JOURNAL OF APPLIED PHYSICS VOLUME 84, NUMBER 11 1 DECEMBER 1998 Characterstcs of populatons and gans n neon-lke argon Ar IX Dong-Eon Km a) Department of Physcs, Pohang Unversty of Scence and Technology, Pohang, 790-784, Korea Dae-Soung Km Department of Offce Automaton, Youngwol Insttute of Technology, Youngwol, Kangwon-Do 230-800, Korea Albert L. Osterheld Lawrence Lvermore Natonal Laboratory, L-41, P. O. Box 808, Lvermore, Calforna 94551 Receved 21 January 1998; accepted for publcaton 20 August 1998 The characterstcs of the populatons and gan coeffcents n Ne-lke Ar have been nvestgated consderng the 37 levels of the 2p 6,2p 5 3s,2p 5 3p,2p 5 3d,2s2p 6 3s,2s2p 6 3p, and 2s2p 6 3d confguratons. It was found that large gans on the 3p 1 S 0 3s 1 P 1, 3p 3 P 2 3s 1 P 1, and 3p 3 D 2 3s 3 P 1 transtons are formed for the densty between 10 18 and 10 19 cm 3. The effect of the opactes of the 2p 1 S 0 3s 3 P 1,2p 1 S 0 3s 1 P 1,2p 1 S 0 3d 3 D 1,2p 1 S 0 3d 1 P 1, and 2p 1 S 0 2s3p 1 P 1 transtons becomes mportant for a densty hgher than 10 18 cm 3 and even ncreases the gan on the 3p 1 S 0 3s 1 P 1 transton for the opacty of the 2p 1 S 0 3s 1 P 1 transton of up to 2. 1998 Amercan Insttute of Physcs. S0021-8979 98 02423-2 I. INTRODUCTION Snce Zherkhn et al. 1 recognzed that the populatons of some levels n the 2p 5 3p confguraton can be larger than those n the 2p 5 3s confguraton for proper electron denstes and temperatures, the x-ray laser schemes usng transtons between the 2p k 3p and 2p k 3s confguratons (0 k 6) n onc systems have been studed both theoretcally and expermentally. 2 4 Palumbo and Elton 5 predcted the results of gans on the 3s 3p transtons n C-lke ons. Vnogradov and Shlyaptsev 6 have shown that the steady-state populaton nversons for a number of ons between Mg III and Fe XVII can be found. Feldman et al. 7 have done the calculatons for Be-lke, B-lke and Ne-lke onc systems, showng that gans could be produced n transtons between the 2p k 3s and 2p k 3p confguratons. Rosen et al. 8 theoretcally predcted the ntegrated gans on several transtons n Nelke Se. Experments usng laser-produced plasmas have demonstrated the amplfcaton of expected transtons n Nelke ons. 2 4,9 In recent years, Rocca and co-workers 10 has reported the observaton of soft-x-ray amplfcaton and saturaton of the J 0 1 transton n Ne-lke Ar usng a fast capllary dscharge. The soft x-ray amplfcaton of the J 0 1 transtons has also been demonstrated n other low-z Ne-lke ons such as T, 11 13 Cr, Mn, 12 Fe, 12,14 Co, N, Cu, and Zn. 14 These works have motvated us to study the gan characterstcs of transtons between the 2p 5 3s and 2p 5 3p confguratons n low-z Ne-lke ons, especally argon ons. In ths work, n order to nvestgate the characterstcs of gans n Ne-lke Ar, we have calculated level populatons and gans on varous transtons, usng the collsonalradatve model for electron densty between n e 10 16 and 10 20 cm 3 and electron temperature of T e 50 and 200 ev. Ths temperature range has been consdered because the a Electronc mal: kmd@vson.postech.ac.kr abundance of the Ne-lke onzaton stage s expected to be hghest for an electron temperature near a half the onzaton potental. The collsonal-radatve model used here has been used for analyzng atomc populaton knetcs n non-localthermodynamc-equlbrum non-lte plasmas at these electron denstes. In the Sec. II, the model and approach n our smulaton are descrbed:.e., the quas-steady-state collsonal-radatve model, atomc model, and the treatment of the opacty. The smulaton results are presented and dscussed n the Sec. III. II. MODEL FOR CALCULATION The rate equaton for the populaton N l of an excted level l n the th onzaton stage s gven by dn l dt u wth R ul N u n e I l N l n e Q 1 l N 1 1, R ul n e C e ul, l u, n e C d ul A ul, l u, 2 R ll n e C lm m l d e A lm n e C lm m l where n e s the electron densty, C e(d) ul the electron exctaton deexctaton rate coeffcent, and A ul the radatve transton rate from a state u to a state l. The terms I l and Q 1 l represent the electron collsonal-onzaton rate coeffcent from a level l and the total recombnaton rate coeffcent to the level, respectvely. Snce an equlbrum between the excted levels s readly establshed due to ther short lfetmes, the quassteady-state approxmaton has been adopted n our calculaton:, 1 0021-8979/98/84(11)/5862/5/$15.00 5862 1998 Amercan Insttute of Physcs Downloaded 11 Nov 2002 to 141.223.90.45. Redstrbuton subject to AIP lcense or copyrght, see http://ojps.ap.org/japo/japcr.jsp
J. Appl. Phys., Vol. 84, No. 11, 1 December 1998 Km, Km, and Osterheld 5863 dn l dt 0. 3 Furthermore, the recombnaton and onzaton processes between neghborng onzaton stages can be gnored because these processes are neglgbly slow compared to the exctaton and deexctaton processes wthn the Ne-lke onzaton stage under the denstes and temperatures of our nterest. Then, the level populatons n a sngle onzaton stage of nterest are determned by the balance of collsonal and radatve transtons. In the present calculaton the level populatons are normalzed n such a way that m l N l, N I m l 1, 4 l where N I s the total populaton of all the levels consdered. The relatve level populatons are calculated by solvng the coupled rate equatons Eq. 1 under the condtons of Eqs. 3 and 4. The coupled rate equatons can then be rewrtten n the matrx form: m1 m 2 n 1 1 1 1 R 21 R 22 R 23 R 2n m 3 R 31 R 32 R 33 R 3n m R n1 R n2 R n3 R nn 1 1 0 0 0. 5 In ths study, the 37 levels of the 2p 6, 2p 5 3s, 2p 5 3p, 2p 5 3d,2s2p 6 3s,2s2p 6 3p, and 2s2p 6 3d confguratons n Ne-lke Ar have been consdered. All the radatve and collsonal processes between the 37 levels have been taken nto account. The necessary atomc data were obtaned from the Hebrew Unversty-Lawrence Lvermore Atomc Code package HULLAC. In the atomc structure calculaton, the wavefunctons, energy levels, and all the mportant radatve decay rates were obtaned from a relatvstc confguraton nteracton treatment based on a parametrc potental model. 15 Electron mpact exctaton cross sectons were calculated n the semrelatvstc, dstorted-wave approxmaton usng the technques of Ref. 16. In ths approach, the cross secton s factorzed nto a radal part nvolvng wavefunctons only, and an angular factor dependng only on the target states. The semrelatvstc approxmaton treats the contnuum electron nonrelatvstcally, whle retanng a fully relatvstc treatment of the bound electrons. The cross sectons were averaged over a Maxwellan velocty dstrbuton wth a temperature T e to obtan exctaton rate coeffcents. The electron exctaton rate coeffcents were approxmated by the fve-parameter ft: 17 C e lu v 1.58 10 p b 5 lu b lu e b lu cm 3 /s, where b lu E lu, and 6 7 ln p b lu c 0 c 1 ln b lu c 2 ln 2 b lu c 3 ln 3 b lu, where E lu s the transton energy n ev and T e the electron temperature n ev. The deexctaton rate coeffcent s obtaned usng the detaled-balance relaton C d ul g l e b luc e g lu. u For a Doppler-broadened spectral lne, the gan coeffcent on a radatve transton between an upper level u and a lower level l s gven by 8 G 1 M 2 kt on gan g u N u g u N l g l, 1/2 A ul 3 ul g u N u N l g u g l 8 9 10 where M s the atomc mass of the on, k the Boltzmann constant, T on the on temperature, ul the wavelength of the transton, g u and g l the statstcal weghts of the levels, N u, N l the populaton of the levels u and l, respectvely, and gan the gan cross secton. Ion temperature T on were chosen to be approxmately equal to electron temperature T e ; however, T on may be smaller than T e. Usng the relatve populatons, Eq. 10 can be rewrtten as follows: cm 4, 1 G ul /N I 9.98 10 32 3 A ul ul T g 1/2 e u m u m l g u g l 11 where s gven n angstrom, T e n ev. The rght-hand sde of Eq. 11 conssts of terms only related to Ne-lke Ar. Hence, t represents nherent characterstcs of gan on a transton n Ne-lke Ar for a gven electron temperature and densty. A rapd radatve decay of the lower level s crucal to the formaton of the populaton nverson between the two levels. Thus, the radaton from the lower level should freely escape from a plasma. In other words, the plasma must be optcally thn to the transton from the lower level, at least n the transverse drecton. We have also nvestgated the opacty effect on gans. The opacty on a radatve transton between the levels l and m s defned by lm k lm L, 12 where L s the effectve length of the plasma medum and k lm ( ) the absorpton coeffcent, whch s gven by k lm lm e2 1/2 f lm Mc2 m e c2 2kT on N l lm, 13 where f lm s the absorpton oscllator strength between the levels l and m, m e the electron mass, and e the electron charge. The opacty has the effect of ncreasng the populaton of the upper level nvolved. Hgh opacty on the resonance transtons between the 2p 6 and 2p 5 3s confguratons eventually results n the ncrease of the level populatons n the 2p 5 3s confguraton, leadng to the decrease of gans on the transtons between the 2p 5 3s and 2p 5 3p confguratons. Downloaded 11 Nov 2002 to 141.223.90.45. Redstrbuton subject to AIP lcense or copyrght, see http://ojps.ap.org/japo/japcr.jsp
5864 J. Appl. Phys., Vol. 84, No. 11, 1 December 1998 Km, Km, and Osterheld FIG. 1. A schematc dagram of several energy levels under nterest n Ne-lke Ar not to scale. The J 0 1 transton I was observed to have gan and be strongly amplfed at proper condtons Ref. 10. The escape probablty method used n our calculaton approxmates ths by the effectve reducton of the radatve decay rates. In our case, the radatve decay rate of a relevant transton was replaced by the effectve rate gven by E( )A, where E( ) s the escape probablty for a gven and A the orgnal radatve transton rate. For the escape probablty E( ), we have used the followng polynomal ft to tabulated values of E( ) calculated by Zemansky: 18 for 0 4.5, E 0.999003 0.697745 0.266529 2 0.061906 3 0.008106 4 0.000453 5, 14 and for 4.5, 1 E ln. 15 III. RESULTS AND DISCUSSION The relatve sublevel populatons (m /g ) have been calculated by solvng the 37 coupled rate equatons gven by Eq. 5. The gans have then been calculated for the followng transtons between the 2s 2 2p 5 3p and 2s 2 2p 5 3s levels that have large radatve transton rates see Fg. 1 : A 2p 5 3p 3 D 2 2p 5 3s 3 P 1, B 2p 5 3p 3 D 1 2p 5 3s 3 P 1, C 2p 5 3p 1 D 2 2p 5 3s 3 P 1, D 2p 5 3p 3 P 0 2p 5 3s 3 P 1, E 2p 5 3p 1 S 0 2p 5 3s 3 P 1, FIG. 2. Relatve sublevel populatons as a functon of electron densty for gven temperatures: a and c for T e 50 ev; b and d for T e 200 ev. F 2p 5 3p 3 P 1 2p 5 3s 1 P 1, G 2p 5 3p 3 P 2 2p 5 3s 1 P 1, H 2p 5 3p 1 P 1 2p 5 3s 1 P 1, I 2p 5 3p 1 S 0 2p 5 3s 1 P 1, In Fg. 2, the relatve sublevel populatons of the relevant levels are presented as functons of electron denstes at gven two-electron temperatures of 50 and 200 ev. As shown n Fg. 2, the relatve sublevel populatons grow wth the electron densty and the temperature due to the ncrease of collsonal exctaton rates. For a gven temperature, n the regon of low electron densty, the populatons of the 3s 3 P 1 and 1 P 1 levels are smaller than those of other levels under consderaton due to ther strong radatve decay; however, n the regon of hgh electron densty, the populatons become dstrbuted accordng to Boltzmann dstrbuton because the collsonal deexctaton rates exceed the radatve transton rates. Due to the close collsonal couplng of the 2p 5 3p 1 S 0 level to the ground level, the level s most largely populated; however, ts nverson aganst the 2p 5 3s 3 P 1 level destroyed around N e 10 19 cm 3. The densty dependence of the populaton dstrbuton s mantaned for temperatures between 50 and 200 ev. The populatons of the 2p 5 3p 1 P 1, 3 P 1, and 3 D 1 levels are very close to one another as those of the 2p 5 3p 1 P 2 and 3 D 2 level are due to the balance of radatve decay and collsonal processes. The populaton of the 3p 3 P 0 level s also very close to that of the 3s 3 P 1 level and ts populaton nverson wth respect to the 3s 3 P 1 level s destroyed above the mddle of 10 17 cm 3. Ths can be Downloaded 11 Nov 2002 to 141.223.90.45. Redstrbuton subject to AIP lcense or copyrght, see http://ojps.ap.org/japo/japcr.jsp
J. Appl. Phys., Vol. 84, No. 11, 1 December 1998 Km, Km, and Osterheld 5865 FIG. 3. Gan coeffcents per on densty as a functon of electron densty for gven temperatures: a and c for T e 50 ev; b and d for T e 200 ev. The desgnaton for the transtons are gven n Fg. 1. explaned by the fact that the collsonal couplng of the 3p 3 P 0 level to the ground level s not so strong as that of the 3p 1 S 0 level but the 3p 3 P 0 level decays radatvely faster than the 3p 3 D 2, 1 D 2, and 3 P 1 levels. Fgure 3 shows the gans on the transtons mentoned above dvded by the total Ne-lke Ar on densty, Eq. 11. Note that the transtons A, G, and I have potental of hgh gan. For the electron densty below 10 18 cm 3, the gans ncrease wth the electron densty due to the ncrease of the collsonal exctaton rates. The gans reach ther maxmum n the densty regon of a few tmes 10 17 to a few tmes 10 18 cm 3 where the collsonal deexctaton rates of each transton become comparable to ts radatve decay rate. For an electron densty hgher than about a few tmes 10 18 cm 3 for T e 50 ev and 10 19 cm 3 for T e 200 ev, the gans rapdly decrease because the collsonal deexctaton rates exceed the radatve decay rates and the populaton dstrbuton approaches Boltzmann dstrbuton. The gans on the transtons E and I are sustaned at hgher electron densty than on other transtons. Even though the 3p 1 S 0 3s 3 P 1 transton E shares the 3p 1 S 0 level wth the 3p 1 S 0 3s 1 P 1 transton I, ts gan s much smaller than that of transton I due to ts smaller gan cross secton: the gan cross secton of transton I s 4.2 tmes as large as that of the transton E. The larger gan cross sectons of the 3p 3 P 2 3s 3 P 1 G and 3p 3 D 2 3s 1 P 1 A transtons make ther gans comparable to that of the 3p 1 S 0 3s 1 P 1 transton I even though ther upper level populatons are several tmes smaller than that of the 3p 1 S 0 level. Transtons A and G have gan cross sectons 3.1 and 4.6 tmes larger than transton I, respectvely. In Rocca s experment, the dscharge devce was operatng at 700 mtorr of Ar, whch corresponds to the densty of about 2.3 10 16 cm 3 at room temperature. The gan on the transton I was observed at the electron densty of about or less than 1 10 19 cm 3. The present calculaton shows a gan per partcle densty on transton I of 1.5 1.7 10 17 cm 4 for an electron densty of 1 10 18 1 10 19 cm 3 Fg. 3 d and results n a gan of 0.35 0.4 cm 1 for Rocca s expermental condtons, whch s n farly good agreement wth the observed value of 0.6 cm 1 n that the present calculaton dd not take nto account hydrodynamcs. The present calculaton also shows a reasonable gan on transton G, the amplfcaton of whch was also notced by Rocca but ts gan measurement was not pursued due to the complcaton of the spectral lne beng blended wth other lnes from dfferent onzaton stages. We have also studed the opacty effect. In the calculaton, the opactes of the followng fast transtons were taken nto account: 2p 1 S 0 3s 3 P 1, 2p 1 S 0 3s 1 P 1, 2p 1 S 0 3d 3 D 1,2p 1 S 0 3d 1 P 1, and 2p 1 S 0 2s3p 1 P 1. These transtons have larger opactes than other transtons. For a gven opacty of a transton, the opacty of another transton can be related by f, 16 f whch only depends on atomc propertes. In the calculaton, the 2p 1 S 0 3s 1 P 1 transton was used as a reference: the values of gven n Fg. 4 are for the transton. The opactes of the other four transtons were calculated accordng to Eq. 16. Fgure 4 shows the opacty effect on the transtons 3p 1 S 0 3s 1 P 1 and 3p 3 D 2 3s 3 P 1. Below the electron densty of 10 18 cm 3, the opacty effect s rather small for up to 2. The opacty effect becomes mportant for an electron densty larger than 10 18 cm 3 or opacty larger than 4. For a densty hgher than 10 18 cm 3, the opacty up to 2 ncreases the gans. The ncrease s larger for hgher temperature. The opactes n transtons 2p 1 S 0 3d 3 D 1,2p 1 S 0 3d 1 P 1, and 2p 1 S 0 2s3p 1 P 1 reduce ther radatve decay rates, leadng to the effectve ncrease of the populaton of the 3p 1 S 0 level va collsonal and radatve cascades. When ther opactes are ntentonally neglected, the gan n the 3p 1 S 0 3s 1 P 1 transton expectedly decreases as shown n Fg. 4 c. The opacty has a larger effect n low temperature than n hgh temperature:.e., when the pumpng rate s small as n low temperature, even a slght reducton of the decay rate leads to a sgnfcant change of the populaton dstrbuton. It s noted that the 3p 3 D 2 3s 3 P 2 transton s less affected than the 3p 1 S 0 3s 1 P 1 transton. IV. CONCLUSION We have studed the characterstcs of the populatons of the levels n the 2s 2 2p 5 3s and 2p 2 2p 5 3p confguratons and the gans on the fast transtons between them. The 3p 3 D 2 3s 3 P 1, 3p 3 P 2 3s 1 P 1, and 3p 1 S 0 3s 1 P 1 Downloaded 11 Nov 2002 to 141.223.90.45. Redstrbuton subject to AIP lcense or copyrght, see http://ojps.ap.org/japo/japcr.jsp
5866 J. Appl. Phys., Vol. 84, No. 11, 1 December 1998 Km, Km, and Osterheld the electron densty s smaller than 10 18 cm 3. For a hgher densty than 10 18 cm 3, the opacty even ncreases the gans because the reduced radatve decay rates of the transtons between the 2s 2 2p 5 3d, 2s2p 6 3p, and 2s 2 2p 6 confguratons eventually result n the ncrease of the populaton of the 3p 1 S 0 level va collsonal and radatve cascades. ACKNOWLEDGMENTS Ths work has been supported n part by POSTECH/ BSRI specal fund, the Basc Scence Research Insttute Program, Mnstry of Educaton Project No: BSRI-97-2439, Korea Research Foundaton n 1997 and Korea Scence Engneerng Foundaton Contract No: 951-0205-01702. FIG. 4. The opacty effect on gans. The opactes of the 2p 1 S 0 3s 3 P 1, 2p 1 S 0 3s 1 P 1, 2p 1 S 0 3d 3 D 1, 2p 1 S 0 3d 1 P 1, and 2p 1 S 0 2s3p 1 P 1 transtons are consdered. The values of are for the 3s 1 P 1 2p 1 S 0 transton. a The opacty effect on the gan of the 3p 1 S 0 3s 1 P 1 transton for T e 50 ev. b The same as n a but wth T e 200 ev. c The same as n b but n ths case the opactes of the 3s 1 P 1 2p 1 S 0 and 3s 3 P 1 2p 1 S 0 transtons are only taken nto account. d The opacty effect on the gan of the 3p 3 D 2 3s 3 P 1 transton for T e 50 ev. transtons have large gans. The calculated gan on the 3p 1 S 0 3s 1 P 1 transton was found to be n good agreement wth the expermental observaton. The maxmum of the gan on each transton occurs around N e 10 18 cm 3. The densty dependences of the gans at low and hgh temperatures are smlar. The effect of the gans due to reabsorpton seems neglgble f the opacty on the 2p 1 S 0 3s 1 P 1 s less than 2 and 1 A. Zherkhn, K. Koshelev, and V. Letokhov, Sov. J. Quantum Electron. 6, 82 1976. 2 R. C. Elton, X-ray Laser Academc, San Dego, CA, 1990. 3 B. J. MacGowan et al., J. Appl. Phys. 61, 5243 1987. 4 C. Sknner, Phys. Fluds B 3, 2420 1991. 5 L. J. Palumbo and R. C. Elton, J. Opt. Soc. Am. 67, 480 1977. 6 A. V. Vnogradov and V. N. Shlyaptsev, Sov. J. Quantum Electron. 1, 754 1980. 7 U. Feldman, A. K. Bhata, and S. Suckewer, J. Appl. Phys. 54, 2188 1983 ; U. Feldman, J. F. Seely, and A. K. Bhata, bd. 56, 2475 1984 ; U. Feldman, G. A. Doschek, and J. F. Seely, bd. 58, 2909 1985. 8 M. D. Rosen et al., Phys. Rev. Lett. 54, 106 1985. 9 D. L. Matthews et al., Phys. Rev. Lett. 54, 110 1985. 10 J. J. Rocca, V. Shlyaptsev, F. G. Tomasel, O. D. Cortazar, D. Hartshorn, and J. L. A. Chlla, Phys. Rev. Lett. 73, 2192 1994 ; J. J. Rocca, F. G. Tomasel, M. C. Marcon, V. N. Shlyaptsev, J. L. A. Chlla, B. T. Szapro, and G. Gudce, Phys. Plasmas 2, 2547 1995 ; J. J. Rocca, D. P. Clark, J. L. A. Chlla, and V. N. Shlyaptsev, Phys. Rev. Lett. 77, 1476 1996. 11 T. Boehly, M. Russotto, R. S. Craxton, R. Epsten, and B. Yaakob, Phys. Rev. A 42, 6962 1996. 12 J. Nlsen, B. J. MacGowan, L. B. Da Slva, and J. C. Moreno, Phys. Rev. A 48, 4682 1993. 13 J. Dunn, A. L. Osterheld, R. Shepherd, W. E. Whte, V. N. Shlyaptsev, A. B. Bullock, and R. E. Stewart, Proc. SPIE 3156, 114 1997. 14 E. E. Fll, Y. L, G. Preztler, D. Schogl, J. Stengruber, and J. Nlsen, Phys. Scr. 52, 158 1995. 15 M. Klapsch, J. L. Schwob, B. S. Fraenkel, and J. Oreg, J. Opt. Soc. Am. 67, 148 1977. 16 A. Bar-Shalom, M. Klapsch, and J. Oreg, Phys. Rev. A 38, 1773 1988. 17 S. Dalhed, J. Nlsen, and P. Hagelsten, Phys. Rev. A 33, 264 1986. 18 R. W. P. McWhrter, n Plasma Dagnostc Technques, edted by R. H. Huddlestone and S. I. Leonard Academc, New York, 1965 ; M.W. Zemansky, Phys. Rev. 36, 219 1930. Downloaded 11 Nov 2002 to 141.223.90.45. Redstrbuton subject to AIP lcense or copyrght, see http://ojps.ap.org/japo/japcr.jsp