to uranium are planned at the ESR and NESR. This re- ..... tron in the nuclear rest frame is larger than the ground- .... electron energy remains almost constant.
- 127 -
Nuclear Sizes and Isotopic Shift M. Tomaselli1,2 , L.-C. Liu3 , T. K¨ uhl1,4, S. Fritzsche5 , D. Ursescu1,4 , and P. Neumayer1,2 1
Gesellschaft f¨ ur Schwerionenforschung mbH, Planckstr. 1, D-64291 Darmstadt, Germany; 2 TU Darmstadt, Institut f¨ ur Kernphysik, Schlossgartenstr. 9, 64289 Darmstadt; 3 Los Alamos National Laboratory, USA; 4 Johannes-Guttenberg-Universit¨ at Mainz,Germany; 5 Universit¨ at Kassel, Heinrich-Plett-Str.40 D-34132, Germany
We study the charge radii of exotic nuclei through nuclear calculations and by non-perturbative isotopic-shift (IS) evaluations. Beside nuclear masses (or binding energies), nuclear charge radii, spins, and nuclear moments are informations on ground state properties of atomic nuclei. The latter can be obtained by atomic spectroscopy. In fact, most information on the static properties of exotic systems have been determined in this way [1]. For a given electronic transition, the IS is the sum of: A) the mass shift (MS) originating from the finite mass of the nucleus and the electron-electron correlation, and B) the field shift (FS) that reflects the differences in the nuclear charge distributions. Although the information on nuclear ground-state properties extracted from a study of hyperfine structure and isotope shift is model-independent, it is hampered in complex neutral atoms by the accuracy with which the electron wave functions are known at the site of the nucleus. However, in the case of simple few-electron systems the electron wave function can be precisely calculated. Recent advances in variational calculations for lithium and lithium-like ions using multiple basis set in Hylleraas coordinates [2], made possible to calculate the MS in the 2S-3S and 2S-2P transitions of lithium with a very good accuracy. Therefore, if the overall isotopic shift can be measured with a comparable precision, the rms charge radius can be extracted. Accordingly [2], the charge radius of the isotopes is given by: δ(IS)exp = δ(M S)the +
2πZ δ|ψ(0)|2 δ < r2 > 3
(1)
In this way, absolute charge radii can be determined. Furthermore, in combination with measurements of the matter radius, neutron radii can be extracted. For the future, experiments on stable and long-lived lithium-like ions up to uranium are planned at the ESR and NESR. This requires the availability of a reliable soft-x-ray laser, which was recently demonstrated at PHELIX [3]. A recent experiment on radioactive lithium isotopes [4], aiming for a determination of the charge and the neutron radius of 11 Li, provides an excellent example for a test of nuclear and atomic theories. Our nuclear computations of the charge radius are performed in the framework of the dynamiccorrelation model DCM for nuclei with an odd number of valence particles, and in the boson dynamic-correlation model (BCDM) for those with an even number of valence particles [5]. These nuclear models take fully into consideration the correlation between valence particles as well as between valence and core particles. Consequently, these computations may reveal feature physics which is associated to the strong correlation between the valence and the core polarized states. Moreover, we propose to test the derived charge radii within the isotopic shift theory in which the electronic transitions for lithium and lithiumlike ions are calculated by considering the three corre-
lated electrons described by a model similar to the nuclear DCM. Within this nonlinear and non-perturbative model, the treatment of the halo of the proton distribution can be performed self-consistently. The proposed theoretical method is applied to two specific problems: a) highresolution isotope shift calculation on unstable lithium isotopes, and b) measurement of the 2p-2s transition in lithium-like uranium. In both ranges of isotopes is performed non-perturbative IS calculation which is appropriate for calculating charge radii in halo- and exotic nuclei. Nuclear-model-independent rms charge radii can be then obtained from IS calculations. The IS is evaluated as in [6] in terms of the rms radius using an electron distribution calculated selfconsistently. At this stage of the calculation, in order to test from one side the non perturbative electron dynamic-correlation model (eDCM) presented in Ref. [7] and from the other side the charge radii as derived from the charge distributions [8], we propose to insert these calculated values in Eq. 1. The calculated MS and the theoretical charge radii should then reproduce the experimental IS. Preliminary results calculated using eDCM for the electron space are given in table 1. One example is the binding energy in the atomic lithium system without nuclear corrections, the other the 2s-2p transition energy in lithiumlike 235 U91+ . Energies in lithium (au) 1s2 2s Drake [9] -7.47806032310 (31) eDCM this work -7.478060733 Transitions in 235 U91+ (eV) 2s-2p Yerokhin [10] 288.44(20) eDCM this work 288.33
References [1] E.W. Otten, Treatise on Heavy-ion Science, D.A. Bromley ed. Vol. 8 (1988) 515, Plenum Press, New York 1988 [2] G.W.F. Drake, Z.C. Yan, Phys. Rev.A 46 (1992) 2378 [3] S. Borneis, Th. K¨ uhl et al., Hyperfine Interactions 127 (2000) 315. [4] W. N¨ ortersh¨ auser et al., Nucl. Instr. and Met. in Phys. Res. B204 (2003) 644; W. N¨ ortersh¨ auser et al., to be published. [5] M. Tomaselli et al., J. Opt. B5 (2003) 395. [6] E.C. Seltzer, Phys. Rev. 188 (1969) 1916. [7] M. Tomaselli et al., Can.J. Phys. 80 (2002) 1347. [8] M. Tomaselli et al., Dynamical Aspect of Nuclear Fission, J. Kliman et al. eds. (2002) 445, World Scientific, New Jersey 2002. [9] Z.-C. Yan and G.W.F Drake, Phys. Rev. A 61 (2000) 022504. [10] V.A. Yerokhin, A.N. Artemyev, V.M. Shabaev et al., Phys. Rev. Lett. 85 (2000) 4699.
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Ab-initio QED Treatment of Electron-Correlation Effects and Deduction of Nuclear Parameters in Highly-Charged Ions I. Bednyakov1 , T. Beyer1 , F. Erler1 , G. Schaller1 , S. Schumann1 , J. Winter1 , G. Plunien1 , G. Soff1 , A.N. Artemyev2 , K.V. Koshelev2 , L.N. Labzowsky2 , V.M. Shabaev2 , and V.A. Yerokhin2 1
Institut f¨ ur Theoretische Physik, TU Dresden;
2
Department of Physics, St. Petersburg State University
The accuracy reached in experimental and theoretical investigations of the low-lying states in lithiumlike ions provides a promising tool for probing QED corrections up to second order in the finestructure constant α and, in principle, for the determination of nuclear parameters via atomic spectroscopy. Theoretical uncertainties in the description of electron-correlation effects as they are inherently generated in evaluations based on relativistic manybody perturbation theory (MBPT) can be improved by performing ab-initio QED calculations up to order α2 or higher. Within the framework of bound-state QED interelectron-interaction effects are mediated by the exchange of virtual photons described by the fully relativistic photon propagator. In order to achieve proper renormalization and to provide gauge-invariant and consistent results for the energy shift one has to evaluate simultaneously a suitable set of Feynman diagrams up to a given order in α. In recent experiments performed at the ESR facility at GSI the 2p1/2 − 2s splittings in very heavy lithiumlike ions have been determined with high precision utilizing low-energy dielectric recombination [1]. A comparison between experimental data and theoretical predictions based on rigorous QED calculations [2] is presented in Table 1. The theoretical values account for various corrections: finite-nuclear size and nuclear recoil, one-, twoand three-photon exchange corrections, one-electron selfenergy and vacuum-polarization corrections of order α as well as for screened self-energy and vacuum-polarization effects. So far an excellent agreement between theory and experiment can be stated. Also the 2p3/2 − 2s transition energy in lithium-like bismuth has been determined experimentally with an accuracy of 0.04 eV [3]. Provided that QED corrections have been calculated with sufficient accuracy one can utilize the knowledge about the atomic structure to probe nuclear physics (determination of nuclear parameters, test of specific nuclear models, etc.). E.g., the magnetic dipole, octupole and the electric quadrupole moments of the 209 83 Bi nucleus can be deduced from the hyperfine-structure splittings (HFS) of the 2p3/2 state. Preliminary studies have been performed recently [4] utilizing the dynamic proton model. It describes HFS as due to the interaction between the electron and the valence proton and takes into account simultaneously the electric and magnetic nuclear moment distributions. Extensive ab-initio QED calculations for the 2p3/2 − 2s transition energy in lithium-like ions with nuclear charge numbers 20 ≤ Z ≤ 100 (see [5] and references therein) have been performed recently. The results for the various contributions of the two-photon exchange corrections to the energy shift of the (2s)2 2p3/2 state in Li-like ions are presented in Table 2. The subscripts ”dir” and ”exch” indicate direct and exchange contribution, respectively, while
Table 1: Experimental [1] and theoretical results [2] for the 2p1/2 − 2s splittings in Li-like ions (in eV). Ion Experiment Total theory 197 76+ Au 216.134(29)(39)(28) 216.17(13)(11) 79 208 79+ 230.650(30)(22)(29) 230.68(6)(13) 82 Pb 238 89+ 280.516(34)(22)(43) 280.64(11)(21) 92 U Table 2: Two-photon exchange correction for the (2s)2 p3/2 state in Li-like ions (in atomic units) [5]. 2el 2el Z −∆Edir −∆Eexch −∆E 3el Total 20 0.03876 0.03902 -0.45509 -0.37731 30 -0.12453 0.03715 -0.29807 -0.38545 40 -0.18304 0.03465 -0.24844 -0.39683 50 -0.21185 0.03156 -0.23126 -0.41156 60 -0.22960 0.02795 -0.22801 -0.42967 70 -0.24292 0.02392 -0.23233 -0.45133 83 -0.25831 0.01822 -0.24511 -0.48519 92 -0.26936 0.01406 -0.25752 -0.51281 100 -0.28022 0.01030 -0.27070 -0.54061
the superscripts ”2el” and ”3el” refer to the two- and three-electron contributions, respectively. The difference between QED and MBPT results can be envisaged as the ”nontrivial” QED contribution to the interelectron interaction. The complete MBPT results can be obtained from QED calculations performed within the Coulomb gauge but restricting the summations over intermediate Dirac states to the positive-energy spectrum only and separating out the contributions due to the exchange of Coulomb and Breit photons, respectively. For the case under consideration the nontrivial QED contribution is essentially larger than that for the 2p1/2 − 2s transition [2]. Moreover, for the 2p3/2 − 2s transition the total correction changes its sign in the region between Z = 92 and Z = 100. The detailed studies in [5] represent an important step towards the evaluation of all two-electron QED corrections of order α2 to the 2p3/2 − 2s transition energy for the Li isoelectronic sequence
References [1] [2] [3] [4] [5]
C. Brandau et al., Phys. Rev. Lett. 91 (2003) 073202. V.A. Yerokhin et al., Phys. Rev. A64 (2001) 032109. P. Beiersdorfer et al., Phys. Rev. Lett. 80 (1998) 3022. K.V. Kosholev et al., Phys. Rev. A68 (2003) 052504. A.N. Artemyev et al., Phys. Rev. A67 (2003) 062506.
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Electron interaction and isotope effects studied by dielectronic recombination with heavy few-electron ions Zolt´an Harman1 , Roxana Schiopu2 , Norbert Gr¨ un1, and Werner Scheid1 1
Institut f¨ ur Theoretische Physik, Justus-Liebig-Universit¨ at Giessen; Johannes-Gutenberg-Universit¨ at Mainz
The dynamics of electrons and their interaction are strongly influenced by relativistic effects in very heavy atomic systems. The investigation of dielectronic recombination (DR), or, the analogous process of resonant transfer and excitation, has proved to be a suitable tool to study these phenomena in highly charged ions [1, 2, 3]. The electrons interact by exchanging virtual photons. Thus, in transversal gauge of the photon field and in low order, the operator responsible for capture is the sum of the Coulomb and generalized Breit operators [4]. The latter has been shown to give an important contribution to dielectronic capture rates in highly charged very heavy ions [1, 2, 3], especially for transitions where inner-shell electrons are involved. Therefore, here we consider the effects of higher-order perturbative terms to the interaction. The expansion of the transition operator yields twophoton exchange corrections, namely, ladder and crossedphotons diagrams. We evaluate these terms in the limit when the frequencies of both transversal photons approach zero. Figure 1 shows the differential cross section for DR into U91+ at an electron energy of 68556 eV within the KL1/2 L3/2 resonance group as a function of the angle of the hypersatellite photon, which is emitted in the transition from the intermediate state to a singly-excited state. The effects of the two-photon exchange are small relative to the one-photon exchange.
diff. cross section (barn/sr)
30 25 20 15 10 one-photon ex. with two-photon ex.
5 0 0
20
40
60
80 100 120 140 160 180 theta (deg)
Figure 1: Differential cross section in the ionic frame as a function of the emission angle of the hypersatellite photon for recombination into U91+ at an energy of 68556 eV. The full curve is calculated with one-photon exchange, the dashed curve includes the two-photon exchange correction. The investigation of the DR process can also provide a new approach to obtain information about the charge distribution of nuclei. At the maxima of the resonances in the total DR cross section the energy of the continuum electron is equal to the difference of the energies of the initial and final bound atomic states. By performing experiments with different isotopes, the resonances are shifted due to the change of the charge distribution. We made calculations of the resonance energies using the
2
Institut f¨ ur Physik,
multiconfiguration Dirac-Fock package GRASP of Dyall et al. [5] for relativistic elements with Z ranging from 54 (Xe) to 94 (Pu). The spherical Fermi distribution was taken with parameters obtained from a fit to experimental values [6]. Figure 2 shows the dependence of resonance energy shifts on the charge number Z. We compare two different scenarios, namely, recombination into H-like ions with the excitation of the bound K-shell electron and recombination into Li-like ions evoking transitions within the L-shell.
Figure 2: Difference between the resonance energies of two different isotopes with A and A − 5 in the case of DR into H-and Li-like ions as a function of the charge number Z. As one can see in figure 2 for the case of initially Hlike ions, the resonance with both electrons in the 2p3/2 state is shifted the most and the 2s21/2 resonance the least. For recombination into Li-like ions (lower curves), the isotopic variation of the resonance energies is even smaller. This is due to the fact that the electrons in the n = 2 and n = 6 shells, actively involved in the reaction, have a smaller overlap with the nucleus. Even though the shifts are smaller for initially Li-like ions with intra-shell excitations than in H- and He-like ions, the better precision in measuring the resonance positions makes these systems the most promising candidates for the experimental observation of nuclear volume effects [7].
References M. Gail et al., J. Phys. B 31 4645 (1998) X. Ma et al., Phys. Rev. A 68 042712 (2003) S. Zakowicz et al., Phys. Rev. A 68 042711 (2003) J.B. Mann and W.R. Johnson, Phys. Rev. A 4 41 (1971) [5] K.G. Dyall et al., Comput. Phys. Commun. 55 425 (1989) [6] W.R. Johnson and G. Soff, At. Data Nucl. Data Tables 33 405 (1985) [7] A. M¨ uller, private communication [1] [2] [3] [4]
- 130 -
Negative-continuum dielectronic recombination into n=2 states A. N. Artemyeva,b , A. E. Klasnikovb,c , T. Beiera , J. Eichlerc , C. Kozhuharova , V. M. Shabaeva,b,c , T. St¨ohlkera , and V. A. Yerokhina,b Gesellschaft f¨ ur Schwerionenforschung, Planckstr. 1, 64291 Darmstadt, Germany St. Petersburg State University, Oulianovskaya 1, St. Petersburg 198504, Russia c Abteilung Theoretische Physik, Hahn-Meitner Institut, 14109 Berlin, Germany
In a collision between an electron and a bare heavy nucleus, radiative recombination (RR) is the main reaction channel in a wide range of collision energies. In a recent work [1], we have investigated the so-called negativecontinuum dielectronic recombination (NCDR) into the ground state of a He-like ion where the incident electron is captured into the 1s state with simultaneous creation of a free-positron–1s-electron pair: X
Z+
−
+e →X
(Z−2)+
+
+e .
This process may occur if the energy of the incident electron in the nuclear rest frame is larger than the groundstate energy of the corresponding He-like ion plus the positron rest energy. The maximum total cross section for this process was found to be about 27µb for U92+ and about 11µb for Pb82+ , corresponding to projectile energies of about 2 GeV/u in the electron-rest frame. The signature of positron emission together with a twofold change of the projectile ion’s charge forms a very distinct signature for the NCDR. The differential cross section for the NCDR process is given by (¯ h = me = c = 1) 3 dσ 4π |pf |N 2 = (−1)P ili +lf 2 dΩf εf pi JM ma ,mb mi ,mf P,κi ,κf ,Mf × exp(i∆κi + i∆κf ) 2li + 1 j M
f f ∗ i ×Cljii 0,m(1/2) mi Cl m , (1/2) −mf Yl ,ml (−pf /|pf |) f
l
f
f
f
2 ×P aP b|I(εi − εP a )|(εi , κi , mi )(−εf , κf , Mf ) ,
where (ε, κ, m) is the electron wave function with energy ε, angular momentum and parity determined by κ, and angular momentum projection m. P is a permutation operator, Y and C denote spherical harmonics and Clebsch-Gordan coefficients, εi , pi and εf , pf denote energy and momentum of the incoming electron and outgoing positron, ∆κ is a phase shift, and I is an expression related to the photon propagator [2]. J and M refer to total angular momentum and its projection of the formed He-like ion, and a and b denote the one-electron wavefunctions. N indicates a normalization factor (cf. [1]). The mutual interactions of the two electrons is of order 1/Z and not considered here. We have now enlarged our investigations to captures of one of the electrons into exited states and present the first results for the capture of one of the electrons in an n = 2 state in Figs. 1 and 2. Detailed results will be published elsewhere. In particular, it is clear from Fig. 2 that there is a range of angle, into which the positron is emitted only in case of both electrons bound in the ground state, which experimentally even allows to distinguish this from a capture of one of the electrons into an n = 2 state. Also, it is possible to estimate
the background to the process by measuring the positron yield outside the angular range of the NCDR positrons. Pb
U 8
3 2,5
6
(1s2s)
2
Cross section [µbarn]
b
(1s2s)
4
1,5 1
0
(1s2p3/2)
1000
2000
1500
(1s2p1/2)
2
(1s2p1/2)
0,5
3000
2500
0
(1s2p3/2)
1000
1500
2000
2500
3000
40
15 2
(1s) +(1s2v)
12 (1s)
9
2
(1s) +(1s2v)
30
2
(1s)
20
2
6 (1s2v)
0
1000
1500
(1s2v)
10
3 2000
3000
2500
0
1000
1500
2000
2500
3000
Kinetic energy of the electron [kev]
Fig. 1: Total cross section of NCDR into various bound states and the sum of cross sections into the (1s)2 and (1s2v) states as function of the energy of the electron. The energies are equivalent to about 1.6 GeV/u up to 5.5 GeV/u of the projectile in the electron-rest frame. Pb
Differential cross section [µbarn/sr]
a
U
1
1
0,01
0,01
(1s)
0,0001
(1s)
2
0,0001
2
(1s2s)
(1s2s) (1s2p1/2) (1s2p1/2)
1e-06
1e-06
(1s2p3/2)
(1s2p3/2)
0
5
10
15
20
25
0
5
10
15
20
25
30
Angle of positron emission [deg]
Fig. 2: Differential cross section of NCDR in the nucleusrest frame for kinetic energies of the electrons of 1200 keV. The cross section folded by any finite angular resolution remains finite at the maximum scattering angle (cf. [1]). This work was supported by the DFG (Grant No. 436 RUS 113/616), by RFBR (Grant No. 01-02-04011), and by the Russian Ministry of Education (Grant No. PD02-1.2-79). A. E. K. acknowledges support by the Russian Ministery of Education (grant No. A03-2.9-219).
References [1] A. Artemyev et al., Phys. Rev. A 67, 052711 (2003). [2] V. A. Yerokhin et al., Phys. Rev. A 62, 042712 (2000).
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Cooling of Ions with Magnetized Electrons in Traps B. M¨ollers, C. Toepffer, G. Zwicknagel Institut f¨ ur Theoretische Physik II, Universit¨at Erlangen
- Energy loss of the ions to the electrons
time. Initially the electrons are heated very fast by the highly charged ions until an equilibrium with the cooling by emission of synchrotron radiation is reached and the electron energy remains almost constant. Because of the high ion charge a test of the perturbation treatment underlying these results is desirable. We currently calculate the cooling force in the framework of the Vlasov-Poisson equation, which accounts for all nonlinearities as well as collective response [4,5]. - This work has been supported by a GSI collaboration contract. Z=92 12
ni / ne → 0 ni / ne = 10-4
10
ion energy (keV/Z)
In precision experiments like the planned QED-tests with highly charged ions in HITRAP [1] it is necessary to work with cool ions in a trap. One possibility is electron cooling: The ions are mixed with cold electrons and lose their energy because of the Coulomb interaction. This method is well established in storage rings, and it is also used in traps. Because of the presence of a strong mag in the trap the cooling force F on netic guiding field B the ion cannot be calculated analytically. We developed several methods to calculate the cooling force on ions in a magnetized electron plasma. For the calculation of cooling times of the ions we use the binary collision model, where the energy transfer from individual collisions of ions and magnetized electrons are accumulated. The energy transfer is calculated by treating the Coulomb interaction as a perturbation to the helical motion of the electrons up to second order including a correction for hard collisions [2,3]. To estimate cooling times we calculated the energy of the ions and the electrons as a function of time. For that calculation three effects are taken into account [1]:
ni / ne = 10-3
8 6 4 2
- Heating of the electrons due to the energy transfer from the ions
0 0
- Cooling of the electrons by emission of synchrotron radiation
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.7
0.8
time (s)
Figure 1 This leads to three coupled differential equations for the ion velocity parallel (vi ) and transverse (vi⊥ ) to the magnetic field and the electron temperature Te which have to be solved numerically: =
1 F⊥ (vi⊥ , vi , Te ) M
dvi dt
=
1 F (vi⊥ , vi , Te ) M
dTe dt
=
−
2 ni dEi (vi⊥ , vi , Te ) vi 3kB ne ds 1 − (Te − Te,0 ) . τe
M is the ion mass, ni the ion density, ne the electron density, Te,0 the temperature to which the trap is cooled and τe the time constant for cooling of the electrons by the emission of synchrotron radiation. Fig. 1 shows the ion energy Ei = 12 M vi2 for U92+ ions as a function of time with ne = 107 cm−3 , B = 6 T, = 30◦ and an initial electron temperature α : = (vi , B) Te = 4 K for different ratios ni /ne . If the heating of the electrons can be neglected (ni /ne → 0) the cooling time is about 0.35 seconds. The cooling time increases with growing ion density since the increasing electron temperature results in a reduction of the cooling force. Fig. 2 shows the electron energy Ee = 32 kB Te as a function of
-4
ni / ne = 10 -3 ni / ne = 10
30
electron energy (eV)
dvi⊥ dt
Z=92 35
25 20 15 10 5 0 0
0.1
0.2
0.3
0.4
0.5
0.6
time (s)
Figure 2
References [1] W. Quint et al., Hyp. Int. 132 (2001) 457. [2] C. Toepffer, Phys. Rev. A 66, 022714 (2002). [3] B. M¨ollers et al., Nucl. Instr. and Meth. B 205, 285 (2003) [4] B. M¨ollers et al., Nucl. Instr. and Meth. B 207, 462 (2003) [5] M. Walter et al., Nucl. Instr. and Meth. B 168, 347 (2000)
- 132 -
Polarization transfer in heavy hydrogen–like ions following the radiative capture of electrons Andrey Surzhykov1 , Stephan Fritzsche1 , Thomas St¨ohlker2 , and Andreas Orˇsi´c Muthig2 Universit¨ at Kassel, D–34132 Kassel;
2
Gesellschaft f¨ ur Schwerionenforschung (GSI), D–64291 Darmstadt
A large number of experiments have been performed recently at the GSI storage ring in order to explore the radiative electron capture (REC) by bare highly–charged ions. In these experiments, attention was placed, in particular, on the electron recombination into the excited ion states and to their subsequent radiative decay. For instance, from the measurements of the angular distribution of the characteristic Lyman–α1 (2p3/2 → 1s1/2 ) radiation, the strong alignment of the 2p3/2 state was found for hydrogen–like uranium ions U91+ following electron capture [1]. One may expect, of course, that such an alignment of the excited ion may arise not only due to the direct capture of electrons into the 2p3/2 state but also due to the cascade feeding from the higher–lying levels. Both of these population mechanisms are now well understood within the theoretical approach, based on Dirac’s relativistic theory [2, 3]. So far, however, theoretical studies on the alignment of the excited ion states have dealt with ion beams and atomic (or electronic) targets which are both spin– unpolarized. While such (theoretical) assumptions are appropriate for the present–days experimental set–up of ”spin–independent” ion–atom collisions, there become new experiments likely to be carried out in the near future in which use is made of either spin–polarized projectile ions and/or target electrons. In this contribution, therefore, we like to address the question: How the alignment of the excited ion states is affected by the ion or electron spin– polarization? In this report, we present theoretical studies for the magnetic sublevel population of the 2p3/2 state of hydrogen– like heavy ions following the radiative capture of free polarized electrons. Similar to the previous studies, we consider the mechanisms for the population of the excited ion states, following both the direct electron capture in the given levels as well as (cascade) feedings from the upper states. For the (direct) electron capture into magnetic sublevels |nb jb µb , the computation of the cross sections σnRR has been performed within the exact relativistic apb jb µb proach and discussed in detail elsewhere [2, 4]. Then, by utilizing these partial cross sections as the initial populaa system tion of the excited states Nnb jb µb (0) = C · σnRR b jb µb of so–called rate equations [5]
dNi =− λij Ni + λki Nk dt j
1.0
Degree of circular polarization
1
0.5
0.0
P = 1.0 P = 0.7 P = 0.3
-0.5
-1.0
0
30
60 90 120 150 Observation Angle (deg)
180
Figure 1: Circular polarization of the Lyman–α1 photons following radiative electron capture by bare uranium ions with energy Tp = 1 MeV/u. Calculations are presented for three different polarizations of the incident electrons. state and, hence, evaluate the alignment parameters A1 and A2 . In general, these parameters depend on the projectile ion energy Tp and its nuclear charge Z [4]. They behave, however, in a rather different way as function of the polarization of the incident electrons. For example, the second–rank parameter A2 , which completely determines the angular distribution as well as the linear polarization of the Lyman–α1 characteristic decay [5], is not affected by the spin–polarization of the incident electrons. Therefore, the (future) angular–distribution measurements on the characteristic radiation will not bring any additional information on the polarization properties of particles. The first–rank (orientation) parameter RR RR RR RR 1 3σ3/2 − 3σ−3/2 + σ1/2 − σ−1/2 , A1 = √ RR + σ RR + σ RR + σ RR 5 σ3/2 1/2 −1/2 −3/2
(2)
in contrast, is proportional to the (degree of) polarization of the incident electrons: A1 ∝ P. The orientation (2) determines the circular polarization of the Lyman–α1 photons which, as seen from the Figure 1, may serve as a ”detector” for the spin–polarization of target electrons or projectile ions. In practice, however, such polarization studies seems to be hardly possible to be performed in the near future since the detection of the circular polarization of hard x–rays still remains an unsolved problem.
(1)
k
is solved which describes the decay dynamics of the ion. In the equations (1), λij is the decay rate for the |i → |j transition and Λ is the total number of (excited) sublevels which are considered in the decay cascade; the index j runs over all those with Ek > Ei . By performing an integration of the system (1), we may find the occupation of the magnetic sublevels of the 2p3/2
References [1] Th. St¨ ohlker et al., Phys. Rev. Lett. 79, 3270 (1997). [2] A. Surzhykov et al., Phys. Rev. Lett. 88, 15300 (2002). [3] A. Orˇsi´c Muthig et al., GSI Sci. Report, 90 (2002). [4] J. Eichler et al., Phys. Rev. A 58, 2128 (1998). [5] K. Blum, Density Matrix Theory and Appl., (1981).
- 133 -
On the measurement of the spin–polarization of highly–charged ions Andrey Surzhykov1 , Stephan Fritzsche1 , Thomas St¨ohlker2 , and Stanislav Tachenov2 1
Universit¨ at Kassel, D–34132 Kassel;
2
Gesellschaft f¨ ur Schwerionenforschung (GSI), D–64291 Darmstadt
During the last decade, ion–atom and ion–electron collisions have been the subject of intense studies at the GSI storage ring. A large number of measurements were performed in order to explore, for example, relativistic as well as quantum electrodynamic (QED) phenomena in energetic collisions of high–Z projectile ions with low–Z targets. Until now, however, most of the collision experiments have dealt with ion beams and target atoms (or free electrons) which are both spin–unpolarized. While, of corse, such ”spin–independent” measurements have brought a great deal of information on the structure and dynamics of heavy atomic systems, more details may be obtained from experiments with spin–polarized ions. Very recently, a number of such polarization experiments have been proposed for studying parity nonconservation phenomena in few– electron systems [1] or spin–dependent effects in electron capture processes [2]. Obviously, however, any practical realization of ”spin– dependent” collision experiments will require the solution of two key problems: (i) how to produce beams of polarized heavy ions and (ii) how their polarization can be measured. The method for producing of polarized hydrogen– like heavy ions was recently discussed by Prozorov and co–workers [3]. In particular, it was proposed to apply the optical pumping of the hyperfine ground–state levels of hydrogen–like europium ion Eu62+ with a nuclear spin I = 5/2 in order to obtain a predominant population of the state |F = 2, MF = 2. Since the ion state |F MF results from the coupling of an electron in the (one–particle) state |jb µb with the nuclear spin F = I+ jb , a fully polarized F = 2 ground state may lead to a polarization of the nuclear spin of about 93 %. However, as mentioned in [3], the measurement and, hence, the control of this polarization remained up to the present a unresolved problem. In this contribution, we suggest to utilize the radiative capture of a target electron into a bound state of the projectile ion as a ”probe” process for measuring the spin–polarization of ion beam. As recently shown, for example, the linear polarization of the recombination x–ray photons is strongly affected by the spin–polarization of the target atoms [4]. Since, however, the electron and ion occur rather symmetrical in the collision process, a similar effect on the polarization of recombination light can therefore be expected if the projectile ions are themselves polarized. In order to investigate such polarization effects we calculated the linear polarization of the photons as emitted in the radiative capture of free electrons into the ground state of spin–polarized hydrogen–like heavy ions [5]. Most naturally, the polarization of the recombination photons is described in terms of the Stokes parameters, which are simply determined by the intensities of the light Iχ , as measured under the different angles with respect to the reaction plane [4, 5]. While the parameter P1 = (I0 − I90 )/(I0 + I90 ) is obtained from intensities within and perpendicular to the reac-
Figure 1: The Stokes parameter P2 of the photons which are emitted in the electron capture into the K–shell of completely polarized hydrogen–like europium ions. tion plane, the parameter P2 follows a similar intensity ratio which is taken at χ = 45◦ and χ = 135◦ , respectively. As shown by the theoretical analysis [5], the two Stokes parameters P1 and P2 behave in rather different ways with respect to the spin–polarization of (hydrogen–like) projectile ions. While the parameter P1 does not depend on beam polarization and, hence, can not be used for polarization studies, the second Stokes parameter P2 appears to be proportional to the degree of the beam polarization P2 (θ) ∝ λF · f (θ) .
(1)
In the Eq. (1), the beam polarization is defined by [3]: λF = nF,MF MF /F (2) MF
as the sum over the magnetic sublevels, where nF,MF refers to the corresponding population. The Stokes parameter P2 may serve, therefore, as a valuable tool for ”measuring” the polarization properties of the heavy ion beams at storage rings. Figure 1 displays the parameter P2 as calculated, for example, for radiative capture of electrons into the ground state of completely polarized (λF = 1) hydrogen–like europium ions with energies in the range 200 MeV/u ≤ Tp ≤ 400 MeV/u. The effect of the ion polarization becomes particularly remarkable around a photon emission angle of θ = 18◦ , where the second Stokes parameter decreases from the P2 = -0.05 for Tp = 200 MeV/u to almost -0.16 for Tp = 400 MeV/u.
References [1] [2] [3] [4] [5]
L. Labzowsky et al., Phys. Rev. A 63, 054105 (2001). A. Klasnikov et al., Phys. Rev. A 66, 042711 (2002). A. Prozorov et al., Phys. Lett. B 574, 180 (2003). A. Surzhykov et al., Phys. Rev. A 68, 022719 (2003). A. Surzhykov et al., Phys. Rev. Lett. to be submitted.
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