Photoionization accompanied by excitation at intermediate photon energies
E. G. Drukarev, E. Z. Liverts, M. Ya. Amusia, R. Krivec, V. B. Mandelzweig
aa r X i v : . [ qu a n t - ph ] N ov Photoionization accompanied by excitation at intermediate photon energies
E. G. Drukarev , E. Z. Liverts , M. Ya. Amusia , , R. Krivec andV. B. Mandelzweig Petersburg Nuclear Physics Institute,
St. Petersburg, Gatchina 188300, Russia Racah Institute of Physics,
The Hebrew University, Jerusalem 91904, Israel A. F. Ioffe Physical-Technical Institute,
St. Petersburg 194021, Russia Department of Theoretical Physics, J. Stefan Institute
P. O. Box 3000, 1001 Ljubljana, Slovenia (Dated:)We calculate the photoionization with excitation- to photoionization ratios R nℓ and R n = Σ ℓ R nℓ for atomic helium and positive heliumlike ions at intermediate values of the photon energies. Thefinal state interactions between the electrons are included in the lowest order of their Sommerfeldparameter. This enables us, in contrast to purely numerical calculations, to investigate the rolesof various mechanisms contributing beyond the high-energy limit. The system of the two boundelectron is described by the functions obtained by the Correlation Function Hyperspherical HarmonicMethod. For the case of heliumlike ions we present the high energy limits as power expansion ininverse charge of the nucleus. We analyze the contribution of excitation of states with nonzeroorbital momenta to the ratios R n . In the case of helium our results for R n are in good agreementwith those of experiments and of previous calculations. I. INTRODUCTION
Atomic photoionization accompanied by excitation and double photoionization are much studied in experi-mental and theoretical works in connection with many-electron problem. In the case of two-electron systemswhich is considered in the present paper, the three-body Coulomb problem is the subject of investigation.Most attention in experimental studies was focused on the process of double photoionization. Photoionizationwith additional excitation have not been investigated in detail. Energy dependence of the cross section ratios ina broad interval of the photon energies ω , dependence of these ratios on the value of nuclear charge Z , branchingratios for excitations of nℓ subshells of a shell with the principle quantum number n are still the subjects offuture experiments. As it stands now, there are experimental data only for atomic helium. The intensity ofexcitation of n -th shell relative to the main photoline n = 1 was measured in [1] for the photon energies up toseveral hundreds eV for the values of n ≤
6. A few measurements of 2 s and 2 p excitations at smaller values ofthe photon energies have been carried out earlier – see [1] for references.Theoretical investigation of the process requires the knowledge of the wave functions describing two electronsin the field of the nucleus. In initial state both electrons are bound by the nucleus. In the final state one ofthe electrons belongs to continuum, while the second one is in excited bound state. Certain approximations(models) for the wave functions are required. Somewhat different approximations are reasonable in differentregions of the photon energy ω .We use the terminology which is similar to the one employed for much studied double photoionization [2]. Itis known that the ratios R n ( ω ) = σ + ∗ n ( ω ) σ +1 ( ω ) (1)of the cross sections σ + ∗ n ( ω ) for ionization with excitation of the second electron to the n -th level, to thosewithout excitation σ ( ω ) do not depend on the photon energy in high energy limit [3, 4] R n ( ω ) = const (2)for ω → ∞ . This requires anyway that ω exceeds strongly the values of single-particle ionization potentials Iω ≫ I . (3)Analysis of [3, 4] have been carried out by employing the nonrelativistic functions for description of theoutgoing electrons. It was shown in [5] that asymptotics of the ratio R ( ω ) remains the same in the whole region(3) including the photon energies corresponding to relativistic outgoing electrons. Recall that this is not truefor the double-to-single photoionization ratios [6].By high energies we mean that part of the region (3) where the ratios exhibit behavior described by Eq. (2).At low energies the ratio I/ω cannot be treated as a small parameter. By intermediate energies we mean thevalues of the photon energies, where inequality (3) is true, while deviations of the cross section ratios from thehigh energy limit are noticeable (with the relative deviations exceeding 10 percent). For atomic helium this isthe region from 300–400 eV till 2 keV. In the systems bound by the nucleus with the charge Z the limits of theinterval are proportional to Z .Since the ionization with excitation is a three-body problem, certain approximated wave functions for bothinitial and final states are required. It was shown in [3] and [4] that in the high energy limit the final stateinteractions (FSI) between the electrons can be neglected. This simplifies the problem of the description of thefinal state (under a proper choice of gauge of gauge interactions of the outgoing electron with the nucleus canbe neglected as well). In [3] the high energy limit of the process was expressed in terms of the initial state wavefunction Ψ i ( r , r ). The high energy limits of the ratios R n for atomic helium were calculated in [7, 8] anddependence on the choice of the approximate function Ψ i was traced. The calculations of [6] include also the Z dependence of the high energy limits of R n . Results of the high energy calculations for Li + are presented in [9]and [10].At low energies there is no small parameter. All the interactions involved should be treated as accurately aspossible. In this energy region one must make a choice of both initial and final state wave functions. The lowenergy calculations of the cross sections σ + ∗ n ( ω ) have been carried out in [14, 15] for He and in [14, 16] for Li + .The paper [16] contains also results for partial cross sections σ + ∗ nℓ ( ω ) of ionization accompanied by excitation ofthe remaining bound electron to the subshells with quantum numbers n and ℓ . Low energy calculations for thetwo-electron ions with larger values of Z were carried out in [14].In the papers [13, 14] the intermediate energy region was approached by extension of the low energy calcula-tions to this energy interval. In the present paper we move from the high energy region by including next toleading order of expansion in powers of ω − . This is achieved by inclusion of the interaction between the finalstate electrons in the lowest order of perturbation theoryWe find several attractive points in such approach. It provides the possibility to clarify the role of variousmechanisms (in a fixed form of electron-photon interactions) representing their contributions in terms of certaincharacteristics of the initial wave function. Within the framework of the approach one can estimate the magni-tude of the neglected terms, thus controlling the accuracy. At the lower limit of the intermediate energy regionnumerical calculations with certain models for the final state wave functions are more precise. The discrep-ancy between the results obtained in numerical and perturbative approaches should diminish as ω increases.Hence these two approaches should supplement each other. Similar analysis of the intermediate energy doublephotoionization have been carried out earlier [17].We expect the approach developed in the present paper to be useful also because of certain discrepanciesbetween experimental data for helium [1] and theoretical results. The high energy limit of the ratio R n ex-trapolated from the data obtained in [1] is in perfect agreement with the calculated one only for n = 2. Thedisagreement increases with n reaching a factor of about 2 for n = 5. There is also visible disagreement betweentheoretical and experimental results for R n ( ω ) at ω ∼ −
400 eV for n = 2 , , R matrixapproach, widely used in the low energy calculations becomes unstable at the high energies. Finally, studies of Z dependence of the ratios R n may be of interest in connection with increasing attention devoted to physics ofthe multicharged ions.We calculate ratios (1) of photoionization accompanied by excitation of the residual ion for helium atom andlight heliumlike positive ions. We obtain also more detailed characteristics R nℓ ( ω ) = σ + ∗ nℓ ( ω ) σ +10 ( ω ) (4)Such ratios are also detected in the low energy experiments [18]. The ratios defined by Eq. (1) can be representedas R n ( ω ) = X ℓ R nℓ ( ω ) . In this paper the calculations are carried out with inclusion of next-to-leading order terms of expansion ofthe ratios (1) in powers of ω − . This means that for ns states we calculate the high energy limits of the ratios(1) and the correction of the order 1 /ω . For nℓ states with ℓ ≥ /ω .In the limit (3) all the interactions of the outgoing electron can be treated perturbatively [3]. In the highenergy limit of σ + ∗ n final state interactions (FSI) of the outgoing electron with the electron bound in the residualion can be neglected. The excitation following photoionization is due to the specific correlation in initial stateknown as shake-up (SU). Only s states can be excited by this mechanism. Excitations of the states with nonzerovalues of angular momentum ℓ are quenched by a small factor of the order I/ω .We can present the ratios (4) as R ns ( ω ) = A n + I ω B n (5)(with I the electron binding energy in hydrogen) for ℓ = 0, while for ℓ ≥ R nℓ ( ω ) = I ω B nℓ , (6)In the atomic system of units used through the paper ( e = m = ~ = 1, c = 137) I = 1 / A n and B ℓn with ℓ ≥ B n containsa smooth dependence on ω . Now we can present the ratios R n defined by Eq. (1) as R n ( ω ) = A n + 12 ω B n ; B n = X ℓ B nℓ . (7)In the SU mechanism the interactions of the outgoing electron with nucleus can be treated perturbatively [3].Excited electrons can be described by the Coulomb field wave functions. Thus, all the specifics of this three-body problem is contained in the wave function of the initial state. The ionized electron approaches the nucleusat the distances which are much smaller than the size of the atom. The SU cross section is thus determined byinitial state wave function Ψ i ( r , r ) at electron–nucleus coalescence point, i. e. by Ψ i ( r = 0 , r ).The SU probabilities depend on n in terms of the wave function and of momentum p n of the outgoing electron p n = 2 ε n with ε n being the energy of the outgoing electron. In the lowest order of expansion in powers of I/ω we can put p n = p = p = 2 ω . (8)The SU terms with this value of p determine the high energy limit of the ratioslim ω →∞ R nℓ ( ω ) = lim ω →∞ R ns ( ω ) δ ℓ = A n . (9)Now we consider three types of contribution beyond the high energy limit, in the same way as it was done in[17] for the double photoionization. The kinematical corrections to SU ratios are caused by taking into account n dependence of momentum p n in the SU amplitudes. This provides the contributions to the terms B n onthe RHS of Eq. (5). Note that these corrections are proportional to the small parameter I/ω , containing alsodependence on specific parameter πξ Z with ξ Z = Zp . (10)One has to have in mind that corrections of the order πξ Z /ω drop as ω − / but contain a numerically largecoefficient. We shall not treat πξ Z as a small parameter, but include it exactly. The dependence of the crosssections on πξ Z known to be presented by the Stobbe factor S ( πξ ) = exp ( − πξ ) which is common for thephotoionization processes [19, 20]. These corrections are expressed in terms of SU contributions A n to theratios (5), which appear only in the ratios R ns .In the next to leading order the excitation energy can be transferred to the second electron also by the initialstate interactions (ISI) beyond the SU. In this case the terms of the order 1 /ω and Z /ω come from the higherterms r /r and ( r r ) /r of the expansion of initial state function Ψ i ( r , r ) at r →
0. Thus the contributionwill be presented in terms of the derivatives of the initial state wave function, integrated with the Coulomb fieldfunction of the bound state.The excitation energy can be transferred also by the final state interactions (FSI) between the final stateelectrons. We include the FSI by perturbative method developed in [21]. The FSI amplitude is presented aspower series of the Sommerfeld parameter of the interaction between the fast outgoing electron and that of theresidual ion ξ = 1 v , (11)while v is their relative velocity. Thus, the square of the amplitude is presented as power series in ξ = 1 / ε with ε being the energy of the outgoing electron. Looking for the terms of the relative order ω − in the crosssections, we must include the lowest correction of the order ξ , putting ξ = 12 ω . (12)The FSI contributions are presented in terms of matrix elements of relatively simple operators sandwiched bythe function Ψ i ( r , r ) and the Coulomb function of the electron in residual ion. The states with any angularmomenta ℓ can be excited by the FSI in the next to leading order of ω − expansion.Thus, all the contributions up to the order ω − will be presented in terms of certain characteristics ofthe initial state wave functions. We employ the functions obtained by Correlation Function HypersphericalHarmonic Method (CFHHM), obtained in [22]. The CFHHM functions have been employed successfully forinvestigation of the parameters of the bound two-electron systems [23] and of some characteristics of the doublephotoionization. Also, the method of inclusion the FSI [21] enabled earlier to remove the discrepancy betweenexperimental and theoretical results in creation of vacancies in electronic shells during nuclear transitions andin single photoionization of the p states. In [17] it was used for investigation of the double photoionization atintermediate energies. In the present paper we use the CFHHM functions and the perturbative treatment ofFSI for investigation of photoionization with excitation.Note that for the system containing larger number of electrons the picture is more complicated. Consideringionization with excitation of the subshell with ℓ = 1 we find for ionization without excitations σ + ∼ ω − / , whilefor ionization accompanied by excitation to an s state σ + ∗ ∼ ω − / (the ISI provides admixture of two s stateelectrons to the system containing two p state ones). Hence the corresponding ratio increases proportionally to ω .Our analysis is completely nonrelativistic. We neglect the terms of the order ω/m in the wave function ofthe final state, and in the operator of the photon–electron interaction. The latter means that we are using thedipole approximation. We assume also ( Z/ ≪
1, to neglect relativistic effects in the initial bound system.
II. GENERAL EQUATIONS
The cross section of photoionization accompanied by excitation of the residual ion into a state with thequantum numbers n, ℓ, m can be written as dσ + ∗ nℓ = 2 πωc X m | F nℓm | δ ( ω − ε n − I i ) d p n (2 π ) . (13)Here I i denotes the ionization potential of K electron in initial state atom. The factor 2 is due to two electronsin the K shell. The overline shows that the averaging over the photon polarizations have been carried out. Theangular dependence of the amplitudes can be written explicitly due to the dipole approximation employed Theamplitude F nℓm = h Φ nℓm | γ | Ψ i , (14)with γ being the operator of interaction between the photon and an electron, while Ψ and Φ nℓm describe theinitial and final two-electron states, can be represented as F nℓm = (4 π ) / ( e · p n ) c T nℓm . (15)After averaging over the photon polarizations one obtains σ + ∗ nℓ = 43 · p n c ω X m | T nℓm | . (16)If the FSI are neglected, the final state function isΦ nℓm ( r , r ) = ψ f ( p n ; r ) ψ nℓm ( r ) (17)with the functions ψ f and ψ n,ℓ,m being just the continuum and bound state single-particle wave functions inthe Coulomb field. If condition (3) is valid, the interactions of the outgoing electron with the nucleus can beincluded perturbatively. Using the velocity gauge for the operator γ , i.e. γ ( r ) = − i ( e · ∇ )with e standing for the photon polarization, we obtain the leading contribution of expansion in powers of p − as coming from the plane waves. Following [17] we can separate two scales in the interactions between the fastoutgoing electron and the nucleus. Those taking place at small distances of the order p − ≪ r c with r c = 1 /Z being the characteristic size of the atom are expressed in terms of the parameter πξ ( n ) Z ( ξ ( n ) Z = Z/p ( n ) . Suchcontributions can be calculated explicitly, producing the factor N ( ξ ( n ) Z ) = N r ( ξ ( n ) Z ) e − πξ ( n ) Z (18)with N r ( ξ ( n ) Z ) = (cid:16) πξ ( n ) Z / (1 − exp ( − πξ ( n ) Z )) (cid:17) / being the normalization factor of the nonrelativistic Coulombcontinuum wave function. The interactions which take place at the distances of the order r ∼ r c can be presentedas p − series thus cancelling in the ratios (1) and (2). Thus we can putΦ (0) nℓm ( r , r ) = N ( ξ ( n ) Z ) e i ( p n r ) ψ nℓm ( r ) . (19)Following [22] we present the factors N ( ξ ( n ) Z ) in the cross sections as N (cid:16) ξ ( n ) Z (cid:17) = h ( πξ ( n ) Z ) e − πξ ( n ) Z (20)with the function h ( ξ ( n ) Z ) = 2 πξ ( n ) Z / (exp ( πξ ( n ) Z ) + exp ( − πξ ( n ) Z )) containing only weak dependence on parameter πξ ( n ) Z . Thus we can put h ( πξ ( n ) Z ) = h ( πξ Z ), with ξ Z defined by Eq. (10). Hence, N (cid:16) ξ ( n ) Z (cid:17) = h ( πξ Z ) e − πξ ( n ) Z . (21)The second factor on the RHS of Eq. (21) is the Stobbe factor mentioned above.We shall present the perturbative FSI contributions also in terms of the function (17). Thus the ratios (1) willbe presented in terms of the matrix elements of initial state two-electron function and the Coulomb function ofthe excited electron. III. AMPLITUDES BEYOND THE SHAKE-UP
Following the analysis given above, we present the amplitudes for ionization with excitation beyond the SUas F nℓm = F ( s ) nℓm + F ( i ) nℓm + F ( f ) nℓm (22)with F ( s ) nℓm standing for SU amplitude, which includes kinematical corrections to the high energy limit, while F ( i ) nℓm and F ( f ) nℓm are the contributions caused by correlations in ISI and FSI correspondingly. A. Amplitudes without inclusion of final state interactions
Since in this subsection we neglect the interactions between the outgoing electrons, we can use Eq. (14) forthe amplitude with the final state wave function presented by Eq. (17). Recall that we use the operator γ inthe velocity form. This provides F nℓm = (4 π ) / ( e · p n ) N ( ξ ( n ) Z ) Z d r ψ ∗ nℓm ( r ) e Ψ i ( p n ; r ) . (23)Here e Ψ i ( p n ; r ) = Z d r Ψ i ( r , r ) e − i ( p n · r ) (24)is the partial Fourier transform of the initial state wave function in variable r .Since the integral over r on the RHS of Eq. (23) is saturated at r ∼ r c , while p n ≫ r − c , we need expansionof the function e Ψ( p n ; r ) in inverse powers of p n . It is convenient to employ the Lippman–Schwinger equation e Ψ i ( p n ; r ) = 2 Zp n J ( p n , r ); J ( p n , r ) = Z d rr e − i ( p n · r ) Ψ i ( r , r ) . (25)The integral on the RHS is dominated by r ∼ p − n ≪ r . Thus the expansion in p − n can be carried out byexpanding the function Ψ( r , r ) in powers of r in the integrand on the RHS of Eq. (25).
1. Shake-up with kinematical corrections
Presenting J ( p n , r ) = lim δ → Z d rr e − i ( p n r ) − δr Ψ i ( r , r ) , (26)we obtain for the leading order contribution J ( p n , r ) = 4 πp n Ψ(0 , r ) , (27)which enable to write for the SU amplitudes F ( s ) nℓm = a ( p n ) S n δ ℓ δ m ; S n = (4 π ) / Z dr r ψ ( r ) n ( r )Ψ(0 , r ) (28)with a ( p n ) = ( ep n ) N ( πξ ( n ) Z ) 8 πZp p n , (29)while the upper index ( r ) in Eq. (28) labels the radial part of the Coulomb function ψ n . In the leading orderwe should neglect the dependence of p n on n , putting p n = p , just as in Eq. (8). The high energy limit of theamplitude (28) is thus F (0) n = a ( p ) S n . (30)In the next to leading term we must include the n dependence of p n . Since the residual ion contains only oneelectron, the latter is described by the Coulomb wave function, and thus p n = p − δ n , (31)where δ n = Z − n ) (32)is the excitation energy of the electron in the final state ion.
2. Contributions of correlations in the initial state
Now we return to Eqs.(24), (25), looking for higher order terms of expansion of the function Ψ i ( r , r ) at r →
0. Since the CFHHN functions are expressed in terms of variables r = | r | , r = | r | , ρ = | r − r | , wepresent the expansion in terms of the function Ψ( r, r , ρ ) = Ψ i ( r , r )Ψ( r, r , ρ ) = (1 + r i ∇ i + 12 r i r j ∇ i ∇ j )Ψ( r, r , | r − r | ) . (33)Here we put Ψ( r, r , | r − r | ) = Ψ(0 , r , r ) after the derivatives are calculated. Using Eq. (33) we find that atsmall r Ψ( r, r , ρ ) = Ψ(0 , r , r ) + r Ψ ′ r ( r, r , r ) − rτ Ψ ′ ρ (0 , r , ρ ) ++ r ′′ r ( r, r , r ) + r (1 − τ ) r Ψ ′ ρ (0 , r , ρ ) ++ r τ ′′ ρ (0 , r , ρ ) − r τ Ψ ′ rρ ( r, r , ρ ) . (34)Here τ = ( r · r ) /r · r . The derivatives Ψ ′ r and Ψ ′ ρ (and those of the second order) are taken at the points r = 0 and ρ = r . The higher terms of expansion in r contribute to the higher order corrections in 1 /p to theamplitudes. Thus, they are neglected. While evaluating the next to leading order terms we must put p n = p .Using Eqs. (26) and (34) we find nonzero contributions to the amplitudes with the angular momenta ℓ = 0and ℓ = 1. For excitation to s states we obtain F (2) n = a ( p ) Q n p ; Q n = − (4 π ) / Z dr r ψ ( r ) n ( r ) ×× (cid:20) Ψ ′′ r ( r, r , r ) + 13 Ψ ′′ ρ (0 , r , ρ ) + 23 r Ψ ′ ρ (0 , r , ρ ) (cid:21) r (35)with the upper index ( r ) labelling the radial part of the single-particle Coulomb field function as in Eq. (28).The function a ( p ) is determined by Eq. (29). Other notations are explained in the text below Eq. (26). Forexcitation into p states, choosing the direction of the outgoing electron momentum as the axis of quantizationof the angular momentum, we obtain F (2) n m = ia ( p ) P n δ m p ; P n = (4 π ) / √ Z dr r ψ ( r ) n ( r )Ψ ′ ρ (0 , r , ρ ) r . (36)Thus interactions in the initial state provide corrections of the order p − to the cross sections of excitationsinto s states. The dependence of the wave function on the interelectron distances ρ = | r − r | , which describesthe electron correlations enables also excitations into p states. Excitation to the states with higher orbitalmomenta due to the ground state correlations only are still impossible. B. Contribution of the final state interactions
Now we include the final state interactions. Following [21, 24] we present the final state wave function asΦ ( f ) = (1 + GV ee + GV ee GV ee )Φ (0) (37)with Φ (0) being the wave function (17), where the FSI have been neglected (here we omit lower indices), V ee is the electron–electron interactions, G is the propagator of the system of two non-interacting electrons in theCoulomb field of the nucleus. The second and third terms on the RHS of Eq. (37) correspond to one and twointeractions between the final state electrons, thus being proportional to the powers of the parameter ξ definedby Eq. (11).The two last terms on the RHS contain infrared divergent contributions caused by the Coulomb interactions V ee . It was shown in [21] that the infrared divergent terms cancel in each order of the expansion of the squareof the amplitude | F | . The situation is similar to that with the infrared singularities in the e − N scatteringanalyzed in [25]. The cancellation can be illustrated by assuming the electron interactions to be defined as V ee ( r ) = lim ν → e − νr /r . The contributions ln ν emerge in intermediate steps but vanish in the final expressionfor | F | .Explicit expressions which include the FSI in process with the fast outgoing electron up to the terms of theorder ξ have been obtained in [21, 24]. The first order amplitude F ( f , corresponding to the second term onthe RHS of Eq. (37) is mostly imaginary. The real part of F ( f is suppressed by additional power of p − andthus can be written as being proportional to ξ − . The second order amplitude is mostly real. Thus Im F ( f ∼ ξ ,Re F ( f ∼ ξ , Re F ( f ∼ ξ , Im F ( f ∼ ξ (we do not trace the dependence on Z here). The FSI amplitudescan be presented as [21] F ( f nℓm = a ( p ) (cid:18) iξ h ψ nℓm | ln( r − r z ) ν | Ψ i i + ξ h ψ nℓm | r ddr | Ψ i i (cid:19) ; (38) F ( f nℓm = − a ( p ) ξ h ψ nℓm | ln ( r − r z ) ν | Ψ i i . (39)Here Ψ i ≡ Ψ i ( r = 0 , r ), is a function of r = | r | , z is the direction of the momentum of the outgoing electron.Recall that z axis is chosen for the quantization of angular momentum, r is the Bohr radius. Thus all thecontributions to F ( f ) nℓm have nonzero values only for m = 0.For s states both amplitudes F f and F f are important, since the terms containing the factor ξ interferewith the SU amplitude. For s states F ( f ns = a ( p )( iξU n + ξ V n ); F ( f ns = − a ( p ) ξ W n , (40)with U n = (4 π ) / Z dr r ψ ( r ) n ( r ) ln r ν Ψ i (0 , r ); V n = (4 π ) / Z dr r ψ ( r ) n ( r ) d Ψ i (0 , r ) dr r ; W n = − (4 π ) / Z dr r ψ ( r ) n ( r ) ln ( r ν )Ψ i (0 , r ) . (41)For the states with ℓ = 0 there is no interference with the SU amplitude. Thus only the first term of theamplitude F ( f is important. We can present F ( f nℓ = iaξS nℓ c ℓ + 0( ξ ) (42)with S nℓ = (4 π ) / Z dr r ψ ( r ) nℓ ( r )Ψ(0 , r ) , (43)while c ℓ = √ ℓ + 12 Z − dt ln(1 − t ) P ℓ ( t ) = − √ ℓ + 1 ℓ ( ℓ + 1) , (44)and P ℓ is the Legendre polynomial. IV. THE RATIOS
Now we can calculate the cross sections and the ratios (1) and (4). The cross sections are related to thesquares of the amplitudes | F | by Eqs. (13) and (16). We start with calculation of the values of | F | . A. Excitation of s states Expressions for excitation of s states have the most complicated structure | F ns | = a ( p n ) S n + a ( p )2 ω (cid:2) S n Q n + 2 S n ( V n + W n ) + U n (cid:3) . (45)0Here the first term on the RHS stands for SU contribution with account of kinematical corrections – Eqs. (28)and (29). The first term in the squarebrackets comes from interference between SU and ISI amplitudes – Eqs. (30) and (35). The second term inbrackets is caused by interference of SU amplitude presented by Eq. (30) with the first and second order FSIamplitudes presented by Eqs. (40) and (41). The last term in brackets is a purely FSI contribution.In order to obtain contribution of the first term on the RHS of Eq. (45) to the ratio (1) we must include the n dependence of the phase volume in Eq. (13) for the cross section. As a result, for purely SU ratio we find R SUns = S n S pp n e − π ( ξ ( n ) Z − ξ ) , (46)with p n defined by Eq. (31). In the lowest order of expansion in powers of I/ω the dependence of R SUn onparameter πξ Z is just the same as it would result in expansion of the RHS of Eq. (46) in powers of πξ Z .However this is not true for the higher order terms of I/ω expansion.The contribution of the other terms to the ratios R n can be found as their ratios to squared amplitude ofphotoionization without excitation | F s | , where the corrections of the order I/ω also should be included. Thisgives R ns ( ω ) = S n S + 12 ω (cid:20) S n S Z − n )(1 − πξ Z ) ++ 2 S n S (cid:18) Q n + V n + W n − S n S ( Q + V + W ) (cid:19) + U n − S n S U (cid:21) . (47)The first term on the RHS is the high energy limit of the ratio. Note that Eq. (47) provides exact dependence onparameter πξ Z in next to leading order of I/ω expansion. Expression in square brackets on the RHS of Eq. (47)should be identified with parameter B n introduced by Eq. (5). Separating energy independent contributionsand the terms, which depend on the photon energy through parameter πξ Z we write B n = d n + πξ Z f n , (48)with d n = S n S Z − n ) + 2 S n S (cid:18) Q n + V n + W n − S n S ( Q + V + W ) (cid:19) + U n − S n S U ; f n = − S n S Z (cid:18) − n (cid:19) . (49) B. Excitation of p states Excitations of the states with ℓ = 0 can take place only beyond the SU approximation. The contributions ofISI and FSI are expressed by Eq. (36) and by the first term in brackets on the RHS of Eq. (38) correspondingly. F n = i p ( P n + S n ) , (50)leading to R n ( ω ) = 12 ω ( P n + S n ) S . (51)1 C. Excitation of the states with ℓ ≥ In this case only the FSI contribute. The amplitude is expressed by Eq. (43) giving R nℓ ( ω ) = 12 ω c ℓ S nℓ S (52)with c ℓ defined by Eq. (44).Now we turn to analysis of particular cases. V. THE CASE OF HELIUMA. High energy limit
In Table I we compare our results for the high energy limits with those extrapolated from the experimentaldata [1] and with the results of previous calculations [7, 9, 13, 14]. There were also several publications of thehigh energy limit for R s only. The pioneering calculation [26] provided R s = 4 . · − , while the latestavailable result is R s = 4 . · − [27].One can see that various theoretical approaches provide very close values for high energy limits at n ≤ n = 2. Discrepancy between theoretical and experimental results increases with n rapidly. B. Beyond the high energy limit
Now we consider the contributions beyond the limit (9). Let us start with excitation of ns states. As weshowed above, the contributions beyond the high energy limits come from kinematical corrections to the SUterm and from the initial and final states electron–electron interactions. One can see from Table 2 that the FSIand ISI contributions are positive, with the FSI term being about 3 times larger than the ISI one for all valuesof n . The kinematical corrections are negative at µ >
1, corresponding to ω <
540 eV. At larger ω values theybecome positive. Their contribution to the parameter B n defined by Eq. (5) does not exceed 10 percent for ω . ω , becoming as large as 25-30 % in the limit µ ≪ ω ≫
500 eV). In this limitthe FSI contribution determines about one half of the parameter B n . Note that the relative role of the threecontributions does not vary much with n .Contributions to excitation of p states come from FSI and ISI. Actually the former one dominates, providingmore than 4/5 of the total contribution to B n . The result of calculations are presented in Table 3. One cansee that B n and B n provide contributions of the same order of magnitude to the energy dependent part ofthe ratio R n ( ω ) defined by Eqs. (1) and (7).Thus the energy dependent parts of the ratios R n ( ω ) determined by parameters B n , are dominated bycontributions of s and p states. The d states provide corrections of about 10%, while contributions of stateswith larger values of orbital momenta are negligibly small. The coefficients B n are dominated by FSI whichprovide more than 70% of the values.Excitation of the states with ℓ ≥ d states excitations are severaltimes smaller than those of p states still providing noticeable contributions to parameter B n (Eq. (71)) for n ≥
3. The relative role of excitation of d states slowly increases with increasing n – see Table 3. Excitation ofstates with higher values of ℓ drops rapidly with increasing of ℓ . For example, the cross section for excitation2of 4 f state is about twenty times smaller than that of 4 d state. The values of B nℓ and B n are presented inTable 3.Energy dependence of the relative role of excitations to the states with ℓ > σ + ∗ nℓ /σ + ∗ = R nℓ /R n is shown in Fig. 1.The ratios B n /A n converge to certain limiting values while n increases. This is due to similar n − behaviorof both characteristics at large n . C. Comparison with earlier results
Now we compare our result with experimental and theoretical data obtained by the others. In Fig. 2 wepresent the ratios R n for 2 ≤ n ≤
6, calculated in the present work and measured in Ref. [1]. We show also theresults of calculations carried out in [14], where the intermediate energy region was reached by moving from thelow energies.We see that for n = 2 our results are in good agreement with those of [1, 14] for ω ≥
400 eV. As expected,there are noticeable deviations from results of [1, 14] at smaller values of ω . We do not show the results ofcalculations for ω ∼
900 eV obtained in [11, 27], which are also in good agreement with those of the presentpaper. For n = 3 we find a good agreement with experimental and theoretical results at all values of ω . For thecases n = 4 and n = 5 our results are close to those of [14], with both sets of the calculated values exceeding theexperimental data at ω ∼ −
400 eV. For n = 6 the experimental results are available only for ω ≤
160 eV,where the accuracy of our approach is poor since ξ Z ≥ .
34. However the deviations between our results andexperimental data is are not large for n = 6, as well as for the other values of n in this energy region. VI. Z DEPENDENCEA. High energy limit
It was shown in [6] that the SU ratios A n drop as Z − at Z ≫
1. The tendency is illustrated by the resultsof calculations for Z ≤
10 presented in Table 4. One can see that the convergence to Z − law becomes betterfor larger values of n . In [28] Z dependence of high energy limits for double-to-single photoionization ratiowas traced and presented as a Z − series. We can write similar presentation for the ionization followed byexcitation. Assuming that A n can be approximated by two terms of the series one has A n = a n Z + b n Z . (53)The values of a n and b n are given in Table 5. The convergence of Z − series is faster than in the case of doublephotoionization [28]. Also, in contrast to the double photoionization case, the leading Z − terms underestimatethe values of the ratios. Note that in approach of [6], where all the interactions between the electrons wheretreated perturbatively, A n = c n Z (54)with c n also presented in Table 5 (a numerical error was corrected in [10]). As expected, our values of a n areclose to c n .The results presented in Table 5 illustrate also the tendency to n − behavior. The values of the product n · R (0) ns for n = 5 and n = 6 differ by 5% for Z = 2 and by 4% for Z = 10.3 B. Beyond the high energy limits
In this case Eq. (3) can be written as ω ≫ Z . In order to trace Z dependence of the characteristics, we consider the limit Z ≫
1. Let us start with excitationsof s -states. One can see that in Eqs. (48) and (49) the ratio s /S n ∼ /Z , while Q n /S ∼ Z , V n /Z ∼ Z , W n /S ∼ U n /S ∼
1. Thus in Eq. (5) A n ∼ /Z , B ns ∼
1. Hence, we can present R n = (cid:18) Z ω r n (cid:19) A n , (55)with r n = B n /Z A n . Using Eqs. (48) and (49) we can write r n = r dn + πξ Z r fn (56)with r dn = d n /Z A n and r fn = f n Z A n = − n − n . (57)In similar way we can present for ℓ ≥ R nℓ = Z ω r nℓ A n , (58)with r nℓ = B nℓ /Z A n , while parameters B nℓ are introduced by Eq. (6).For the cross section ratios R n defined by Eq. (1) we present R n = (cid:18) Z ω r n (cid:19) A n ; r n = r dn + πξ Z r fn ; r dn = r d + X ℓ> r nℓ ; r fn = r fn . (59)For illustration we present characteristics of the process for Z = 10. Interplay of the three types of contribu-tions to the parameter B n describing excitation of s states is shown in Table 6. As well as in the case Z = 2the FSI provides the main contribution. However the domination is less pronounced than in the case of helium.As one can see from Eq. (57), the ratios r fn do not depend on Z . Dependence of parameters r nℓ and r n on Z is illustrated by the results presented in Table 7. The values of r dn for Z = 10 are somewhat larger than for Z = 2. This is mainly due to larger contribution for excitation of p states in the case Z = 10. On the other handthe role of excitation of d states becomes smaller – see Fig. 3. The ratio of r dn for Z = 2 and Z = 10 exhibitsvery weak dependence on n . The values of r nℓ and r n converge to certain limiting values while n increases.This is due to similar n − behavior of the parameters B nℓ and A n at large n .To estimate the limiting behavior of the ratios R n for Z ≫
1, note that the first term on the RHS of Eq. (7)for R n depends on the nuclear charge as Z − . Since the values of r fn are several times smaller than r dn they canbe neglected for πξ Z .
1. At these energies the second term contains only weak dependence on Z . VII. SUMMARY
We have considered photoionization accompanied by excitation for helium atoms and positive two-electronions. We focused on the case of intermediate photon energies, for which expansion of the amplitudes in powers4of ω − is possible, while account of the lowest term only is not sufficient. We included the final state interactionsbetween the electrons in the lowest order of their Sommerfeld parameter. This enabled us to analyze the roleof various mechanisms of transferring the excitation energy.We calculated the ratios R nℓ of the cross sections σ + ∗ nℓ for ionization, accompanied by transition of the secondelectron to the bound state with quantum numbers n, ℓ to the cross section for ionization without excitations σ +10 , and also found the sums R n = P ℓ R nℓ – Eq. (41).Following [17], we separated three types of contributions beyond the high-energy limit. These are the kine-matical correction to the shake-up (SU) terms, and the contributions describing the transfer of excitation energyby initial state and final state interactions (ISI and FSI correspondingly). The FSI were included by the pertur-bative approach developed in [21] and employed in [24]. This enabled us to extract the energy dependent factors,presenting the amplitudes in terms of the matrix elements containing the initial state wave functions. The latterhave been obtained in [22] by Correlation Function Hyperspherical Harmonic Method. These functions wereemployed in atomic physics calculations earlier [23].We carried out the calculations, taking into account the next-to-leading terms of expansion in powers of ξ Z .The kinematical corrections to the SU terms depend also on the parameter πξ Z . Dependence on this parameterwas included exactly.The cross sections for excitation of ns states have the most complicated structure. In this case we had toinclude kinematical corrections to SU terms. The ISI amplitudes are proportional to ξ Z , and we included theirinterference with the SU amplitudes. The first and second order FSI amplitudes contain the factors iξ and ξ . Thus we had to include the interference between SU and FSI amplitudes and a purely FSI term. All thecorrections should be included in the expressions for ionization cross sections without excitation σ +10 as well.The cross section for excitation of p states was determined by ISI and FSI mechanism, with both amplitudesbeing proportional to iξ . Ionization accompanied by excitations to the state with ℓ ≥ n ≤ ns states we found the FSI to provide the largestcontributions. Excitations of np and ns states, provide the contributions of the same order of magnitude to theenergy dependent parts of the ratios R n . Excitations of nd states determine a corrections of about 10% to R n .Excitation of states with ℓ ≥ ω ≥
400 eVfor n = 2 and even at smaller values of ω for the larger values of n – Fig. 2.For larger Z we found approximate formula (53), which presents the high energy limits in the form of Z − series, with the leading terms of expansion being consistent with the earlier results [6, 10]. Excitation of ns states beyond the high energy limit is still dominated by the FSI. The role of excitation of the states with ℓ = 1increases, e. g. for Z = 10 transitions to np states provide the largest contribution to the energy dependentpart of R n . The role of excitation of nd states drops with Z . These results are illustrated by Table VII. In thelimit Z ≫ R n can be presented as the sums of two terms. The high energy limit term does notdepend on ω , dropping with Z as Z − . The second term drops as ω − , varying with Z slowly.For the case of helium, as well as for the ions with larger values of Z , the contribution of ISI to the ratios R n is about 10%. Hence, the ratios R n are determined by the kinematical corrections to SU and by the FSI.5 Acknowledgements
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Fig.1. Energy dependence of the relative role of excitations to the states with ℓ > X nℓ = σ + ∗ nℓ /σ + ∗ n = R nℓ /R n for the case of helium ( Z = 2).Fig.2. Energy dependence of the ratios R n in 10 − units for the case of helium. The dots stand for theexperimental data of [1]. The solid lines show the results of the present work. The dashed lines show the resultsof the calculations carried out in [14].Fig.3. Energy dependence of the relative role of excitations to the states with ℓ >
0, expressed by the ratio X nℓ = σ + ∗ nℓ /σ + ∗ n = R nℓ /R n for the case Z = 10.7 TABLE I: The high energy limits of the ratios R ns in percent for helium atom. The column “Theory-I” [7] stands forearly high energy calculation with a variational initial state function, “Theory-II” [9] results obtained using many-bodyperturbation theory, Theory-III [13] and Theory-IV [14] are the extensions of the low energy results with multiconfig-uration Hartree–Fock and variational ground state wave functions correspondingly. The last column shows the resultobtained in [1] by extrapolation of their experimental data of [1] with statistical errors given in parentheses n Theory-I Theory-II Theory-III Theory-IV This work Experiment2 5.34 4.79 4.78 4.79 4.80 4.80 (13)3 0.66 0.592 0.605 0.596 0.590 0.543 (33)4 0.21 0.19 0.200 0.197 0.195 0.118 (37)5 0.100 0.09 0.092 0.091 0.0900 0.048 (30)6 0.055 0.05 0.050 0.0515 0.0493 ...TABLE II: Contributions of various mechanisms to the value B n defined by Eq. (5) for the case of helium. Here µ = πξ Z ; µ = 1 .
04 or ω = 500 eV, µ = 0 .
73 for ω = 1 keV. n Kinematical corrections ISI FSI B n . − µ ) 0.027 0.094 0 . − . µ . − µ )( −
1) 0.50(-2) 1.46(-2) (3 . − . µ )( − . − µ )( −
1) 1.76(-3) 5.00(-3) (1 . − . µ )( − . − µ )( −
2) 0.85(-3) 2.40(-3) (4 . − . µ )( − . − µ )( −
2) 0.47(-3) 1.35(-3) (2 . − . µ )( − B nℓ and B n of the energy dependent contributions to the ratios R nℓ and R n defined by Eqs.(5)–(7). States B nℓ B n s . − . µ p . − . µ s (3 . − . µ )( − p . −
2) (5 . − . µ )( − d . − s (1 . − . µ )( − p . −
2) (1 . − . µ )( − d . − f . − s (4 . − . µ )( − p . −
3) (0 . − . µ )( − d . − s (2 . − . µ )( − p . −
3) (4 . − . µ )( − d . − TABLE IV: The values A n Z · , with A n defined by Eq. (9). n Z = 2 Z = 3 Z = 4 Z = 6 Z = 102 19.1 14.9 12.8 11.4 10.43 2.36 2.18 2.06 1.93 1.844 0.781 0.749 0.722 0.692 0.6605 0.360 0.351 0.340 0.327 0.3166 0.197 0.193 0.188 0.182 0.176TABLE V: The values of coefficients a n and b n in Eq. (53) and of coefficients c n in Eq. (54). n a n ( − b n ( − c n ( − B n defined by Eq. (5) for the case Z = 10, µ = πξ Z . n Kinematical corrections ISI FSI B n . − µ )( −
1) 1 . −
2) 4 . −
2) 0 . − . µ . − µ )( −
2) 0 . −
3) 1 . −
2) (2 . − . µ )( − . − µ )( −
2) 1 . −
3) 3 . −
3) (8 . − . µ )( − . − µ )( −
2) 0 . −
3) 1 . −
3) (4 . − . µ )( − . − µ )( −
2) 0 . −
3) 1 . −
3) (2 . − . µ )( − TABLE VII: The values of characteristics r dn for ℓ = 0 and r nℓ for ℓ > r dn defined by Eqs. (58), (55) and (56) for Z = 2 and Z = 10. Z = 2 Z = 10State r dn , r nℓ r dn r dn , r nℓ r dn s p s p d s p d f . − s p d s p d
00 400 600 800 1000 Ω H eV L nl H Ω L X X X R () photon energy (eV) R () R () R () R6