Multipole effects to the opacity of hot dense gold plasma

Transcription

1 Front. hys. China (6) 4: DOI.7/s RESEARCH ARTICLE ZENG Li, JIN Fng-tao, YUAN Jian-min Multipol cts to th opacity o hot dns gold plasma Highr Education rss and Springr-Vrlag 6 Astract Th contriutions o th multipol transitions to th opacity o hot dns gold plasma ar takn into account y using an avrag-atom modl. Th inluncs o th E, E3 and E4 transitions on th Rossland opacity ar studid, rspctivly. Comparisons with Miao s calculation hav n mad. It shows that using th Taylor sris to account or th multipol transitions is no longr valid sinc ik. r is not much smallr than th unit whn th photon nrgy gos vry high. Kywords plasma, opacity, multipol cts ACS numrs 5.5.Os, 5..- o hot and dns mattrs ar considral. In th prsnt study, avrag-atom modl [, 3] is usd to display th changs o th opacity causd y E transitions. And th lctron oritals ar otaind y a widly-usd schm o a ull sl-consistnt Dirac-Slatr modl [3 9]. Sinc th avrag-atom modl is a on-lctron approximation, thr is no M transition twn dirnt lctron oritals, though th M transitions twn th stats o an actual atom within on lctronic coniguration can giv comparal cross sctions and intrrnc cts with th E procsss. Howvr, th nglct o th M transitions would not induc loss in th radiativ asorptions or high photon nrgis. Introduction Gnrally, th magnituds o th so-calld highr ordr transitions, such as th magntic dipol (M) and th lctric quadrupol (E), ar iv to ight ordrs lss than th lctric dipol (E) transitions among th valnc stats o atoms or ions. Howvr, in a vry hot and dns plasma, th innr-shll radiativ transitions play an important rol or th radiativ transr. For ths kinds o innr-shll procsss, transitions du to th highr ordr intractions than th E trm cannot nglctd. A dtaild study on th multipol cts in photoionization cross sctions o isolatd atoms was carrid out y Ron t al. [] within th indpndnt particl modl. For photoionization procsss causd y highr nrgy photons aov kv, th cross sctions du to th E transition ar comparal to, or vn largr than thos o th E transitions. For many cass, th contriutions o th ound-r procsss to th total opacity ZENG Li, JIN Fng-tao, YUAN Jian-min ( ) Dpartmnt o hysics, National Univrsity o Dns Tchnology, Changsha 473, China myuan@nudt.du.cn Rcivd August, 6 Mthod o calculation In our AA modl, th inlunc o th nvironmnt on th atom is assumd to hav sphrical symmtry on avrag. Th movmnt o an lctron undr th intractions o th nuclus and othr lctrons is approximatd y a cntral-ild, which is dtrmind with th standard sl- consistnt calculation. In th cntral-ild, th radial part o th Dirac quation has th orm: () r χkm () rˆ ϕ () r i Q ( r) ( rˆ ) χ km whr (r) and Q (r) ar rspctivly th larg and small componnt o th radial wav unctions, and χ km ( ˆr ) is th spinor sphrical harmonics. Th movmnt o an lctron undr th intractions o th nuclus and othr lctrons is approximatd y a cntral-ild, which is dtrmind with th standard sl-consistnt calculation. In th cntral-ild, th radial part o th Dirac quation has th orm: d k + () r ε c U () r Q () r dr r c d k Q () r ε c U () r () r dr r c ()

2 49 whr ε is th lctron nrgy. Th sl-consistnt potntial U (r) is composd o th static part, th xchang part and th corrlation part, i.., U (r) U s (r) + U x (r) + U corr (r). In th calculation th radial wav unctions satisy th oundary conditions: r l+ () r ar ( r ) or ound lctrons, and ln ( r) dr r r (3) (4) or r lctrons, whr r is th radius o th atom sphr. Th ound lctron dnsity is otaind according to ρ () ( () + ()) r (5) r p r Q 4πr whr is th occupation numr o th stat. In th avrag-atom modl, th occupation numr is dtrmind y th Frmi-Dirac distriution: κ xp[( ε µ ) / T ] + Th r lctron is considrd much mor simply with an assumption o th Thomas-Frmi tratmnt, and th local r lctron dnsity is calculatd with a Frmi-Dirac distriution o th local r lctrons in th plan wav momntum k spac, which can writtn as: k dk ρ () r π k r k c c c V r / T 4 ( ) ( + ( ) µ ) + whr k () r [ U () r c + U ()] r c and µ is th so-calld chmical potntial. Th total lctron dnsity is th sum o ρ (r) and ρ (r), (6) (7) ρ() r ρ () r + ρ () r (8) Th chmical potntial µ is dtrmind so that th lctrical nutrality is satisid 4π r ρ ()d r r Z (9) whr Z is th nuclar charg [3]. In hot dns plasmas, with th incras o th dnsity, th spac occupid y an atom dcrass and som high xcitd lctrons will no longr ound lctrons. Thus th wav unctions o an atom in high dnsity plasma will much dirnt rom that o r atoms. Th siz cts on th lctronic structurs o atoms and ions ar considral and must includd in th calculations. For pur mattr th avrag atomic siz is takn to R 3V 4π /3 whr V is th avrag atomic volum. () Contriutions to th opacity consists o our parts [, ]: NA κν σ ( hν) σ ( hν) σ ( h ) A + + ν hν xp + κs kt B () whr N A is th Avogadro constant, A is th atomic wight, k B is th Boltzmann constant. σ (hν) and σ (hν) ar th ound-ound and ound-r asorption cross sctions at th photon nrgy o hν, rspctivly. Th r-r, σ (hν), contriution taks Kramrs hydrogn-lik approximation: 5/ *3 Z Nion / 3 B hν 3π σ ( hν) g ( h ν) () 3 6 ckt ( ) ( ) whr g (hν) is th Gaunt actor. Th scattring trm, κ s, is approximatd y using Thompson scattring cross sction. Th ound-ound and ound-r asorption cross sctions ar calculatd rspctivly y: and πh σ ( hν) (i + ) nc i, p ( p ) ( hνϕ ) ( hν) i i i i (3) πh di ε σ ( hν) (i + ) pi (4) mc d( hν ) whr ϕi ( hν ) is th lin proil takn as th Gaussian unction. Th transition proaility is proportional to th squar o th matrix lmnt ϕ xp(i k r) α ε ϕi, whr k is th photon wav numr, r is th atomic radius, α is th 4 4 Dirac matrix and ε is th polarization vctor o th photon. Whn ik. r is much smallr than th unit, th trm ik. r can xprssd in th Taylor sris: i kr (i k r) + i k r+ +, k r (5)! whr th trms in th right sid ar corrsponding to th E, M, E transitions t al. Howvr, this xprssion is no longr valid whn th photon nrgy is suicintly high, that is to say ik. r is not much smallr than th unit. Thn, ik. r can xpandd in a sris o th sphrical Bssl unctions []: ikr M ( i) π(l+ ) L L ( L ) L( kr) YLM( θ, ϕ) (6) k L( L+ ) and th lctric multipol transition proailitis oth or ound-ound and ound-r transitions ar writtn as:

3 49 W 4π (L+ ) i Ri L (,, L+ c v B i L ) (7) in which i and corrspond to th initial and inal stat o th transition, and r dr Ri L [ ( ) ( ) ( ) ( )]( ) i r Q r + r Qi r κi κ krl ( kr) kr + [( i( r) Q ( r) + ( r) Qi( r)) L( L+ ) and ( κ κi)( i( r) ε( r) + Qi( r) Qε( r)) L] L( kr) B (, i, L) (l + )( + ) i (8) C l l i i, ; i L W l l L (9) C l li L and W i i whr l, l ; L ar rspctivly th Clsch-Gordan coicint and Racah coicint, L (kr) is th sphrical Bssl unction, and k is th wav vctor. With th transition proaility W i th oscillator strngth is dind as [3]: 3 mc i ( ν ) 3 i h W 8π v mc L + Ri L B ( i,, L ) () π ν L + In practical applications, Rossland and lanck man opacity ar usually rquird. Thy ar rspctivly dind y: κ R κ [ { WR ( u)du K K W ( u)du ν ν () whr u hν /kt, W R and W ar th Rossland and lanck wighting unctions: R u W ( u) 4π ( ) 5 u W ( u) 4 π 3 Rsults and discussions () Firstly, w calculatd th opacitis only including th multipol cts o th ound-ound transitions. Tal shows th Rossland man opacitis o Au at a dnsity o g. cm 3 whn th maximum ras o th lctric multipol transitions ar considrd, rspctivly. From this tal on can asily s that th Rossland man opacitis hav a prcptil incras whn E transitions ar includd }] in th calculations. Howvr, th contriutions o th E3 and E4 transitions to th opacity ar ngligil. Figur shows th spctrally rsolvd opacity o Au at th tmpratur o 5 V and th dnsity o g. cm 3, in which th multipol cts o ound-ound and ound-r transitions hav oth n takn into account. Th asorptions aov 4 a.u. ar rom th photoxcitations and photoionizations o th s orital, and th asorptions low ar rom th highr oritals. With th incras o th photon nrgy, th E transitions hav mor contriutions to th total opacitis. From Eq. () and Eq. () w can ind that th Rossland man opacity is mainly dtrmind y th ottom o th spctrum and th contriutions rom photoionizations com mor important with th incras o th tmpraturs. For comparison, Miao s calculation [3] o Au plasma at th sam conditions using th avrag atom modl is plottd in Fig., in which th E transitions hav n includd. Unlik th prsnt work, Miao t al. uss th irst two trms in th Taylor sris, i.., Eq. (5), to account or th E and E transitions. In this igur on can s that th ound-r asorption has wak wiggls with th E transitions whn th photon nrgy gos highr and highr. In th avrag atom modl ths kind o wiggls ar du to th tratmnt o th lctron radial wav unctions in th plasma nvironmnt. In contrast to th E transition, such oscillations ar mor ovious or E transition du to th r actor in th transition matrix rathr than th r actor in E transition. Howvr, such wiggls do not appar in Fig. caus th prsnt work xpand ik. r using th sphrical Bssl unctions. Tal Rossland man opacity o Au plasma at a dnsity o g. cm 3 whn th maximum ras o th lctric multipol transitions ar considrd. κ, κ, κ 3, κ 4 rr th Rossland man opacitis, in which up to E, E, E3, E4 transitions ar considrd, rspctivly. T/V κ κ κ 3 κ From Eq. (), w know that th Rossland man opacity is snsitiv to th ottom o th spctrum and th Rossland wighting unction rachs its maximum at 3.83 tims th tmpratur. Thus on can xpct that th ound-r asorptions would hav main contriutions to th Rossland man opacity whn th tmpratur gos highr and highr. In Fig. 3, th Rossland man opacitis o Au at th dnsity o g. cm 3 and tmpraturs rom V ar plottd. It is ovious that th multipol cts com im-

4 493 portant only at th tmpraturs gratr than 5 V. In act, at th tmpratur o V thr is only a.78 % dirnc twn th Rossland man opacitis with and without E and E3 transitions in th calculation. As th tmpratur incrass to V, th rlativ dirnc twn th Rossland man opacity with E and E3 transitions incrass to 4.7 %. From Fig. 3 w can also s that th E3 transitions hav littl ct to th opacitis at th tmpraturs having n studid. E3 transitions only hav.7 % contriutions to th total Rossland man opacity at a tmpratur o V and incrass to.7 % at a tmpratur o V. Consquntly, th lin with E transitions in Fig. 3 is almost ovrlappd y th lin with E3 transitions. In th work o Miao t al., th contriutions o E transitions only hav.7 % dirnc to th total Rossland man opacity at a tmpratur o V ut incrass to 85 % [3] at a tmpratur o V. W thi that this work has ovrstimatd th contriutions o th multipol cts sinc th Taylor sris is no longr valid whn th photon nrgy is vry high and ik. r is thror not smallr than th unit. Fig. 3 Rossland man opacity vrsus tmpratur o Au at a dnsity o g. cm 3. In summary, w hav studid th multipol cts to th opacitis o hot gold plasmas using th avrag atom modl. It shows that at highr tmpraturs th E transition has contriutions to oth th spctrally rsolvd opacity and th Rossland man opacity, ut th contriutions rom E3 and E4 transitions ar vry small. Comparisons with th rsults otaind y Miao t al. show that th contriutions o E transitions may ovrstimatd y th approachs o th Taylor sris sinc ik. r is not valid at vry high photon nrgy. Fig. Th opacity o Au at T 5 V and ρ g. cm 3. Th thick dashd and solid lins rr to th cass with and without E transitions considrd, rspctivly. Th thin dashd and solid lins rr to th ound-r opacity with and without E transition considrd, rspctivly. Acknowldgmnts This work was supportd y th National Scinc Fund or Distinguishd Young Scholars undr Grant No. 546, th National Natural Scinc Foundation o China undr Grant No and 44, th National High-Tch ICF Committ in China, and th China Rsarch Association o Atomic and Molcular Data. Rrncs Fig. Spctrally rsolvd opacity o Au Calculatd y Miao t al. [3] at a tmpratur o 5 V and a dnsity o g. cm 3. Th long dashd lin rrs to th rsult o including E transition only. Th long solid lin rrs to th rsult o including E and E transitions. Th shot dashd lin rrs to th opacity including E transition only, and th shot solid lin rrs to rsult o including E and E transitions.. Ron A., Goldrg I. B., Stin J., Manson S. T., ratt R. H., and Yin R. Y., Rlativistic, rtardation, and multipol cts in photoionization cross sctions: Z, n, and l dpndnc, hys. Rv. A, 994, 5: 3. Yuan J. M., Sl-consistnt avrag-atom schm or lctronic structur o hot and dns plasmas o mixtur, hys. Rv. E,, 66: Miao J. S. and Yuan J. M., Contriution o th E transitions to th opacity o hot and dns plasmas o havy lmnts y an avragatom approach, hys. Rv. E, 4, 69: Rozsnyai B. F., Rlativistic Hartr-Fock-Slatr calculations or aritrary tmpratur and mattr dnsity, hys. Rv. A, 97, 5: Faussurir G., Blancard C., and Dcostr A., Statistical mchanics o highly chargd ion plasmas in local thrmodynamic quilirium, hys. Rv.E, 997, 56:3474

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