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Correlated double-electron capture with emission of a single photon
Physics Letters A 328 (2004) 350–356 www.elsevier.com/locate/pla
Correlated double-electron capture with emission of a single photon A.I. Mikhailov a,b , I.A. Mikhailov a , A.V. Nefiodov a,b,∗ , G. Plunien b , G. Soff b a Petersburg Nuclear Physics Institute, 188300 Gatchina, St. Petersburg, Russia b Institut für Theoretische Physik, Technische Universität Dresden, Mommsenstraße 13, D-01062 Dresden, Germany
Received 4 June 2004; accepted 12 June 2004
Communicated by V.M. Agranovich
Abstract We investigate the correlated double-electron capture into the K shell of bare ions with emission of a single photon. In the energy region corresponding to the double-photoionization threshold, the total cross section of the process and the ratio of cross sections for the double-to-single electron capture are calculated for light ions, taking into account the leading orders of the 1/Z and αZ expansions. Available theoretical results and experimental perspectives are discussed. 2004 Published by Elsevier B.V. PACS: 34.80.Lx; 32.80.Fb; 32.80.-t; 31.25.-v; 32.30.Rj
Double photoionization of an atom caused by the absorption of a single photon, the so-called double photoeffect, is a fundamental phenomenon, in which correlation effects play a crucial role [1–3]. Since the photon interacts only with a single electron, the simultaneous ejection of two electrons is exclusively caused by the electron–electron interaction. So far experiments have been performed with helium [4–6] and few neutral atoms with moderate values of the nuclear charge number Z [7–11]. The majority of investiga-
* Corresponding author.
E-mail address: [email protected] (A.V. Nefiodov). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.06.042
tions concerns with the energy dependence of the ratio of double-to-single photoionization cross sections. In contrast to neutral atoms, the study of the double photoeffect in multicharged ions is of particular interest. The photoionization processes on the simplest many-electron systems such as heliumlike ions allow for a rigorous treatment within the framework of QED perturbation theory [12–15]. This problem could also serve as a testing ground for numerous theoretical approaches developed recently for the description of the complete photofragmentation of helium. An experimental study of the double photoeffect via the timereversed process has been undertaken in fast collisions of bare ions with light target atoms [16–18]. To iden-
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351
Fig. 1. Feynman diagrams for the double-electron capture into the atomic K shell with the emission of a single photon. Solid lines denote electrons in the Coulomb field of the nucleus, dashed line denotes the electron–electron Coulomb interaction, and the wavy line denotes an outgoing photon. The line with a heavy dot corresponds to the Coulomb Green’s function. Diagram (a) takes into account the electron–electron interaction in the initial state, while diagram (b) accounts for it in the final state.
tify the correlated simultaneous capture of two electrons, the X-ray photon emission has been observed in coincidence with double charge exchange. However, several measurements, which have been performed at GSI in Darmstadt, have provided only upper limits for the total cross sections of the process under investigation. Moreover, the experiments carried out with bare uranium ions are in contradiction with presently available theoretical predictions [18,19]. The advantage of studying of the double photoeffect via its timereversed version utilizing beams of multicharged ions is that it opens the possibility to investigate the process involving excited electron states. In fast ion–atom collisions, the most dominant charge–exchange mechanism is the radiative electron capture with emission of a single photon [20]. The latter can be viewed as the time-reversed single photoionization. Employing the principle of detailed balance, the cross section for the direct process can be related with that for the converse one [21]. In the case of collisions of fast multicharged ions with light target atoms, the characteristic velocity of target K-shell electrons is supposed to be small with respect to that of the incident ion, that is, the target electrons captured by the projectile can be treated as quasifree. The peculiarity of the kinematics for double-electron capture with emission of a single photon studied experimentally in Refs. [16–18] is that initially both electrons have identical momenta, both in magnitude and direction. In the double photoeffect both photoelectrons can be ejected with arbitrary energy sharing in different solid angles. In this Letter, we investigate the correlated doubleelectron capture into the K shell of a bare ion with the emission of a single photon. The study is performed for the nonrelativistic domain of photon energies ω characterized by I2K ω m, where I2K is
the threshold energy for double ionization from the K shell and m is the electron mass (h¯ = c = 1). Since the nucleus of an ion is sufficiently heavy, it is assumed to be a source of an external Coulomb field. All electrons involved in the process are considered as being nonrelativistic. This implies the smallness of the Coulomb parameter, that is, αZ 1, where α is the fine-structure constant. We employ perturbation theory with respect to the electron–electron interaction. Our approach is presented in more details in Ref. [15]. In zeroth-order approximation, the wave functions of the initial and final electron states are taken as products of hydrogenlike Coulomb wave functions, while the intermediate electron is described by the Coulomb Green’s function [22,23]. The dominant contribution to the process under investigation is indicated by the Feynman graphs depicted in Fig. 1. We shall work in the reference frame connected with the nucleus of the incident ion. The probability for double-electron capture into the K shell of a bare ion with emission of a single photon per unit time is given by dW =
dk 2π |A|2 δ(2Ep − ω + I2K ), V2 (2π)3
(1)
where Ep denotes the one-electron energy in the initial continuum state, ω = |k| = k is the energy of the emitted photon, I2K = 2I with I = η2 /(2m) being the Coulomb potential for single ionization, and η = mαZ is the characteristic momentum of a K-shell electron. Expression (1) assumes summation over the polarizations e of the photon. The delta function ensures energy conservation. The one-electron wave functions of the continuous spectrum imply the normalization condition for one particle per volume V . The amplitude A is obtained from that for the double K-shell photo-
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effect by reversing the signs of the momenta of the photon (k → −k) and of the electrons (p → −p), together with complex conjugation of the polarization vector (e → e∗ ) [15]. Due to the symmetry with respect to time reversal |A|2 remains the same for the direct and the converse processes [21]. In accordance with the definition of the effective cross section, Eq. (1) should be divided by the current flux of incident electrons. The latter is given by j = v/V , where v = p/m is the absolute magnitude of the initial velocity of the incident electrons before their collision with the nucleus. Accordingly, the differential cross section for the correlated double-electron capture can be expressed as dσ (2) = 2π
ω2 dΩk , |A|2 vV (2π)3
(2)
and defines the angular distribution of photons emitted into an element of the solid angle dΩk . The normalization volume V remaines in the expression (2), since we are dealing with a three-body collision problem. As it follows from the theoretical analysis, the total cross section of the double K-shell photoeffect reaches its maximum in the near-threshold domain, while the emission of electrons with arbitrary energy sharing is of comparable importance there [15]. Far beyond the double-photoionization threshold, the photon energy becomes distributed very nonuniformly among the outgoing electrons [12]. Accordingly, the region of the photon energies ω ∼ I2K seems to be the most favourable for the study of the time-reversed process. In addition, since ω η, the total cross sections can be fairly well calculated within the framework of the electric dipole approximation (k = 0) [14,15]. The angular dependence can be merely reduced to the common factor (e∗ · p) in the amplitude A. Within the dipole regime, the law of momentum conservation implies 2p q, where q is the recoil momentum transferred to the nucleus. The nucleus absorbs the change of momenta of both captured electrons. For electrons with momenta p ∼ q ∼ η, the process of correlated double-electron capture into the K shell of the bare ion occurs most efficiently at atomic distances of the order of the K-shell radius. The case of fast collisions of a multicharged ion with a light target atom implies also that Z Zt holds, where Zt is the nuclear charge number of the target atom.
In ion–atom collisions the normalization volume V is characteristic for the target atom and corresponds to an effective localization volume of the two electrons captured by the incident ion. The volume V can −2 |ψ(r)|2 dr, where ψ(r) is be defined via V = ψmax the wave function of a single target electron, which reaches the maximum value ψmax . For a hydrogenlike ion in its ground state one has V = π(a0 /Zt )3 , where a0 = 1/(mα) denotes the Bohr radius. Usually, one evaluates the partial contributions to the cross section resulting from different electron configurations of the target atom. Since the triplet–singlet transition amplitudes turn out to be strongly suppressed, it is sufficient to focus on the singlet–singlet transitions only [15]. In many-electron atoms, the density distributions of electrons in different shells vary considerably. The (1s)2 configuration is packed most closely. Accordingly, one can assume that correlated doubleelectron capture into the K shell of an incident bare ion with emission of a single photon occurs predominantly from the 11 S0 state of the light target atom. In a first approximation, we neglect the electron screening and the partial contributions from electrons in a target atom with principal quantum numbers n 2. The normalization volume V can be calibrated in units of V0 = (a0 /Zt )3 . For the K-shell, the dimensionless volume V = V /V0 is given by V = π . Since both K-shell target electrons are prepared in the appropriate spin state, there is no need to average of the cross section (2) over the initial electron states. Integration over angles and summation over polarizations of the ejected photon yields σ (2) = σ0
219 Zt3 Q(ξ ), 3VZ 5
(3)
where ξ = η/p, σ0 = α 3 a02 , and V = π . Within the dipole approximation, Q is a universal function of the dimensionless variable ξ , which is obtained by numerical integration only (see Fig. 2). The near-threshold domain corresponds to the values ξ ∼ 1. In the case of fast collisions (ξ 1), the cross section (3) decreases rapidly. This is connected with rather unfortunate kinematics of the process. In the direct double photoeffect, it corresponds to the emission of both electrons with equal-energy sharing due to absorption of a highenergy photon. For slow collisions (ξ 1), the cross section σ (2) increases. However, then the consistent
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incident ion can capture one singlet pair in the volume V1 and 7 singlet pairs in the volume V2 . As a result, we obtain V1−1 + 7V2−1 = (0.857 + 0.139)V −1 V −1 . This means that the cross section under consideration coincides with that of the correlated double-electron capture from the K shell of heliumlike carbon, which is calculated without taking into account the electron screening effects, that is, for γ = 0. Another quantity, which is of experimental interest, is the ratio R = σ (2) /σ (1) of cross sections for the double-to-single electron capture with emission of a single photon. In collisions of bare ions with target atoms, the cross section σ (1) of the radiative electron capture is related with that σ + for the single photoeffect via the principle of detailed balance yielding [25] σ (1) = Zt
Fig. 2. The universal quantity Q(ξ ) is calculated within the dipole approximation as a function of the dimensionless variable ξ = η/p. Here η = mαZ and p is the absolute magnitude of the momentum of the electrons at large distances from the bare ion.
consideration requires a rigorous description of the influence of the nuclear field of the target atom. Considering different electron shells of the target atom with taking into account screening effects, one can define the corresponding effective volumes Vn employing Slater’s wave functions according to Vn = π(a0 n/Zeff )3 , where the effective nuclear charge is given by Zeff = Zt − γ [24]. The screening correction γ 0 depends on both the electron state and the electron configuration. The dimensionless volume reads as Vn = V(nZt /Zeff )3 . For n 4 one should introduce an effective principal quantum number. If the electrons are captured from different electron shells, the normalization volume Vn is defined as an effective volume for localization of that electron, which is farther from the nucleus. Although an incident ion can capture a rather large amount of different singlet pairs of target electrons, the corresponding contributions to the cross section turn out to be much smaller than that of the (1s)2 configuration. For example, in a carbon (target) atom, there are two singlet pairs, (1s)2 and (2s)2 , and one triplet pair, (2p)2 , with screening corrections γ1s = 0.3 and γ2s = γ2p = 2.75, respectively [24]. The
2ω12 + σ , p2
(4)
where p is the momentum of the incident electron. The factor Zt takes into account the possibility for capture of any electron of the target atom. The cross section σ + corresponds to the photoabsorption by one electron only. The energy ω1 of the emitted photon is half of that in the correlated double-electron capture, that is, ω1 = ω/2 = Ep + I . In the dipole regime, the nonrelativistic expression for σ + is known analytically [26]. For the radiative electron capture into the K shell of the bare ion, we have 210 2 π σ0 Zt H (ξ ), 3 ξ 2 exp(−4ξ cot−1 ξ ) , H (ξ ) = 2 εγ [1 − exp(−2πξ )] σ (1) =
(5) (6)
where the dimensionless energy εγ = ω/I = 2(1 + ξ −2 ) has been introduced. Then the ratio R = σ (2) / σ (1) is given by R=
29 Zt2 Q(ξ ) . π 2 VZ 5 H (ξ )
(7)
The function Q(ξ )/H (ξ ) exhibits a universal dependence on the dimensionless parameter ξ (see Fig. 3). This is due to employing the dipole approximation and taking into account the leading orders of perturbation theory. In Table 1, we present a comparison of our calculations with available experimental data. This reveals a significant disagreement between our predictions and
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Table 1 For bare ions with nuclear charge numbers Z, the kinetic energies per unit mass, the dimensionless parameters ξ , the nuclear charge numbers Zt of target atoms, the cross sections σ (1) for the radiative electron capture [according to Eq. (5)], the cross sections σ (2) for the correlated double-electron capture with the emission of a single photon [according to Eq. (3)], and the ratios R [according to Eq. (7)] are tabulated Z 18 92
Energy (MeV/u) 11.4 297
ξ 0.840 0.841
Zt 6 18
Fig. 3. The universal quantity Q(ξ )/H (ξ ) is calculated as a function of the dimensionless variable ξ = η/p.
those reported in Ref. [19], where the cross section σ (2) does not depend on the particular choice of target atoms. In Ref. [19], it was also argued that the total cross section grows rapidly in the relativistic domain due to small characteristic distances Rch involved in the process of the correlated double-electron capture. The nonrelativistic wave functions are enhanced by the 2 factor F = (a0 /Rch )(αZ) /2 , while the characteristic spatial scale is defined by the minimal recoil momen−1 . tum qmin transferred to the nucleus, that is, Rch ∼ qmin For the process under consideration, it can be shown that qmin = |2p − ω| ∼ m even in the ultrarelativistic limit. For heavy multicharged ions, this is also rather close to the average momentum η of a K-shell elec-
σ (1) (kb) 0.36 1.08
σ (2) (mb)
R
This work
Experiment
3.2 2.5 × 10−2
5.2 [16] 10 [18]
0.9 × 10−5 2.3 × 10−8
tron. Accordingly, the characteristic distances Rch are of the order of the Compton wave length of the elec2 tron. Then for Z = 92 the factor F = α −(αZ) /2 ∼ 3 gives rise to the “enhancement” of the total cross section σ (2) by a factor F 8 ∼ 6 × 103 , which is at least three orders of magnitude less than the value obtained in Ref. [19]. Moreover, it is difficult to expect even such an enhancement merely by analogy with the single photoeffect. Indeed, according to Ref. [19], the cross section σ (1) would be multiplied by F 4 ∼ 80. However, the relativistic enhancement is known to be absent. For example, the measurement of the radiative electron capture by bare uranium ions at an energy of 295 MeV/u in collisions with a gaseous Ar target leads to σ (1) = 1069 ± 321 b [20], which is in fair agreement with the nonrelativistic prediction according to Eq. (5) (see Table 1). The agreement becomes even better for lower-energy collisions and for ions with moderate values of Z. In contrast to ion–atom collisions, one can expect much larger values for the cross section σ (2) in the case of slow collisions of multicharged ions with the solid-state target. The electrons of the valence zone behave itself as quasifree particles with a characteristic velocity of motion determined by the temperature of the target. This velocity is much smaller than that of the projectile even for ξ ∼ 10. Therefore, in the reference frame connected with the nucleus of the incident ion, the electrons of the valence zone appear as an electron beam characterized by the absolute magnitude of the velocity v and the concentration ne = κρt NA /Mt , where κ is the number of valence electrons in the target atom, NA is Avogadro’s number, and ρt and Mt are the density and the molar mass of the target, respectively. The cross section σ (2) for the correlated double-electron capture from the valence zone looks similar to Eq. (2) with the substitution of the nor-
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malization volume V by n−1 e . Finally, we obtain 219 σ0 σ (2) = ne a03 (8) Q(ξ ). 3Z 5 Let us estimate the cross section (8) for collisions of bare Ar ions with the 9 Be target at ξ = 10. In this case, we have κ = 2, ρt = 1.85 g/cm3 , Mt = 9 g/mol, ne = 2.46 × 1023 cm−3 , Q(ξ ) = 1.63, and σ (2) = 0.12 b. The knowledge of the function Q(ξ ) given in Fig. 2 provides a rather wide range for a choice for optimal experimental conditions. Concluding, we have performed nonrelativistic calculations of the total cross section for the correlated double-electron capture into the K shell of a bare ion with emission of a single photon. At present, the experimental observation of the process on beams of heavy multicharged ions still could not be realized. However, for light ions, the cross sections are expected to be close to the upper experimental limit. Therefore, it is highly desirable to measure again precisely the total cross section. For example, for the experiment described in Ref. [16], the observation of the process can become feasible, if one slightly decelerates the beam of bare Ar ions. The experimental conditions of Ref. [16] also allow for the investigation of correlated double-electron capture into excited electron states of the bare ion. Calculations of total cross sections for the double photoeffect [27–29] suggest that the cross sections for the correlated two-electron capture into the singlet n1 S0 configurations with n 2 are a few times larger than that into the ground 11 S0 state. The novel technique of deceleration of multicharged ions planned at GSI can be applied in order to perform such experiments in the low-energy regime.
Acknowledgements We thank A. Warczak for useful communications. This research was supported in part by INTAS (Grant No. 03-54-3604). A.M. and A.N. are grateful to the Dresden University of Technology for the hospitality and for financial support from GSI. G.S. and G.P. acknowledge financial support from BMBF, DFG, and GSI. A.M., I.M., and A.N. are supported by RFBR (Grants Nos. 01-02-17246 and 00-15-96610). A.N. acknowledges support from the Alexander von Humboldt Foundation.
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Correlated double-electron capture with emission of a single photon A.I. Mikhailov a,b , I.A. Mikhailov a , A.V. Nefiodov a,b,∗ , G. Plunien b , G. Soff b a Petersburg Nuclear Physics Institute, 188300 Gatchina, St. Petersburg, Russia b Institut für Theoretische Physik, Technische Universität Dresden, Mommsenstraße 13, D-01062 Dresden, Germany
Received 4 June 2004; accepted 12 June 2004
Communicated by V.M. Agranovich
Abstract We investigate the correlated double-electron capture into the K shell of bare ions with emission of a single photon. In the energy region corresponding to the double-photoionization threshold, the total cross section of the process and the ratio of cross sections for the double-to-single electron capture are calculated for light ions, taking into account the leading orders of the 1/Z and αZ expansions. Available theoretical results and experimental perspectives are discussed. 2004 Published by Elsevier B.V. PACS: 34.80.Lx; 32.80.Fb; 32.80.-t; 31.25.-v; 32.30.Rj
Double photoionization of an atom caused by the absorption of a single photon, the so-called double photoeffect, is a fundamental phenomenon, in which correlation effects play a crucial role [1–3]. Since the photon interacts only with a single electron, the simultaneous ejection of two electrons is exclusively caused by the electron–electron interaction. So far experiments have been performed with helium [4–6] and few neutral atoms with moderate values of the nuclear charge number Z [7–11]. The majority of investiga-
* Corresponding author.
E-mail address: [email protected] (A.V. Nefiodov). 0375-9601/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physleta.2004.06.042
tions concerns with the energy dependence of the ratio of double-to-single photoionization cross sections. In contrast to neutral atoms, the study of the double photoeffect in multicharged ions is of particular interest. The photoionization processes on the simplest many-electron systems such as heliumlike ions allow for a rigorous treatment within the framework of QED perturbation theory [12–15]. This problem could also serve as a testing ground for numerous theoretical approaches developed recently for the description of the complete photofragmentation of helium. An experimental study of the double photoeffect via the timereversed process has been undertaken in fast collisions of bare ions with light target atoms [16–18]. To iden-
A.I. Mikhailov et al. / Physics Letters A 328 (2004) 350–356
351
Fig. 1. Feynman diagrams for the double-electron capture into the atomic K shell with the emission of a single photon. Solid lines denote electrons in the Coulomb field of the nucleus, dashed line denotes the electron–electron Coulomb interaction, and the wavy line denotes an outgoing photon. The line with a heavy dot corresponds to the Coulomb Green’s function. Diagram (a) takes into account the electron–electron interaction in the initial state, while diagram (b) accounts for it in the final state.
tify the correlated simultaneous capture of two electrons, the X-ray photon emission has been observed in coincidence with double charge exchange. However, several measurements, which have been performed at GSI in Darmstadt, have provided only upper limits for the total cross sections of the process under investigation. Moreover, the experiments carried out with bare uranium ions are in contradiction with presently available theoretical predictions [18,19]. The advantage of studying of the double photoeffect via its timereversed version utilizing beams of multicharged ions is that it opens the possibility to investigate the process involving excited electron states. In fast ion–atom collisions, the most dominant charge–exchange mechanism is the radiative electron capture with emission of a single photon [20]. The latter can be viewed as the time-reversed single photoionization. Employing the principle of detailed balance, the cross section for the direct process can be related with that for the converse one [21]. In the case of collisions of fast multicharged ions with light target atoms, the characteristic velocity of target K-shell electrons is supposed to be small with respect to that of the incident ion, that is, the target electrons captured by the projectile can be treated as quasifree. The peculiarity of the kinematics for double-electron capture with emission of a single photon studied experimentally in Refs. [16–18] is that initially both electrons have identical momenta, both in magnitude and direction. In the double photoeffect both photoelectrons can be ejected with arbitrary energy sharing in different solid angles. In this Letter, we investigate the correlated doubleelectron capture into the K shell of a bare ion with the emission of a single photon. The study is performed for the nonrelativistic domain of photon energies ω characterized by I2K ω m, where I2K is
the threshold energy for double ionization from the K shell and m is the electron mass (h¯ = c = 1). Since the nucleus of an ion is sufficiently heavy, it is assumed to be a source of an external Coulomb field. All electrons involved in the process are considered as being nonrelativistic. This implies the smallness of the Coulomb parameter, that is, αZ 1, where α is the fine-structure constant. We employ perturbation theory with respect to the electron–electron interaction. Our approach is presented in more details in Ref. [15]. In zeroth-order approximation, the wave functions of the initial and final electron states are taken as products of hydrogenlike Coulomb wave functions, while the intermediate electron is described by the Coulomb Green’s function [22,23]. The dominant contribution to the process under investigation is indicated by the Feynman graphs depicted in Fig. 1. We shall work in the reference frame connected with the nucleus of the incident ion. The probability for double-electron capture into the K shell of a bare ion with emission of a single photon per unit time is given by dW =
dk 2π |A|2 δ(2Ep − ω + I2K ), V2 (2π)3
(1)
where Ep denotes the one-electron energy in the initial continuum state, ω = |k| = k is the energy of the emitted photon, I2K = 2I with I = η2 /(2m) being the Coulomb potential for single ionization, and η = mαZ is the characteristic momentum of a K-shell electron. Expression (1) assumes summation over the polarizations e of the photon. The delta function ensures energy conservation. The one-electron wave functions of the continuous spectrum imply the normalization condition for one particle per volume V . The amplitude A is obtained from that for the double K-shell photo-
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effect by reversing the signs of the momenta of the photon (k → −k) and of the electrons (p → −p), together with complex conjugation of the polarization vector (e → e∗ ) [15]. Due to the symmetry with respect to time reversal |A|2 remains the same for the direct and the converse processes [21]. In accordance with the definition of the effective cross section, Eq. (1) should be divided by the current flux of incident electrons. The latter is given by j = v/V , where v = p/m is the absolute magnitude of the initial velocity of the incident electrons before their collision with the nucleus. Accordingly, the differential cross section for the correlated double-electron capture can be expressed as dσ (2) = 2π
ω2 dΩk , |A|2 vV (2π)3
(2)
and defines the angular distribution of photons emitted into an element of the solid angle dΩk . The normalization volume V remaines in the expression (2), since we are dealing with a three-body collision problem. As it follows from the theoretical analysis, the total cross section of the double K-shell photoeffect reaches its maximum in the near-threshold domain, while the emission of electrons with arbitrary energy sharing is of comparable importance there [15]. Far beyond the double-photoionization threshold, the photon energy becomes distributed very nonuniformly among the outgoing electrons [12]. Accordingly, the region of the photon energies ω ∼ I2K seems to be the most favourable for the study of the time-reversed process. In addition, since ω η, the total cross sections can be fairly well calculated within the framework of the electric dipole approximation (k = 0) [14,15]. The angular dependence can be merely reduced to the common factor (e∗ · p) in the amplitude A. Within the dipole regime, the law of momentum conservation implies 2p q, where q is the recoil momentum transferred to the nucleus. The nucleus absorbs the change of momenta of both captured electrons. For electrons with momenta p ∼ q ∼ η, the process of correlated double-electron capture into the K shell of the bare ion occurs most efficiently at atomic distances of the order of the K-shell radius. The case of fast collisions of a multicharged ion with a light target atom implies also that Z Zt holds, where Zt is the nuclear charge number of the target atom.
In ion–atom collisions the normalization volume V is characteristic for the target atom and corresponds to an effective localization volume of the two electrons captured by the incident ion. The volume V can −2 |ψ(r)|2 dr, where ψ(r) is be defined via V = ψmax the wave function of a single target electron, which reaches the maximum value ψmax . For a hydrogenlike ion in its ground state one has V = π(a0 /Zt )3 , where a0 = 1/(mα) denotes the Bohr radius. Usually, one evaluates the partial contributions to the cross section resulting from different electron configurations of the target atom. Since the triplet–singlet transition amplitudes turn out to be strongly suppressed, it is sufficient to focus on the singlet–singlet transitions only [15]. In many-electron atoms, the density distributions of electrons in different shells vary considerably. The (1s)2 configuration is packed most closely. Accordingly, one can assume that correlated doubleelectron capture into the K shell of an incident bare ion with emission of a single photon occurs predominantly from the 11 S0 state of the light target atom. In a first approximation, we neglect the electron screening and the partial contributions from electrons in a target atom with principal quantum numbers n 2. The normalization volume V can be calibrated in units of V0 = (a0 /Zt )3 . For the K-shell, the dimensionless volume V = V /V0 is given by V = π . Since both K-shell target electrons are prepared in the appropriate spin state, there is no need to average of the cross section (2) over the initial electron states. Integration over angles and summation over polarizations of the ejected photon yields σ (2) = σ0
219 Zt3 Q(ξ ), 3VZ 5
(3)
where ξ = η/p, σ0 = α 3 a02 , and V = π . Within the dipole approximation, Q is a universal function of the dimensionless variable ξ , which is obtained by numerical integration only (see Fig. 2). The near-threshold domain corresponds to the values ξ ∼ 1. In the case of fast collisions (ξ 1), the cross section (3) decreases rapidly. This is connected with rather unfortunate kinematics of the process. In the direct double photoeffect, it corresponds to the emission of both electrons with equal-energy sharing due to absorption of a highenergy photon. For slow collisions (ξ 1), the cross section σ (2) increases. However, then the consistent
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353
incident ion can capture one singlet pair in the volume V1 and 7 singlet pairs in the volume V2 . As a result, we obtain V1−1 + 7V2−1 = (0.857 + 0.139)V −1 V −1 . This means that the cross section under consideration coincides with that of the correlated double-electron capture from the K shell of heliumlike carbon, which is calculated without taking into account the electron screening effects, that is, for γ = 0. Another quantity, which is of experimental interest, is the ratio R = σ (2) /σ (1) of cross sections for the double-to-single electron capture with emission of a single photon. In collisions of bare ions with target atoms, the cross section σ (1) of the radiative electron capture is related with that σ + for the single photoeffect via the principle of detailed balance yielding [25] σ (1) = Zt
Fig. 2. The universal quantity Q(ξ ) is calculated within the dipole approximation as a function of the dimensionless variable ξ = η/p. Here η = mαZ and p is the absolute magnitude of the momentum of the electrons at large distances from the bare ion.
consideration requires a rigorous description of the influence of the nuclear field of the target atom. Considering different electron shells of the target atom with taking into account screening effects, one can define the corresponding effective volumes Vn employing Slater’s wave functions according to Vn = π(a0 n/Zeff )3 , where the effective nuclear charge is given by Zeff = Zt − γ [24]. The screening correction γ 0 depends on both the electron state and the electron configuration. The dimensionless volume reads as Vn = V(nZt /Zeff )3 . For n 4 one should introduce an effective principal quantum number. If the electrons are captured from different electron shells, the normalization volume Vn is defined as an effective volume for localization of that electron, which is farther from the nucleus. Although an incident ion can capture a rather large amount of different singlet pairs of target electrons, the corresponding contributions to the cross section turn out to be much smaller than that of the (1s)2 configuration. For example, in a carbon (target) atom, there are two singlet pairs, (1s)2 and (2s)2 , and one triplet pair, (2p)2 , with screening corrections γ1s = 0.3 and γ2s = γ2p = 2.75, respectively [24]. The
2ω12 + σ , p2
(4)
where p is the momentum of the incident electron. The factor Zt takes into account the possibility for capture of any electron of the target atom. The cross section σ + corresponds to the photoabsorption by one electron only. The energy ω1 of the emitted photon is half of that in the correlated double-electron capture, that is, ω1 = ω/2 = Ep + I . In the dipole regime, the nonrelativistic expression for σ + is known analytically [26]. For the radiative electron capture into the K shell of the bare ion, we have 210 2 π σ0 Zt H (ξ ), 3 ξ 2 exp(−4ξ cot−1 ξ ) , H (ξ ) = 2 εγ [1 − exp(−2πξ )] σ (1) =
(5) (6)
where the dimensionless energy εγ = ω/I = 2(1 + ξ −2 ) has been introduced. Then the ratio R = σ (2) / σ (1) is given by R=
29 Zt2 Q(ξ ) . π 2 VZ 5 H (ξ )
(7)
The function Q(ξ )/H (ξ ) exhibits a universal dependence on the dimensionless parameter ξ (see Fig. 3). This is due to employing the dipole approximation and taking into account the leading orders of perturbation theory. In Table 1, we present a comparison of our calculations with available experimental data. This reveals a significant disagreement between our predictions and
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Table 1 For bare ions with nuclear charge numbers Z, the kinetic energies per unit mass, the dimensionless parameters ξ , the nuclear charge numbers Zt of target atoms, the cross sections σ (1) for the radiative electron capture [according to Eq. (5)], the cross sections σ (2) for the correlated double-electron capture with the emission of a single photon [according to Eq. (3)], and the ratios R [according to Eq. (7)] are tabulated Z 18 92
Energy (MeV/u) 11.4 297
ξ 0.840 0.841
Zt 6 18
Fig. 3. The universal quantity Q(ξ )/H (ξ ) is calculated as a function of the dimensionless variable ξ = η/p.
those reported in Ref. [19], where the cross section σ (2) does not depend on the particular choice of target atoms. In Ref. [19], it was also argued that the total cross section grows rapidly in the relativistic domain due to small characteristic distances Rch involved in the process of the correlated double-electron capture. The nonrelativistic wave functions are enhanced by the 2 factor F = (a0 /Rch )(αZ) /2 , while the characteristic spatial scale is defined by the minimal recoil momen−1 . tum qmin transferred to the nucleus, that is, Rch ∼ qmin For the process under consideration, it can be shown that qmin = |2p − ω| ∼ m even in the ultrarelativistic limit. For heavy multicharged ions, this is also rather close to the average momentum η of a K-shell elec-
σ (1) (kb) 0.36 1.08
σ (2) (mb)
R
This work
Experiment
3.2 2.5 × 10−2
5.2 [16] 10 [18]
0.9 × 10−5 2.3 × 10−8
tron. Accordingly, the characteristic distances Rch are of the order of the Compton wave length of the elec2 tron. Then for Z = 92 the factor F = α −(αZ) /2 ∼ 3 gives rise to the “enhancement” of the total cross section σ (2) by a factor F 8 ∼ 6 × 103 , which is at least three orders of magnitude less than the value obtained in Ref. [19]. Moreover, it is difficult to expect even such an enhancement merely by analogy with the single photoeffect. Indeed, according to Ref. [19], the cross section σ (1) would be multiplied by F 4 ∼ 80. However, the relativistic enhancement is known to be absent. For example, the measurement of the radiative electron capture by bare uranium ions at an energy of 295 MeV/u in collisions with a gaseous Ar target leads to σ (1) = 1069 ± 321 b [20], which is in fair agreement with the nonrelativistic prediction according to Eq. (5) (see Table 1). The agreement becomes even better for lower-energy collisions and for ions with moderate values of Z. In contrast to ion–atom collisions, one can expect much larger values for the cross section σ (2) in the case of slow collisions of multicharged ions with the solid-state target. The electrons of the valence zone behave itself as quasifree particles with a characteristic velocity of motion determined by the temperature of the target. This velocity is much smaller than that of the projectile even for ξ ∼ 10. Therefore, in the reference frame connected with the nucleus of the incident ion, the electrons of the valence zone appear as an electron beam characterized by the absolute magnitude of the velocity v and the concentration ne = κρt NA /Mt , where κ is the number of valence electrons in the target atom, NA is Avogadro’s number, and ρt and Mt are the density and the molar mass of the target, respectively. The cross section σ (2) for the correlated double-electron capture from the valence zone looks similar to Eq. (2) with the substitution of the nor-
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malization volume V by n−1 e . Finally, we obtain 219 σ0 σ (2) = ne a03 (8) Q(ξ ). 3Z 5 Let us estimate the cross section (8) for collisions of bare Ar ions with the 9 Be target at ξ = 10. In this case, we have κ = 2, ρt = 1.85 g/cm3 , Mt = 9 g/mol, ne = 2.46 × 1023 cm−3 , Q(ξ ) = 1.63, and σ (2) = 0.12 b. The knowledge of the function Q(ξ ) given in Fig. 2 provides a rather wide range for a choice for optimal experimental conditions. Concluding, we have performed nonrelativistic calculations of the total cross section for the correlated double-electron capture into the K shell of a bare ion with emission of a single photon. At present, the experimental observation of the process on beams of heavy multicharged ions still could not be realized. However, for light ions, the cross sections are expected to be close to the upper experimental limit. Therefore, it is highly desirable to measure again precisely the total cross section. For example, for the experiment described in Ref. [16], the observation of the process can become feasible, if one slightly decelerates the beam of bare Ar ions. The experimental conditions of Ref. [16] also allow for the investigation of correlated double-electron capture into excited electron states of the bare ion. Calculations of total cross sections for the double photoeffect [27–29] suggest that the cross sections for the correlated two-electron capture into the singlet n1 S0 configurations with n 2 are a few times larger than that into the ground 11 S0 state. The novel technique of deceleration of multicharged ions planned at GSI can be applied in order to perform such experiments in the low-energy regime.
Acknowledgements We thank A. Warczak for useful communications. This research was supported in part by INTAS (Grant No. 03-54-3604). A.M. and A.N. are grateful to the Dresden University of Technology for the hospitality and for financial support from GSI. G.S. and G.P. acknowledge financial support from BMBF, DFG, and GSI. A.M., I.M., and A.N. are supported by RFBR (Grants Nos. 01-02-17246 and 00-15-96610). A.N. acknowledges support from the Alexander von Humboldt Foundation.
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