Calculation of Tin Atomic Data and Plasma Properties

Transcription

1 AN-ET-04/4 Cacuaton of Tn Atomc Data and Pasma Propertes prepared by Energy Technoogy Dvson Argonne Natona aboratory Argonne Natona aboratory s managed by The Unversty of Chcago for the U. S. Department of Energy

2 About Argonne Natona aboratory Argonne s managed by The Unversty of Chcago for the U.S. Department of Energy under contract W-3-09-Eng-38. The aboratory s man facty s outsde Chcago, at 9700 South Cass Avenue, Argonne, Inos For nformaton about Argonne and ts poneerng scence and technoogy programs, see Avaabty of Ths Report Ths report s avaabe, at no cost, at It s aso avaabe on paper to U.S. Department of Energy and ts contractors, for a processng fee, from: U.S. Department of Energy Offce of Scentfc and Technca Informaton P.O. Box 6 Oak Rdge, TN phone (865) fax (865) reports@adons.ost.gov Dscamer Ths report was prepared as an account of work sponsored by an agency of the Unted States Government. Nether the Unted States Government nor any agency thereof, nor The Unversty of Chcago, nor any of ther empoyees or offcers, makes any warranty, express or mped, or assumes any ega abty or responsbty for the accuracy, competeness, or usefuness of any nformaton, apparatus, product, or process dscosed, or represents that ts use woud not nfrnge prvatey owned rghts. Reference heren to any specfc commerca product, process, or servce by trade name, trademark, manufacturer, or otherwse, does not necessary consttute or mpy ts endorsement, recommendaton, or favorng by the Unted States Government or any agency thereof. The vews and opnons of document authors expressed heren do not necessary state or refect those of the Unted States Government or any agency thereof, Argonne Natona aboratory, or The Unversty of Chcago.

3 AN-ET-04/4 Cacuaton of Tn Atomc Data and Pasma Propertes by V. Morozov, V. Tokach, and A. Hassanen Energy Technoogy Dvson, Argonne Natona aboratory September 004 Argonne Natona aboratory s managed by The Unversty of Chcago for the U. S. Department of Energy

4 CONTENTS ABSTRACT... INTRODUCTION.... HFS ENERGY EVES AND OTHER DATA.... ION STATES AND POPUATIONS OF ATOMIC EVES THERMODYNAMIC PASMA PROPERTIES AND EQUATION OF STATE DETAIED SET OPACITIES IN WIDE ENERGY RANGE PANCK GROUP AVERAGED OPACTITES IN WIDE ENERGY RANGE HF ENERGY EVES AND OTHER DATA FOR EUV EMITTING TIN ION SPECIES DETAIED TIN EUV TRANSITIONS DETAIED TIN EUV OPACITIES... CONCUSION...3 ACKNOWEDGMENTS...4 REFERENCES...4 APPENDIX A...6 APPENDIX B...9 APPENDIX C...3

5 FIGURES Fg. : Concentratons of tn ons for varous denstes...5 Fg. : Cacuatons from equaton of state for tn ons at varous denstes...7 Fg. 3: Optca coeffcents for tn pasma at T = 6 ev and N = 0 6 cm Fg. 4: Panck mean optca coeffcents for tn pasma...0 Fg. 5: Expermenta data vs. cacuaton for Sn VIII 4 p 7 6 d 4d 5 transton...9 Fg. 6: Cacuaton of tn EUV transtons by the HEIGHTS-ATOM code...0 Fg. 7: Cacuaton of tn EUV transtons by the Cowan code... Fg. 8: Tn pasma optca coeffcents n the EUV range...3 TABES Tabe : Concentraton (n percent) of xenon ons at 0 7 cm Tabe : Concentraton (n percent) of tn ons at 0 7 cm Tabe 3: Energes of Xe XI nner shes by SD HF and average term HF...3 Tabe 4: Energes of Sn VIII nner shes by SD HF and average term HF...3 Tabe 5: O I p 3 3s excted state energy eves...4 Tabe 6: Ar II 3p 4 3d excted state energy eves...4 Tabe 7: Sn VIII 4d 7 ground state energy eves...5 Tabe 8: Xe XI 4d 8 ground state energy eves (n cm - )...6 Tabe 9: Xe XI 4d 7 5p excted state energy eves (n cm - )...6 Tabe 0: Summary of the tn EUV transtons...

6 ABSTRACT Ths report revews the major methods and technques we use n generatng basc atomc and pasma propertes reevant to extreme utravoet (EUV) thography appcatons. The bass of the work s the cacuaton of the atomc energy eves, transtons probabtes, and other atomc data by varous methods, whch dffer n accuracy, competeness, and compcaton. ater on, we cacuate the popuatons of atomc eves and on states n pasmas by means of the coson-radaton equbrum (CRE) mode. The resuts of the CRE mode are used as nput to the thermodynamc functons, such as pressure and temperature from the nterna energy and densty (equaton of state), eectrc resstance, therma conducton, and other pasma propertes. In addton, optca coeffcents, such as emsson and absorpton coeffcents, are generated to resove a radaton transport equaton (RTE). The capabtes of our approach are demonstrated by generatng the requred atomc and pasma propertes for tn ons and pasma wthn the EUV regon near 3.5 nm. INTRODUCTION To meet the requrements of the Inte thography Roadmap goas for hgh voume manufacturng n the future [] and Internatona SEMATECH s EUV Source Program goas [], the EUV source s requred to have a power of 80-0 W at 3.5 nm (% bandwdth). Varous aser produced pasma (PP) and gas dscharge produced pasma (DPP) devces are under deveopment and nvestgaton by dfferent research groups. Both types of EUV sources have advantages and dsadvantages. At present, none of the current EUV sources can dever enough power eves demanded by commerca chp manufacturers. The effcency of generatng EUV radaton s the key factor n successfu deveopment of the source. The eadng EUV groups are usng a varety of scentfc and engneerng approaches to maxmze EUV brghtness from ther devces. A common technque s the ncrease of the optmzed overa converson effcency of a devce, because t mnmzes the requred nput power for a requred EUV output. Because many physca processes are nvoved, and many technca probems need to be soved, when optmzng a partcuar EUV devce, ony computer modeng can generate a compete pcture at a reasonabe pace. We are deveopng an ntegrated mode to smuate the envronment of the EUV source and optmze the output of the source. The mode descrbes the hydrodynamc and optca processes that occur n EUV devces. It takes nto account pasma evouton and magnetohydrodynamc (MHD) processes as we as photon radaton transport. It uses the tota varaton dmnshng scheme n the ax-fredrch formuaton for the descrpton of magnetc compresson and dffuson n a cyndrca geometry. Aso under deveopment are modes for opacty cacuatons: a cosona radaton equbrum mode, a sef-consstent fed mode wth Auger processes, and a nonstatonary knetc mode. Radaton transport for both contnuum and nes wth detaed spectra profes s taken nto account. The deveoped modes are beng ntegrated nto the HEIGHTS-EUV computer smuaton package [3, 4]. Beng sef-consstent,

7 the HEIGHTS-EUV package can generate a requred nformaton and s competey ndependent of the data from other externa packages. Furthermore, expermentay (or numercay) obtaned reabe data can be ncorporated nto HEIGHTS-EUV to ncrease the overa accuracy and effcency of the smuaton resuts. The focus of ths report s the major methods and technques we use n our HEIGHTS- ATOM code to generate basc atomc and pasma propertes. Based upon accuratey generated atomc energy eves, transton probabtes, and other atomc data, we can cacuate the popuatons of atomc eves and on states n pasma by means of the CRE mode. In turn, the resuts of the CRE mode are used as nput to thermodynamc functons, such as those for pressure and temperature from the nterna energy and densty (equaton of state), eectrc resstance, therma conducton, and other pasma propertes. Optca coeffcents are aso generated to resove the RTE. Combnng the generated propertes wth an approprate descrpton of an EUV devce n terms of the MHD boundary condtons, we are abe to smuate the dynamcs of the devce and ts tota radaton output. The accuracy and competeness of atomc data are key factors n the successfu numerca smuaton of the EUV devce. Ths report contans the resuts of our computer smuaton of the tn atomc data and pasma propertes n tabuated form. The cacuaton of atomc propertes by the Hartree-Fock- Sater (HFS) approxmaton s descrbed n the frst secton. Secton s devoted to the descrpton of the CRE mode, whch s used to obtan the popuatons of the atomc eves and the on composton of the pasma. The cacuaton of thermodynamc pasma propertes and the equaton of state are descrbed n the thrd secton. The resouton of the radaton transport equaton n a wde spectra range depends upon the competeness of the optca coeffcents, whch are dscussed n Secton 4. The Panck mean opactes are presented n Secton 5. The next three sectons are prmary dedcated to cacuatng hghy accurate EUV nformaton for tn. Secton 6 dscusses the mtatons of usng the smpfed methods for generatng EUV atomc data and descrbes the advanced S-dependent Hartree-Fock (SD HF) atomc method. Detaed EUV transtons are presented n Secton 7. Fnay, Secton 8 presents cacuatons of tn EUV opactes wth very hgh resouton.. HFS ENERGY EVES AND OTHER DATA In modeng the dynamcs and the output of an EUV source, one needs to dstngush two aspects. Frst, n sovng the hydrodynamc part of the probem, the pasma nterna energy must be corrected and re-dstrbuted accordngy to the radaton transport. A key eement n ths process s the competeness of the radant energy redstrbuton n the whoe pasma doman wthn the very broad spectra range of partcpatng photons. The second aspect s the detaed cacuaton of the effectve radaton of the EUV source wthn the operatng energy range of 3.5 ± % nm. In ths case, the cacuaton of the radaton transport must be orented to the accurate accountng of ony those photons, whose energes are wthn the narrow EUV range. The smuaton of the dynamcs of the pasma evouton typcay nvoves a wde range of temperature and densty vaues and a very compcated onc structure. From the atomc physcs vewpont, detaed resouton of each possbe eve for each possbe on (and

8 consequenty, each eectronc transton n the on) n a wde range of temperatures, denstes, and energes s enormousy aborous, especay when one s accountng for the fne structure of each spt eve. The dffcutes come from the very arge number of atomc terms and spt 5 eves. For exampe, to spt a d she, one needs to account for 6 terms and 37 eves. The 7 f she w have 9 terms and 37 eves. In approxmaton of confguratons, the transton n n 4d 4d 4 f w ony have one strong ne, whe spttng the shes nto eves repaces ths ne by severa hundred weak nes. Knowng that the tota transton strength s unchanged, the radaton n strong nes may be coapsed for a dense pasma, whe the radaton n weak nes stays optcay transparent. The ne spttng may dramatcay nfuence the tota hydrodynamc behavor of a pasma through the radaton transport mechansm [5]. The resuts of theoretca approxmatons of atomc data strongy depend on the chosen theoretca modes [6-8], and ths s partcuary true for ntermedate- and hgh-z materas. To descrbe a sphercay symmetrca quantum system, the sef-consstent fed methods, such as Drac-Fock (DF) or Hartree-Fock (HF) methods, are beeved to be the most effectve. However, the very arge number of ons and ther atomc eves and transtons nvoved nto the pasma computaton, mts ther appcabty. In consstenty sovng the MHD equatons and the RTE, the pasma propertes, opactes, equaton of state, and on concentratons can be obtaned from the structures of atomc eves and transton probabtes, cacuated by smpfed methods. One such method s the HFS approxmaton [9], whch aows the determnaton of the energes and other atomc characterstcs for each n-confguraton of each on that mght exst n the pasma. The smpfcaton of the method normay resuts n the shft of some spectra nes from ther true paces by severa percent, whch s not probematc n determnng the ntegra radaton fux. Nevertheess, we cannot negect the fne structure of open shes, and have thus combned the HFS methods wth the spttng of the energy eves and spectra nes by means of the Racah theory of anguar momentum wthn the framework of perturbaton theory [0,, 5]. The accuracy of the HFS mode s typcay wthn severa percents for the spt eves. However, ths accuracy s not good enough for the second part of our project, whch nvoves determnng the EUV output of the source wthn the very narrow % bandwdth. [To obtan a spectroscopc accuracy of the EUV optca coeffcents, the atomc data are cacuated by the Sdependent (SD) HF method, as dscussed n Secton 6.] In the HFS method the potenta of drect eectron nteracton s cacuated from the rada wavefunctons of partcpatng eectrons, and exchange nteractons are averaged n the form of the exchange potenta V ex. The rada wavefuncton of an atom can be represented as the product of rada wavefunctons P n (r) of the eectrons. It s assumed that a q n equvaent eectrons have the same rada wavefuncton: d dr Z( r) + + V r r ( + ) ( r) + ε n P r ( r) = 0, Z0 ( ) ρ ( x) dx Z r = dr, Vex ( r) = ρ, ρ ( r) = r q r π n r ex n 3 n P n (). r (.) 3

9 Tradtonay, n atomc physcs, the energes are expressed n Ry, and dstances are n a 0 or Bohr unts. In the above equatons, ε n s bndng energy of the eectron, Z (r) s the effectve charge of the on fed, Z 0 s the nucear charge, and ρ (r) s the eectron densty. From souton of the HFS equatons, one may fnd one rada wavefuncton for each she of the atom or on. We normay cacuate the functons for each on, startng from the neutra and endng wth the totay onzed core. Obtaned wavefunctons are used to ater cacuate reatvstc correctons, transton waveengths, and dpoe transtons from the ground state to the hghy excted state (prncpa quantum number may reach up to 0), aowed by the seecton rues. Dscrete transtons from nner shes wth energes mted to the vaue dependent on onzaton potenta were aso taken nto account. Photoonzaton crosssectons were cacuated for a nner and excted states. Detaed spn-orbt spttng of non-fed shes was mpemented k k outsde the HFS, but wthn the CRE mode, by usng Sater ntegras F, G, and constants of k spn-orbt spttng ξ as descrbed n Ref. 5.. ION STATES AND POPUATIONS OF ATOMIC EVES To descrbe the popuatons of atomc eves, we utze the CRE mode, whch s equay appcabe for ow, medum, and hgh temperature ranges. The CRE mode accounts for the exctaton and onzaton processes that can take pace n a pasma. The fact that the CRE mode consders the transtons between a atomc eves s of partcuar mportance. Nonoca effects are accounted for n the form of an escape factor [, 3], whch negects by photoexctaton n contnuum and reduces the strength of spontaneous transtons. Such an approxmaton fary descrbes the pasma behavor n the condton of absence of an externa source of hard radaton. The pasma onzaton state and popuaton eves n for a prescrbed set of temperatures and denstes are cacuated accordng to the system of knetc equatons n statonary form: d n = n Kj + n jk dt j j j = 0. (.) The popuaton of atomc eve s determned by the set of transtons from ths eve to other eves j wth transton rates K j, as we as transtons from other eves j to ths eve wth transton rates K j. One equaton s wrtten for each atomc eve. If eve defnes the ground state, then the popuaton of ths atomc eve gves the concentraton of the on n the pasma. Impact-eectron exctaton and de-exctaton, mpact eectron onzaton, three-body recombnaton, spontaneous transton, and photo- and d-eectronc recombnaton are ncuded n cacuaton of tota rates of eectron transton. From the known popuaton of eves, on and eectron concentratons N and N e are defned for a gven temperature. A numerca smuaton of concentraton of tn ons cacuated by the CRE mode wth spttng HFS atomc eves s presented n Fg. for varous pasma denstes. Due to onzaton of the d-she, the ons change very qucky when temperature ncreases. Ths condton resuts n a ow concentraton of the major on, hardy arger than 50% at best. At the same tme, the cear advantage of tn as an EUV workng eement s that severa ons are EUV productve, 4

10 whch notceaby wden the range of the requred temperature n the devce. Ths stuaton s especay true at ow densty, when EUV emttng ons are appearng at sghty hgher temperature and not changng as fast. Fg. : Concentratons of tn ons for varous denstes Our cacuatons can be vadated and benchmarked by comparson to the resuts of other authors. Accordngy to the resuts cacuated by the CRE mode and reported n [4], the 3 concentratons of xenon ons at n = 0 7 cm e and T e = 3 ev were 0% Xe+9, 5% Xe+0, 0% Xe+, and 3% Xe+, as shown by gray shadng n the second coumn of Tabe. As seen from the other data n the tabe, our cacuatons generay agree wth these fgures, except that we obtaned a twce arger vaue at T e = 3 ev for the concentraton of Xe+0 ons. We tend to rey on neary 45% concentraton of Xe+0, whch woud fx the tota concentraton of a ons to 00% and be more reastc n terms of the reatve concentraton of the major on to the others. The authors of Ref. 4 aso reported on concentratons at smar temperature and densty 5

11 for a tn pasma to be 5% Sn+8, 40% Sn+9, 30% Sn+0, and 5% Sn+. As shown n coumn of Tabe, these resuts are smar to our computatons for T e = 9 ev. Resuts of the tabes generay confrm the very compex composte nature of xenon and tn pasmas, where at the same condtons one may expect to fnd up to seven on speces. Tabe : Concentraton (n percent) of xenon ons at 0 7 cm -3. Top two rows gve temperature and pasma densty. Shaded data from Ref. 4. T e, ev ρ e, g/cm Xe Xe Xe Xe Xe Xe Xe Tabe : Concentraton (n percent) of tn ons at 0 7 cm -3. Top two rows gve temperature and pasma densty. Shaded data from Ref. 4. T e, ev ρ e, g/cm Sn Sn Sn Sn Sn Sn

12 3. THERMODYNAMIC PASMA PROPERTIES AND EQUATION OF STATE After eectron N e and on N concentratons are found by the CRE mode, they can be substtuted nto a set of equatons of state. Generay, the two-temperature approxmaton for pressure p contans the correspondent terms for knetc energy of ons and eectrons. The equaton for nterna energy e nt contans the terms for onzaton and exctaton of eectrons: P = kt e nt 3kTe N = ρ N + kt N, e e e 3kT N + ρ + ρ N j j I + j N ε N Here we use tradtona notaton for Botzmann s constant k, onzaton potenta I of the atomc eve, pasma densty ρ, exctaton energy of eve j to eve ε, eectron and on temperatures T e, T. Resuts of these cacuatons are presented n Fg... j (3.) Fg. : Cacuatons from equaton of state for tn ons at varous denstes 7

13 Recproca to the resstvty η, the eectrca conductvty σ s found as the sum of conductvtes defned by the eectron scatterng σ and σ on charged and neutra partces. ( kt ) 3 c η 4 e β 3 N ee = = + ; σ c = ; σ. n σ σ σ π π e Z = (3.) Λ π m kt N s c n Here we use standard notaton for Couomb ogarthm Λ, concentraton of neutra atom N, eectron charge e, and mean on charge Z. Parameter β s an eectron-eectron scatterng 0 correcton [5]. Emprca vaues of transport crosssectons s 0 are taken from [6]. n e e DETAIED SET OPACITIES IN WIDE ENERGY RANGE The eectronc transtons and ther accompanyng absorpton and emsson of photons are subdvded nto three types: bremsstrahung; photoonzaton from ground, excted, and nner eves; and dscrete transtons. The atter s approxmated n the form of dpoe transtons and ncudes transtons between ground and excted states, transtons between excted states, and party the transtons from nner shes. Because of ts mportance, the profes of spectra nes are processed very carefuy by means of a major broadenng mechansms, such as radaton, Stark, Dopper, and resonance broadenngs [7]. The tota absorpton coeffcent κ abs s cacuated as a sum of absorpton coeffcents for free-free κ ff, bound-free κ bf the popuaton eves: κ ( T, ρ, hω) = ff κ ( T, ρ, hω) = bf κ ( T, ρ, hω) = bb σ j j, k, and bound-bound κ bb radaton transtons weghed to the vaue of ( T, hω ) n ( T, ρ ) n ( T, ρ )( exp( hω kt )) σ ( hω) N j π e f m c e jk j ( T, ρ), Φ( T, ρ, ω) N e j ( T, ρ ). Knowng the crosssectons of nverse processes, the tota emsson coeffcent κ em s cacuated by smar formuae. The vaues of eectron densty n e and on densty n are cacuated as descrbed n prevous secton. The oscator strengths f jk, crosssectons for photo absorpton σ, crosssectons for photoonzaton σ j, and ne profe Φ are gven esewhere [5, 6, 8]. As shown earer [5, 8], accountng for eectrostatc and spn-orbt spttng of shes and spectra nes consderaby nfuences the dynamcs of the energy baance n the pasma. A method of cacuatng the optca coeffcents for hgh-z pasma was deveoped and mpemented [5, 9] by means of the CRE mode. The mportant feature of ths method s the jont use of HFS atomc data and Racah technques of anguar moments. Ths feature makes possbe the use of comparatvey unsophstcated methods to consder the compex eectronc structure of each, (4.) 8

14 partcpatng on, and the compcated spttng of each confguraton nto terms (usng the Scoupng approxmaton), over a wde range of spectra frequences, and n the expected range of temperatures and denstes. By ths technque, the detaed emsson and absorpton spectra are ntay cacuated over a very compete spectra frequency scae (up to 00,000 ponts) for the expected range of MHD vaues. The wdth of the frequency nterva was comparabe to the Dopper wdth of the strongest spectra nes, whch provdes a satsfactory resouton of the ne profes. In Fg. 3 we present cacuated emsson and absorpton coeffcents for the tn pasma at 6 typca temperature and densty (6 ev and 0 cm -3 ). 5. PANCK GROUP AVERAGED OPACTITES IN WIDE ENERGY RANGE Because of the arge sze of the generated opacty tabes, the practca use of such detaed data s not convenent, and the emsson and absorpton coeffcents are thus averaged n spectra groups. A rgorous theory for averagng the opactes wthn a group of frequences does not exst. Such an averagng procedure s consdered correct ony when the absorpton coeffcent s constant wthn the group, or the optca thckness of each ne s very sma, and the absorpton becomes a near-ke functon from the frequency. Ths stuaton s ony possbe for a contnuum, but even n that case, every photoonzaton threshod must become a boundary of a group. It becomes even more compcated for the ned spectrum, when the absorpton coeffcent often drops severa orders of magntude wthn a very mted frequency nterva, say, from the center of a very strong spectra ne to the wngs of the same ne, and the center of the ne s optcay thck. Moreover, the temperature and densty vaues may vary. Ths change eads to spectra nes for the other ons, aong wth changes n the wdth of the exstng nes. From a practca vewpont, an organzed seecton of the strongest nes s a reasonabe way to descrbe the optca coeffcents wthn the most mportant hydrodynamc areas for typca temperature and densty vaues. The other nes are averaged wthn broad groups. Unfortunatey, the prmary goa of the numerca smuaton s the determnaton of the typca hydrodynamc parameters wthn the mportant areas of the pasma doman! For a unform sotherma pasma, the optca thckness of a spectra ne s determned by mutpyng the absorpton coeffcent of the ne by the near dmenson of the pasma τ ε = κ abs ε. In the case of a nonunform nonsotherma pasma, ths defnton s generazed ( ) ( ) as ( ε ) κ ( τ = T, ρ, ε ) d over the nterva, where the on exsts emttng wth frequency ε. abs The borders of the groups are cacuated from the foowng consderatons. Frst, the wdth of a group ε shoud never exceed α T, where T s the pasma temperature, and α s a chosen parameter for averagng. Ths condton provdes the smooth averagng of opactes n the contnuum regon, where the averagng s normay performed wthn the broad groups, and the optca thckness s much ess than one. Second, nvarabty of the optca coeffcent s requred for a chosen vaue β wthn the group. Ths condton provdes a very specfed resouton of the spectra nes wth the optca thckness neary at unty, and the wngs of the nes wth the optca thckness greater than unty. A fna consderaton s that a those 9

15 frequences whch beong n the detaed spectrum to a ne wth ( ε ) τ are aso ncuded n the doman of the groups. Ths condton provdes a very thorough resouton of the strong nes n the averagng spectrum. By varaton of the parameters α and β, severa group mean opactes are generated wth dfferent eves of competeness and deta. Fg. 3: Optca coeffcents for tn pasma at T = 6 ev and N = 0 6 cm -3 Fg. 4: Panck mean optca coeffcents for tn pasma, averaged over 69 (eft) and 340 (rght) groups 0

16 Based upon severa recent studes [3, 4], t s supposed that the maxma radaton fux corresponds to the moment of pnch formaton. Typca tn pnch parameters are the foowng: 7 the temperature s cose to 5 ev, the densty s 3 0 cm -3, the spectra range n radaton energes vares from 5 ev to 50 ev, and the average optca pasma thckness s cm. Usng these parameters, we have generated a basc set of opactes, averaged wthn 69 spectra groups. Takng a wder set of temperature and densty pars, we have generated the optma scae of spectra groups. Combnng a scaes, the resutng energy scae has a tota of 340 energy groups. Resuts of the Panck mean absorpton and emsson coeffcents for the mentoned temperature and densty vaues are shown n Fg. 4 above. As the next step, the output of the sef-consstent hydrodynamc/radaton transport cacuatons wth these opactes can be used for further mprovements n the quaty of the coeffcents. 6. HF ENERGY EVES AND OTHER DATA FOR EUV EMITTING TIN ION SPECIES The accuracy of the HFS mode s typcay wthn severa percents for the spt eves, whch s nsuffcent for the narrow % bandwdth. To obtan a hgher accuracy of the EUV optca coeffcents, we use the SD HF method, whch s more accurate but sgnfcanty more advanced and dffcut to mpement. Strcty speakng, the HF method encompasses severa methods for cacuaton of varous atomc structures. Beng apped to the same atomc system, each modfcaton of the HF method can produce dfferent resuts, so the choce of the approprate method s very mportant n terms of the overa accuracy of the cacuatons. In the HF method the wavefunctons, egenvaues, and tota energy of an atom are found from the varatona prncpe for the we-known Schrödnger equaton: Ĥ Ψ = EΨ, (6.) where Ĥ s a Hamtonan of the atom, ncudng the nteracton of each eectron wth the nuceus and the other eectrons; E s the tota energy of an atom; and Ψ s the atomc wavefuncton, whch s expressed through the atomc orbtas ϕ r ) ϕ ( r ) as ϕ ϕ n ( r ) ϕ( r ) ϕ( rn ) ( r ) ϕ ( r ) ϕ ( r ) ( r ) ϕ ( r ) ϕ ( r ) n n ( j n j n Ψ =. (6.) ϕ The number of the atomc orbtas s determned by the number of shes n the atom. For nstance, the tn atom has s s p 6 3s 3p 6 3d 0 4s 4p 6 4d 0 5s 5p ground state, or orbtas. Each orbta s found from the system of ntegra-dfferenta equatons for functons ϕ n :

17 d dr N n n ( + ) + r x β x Z r + N N x x ( n) y ( r) + α ( n, n ) y ( r) ε ϕ ( r) x ( n, n ) y () r ϕ () r ε ϕ () r = 0. n, n n x f n x n, n n n, n n n n x x n, n The expressons for the rada ntegras y x x n, n ( r), as we as the potentas f yn n ( r) α y x () r, and β y x () r x n, n x n, n n n x,,, are omtted here and can be found esewhere [6]. Here, we just note that they defne the nteracton of the eectrons of the n she wth the other eectrons of the same she, averaged over a anges, and wth the eectrons of the other shes, ncudng both the usua and the exchange nteractons. In genera, the coeffcents f x, α x, and β x depend on the whoe set of quantum numbers, defnng the atomc eve under consderaton, and partcuary on and S. Consequenty, dfferent orbtas (or rada wavefunctons) ϕ, ϕ n n, K correspond to dfferent terms of varous eectron confguratons. In a we-known monograph [8], Cowan ntroduces the concept of the S-dependent Hartree-Fock (SD HF) cacuatons. Hs approach uses a dfferent set of rada wavefunctons for each S term, assumng the pure S-coupng scheme. To be precse, the method shoud be caed SνD HF, because the tota energy for d-shes aso depends on the senorty number ν. The number of terms can be arge, especay for d- and f-shes. Sgnfcant smpfcaton can be ganed by assumng that for a terms the wavefunctons and egenvaues are dentca and cacuated for the center of gravty of the she. The so-caed average term approxmaton works very we for the hghy excted states, but s qute unpredctabe for the outer and nner shes. In contrast to the average term HF, a SD HF wavefunctons, egenfunctons, and Sater ntegras depend on the and S quantum numbers for nner and outer shes. Ths condton requres onger, but more accurate computatons, and very mportant for cacuaton the tota energes of the atomc eves. Tabe 3 presents our cacuatons of the energes of the Xe XI nner shes by the SD HF method. These are compared to the resuts obtaned by the weknown Cowan HF average term code, whch s wdey used n spectroscopc research for dentfcaton of atomc eves. The correspondence s wthn -5%, athough our resuts are consstenty ower than those from the Cowan code for the nner shes, and sghty hgher for the open she. Smar resuts for Sn VIII are presented n Tabe 4, and the energes for the Sn VIII Sn XIII, cacuated by varous methods are shown n Appendx A. The anguar wavefunctons are normay cacuated separatey to the rada wavefunctons by the summaton of the eectron momentums. As a rue, the two mted cases are rarey reazed n practce, that s, when eectrostatc nteracton s consdered predomnant (Scoupng) or spn-orbt nteracton exceeds the eectrostatc nteracton (jj-coupng). In such ntermedate cases, the Hamtonan matrx cannot be wrtten dagona n any coupng schemes. Therefore, the compete matrx s wrtten by transformng the Couomb matrx from Srepresentaton to jj-representaton. The egenvaues of the Hamtonan matrx are found ater by numerca dagonazaton, and the egenvector (purty vector) defnes the composton of the eve, correspondng to ths egenvaue. The eve s normay assgned accordng to the hghest

18 contrbuton of the bass term from the purty vector. Note that the energy eves, found wthn the ntermedate coupng, never have 00 percent pure S- or jj-coupng, and the dfference aways exsts between the eve assgned wthn the ntermedate coupng or the eve cacuated wthn the pure coupng scheme and accordngy assgned as requred by the scheme. The reatvstc effects can be neggbe for the ow-z eements, but apparenty become evdent for ntermedate-z and hgh-z eements. The most wdespread and reatvey easy way to account for them wthn the HF method s by one-eectron reatvstc correctons wthn the framework of perturbaton theory. The more strct and accurate way to account for reatvstc effects s to use the Drac-Fock approxmaton, but ths sgnfcanty compcates the probem, whe the gan from t woud be pronounced ony for the hgh-z eements. Tabe 3: Energes of Xe XI nner shes by SD HF and average term HF. HEIGHTS-ATOM Cowan Conf S P D 3 F G E av s s p s p d s p d Tabe 4: Energes of Sn VIII nner shes by SD HF and average term HF. Conf P 4 P D HEIGHTS-ATOM 3 D F 4 F G Cowan H E av s s p s p d s p d As foows from Tabes 3 and 4, t s very hard to benchmark and verfy the accuracy of the computaton of such compcated eements as xenon or tn wthout expermenta resuts. Before proceedng wth our tn cacuaton, we checked the accuracy of our code on we-known eements wth avaabe expermenta measurements of the eves for oxygen (Tabe 5) and argon (Tabe 6). The NIST tabes are pubshed n Ref. 0. The smpcty of oxygen and argon comes from the pronounced S-coupng approxmaton, whch aows us to unquey dentfy and assgn the eves once the ntermedate coupng s apped. The average term HF method presents smar, but sghty ess accurate resuts, despte the fact that the average term approxmaton s expected to work very we for the excted states [8]. As mentoned above, the ntermedate-z eements, such as tn, do not have any pure 3

19 coupng scheme, and the ntermedate coupng approxmaton s the ony reasonabe choce n computer cacuaton of atomc eves. However, the assgnment of the cacuated eves s arbtrary, preservng ony the exact quantum number of tota momentum J. In coumn 3 of the Tabe 7, we present our resuts for the Sn VIII energy eves, and n the coumn 7 of the same tabe are the resuts of the Cowan code. We have ordered the eves by ncreasng the tota moment and arranged the Cowan code eves accordngy to our dentfcaton. Snce dfferent ntermedate coupng codes mght have dfferent namng schemes, the reasonabe way of presentng the resuts s arrangng the eves accordngy to the tota moment of the eve. The expermenta energy eves are avaabe thanks to the Ref. 3, and we can benchmark the accuracy of these cacuatons. As one can see, the energy eves computed by us are cose to the Cowan code resuts and dffer from the expermentay defned vaues from 3% to %. Tabe 5: O I p 3 3s excted state energy eves (n cm - ). Excted eve HEIGHTS-ATOM Cowan NIST eve Acc,% eve Acc Tabes p 3 ( P)3s( S) 3 P p 3 ( 4 S)3s( S) 3 S p 3 ( P)3s( S) P p 3 ( P)3s( S) 3 P p 3 ( D)3s( S) 3 D p 3 ( 4 S)3s( S) 5 S p 3 ( P)3s( S) 3 P p 3 ( D)3s( S) D p 3 ( D)3s( S) 3 D p 3 ( D)3s( S) 3 D Tabe 6: Ar II 3p 4 3d excted state energy eves (n cm - ). Excted eve HEIGHTS-ATOM Cowan NIST eve Acc,% eve Acc Tabes 3p 4 ( D)3d( D) S p 4 ( D)3d( D) P p 4 ( 3 P)3d( D) P p 4 ( 3 P)3d( D) 4 P p 4 ( 3 P)3d( D) 4 D p 4 ( S)3d( D) D p 4 ( D)3d( D) P p 4 ( D)3d( D) D p 4 ( 3 P)3d( D) P p 4 ( 3 P)3d( D) 4 P p 4 ( 3 P)3d( D) D p 4 ( 3 P)3d( D) 4 D p 4 ( 3 P)3d( D) 4 F p 4 ( S)3d( D) D p 4 ( D)3d( D) D p 4 ( D)3d( D) F p 4 ( 3 P)3d( D) 4 P p 4 ( 3 P)3d( D) D p 4 ( 3 P)3d( D) 4 D p 4 ( 3 P)3d( D) F p 4 ( 3 P)3d( D) 4 F p 4 ( D)3d( D) F p 4 ( D)3d( D) G p 4 ( 3 P)3d( D) 4 D p 4 ( 3 P)3d( D) F p 4 ( 3 P)3d( D) 4 F p 4 ( D)3d( D) G p 4 ( 3 P)3d( D) 4 F

20 From a theoretca standpont, the agreement of the cacuated atomc energy eves wth the expermentay measured vaues wthn 3% to % may be consdered encouragng, but n reaty, the requred EUV bandwdth has much greater restrctons. The we-known effect of overestmaton of the theoretca energy-eve spttng resuts n sghty arger vaues of Couomb eectron-eectron nteractons [8, ]. The eectron correaton cannot be determned scaed-down theoretca vaues of the snge-confguraton Sater ntegras. Any expermenta data can actuay be a great hep n determnng the exact vaues of the scang factors, as demonstrated n Tabe 8. The expermenta vaues n the second coumn are taken from Ref.. As expected, the purey theoretca vaues for the ground energy eves of Xe XI, shown n coumn 3, are sghty overestmated, by approxmatey 0%, but scang the Sater ntegras to a factor of 0.8 yeds a very accurate resut, wthn ess than % for most ground eves. Thanks to the Ref., we can aso benchmark the accuracy of our cacuaton for the Xe XI 4d 7 5p excted eves. Addtonay, we have cacuated the same energy eves by the Cowan code. The cacuated resuts for Xe XI 4d 7 5p excted eves n Tabe 9 show exceent agreement wth the expermenta vaues, as accurate as -% for most eves. As evdent from the tabe, the Cowan code produces sghty hgher energy vaues, whch woud shft the 4d 8 4d 7 5p transton array toward onger waveengths. Tabe 7: Sn VIII 4d 7 ground state energy eves. Exp HEIGHTS-ATOM (th) HEIGHTS-ATOM (ft) Cowan cm - Energy, cm - Acc Energy, cm - Acc Energy, cm - Acc 4d 7 3P % % % 4d 74 3P % % % 4d 7 3P % % % 4d 74 3P % % % 4d 7 D d 7 3D % % % 4d 74 3F % % % 4d 74 3P % % % 4d 7 D % % % 4d 7 3D % % % 4d 7 3F % % % 4d 74 3F % % 00.3% 4d 7 3F % % % 4d 74 3F % % % 4d 7 3G % % % 4d 74 3F d 7 3G % % % 4d 7 3H % % % 4d 7 3H % % % 5

21 Tabe 8: Xe XI 4d 8 ground state energy eves (n cm - ). Exp Theor Acc,% Modfed Acc,% 4d 8 3 F d 8 3 F d 8 3 F d 8 3 P d 8 3 P d 8 3 P d 8 G d 8 D d 8 S Tabe 9: Xe XI 4d 7 5p excted state energy eves (n cm - ). J HEIGHTS Exp Acc,% Cowan Exp Acc,% J HEIGHTS Exp Acc,% Cowan Exp Acc,% J = 0 J = J = J = 3 J = 3 J = 4 J = 5 J = 6 6

22 The resuts, presented n Tabe 7 above were cacuated n two steps. In the frst step, we obtaned the tota energes of the on by a pure anaytca method. Resuts of ths cacuaton are presented n coumn 3, and ther reatve accuracy compared to the measurements n Ref. 3 s shown n coumn 4. As expected, the pure theoretca method does not aow us to obtan an accuracy of % for the atomc eves of such a compcated on as Sn VIII. We have shaded those eves for whch the cacuatons are not accurate enough. In the second step, we used the 4 F dd = 0.8 F dd, F f ( dd ) = expermentay defned vaues and found that at f ( ) t ( ) F t ( dd ), and ζ ( d ) =. 96 ζ ( d ) f 0 our vaues agree notceaby better, as shown by coumns t 5 and 6 n Tabe 7. In the formuas above, the subscrpt f means ft vaue, and the subscrpt t means theoretca vaue. At ast, we have cacuated the same eves by the Cowan code, whch despte beng purey theoretca, s st utzng scang factors near 0%, dependng upon the nucear charge of the eement [8]. Our cacuatons for the excted Sn VIII 4d 6 5p energy eves are presented n Appendx B. Expermenta vaues are taken from Ref. 3, and the ft vaues were 4 4 F dd 0.57 F dd F dd. F dd ζ d = ζ d, determned as f ( ) = t ( ), f ( ) = t ( ), f ( ) t ( ) 3 3 ζ ( p) = ζ ( p), F ( pd ) = 0.76 F ( pd ), and G ( pd ).4 G ( pd ) f t f t f =. As before, the theoretca vaues agreed wth the expermenta data wthn 5%, and once the expermenta vaues are taken nto account, the accuracy of our numerca smuaton becomes ess than % n most cases. In sum, our cacuated resuts compare favoraby wth avaabe expermenta data and a Cowan-average-term Hartree-Fock code, as we as other atomc codes wth reatvstc correctons. At the same tme, the accuracy of the purey theoretca methods s not wthn the requred %, so wthout the expermenta vaues, one cannot guarantee the cacuaton of accurate EUV transtons. We have attaned an accuracy wthn 3-7% of our purey theoretca cacuatons for Sn ons whch ack expermenta energy vaues. Once the energy eves are obtaned, we can sgnfcanty mprove our cacuatons to practcay as accurate as the measured data. t 7. DETAIED TIN EUV TRANSITIONS Emsson of ght takes pace when an atom changes from a state of hgher energy to a state of ower energy. Smary, absorpton resuts n an upward transton that s caused by the acton of the radaton fed on an atom. The energy of a transton from state to state j s formay defned as the dfference of tota energes of the atom or the on n states and j. As we known from the terature [6-8], the foowng are equvaent measures of tota strength of the spectrum ne: the probabty of a transton W j, the radaton ntensty of a spectrum ne I j, and the weghted spontaneous-emsson transton probabty g j Aj, expressed through the statstca weght of the eve g j = J + and the Ensten spontaneous emsson transton probabty rate A j. In our study, we normay use the non-dmensona (absorpton) oscator strength, whch s reated to the ne strength S by the foowng expresson: 7

23 8 ( ). 3 S J E E f j j + = (7.) As above, uness otherwse stated, we use atomc unts. The energy of the transton ( ) j E E s gven n Rydberg unts, + J s the degeneracy of the nta eve, and the ne strength s defned by the matrx eement of the wavefunctons of the nta and fna states wth the eectrc dpoe operator. Usng as before the ntermedate coupng approxmaton, one can determne the wavefunctons of both states n the form of a near combnaton of S-couped basc functons, where the coeffcents of the components are obtaned from the energy egenvectors for the correspondent states. Ths aows us to represent the ne strength n the form of the two dot products of the two component vectors to the dpoe-transton matrx. Such a technque reduces the probem of cacuatng the ne-strength matrx eements wth uncouped states to the matrx eements wth the S-couped bass functons. The mathematca expressons for the matrx eements are rather compex and engthy, so we drect the reader to the more compete theoretca pubcatons [7, 4, 5]. Usng tradtona atomc theory notaton for 6j- and 9j-symbos, fractona parentage coeffcents, and square brackets, we ony present the fna expressons for cacuaton of the ne strengths of typca transtons: ( ) ( ) [ ] ( ) ( )( ), { {,,,,,,, () 0.5 k k n n J S S k SS k n k n S P S S S S s S S S S S J J S J J S S nk D α α α α δ = (7.) ( ) ( ) [ ] ( ) ( ). {,,, () 0.5 n n J S S S n n S P S S J J S J J n D α α δ = (7.3) In ths study, we do not have more than two open shes, the acute symbo s used for the core quantum numbers, the δ -functon expresses the seecton rue of nonchangeabe spn momentum, and the rada ntegra () P s defned by the matrx eement of the rada wavefunctons of partcpatng shes: ( ) ( ) ( ) + ± = 0, max, ()., max dr rp P P n n δ (7.4) Eectrc dpoe transtons can occur ony when the oscator strength (and, correspondenty, the ne strength) n (7.) s non-zero. From the propertes of the matrx eement n the defnton of the ne strength S, t foows that a transton can occur ony when the partcpatng states have opposte party, and. 0, ± = = J J J The transton 0 = J = J s not aowed. In cacuatng the EUV radaton from tn or xenon pasma, another dffcuty may take

24 pace, such as accountng for the transtons from the nner shes. As reported n recent studes [4], some nner eves of the ons wth ntermedate and heavy atomc numbers may have energes comparabe to the onzaton potenta, whch sgnfcanty decreases the accuracy of the HF method. For exampe, n xenon or tn pasma wth major radatng waveength regon of q q nterest near 3 nm, transtons from outer shes 4d 4d 5p must be consdered n q q combnaton wth the transtons 4d 4d 4 f and the transtons from the nner shes, such as 4 6 q 5 q+ p 4d 4 p 4d. The exstence of sem-emprca or expermenta nformaton on atomc energy eves may greaty hep n mprovng the accuracy of the ab nto smuaton. Unfortunatey, such nformaton for the ntermedate and hgh-z eements, especay ther hghy excted ons, s not fuy presented n the terature [6]. For exampe, no Sn expermenta data are avaabe for the EUV regon, except the work of Azarov and Josh [3], whch s not actuay dedcated to the nm range and ony deas wth the on Sn VIII and ts transton 4d 4d 5p. Accordng to them, the transton has a very wde range of spttng (near 6 nm), wth the major nes beng concentrated around -3 nm (eft fgure n Fg. 5). As shown by the rght fgure n Fg. 5, resuts of our cacuaton show very good agreement n shape, wdth, and pace of the transton. Fgure 6 presents the resuts of our cacuaton of the EUV transtons for sx tn ons, from Sn VIII to Sn XIII. Detas of these cacuatons are presented n Appendx C, and the summary s shown n Tabe 0. V.I. Azarov and Y.N. Yosh [3] Fg. 5: Expermenta data vs. cacuaton for Sn VIII HEIGHTS-ATOM cacuaton 4 p 7 6 d 4d 5 transton We have determned that the 4d - 4f and 4d - 5p transtons ony party cover the EUV range of nterest. Among the fve ons startng from Sn IX, the hghest EUV emsson shoud be from the Sn XI 4d 4 3 4d 4 f transton and from the Sn XII 4d 3 4d 5 p transton. Smar resuts were obtaned by means of the Cowan code and are presented n Fg. 7. Despte the dfference n the hghest vaues of the oscator strength, the code aso predcts the parta coverage of the EUV range by severa transtons, when the hghest emsson corresponds to the Sn X 4d 5 4 4d 4 f and Sn XI 4d 4 3 4d 4 f transtons. 9

25 Fg. 6: Cacuaton of tn EUV transtons by the HEIGHTS-ATOM code 0

26 Fg. 7: Cacuaton of tn EUV transtons by the Cowan code

27 Tabe 0: Summary of the tn EUV transtons. Sn VIII Sn IX Sn X Sn XI Sn XII Sn XIII Ground Confguraton 4d 7 4d 6 4d 5 4d 4 4d 3 4d # eves n GC Excted 4f Confguraton 4d 6 4f 4d 5 4f 4d 4 4f 4d 3 4f 4d 4f 4d 4f # eves n 4f EC Transton array 4d 7-4d 6 4f 4d 6-4d 5 4f 4d 5-4d 4 4f 4d 4-4d 3 4f 4d 3-4d 4f 4d -4d 4f Tota oscator strength Spttng, nm Tota transtons EUV transtons Excted 5p Confguraton 4d 6 5p 4d 5 5p 4d 4 5p 4d 3 5p 4d 5p 4d 5p # eves n 5p EC Transton array 4d 7-4d 6 5p 4d 6-4d 5 5p 4d 5-4d 4 5p 4d 4-4d 3 5p 4d 3-4d 5p 4d -4d 5p Tota oscator strength Spttng, nm Tota transtons EUV transtons DETAIED TIN EUV OPACITIES As dscussed n Secton 4, the optca emsson and absorpton coeffcents are cacuated wthn the CRE mode wth ne spttng. Ths mode s based on the Hartree-Fock-Sater (HFS) method wth both eectrostatc and spn-orbt spttng of confguratons and spectra nes. Instead of the Racah theory of anguar moments for spttng the HFS confguraton average energes nto eves, we use the better approxmaton for the spt energy eves nstead, obtaned by the reaxed core Hartree-Fock method n ntermedate coupng. In ths way, the HFS data for the EUV ons, confguratons and transtons are removed, and substtuted the mproved data drecty n the CRE. Ths guarantees that the EUV range w be covered ony by those transtons that we have cacuated accuratey by the HF method, whe the other energy ranges w st be cacuated as prevousy. The sze of the generated EUV optca tabes s not very arge (around 0-40 MB of dsk space) due to the very narrow energy range and reatvey sma number of transtons wthn that range. Therefore, t s unnecessary to provde addtona modfcatons of the resut tabes, such as group averagng, as we dd prevousy to reduce the number of spectra ponts and the amount of requred dsk space. In Fg. 8 we present resuts of our computaton of the Sn emsson and absorpton coeffcents n the EUV range of nm or ev, for the typca EUV temperature and densty ranges, such as 0-30 ev and cm -3. The coeffcents are very dense, when densty s around 0 7 cm -3, and become sparse when the densty s hgher (up to 0 9 cm -3 ) or ower (beow 0 6 cm -3 ). The resuts of ths cacuaton ndcate that, to maxmze the EUV output, the EUV source needs to operate wthn the named densty range.

28 CONCUSION Fg. 8: Tn pasma optca coeffcents n the EUV range The report revewed the major atomc and pasma methods we use wthn the comprehensve HEIGHTS-EUV package. The methods dffer n accuracy, competeness, and compcaton. The on states, popuatons of atomc eves, and optca coeffcents, such as emsson and absorpton coeffcents, are cacuated by means of the combnaton of the HFS atomc mode and CRE pasma mode wth spttng atomc eves. The accuracy of these methods s satsfactory to smuate the pasma magnetohydrodynamc behavor, but nsuffcent to generate the pasma spectroscopc characterstcs. Sgnfcant accuracy mprovement s acheved by usng the advanced SD HF n ntermedate coupng atomc method. However, emprca correcton of atomc energy eves s necessary to obtan the spectroscopc accuracy of the data and meet the % bandwdth requrement of eadng EUV source manufacturers wthn the 3.5 nm range. The appcabty of the methods s demonstrated for the tn ons and pasma. Deta atomc and pasma propertes, such as energy eves, oscator strengths, equaton of state, reatve on concentratons, wde range opactes, and EUV range opactes near 3.5 nm are cacuated, and compared wth the avaabe expermenta resuts. The presented resuts are used by the authors n smuaton the dynamcs and characterstcs of varous EUV sources. 3