Correlation and recombination heating in an ultracold plasma. Analitic estimations.

Transcription

1 Corrlaton and rcombnaton hatng n an ultracold plasma. Analtc stmatons. Anatol P. Gavrlyuk Insttut of Computatonal Modlng of Sbran Branch of Russan Acadmy of Scncs Krasnoyarsk , Russa Abstract. In th artcl an ultracold lctron-on plasma cratd by photoonzaton of coold atoms s nvstgatd. W obtand analtcal xprssons for non-dalty paramtrs whch stablsh du to corrlaton hatng. In th work th narst nghbour and th Wgnr-Stz cll approxmatons wr usd, th rcombnaton hatng of lctrons was takn nto account. W hav got a good agrmnt wth th xprmnt rsults. Th possblty of craton of strongly non-dal lctron subsystm has bn shown and condtons of ths procss hav bn dtrmnd. Th xprmnts [1,] rsultd n growng ntrst to an ultracold plasma (UP) wth tmpraturs of partcl T, 100 K. In ths works a quasnutral ultracold lctron-on plasma was frst obtand by mans of a narthrshold photoonzaton of coold atoms. It was xpctd that by ths way strongly non-dal plasma could b producd wth, >>1, whr and ar th paramtrs of non-dalty of lctron and on subsystms rspctvly. But n th postror works [3-6] basd on usng of molcular dynamcs and Mont Carlo tchnqus t was shown that th partcls formd du to photoonzaton wr hatng up quckly. Ths so-calld corrlaton hatng (or dsordr-nducd hatng) s causd by stablshng of spatal dstrbuton of partcls corrspondng to th mnmum potntal nrgy. Th last mans that th dgr of corrlaton of spatal dstrbuton grows fast n ntally non-corrlatd partcl nsmbl. Th scond mpotant caus s hatng of lctrons at a thr-body rcombnaton. Th causs lad to sgnfcant dcras of non-dalty of both lctron and on subsstms vn bfor th plasma cloud bgns to broadn out. Th am of th work s to gt rathr smpl and adquat xprssons dscrbng th abov mntond procsss and th stat of plasma at tms 1, 1 t (ω, ω th lctron and on plasma frquncs rspctvly). It would mak much asy to analz th possblty and condtons of craton strongly non-dal UP. Corrlaton and rcombnaton hatng of lctrons. Whn UP s cratd by photoonzaton thr ar two man procsss (bs ds th photoonzaton) dtrmnng th tmpratur of lctrons: corrlaton hatng and thr-body rcombnaton.

2 Lt s consdr frst th corrlaton hatng of th lctron subsstm and stmat th tmpratur and th paramtr of non-dalty, whch stablsh n UP producd by fast photoonzaton of cold atoms. Th non-dalty paramtr s gvn by th xprsson: whr th lmntary charg, z / kt, a 4 a 3 n 1 3, (1) z th partcl charg (furthr z =1), a th Wgnr-Stz radus, k th Boltzmann constant, T th tmpratur of partcls, n th concntraton of partcls. So coold atom gas wth nglgbly small kntc nrgy of th partcls s xposd to photoonzng radaton wth photon nrgy I 0, I 0 th onzaton potntal of atom, Δ th ntal kntc nrgy of formd fr lctrons and I 0. Thus whn photoonzng th atom, th nrgy s addd to ts ntal nrgy (- I 0 ) and th nrgy of plasma bcoms qual to Ep NI0 NI0 N N, () whr N th full numbr of ons producd. From th othr sd, nglctng th moton of ons th Hamltonan of ths plasma (classcal) can b wrttn as whr k k H p 1 1 N N N N k, (3) kj rk rj k j rk rj k j rk r j k rk r k r, r th radus-vctors of th k -th lctron and on rspctvly, k th kntc nrgy of th k th lctron. Th frst thr sums dscrb th Coloumb ntracton nrgy. In th last sum th frst trm s th nrgy of Coloumb ntracton of th on wth th narst lctron., For th narst nghbour approxmaton (gnorng th ntracton btwn th chargd partcl and all othr partcls xcpt th narst on wth th oppost charg) th total contrbuton of th frst thr trms s zro. Thus th last sum gvs th man contrbuton. Avragng ovr all partcls, w hav for th plasma nrgy: Ep N N N, (4) r r whr / r r U, - th man valus of th lctron-on ntracton nrgy and th lctron kntc nrgy, rspctvly. Th frst on w stmat to th narst nghbour

3 approxmaton ( as n [7-9]). Th dnsty of probablty of fndng th narst lctron on th dstanc R r r from th on s 3 F(R) R a a 3 R xp 3 3, (5) and th nrgy of thr ntracton s U / R. Usng (5) w gt an stmaton: U R xp dr am 1,35. (6) 3 3 R R 3 3 a 3 a 0 a a Hr Гam th gamma-functon. From (4) and (6) w gt U (7) a Thn th non-dalty paramtr of th lctron subsystm s qual to / a / a ( / 3)[ U ] ( / 3)[ 1.35( / a)] Whn 0, th maxmum valu of th non-dalty paramtr s. (8) whr , 0, (9) a 3 th non-dalty paramtr, formally calculatd for th ntal nrgy of photolctrons. Though th valu of 0 can b as bg as s wshd, on can s from (9) that whn rlaxng th lctrom subsystm, th ral non-dalty paramtr stablshs at th lvl ~ 1, that s th lctrons can t b strongly non-dal at 0. Th tm of rlaxaton 1 ~, th lctron plasma frquncy. Ths rsult s n a good agrmnt wth th data [5] obtand on bass of molcular dynamcs mthod. Morovr t can b sn from (8) that t s possbl to rach th rang 1only f 0 U. (10) Undr ths condton th avrag kntc nrgy of lctrons 0 ( Fg.1), thy ar dlocalzd [9]. Ths can b outrmost lctrons of Rdbrg atoms wth crossng aras of localzaton. At U w hav 0 from (7) that contradcts th phscal sns. Indd undr ths condton fr ( dlocalzd ) lctrons ar absnt. So th craton of strongly non-dal lctron subsystm s possbl only f th condton (10) s satsfd.

4 Fg. 1. Th dagram for ponts th ons. E nrgs n plasma: whn U thr ar no fr lctrons, To stmat an nflunc of hatng du to th thr-body rcombnaton on th lctron tmpratur w wll us th dscrpton of rcombnaton suggstd n [10]. Accordng to th artcl th rcombnaton s tratd as a dffuson of lctrons on atom nrgy stats and s dscrbd by th Fokkr-Planck quaton. To a statonary ( on th nrgy stats) rcombnaton flow approxmaton t can b wrttn whr 7 3 dn 1, 10 n j, (11) dt 9/ ( / k) th avrag kntc nrgy of fr lctrons. W assum that ts ntal valu s dfnd by (7). At ths approach th rcombnaton rprsnts as a stppd squnc of lctron transtons btwn bound stats du to collsons wth fr lctrons. Whn th bound lctrons shft down th nrgy axs thr nrgy s transmttd to fr lctrons, th rcombnaton hatng of th last occurs. Bsds th stppd procss rsultng n th hatng of fr lctrons thr ar down transtons du to spontanous dcay. For hgh xctd ( Rdbrg) stats th rat of such procss s small n comparson wth th stppd on but whn lowrng of a stat th stppd procss rat dcrass and th rat of th spontanous on ncrass. Th boundary nrgy can b dfnd [11] 1/ 4 R 1/8 ER 15 n E / k(k), (1) ( / k) blow whch th spontanous dcays domnat. So t can b consdrd that whn th rcombnaton of on lctron occurs th fr lctrons ar gvn th nrgy Wr ER. Thn to a statonary rcombnaton flow approxmaton th chang of kntc lctron nrgy s dscrbd by th quaton:

5 As at n ~ d dt E j. (13) n R 10 cm and / k 100K, E R / k ~ K so w nglct n (13). W nglct also th chang n durng th tm t r. Aftr substtuton for th xprssons for j and ER n (13) w gt: Intgratng (14) from t t0 to t tr w hav: 6.5 d( / k) 1,8 10 n. (14) dt 4.65 ( / k) r 0 0 / k 10 n(t t)( / k), (15) whr 0 th ntal kntc lctron nrgy whch can b qual to n (7). It s asy to s that whn 0 / k n(t t) r th kntc nrgy dpnds wakly on ts ntal valu 0. Exprmntal rsults from [] can b usd to mak sur n adquacy of (15). In th fgur takn from ths work th xprmntal data for plasma clouds xpandng vlosts v 0 dpndng on 0 for dffrnt ntal concntratons of X plasma ar prsntd. On th fgur thr ar also thortcal dpndncs v() 0 0 obtand from (15) and th followng formular n []: v() / M. 0 0 Hr α=1.7 a trmmng paramtr, М th on mass. Fg. Th vlosty of plasma xpanson v0 for dffrnt ntal concntratons (n cm -3 ). Dashd lns th calculatons on bass of (15).

6 Th calculaton of was don for t 0 0. Th valu t r was chosn th xpanson of th cloud to b nsgnfcant. As t follows from [] (s Fg.) ths tm s t r 3 мкс. It s sn from th Fg. that thr s a good agrmnt btwn th thortcal and xprmntal data. Now usng (15) w can stmat th nrgy of lctrons and thr non-dalty paramtr whch stablsh for tm not xcdng th rlaxaton tm of th on subsystm shft of ons s nsgnfcant. Dsrgardng n (15) w gt: 1 0 t ~ (n comparson wth ) and mn 1 ~ whn 0 / k (at Δ=0) ( / k) 0.13n n A, (16) whr А s th atom mass of th on. Thn th non-dalty paramtr taks on a valu Thus th maxmum of 0.0 n () max0.11. (17) 0.09 A whch stablshs n th lctron subsystm bfor ons start movng s wakly dpnds on th plasma concntraton and th on mass and s approxmatly qual to 0.1. Not that dcrass whn Δ (or 0 ) grows. It s asy to mak sur that th quantty (Δn) of rcombnd at ths procss lctrons s a small part of ntally formd ons. Snc th nrgy mttd undr rcombnaton s qual to th nrgy rcvd by th fr lctrons w hav: E n()n. (18) R 0 Th rcombnaton spd s maxmum whn th kntc nrgy s mnmum dscrbd by (16). Thn from (18) and (16) w gt n (19) n E R (nA) 10 3 Evn f n 10 cm and A 100 t lads to only n / n That s th rlatv part of rcombnd lctrons (ons) for th tm t 1 s rally vry small. Ths fact valdats th dsrgardng of chang of concntraton usd abov undr th condtons consdrd. Th corrlaton hatng of ons. As t follows from th prcdng rsults, aftr photoonzaton th lctron subsystm rapdly bcoms wakly non-dal. At th sam tm th ntal spatal dstrbuton of formd ons s homognous and random. That s for tms t th on-on corrlaton s absnt. Takng nto account th wak non-dalty of lctrons w can nglct both lctron-lctron and lctronon corrlatons. In ths cas th nsmbl avrag nrgy of lctrostatc ntracton pr on on

7 s U 0 quals zro. Latr th rlaxaton of spatal on dstrbuton to mor odrd on occurs rsultng n growng of kntc nrgy of ons ( th corrlaton hatng). To stmat th magntud of ths hatng w us a on componnt plasma modl (ОCP). Th rol of background whch nutralzs ons s playd by th wakly non-dal lctron subsystm. Not that th corrlaton hatng of ons n OCP for ntally dsordrd spatal dstrbuton was frst consdrd n [1]. To dtrmn th man kntc nrgy of ons thy hav aftr th corrlaton hatng w us an approxmaton of on sphrs [13] or Wgnr-Stz cll (Fg.3). Th approxmaton s vald f th non-dalty paramtr of ons stablshs n th rang 1 (ths s tru as w wll s blow). Th charg of unform background of th cll s qual to th on charg wth oppost sgn. Th lctrostatc nrgy of ntracton btwn on and background of th cll s [13]: U 3 r a a a, (0) whr th frst trm s th nrgy of th on whn t s n th cntr of th cll and th scond on taks account of th shft of th on on th dstanc r from th cll cntr. Fg.3. Th modl of on sphr. dstanc btwn on and th cll cntr. Th rspctv nrgy of th cll background s: U - th charg dnsty of th unform background, r - th 3. (1) 5 a Thn th full nrgy of lctrostatc ntracton nto th cll contanng an on quals to: s r U U U 0.9 a a a. ()

8 Not that th scond trm n th rght sd of (0) dscrbs th potntal nrgy of harmonc oscllator. Its avrg on nsmbl manng s qual to th avrg kntc nrgy of ons: r 3 kt. (3) a a In UP [1,] th ntal kntc nrgy of ons s nglgbly small ( 0 / k ~ 1 K ) that s why w assum t s zro. Th ntal lctrostatc nrgy s also zro. Th systm s closd, so th full nrgy s constant. Thn w can wrt down for th man full nrgy: E s s U U 0. (4) 0 0 Havng rgard to () and (3) th xprsson (4) taks on a form: E 0.9 3kT 0.9 =0. (5) a a From hr w gt th on tmpratur and th non-dalty paramtr thos th corrlaton hatng has ld to kt 0.3, a / a 3.3. (6) kt Blow n th tabl th thortcal ( dfnd by (6)) and xprmntal tmpraturs of ons aftr th corrlaton hatng ar prsntd. For th thortcal tmpratur of lctrons th lowr lmt obtand from (16) has bn gvn. It s sn that th lctron tmpratur n xprmnts xcdd th thortcal mnmum. That s ntally th lctrons wr wakly non-dal and th rol of thr-body rcombnaton for tms ~ s nglgbl. Th xprmntal tmpraturs of ons prsntd n th tabl rlat mostly to th cntral part of th plasma cloud. In ths ara plasma s mor unform and ts dnsty s clos to th pak valu gvn n th tabl. Th thortcal and xprmntal tmpraturs of ons Sr + whn th corrlaton hatng n UP has occurd. n, cm -3 T, K (thor.) T, K (thor.) T, K (xp.) T, K (xp.) sourc [14] ± [15] [16] [16] As w s from th tabl thr s a good agrmnt btwn th thortcal and xprmntal valus of on tmpratur got as a rsult of th corrlaton hatng.

9 Concluson. Th stmatons prformd hav allowd to obtan smpl and adquat xprssons for tmpraturs ( non-dalty paramtrs) of lctrons and ons thos stablsh du to th corrlaton hatng of th partcls. Ths xprssons ar unvrsal, thy do not dpndnt on sort of ons, and to fnd th mntond paramtrs w do not nd to us any tabl valus or to prform addtonal calculatons. Th dscovrd possblty to gt a strongly non-dal lctron subsystm whn xctng atoms to som spcfd Rdbrg stats s of partcular ntrst. It can b xpctd that usng of odrd («crstall») ntal atom dstrbuton wll b found most advantagous. It has bn shown that th rcombnaton hatng s th major factor dtrmnng th lctron tmpratur for tms. Th approach usd [9] dscrbs th procss qut wll. Th rcombnaton hatng lads to th fact that moton of ons n UP s always gong on aganst th background of wakly non-dal lctrons (namly ths allows to us th OCP modl). That s why usng of UP modls [6], whr th strong shldng of ons by lctrons ( a / 1, - th lctron Dby radus) s consdrd, s probably not vald. Th xprsson drvd for Γ not dpndnt on th tmpratur of lctrons. Howvr such dpndnc was obtan n th work [15]. It can b partly xpland by shldng ( vn wak) of ons by lctrons. It can also b rffrd to th nhomognty of th plasma cloud whch must lad to nhomognous rcombnaton and corrlaton hatng of th partcls gvng rs to addtonal flows of nrgy and partcls. Rfrncs. [1] T. C. Klan, S. Kuln, S. D. Brgson, L. A. Orozco, C. Orzl, and S. L. Rolston, Phys. Rv. Ltt. 83, 4776 (1999). [] S. Kuln, T. C. Klan, S. D. Brgson and S. L. Rolston, Phys. Rv. Ltt. 85, 318 (000). [3] A. N. Tkachv, S. I. Yakovlnko, Quantum Elctroncs 31, 1084 (001). [4] F. Robchaux, J. D. Hanson, Phys. Rv. Ltt. 88, (00). [5] S. G. Kuzmn and T. M. O`Nl, Phys.Rv.Ltt. 88, (00). [6] M. S. Murllo, Phys.Rv.Ltt. 87, (001). [7] S. Chandraskhar, Rv. Mod. Phys. 15, 1 (1943). [8] V. S. Vorob`v, A. L. Khomkn, Thortcal and Mathmatcal Physcs 6, 46 (1976). [9] V. S. Vorob'v and A. A. Lkal'tr, Chmstry of Plasma, Moscow: Enrgoatomzdat 1989 (n Russan). [10] A. V. Gurvch and L. P. Ptavsk, Sov. Phys. JETP 19, 870 (1964). [11] L. M. Bbrman, V. S. Vorob'v and I. T. Yakubov, Kntcs of Nonqulbrum Low- Tmpratur Plasmas, Nw York: Consultants Burau dos

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