WEST VIRGINIA UNIVERSITY

WEST VIRGINIA UNIVERSITY PLASMA PHYSICS GROUP INTERNAL REPORT PL - 045 Mea Optical epth ad Optical Escape Factr fr Helium Trasitis i Helic Plasmas R.F. Bivi Nvember 000 Revised March 00

TABLE OF CONTENT.0 Itrducti.0 Light absrpti i plasmas.0 Absrpti ad emissi i plasmas 4.0 Optical Escape Factr ad Mea Optical epth 5.0 Evaluati f Optical Escape Factrs ad Mea Optical epths fr several He Trasitis 5. Evaluati f the lwer excited level ppulatis 5. Evaluati f the Optical Escape factrs ad Mea Optical epth 6.0 Iterpretati 7.0 Cclusi 8.0 Refereces Table - 7 Figure Figure Lwer level excited ppulati, Mea Optical epth, atteuati fracti ad Optical Escape Factr fr the differet S P ad S P trasitis as a fucti f the electr desity Optical Escape Factr as a fucti f the Mea Optical epth He Grtria diagram with imprtat visible trasiti Figure Ppulati f the P level accrdig t the Cllisial Radiative Mdel [0] Figure 4 Ppulati f the P level accrdig t the Cllisial Radiative Mdel [0]

.0 Itrducti Emissi ad absrpti prcesses i plasmas are cmplex matters. Mre specifically, lie itesities assciated with trasitis f plasma species eutral r i are the result f sptaeus emissi, stimulated emissi ad absrpti. I rder t use lie itesity fr ay diagstic, e must first examie if the plasma ptically affects this specific trasiti. A lie is ptically thi if bth stimulated emissi ad absrpti are egligible cmpare t sptaeus emissi. I this case, the lie itesity I f that trasiti is give by: I 4π - N k A k dv where N k is the ppulati f the excited ppulati respsible fr the trasiti, A k is the sptaeus trasiti prbability assciated with that specific trasiti ad dv is the uit plasma vlume. The itegrati is carried ut ver the spectrscpy defied vlume []. The plasma is ptically thick r paque, if the absrpti prcess is dmiat ad that ly a small fracti f the emitted light escapes the plasma. The plasma is fte ptically thi fr lw-desity plasmas 0 0 cm - ad becmes icreasigly paque as desity icrease. Geerally, the plasma is paque t resace lies trasitis ivlvig the grud state r metastables but remais ptically thi fr trasitis betwee excited levels eve at mderately high desity 0 4 cm -. The tw imprtat parameters t quatify the pacity f the plasma are the Optical Escape Factr OEF ad the Mea Optical epth MO. The OEF crrespds t the fracti f light that ca escape the plasma; fr OEF the plasma is ptically thi. The MO is the etial absrpti cefficiet assciated t a give trasiti. A MO smaller r equal tha.0 crrespds t ptically thi plasmas..0 Light absrpti i plasma The chage f lie itesity I related t absrpti ca be writte as: di - I χ ρ ds - I d where χ is the atmic absrpti cefficiet uits cm, ρ is the desity cm -, ds is the uit travel distace i the absrbig medium ad is the ptical depth r the ptical thickess f the medium uitless. χ ρ ds Alteratively, χ ρ ds ca be writte as α ds, where α cm - is the absrpti cefficiet f the medium. Re-writig equati i terms we have: The sluti f this differetial equati is: di / I - d 4 I I - I -α x 5 Where I is the -atteuated lie itesity ad x the absrbig medium legth. Thus, equati 5 describe the itesity f a light beam f iitial itesity I gig thrugh a absrbig medium.

.0 Absrpti ad Emissi i plasmas Ufrtuately, fr a plasma which is bth a emittig ad absrbig medium the situati is mre cmplex. Equati must be re-writte t accut fr that situati ad we have []: di - I χ ρ ds + J ρ ds 6 where J ρ ds is the surce term f the ew equati. ividig each side f the equati d χ ρ ds equati 6 becmes: di / d - I + J / χ 7 Equati 7 is the radiative trasfer equati. The sluti f this differetial equati is give by: [ { χ ρ ] J I 8 χ Where I averaged itesity ad is a characteristic legth f the plasma depeds the gemetry, ex: at the ceter f a sphere, is the radius. Csiderig the spectrum regi ear the trasiti betwee level lwer level; grud r excited ad level higher excited level J ca be writte as: f h A I B J + 9 ρ 4π c where is the higher level ppulati, f is the shape fucti f the emissi lie, A is the sptaeus emissi prbability ad B is the trasiti prbability f iduced emissi. Sice the emissi is t cheret, the stimulated emissi term ca be eglected ad J becmes: A f h J 0 4πρ Meawhile, the atmic absrpti cefficiet χ ca be writte as: B f h χ ρ c where B is the trasiti prbability f absrpti, f is the shape fucti gverig the absrpti ad is the ppulati f the lwer level. We will w suppse that bth emissi ad absrpti prfiles are dmiated by ppler bradeig []. 0 f f f π where the ppler frequecy kt/mc / v th /c, while is the cetral frequecy ad v th is the thermal velcity f the gas kt/m /. The average itesity ca be writte as: 4

c A I [ χ ρ ] 4π B Equati describe the average itesity takig accut bth the emissi ad absrpti prcesses i the plasma. Neglectig stimulated emissi, the ppulati f the excited level is give by: d dt 4 π B electr cllisial ad A + I f d c + 4 radiative trasfer terms where the tw first terms are related t the sptaeus emissi ad t the re-absrpti, respectively. The ther terms are assciated t the electr impact excitati frm grud state ly term fr the Cra mdel ad secdary prcesses such electr impact frm metastables, excitati trasfer prcesses frm eighbrig levels etc Cllisial Radiative mdel. Oly the first tw terms are ivlved i the emissi ad absrpti f light emitted by the plasma. Re-writig equati 4 we have: d dt 4π B A I f d A c + electr cllisial ad 5 radiative trasfer terms 4.0 Optical Escape Factr ad Mea Optical epth Itrducig the Optical Escape Factr, equati 5 ca be writte as: d dt A Λ + electr + cllisial ad 6 radiative trasfer terms where the ptical escape factr is defied as: 4π B Λ I f d A c 7 Usig the average itesity btaied i equati, the ptical escape factr ca be writte as: Λ 4π B c A [ χ ρ ] f d A c 4π B 0 Λ [ χ ρ ] f d 0 8 9 Substitutig the f fucti as defied i equati 9 we btai: 5

6 [ ] ρ χ π d Λ 0 0 Nw sice ρ χ ad c h f B ρ χ we have: c h f B Substitutig the f fucti ce mre: c h B π 4 At we have: c h B π 5 6 where is the mea ptical depth f the plasma. Replacig i equati 0 we have: π d Λ 0 7 Chage f variable: x d dx 8 { } dx x x π Λ 9 This itegral has bee slved by Mitchell ad Zemasky [4]: + + Λ!! 4! 0

Fr large values, rawi [5] has shw that: Λ / π l which is.6 times smaller tha the value btaied by Hlstei i a differet calculati [6, 7]. Startig frm equati 5 we ca re-write i terms f bservable quatities. First by replacig the value defied earlier: B h π v th Secd by utilizig the relatiship betwee the differet trasiti prbabilities: B A c g g B g g 8 π h I terms f wavelegth the ressi simplifies t: g A λ 4 / 8 g π v The Optical Escape Factr Λ ca w be calculated by first usig equati 4 t btai the Mea Optical epth ad, the equati 0. The OEF values are shw i table ad i figure as a fucti f the MO. As metied earlier, OEF is clse t uity fr MO values smaller tha 0.0. th 5.0 Evaluati f Optical Escape Factrs ad Mea Optical epths fr several He trasitis 5. Evaluati f the lwer excited level ppulatis Sme imprtat trasitis f the helium atm are shw i the Grtria diagram see figure. Of particular iterest fr a lie rati diagstic [8] are the siglet trasitis: S P at 44.8, 504.8, ad 78. m ad, the triplet trasitis: S P at 4.0, 47. ad 706.5 m. Accrdig t equati 4 the Mea Optical epth depeds the lwer excited level t the ppulati f the upper level. I rder t evaluate the lwer excited ppulatis P ad P, we must first evaluate the eutral ppulati i the plasma clum. We csider the average plasma cditi with a fillig pressure f 0 mtrr real pressure. At this pressure, the electr temperature is abut 0 ev Lagmuir Prbe ad the average eutral temperature withi the plasma measured by LIF is abut 0.05 ev 580 K [9]. Thus, the eutral desity i plasma clum is abut.7 x 0 4 cm -. The average plasma radius is abut.5 cm. We must the use a mdel t predict what fracti f the eutral ppulati is withi a give excited state. Fr eutral desity lwer tha 0 cm -, the simpler steady cral mdel ca be used, while fr higher desity 0 < e 0 4 cm - the cllisial radiative mdel is a better chice. T be csistet, we used Brda's cllisial radiative mdel [0] t evaluate all the ppulati f the lwer excited levels ivlved i the selected trasitis. I this wrk, all the 7

trasitis eded i the P ad the P levels. Usig T e 0 ev ad figures ad 4 frm Brda [0], the ppulati f the P ad P excited levels are btaied. These ppulati umbers are shw i tables ad. 5. Evaluati f the Optical Escape factrs ad Mea Optical epth The statistical weights g ad g, the wavelegths ad the trasiti prbabilities A ji are give i tables []. The eutral thermal velcity is btaied thrugh the LIF measuremet [9]. Usig equati 4 ad equati 0, the Mea Optical epths ad Optical Escape Factrs are calculated fr the He trasitis listed earlier. The MO ad OEF are shw i tables t 7. A similar calculati ca be used t evaluate the MO ad OEF fr the remaiig He trasitis t listed here. Is the plasma cditi used fr the calculati i secti 5. represetative f all ptical situatis that ca be fud i HELIX plasmas? Yes, ther plasma cditis will yield similar MO ad OEF cefficiets. A thier plasma He smaller will yield cmparable ppulati sice T e is ected t be larger fr this cditi. A high temperature, high desity plasma high pwer discharge will iduce a large ppulati but will als geerates a large v th values ad s. I all cases, the differet T e, ad v th values acts as check ad balace factrs i equati 4 resultig i MO values cmparable t thse listed i tables t 7. 6.0 Iterpretati Fr all siglet trasitis, the OEF is essetially uity eve fr electr desity up t 0 4 cm -. This meas that the plasma is essetially thi with respect t these siglet trasitis. Fr the triplet trasitis, the situati is smewhat less favrable. Essetially, trasitis frm the upper levels 5 S ad 4 S 4.0 ad 47. m, respectively ca be csidered as ptically thi fr plasma with electr desity up t 0 4 cm - less tha % f re-absrpti i each case. Meawhile, the trasiti frm the S level 706.5m is largely re-absrbed by the plasma 5% at 0 cm -, 40% at 0 4 cm -. Here, a strg trasiti large A ji, a large ppulati the presece f the lg-lived S metastable ear the P level is essetially respsible fr this large ppulati ad, a lg wavelegth λ all ctribute t make the MO large eugh 0.065 0 cm - < < 0.7 0 4 cm - s that the plasma is lger trasparet t this trasiti. Thus, the 706.5 m lie itesity must be crrected fr re-absrpti. Sice the re-absrpti is fucti f the eutral desity prfile suppsed uifrm i this calculati, the crrecti is t a simple perati. If pssible, the lie rati diagstic shuld t use this trasiti. 7.0 Cclusi We have shw hw t evaluate the Optical Escape Factr ad the Mea Optical epth fr He trasitis i plasmas. This calculati reveals that mst trasitis are ptically thi with respect t the plasma fr electr desity up t 0 4 cm -. This is i part due t the small relative umbers f the lwer excited ppulatis. Prvide the He trasiti des t ivlved the grud state r ay f the S, S r smetimes P levels, the trasiti is ptically thi with respect t the plasma fr electr desity up t 0 4 cm -. Fr resace trasiti, the MO ad OEF calculatis are ecessary sice these trasitis will be re-absrbed by the plasma. The methd culd be exteded t ther gases prvide a cllisial radiative mdel ca be btaied t evaluate the lwer excited ppulatis. 8

8.0 Refereces [] Bivi R. F., Spectrscpy System fr the WVU Helic Plasma evices, Iteral Reprt WV- PL- 046, West Virgiia Uiversity 000 [] McWhirter R. W. P. i Plasma iagstic Techiques, Chapter 5, R. H. Huddleste ad S. L. Leard, Academic Press, 965 [] Bivi R. F., Study f the ifferet Lie Bradeig Mechaism fr the Laser Flurescece iagstic fr the HELIX ad LEIA Plasmas, WV-PL-09, West Virgiia Uiversity, 998 [4] Mitchell A. C. ad M. W. Zemasky, Resace Radiati ad Excited Atms, Cambridge Press, 97 [5] rawi H. W. ad F. Emard, Beitr. Plasmaphysik 0a, 4 97 [6] Hlstei T., Phys. Rev. 7, 947 [7] Hlstei T., Phys. Rev. 8, 59 95 [8] Bivi R. F., J. L. Klie ad, E. E. Scime, Electr Temperature Measuremets i Helic Plasmas by Helium Lie Itesity Ratis, T be submitted t Physics f Plasmas 00 [9] Scime, E. E., P. A. Keiter, M. W. Zitl, M. M. Balkey, J. L. Klie, ad M. E. Kepke, Plasma Surces Sci. Techl. 7, 86 998 [0] Brsda B. Ph.. Thesis, Ruhr-Uiversitat, Bchum, 99 [] Wiese W. L., M. W. Smith, ad B. M. Gle, Atmic Trasiti Prbabilities, Vl.. Natial Stadard Referece ata Series 966 9

Table. Numerical values fr the Mea Optical epth ad Optical Escape Factrs 6 terms i the summati, see equati 0 Optical epth Λ Escape Factr E-6.0 5E-6.0 E-5.0 5E-5.0 E-4 0.9999 5E-4 0.9997 E- 0.999 5E- 0.9965 E- 0.990 5E- 0.9654 E- 0.9 5E- 0.709.00 0.59 5.00 0.0895 0.00 0.0405 Table - 7 Lwer level excited ppulati, Mea Optical epth, atteuati fracti I/I ad Optical Escape Factr Λ fr the differet S P ad S P trasitis as a fucti f plasma desity. Table. Mea Optical depth ad ptical escape factr fr the He I 4.0 m trasiti e cm - cm - I/I Λ 0 6.5 x 0 9.85 x 0 -.998.9987 0 5.0 x 0 0.45 x 0 -.986.9898 0.05 x 0.00 x 0 -.970.9790 0 4 7.9 x 0 0.5 x 0 -.978.984 0

Table. Mea Optical depth ad ptical escape factr fr the He I 44.8 m trasiti e cm - cm - I/I Λ 0 7.09 x 0 6.79 x 0-6.00.000 0 6.80 x 0 7.7 x 0-5.00.000 0 6.80 x 0 8.7 x 0-4.00.9999 0 4 5.95 x 0 9.50 x 0 -.999.9989 Table 4. Mea Optical depth ad ptical escape factr fr the He I 47. m trasiti e cm - cm - I/I Λ 0 6.5 x 0 9.00 x 0 -.998.9986 0 5.0 x 0 0.55 x 0 -.985.9890 0.05 x 0.0 x 0 -.969.9777 0 4 7.9 x 0 0.45 x 0 -.976.985 Table 5. Mea Optical depth ad ptical escape factr fr the He I 504.8 m trasiti e cm - cm - I/I Λ 0 7.09 x 0 6.90 x 0-6.00.000 0 6.80 x 0 7.80 x 0-5.00.000 0 6.80 x 0 8.80 x 0-4.00.9999 0 4 5.95 x 0 9.60 x 0 -.998.9989 Table 6. Mea Optical depth ad ptical escape factr fr the He I 706.5 m trasiti e cm - cm - I/I Λ 0 6.5 x 0 9 6.65 x 0 -.96.954 0 5.0 x 0 0 4.6 x 0 -.6.785 0.05 x 0 8.45 x 0 -.89.5666 0 4 7.9 x 0 0 7.5 x 0 -.489.658 Table 7. Mea Optical depth ad ptical escape factr fr the He I 78. m trasiti e cm - cm - I/I Λ 0 7.09 x 0 6 4.55 x 0-5.00.000 0 6.80 x 0 7 4.40 x 0-4.00.9997 0 6.80 x 0 8 4.40 x 0 -.996.9969 0 4 5.95 x 0 9.85 x 0 -.96.97

Fig.. Optical Escape Factr as a fucti f the Mea Optical epth. Optical Escape Factr Λ 0.8 0.6 0.4 0. 0 0-6 0-5 0.0000 0.000 0.00 0.0.0 0 Mea Optical epth

E ev 5 Figure. He Grtria iagram S P F S P F Siglet Triplet Iizati 4.580 ev 4 5 4 504.8 78. 44.8 50.6 48.8 49. 667.8 5 4 47. 706.5 4.0 40.6 447. 587.6 M 88.9 08.9 m 0 M 58.4 m M: Metastable States 0 Grud State

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