- Email: [email protected]
Double K-photoionization of heavy atoms
ARTICLE IN PRESS
Radiation Physics and Chemistry 75 (2006) 1529–1533 www.elsevier.com/locate/radphyschem
Double K-photoionization of heavy atoms E.P. Kanter, R.W. Dunford, B. Kra¨ssig, S.H. Southworth, L. Young Argonne National Laboratory, Argonne, IL 60439, USA Accepted 8 December 2005
Abstract The double K-photoionization of heavy atoms is a rare process which produces a ‘‘hollow atom’’. We have recently conducted a comprehensive study of the energy-dependence of double K-photoionization in Ag ðZ ¼ 47Þ aimed at elucidating the processes contributing. These measurements have served to isolate the effects of the dynamic electron–electron scattering term from the shake-off contribution and demonstrate a significantly larger scattering contribution than in lighter atoms. Based on those results and other measurements in lighter systems, we have developed a quantitative model to predict the energy-dependence of double K-photoionization in other atoms. r 2006 Elsevier Ltd. All rights reserved. PACS: 32.80.Fb; 32.80.Hd; 32.30.Rj; 23.20.Nx Keywords: X-ray photoionization; Double-K ionization
1. Introduction The double K-photoionization of a heavy atom is a rare process (approximately 104 of the probability of single K-photoionization) which produces a ‘‘hollow atom’’. Because of the single-particle nature of the photon–electron dipole operator (Madden and Codling, 1963), double ionization would not occur in the absence of the e–e interaction and thus this process serves as an extremely sensitive probe of electron correlations within atoms (McGuire et al., 1995). Beyond such fundamental interest, double photoionization has several practical applications including a possible source of entangled electrons (Chandra and Chakraborty, 2002). With the advent of third generation X-ray sources, several groups have studied this process in progressively heavier systems. We have recently conducted a comprehensive
study of the energy-dependence of double-K photoionization in Ag ðZ ¼ 47Þ aimed at elucidating the processes contributing. Those measurements have served to isolate the effects of the dynamic electron–electron scattering term from the shake-off (SO) contribution and demonstrate a significantly larger scattering contribution than in lighter atoms. The measured ratio (double/single ionization) in the peak region was found to agree well with the Z-dependence we had found from fitting previous measurements and much slower than the characteristic 1=Z2 -dependence of SO, further confirming the large scattering contribution in the peak region. Based on those results, we have developed a quantitative model to predict the energy-dependence of double K-photoionization in other atoms.
2. Theoretical background Corresponding author. Tel.: +1 630 252 4050;
fax: +1 630 252 2864. E-mail address: [email protected] (E.P. Kanter).
It has long been recognized that the double excitation and ionization of He-like systems by photoabsorption
0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.12.044
ARTICLE IN PRESS E.P. Kanter et al. / Radiation Physics and Chemistry 75 (2006) 1529–1533
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provides a precise test of electron correlations and dynamics in atoms. Experimentally however, this has been difficult to achieve in all but the lightest systems ðZp3Þ (Lubell, 1995). Because double ionization proceeds through the Z-independent e–e interaction, the effect of electron–electron correlations has in fact been found to be persistent with increasing Z (Briand et al., 1981) and consequently more challenging to treat in high-Z systems where relativistic effects simultaneously become important (Berry et al., 1993). Because of the relative isolation of the K-shell electrons from the outer shells in heavy atoms, the double-K ionization of a heavy atom, producing a hollow atom, is comparable to the double ionization of the same heliumlike system, stripped of the outer shell electrons. In many-body perturbation theory there are three terms which contribute in first order: so-called ‘‘two step one’’ (TS1), SO, and ground state correlations (GSC) as shown in Fig. 1. In the latter (GSC), electronic interaction precedes photoabsorption and thus is sensitive to electron correlations in the initial state. For the other two terms, the electronic interaction is after photoabsorption and thus they are sensitive to correlations in the intermediate state with a K-vacancy. In TS1, the final interaction is between an electron and the other electron, essentially representing electron–electron scattering. For SO, the final interaction is between the hole and the remaining electron, and thus represents the relaxation of the vacancy excitation. The relative contributions of these terms has been shown to be gauge-dependent (Hino et al., 1993). In the ‘‘sudden
1s
ε1 1s
ε2
1s
TS1 1s
ε1 1s
ε2
Shake-off ε1
1s
ε2
Ground State Correlations Fig. 1. Schematic diagrams showing the three first-order terms contributing to double-K photoionization: ‘‘Two-step-one’’ (TS1), shake-off, and ground state correlations. Incident photons are denoted by wiggly lines, electromagnetic interactions by dashed lines, and time increases moving from bottom to top in each diagram. Continuum electrons are labelled 1;2 and the corresponding vacancies are labelled 1s. Further details are given in text.
approximation’’ the SO and GSC are lumped together as shake processes (Sudden Shake-off (SSO)) while the dynamical scattering process described by TS1 is excluded. As was recently pointed out in the case of He doubleionization (Schneider et al., 2002), the scattering (or knockout (KO) in their parlance) and shake (SSO as described above in the sudden approximation) contributions can be separated in a gauge-independent way by exploiting the very different classical/quantum natures of those processes. Rost and co-workers have argued that, while electron–electron scattering can be described classically, there is no classical analog of sudden shake processes which must be treated quantum mechanically (Schneider and Rost, 2003). They further showed that KO and SSO can be distinguished because of their very different energy dependences. Because of the strong energy-dependence of electron-electron scattering, KO is most prominent at lower energies, falling rapidly with increasing energy. In contrast, at higher energies (where KO is negligible), double photoionization of the K shell has been discussed extensively in terms of shake processes where a fast photoelectron leaves with most of the available energy and then as the core-excited atom relaxes the second electron is shaken up/off leading to the concept of an energy-independent asymptotic limit of the ratio between double and single ionization (A˚berg, 1970). The studies of double K ionization following electron capture (EC) decay of radioactive nuclei provide a useful resource in this endeavor. Because one of the electrons is absorbed in the nucleus, there is only a single free electron in the final state following EC, and thus double K-ionization proceeds by a pure SO process in the extreme limit of the sudden approximation. Hence, the asymptotic limit is determined independently by these EC radioactive source measurements.
3. Synchrotron experiments In 1999, our group first observed the double-ionization of the K shell of a heavy atom (molybdenum, Z ¼ 42) with synchrotron radiation (Kanter et al., 1999). The experiment was performed on the BESSRC 12BM bending magnet beamline at the APS. We used the method of satellite–hypersatellite coincidences, pioneered by Briand et al. (1971), with 50-keV incident photons. This energy is close to that corresponding to the predicted maximum ð53 keVÞ in the double photoionization cross section for He-like Mo (Kornberg and Miraglia, 1994). From these measurements, the ratio PKK (the Bayesian conditional probability of producing two K vacancies when at least one K-electron is removed) was determined to be 3:4ð6Þ 104 for 50 keV incident photon energy.
ARTICLE IN PRESS E.P. Kanter et al. / Radiation Physics and Chemistry 75 (2006) 1529–1533
Peak power law fit Samson et al. Ahopelto et al. Diamant et al. Oura et al. ANL Expts.
10-1
PKK
10-2
ut ko oc Kn
In an effort to experimentally separate the dynamical scattering (KO) and sudden shake terms (SSO) in a heavy atomic system, we recently measured double K ionization of silver ðZ ¼ 47Þ for several energies from threshold (51.782 keV) to the expected maximum ð100 keVÞ using the coincidence method at the APS (Kanter et al., 2006). We chose to study silver because of extensive measurements which have been previously carried out using the EC decay of 109Cd in a radioactive source (Ko et al., 1998) producing hollow 109Ag. Because double K-ionization produced by EC proceeds by a pure SO process, the asymptotic limit is determined independently by the radioactive source measurements. As discussed above, Rost and his collaborators have suggested a scalable method to treat this problem (Schneider et al., 2002; Schneider and Rost, 2003). They calculate shake-off (SSO) quantum mechanically. The dynamical KO is treated quasiclassically and then added incoherently to the SSO result. Taking their result for He and independently scaling the SSO and KO results before adding them together, we obtained an excellent fit to the data. For the energy axis, we scaled the excess excitation energy by Z 2 (Kornberg and Miraglia, 1994). Scaling of PKK was treated differently. Scaling the SSO and KO contributions by 1=Z2 produces an asymptotic value of 2:98 105 , which is somewhat smaller than the experimental EC value of 4:2ð5Þ 105 described earlier. We therefore rescaled both SSO and KO by a factor of 1.41 to exactly match the asymptote. Finally, an additional arbitrary rescaling factor for the KO term was then applied to best-fit our data. This factor was found to be 5.8(7) with a normalized w2 ¼ 1:2. The observation that this factor is so much larger than unity implies a much larger KO contribution at higher Z than was previously assumed. Because of the dominance of the KO term in the region of the broad maximum, it is instructive to plot the trend of PKK in the peak region with Z as we suggested a few years ago (Kanter et al., 1999). Combining all of the available data, including our measurements on Ne (Southworth et al., 2003), Mo (Kanter et al., 1999), and Ag, we show that result in Fig. 2. We also show in Fig. 2 the earlier photoionization results (Ahopelto et al., 1979) in lighter atoms ð22pZp28Þ. Recently reported new measurements by Oura et al. (2002) and Diamant et al. (2000) are also included. Those measurements were all carried out somewhat above threshold but well below the asymptotic regime and thus are sensitive to the KO contribution. Shake processes lead to a 1=Z 2 fall-off in the doubleK ionization probability (Levinger, 1953) and that is a predominant characteristic of all of the Z-scaling laws suggested for He-like ions. For comparison, we show in Fig. 2 one of those scaling laws (Forrey et al., 1995). It shows excellent agreement with the asymptotic ratios measured in He by Spielberger et al. (1995) and by the
1531
10-3
Asymptotic Forreyetetal.al. Forrey Spielbergeret et Spielberger al.al. ECdecay decay EC
10-4
10-5 1
10 Atomic Number
100
Fig. 2. The ratio of double to single K-ionization as a function of atomic number. The solid symbols are used for experiments carried out in the region of the predicted peak in the double ionization cross section while the open symbols are used for measurements of the ratio and the corresponding theoretical prediction (dashed curve) for the asymptotic energy limit (Forrey et al., 1995). Also shown (solid line) is our 1=Z1:61 power-law fit.
EC measurements in Ag (Ko et al., 1998), Cl ðZ ¼ 17Þ (Miskel and Perlman, 1954), Mn ðZ ¼ 25Þ (Briand et al., 1974), and Rb ðZ ¼ 37Þ (Schupp and Nagy, 1984). In contrast, those measurements in the peak region, besides being obviously systematically higher, also fall off significantly slower than the 1=Z2 -dependence of SO. Surprisingly, all of the newer data show good agreement with the somewhat weaker 1=Z 1:61 -dependence which we reported earlier (Kanter et al., 1999). This result and our Ag findings described above, requiring rescaling of the Rost group’s KO contribution, suggest that KO may in fact become more important with increasing Z than previously assumed.
4. Simple model What is the origin of this Z-dependence? As was first pointed out by Samson (1990) in his investigation of double ionization of atoms, the knockout contribution is proportional to the electron-impact ionization cross section ðsee Þ. For our case of double K-ionization, the
ARTICLE IN PRESS E.P. Kanter et al. / Radiation Physics and Chemistry 75 (2006) 1529–1533
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peak value of see can be estimated from the phenomenological Lotz formula (Lotz, 1970) to vary with Z4 . More recently, Santos et al. (2003) reported the results of calculations of see with the relativistic binaryencounter Bethe model (BEB). Fitting the peak values they report, we find a somewhat steeper Z4:3 -dependence. Following the suggestion of Kheifets (2001) we can express the proportionality between KO and electron impact ionization as PKK ðKOÞ ’ see
sphoto , pr2K
(1)
where sphoto is the single electron K-photoionization cross section and rK the radius of the K shell. The photoionization cross section (Agarwal, 1991) scales as , sphoto / Z 5 E 7=2 e
(2)
where E e is the excess electron energy above the K-edge. For Zp50 the hypersatellite shifts, and consequently E e , scale roughly as Z 1:1 (Chen et al., 1982). Using the hydrogenic Z-dependence for the radius and see as above we find an overall Z0:85 scaling for KO/SSO, demonstrating a slow, monotonic increase in this ratio with increasing Z as suggested by the data. This dependence is compared to our data in Fig. 3. We use our measurements of Ne (Southworth et al., 2003), Mo (Kanter et al., 1999), and Ag in the peak region to define a peak yield function Y p ðZÞ. The asymptotic-dependence of Forrey et al. (1995) is similarly represented by Y a ðZÞ for the asymptotic yield. Since the former includes both KO and SSO contributions while the latter is only SSO, then the ratio (Y p ðZÞ–Y a ðZÞ)/Y a ðZÞ is a measure of the KO/SSO relative contributions. That ratio is plotted in Fig. 3. We 10 ANL data Z0.85
(Yp/Ya)-1
8 6
have best-fitted that ratio to a power law Z 0:85 (solid line in the figure) as suggested above and the results appear to be consistent. The energy-dependence of PKK in this model is also illuminating. The peak in the energy-dependence of PKK is determined by see . Using the Lotz formula, we expect the peak of see at an electron energy E e such that E e =I ¼ e, where I is the binding of the residual K electron and e is the natural logarithm base. Assuming the binding energies of the two K electrons are roughly equal, then the corresponding peak in photon energy E g would occur at Eg ¼ Ee þ I ¼ Ið1 þ eÞ ’ 50:6 Z2 eV,
ð3aÞ ð3bÞ ð3cÞ
which for Ag corresponds to 112 keV, in good agreement with the trend of the data. Because the single ionization cross section asymptotically falls as E 7=2 , the e peak of PKK should be shifted to higher energy than the predicted peak of sKK (Kornberg and Miraglia, 1994) and that is indeed what we find.
5. Summary This work has demonstrated that it is now possible, with modern synchrotron sources, to explore double Kionization in heavy atoms in far greater detail than has previously been possible. Although this problem has been studied extensively in He, little attention has been paid to heavier systems until quite recently. We showed that the new experimental data indicates that the electron scattering contribution to double K-ionization becomes significantly more important relative to shakeoff with increasing Z and we have suggested a simple model of that dependence. There is clearly need for a more comprehensive theoretical description which simultaneously treats both electron–electron correlations and relativity in these heavy atoms.
4
Acknowledgements
2
We are grateful to have collaborated in various aspects of this work with I. Ahmad and D.S. Gemmell. We thank J. Greene for preparation of the target samples and the BESSRC staff (M. Beno, M. Engbretson, G. Jennings, G. Knapp, C.A. Kurtz, J. Linton, P.A. Montano, U. Ru¨tt, and C. Wiley) for assistance with the beamlines. This work was supported by the Chemical Sciences, Geosciences, and Biosciences Division and the Advanced Photon Source by the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, under Contract No. W-31-109-Eng-38.
0
0
10
20 30 Atomic Number
40
50
Fig. 3. The dimensionless ratio of ðY p Y a Þ=Y a as described in text. The data points show that ratio computed using our previous measurements in Ne, Mo, and Ag. The asymptotic expression of Forrey et al. (1995) is used for Y a . The line shows a best fit to those points using the Z0:85 power law derived in the text.
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References A˚berg, T., 1970. Asymptotic double-photoexcitation cross sections of the Helium atom. Phys. Rev. A 2, 1726–1729. Agarwal, A., 1991. X-ray Spectroscopy, second ed. Springer, Berlin. Ahopelto, J., Rantavuori, E., Keski-Rahkonen, O., 1979. Kah hypersatellite spectra in photon excitation and K shell double photoionization cross section for transition metals Ti, Cr, Fe, Ni. Phys. Scr. 20, 71–74. Berry, H.G., Dunford, R.W., Livingston, A.E., 1993. Comparisons of the QED and relativistic parts of the triplet-state energies in the heliumlike sequence. Phys. Rev. A 47, 698–701. Briand, J.P., Chevallier, P., Tavernier, M., Rozet, J.P., 1971. Observation of K hypersatellites and KL satellites in the X-ray spectrum of doubly K-ionized gallium. Phys. Rev. Lett. 27, 777–779. Briand, J.P., Chevallier, P.J., Johnson, A., Rozet, J.P., Tavernier, M., Touati, A., 1974. Experimental determination of the energy of K hypersatellite lines for various elements. Phys. Lett. A 49, 51–53. Briand, J.P., Chevallier, P., Chetioui, A., Rozet, J.P., Tavernier, M., Touati, A., 1981. Correlation effects in double-Kvacancy production. Phys. Rev. A 23, 39–45. Chandra, N., Chakraborty, M., 2002. Entanglement in double photoionization of atoms. J. Phys. B: At. Mol. Opt. Phys. 35, 2219–2238. Chen, M.H., Crasemann, B., Mark, H., 1982. Effect of the Breit interaction on K X-ray hypersatellite spectra. Phys. Rev. A 25, 391. Diamant, R., Huotari, S., Ha¨ma¨la¨inen, K., Kao, C.C., Deutsch, M., 2000. Cu Kh a1;2 hypersatellites: suprathreshold evolution of a hollow-atom X-ray spectrum. Phys. Rev. A 62, 052519. Forrey, R.C., Sadeghpour, H.R., Baker, J.D., Morgan III, J.D., Dalgarno, A., 1995. Double photoionization of excited 1S and 3S states of the helium isoelectronic sequence. Phys. Rev. A 51, 2112–2116. Hino, K., Ishihara, T., Shimizu, F., Toshima, N., McGuire, J.H., 1993. Double photoionization of helium using manybody-perturbation-theory. Phys. Rev. A 48, 1271–1276. Kanter, E.P., Dunford, R.W., Kra¨ssig, B., Southworth, S.H., 1999. Double K-vacancy production in molybdenum by X-ray photoionization. Phys. Rev. Lett. 83, 508–511. Kanter, E.P., Ahmad, I., Dunford, R.W., Gemmell, D.S., Kra¨ssig, B., Southworth, S.H., Young, L., 2006. Double K-shell photoionization of silver. Phys. Rev. A 73, 022708. Kheifets, A., 2001. On different mechanisms of the two-electron atomic photoionization. J. Phys. B: At. Mol. Opt. Phys. 34, L247–L252.
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Ko, S.K., Cho, H.J., Nha, S.K., 1998. X-ray spectroscopy in the EC nucleus. Nucl. Phys. A 641, 49–63. Kornberg, M.A., Miraglia, J.E., 1994. Scaling laws in double photoionization. Phys. Rev. A 49, 5120–5123. Levinger, J.S., 1953. Effects of radioactive disintegrations on inner electrons of the atom. Phys. Rev. 90, 11–25. Lotz, W., 1970. Electron-impact ionization cross sections for atoms up to Z ¼ 108. Z. Phys. 232, 101–107. Lubell, M.S., 1995. The case for synchrotron radiation studies of two-electron ions, atoms, and molecules at the ALS. Nucl. Instr. Meth. B 99, 177–181. Madden, R.P., Codling, K., 1963. New autoionizing atomic energy levels in He, Ne, and Ar. Phys. Rev. Lett. 10, 516–518. McGuire, J.H., Berrah, N., Bartlett, R.J., Samson, J.A.R., Tanis, J.A., Cocke, C.L., Schlachter, A.S., 1995. The ratio of cross sections for double to single ionization of helium by high energy photons and charged particles. J. Phys. B: At. Mol. Opt. Phys. 28, 913–940. Miskel, J.A., Perlman, M.L., 1954. Double vacancies in the k shell associated with K-electron capture in A37. Phys. Rev. 94, 1683–1687. Oura, M., Yamaoka, H., Kawatsura, K., Takahiro, K., Takeshima, N., Zou, Y., Hutton, R., Ito, S., Awaya, Y., Terasawa, M., Sekioka, T., Mukoyama, T., 2002. Correlative multielectron processes in K-shell photoionization of Ca, Ti, and V in the energy range of 8–35 keV. J. Phys. B: At. Mol. Opt. Phys. 35, 3847–3863. Samson, J.A.R., 1990. Proportionality of electron-impact ionization to double photoionization. Phys. Rev. Lett. 65, 2861–2864. Santos, J.P., Parente, F., Kim, Y.-K., 2003. Cross sections for K-shell ionization of atoms by electron impact. J. Phys. B: At. Mol. Opt. Phys. 36, 4211–4224. Schneider, T., Rost, J.-M., 2003. Double photoionization of two-electron atoms based on the explicit separation of dominant ionization mechanisms. Phys. Rev. A 67, 062704. Schneider, T., Chocian, P.L., Rost, J.-M., 2002. Separation and identification of dominant mechanisms in double photoionization. Phys. Rev. Lett. 89, 073002. Schupp, G., Nagy, H.J., 1984. Double K-shell ionization in the electron capture decay of 85Sr. Phys. Rev. C 29, 1414–1419. Southworth, S.H., Kanter, E.P., Kra¨ssig, B., Young, L., Armen, G.B., Levin, J.C., Ederer, D.L., Chen, M.H., 2003. Double K-shell photoionization of neon. Phys. Rev. A 67, 062712. Spielberger, L., Jagutzki, O., Do¨rner, R., Ullrich, J., Meyer, U., Mergel, V., Damrau, M., Vogt, T., Ali, I., Khayyat, K., Bahr, D., Schmidt, H.G., Frahm, R., Schmidt-Bo¨cking, H., 1995. Separation of photoabsorption and Compton scattering contributions to He single and double ionization. Phys. Rev. Lett. 74, 4615–4618.
Radiation Physics and Chemistry 75 (2006) 1529–1533 www.elsevier.com/locate/radphyschem
Double K-photoionization of heavy atoms E.P. Kanter, R.W. Dunford, B. Kra¨ssig, S.H. Southworth, L. Young Argonne National Laboratory, Argonne, IL 60439, USA Accepted 8 December 2005
Abstract The double K-photoionization of heavy atoms is a rare process which produces a ‘‘hollow atom’’. We have recently conducted a comprehensive study of the energy-dependence of double K-photoionization in Ag ðZ ¼ 47Þ aimed at elucidating the processes contributing. These measurements have served to isolate the effects of the dynamic electron–electron scattering term from the shake-off contribution and demonstrate a significantly larger scattering contribution than in lighter atoms. Based on those results and other measurements in lighter systems, we have developed a quantitative model to predict the energy-dependence of double K-photoionization in other atoms. r 2006 Elsevier Ltd. All rights reserved. PACS: 32.80.Fb; 32.80.Hd; 32.30.Rj; 23.20.Nx Keywords: X-ray photoionization; Double-K ionization
1. Introduction The double K-photoionization of a heavy atom is a rare process (approximately 104 of the probability of single K-photoionization) which produces a ‘‘hollow atom’’. Because of the single-particle nature of the photon–electron dipole operator (Madden and Codling, 1963), double ionization would not occur in the absence of the e–e interaction and thus this process serves as an extremely sensitive probe of electron correlations within atoms (McGuire et al., 1995). Beyond such fundamental interest, double photoionization has several practical applications including a possible source of entangled electrons (Chandra and Chakraborty, 2002). With the advent of third generation X-ray sources, several groups have studied this process in progressively heavier systems. We have recently conducted a comprehensive
study of the energy-dependence of double-K photoionization in Ag ðZ ¼ 47Þ aimed at elucidating the processes contributing. Those measurements have served to isolate the effects of the dynamic electron–electron scattering term from the shake-off (SO) contribution and demonstrate a significantly larger scattering contribution than in lighter atoms. The measured ratio (double/single ionization) in the peak region was found to agree well with the Z-dependence we had found from fitting previous measurements and much slower than the characteristic 1=Z2 -dependence of SO, further confirming the large scattering contribution in the peak region. Based on those results, we have developed a quantitative model to predict the energy-dependence of double K-photoionization in other atoms.
2. Theoretical background Corresponding author. Tel.: +1 630 252 4050;
fax: +1 630 252 2864. E-mail address: [email protected] (E.P. Kanter).
It has long been recognized that the double excitation and ionization of He-like systems by photoabsorption
0969-806X/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.radphyschem.2005.12.044
ARTICLE IN PRESS E.P. Kanter et al. / Radiation Physics and Chemistry 75 (2006) 1529–1533
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provides a precise test of electron correlations and dynamics in atoms. Experimentally however, this has been difficult to achieve in all but the lightest systems ðZp3Þ (Lubell, 1995). Because double ionization proceeds through the Z-independent e–e interaction, the effect of electron–electron correlations has in fact been found to be persistent with increasing Z (Briand et al., 1981) and consequently more challenging to treat in high-Z systems where relativistic effects simultaneously become important (Berry et al., 1993). Because of the relative isolation of the K-shell electrons from the outer shells in heavy atoms, the double-K ionization of a heavy atom, producing a hollow atom, is comparable to the double ionization of the same heliumlike system, stripped of the outer shell electrons. In many-body perturbation theory there are three terms which contribute in first order: so-called ‘‘two step one’’ (TS1), SO, and ground state correlations (GSC) as shown in Fig. 1. In the latter (GSC), electronic interaction precedes photoabsorption and thus is sensitive to electron correlations in the initial state. For the other two terms, the electronic interaction is after photoabsorption and thus they are sensitive to correlations in the intermediate state with a K-vacancy. In TS1, the final interaction is between an electron and the other electron, essentially representing electron–electron scattering. For SO, the final interaction is between the hole and the remaining electron, and thus represents the relaxation of the vacancy excitation. The relative contributions of these terms has been shown to be gauge-dependent (Hino et al., 1993). In the ‘‘sudden
1s
ε1 1s
ε2
1s
TS1 1s
ε1 1s
ε2
Shake-off ε1
1s
ε2
Ground State Correlations Fig. 1. Schematic diagrams showing the three first-order terms contributing to double-K photoionization: ‘‘Two-step-one’’ (TS1), shake-off, and ground state correlations. Incident photons are denoted by wiggly lines, electromagnetic interactions by dashed lines, and time increases moving from bottom to top in each diagram. Continuum electrons are labelled 1;2 and the corresponding vacancies are labelled 1s. Further details are given in text.
approximation’’ the SO and GSC are lumped together as shake processes (Sudden Shake-off (SSO)) while the dynamical scattering process described by TS1 is excluded. As was recently pointed out in the case of He doubleionization (Schneider et al., 2002), the scattering (or knockout (KO) in their parlance) and shake (SSO as described above in the sudden approximation) contributions can be separated in a gauge-independent way by exploiting the very different classical/quantum natures of those processes. Rost and co-workers have argued that, while electron–electron scattering can be described classically, there is no classical analog of sudden shake processes which must be treated quantum mechanically (Schneider and Rost, 2003). They further showed that KO and SSO can be distinguished because of their very different energy dependences. Because of the strong energy-dependence of electron-electron scattering, KO is most prominent at lower energies, falling rapidly with increasing energy. In contrast, at higher energies (where KO is negligible), double photoionization of the K shell has been discussed extensively in terms of shake processes where a fast photoelectron leaves with most of the available energy and then as the core-excited atom relaxes the second electron is shaken up/off leading to the concept of an energy-independent asymptotic limit of the ratio between double and single ionization (A˚berg, 1970). The studies of double K ionization following electron capture (EC) decay of radioactive nuclei provide a useful resource in this endeavor. Because one of the electrons is absorbed in the nucleus, there is only a single free electron in the final state following EC, and thus double K-ionization proceeds by a pure SO process in the extreme limit of the sudden approximation. Hence, the asymptotic limit is determined independently by these EC radioactive source measurements.
3. Synchrotron experiments In 1999, our group first observed the double-ionization of the K shell of a heavy atom (molybdenum, Z ¼ 42) with synchrotron radiation (Kanter et al., 1999). The experiment was performed on the BESSRC 12BM bending magnet beamline at the APS. We used the method of satellite–hypersatellite coincidences, pioneered by Briand et al. (1971), with 50-keV incident photons. This energy is close to that corresponding to the predicted maximum ð53 keVÞ in the double photoionization cross section for He-like Mo (Kornberg and Miraglia, 1994). From these measurements, the ratio PKK (the Bayesian conditional probability of producing two K vacancies when at least one K-electron is removed) was determined to be 3:4ð6Þ 104 for 50 keV incident photon energy.
ARTICLE IN PRESS E.P. Kanter et al. / Radiation Physics and Chemistry 75 (2006) 1529–1533
Peak power law fit Samson et al. Ahopelto et al. Diamant et al. Oura et al. ANL Expts.
10-1
PKK
10-2
ut ko oc Kn
In an effort to experimentally separate the dynamical scattering (KO) and sudden shake terms (SSO) in a heavy atomic system, we recently measured double K ionization of silver ðZ ¼ 47Þ for several energies from threshold (51.782 keV) to the expected maximum ð100 keVÞ using the coincidence method at the APS (Kanter et al., 2006). We chose to study silver because of extensive measurements which have been previously carried out using the EC decay of 109Cd in a radioactive source (Ko et al., 1998) producing hollow 109Ag. Because double K-ionization produced by EC proceeds by a pure SO process, the asymptotic limit is determined independently by the radioactive source measurements. As discussed above, Rost and his collaborators have suggested a scalable method to treat this problem (Schneider et al., 2002; Schneider and Rost, 2003). They calculate shake-off (SSO) quantum mechanically. The dynamical KO is treated quasiclassically and then added incoherently to the SSO result. Taking their result for He and independently scaling the SSO and KO results before adding them together, we obtained an excellent fit to the data. For the energy axis, we scaled the excess excitation energy by Z 2 (Kornberg and Miraglia, 1994). Scaling of PKK was treated differently. Scaling the SSO and KO contributions by 1=Z2 produces an asymptotic value of 2:98 105 , which is somewhat smaller than the experimental EC value of 4:2ð5Þ 105 described earlier. We therefore rescaled both SSO and KO by a factor of 1.41 to exactly match the asymptote. Finally, an additional arbitrary rescaling factor for the KO term was then applied to best-fit our data. This factor was found to be 5.8(7) with a normalized w2 ¼ 1:2. The observation that this factor is so much larger than unity implies a much larger KO contribution at higher Z than was previously assumed. Because of the dominance of the KO term in the region of the broad maximum, it is instructive to plot the trend of PKK in the peak region with Z as we suggested a few years ago (Kanter et al., 1999). Combining all of the available data, including our measurements on Ne (Southworth et al., 2003), Mo (Kanter et al., 1999), and Ag, we show that result in Fig. 2. We also show in Fig. 2 the earlier photoionization results (Ahopelto et al., 1979) in lighter atoms ð22pZp28Þ. Recently reported new measurements by Oura et al. (2002) and Diamant et al. (2000) are also included. Those measurements were all carried out somewhat above threshold but well below the asymptotic regime and thus are sensitive to the KO contribution. Shake processes lead to a 1=Z 2 fall-off in the doubleK ionization probability (Levinger, 1953) and that is a predominant characteristic of all of the Z-scaling laws suggested for He-like ions. For comparison, we show in Fig. 2 one of those scaling laws (Forrey et al., 1995). It shows excellent agreement with the asymptotic ratios measured in He by Spielberger et al. (1995) and by the
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10-3
Asymptotic Forreyetetal.al. Forrey Spielbergeret et Spielberger al.al. ECdecay decay EC
10-4
10-5 1
10 Atomic Number
100
Fig. 2. The ratio of double to single K-ionization as a function of atomic number. The solid symbols are used for experiments carried out in the region of the predicted peak in the double ionization cross section while the open symbols are used for measurements of the ratio and the corresponding theoretical prediction (dashed curve) for the asymptotic energy limit (Forrey et al., 1995). Also shown (solid line) is our 1=Z1:61 power-law fit.
EC measurements in Ag (Ko et al., 1998), Cl ðZ ¼ 17Þ (Miskel and Perlman, 1954), Mn ðZ ¼ 25Þ (Briand et al., 1974), and Rb ðZ ¼ 37Þ (Schupp and Nagy, 1984). In contrast, those measurements in the peak region, besides being obviously systematically higher, also fall off significantly slower than the 1=Z2 -dependence of SO. Surprisingly, all of the newer data show good agreement with the somewhat weaker 1=Z 1:61 -dependence which we reported earlier (Kanter et al., 1999). This result and our Ag findings described above, requiring rescaling of the Rost group’s KO contribution, suggest that KO may in fact become more important with increasing Z than previously assumed.
4. Simple model What is the origin of this Z-dependence? As was first pointed out by Samson (1990) in his investigation of double ionization of atoms, the knockout contribution is proportional to the electron-impact ionization cross section ðsee Þ. For our case of double K-ionization, the
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peak value of see can be estimated from the phenomenological Lotz formula (Lotz, 1970) to vary with Z4 . More recently, Santos et al. (2003) reported the results of calculations of see with the relativistic binaryencounter Bethe model (BEB). Fitting the peak values they report, we find a somewhat steeper Z4:3 -dependence. Following the suggestion of Kheifets (2001) we can express the proportionality between KO and electron impact ionization as PKK ðKOÞ ’ see
sphoto , pr2K
(1)
where sphoto is the single electron K-photoionization cross section and rK the radius of the K shell. The photoionization cross section (Agarwal, 1991) scales as , sphoto / Z 5 E 7=2 e
(2)
where E e is the excess electron energy above the K-edge. For Zp50 the hypersatellite shifts, and consequently E e , scale roughly as Z 1:1 (Chen et al., 1982). Using the hydrogenic Z-dependence for the radius and see as above we find an overall Z0:85 scaling for KO/SSO, demonstrating a slow, monotonic increase in this ratio with increasing Z as suggested by the data. This dependence is compared to our data in Fig. 3. We use our measurements of Ne (Southworth et al., 2003), Mo (Kanter et al., 1999), and Ag in the peak region to define a peak yield function Y p ðZÞ. The asymptotic-dependence of Forrey et al. (1995) is similarly represented by Y a ðZÞ for the asymptotic yield. Since the former includes both KO and SSO contributions while the latter is only SSO, then the ratio (Y p ðZÞ–Y a ðZÞ)/Y a ðZÞ is a measure of the KO/SSO relative contributions. That ratio is plotted in Fig. 3. We 10 ANL data Z0.85
(Yp/Ya)-1
8 6
have best-fitted that ratio to a power law Z 0:85 (solid line in the figure) as suggested above and the results appear to be consistent. The energy-dependence of PKK in this model is also illuminating. The peak in the energy-dependence of PKK is determined by see . Using the Lotz formula, we expect the peak of see at an electron energy E e such that E e =I ¼ e, where I is the binding of the residual K electron and e is the natural logarithm base. Assuming the binding energies of the two K electrons are roughly equal, then the corresponding peak in photon energy E g would occur at Eg ¼ Ee þ I ¼ Ið1 þ eÞ ’ 50:6 Z2 eV,
ð3aÞ ð3bÞ ð3cÞ
which for Ag corresponds to 112 keV, in good agreement with the trend of the data. Because the single ionization cross section asymptotically falls as E 7=2 , the e peak of PKK should be shifted to higher energy than the predicted peak of sKK (Kornberg and Miraglia, 1994) and that is indeed what we find.
5. Summary This work has demonstrated that it is now possible, with modern synchrotron sources, to explore double Kionization in heavy atoms in far greater detail than has previously been possible. Although this problem has been studied extensively in He, little attention has been paid to heavier systems until quite recently. We showed that the new experimental data indicates that the electron scattering contribution to double K-ionization becomes significantly more important relative to shakeoff with increasing Z and we have suggested a simple model of that dependence. There is clearly need for a more comprehensive theoretical description which simultaneously treats both electron–electron correlations and relativity in these heavy atoms.
4
Acknowledgements
2
We are grateful to have collaborated in various aspects of this work with I. Ahmad and D.S. Gemmell. We thank J. Greene for preparation of the target samples and the BESSRC staff (M. Beno, M. Engbretson, G. Jennings, G. Knapp, C.A. Kurtz, J. Linton, P.A. Montano, U. Ru¨tt, and C. Wiley) for assistance with the beamlines. This work was supported by the Chemical Sciences, Geosciences, and Biosciences Division and the Advanced Photon Source by the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, under Contract No. W-31-109-Eng-38.
0
0
10
20 30 Atomic Number
40
50
Fig. 3. The dimensionless ratio of ðY p Y a Þ=Y a as described in text. The data points show that ratio computed using our previous measurements in Ne, Mo, and Ag. The asymptotic expression of Forrey et al. (1995) is used for Y a . The line shows a best fit to those points using the Z0:85 power law derived in the text.
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