The present work provides a reliable interpretation of the  Khα₁/Khα₂ intensity ratios and an  explanation of the lifetime values for K-shell hollow atoms based on an advanced  theoretical analysis (using extensive multiconfiguration Dirac–Fock calculations with  the inclusion of the transverse Breit interaction and quantum electrodynamics  corrections). It was found that, as a result of closing the Khα₁  de-excitation channel in the pure LS coupling scheme, the  Khα₁/Khα₂ intensity ratio changes with the  atomic number from small values (for the LS coupling limit at low Z) to about 1.5– 1.6 (for the j–j coupling limit at high Z). However, closing the  Khα₁ de-excitation channel (due to the domination of the  pure LS coupling for the low-Z atoms) does not enlarge the lifetimes of hollow  atoms

Dousse, Jean-Claude

Hoszowska, Joanna

Kozioł, K.

Polasik, M .

Rzadkiewicz, J.

Starosta, J.

Słabkowska, K.

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Lifetimes of doubly K-shell ionized statesM Polasik1, K Słabkowska1, K Kozioł1, J Starosta1, J-Cl Dousse2,J Hoszowska2 and J Rzadkiewicz3,41 Faculty of Chemistry, Nicholas Copernicus University, 87-100 Torun´, Poland2 Institute of Physics, Jan Kochanowski University, 25-406 Kielce, Poland3 The Andrzej Soltan Institute for Nuclear Studies, 05-400b Otwock- ´Swierk, Poland4 Institute of Plasma Physics and Laser Microfusion, Hery 23, 01-497 Warsaw, PolandE-mail: mpolasik@uni.torun.plAbstractThe present work provides a reliable interpretation of the K hα1/K hα2 intensity ratios and anexplanation of the lifetime values for K -shell hollow atoms based on an advanced theoreticalanalysis (using extensive multiconﬁguration Dirac–Fock calculations with the inclusion of thetransverse Breit interaction and quantum electrodynamics corrections). It was found that, as aresult of closing the K hα1 de-excitation channel in the pure LS coupling scheme, theK hα1/K hα2 intensity ratio changes with the atomic number from small values (for the LScoupling limit at low Z ) to about 1.5–1.6 (for the j– j coupling limit at high Z ). However,closing the K hα1 de-excitation channel (due to the domination of the pure LS coupling for thelow-Z atoms) does not enlarge the lifetimes of hollow atoms.PACS numbers: 32.30.Rj, 32.70.Cs, 32.80.Fb, 32.70.Jz1. IntroductionIn recent decades, the hypersatellite lines in x-ray spectrawere measured by using various experimental techniques.High-resolution measurements of such spectra [1–4] givedirect access to information on the natural width of thehypersatellite lines and the lifetimes of the K -shell hollow (i.e.doubly K -shell ionized) atoms.Recently, the K hα1,2 hypersatellite lines in x-ray spectraof Zr, Nb, Mo and Pd targets bombarded with 250MeVcarbon and 360MeV oxygen ions have been measuredwith high-resolution diffraction spectrometry by Rzadkiewiczet al [2]. The de-excitation of the studied doubly K -shellionized state via the spin ﬂip in the K hα1 transition isforbidden in the pure LS coupling scheme (1S0 →3P1) butallowed in j– j coupling (see ﬁgure 1). As a result, theK hα1/K hα2 intensity ratio changes with the atomic number(see ﬁgure 2) from small values (for the LS coupling limit atlow Z ) to about 1.5–1.6 (for the j– j coupling limit at high Z ).In a work by Rzadkiewicz et al [2], the experimentalwidths of K hα1,2 lines have been compared with twoformulae: the ﬁrst one—statistical (proposed by Mosséet al [7])—is based on the assumption K K ≈ 2K andthe second one (phenomenological—proposed in [2]) takesinto account the possibility of enlarging the lifetime of theK−2 hole state (and correspondingly decreasing the naturalwidth of the hypersatellite line) as a result of closing theK hα1 de-excitation channel in the pure LS coupling scheme.However, it was found that the experimental natural widthsof the K hα1 and K hα2 lines signiﬁcantly exceed the valuesobtained from the formula proposed by Rzadkiewicz et al [2],but are in good agreement with the simple statistical formula.Moreover, the experimental lifetimes of the double K -holestates (deduced from their measured natural widths) are closeto those predicted by the statistical formula, but signiﬁcantlyshorter than the values determined by means of the formulaproposed by Rzadkiewicz et al [2].The present work provides a reliable interpretationof the K hα1/K hα2 intensity ratios and an explanationof the lifetime values for the K -shell hollow atomsbased on the advanced theoretical analysis (using extensivemulticonﬁguration Dirac–Fock (MCDF) calculations with theinclusion of the transverse Breit interaction and quantumelectrodynamics (QED) corrections [8]).2. Results and discussionThe proper description of the atomic state requires applyingan intermediate coupling (i.e. mixing of conﬁguration state1Published in "Physica Scripta 2011(T144): 014021, 2011"which should be cited to refer to this work.Table 1. Coefﬁcients for (1s−12p−1)1P1 and (1s−12p−1) 3P1 states in the upper and lower ﬁnal (1s−12p−1) states (see equations (1) and (2)),(1s−12p−1) 1P1 contributions for upper and lower ﬁnal states, and K hα1/K hα2 intensity ratios for selected atoms.Atoms Z Coefﬁcients for 1P1 state 1P1 state 1P1 state K hα1/K hα2the ﬁnal (1s−12p−1) states contribution contribution contributions intensitya b for the upper state for the lower state ratio ratio– 0 1.0 0.0 1.0 0.0 0.0 –Ne 10 0.999 936 0.011 309 0.999 872 0.000 128 0.000 128 –Ar 18 0.9925 0.1220 0.9851 0.0149 0.015 0.014Zn 30 0.8509 0.5254 0.7240 0.2760 0.381 0.377Kr 36 0.7500 0.6614 0.5626 0.4374 0.778 0.793Zr 40 0.7034 0.7108 0.4948 0.5052 1.021 1.041Mo 42 0.6856 0.7279 0.4701 0.5299 1.127 1.145Pd 46 0.6584 0.7527 0.4334 0.5666 1.307 1.313Rn 86 0.5867 0.8098 0.3442 0.6558 1.906 1.610No 102 0.5819 0.8132 0.3386 0.6614 1.953 1.490– ∞ 0.5773 0.8165 0.333 0.6667 2.0 –K−1L−1K−2Khα2 Khα1LS[1S0][1P1][3P1]j-j[1s−21/2]J=0[1s−11/22p−11/2]J=1[1s−11/22p−13/2]J=1Figure 1. K hα1,2 hypersatellite x-ray transitions.Figure 2. Theoretical predictions for K hα1/K hα2 intensity ratioscompared with some experimental data [1, 3–6].functions of a certain J ). Therefore, to describe the K−1L−1hole states (see ﬁgure 1) for all the studied atoms, wecan apply the intermediate coupling for j– j or LS basis.For the j– j basis (typical in MCDF calculations) the ﬁnalupper and the ﬁnal lower states are given as a combinationof [1s−11/22p−11/2]J=1 and [1s−11/22p−13/2]J=1 states. In this case,transitions from [1s−21/2]J=0 to both states ([1s−11/22p−11/2]J=1and [1s−11/22p−13/2]J=1) are allowed. For heavy atoms theintermediate coupling is well enough approximated by purej– j coupling. In this coupling, the K hα1 transition shouldbe two times stronger than the K hα2 transition (two timesmore 2p3/2 electrons than 2p1/2). However, the energy ofthe K hα1 transition is signiﬁcantly higher than that of theK hα2 transition, and moreover, 2p3/2 and 2p1/2 orbitalsdiffer signiﬁcantly enough. Therefore, for heavy atoms theK hα1/K hα2 intensity ratio does not amount to 2, but is abit smaller, i.e. it amounts to about 1.5–1.6 (see column 8 intable 1). For the LS basis in the intermediate coupling, theﬁnal upper state (commonly labeled as 1P1, see ﬁgure 1) isgiven asa · |(1s−12p−1) 1P1〉+ b · |(1s−12p−1) 3P1〉 (1)and the ﬁnal lower state (commonly labeled as 3P1) is givenasb · |(1s−12p−1) 1P1〉− a · |(1s−12p−1) 3P1〉. (2)The coefﬁcients a and b for selected atoms are presentedin columns 3 and 4 in table 1. As can be seen, for lightatoms the intermediate coupling is well approximated bythe LS coupling (i.e. the mixing coefﬁcients differ fromeach other, e.g. for Ar). However, in the LS couplingonly (1s1/2)−2 1S0 → (1s−12p−1) 1P1 transition is allowed,but (1s1/2)−2 1S0 → (1s−12p−1) 3P1 is forbidden. Thanks tothe intermediate coupling in the LS basis, the K hα1/K hα2intensity ratio can be estimated as a contributions ratio of thepure (1s−12p−1) 1P1 state in the ﬁnal lower state (responsiblefor K hα1 transition) and in the ﬁnal upper state (responsiblefor K hα2 transition)—see columns 7 and 8 in table 1.From table 1, one can also see that going from heavyto light atoms closing the K hα1 channel is accompanied bystrengthening of the K hα2 channel (with fully seizing thecontribution of the closed K hα1 channel). It is a fact that thesum of the pure (1s−12p−1) 1P1 state contributions for bothﬁnal states is constant and amounts to 1.0 (see the sums ofvalues in columns 5 and 6 of table 1).Based on the results of MCDF calculations [10, 11],the lifetimes (and radiative and total natural widths) of K−2and K−1 hole states and widths of the K hα1 and K hα2hypersatellite lines have been evaluated (see table 2). It wasfound that with a change of the atomic number Z only therelative roles of both channels are changed and closing theK hα1 channel for lighter atoms does not lead to increasing2Table 2. The K−2 and K−1 hole state lifetimes, the K−2/K−1 natural width ratios and the natural linewidths for K hα lines for selectedatoms.Atom Z K−2/K−1 LinewidthLifetimes (s) natural width ratio for K hα lines (eV)K−2 K−1 Radiative Total K hα1 K hα2Ne 10 4.491×10−14 1.725×10−13 3.336 2.04 0.89 0.89Ar 18 4.258×10−16 1.010×10−15 2.67 2.37 2.32 2.33Zn 30 1.835×10−16 4.105×10−16 2.31 2.22 5.88 6.26Kr 36 1.102×10−16 2.477×10−16 2.27 2.25 9.82 9.88Zr 40 7.871×10−17 1.743×10−16 2.21 (2.2)a 13.65 13.77Mo 42 6.684×10−17 1.472×10−16 2.20 (2.2)a 16.01 16.15Pd 46 4.882×10−17 1.076×10−16 2.20 (2.2)a 21.65 21.83Rn 86 4.425×10−18 9.353×10−18 2.11 (2.1)a 226.0 225.9No 102 2.291×10−18 4.817×10−18 2.10 (2.1)a – –aPossible changes in the ﬂuorescence yields have been considered by Chen [9] and found tobe relatively small.lifetimes (and decreasing the natural width) of K−2 states(see the values in columns 3–6 of table 2). Moreover, fromcolumns 3 and 4 of table 2, one can see that the lifetimes ofthe K−2 state are always more than two times shorter than thelifetimes of the K−1 state.3. Summary and conclusionsAs a result of closing the K hα1 de-excitation channel in thepure LS coupling scheme, the K hα1/K hα2 intensity ratiochanges with atomic number from small values (for the LScoupling limit at low Z ) to about 1.5–1.6 (for the j– j couplinglimit at high Z ). However, it was found that closing the K hα1de-excitation channel (due to the domination of the pure LScoupling for the low-Z atoms) does not enlarge the lifetimesof hollow atoms. The lifetimes of doubly K -shell ionizedstates are a bit more than two times shorter than for singlyK -shell ionized ones for all of the atomic number range. Theobtained theoretical results are in agreement with measuredK hα1/K hα2 intensity ratios and the lifetimes of the doublyK -shell ionized states.AcknowledgmentsThis work was supported by the Polish Ministry of Scienceand Higher Education through grant no. N N202 1465 33 andby the Swiss National Science Foundation.References[1] Rzadkiewicz J et al 1999 Phys. Lett. A 264 186[2] Rzadkiewicz J et al 2005 Nucl. Instrum. Methods B 235 110[3] Boschung B et al 1995 Phys. Rev. A 51 3650[4] Diamant R et al 2009 Phys. Rev. A 79 062511–1Diamant R et al 2009 Phys. Rev. A 79 062512–1[5] Salem S I et al 1984 Phys. Rev. A 29 2634Salem S I et al 1982 Phys. Rev. Lett. 49 1240[6] van Eijk C W E, Wijnhorst J and Popelier M A 1981 Phys.Rev. A 24 854[7] Mossé J P, Chevallier P and Briand J P 1985 Z. Phys. A322 207[8] Grant I P 1984 Int. J. Quantum Chem. 25 23[9] Chen M H 1991 Phys. Rev. A 44 239[10] Dyall K G, Grant I P, Johnson C T, Parpia F A and PlummerE P 1989 Comput. Phys. Commun. 55 425[11] Polasik M 1995 Phys. Rev. A 52 2273

Lifetimes of doubly K -shell ionized states

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