. 8 and 9 correspond to calculations in which the Breit Z the contribution of the resonant scattering decreases faster interaction and the radiative corrections were neglected. The than the contribution from the interference term. Whereas energy shift of the resonances is clearly visible and roughly for relatively small Z all the three parts of the cross section equals to ∼4 and ∼10 eV for iron and krypton, respectively. are equally important, for higher Z the purely resonant part A slight decrease in the differential cross section due to the becomes of minor importance and the resonance structure contribution of the Breit interaction is noticeable but turns out in the cross section is determined solely by the interference to be rather small even for krypton. term. In the top panels of Figs. 7–10 we present results for We also note that the order of resonances also depends on collisions of highly charged ions with H2. In order to evaluate Z, in particular, it leads to different orders of maxima and minima for different Z. The resonant energies of the incident electron as well as the energies and widths of the corre- TABLE I. The parameters characterizing the resonant structure sponding autoionizing states for various ions are presented in of the cross section for various autoionizing (LL) states of helium- Tables I–IV. like calcium are presented. The second and third columns present 2 the energies (E = E(LL) − 2mc ) and widths () of the autoionizing states. The fourth column presents the resonant energies of the scattered electron (εres = E(LL) − ε1s). The data are given in the rest frame of the ion.

Autoionizing state E (keV) (eV) εres (keV) 2 − (2s)0 2.6710 0.25 2.7989 (2s2p1/2 )0 −2.6670 0.08 2.8030 (2s2p1/2 )1 −2.6654 0.08 2.8045

(2s2p3/2 )2 −2.6600 0.07 2.8100 2 − (2p1/2 )0 2.6465 0.13 2.8234 (2p1/22p3/2 )1 −2.6432 0.13 2.8267

(2p1/22p3/2 )2 −2.6405 0.16 2.8294 (2s2p3/2 )1 −2.6313 0.19 2.8386 2 − (2p3/2 )2 2.6295 0.33 2.8405 2 − (2p3/2 )0 2.5995 0.12 2.8705 FIG. 10. The same as in Fig. 7 but for 38.493 MeV/uXe53+(1s).

013087-10 QED THEORY OF ELASTIC ELECTRON SCATTERING ON … PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

TABLE II. Same as in Table I but for helium-like iron. TABLE IV. Same as in Table I but for helium-like xenon.

Autoionizing Autoionizing state E (keV) (eV) εres (keV) state E (keV) (eV) εres (keV) 2 − − (2s)0 4.5597 0.31 4.7179 (2s2p1/2 )0 20.669 3.638 20.632 − 2 − (2s2p1/2 )0 4.5560 0.21 4.7216 (2s)0 20.669 2.265 20.632 (2s2p1/2 )1 −4.5516 0.20 4.7260 (2s2p1/2 )1 −20.646 3.617 20.655 − 2 − (2s2p3/2 )2 4.5356 0.19 4.7421 (2p1/2 )0 20.574 5.166 20.726 2 − − (2p1/2 )0 4.5242 0.35 4.7534 (2s2p3/2 )2 20.250 3.372 21.051 (2p1/22p3/2 )1 −4.5138 0.37 4.7639 (2p1/22p3/2 )1 −20.205 6.905 21.096 (2p1/22p3/2 )2 −4.5082 0.45 4.7695 (2p1/22p3/2 )2 −20.189 7.050 21.112 (2s2p3/2 )1 −4.5001 0.31 4.7776 (2s2p3/2 )1 −20.185 3.463 21.116 2 − 2 − (2p3/2 )2 4.4872 0.53 4.7905 (2p3/2 )2 19.777 6.784 21.524 2 − 2 − (2p3/2 )0 4.4513 0.34 4.8264 (2p3/2 )0 19.726 6.695 21.575 the doubly differential scattering cross section for collisions Comparing the cross sections for collisions with free and with molecular hydrogen, we use the impulse approximation quasifree electrons we can conclude that the resonance struc- where the electrons, which are initially bound in hydrogen, ture remains basically the same and the only difference is are considered as quasifree. Using this approximation one can the bending of the background caused by the convolution show that the cross section in collisions with hydrogen can be with the Compton profile. The bending is more prominent expressed via the cross section in collisions with free electrons for collisions with heavier ions since the energy interval according to considered scales with Z as Z2. 2σ ε ,θ σ ε ,θ ε In our approach, the Coulomb interaction of the incident d ( f f ) = d ( f f ) f . 2 J(pz ) (72) and scattered electron with the nucleus and partly with the dε f d f d f γ p f H2 free bound 1s electron is taken into account within the Furry √ picture. In Eq. (60) the interaction of the electron in the Here, γ = 1/ 1 − V 2 and p = γ (p − V ε ), where V col z f col f col continuum with the 1s electron is considered, in the first order is the collision velocity. Further, of the perturbation theory, as the Coulomb interaction with a pointlike charge located in the origin. This is consistent J(p ) = d2p |ψ (p , p )|2 z ⊥ 1s ⊥ z with the Furry picture for the continuum electrons, in which they are regarded as moving in the field V F =−α(Z − 1)/r. 8 α5 cont = (73) Equation (61) represents the difference between the long- 3 π α2 + 2 3 pz range term of the Coulomb interaction of the continuum electron with the 1s electron, given by Eq. (55) [contribution is the Compton profile for the 1s electron of hydrogen atom of the term with K = 0inEq.(54)], and the interaction of with the corresponding wave function ψ (p ), p = (p , p ) 1s ⊥ z the continuum electron with the pointlike charge placed in the is the momentum of the incident electron in the rest frame origin. Hence, Eq. (61) describes the interaction with a short- of hydrogen, α is the hyperfine structure constant (which range potential in the first order of the perturbation theory. We corresponds to the characteristic orbiting momentum of the note that all numerical results presented in Figs. 7–10 were 1s electron in hydrogen expressed in relativistic units), and dσ (ε f ,θ f ) obtained using the Furry picture mentioned above. ( ) is the cross section for collision with a free d f free In order to investigate the importance of the higher orders electron given by Eq. (71). of the perturbation theory corresponding to the interaction with this potential, we replaced in Eqs. (60) and (61)the TABLE III. Same as in Table I but for helium-like krypton. interaction with a pointlike charge by the interaction with a charge density corresponding to the 1s electron wave function Autoionizing + state E (keV) (eV) ε (keV) ψ (r )ψ1s(r ) res V =−α dr 1s . cont | − | (74) 2 − r r (2s)0 8.8807 0.551 9.0555 (2s2p1/2 )0 −8.8782 0.724 9.0579 Since the 1s electron wave function is independent of angular (2s2p1/2 )1 −8.8670 0.725 9.0691 variables, with employment of the decomposition (54)the 2 − potential (74) can be deduced to (2p1/2 )0 8.8237 1.133 9.1124 − ∞ (2s2p3/2 )2 8.8008 0.687 9.1354 ρ =−α (r ), (2p1/22p3/2 )1 −8.7705 1.366 9.1657 Vcont (r) dr (75) 0 r> (2p1/22p3/2 )2 −8.7593 1.494 9.1769 ρ (2s2p / ) −8.7555 0.792 9.1806 where (r ) is the probability density function of the 1s 3 2 1 2 − electron r> = max(r, r ). (2p3/2 )2 8.6856 1.453 9.2505 2 − With this replacement the Furry picture should be changed (2p3/2 )0 8.6459 1.316 9.2902 accordingly: now the continuum electron is considered to be

013087-11 K. N. LYASHCHENKO et al. PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

4+ FIG. 11. Same as in Fig. 7 but for 359.1 keV/uB (1s). The solid black curves represent results of calculations with potential Vcont = −α/r, the dotted red curve corresponds to the calculation with the potential Vcont given by Eq. (74). moving in the potential Gaussian function 2σ ε ,θ − ε − ε 2 α ψ+ ψ d ( f f ) 1 ( f ) F Z 1s(r ) 1s(r ) = √ dε exp V =− + α dr . (76) 2 cont dε f d f , 2πσ 2σ r |r − r| H2 expt r r d2σ ε, θ × ( f ) , Employment of the potential (76) instead of the potential ε (77) F =−α − / d d f H Vcont (Z 1) r yields a correction to the scattering 2 amplitude. This correction is scaled with Z as 1/Z2. where σr = 0.56 eV is the experimental resolution. The result In [5] an experimental-theoretical investigation of resonant of the convolution is presented in Fig. 12. The resonance 4+ electron scattering in collisions of hydrogen-like ions B (1s) with the (2s)2 state is clearly seen in our calculations and of boron with H2 targets was reported. In particular, in [5] results of nonrelativistic calculations within the R-matrix approach were presented. In order to make a certain test of our method, we performed calculations for the same scattering system. In Fig. 11 the differential cross section of elastic electron scattering on B4+(1s) is presented in the rest frame of the ion. The solid black curve shows our results obtained F =−α − / by using the potential Vcont (Z 1) r, the dashed red curve displays the results calculated with the potential (76). By comparing them one can conclude that the higher orders of the perturbation theory corresponding to the interaction (61) give quite a small correction in the case of boron ions and, hence, can be neglected for heavier ions as well. Comparing our results with those obtained using the nonrelativistic R-matrix approach of [5] we see that overall there is a reasonably good agreement between them. Nevertheless, one substantial disagreement should be mentioned: our results show the presence of clear resonances due to the (2s, 2p1/2 )1 2 and (2p3/2 )2 autoionizing states which are absent in the calculation [5]. It can be explained by the difference between relativistic and nonrelativistic description of the electron states FIG. 12. Double-differential cross section from the top panel which remains noticeable even for light atomic systems. of Fig. 11 convoluted with Gaussian [see Eq. (77)]. The solid For a comparison of our results with the experimental data black curves represent results of calculations with potential Vcont = of [5], the doubly differential cross section for collision with −α/r, the dotted red curve corresponds to the calculation with the hydrogen (the upper panel of Fig. 11) was convoluted with a potential Vcont given by Eq. (74).

013087-12 QED THEORY OF ELASTIC ELECTRON SCATTERING ON … PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

TABLE V. Same as in Table I but for helium-like boron. part of electron scattering on a hydrogen-like uranium ion was calculated in [9]. We also considered electron scattering on Autoionizing hydrogen-like uranium taking into account not only the reso- ε state E (eV) (eV) res (eV) nant part, but also the Coulomb scattering and the interference 2 between them. In this case, the Coulomb scattering turned out (2s )0 −154.5 0.195 185.8 (2s2 )1S 0.194a 186.14a to be by far the dominant one. However, our results obtained by neglecting the Coulomb scattering (and the interference) (2s2p / ) −153.3 0.008 187.0 1 2 1 are in a good agreement with those of Kollmar et al. [9]. (2s2p1/2 )0 −152.9 0.008 187.4 (2s2p3/2 )2 −152.8 0.008 187.4 (2s2p)3P 0.015a 187.44a IV. CONCLUSION p2 − < (2 1/2 )0 149.5 0.001 190.8 We have considered elastic scattering of an electron on a 2 − (2p3/2 )2 149.4 0.002 190.9 hydrogen-like highly charged ion. The focus of the study has − < (2p1/22p3/2 )1 148.9 0.001 191.3 been on electron impact energies where autoionizing states − (2p1/22p3/2 )2 147.2 0.192 193.0 of the corresponding helium-like ion may play an important (2p2 )1D 0.188a 193.89a role in the process. Compared to electron scattering on light (2s2p3/2 )1 −146.5 0.103 193.8 ions, the main difference in the present case is a strong (2s2p)1P 0.088a 194.20a field generated by the nucleus of the highly charged ions 2 − (2p3/2 )0 139.7 0.007 200.6 which makes it necessary to take into account the relativis- (2p2 )1S 0.010a 200.90a tic and QED effects. To this end, we have developed ab initio relativistic QED theory for elastic electron scattering a Reference [5]. on hydrogen-like highly charged ions which describes in a unified and self-consistent way both the direct (Coulomb) in the calculations of [5]; however, it is not observable in scattering and resonant scattering proceeding via formation the experimental data of [5]. Also, we note that our results and consequent decay of autoionizing states. obtained using Eq. (76) (the dashed red line in Fig. 11)arein Using this theory we have calculated scattering cross sec- a somewhat better agreement with the experiment. tions for a number of collision systems ranging from relatively The resonant energies of the incident electron as well as light to very heavy ones. As one could expect, with increasing the energies and the widths of the corresponding autoionizing the charge of the ionic nucleus, the role of the resonant states for helium-like boron are presented in Table V and scattering decreases. However, even for ions with Z ≈ 50 compared with data taken from [5]. A good agreement of the resonances in the cross section remain clearly visible for our results with the theoretical and experimental data of [5] backward scattering. shows that our approach can also be applied to relatively light Although the presented theory has been developed first systems. of all for the description of collisions with highly charged − + It is worth noting that experimental data for elastic res- ions, its application for such a light system as (e + B4 ) onant electron scattering are available only for rather light demonstrates that it can be successfully used for an accurate ions: B4+,C5+,N+6,O7+, and F8+ [5,25]. Consequently, the description of a very broad range of colliding systems. considered impact energies were relatively modest and such scattering processes could be successfully described even by ACKNOWLEDGMENTS a nonrelativistic theory. To our knowledge, experiments on electron resonant scattering on ions with a (much) larger The authors are grateful to D. Yu, S. Hagmann, and P.-M. charge are still absent. However, recently such experiments Hillenbrand for helpful discussions. The work of D.M.V. and have become feasible [26]. The most appropriate means for O.Y.A. on the calculation of the differential cross sections measuring the cross sections of resonant electron scattering was supported solely by the Russian Science Foundation on heavy highly charged ions are storage rings of heavy ions. under Grant No. 17-12-01035. The work of O.Y.A. was Two types of targets can be employed: a beam of free electrons partly supported by Ministry of Education and Science of the or a gas of light atoms which serve as a source of quasifree Russian Federation under Grant No. 3.1463.2017/4.6. A.B.V. electrons. For instance, the former experimental setup was acknowledges the support from the Deutsche Forschungsge- used to measure the cross section for dielectronic recombi- meinschaft (DFG, German Research Foundation) under Grant nation with H-like uranium [27]. For registration of scattered No 349581371 (VO 1278/4-1). electrons, an electron detector with a fine energy resolution is necessary. Detectors with the required resolution are available APPENDIX A: COULOMB SCATTERING at the GSI [28,29] and also at some other scientific centers [30,31]. Hence, all the ingredients to perform experiment on The Coulomb scattering amplitude can be calculated as the ψ (−) ψ (+) resonant electron scattering on heavy highly charged ions scalar product of the in and out states ( and ) are already available and the relativistic QED theory of this (−) (+) 3 (−)+ (+) S = ψ μ ψ μ = d r ψ μ (r)ψ μ (r). (A1) process, presented in our paper, may serve as a guidance for if pf f pi i pf f pi i experimental studies. Finally, in this section we would like to note the following. We assume that the z axis is directed along the electron As was already mentioned in the Introduction, the resonant momentum pi in the initial state. Presenting the in and out

013087-13 K. N. LYASHCHENKO et al. PHYSICAL REVIEW RESEARCH 2, 013087 (2020) states in the form given by Eq. (3), the S-matrix element reads Taking into account that as 2 ∗ i(φ jl+φ ) l−l S = N M (θ,ϕ)M μ (0, 0)e j l i if j l m μ f jlm i 2l + 1 jlm j l m Y (0, 0) = δ , (A4) lml 4π ml 0 × 3 ψ+ ψ d r ε jlm (r) ε jlm(r) + χ vμ(0, 0) = δμm , (A5) ms s 2 ∗ 2iφ = N δ(ε − ε ) M (θ,ϕ)M μ (0, 0)e jl , (A2) jlmμ f jlm i we obtain jlm where we introduced l + , = ls ,μ 2 1 ∼ δ . ν = + ν ν M jlmμ(0 0) Cjm(0 ) mμ (A6) M jlmμ( ) jlm( )vμ( ) 4π ls ∗ + = C (ml , ms )Y (θ,ϕ)χ vμ(θ,ϕ) jm lml ms It is convenient to introduce the quantity ml ms = χ + θ,ϕ . ls ∗ + i(m −m)ϕ (vμ)ms m vμ( ) (A7) = C (m , m )Y (θ,0)χ vμ(θ,ϕ)e s . s jm l s lml ms ml ms (A3) Then, the S-matrix element can be written as

∗ 2l + 1 2 ls + i(μi−ms )ϕ ls 2iφ jl Sif = N δ(ε − ε ) C μ (ml , ms )Ylm (θ,0) χ vμ (θ,ϕ) e C μ (0,μi ) e j i l ms f j i 4π jl ml ms i ∗ 2l + 1 2 i(μi−m)ϕ ls ls 2iφ jl = (−2πi)δ(ε − ε )N vμ e C μ (μi − m, m)C μ (0,μi )e Ylμ −m(θ,0). (A8) 2π f m 4π j i j i i m jl

Making use of Eq. (18) we get − Coul 2 ( 1)p ∗ R = N vμ M μ , (A9) if (2π )2 f m m i m

where, following [17], we introduced the matrix Mmμi : 1 2l + 1 i(μi−m)ϕ ls ls 2iφ jl Mmμ = 4π e C μ (μi − m, m)C μ (0,μi )Ylμ −m(θ,0)e . (A10) i 2pi 4π j i j i i jl

In particular, 1 2l + 1 ls ls 2iφ jl M μ = 4π C μ(0,μ)C μ(0,μ)Y (θ,0)e , if m = μ (A11) m 2pi 4π j j l0 jl 1 2l + 1 2iμϕ ls ls 2iφ jl M μ = 4π e C μ(2μ, μ¯ )C μ(0,μ)Y μ(θ,0)e , if m =−μ. (A12) m 2pi 4π j j l2 jl Accordingly, we obtain

−iϕ M+1/2,+1/2(θ,ϕ) = f (θ ), M+1/2,−1/2(θ,ϕ) = g(θ )e , iϕ M−1/2,+1/2(θ,ϕ) =−g(θ )e , M−1/2,−1/2(θ,ϕ) = f (θ ). (A13) Taking into account that + − (2l 1)(l 1)! 1 Y (θ,0) =−Y ,− (θ,0) =− P (cos θ ), (A14) l1 l 1 4π(l + 1)! l 2l + 1 Y (θ,0) = P (cos θ ), (A15) l0 4π l

/ |κ| Cl1 2(0, m) 2 = , (A16) jm 2l + 1

013087-14 QED THEORY OF ELASTIC ELECTRON SCATTERING ON … PHYSICAL REVIEW RESEARCH 2, 013087 (2020)

1 1 Cls (2m, m¯ )Cls (0, m) = l(l + 1), j = l + (A17) jm jm 2l + 1 2 1 1 Cls (2m, m¯ )Cls (0, m) =− l(l + 1), j = l − (A18) jm jm 2l + 1 2

we obtain Then, the amplitudes (A19) and (A20) can be written as 1 1 2iφκ f (θ ) = |κ|(e − 1)P (cos θ ), (A19) f (θ ) = |κ|aκPl (cos θ ) 2pi l 2ip jl jl 1 φ φ θ = 2i κ=−l−1 − 2i κ=l 1 θ , 1 g( ) (e e )Pl (cos ) (A20) = [(l + 1)aκ=− − + laκ= ]P (cos θ ), (B7) 2pi 2ip l 1 l l l l 1 φκ ≡ φ κ 1 where jl and is the Dirac quantum number g(θ ) = (aκ=−l−1 − aκ=l )P (cos θ ). (B8) 2ip l 1 l κ = j + (−1)j+l+1/2. (A21) 2 Now, we introduce the approximate amplitudes 1 Using Eq. (69), the differential cross section for the f˜(θ ) = [(l + 1)˜aκ=− − + la˜κ= ]P (cos θ ), (B9) 2ip l 1 l l Coulomb scattering is obtained to be l 1 3 θ = − 1 θ , ε d p g˜( ) [˜aκ=−l−1 a˜κ=l ]Pl (cos ) (B10) dσ = 2π|R |2 δ(E − E ) , (A22) 2ip if if p i f (2π )3 l

2 where dσ ∗ if = . vμ f Mmμi (A23) = γ =|κ| dν m a˜κ aκ ( ) (B11) m and, correspondingly,

APPENDIX B: NUMERICAL CALCULATION (l + 1 − iν) a˜κ=− − = (l + 1 + iν ) − 1, (B12) OF THE COULOMB AMPLITUDES l 1 (l + 2 + iν) The Coulomb amplitudes are given by series equations (l − iν) a˜κ= = (l − iν ) − 1. (B13) (A19) and (A20). These series are not convenient for a direct l (l + 1 + iν) numerical calculation. However, the leading part of these The series equations (B9) and (B10) can be summed up series can be calculated analytically and the remaining part analytically [32]: can be easily summed up numerically. − ν ν ν − ν The Coulomb phase shifts for the potential V =−αZ/r (1 i ) ν 2 θ/ f˜(θ ) = ei ln sin ( 2) csc2(θ/2) + , read as (1 + iν) 2p 2p 1 π (B14) φκ = arg (γ − iν) + η − πγ + (l + 1), (B1) 2 2 (1 − iν) 2 ν − ν g˜(θ ) = eiν ln sin (θ/2) cot(θ/2) . (B15) where (1 + iν) 2p η −κ + iν As a result, the Coulomb amplitudes can be rewritten as e2i = , (B2) γ + iν 1 ε f (θ ) = f˜(θ ) + |κ|(aκ − a˜κ )Pl (cos θ ) ν = αZ, (B3) 2ip p jl me 1 ν = αZ, (B4) = f˜(θ ) + [(l + 1)(aκ=− − − a˜κ=− − ) p 2ip l 1 l 1 l γ = κ2 − α2 2, Z (B5) + l(aκ=l − a˜κ=l )]Pl (cos θ ), (B16) where α is the ﬁne-structure constant and me is the electron 1 g(θ ) = g˜(θ ) + (aκ=− − − a˜κ=− − − aκ= mass. Following [32] we introduce 2ip l 1 l 1 l l 2iφκ aκ (γ ) = e − 1 + 1 θ , a˜κ=l )Pl (cos ) (B17) −κ + iν (γ − iν) = (−1)l+1 e−iπγ − 1. (B6) where the corresponding series can be easily calculated nu- γ + iν (γ + iν) merically.

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