ISSN 0030�400X, Optics and Spectroscopy, 2014, Vol. 117, No. 2, pp. 167–175. © Pleiades Publishing, Ltd., 2014.Original Russian Text © E.P. Ivanova, 2014, published in Optika i Spektroskopiya, 2014, Vol. 117, No. 2, pp. 179–187.

167

INTRODUCTION

Currently, a compact X�ray laser based on the3d94d (J = 0)–3d94p (J = 1) transition in Ni�like kryp�ton with λ = 32.8 nm [1–4], in which plasma is formedin a flow of krypton clusters via ionization by a pump�laser optical field, is being developed. The formationof an inverted state of plasma is considered within theradiation�collisional model, in which gains g(t) aredetermined by solving the kinetic equations for levelpopulations. The figure shows a schematic diagram ofenergy levels and the strongest laser transition

3 4d5/2 (J = 0)–3 4p1/2 (J = 1) in the Kr IX ion.To calculate the wavelengths and gains, one mustknow the spectroscopic characteristics of the Kr IXion: energy levels, probabilities of radiative transitionsbetween levels, and probabilities of transitions inducedby collisions with electrons. The ionization potentialfor the ground state of Kr IX is 1890400 cm–1 ≈234.4 eV. Below, there are levels of the 3d9nl configu�rations and, partially, the levels of configurations withelectron excitation from the 3p6 shell: 3p53d104s and3p53d104p. The 3p53d104d configuration lies above theionization potential, as well as the states of the config�urations with electron excitation from the inner shell3s. The figure shows also the levels of ions in adjacentionization stages: Kr VIII (3d10nl, with one electronbeyond the filled 3d10 shell) and Kr X (3d9, with onevacancy in the filled 3d10 shell). The energy levels ofthe states with one electron above the core (Cu�likeions) or with one vacancy in the core (Co�like ions)have been thoroughly investigated by experimentersand are known with a high accuracy up to Z ∼ 90. Inour approach, high�precision single�particle energiesof these states are used to construct zero�approxima�tion wave functions in order to calculate the spectro�scopic characteristics within the relativistic perturba�

d5/29 d3/2

9

tion theory with a zero�approximation model poten�tial (RPTMP).

The number of experimental studies of the spectro�scopic characteristics of the Kr IX ion is rather small.A semiempirical calculation of the energy levels ofthree lower configurations, 3d94s, 4p, and 4d, by thegeneralized least�squares method was performed in[5]. An exhaustive review of published spectroscopicdata on the Kr IX ion can be found in [6]. In the samestudy, 115 spectral lines due to the transitions betweenthe 3d94s, 3d94p, 3d94d, and 3d94 f configurationswere measured, as a result of which the energy levels ofthese configurations were experimentally determinedfor the first time. An important result obtained in [6]was high�precision measurement of the line via theabove�mentioned laser transition: 3d5/24d5/2 (J = 0)–3d3/24p1/2 (J = 1).

The wavelengths, probabilities, and line strengthsfor the E1, M1, E2, M2, E3, and M3 radiative transi�tions between the excited states 3s23p63d94l,3s23p53d104l, and 3s3p63d104l and the ground state3s23p63d10 for Ni�like ions with nuclear charges Z =30–100 were calculated in [7, 8] within the relativisticmany�body perturbation theory (RMBPT).

In this study, we calculated the energies of 68 lowerexcited Kr IX levels of the 3s23p63d94s, 4p, 4d, 4f and3s23p53d104s, 4p configurations. We also calculated theprobabilities of radiative transitions from these statesto the 3s23p64d10 ground state. The energy levels for thestates with J = 0, 1 are compared with the experimen�tal results [6] and theoretical data of other researchers[7, 8]. The probabilities of electric (E1) and magnetic(M1) dipole transitions to the ground state arereported for the states with J = 1; they are comparedwith the corresponding results of [7]. The reason forthe large discrepancy between the experimental and

SPECTROSCOPY OF ATOMS AND MOLECULES

Energy Levels and Probabilities of Radiative Transitionsin the Kr IX Ion

E. P. IvanovaInstitute for Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow, 142190 Russia

e�mail: eivanova@isan.troitsk.ruReceived September 4, 2013

Abstract—The energy levels of the 3d94l (l = 1–3) and 3p53d104l (l = 0, 1) configurations in the Kr IX ionand the probabilities of radiative transitions to the ground state from these excited states have been calculatedwithin the relativistic perturbation theory using a zero�approximation model potential. The results are shownto be stable within this approximation. The well�known problem of anomalously low accuracy of the calcu�lations of some higher lying singlet levels is considered.

DOI: 10.1134/S0030400X1408013X

168

OPTICS AND SPECTROSCOPY Vol. 117 No. 2 2014

IVANOVA

theoretical values of energy levels for the 3d5/24d5/2

(J = 0) state (the upper working level of laser transi�tion) is analyzed.

THEORETICAL APPROACH

Successive consideration of relativistic effects isinevitably based on the electrodynamic theory, and

3p51/2

3p53/2

3d93/2

3d95/2

3p53d104p

3p53d104s

Kr X Kr IX Kr VIII

3d94f

3d94d

3d94p

3d94s

1S0

1P1

1s22s22p63s23p63d10

3d104f7/2

3d104f5/2

3d104d5/2

3d104d3/2

3d104p3/2

3d104p1/2

3d104s1/2

Energy�level diagram of the Kr IX ion and ions in adjacent ionization stages (Kr VIII and Kr X). The arrow shows the strongest

laser transition: 3 4d5/2 (J = 0) – 3 4p1/2 (J = 1) [3d94d 1S0 – 3d94p 1P1].d5/29

d3/29

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ENERGY LEVELS AND PROBABILITIES OF RADIATIVE TRANSITIONS 169

a specific calculation scheme must be built as a justi�fied approximation of the electrodynamic approach.The theory of nonrelativistic atom applies a well�known convenient field procedure to calculate theenergy shifts ΔE of degenerate states, which is relatedto diagonalization of secular matrix [9]. The latter isconstructed using the Gell�Mann–Low adiabatic for�mula. A similar approach, based on the Gell�Mannand Low formula and electrodynamic scatteringmatrix, is applied in relativistic�atom theory [10]. Theperturbation theory is built proceeding from the elec�tron–field interaction, with the electron–nucleusinteraction potential included in the zero approxima�tion. A secular matrix is calculated for a group ofdegenerate or almost degenerate states, and its diago�nalization yields a set of energies and vectors of states.

An energy approach within the RPTMP methodfor calculating the relativistic radiative decay of atomicstate was reported by Driker and Ivanov [11]. Theyused the Gell�Mann–Low adiabatic formula with arelativistic scattering matrix to derive an analyticalexpression for the imaginary part of the electronicenergy of atom at which the ground state is a filledshell. The individual terms of the sum over virtualstates are interpreted as the probabilities of single�photon, two�photon, and autoionizing decays throughdifferent physical channels. The angular symmetry forthe aforementioned transitions was taken into accountin [12].

We have used the RPTMP method many times inour previous calculations of the real part of the energylevels of many�electron atomic systems with one, two,or three quasi�particles above the filled�shell core[13–20] (here, a “quasi�particle” is an electron abovethe core or a vacancy in the core). Recently, thismethod was used to calculate with a good accuracy theenergy levels of Ag�like ions and the resonant states ofPd�like ions with Z ≤ 86 [21, 22]. A specific feature ofthe RPTMP method is the use of empirical energy lev�els to determine the model potential and the basis ofsingle�quasi�particle wave functions. The atomic sys�tems in which the ground state is a filled shell (Ne�,Ni�, and Pd�like ions) have one electron above thecore and one vacancy in the core in the lower excitedstates. We considered the imaginary part of the energyof a Ne�like ion, which is the probability of radiativedecay to the ground state, in [23]. Below, we brieflydescribe the RPTMP method as applied to calculationof the energy levels and radiative transition probabili�ties (RTPs) from excited states to the ground state inions of the aforementioned type.

We assume that the state of many�electron atomicsystem can be presented by the Dirac equation withthe Hamiltonian

(1)H h ri( )i

∑ V rirj( ).i j>

∑+=

in atomic units; h(r) is the single�electron DiracHamiltonian for an electron moving in the Coulombpotential of the nucleus, Z/r, and V(rirj) describes theelectron–electron interaction. Summation in (1) isover all core electrons and electrons above the core. Inthe model�potential method, the zero approximationincludes, along with the electron–nucleus interaction,some seed potential imitating the electron–electroninteraction; in the successive perturbation theory, thisinteraction must be subtracted in each order. Themany�electron equation will be solved within the per�turbation theory, taking the Hamiltonian

(2)

where Vc(r|b(Z)) is the central single�electron poten�tial of the core, as a zero approximation. The pertur�bation is the difference between the total and zeroHamiltonians:

Vint = H – H0 = − (ri |b(Z)) + (rirj). (3)

The interaction between quasi�particles can bepresented in a complex form (one can find all designa�tions in [11, 12])

V(rirj) = exp(iωrij)(1 – )/rij. (4)

The second term in (4) takes into account the mag�netic interaction between electrons and the effect ofinteraction delay; and are Pauli matrices.

The energy matrix will be calculated within the jjcoupling scheme for the momenta of single�quasi�particle states. The transition to the intermediate cou�pling scheme is performed by diagonalizing the energymatrix.

METHOD FOR CALCULATING WAVE FUNCTIONS

The single�quasi�particle wave function is found bysolving the Dirac equation with a model potential,which can be written in the two�component form forthe central field (here, fine�structure constant α = 1):

(5)

The relativistic moment number is

(6)

At sufficiently large κ values, radial functions F and Gchange very rapidly in the region r → 0:

F(r), G(r) → rγ – 1, (7)

where γ = (κ2 – α2Z 2)1/2.

H0 h ri( )i

∑ Vc ri b Z( )( ),i

∑+=

Vc

i

∑ Vi j>

∑

α̂i α̂j

α̂i α̂j

dF/dr 1 κ+( )F/r ε m V–+( )G–+ 0,=

dG/dr 1 κ–( )G/r ε m– V–( )F+ + 0.=

κ l 1+( ), j– 1,>=

κ l, j l.<=

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This circumstance hinders numerical integrationin the vicinity of r → 0: the integration step must besignificantly reduced. To avoid this, we will use wavefunctions with weak power�law dependence:

f = Fr1 – |κ|, g = Gr1 – |κ|. (8)

The Dirac equations are then transformed into

(9)

Here, instead of atomic units (a.u.), we introduced theso�called Coulomb units (C.u.) of length and energy:1 C.u. = (1 a.u.) Z for length and 1 C.u. = 219486Z 2

for energy. The atomic characteristics change slowlywith a change in Z in the Coulomb units. Hence, theseunits are convenient for extrapolation with a changealong isoelectronic sequences. The Enκ value is the sin�gle�electron energy of quasi�particle counted from thecore energy. System of equations (9) has two funda�mental solutions. We use the solution regular at r → 0.The boundary values of the correct solution are deter�mined by the first terms of expansion in a Taylor series:

(10)

At correctly specified model potential V(r), thewave function of one electron or one vacancy abovethe core is determined from the conditions f → 0 andg → 0 at r → ∞. At a correctly specified Enκ value, theasymptotic form of the large and small components ofthe wave function can be written as

(11)

where n* = [1/(2|Enk|)]1/2 is the effective principalquantum number. Equations (9) are solved by theRunge–Kutta method. The end of the integrationrange corresponds to rk ≈ 30n*.

The model potential of the 1s22s22p63s23p63d10 coreincludes the nucleus potential and the potentials of theshells with n = 1, 2, and 3:

(12)

The first term in the braces is the Coulomb potential ofa point nucleus. The second and third terms are thepotentials of the K and L shells of the core. The lastterm is the potential of the M shell of the core, forwhich one general parameter b3(nlj|Z) was introduced.Parameter b2(nlj |Z) has been determined previously inenergy calculations for the neon isoelectronicsequence [19, 20, 23]. The electronic part of the modelpotential has the following asymptotic form:

(13)

ENERGY LEVELS OF Kr IX

A matrix element of the energy of excited state withtotal momentum J and projection MJ is presented inthe form of a perturbation�theory series:

(14)

where (n1l1 j1) and (n2l2 j2) are the exact(empirical) single�particle energies of electron andvacancy, respectively. In calculations of Ni�like ions,

(n2l2 j2) is determined by the ionization potentialfrom the filled shell of Ni�like ion and the energy�level

structure of a Co�like ion, while (n1, l1, j1) is deter�mined by the ionization potential of a Cu�like ion andthe energy�level structure of a Cu�like ion (figure).

Note that single�quasi�particle energies + contribute to only diagonal matrix elements. In Ni�like ions, for almost all states, they amount to ~80% ofthe total energy of the transition from the ground state1s22s22p63s23p63d10 to the excited state. Within thelow�order order perturbation theory, one must calcu�late only the matrix elements of the electron–vacancyinteraction operator:

(15)

f ' κ κ+( )– f/r αZVg– αZEnk 2/αZ+( )g,–=

g' κ κ–( )f/r αZVf– αZEnkf.–=

g r0( ) V 0( ) Enκ–( )rαZ/ 2κ 1+( ),=

f 1, for κ 0,<=

f r0( ) V 0( ) Enκ– 2/α2Z2–( )αZ,=

g 1, for κ 0.>=

f g, r/n*–( )exp ,∼

V r( ) 1/Zr( ) Z 2 1 2r–( ) 1 r+( )exp–[ ]–{–=

– 8 1 b2r–( ) 1 0.75b2r+(exp–[

+ 0.25b2r2 0.0625b23r3 ) ]+

– 18 1 b3r–( ) 1 5/6( )b3r+(exp–[

+ 1/3( )b32r2

5/54( )b33r3 ) ] }.+

Vel 28/r at r ∞,→ →

Vel const, dVel/dr 0 at r 0.→ → →

E n1l1j1n2l2j2 JMJ[ ]( )

= Eel0 n1l1j1( ) Evac

0 n2l2j2( ) ΔE 1( ) ΔE 2( ) …,+ + + +

Eel0 Evac

0

Evac0

Eel0

Evac0 Eel

0

ΔE 1( ) ⟨n1l1j1n2l2j2 JMJ[ ] iωr12( )exp 1 α̂1α̂2–( )/r12 n1' l1' j1' n2' l2' j2' JMJ[ ]⟩=

= 2j1 1+( ) 2j1' 1+( ) 2j2 1+( ) 2j2' 1+( )[ ]1/2

× 1–( )j2 j2' a J+ + +

a

∑aj1j1'

Jj2' j2⎩ ⎭⎨ ⎬⎧ ⎫

Qa 12; 2'1'( ) 1–( )j2' j2+

/ 2J 1+( )Qj 12'; 1'2( )+ ,

OPTICS AND SPECTROSCOPY Vol. 117 No. 2 2014

ENERGY LEVELS AND PROBABILITIES OF RADIATIVE TRANSITIONS 171

where

(16)

The tilde sign ~ above the numerical symbol ofelectron or vacancy state denotes the small componentof the Dirac equation. The first four terms in (16) cor�respond to the Coulomb interaction between the elec�tron and vacancy. The last four terms describe theirmagnetic interaction. Function Sa of angularmomenta can be presented in the form of a product:

(17)

where

(18)

(19)

Here, {l1l2l3} denotes the triangle condition for theangular momenta and the parity condition for theirsum. The two�component fragments in (17) have thefollowing symmetry properties:

(20)

The Coulomb two�component fragment is invariantwith respect to the replacement of the large compo�nent of the Dirac equation with the small one:

(21)

The symbols in the large braces and parentheses are,respectively, 6j and 3j coefficients. The radial Slaterintegrals

(22)

are expressed in terms of Bessel functions of the firstand second kind:

(23a)

(23b)

(24)

The calculation of radial Dirac functions, Besselfunctions, and radial integrals in our method isreduced to the solution of a unified system of differen�tial equations with known conditions at r = 0. The sin�gle�particle energies that correspond to the states ofone electron above the 1s22s22p63s23p63d10 core andone vacancy in the same core (energies of Cu� and Co�like ions) are known from studies [14] and [15],respectively.

The higher�order corrections of perturbation the�ory with respect to electron–electron interaction wereclassified in [24]. For convenience and clarity, energycorrections were presented in the form of Feynmandiagrams. We also selected two types of correctionsmaking the largest contribution: (1) interaction ofelectrons through a polarized core (polarization dia�grams) and (2) mutual screening of quasi�particles(staircase diagrams). We found a potential representa�tion for corrections of both types and reported thegeneral integral expressions that almost completely(efficiently) take into account these corrections. Theintegrals for calculating these corrections were alsoincluded in the general system of differential equationsfor calculating radial integrals. It was shown in [24]that this efficient consideration of the higher orders ofperturbation theory significantly improves the calcula�tion accuracy. These corrections were always takeninto account in the subsequent calculations of spectro�scopic constants. The energy levels of the isoelectronicsequence for nickel were calculated in [13] by theRPTMP method for only resonant states with J = 1.

Qa 12; 2'1'( ) Qa n1l1j1n2l2j2; n2'l2'j2'n1'l1'j1'( )=

= Ra 12; 2'1'( )Sa 12; 2'1'( ) Ra 12; 2'1'( )Sal

12; 2'1'( )+

+ Ra 12; 2'1'( )Sa 12; 2'1'( ) Ra 12; 2'1'( )Sal 12; 2'1'( )+

~~ ~~~ ~ ~ ~

~ ~ ~ ~ ~ ~ ~ ~

+ Rl 12; 2'1'( )Sal 12; 2'1'( )

l

∑~ ~ ~~

+ Ra 12; 2'1'( )Sal 12; 2'1'( )

l

∑

– Rl 12; 2'1'( )Sal 12; 2'1'( )

l

∑

– Ra 12; 2'1'( )Sal 12; 2'1'( ).

l

∑

~~ ~~

~ ~ ~ ~

~ ~ ~ ~

Sa 12; 2'1'( ) Sa 11'( )Sa 22'( ),–=

Sal 12; 2'1'( ) Sa

l 11'( )Sal 2'2( ),–=

Sa 11'( ) l1 l '1a{ } j1 j '1 a

1/2 1/2– 0⎝ ⎠⎜ ⎟⎛ ⎞

,=

Sal 11'( ) l1l 1' l{ } j1' j1 a

1/2 1/2– 0⎝ ⎠⎜ ⎟⎛ ⎞

2a 1+( )1/2=

× 2a a 1+( )( ) 1/2– 2j1' 1 2j1 1+( ) 1–( )j1 j1' a+ +

+ +( ��

× a 1 l

1– 1 0⎝ ⎠⎜ ⎟⎛ ⎞

1–( )l 1' j1' 1/2+ + a 1 l

0 0 0⎝ ⎠⎜ ⎟⎛ ⎞

+

× 1–( )l 1' j1' 1/2+ +

.

Sa 11'( ) Sa 1'1( ),=

Sal 11'( ) Sa

l 1'1( ) 1–( )l a l+ +.=

Sa 11'( ) Sa 11'( ).=~~

Ra 12; 2'1'( ) dr1dr2r12r2

2f1 r1( )f2 r2( )∫∫=

× Wa r1r2( )f2' r2( )f1' r1( )

Wa r1r2( ) π r1r2( ) 1/2– 1–( )a 2a 1+( )=

× Ja 1/2+ αZω r1,( )J a– 1/2– αZω r2,( ), r1 r2,<

Wa r1r2( ) π r1r2( ) 1/2– 1–( )a 2a 1+( )=

× Ja 1/2+ αZω r2,( )J a– 1/2– αZω r1,( ), r1 r2,>

ω E10 E2

0– E2'0 E1'

0–+ .=

172

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RADIATIVE TRANSITION PROBABILITIESTO THE GROUND STATE

Currently, there are no methods for direct experi�mental measurement of RTP in multiply charged ions.Experimenters observe excited�state lifetimes, whichyield the total characteristic of radiative decay rates ofstate over all possible channels. The radiative transi�tion probabilities play a very important role in the cal�culations of level populations and spontaneous emis�sion gains, as well as in the modeling emission spectraof ions in plasma. When calculating RTP, it is impor�tant to perform a detailed comparison of the calcula�tion results with the corresponding data obtained byother atomic�calculation methods.

The RPTMP method is based on the unified energyapproach of quantum electrodynamics. Here, theenergy matrix is presented in complex form. Theimaginary part of the k level is its total width, wherestate decay probabilities Wkn over all possible channels,

(25)

are taken into account. Here, Γk is the total level width

.

The imaginary part of the matrix element is pre�sented as the sum of partial contributions of the prob�abilities of electron transitions to the core state andbetween excited ion states. In the case of ions with afilled shell in the ground state, the imaginary part ofthe matrix element, corresponding to radiative decayto the ground state, can be written as

(26)

Here, the subscripts 1 and 2 refer to the quantumnumbers of the electron and vacancy, respectively.Quantity Qλ(1 2; 1 2) is given by expression (16). It canbe written in the form

(27)

where and are, respectively, the Coulomband magnetic parts of the matrix element. The imagi�nary part of radial integrals is presented in the form ofa product of single integrals. Their determination isrelated to the choice of photon propagator gauge.Here, as in [11, 12], the Lorenz condition is used forthe photon propagator gauge. The total probability ofλ�pole transition is presented as the sum of the contri�

butions of electric ( ) and magnetic ( ) radia�tion:

(28)

Ek EkRe( ) iΓk/2,–=

k km

M

WΓ = ∑

ImE n1l1j1n2l2j2[ ]J 1s22s22p63s23p63d10( )

= 2j1 1+( )– 2j2 1+( ) 2J 1+( ) 1–ImQλ 12; 12( )δJλ.

λ

∑

Qλ Qλ

Coul Qλ λ 1–,

Br Qλ λ,

Br Qλ·λ 1+

Br,+ + +=

CoulQλ

Br,JQλ

EWλ

MWλ

Wλ

E Qλ

Coul Qλ λ 1–,

Br Qλ λ 1+,

Br,+ +=

(29)

In the case of λ�pole radiation of electric type, thestate parity changes by (–1)λ, while, in the case ofmagnetic λ�pole radiation, the state parity changes by

(–1)λ+1. We used this principle to divide and into electric and magnetic parts. RTP calculations areoften performed using an approximation: expansion inseries in parameter αω or αZ. The asymptotic terms ofexpansions of expressions (28) and (29) have the form

(30)

(31)

To pass to the nonrelativistic limit, one must expandthe radial integrals in parameter α and leave the mainterm. The nonrelativistic limit for the photon propa�gator in the Lorenzian form corresponds to the radia�tive�transition operator in the form of length. For thedipole�transition operator, we have

(32)

The probabilities of electric� and magnetic�dipoleradiative transitions (E1 and M1, respectively) to theground state of Kr IX are compared with the corre�sponding RMBPT calculation data in Table 1. Theresults for the strongest resonant transitions are con�sistent within ~10–30%.

RESULTS AND DISCUSSION

Table 1 presents the results of our calculation of theenergy levels with J = 0, 1 in the Kr IX ion within threeapproximations: zero (E0), first (E0 + ΔE1), andapproximation with partial consideration of higher

orders of perturbation theory (E0 + ΔE1 + Δ ). Thesixth and seventh columns contain, respectively, theRMBPT data and the high�precision experimentalvalues obtained in [6]. In our calculation, the zero�approximation energy is the sum of exact single�parti�cle energies: the energy of one electron above the 3d10

core and one vacancy in the same core, which wasassumed to be zero reference for all energies (see fig�ure). In this approach, the zero single�particle ener�gies and zero�approximation basis wave functions takeinto account most of the interactions of externalquasi�particles with inner�core electrons. For eachlevel, the zero�approximation value amounts to ~80%of the exact energy of excited state of the Kr IX ion.Nevertheless, our calculation in the first order of per�turbation theory, as well as the calculation with partialconsideration of higher orders of perturbation theory,ignores the contributions of the interactions of outerquasi�particles with inner�core electrons, as a result ofwhich the shift of all levels calculated by us with

WJM QJ J,

Br.=

λ

CoulQ λ

Br, JQ

Qλ

CoulαωZ( )

2λ 1+,∼

Qλ λ 1–,

BrαωZ( )

2λ 1+, Qλ λ,

BrαωZ( )

2λ 3+,∼ ∼

Qλ λ 1+,

BrαωZ( )

2λ 5+.∼

Q1E

α 0→lim Q1

Coul Q1 0,Br Q1 2,

Br 2Q1Coul

.≈+ +=

E ho

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ENERGY LEVELS AND PROBABILITIES OF RADIATIVE TRANSITIONS 173

respect to the ground level is ≤1% (~6000 cm–1). Thisshift is retained for the higher�lying configurations3p63d94l (l > 0). In this case, it is expedient to comparethe corresponding experimental values with all levelscalculated with partial consideration of higher orders,shifted down by a constant of about 6000 cm–1. Thedeviation of each level from the corresponding exper�imental value then characterizes the accuracy of spec�ified approximation. A deviation of our calculationresults for almost all states is observed in the fourth sig�nificant digit. Thus, we can conclude that the energiesof the levels calculated in the first order of perturbationtheory from formula (15) are in good agreement withthe experimental value found in [6]. A considerationof the higher orders of perturbation theory in accor�dance with the formulas reported in [24] refines onlyslightly the values found in the first order of perturba�tion theory. The data in Table 1 suggest that the pertur�bation�theory series converges rapidly for almost eachlevel.

This holds true for all other states of the Kr IX ionwith J = 2–6, which are not included in Table 1. Anexception is the upper lasing level 3d5/24d5/2 (J = 0)(1S0 state in the LS�coupling scheme). In the zeroorder, the energy of this level is close to that of anotherlevel with J = 0: 3d3/24d3/2. It is noteworthy that in zeroorder the 3d5/24d5/2 (J = 0) level lies lower than the3d3/24d3/2 (J = 0) level. In the first order in electron–electron interaction, the levels change their position(Table 1). The calculation accuracy can be increasedin two ways: (i) by including higher�lying configura�tions in the calculation or (ii) by taking into accountthe higher orders of perturbation theory. We calculatedenergy levels for the 3d94l and 3d95l (l = 0–3) config�urations. The consideration of the 3d95l configura�tions does not change much the energies of the statesunder study. In contrast to other levels, the consider�ation of the higher orders of perturbation theory is sig�nificant for the 3d5/24d5/2 (J = 0) level. This problem of

Table 1. Energy levels (E, cm–1) for states with J = 0, 1 in the Kr IX ion, calculated in the zero and first orders of perturbationtheory and with partial consideration of the higher orders of perturbation theory, in comparison with calculation by the multi�configuration Dirac–Fock method (RMBPT) [7, 8] and the experimental results of [6]; in the last three columns, the probabil�ities of radiative (AJ0, s–1) level decay to the ground state calculated in this study are compared with the RMBPT data of [7]

Level J E0 E0 + ΔE1 E0 + ΔE1 + Δ E Experiment , + AJ0

this study [7, 8] [6] this study [7]

3d10 0 0.00 0.00 0.00

3d94s

3d3/24s1/2 1 885055 696512 706655 690307 700677 7.31–02, 1.82–02 3.35–02

3d94p

3d5/24p3/2 1 1028100 848961 856914 839222 849552 2.88+08, 3.40+08 7.66+08

3d3/24p3/2 0 1038555 855817 863727 856150 0.00–00

3d3/24p1/2 1 1028755 861830 869929 853155 864021 7.51+10, 6.84+10 8.06+10

3d3/24p3/2 1 1038555 867790 875487 859587 869959 2.39+10, 2.54+10 1.83+10

3d94d

3d5/24d3/2 1 1048700 1086517 1089827 1073077 1080459 1.95+06, 8.98+05 4.33+02

3d5/24d5/2 1 1249900 1095709 1099428 1082298 1093047 1.92+06, 8.68+05 3.61+03

3d3/24d3/2 0 1259155 1100940 1104706 1098307 0.00–00

3d3/24d5/2 1 1260355 1103246 1106966 1089502 1100285 4.12+06, 1.99+05 9.59+02

3d3/24d3/2 1 1259155 1106914 1110531 1093518 1104310 3.22+05, 1.54+05 5.98+01

3d5/24d5/2 0 1249900 1230466 1219660 1168299 0.00–00

3d94f

3d5/24f5/2 0 1437300 1312667 1309917 1301331 0.00–00

3d5/24f5/2 1 1437300 1313768 1311116 1302270 4.62+08, 6.66+08 2.37+09

3d5/24f7/2 1 1438310 1320913 1318914 1310694 7.05+10, 7.21+10 1.11+11

3d3/24f5/2 1 1447755 1333359 1332685 1325293 4.60+11, 6.21+11 5.20+11

Eho AJ0

1( )AJ0

1( )AJ0

ho

174

OPTICS AND SPECTROSCOPY Vol. 117 No. 2 2014

IVANOVA

anomalously large discrepancies between the theoret�ical and experimental values for some singlet states iswell known in the theory of atomic calculations. In ourcase, this is explained by the poor convergence of theperturbation�theory series for the 3d5/24d5/2 (J = 0)level (Table 1). In [25], we considered deviation ΔE ofour theoretical values from the experimental values forthe energy of this level along the isoelectronicsequence of Ni�like ions. The ΔE value is constantwith high accuracy in the range 35 < Z < 80, a fact sug�gesting that the discrepancy is due to the ignored sec�ond�order corrections in the Coulomb electron–elec�tron interaction.

In the RMBPT calculation [7], the levels of the3p63d94s configuration are shifted down with respectto the experimental one by ~10000 cm–1 (they are alsomuch lower than the corresponding levels calculatedby us); for the higher lying configurations, the shift ofthe levels with respect to the experimentally found val�ues changes. For example, in Table 2, which containsthe data for the 3p53d104s and 3p53d104p configura�tions, the energy levels from [7] are close to the corre�sponding levels determined in our calculation or lieabove them.

We calculated the radiative transition probabilitiesto the ground state in two approximations: (i) in thefirst order of perturbation theory and (ii) with partialconsideration of the higher orders of perturbation the�ory [24]. Table 1 contains the probabilities of the elec�tric�dipole transitions from the 3d94p and 3d94f con�figurations and the magnetic�dipole transitions fromthe 3d94s and 3d94d configurations (Table 1, penulti�mate column). The probability of the strongest elec�

tric�dipole transitions are highly stable with respect tothe approximation used: they are consistent with theRMBPT data of [7] within 20–30%. Note that thedecay of the lower lasing state from 3d3/24p1/2 (J = 1) isin agreement with the corresponding RMBPT resultwith an error of ~11%.

A significant discrepancy is observed for the proba�bilities of magnetic�dipole transitions from the 3d94dconfiguration (levels with J = 1) that we calculated andthe RMBPT data [7]. Here, the difference reachesseveral orders of magnitude. Note that the calculation[7] of the probabilities of magnetic dipole decaysincludes not only the 3d4s and 3d4d configurations,but also the 3s4s and 3s4d configurations, which liemuch higher than the ionization potential (figure) andare not bound states of the Kr IX ion. The reason forsuch a large discrepancy must be additionally consid�ered. We should note that the highly lying configura�tions from Table 2 exhibit good stability of the calcu�lated probabilities of resonant decays and good coinci�dence with the results of [7]; however, the probabilitiesof magnetic radiative transitions to the ground statealso differ by several orders of magnitude.

The decay rates of magnetic�dipole states do notplay any significant role in the calculation of the levelpopulation kinetics. Our analysis of the accuracy ofcalculating the energy levels and decay rates of elec�tric�dipole states gives grounds to consider the resultsobtained sufficiently reliable to calculate gains in theKr IX ion. The whole set of spectroscopic constantsand gains obtained in different experimentalapproaches will be reported elsewhere.

Table 2. Energy levels (E, cm–1) and radiative transition probabilities (AJ0, s–1) of states with J = 0, 1 in the Kr IX ion for theconfigurations with electron excitation from the 3p shell in comparison with the results of calculation within the multiconfigu�ration Dirac–Fock method [7, 8] (RMBPT)

Level J E0 E0 + ΔE1 E0 + ΔE1 + Δ E , + AJ0

this study [7, 8] this study [7]

3p53d104s

3p3/24s1/2 1 1821921 1641329 1650875 1648478 1.06+11, 9.79+10 1.43+11

3p1/24s1/2 0 1885826 1699050 1708647 0.00+00

3p1/24s1/2 1 1885826 1702361 1711996 1706048 6.81+10, 6.15+10 4.93+10

3p53d104p

3p3/24p1/2 1 1965621 1785948 1793573 1792789 1.04+06, 5.17+05 1.08+04

3p3/24p3/2 1 1975421 1803790 1810993 1818148 1.62+06, 9.15+05 2.93+03

3p3/24p3/2 0 1975421 1832966 1838276 0.00+00

3p1/24p1/2 1 2029526 1854511 1862064 1865150 1.66+05, 8.69+04 1.85+01

3p1/24p3/2 1 2039326 1867133 1874400 1875574 9.08+04, 8.98+05 1.98+04

3p1/24p1/2 1 2029526 1895279 1896043 0.00+00

Eho AJ0

1( )AJ0

1( )AJ0

ho

OPTICS AND SPECTROSCOPY Vol. 117 No. 2 2014

ENERGY LEVELS AND PROBABILITIES OF RADIATIVE TRANSITIONS 175

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Translated by Yu. Sin’kov