Relativistic theory of the double photoionization of helium-like atoms
aa r X i v : . [ phy s i c s . a t o m - ph ] A ug Relativistic theory of the double photoionization of helium-like atoms
Vladimir A. Yerokhin
1, 2, 3 and Andrey Surzhykov
1, 2 Institute of Physics, University of Heidelberg, Philosophenweg 12, D-69120 Heidelberg, Germany GSI Helmholtzzentum f¨ur Schwerionenforschung GmbH, Planckstraße 1, D-64291 Darmstadt, Germany Center for Advanced Studies, St. Petersburg State Polytechnical University,Polytekhnicheskaya 29, St. Petersburg 195251, Russia
A fully relativistic calculation of the double photoionization of helium-like atoms is presented. Theapproach is based on the partial-wave representation of the Dirac continuum states and accountsfor the retardation in the electron-electron interaction as well as the higher-order multipoles ofthe absorbed photon. The electron-electron interaction is taken into account to the leading orderof perturbation theory. The relativistic effects are shown to become prominent already for themedium- Z ions, changing the shape and the asymptotic behaviour of the photon energy dependenceof the ratio of the double-to-single photoionization cross section. PACS numbers: 32.80.Fb, 31.15.aj, 33.60.+q, 34.10.+x
I. INTRODUCTION
The ejection of two electrons caused by absorbtionof a single photon is one of the fundamental few-bodyprocesses in atomic physics. The process is called dou-ble photoionization or, less frequently, photo-double-ionization. The characteristic feature of this process isthat it proceeds exclusively through the electron-electroninteraction. Because of this, double photoionization haslong been used as a testing ground for understanding ofthe electron correlation phenomena.The traditional system for studying double photoion-ization is the helium atom, for which numerous exper-imental and theoretical investigations have been per-formed during the last four decades (for a recent re-view, see Ref. [1]). Most widely studied is the ra-tio of the double-to-single photoionization cross sections R = σ ++ /σ + as a function of the energy of the incomingphoton ω . Early calculations of this ratio were mainlyconcentrated either at the near-threshold region of thephoton energies ( ω & ω cr ≈
79 eV for helium, with ω cr being the double ionization energy), where Wannier the-ory is shown to be applicable [2, 3], or at the asymptoticalnonrelativistic region ω cr ≪ ω ≪ m [4–6] (where m is theelectron mass). The intermediate region of the photonenergies ( ω ≈
200 eV for helium, where R ( ω ) reaches itsmaximum) turned out to be much more difficult for anaccurate theoretical treatment. Reliable theoretical pre-dictions in this region were obtained only in the end of1990th by means of sophisticated many-body techniques(notably, the close-coupling methods) [7–10].Since the helium atom is an essentially nonrelativisticsystem, any relativistic effects in its double photoion-ization are considered to be of little importance at thepresent level of experimental precision. However, the re-cent experiments on the double K -shell photoionizationof moderately heavy atoms up to silver [11–15] demon-strated significant enhancements (of about factor of fivefor silver) of the cross section as compared with resultsof nonrelativistic calculations. A natural candidate for explaining this enhancement would be relativity, whichobviously cannot be disregarded when dealing with thedeeply bound electron states in silver. Because of therelative isolation of the K -shell electrons from the outerelectrons, the double K -shell ionization of a heavy atomis often compared to the double photoionization of thecorresponding helium-like ion. However, an accurate the-oretical treatment of double K -shell ionization should in-clude both the relativistic effects on the inner-shell elec-trons and the electron-correlation effects induced by theouter shells. Such a calculation is rather difficult and hasnot been performed so far.In contrast to many-electron atoms, the helium-likeion is a relatively simple system for which an ab ini-tio description is feasible. So far, there have not beenany direct measurements of the double photoionizationof helium-like ions. However, with the advent of newpowerful light sources, such as the free-electron laser(FLASH) in Hamburg, the Linear Coherent Light Source(LCLS) at Stanford, and the X-ray Free Electron Laser(XFEL) at Hamburg, the experimental study of variousphotoabsorbtion processes is going to be possible for agreat variety of ions in different charge states [16]. Mea-surements of the double photoionization in moderatelyheavy helium-like ions would allow us to effectively testour understanding of the relativistic electron-correlationeffects.An alternative approach to the investigation of thedouble photoionization of highly charged ions is to studyit in the inverse kinematics, through the radiative dou-ble electron capture. For bare oxygen, such measurementwas recently accomplished in Ref. [17]. The experimen-tal upper bound on this process for bare uranium wasreported in Ref. [18].The goal of the present investigation is to perform an ab initio relativistic calculation of the double photoion-ization of a helium-like atom. The electron-electron cor-relation (both on the initial and the final states) will betaken into account to the leading order of the perturba-tion theory. To the given order of perturbation theory,the treatment is exact , i.e., includes all multipoles of theabsorbed photon, the retardation in the electron-electroninteraction as well as the interaction of the electronswith the nucleus without any expansion in the bindingfield. The treatment is gauge invariant, both with re-spect to the gauge used for the absorbed photon and thegauge of the electron-electron interaction. The higher-order electron-correlation effects omitted are estimatedby comparing the present numerical results with the ex-perimental data available for helium.The remaining paper is organized as follows. In Sec. IIwe present a short summary of the relativistic formulasfor the single photoionization, which forms the basis forour treatment of the double photoionization. In Sec. IIIwe describe the QED theory of the double photoioniza-tion to the lowest relevant order of the perturbation the-ory. Sec. IV presents details of the calculation. Numeri-cal results are presented and discussed in Sec. V.The relativistic units ( ~ = c = m = 1) are usedthroughout this paper. II. SINGLE PHOTOIONIZATION
In this section we summarize the relativistic formulasfor the single photoionization, as they build the basis forthe description of the double photoionization.Differential cross section of the photoionization of aone-electron atom is dσ + = 4 π αω (cid:12)(cid:12) τ + fi (cid:12)(cid:12) δ ( ε i + ω − ε ) d p , (1)where α is the fine-structure constant, ω is the energyof the absorbed photon, ε i is the initial (bound-state)energy, ε and p are the energy and the momentum ofthe emitted electron, respectively, ε = p m + p . Theamplitude of the process τ + fi is given by τ + fi = h p m | R λ | κ i µ i i , (2)where | κ i µ i i denotes the initial Dirac bound state withthe relativistic angular quantum number κ i and its pro-jection µ i , and | p m (cid:11) is the wave function of the emittedelectron. The general relativistic expression for the pho-ton absorption operator R λ is R λ = α · ˆ u λ e i k · r + G (cid:0) α · ˆ k − (cid:1) e i k · r , (3)where α is a three-component vector of the Dirac matri-ces, ˆ u λ is the polarization vector of the absorbed pho-ton, k is the photon momentum, ˆ k = k / | k | , and G is thegauge parameter. In our treatment, all electron statesare the eigenfunctions of the Dirac Hamiltonian, so thegauge-dependent part of R λ vanishes identically. How-ever, we keep the general gauge of the absorbed photonin actual calculations in order to check the numericalprocedure.The wave function | p m (cid:11) ≡ | p m (cid:11) − in Eq. (2) is theone-electron continuum Dirac state with the asymptotic momentum p , helicity m = ± / , and the “ − ” asymptoticbehaviour (i.e., the plane wave modified by the Coulomblogarithmic phase plus the incoming spherical wave) [19] | p m (cid:11) − = u ( p , m )(2 π ) / e i [ pz − η ln p ( r − z )] + f ( θ, φ ) e − i [ pr + η ln 2 pr ] r (4)as | r − z | → ∞ , where η = Zα ε/p is the Sommerfeldparameter. The free-electron 4-spinors u ( p , m ) [20] arenormalized by the condition u u ≡ u + γ u = 1. The ex-plicit expression of the electron wave function is | p m (cid:11) − = 1 √ p ε X κµ i l e − i ∆ l C jµlm l , m Y ∗ lm l ( ˆ p ) | εκµ i , (5)where ∆ l are the scattering phases [21], j = | κ | − / , l = | κ + / | − / , and | εκµ i are Dirac continuum stateswith given relativistic angular momentum κ and the mo-mentum projection µ , normalized on the energy scale[19].Taking into account that d p = p ε dεd Ω, the sin-gle differential cross section of the photoionization of ahydrogen-like atom is written as dσ + d Ω = 4 π αω p ε (cid:12)(cid:12) τ + fi (cid:12)(cid:12) , (6)where the energy of the emitted electron is fixed by theenergy conservation, ε = ε i + ω .In our analysis of the double photoionization, we willneed the cross section of the single photoionization ofa helium-like atom. In the independent-particle approxi-mation, the wave function of the initial two-electron stateis | J M (cid:11) = N X µ a µ b C J M j a µ a j b µ b × √ (cid:0) | κ a µ a (cid:11) | κ b µ b (cid:11) − | κ b µ b (cid:11) | κ a µ a (cid:11)(cid:1) , (7)where N = 1 / √ N = 1otherwise. Averaging the cross section over the momen-tum projection of the initial state M , employing the ex-plicit expression for the continuum Dirac state (5), andintegrating over the angles over the emitted electron, weobtain the total photoionization cross section of a helium-like atom as σ + = 4 π αω (cid:20) j a + 1 X κµµ a |h ε κµ | R λ ( ω ) | κ a µ a i| + 12 j b + 1 X κµµ b |h ε κµ | R λ ( ω ) | κ b µ b i| (cid:21) , (8)where ε = ε a + ω and ε = ε b + ω and we assumed thatthe energy of the photon is sufficient to ionize any of thetwo initial-state electrons. If this is not the case, onlyone of the two terms in the brackets should be retained. (cid:1) (cid:2) p a p ap b p bk k FIG. 1: Feynman diagrams representing the double photoion-ization of a helium-like atom. a and b denote the bound elec-tron states, p and p are the continuum electron states, k denotes the incoming photon. Double lines denote electronspropagating in the binding nuclear field. The electron correlation effects on σ + omitted in theindependent-particle approximation are well studied inthe literature. In particular, the leading nonrelativisticterm of the 1 /Z expansion of the high-energy asymptoticsof σ + was obtained in Ref. [22]. However, since we arepresently interested in the ratio of the double-to-singlephotoionization cross sections σ ++ /σ + , we prefer to treatboth these cross sections on the same footing, i.e., to theleading nonvanishing order of perturbation theory withrespect to the electron-electron interaction. III. DOUBLE PHOTOIONIZATION: GENERALFORMULAS
According to the general rules of quantum field theory[23], the differential cross section of the double photoion-ization is dσ ++ = 4 π αω (cid:12)(cid:12)(cid:12) τ ++ fi (cid:12)(cid:12)(cid:12) δ ( ε i + ω − ε − ε ) d p d p , (9)where ( ε , p ) and ( ε , p ) are the energy and the mo-mentum of the two final-state electrons, respectively, ε i is the energy of the initial bound state, and τ ++ fi is theamplitude of the process. To the leading order of per-turbation theory, the amplitude of the process is repre-sented by the two Feynman diagrams shown in Fig. 1,where the antisymmetrization of the initial and the finaltwo-electron states is assumed.The initial-state wave function is the same as for thesingle photoionization [see Eq. (7)] and the initial-stateenergy is ε i = ε a + ε b . The wave function of the finalstate is | p m , p m (cid:11) − = 1 √ (cid:16) | p m (cid:11) − | p m (cid:11) − −| p m (cid:11) − | p m (cid:11) − (cid:17) . (10)Note that in the present approach, we use the wavefunctions that do not include any electron-correlation ef-fects (apart from the antisymmetrization); the electron-electron interaction enters explicitly into the amplitudeof the process [see Eq. (16) below]. In the nonrelativis-tic case, this approach has been successfully applied to the double photoionization by many authors, notably inRefs. [4, 24]. Although the formal parameter of the per-turbative expansion is 1 /Z , the leading-term approxima-tion was shown to adequately describe the double pho-toionization even for helium. Our present treatment isaimed primarily at the helium-like ions, for which theperturbative expansion converges much faster than forhelium.The triple differential cross section of the double pho-toionization obtained from Eq. (9) is d σ ++ d Ω d Ω dε = 4 π αω p p ε ε × X m m (cid:12)(cid:12) τ ++ λ ( p m , p m ; ω, J M ) (cid:12)(cid:12) , (11)where the energy of the second electron is fixed by theenergy conservation, ε = ε a + ε b + ω − ε . The energydistribution of the emitted electrons can be convenientlyparameterized by the fractional energy sharing parameter x , ε = m + x ( ω − ω cr ) , (12) ε = m + (1 − x )( ω − ω cr ) , (13)where ω cr is the threshold value of the photon energy(the double ionization energy). In our approximation, ω cr = 2 m − ε a − ε b .Let us now consider the single differential cross section dσ ++ /dε . Substituting the explicit expression for thecontinuum wave function (5) into Eq. (11), integratingover the angles, and summing over m and m , we arriveat dσ ++ dε = 4 π αω X κ κ µ µ (cid:12)(cid:12) τ ++ λ ( ε κ µ , ε κ µ ; ω, J M ) (cid:12)(cid:12) , (14)where the amplitude contains only the spherical-waveDirac continuum states.The total cross section is obtained as an integral of thesingle differential cross section over a half of the energysharing interval σ ++ = Z m +( ω − ω cr ) / m dε dσ ++ dε = ( ω − ω cr ) Z dx dσ ++ dε . (15)The other half of the energy interval corresponds to in-terchanging the first and the second electron, which isalready accounted for by the wave function.The general expression for the transition amplitude ofthe double photoionization process can be obtained bythe two-time Green’s function method [23], τ ++ λ ( ε κ µ , ε κ µ ; ω, J M ) = N X µ a µ b C J M j a µ a j b µ b X P Q ( − P + Q × X n (cid:26) h P ε P ε | I (∆ P ε Qb ) | n Qb i h n | R λ | Qa i ε Qa + ω − ε n (1 − i
0) + h P ε | R λ | n i h n P ε | I (∆ P ε Qb ) | Qa Qb i ε P ε − ω − ε n (1 − i (cid:27) . (16)The first term in the brackets corresponds to the electron-electron interaction modifying the final-state electronwave function (the left graph in Fig. 1) and the secondone, the initial-state electron wave function (the rightgraph in Fig. 1). The summation over P and Q corre-sponds to the permutation of the initial and final elec-trons, P ε P ε = ( ε ε ) or ( ε ε ), QaQb = ( ab ) or ( ba ),and ( − P and ( − Q are the permutations sign. Thesummation over n in Eq. (16) runs over the completeDirac spectrum, ∆ ab ≡ ε a − ε b , and I (∆) is the electron-electron interaction operator, I ( ω ) = e α µ α ν D µν ( ω, x ) , (17)where D µν is the photon propagator, α µ = (1 , α ) arethe Dirac matrices. In the Feynman gauge, the electron-electron interaction takes the form I Feyn ( ω ) = α (1 − α · α ) e i | ω | x x , (18)where x = | x − x | . In the Coulomb gauge, theelectron-electron interaction acquires an additional term,which can be expressed as I Coul ( ω ) = I Feyn ( ω )+ α (cid:20) − ( α · ∇ )( α · ∇ ) ω (cid:21) − e i | ω | x x . (19) IV. NUMERICAL CALCULATION
The final formulas for the single and double photoion-ization cross sections are presented in Appendices A andB. The calculation of the single photoionization cross sec-tion is straightforward, in contrast to that of the doublephotoionization. The major difficulty in the numericalevaluation of the double photoionization is the summa-tion over the complete spectrum of the Dirac equation.In this work we use the approach based on the analyti-cal representation of the Dirac Coulomb Green function,which is represented by an infinite sum over the partialwaves. The numerical approach was developed in the pre-vious works [25–27], where it was used for calculating theQED and electron-electron interaction corrections to theradiative recombination of electrons with highly chargedions. For a given value of the relativistic angular momen-tum quantum number κ , the radial part of the DiracCoulomb Green function is represented in terms of thetwo-component solutions of the radial Dirac equation reg-ular at the origin (cid:0) φ κ (cid:1) and the infinity ( φ ∞ κ ), G κ ( E, r , r ) = − φ ∞ κ ( E, r ) φ T κ ( E, r ) θ ( r − r ) − φ κ ( E, r ) φ ∞ T κ ( E, r ) θ ( r − r ) , (20)where E denotes the energy argument of the Green func-tion, r and r are the radial arguments, and θ is the stepfunction. For the point Coulomb potential, the regularand irregular solutions of the radial Dirac equation areexpressed analytically in terms of the Whittaker func-tions of the first and second kind (for explicit formulassee, e.g., Ref. [28]).When the energy argument E is real and greater thanthe electron rest mass E > m , the Dirac Green functionis a complex multi-valued function. So, care should betaken in this case to choose the correct branch (i.e., thesign of the imaginary part) of the Green function. Thebranch of the Green function is fixed by the sign of theinfinitesimal imaginary addition i ε Qa + ω , is greater than the electron mass. Forpositive-energy intermediate states, the energy denomi-nator takes the form ε Qa + ω − ε n + i
0, which implies thatthe energy argument of the Green function has a smallpositive imaginary addition and, therefore, the branchcut of the Green function [ m, ∞ ) should be approachedfrom above.A serious problem arises in the numerical evaluation ofthe radial integrals for the left graph in Fig. 1 (with theelectron correlation modifying the final-state wave func-tion). In this case, the continuum-state Dirac wave func-tion has to be integrated together with the Dirac Greenfunction with the energy argument E > m and the spher-ical Bessel function. All three functions are strongly os-cillating and slowly decreasing for large radial arguments.It is practically impossible to accurately evaluate such in-tegral by a straightforward numerical integration. In thiswork, we use the method of the complex-plane rotationof the integration contour, which was previously appliedby us to the evaluation of the free-free transition integralsin the bremsstrahlung [29].We now consider the evaluation of the problematic ra-dial integrals in more details. Let us introduce the radiusof the atom R , which is defined as the smallest distanceat which all the bound-state electron wave functions van-ish. In the inner region r < R , evaluation of each partial-wave expansion term of the amplitude [Eq. (16)] involvesa three-dimensional radial integration. In the outer re-gion r > R , however, all the integrals involving thebound-state wave functions reach their asymptotical val-ues and only an one-dimensional integral of the free-freetype needs to be evaluated. The general form of suchintegral is J R = Z ∞R dr r j L ( k r ) f iκ ( E , r ) φ ∞ j κ ( E , r ) , (21)where j L is the spherical Bessel function originating fromthe photon propagator in the electron-electron interac-tion, f i is the radial component of the Dirac continuumwave function ( i = 1 , φ ∞ j is the radial componentof the irregular solution of the Dirac equation originatingfrom the Green function ( j = 1 , E = E + k , which leads to theinequality p > p + k , where p , = q E , − m arethe electron momenta associated with the correspondingenergies. We now analytically continue the integrand ofEq. (21) into the complex r plane and take into accountthe asymptotical behaviour of individual functions forlarge values of Im( r ), f κ ( E , r ) ∼ exp [ p | Im( r ) | ] , (22) φ ∞ κ ( E , r ) ∼ exp [ p Im( r )] , (23) j l ( k r ) ∼ exp [ k | Im( r ) | ] , (24)where only the leading exponential behaviour is kept.We observe that, rotating the integration contour inthe integral (21) into the lower complex half-plane r →− ir , we transform the strongly oscillating integrandinto an exponentially decreasing one, which falls off asexp [ − ( p − p − k ) r ] for large r . After the rotation ofthe contour, the integral can be easily evaluated numer-ically up to a desired precision.After the radial integrals are successfully evaluated,the remaining problem is the summation of the partial-wave expansions. Altogether there are five partial-waveexpansions to be delt with: the expansions of the both fi-nal electron wave functions, the Green function, the wavefunction of the absorbed photon, and the photon propa-gator of the electron-electron interaction. After the an-gular momentum selection rules are taken into account,only two partial-wave expansions of the two final-stateelectrons remain unbound. The corresponding expansionparameters are κ and κ in Eq. (14). The convergenceof the resulting double partial-wave expansion is goodfor the photon energies near the threshold ω ∼ ω cr anda non-symmetric energy sharing x ≪ / but graduallydeteriorates when ω increases and x approaches / . Inthe most difficult case considered here, ω/ω cr ≈
30 and x = / , up to ( | κ | , | κ | ) = (15 ,
15) partial waves wereincluded into the calculation. The truncated partial waveexpansion was extrapolated into infinity by using the ǫ resummation algorithm (see, e.g., Ref. [30]).In order to check the numerical procedure, we eval-uated the nonrelativistic limit of our calculations. Theeasiest way to do this is to decrease the value of thefine structure constant (or, equivalently, to increase thevalue of the speed of light) by a large factor. However,a straightforward implementation of this scheme leads tonumerical instabilities, since the relativistic operators in-volve products of the upper and the lower components ofthe Dirac wave function and the lower component van-ishes in the nonrelativistic limit. In order to avoid thisproblem, we express the lower component in terms of theupper component as f κ ( r ) = 12 m (cid:18) ddr + 1 + κr (cid:19) g κ ( r ) , (25)which is valid to the leading order in Zα . After this sub-stitution, our relativistic code yielded a stable numericallimit when the fine structure constant was decreased byorders of magnitude, thus giving us the nonrelativisticlimit. V. RESULTS AND DISCUSSION
We start this section with a discussion of the total crosssection of the double photoionization. As explained pre-viously, our approach accounts for the electron correla-tion to the first order of perturbation theory. Withinthis approximation we perform a rigorous relativistictreatment, without any further simplifications. This ap-proach is expected to yield accurate results for heavyhelium-like ions, whereas for light ions, the precision ofthe method gradually deteriorates with decrease of thenuclear charge. For helium, the relativistic effects areweak but the electron correlation is strong, so that inthis case our approach predictably yields worse resultsthan the modern nonrelativistic methods. However, wewill use the helium case for estimating the higher-orderelectron correlation effects omitted, having in mind thatin helium-like ions these effects are suppressed by theinverse power of the nuclear charge.In Fig. 2, we compare the nonrelativistic limit of ourcalculations of the ratio of the double-to-single photoion-ization cross section R ( ω ) ≡ σ ++ /σ + in helium-like ionswith the experimental data available for helium and withthe theoretical high-energy limits. In order to facili-tate the analysis of the results, we plot the scaled ratio Z R ( ω ) as a function of the ratio ω/ω cr .Firstly, we confirm the known statement [31, 32] thatthe nonrelativistic limit of the scaled ratio R ( ω ) ≡ Z R ( ω/ω cr ) is well described by a universal function thatdoes not depend on the nuclear charge number. We willshow later that this universal scaling is strongly violated Levin [34] Doerner [35] Levin [36] Samson [37]
Z=2, NR Z=40, NR Z=54, NR NR limit, 1/z NR limit, corr Z (cid:1) s (cid:2)(cid:0) / s (cid:3) w / w (cid:4)(cid:5) FIG. 2: (Color online) Comparison of the nonrelativistic limitof the present calculations with the experimental results forhelium and the nonrelativistic theoretical high-energy lim-its. The solid (black) line shows our nonrelativistic resultsfor Z = 2; the dashed (blue) line, for Z = 40; the dashed-dotted (red) line, for Z = 54. (The three lines are practicallyindistinguishable on the picture.) The triangles, squares, anddiamonds represent the experimental results for helium ob-tained in Refs. [34–37], respectively. The dotted line showsthe nonrelativistic high-energy limit calculated to the leadingorder of the perturbation theory [4]. The dash-dot-dotted lineshows the nonrelativistic asymptotical limit calculated withthe fully correlated wave functions [5, 6]. by the relativistic effects. Fig. 2 shows that the numer-ical results calculated for different nuclear charges arepractically indistinguishable from each other in the non-relativistic limit.Secondly, we observe good agreement between our non-relativistic results and the leading term of the 1 /Z expan-sion of the asymptotics [4, 33], R , /Z = 0 . /Z . Thedeviation of our numerical results from the experimentaldata [34–37] is consistent with the deviation of the per-turbative asymptotical value R , /Z from the fully corre-lated result R , corr = 0 . /Z [5, 6]. We, therefore, es-timate the higher-order electron-correlation effects omit-ted in the present treatment to be about 2 /Z ×
30% forthe ratio R ( ω ).We now turn to our relativistic calculations. Fig. 3presents our numerical results obtained with a step-by-step inclusion of individual relativistic effects for severalhelium-like ions. The four different treatments comparedare (i) the nonrelativistic calculation (dotted line), (ii)the calculation with the relativistic wave functions butwith neglecting the retardation in the electron-electroninteraction and all multipoles of the absorbed photonhigher than the relativistic E ω/ω cr .
3, the dominant relativistic effect is the retarda-tion in the electron-electron interaction, whereas for highphoton energies, ω/ω cr &
10, the effect of the highermultipoles of the absorbed photon becomes dominant.Altogether, the relativistic effects are large and changequalitatively the shape of the energy dependence of theratio R ( ω ) already for medium- Z ions. In particular, themaximum of the curve located for helium at ω/ω cr ≈ . Z >
20 and R ( ω ) becomes a monotoni-cally growing function (at least, up to the maximal pho-ton energies accessible in our calculations), which is incontrast to the nonrelativistic case where R ( ω ) graduallydecreases to approach a constant high-energy limit fromabove. It is interesting to note that similarly strong rel-ativistic effects were recently reported for the nonelasticelectron scattering from the hydrogen-like ions [38]. Sig-nificant effects caused by the magnetic dipole and electricquadrupole transitions in the double photoionization ofhelium-like ions were previously reported in Ref. [39].In order to check our relativistic calculations, we em-ployed different gauges for the absorbed photon and,separately, for the photon propagator of the electron-electron interaction. We found that the gauge-dependentterm in the photon absorbtion operator [see Eq. (3)]vanishes in the actual calculations. Independently, wedemonstrated that our calculations with the electron-electron interaction operator in the Feynman and theCoulomb gauges [see Eqs. (18) and (19)] yield the sameresults. This was an important cross-check of the nu-merical procedure since the contributions of the two di-agrams in Fig. 1 are separately not gauge invariant. Asan additional test, we calculated the contribution of theright graph in Fig. 1 independently by a different numeri-cal technique based on the B -spline representation of theDirac Coulomb Green function [40].The results of our relativistic calculations for vari-ous helium-like ions are summarized in Fig. 4. It canbe readily seen that the relativistic effects strongly vio-late the nonrelativistic scaling rule stating that the func-tion R ( ω ) ≡ Z R ( ω/ω cr ) does not depend on the nu-clear charge. The relativistic enhancement of the ratioof the double-to-single photoionization cross sections canbe conveniently parameterized as R ( ω, Z ) = R NR ( ω, Z ) (cid:20) Zα ) ωω cr f rel ( ω, Z ) (cid:21) , (26)where R NR is the nonrelativistic limit of the ratio σ ++ /σ + and f rel is a smooth function of the nuclearcharge Z and photon energy ω . This function is plot-ted for several nuclear charges in Fig. 5. We observethat the numerical values of f rel lay within the intervalof (1 . , .
5) for the wide range of nuclear charge numbersand photon energies. We also found that for Z . f rel ≈ Z -independent. In particular, forhelium and 1 keV photon energy ( ω/ω cr ≈ Z ( Zα ) /σ + and not by Z /σ + as for the total cross section; the additional factoris due to the fact that the integration interval in Eq. (15)is proportional to ( Zα ) . We also confirm the knownstatement [4] that the highly asymmetric energy sharingbetween the two emitted electrons dominates at the highphoton energies, with one of the electrons carrying awaymost of the photon energy. VI. CONCLUSION
In this paper, we presented a relativistic calculation ofthe double photoionization of helium-like atoms. Our ap-proach is based on the partial-wave representation of theDirac continuum states and accounts for the retardationin the electron-electron interaction, the higher-order mul-tipoles of the absorbed photon as well as the interactionof the electrons with the nucleus without any expansionin the binding field. The electron-electron interactionis taken into account to the first order of perturbationtheory. The omitted higher-order electron correlation ef-fects are estimated by comparing our numerical resultsfor atomic helium with the experimental results and theavailable nonrelativistic theory. The calculational resultsare shown to be gauge invariant both with respect ofthe gauge of the absorbed photon and the gauge of theelectron-electron interaction.Our calculation shows that the relativistic effects be-come prominent in the double photoionization cross sec-tion already for medium- Z ions. These effects changethe shape of the energy dependence of the ratio of thedouble-to-single photoionization R ( ω ) drastically. In par-ticular, the well-known constant high-energy asymptoticbehaviour of R ( ω ) in helium gives place to the monotoni- cally growing behaviour in the case of helium-like targetswith Z & K -shell photoionization in neutral atoms. In particular,for the photon energy of 90 keV in silver ( ω/ω cr = 1 . Acknowledgement
The work reported in this paper was supported bythe Helmholtz Gemeinschaft (Nachwuchsgruppe VH-NG-421). The computations were partly performed onthe computer cluster of St. Petersburg State Polytechni-cal University.
Appendix A: Single photoionization: calculationformulas
In order to perform integrations over the angular vari-ables in the matrix element of the photon absorptionoperator, we fix the z -axis of our coordinate system tobe directed along the photon momentum k and assumethat the polarization vector ˆ u has the only nonvanishingspherical component u λ ( λ = ± h κ a µ a | α · ˆ u λ e i k · r | κ n µ n i = ( − X JL i L Π L ( − j n − µ n C Jλj a µ a ,j n − µ n C JλL , λ P JL ( an ) , (A1)where Π L = √ L + 1 and the initial and final states are the Dirac states with given relativistic angular momentumquantum number κ and the momentum projection µ . The radial integrals P JL are P JL ( an ) = Z ∞ dr r j L ( ωr ) (cid:2) g n ( r ) f a ( r ) S JL ( κ n , − κ a ) − f n ( r ) g a ( r ) S JL ( − κ n , κ a ) (cid:3) , (A2)where j L is the spherical Bessel function, g i and f i are the upper and the lower radial components of the Dirac wavefunction, respectively, and S JL are the angular coefficients given by Eqs. (C7)-(C9) of Ref. [42]. The matrix element w (cid:6) w (cid:7)(cid:8) w (cid:9) w (cid:10)(cid:11) w (cid:12) w (cid:13)(cid:14)(cid:15)(cid:16) s (cid:17)(cid:18) (cid:19) s (cid:20) (cid:21) (cid:22) (cid:23)(cid:24) (cid:25) (cid:26) (cid:27)(cid:28) (cid:29) (cid:30) (cid:31) FIG. 3: (Color online) Comparison of the results of the fully relativistic calculations (solid line, black) with results of variousapproximate treatments. The dotted (red) line represents the nonrelativistic limit. The dashed-dotted (blue) line is obtainedwith the relativistic wave functions but without retardation and with including the relativistic E E ω cr = 2 .
56 keV for Ne , ω cr = 10 .
60 keV for Ca , and ω cr = 43 .
75 keV for Zr . Z = 54 Z = 47 Z = 40 Z = 30 Z = 20 Z = 10 Z = 2 Z ! s " / s $ w / w %& FIG. 4: (Color online) Relativistic results for the scaled ratioof the double photoionization cross section σ ++ to the sin-gle photoionization cross section σ + , for helium-like ions withdifferent nuclear charges Z , as a function of the photon en-ergy ω divided by the threshold photon energy of the doublephotoionization ω cr . of the gauge-dependent part of the photon absorption operator can be evaluated as h κ a µ a | (cid:0) α · ˆ k − (cid:1) e i k · r | κ n µ n i = ( − X JL i L Π L ( − j n − µ n C J j a µ a ,j n − µ n (cid:20) i C J L , P JL ( an ) + δ J,L C L ( κ a , κ n ) R L ( an ) (cid:21) , (A3) Z = 10 Z = 20 Z = 30 Z = 40 Z = 54 f ’() ( w ) w / w *+ FIG. 5: (Color online) Relativistic enhancement function f rel defined by Eq. (26), for different helium-like ions. ,- w . w /0 1 234 56 a
78 9: s ;< => e ?@ s A B w C w DE F G
H I JKL M NOP Q RST U V w W w XY Z [
FIG. 6: (Color online) Normalized energy distribution of the ejected electrons as a function of the energy sharing parameter x defined by Eqs. (12) and (13), for different nuclear charge numbers and photon energies. where the angular coefficients C L are given by Eq. (C10) of Ref. [42] and R L is the radial integral R L ( an ) = Z ∞ dr r j L ( ωr ) (cid:2) g n ( r ) g a ( r ) + f n ( r ) f a ( r ) (cid:3) . (A4) Appendix B: Double photoionization: calculationformulas
Let us write the amplitude (16) as τ ++ λ = τ ab − τ ab − τ ba + τ ba , (B1)where τ ab corresponds to the part of Eq. (16) with P ε P ε = ε ε and QaQb = ab and the remainingthree terms are obtained by permutations. The contri- butions due to the two terms in the braces of Eq. (16)will be denoted by subscripts A and B , respectively, τ ab = τ ab,A + τ ab,B .The calculation formulas take the simplest form in thecase when the initial electron state is the ground stateof the atom. In this case, the permutation over the a and b electrons yields just a combinatorial factor andall summations over the momentum projections can beevaluated in the closed form. The result for the single0differential cross section is dσ ++ dε = 4 π αω X κ κ J (cid:12)(cid:12)(cid:12)(cid:12) (cid:2)e τ ab + ( − j − j + J e τ ab (cid:3)(cid:12)(cid:12)(cid:12)(cid:12) . (B2) The amplitude is e τ ab = e τ ab,A + e τ ab,B , e τ ab,A = α √ X LL ′ κ n ( − i L Π L C JλL , λ ( − L ′ + J + j n + j √ (cid:26) j j J j n L ′ (cid:27) X n R L ′ (∆ , ε ε nb ) P JL ( na ) ε a + ω − ε n , (B3)and e τ ab,B = α √ X LL ′ κ n ( − i L Π L C JλL , λ δ j n ,j √ j + 1) X n P JL ( ε n ) R L ′ (∆ , nε ab ) ε − ω − ε n , (B4)where R L (∆ , abcd ) is the relativistic generalization of theSlater integral for the electron-electron interaction given by Eqs. (C1)-(C10) of Ref. [42] and ∆ = ε − ε b . [1] J. S. Briggs and V. Schmidt, J. Phys. B , R1 (2000).[2] H. Kossmann, V. Schmidt, and T. Andersen, Phys. Rev.Lett. , 1266 (1988).[3] F. Maulbetsch and J. S. Briggs, Phys. Rev. Lett. ,2004 (1992).[4] M. Y. Amucia, E. G. Drukarev, V. G. Gorshkov, andM. P. Kazachkov, J. Phys. B , 1248 (1975).[5] L. R. Andersson and J. Burgd¨orfer, Phys. Rev. Lett. ,50 (1993).[6] R. C. Forrey, H. R. Sadeghpour, J. D. Baker, J. D. Mor-gan, and A. Dalgarno, Phys. Rev. A , 2112 (1995).[7] J.-Z. Tang and I. Shimamura, Phys. Rev. A , R3413(1995).[8] A. S. Kheifets and I. Bray, Phys. Rev. A , R995 (1996).[9] K. W. Meyer, C. H. Greene, and B. D. Esry, Phys. Rev.Lett. , 4902 (1997).[10] Y. Qiu, J.-Z. Tang, J. Burgd¨orfer, and J. Wang, Phys.Rev. A , R1489 (1998).[11] E. P. Kanter, R. W. Dunford, B. Kr¨assig, and S. H.Southworth, Phys. Rev. Lett. , 508 (1999).[12] S. H. Southworth, E. P. Kanter, B. Kr¨assig, L. Young,G. B. Armen, J. C. Levin, D. L. Ederer, and M. H. Chen,Phys. Rev. A , 062712 (2003).[13] E. P. Kanter, I. Ahmad, R. W. Dunford, D. S. Gemmell,B. Kr¨assig, S. H. Southworth, and L. Young, Phys. Rev.A , 022708 (2006).[14] J. Hoszowska, A. K. Kheifets, J.-C. Dousse, M. Berset,I. Bray, W. Cao, K. Fennane, Y. Kayser, M. Kavˇciˇc,J. Szlachetko, and M. Szlachetko, Phys. Rev. Lett. ,073006 (2009).[15] J. Hoszowska, J.-C. Dousse, W. Cao, K. Fennane,Y. Kayser, M. Szlachetko, J. Szlachetko, and M. Kavˇciˇc,Phys. Rev. A , 063408 (2010).[16] S. W. Epp, J. R. C. L´opez-Urrutia, G. Brenner,V. M¨ackel, P. H. Mokler, R. Treusch, M. Kuhlmann,M. V. Yurkov, J. Feldhaus, J. R. Schneider, M. Wellh¨ofer, M. Martins, W. Wurth, and J. Ullrich, Phys. Rev. Lett. , 183001 (2007).[17] A. Simon, A. Warczak, T. ElKafrawy, and J. A. Tanis,Phys. Rev. Lett. , 123001 (2010).[18] G. Bednarz, D. Sierpowski, T. St¨ohlker, A. Warczak,H. Beyer, F. Bosch, A. Br¨auning-Demian, H. Br¨auning,X. Cai, A. Gumberidze, S. Hagmann, C. Kozhuharov,D. Liesen, X. Ma, P. Mokler, A. Muthig, Z. Stachura,and S. Toleikis, Nucl. Instrum. Methods B 205 , 573(2003), 11th International Conference on the Physics ofHighly Charged Ions.[19] J. Eichler and W. Meyerhof,
Relativistic Atomic Colli-sions , Academic Press, San Diego, 1995.[20] M. E. Rose,
Relativistic Electron Theory , John Wiley &Sons, NY, 1961.[21] J. Eichler and T. St¨ohlker, Phys. Rev. , 1 (2007).[22] A. Mikhailov, A. Nefiodov, and G. Plunien, Phys. Lett.A , 211 (2006).[23] V. M. Shabaev, Phys. Rep. , 119 (2002).[24] A. I. Mikhailov, I. A. Mikhailov, A. N. Moskalev, A. V.Nefiodov, G. Plunien, and G. Soff, Phys. Rev. A ,032703 (2004).[25] V. M. Shabaev, V. A. Yerokhin, T. Beier, and J. Eichler,Phys. Rev. A , 052112 (2000).[26] V. A. Yerokhin, V. M. Shabaev, T. Beier, and J. Eichler,Phys. Rev. A , 042712 (2000).[27] V. A. Yerokhin and A. Surzhykov, Phys. Rev. A ,062703 (2010).[28] P. J. Mohr, G. Plunien, and G. Soff, Phys. Rep. ,227 (1998).[29] V. A. Yerokhin and A. Surzhykov, Phys. Rev. A ,062702 (2010).[30] E. Caliceti, M. Meyer-Hermann, P. Ribeca,A. Surzhykov, and U. Jentschura, Phys. Rep. ,1 (2007).[31] M. A. Kornberg and J. E. Miraglia, Phys. Rev. A , , 231003 (2009).[33] A. I. Mikhailov and I. A. Mikhailov, Zh. Eksp. Teor. Fiz. , 1537 (1998), [Sov. Phys. JETP , 833 (1998)].[34] J. C. Levin, G. B. Armen, and I. A. Sellin, Phys. Rev.Lett. , 1220 (1996).[35] R. D¨orner, T. Vogt, V. Mergel, H. Khemliche, S. Kravis,C. L. Cocke, J. Ullrich, M. Unverzagt, L. Spielberger,M. Damrau, O. Jagutzki, I. Ali, B. Weaver, K. Ullmann,C. C. Hsu, M. Jung, E. P. Kanter, B. Sonntag, M. H.Prior, E. Rotenberg, J. Denlinger, T. Warwick, S. T.Manson, and H. Schmidt-B¨ocking, Phys. Rev. Lett. ,2654 (1996).[36] J. C. Levin, I. A. Sellin, B. M. Johnson, D. W. Lindle,R. D. Miller, N. Berrah, Y. Azuma, H. G. Berry, andD.-H. Lee, Phys. Rev. A , R16 (1993). [37] J. A. R. Samson, W. C. Stolte, Z.-X. He, J. N. Cutler,Y. Lu, and R. J. Bartlett, Phys. Rev. A , 1906 (1998).[38] C. J. Bostock, D. V. Fursa, and I. Bray, Phys. Rev. A , 052708 (2009).[39] A. I. Mikhailov, I. A. Mikhailov, A. N. Moskalev, A. V.Nefiodov, G. Plunien, and G. Soff, Phys. Lett. A ,395 (2003).[40] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plu-nien, and G. Soff, Phys. Rev. Lett. , 130405 (2004).[41] L. Spielberger, O. Jagutzki, R. D¨orner, J. Ullrich,U. Meyer, V. Mergel, M. Unverzagt, M. Damrau,T. Vogt, I. Ali, K. Khayyat, D. Bahr, H. G. Schmidt,R. Frahm, and H. Schmidt-B¨ocking, Phys. Rev. Lett. , 4615 (1995).[42] V. A. Yerokhin and V. M. Shabaev, Phys. Rev. A60