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3 1 Quantum and plasma scrning ffcts on th Wannir thrshold law for th doubl-lctron ionization in strongly coupld smiclassical plasmas Young-a Jung a) National Institut for Fusion Scinc, Toki, Gifu, , Japan and partmnt of Applid Physics, Hanyang Univrsity, Ansan, Kyunggi-o , South Kora aiji Kato National Institut for Fusion Scinc, Toki, Gifu, , Japan Th quantum and plasma scrning ffcts on th Wannir thrshold law for th doubl-lctron ionization ar invstigatd in strongly coupld smiclassical plasmas. Th rnormalizd lctron charg and Wannir xponnt ar obtaind by considring th quation of motion in th Wannir configuration with th scrnd psudopotntial modl as functions of th d Brogli wav lngth, by lngth, and charg of th rsidual ion. It is shown that th rnormalizd lctron charg significantly incrass with an incras of th d Brogli wav lngth, spcially, for small radial distancs. It is also shown that th quantum ffcts nhanc th Wannir xponnt for th doubllctron ionization. In addition, th quantum ffct on th Wannir thrshold law is found to b mor important than th plasma scrning ffct in strongly coupld smiclassical plasmas. Kywords: doubl-lctron ionization, Wannir thrshold law, strongly coupld smiclassical plasma a) Prmannt addrss: partmnt of Applid Physics, Hanyang Univrsity, Ansan, Kyunggi-o , South Kora. Elctronic mail: ydjung@hanyang.ac.kr PACS numbr; 5.0.-j

4 Aftr Wannir s sminal work 1 on th ionization thrshold law in 1953, th doubllctron scap,3 from an atom or ion has rcivd considrabl attntion bcaus th invstigation of th long-rang radial lctron-lctron corrlations has playd an important rol in many aras of physics. In addition, rcntly th lctron ionization procsss 4-7 in plasmas such as th lctron-impact ionization and photoionization hav rcivd much attntion sinc ths procsss provid usful information on various plasma paramtrs as wll as knowldg about th dynamics and structur of atomic systms. It has bn shown that th scrnd particl intraction in wakly coupld plasmas would b charactrizd mostly by th standard by-hückl modl. 8,9 In ths wakly coupld, th avrag nrgy of intraction btwn plasma particls is known to b smallr than th avrag kintic nrgy of a particl in plasmas. Howvr, it is obvious that th physical procsss in such strongly coupld plasmas would b quit diffrnt from thos in wakly coupld classical plasmas. Morovr, th intrst in strongly coupld plasmas has rmarkably incrasd du to rcnt invstigations on various physical proprtis of astrophysical compact objcts and lasr inducd inrtial confinmnt fusion plasmas. In addition, it has bn shown that th intraction potntial in strongly coupld smiclassical plasmas would not b rprsntd by th convntional by-hückl modl bcaus of nonidal particl intractions du to collctiv and quantum ffcts Howvr, th bhavior of th Wannir thrshold law for th doubl-lctron ionization in strongly coupld smiclassical plasmas has not bn spcifically invstigatd as yt. Thus, in this papr w invstigat th quantum and plasma scrning ffcts on th Wannir thrshold law for th doubl-lctron scap from th ion in strongly coupld smiclassical plasmas. Th scrnd rnormalizd lctron is obtaind by using th scrnd psudopotntial modl 10 taking into account th quantum and corrlation ffcts as a function of th d Brogli wav lngth, by lngth, and charg of th rsidual ion. Furthrmor, th quantum and plasma scrning ffcts on th Wannir xponnt for th doubl-lctron ionization is obtaind by considring th scrnd ffctiv charg of th rsidual ion in strongly coupld smiclassical plasmas. Th forc quation for a systm of two lctrons ( r1, r ) in th fild of th ion with nuclar charg Z is rprsntd by

5 3 Fr ( ) = ( r) + ( r, r), [ V V ] 1 i 1 1 r1 (1) whr Fr ( 1) is th forc acting on th lctron at th position r 1, Vi ( r 1) and V ( r, ) 1 r ar, rspctivly, th lctron-ion and lctron-lctron intraction potntials. Rcntly, th usful analytic ffctiv psudopotntial 10 of th particl intractions in strongly coupld smiclassical plasmas has bn obtaind on th basis of th dilctric rspons function formalism taking into account th corrlation btwn th Boltzmann factor and quantum mchanical Slatr sum. Using th ffctiv psudopotntial modl, 10 th scrnd lctron-ion Vi ( r 1) and lctron-lctron V ( r, ) 1 r intraction potntials in strongly coupld smiclassical plasmas would b obtaind by Z 1 V ( r ) = xp ( A ( λ, r ) r ) xp ( B ( λ, r ) r ), () i 1 i i 1 i i λi / r r 1 1 V ( r, r ) = xp ( A ( λ, r ) r r ) xp ( B ( λ, r ) r r ), λ / r r 1 r (3) whr λαβ ( = / πμ kt αβ B ) and μ αβ ar, rspctivly, th thrmal d Brogli wavlngth and rducd mass of th α β pair, k B is th Boltzmann constant, T is th plasma tmpratur, and th scrning paramtrs A αβ and B αβ ar givn by Aαβ (1 1 4 λαβ / r ) /( λαβ) and Bαβ ( λαβ / r ) /( λαβ). Aftr som mathmatical manipulations, th quation of motion of th lctron at r 1 in strongly coupld smiclassical plasmas is thn writtn as

6 4 1/ 1/ d r1 Z λi / r λi / r = r 1 + xp r 3 1 dt m 1 4 / r1 r1 i λi r λ λ i 1/ 1/ λi / r λi / r xp r 3 1 r1 r 1 λ i λ i 1/ 1/ λ / r λ / r + r 1 + xp r 3 1 m 1 4 / r1 r1 λ r λ λ 1/ 1/ λ / r λ / r + xp r 3 1, r1 r 1 λ λ (4) whr m is th lctron mass and r 1 ( r 1 r ) is th intrlctronic distanc. In th Wannir mod, 3,14 th jctd two lctrons would b placd at qual radial distancs from th rsidual ion and also in opposit dirctions with rspct to th rsidual ion in th ridg of th potntial surfac. Hnc, th quation of motion in th Wannir configuration [ r1( r) = r ] for th doubl-lctron scap from th ion bcoms d r dt Z (, r λ, r, Z) V ( r ). r r = i 1 m r1 1 = (5) whr Z ( r, λ, r, Z) rprsnts th rnormalizd lctron charg at th radial distanc r in strongly coupld smiclassical plasmas: 1 η(, r λ, r ) Z(, r λ, r, Z) = Z ξ( r, λ, r ) (6)

7 5 Hr, η( r, λ, r ) / ξ( r, λ, r ) trm shows th corrction du to th quantum and scrning ffcts on th rnormalizd lctron charg, and η( r, λ, r ) and ξ ( r, λ, r ) ar, rspctivly, 1/ 1/ λ / r λ / r η(, r λ, r ) = 1+ r xp r 1 4 λ / r λ λ 1/ 1/ λ / r λ / r 1+ r xp r, λ λ (7) 1/ 1/ λ / r 1 1 λ / r ξ(, r λ, r ) = 1+ r xp r 1 λ / r λ λ 1/ 1/ 1+ 1 λ / r 1+ 1 λ / r 1+ r xp r, λ λ (8) whr λ( = / πmkt) is th thrmal d Brogli wavlngth of th lctron-lctron B pair. According to th Wannir s configuration of ridg propagation, 3 th scaping two lctrons mov indpndntly with th scrnd rnormalizd lctron charg Z (, r λ, r, Z) in th fild of th rsidual ion with th ffctiv chang ff Z (, r λ, r, Z)[ = Z (, r λ, r, Z) Z] in strongly coupld smiclassical plasmas. If th quantum ffcts ar absnt ( λ 0) in th plasma, th scrning paramtrs ar turnd out to b Ai 1/ r, A 1/ r, Bi, and B, so that th rnormalizd lctron charg bcoms cas of th wakly coupld plasma. 15 Z rr Z = + Z + r r + r r, i.., th r/ r (,, ) 1 ( /4 )(1 / )/(1 / ) From Eq. (6), w can show that th individual

8 6 ffctiv scrnd lctron charg is givn by Z 1 + (1/ 4 Z) for th cas of th fr targt atom or ion, i.., λ 0 and r. In addition, th radial lctron corrlation ffct on th rnormalizd lctron charg would b nglctd with incrasing Z so that th ffctiv charg bcoms Z 1. According to Wannir s work, it has bn shown that th nrgy (E) dpndnc of th cross sction ( σ ) nar th ionization thrshold I for a fr atom or ion with nuclar charg Z is rprsntd by a powr law σ ( E I) ζ, whr 1/ ζ ( Z){ = (1/ 4)[(100Z 9) /(4Z 1)] 1/ 4} is th Wannir xponnt 1,14 for th doubl-lctron ionization. For a pur Coulomb cas, th solution of Eq. (5) is known as r /3 t. 14 Howvr, for th psudopotntial cas in plasmas it is xpctd that th scrnd ffctiv charg provids th dtaild information on th proprtis of plasmas as wll as th ionization mchanism. Hnc, in strongly coupld smiclassical plasmas, howvr, th ffctiv charg Z 1/4 of th rsidual ion would b rplacd by th scrnd ffctiv charg Z (,,, ) ff r λ r Z du to th influnc of th quantum and scrning ffcts. Thrfor, th scrnd Wannir xponnt function ζ ( r, λ, r, Z) in strongly coupld smiclassical plasmas bcoms 1/ 1 100Z η( r, λ, r) / ξ( r, λ, r ) ζ(, r λ, r, Z) = Z η( r, λ, r)/ ξ( r, λ, r) (9) Th rang of radial distancs of importanc for th Hamiltonian matrix lmnt for th atomic ionization procss is known as r a Z, 16 i.., th distanc btwn th scaping lctron and rsidual ion is comparabl to th Bohr radius, whr az ( = a0 / Z) is th Bohr radius of th hydrognic ion with nuclar charg Z, and a ( / m ) 0 = is th Bohr radius of th hydrogn atom. Thus, th scrnd Wannir xponnt nar th ionization thrshold for th doubl-lctron scap from th targt ion with nuclar charg Z in strongly coupld smiclassical plasmas is thn obtaind by th rlation ζ ( λ, r, Z)[ = ζ( r, λ, r, Z) ]: = r a Z

9 7 1/ 1/ 1 λ / r λ / r λ r λ 1 ζ ( λ, r, Z) = 100Z / 1/ 1/ λ / r λ / r xp 1+ λ λ 1/ λ / r xp λ 1/ 1/ 1 1 λ / r 1 1 λ / r / 1+ xp λ λ 1/ 1/ 1+ 1 λ / r 1+ 1 λ / r 1+ xp λ λ 1/ 1/ 1/ 1 λ / r λ / r λ / r / 4Z 1+ xp 1 4 λ / r λ λ 1/ 1/ λ / r λ / r 1+ xp λ λ 1/ 1/ 1 1 λ / r 1 1 λ / r + xp / 1 λ λ 1/ 1/ 1+ 1 λ / r 1+ 1 λ / r 1 1+ xp, λ λ 4 1/ (10)

10 8 whr λ( λ/ ) is th scald d Brogli wavlngth and r ( r / a ) is th scald a Z Z by lngth. From Eq. (10), it can b also shown that th Wannir xponnt dcrass with an incras of th charg numbr Z. Figur 1 rprsnts th thr-dimnsional plot of th rnormalizd lctron charg Z for th doubl-lctron scap from th ion in strongly coupld smiclassical plasmas as a function of th scald d Brogli wavlngth λ and scald radial distanc r( r/ a Z ). As it is sn, th rnormalizd lctron charg significantly incrass with an incras of th d Brogli wavlngth, spcially, for small radial distancs. Figur shows th rnormalizd lctron charg Z as a function of th scald radial distanc r for various valus of th d Brogli wavlngth. From this figur, it is found that th quantum ffcts on th rnormalizd lctron charg dcrass with incrasing th radial distanc byond th by radius. Figur 3 rprsnts th thr-dimnsional figur of th Wannir xponnt ζ as a function of th scald by lngth r and scald d Brogli wavlngth λ. As it is sn, th quantum ffcts nhanc th Wannir xponnt for th doubl-lctron ionization. It is also found that th quantum ffct on th Wannir thrshold law is mor important than th plasma scrning ffct in strongly coupld smiclassical plasmas sinc th raction zon is placd insid of th by sphr. Hnc, th thrshold cross sctions for th doubl-lctron scap from th ion in strongly coupld smiclassical plasmas is xpctd to b smallr than thos in classical plasmas sinc th scrnd Wannir xponnt can b writtn as ζ[ = ln σ / ln( E I)]. Hnc, w hav found that th quantum ffct plays an important rol on th Wannir thrshold law for th doubl-lctron ionization in strongly coupld smiclassical plasmas. Ths rsults would provid usful information on th quantum and plasma scrning ffcts on th thrshold bhavior of th doubl-lctron scap from th ion in strongly coupld smiclassical plasmas.

11 9 ACKNOWLEGMENTS On of th authors (Y.-. Jung) gratfully acknowldgs th irctor-gnral Profssor O. Motojima, irctor Profssor M. Sato, irctor Profssor Y. Hirooka, and Profssor I. Murakami for warm hospitality whil visiting th National Institut for Fusion Scinc (NIFS) in Japan. This work was supportd by th Kora Rsarch Foundation Grant fundd by th Koran Govrnmnt (MOEHR, Basic Rsarch Promotion Fund) (KRF C00169).

12 10 REFERENCES 1 Wannir G. H., Phys. Rv. 90, 817 (1953).. Kato and S. Watanab, Phys. Rv. A 56, 3687 (1997). 3 J. Brakdar, Concpts of Highly Excitd Elctronic Systms (Wily-VCH, Winhim, 003) p V. P. Shvlko, Atoms and Thir Spctroscopic Proprtis (Springr, Brlin, 1997), p S. P. Khar, Introduction to th Thory of Collisions of Elctrons with Atoms and Molculs (Kluwr, Nw York, 00), p V. P. Shvlko, H. Tawara, F. Schurmann, B. Fabian, A. Müllr, and E. Salzborn, J. Phys. B 38, 55 (005). 7 A. Müllr, Adv. At. Mol. Opt. Phys. 55, 93 (008). 8 F. B. Baimbtov, Kh. T. Nurknov, and T. S. Ramazanov, Phys. Ltt. A 0, 11 (1995) 9 T. S. Ramazanov and S. K. Kodanova, Phys. Plasmas 8, 5049 (001). 10 T. S. Ramazanov and K. N. zhumagulova, Phys. Plasmas 9, 3758 (00). 11 T. Ramazanov, K. Galiyv, K. N. zhumagulova, G. Röpk, and R. Rdmr, Contrib. Plasma Phys. 43, 39 (003). 1 T. S. Ramazanov, K. N. zhumagulova, and Y. A. Omarbakiyva, Phys. Plasmas 1, 0970 (005). 13 T. S. Ramazanov and K. N. Turkhanova, Phys. Plasmas 1, 1050 (005). 14 G. F. rukarv, Collisions of Elctrons with Atoms and Molculs (Plnum, Nw York, 1987), p Y.-. Jung, Phys. Scr. 7, 37 (005). 16 H. A. Bth and E. E. Salptr, Quantum Mchanics of On- and Two-Elctron Atoms Springr-Vrlag, Brlin, 1957), p. 316.

13 11 FIGURE CAPTIONS FIG. 1 Th thr-dimnsional plot of th rnormalizd lctron charg Z as a function of th scald d Brogli wavlngth λ and scald distanc r whn Z = and r = 0. FIG. Th rnormalizd lctron charg Z as a function of th scald distanc r whn Z = and r = 30. Th solid lin rprsnts th cas of λ = 1. Th dashd lin rprsnts th cas of λ = 4. Th dottd lin rprsnts th cas of λ = 6. FIG. 3 Th thr-dimnsional plot of th Wannir xponnt ζ as a function of th scald by lngth r and scald d Brogli wavlngth λ whn Z =.

14 Figur Z Hr ê,ll ê r 0 4 l 30 0

15 Figur ZHr ê L ê r

16 Figur z Hr ê,ll ê r l 8