Atomic Collisions and Spctra 125 PRINCIPLES OF PLASMA PROCESSING Cours Nots: Prof. J. P. Chang Part B3: ATOMIC COLLISIONS AND SPECTRA I. ATOMIC ENERGY LEVELS Atoms and molculs mit lctromagntic radiation or photons whn th lctrons or nucli undrgo transitions btwn various nrgy lvls of th atomic or molcular systm. Dtaild thory of radiation rquirs quantum lctrodynamics to fully dscrib th intraction btwn matrials and lctromagntic radiation, so it will not b dtaild in this lctur. Howvr, a small st of ruls will b discussd to allow th study of th basic physics of radiation. First w will considr th intraction btwn lctrons in th bound stats of atoms and lctromagntic radiation. Atoms mit lctromagntic radiation or photons whn thir bound lctrons undrgo transitions btwn various nrgy lvls of atomic systm. Each atomic systm has its uniqu nrgy lvls dtrmind by th lctromagntic intraction among various bound lctrons and nuclus. Calculation of atomic nrgy lvls rquirs solving a Schrodingr quation for a many particl systm (nuclus and lctrons) and thr is no xact solution availabl xcpt for th simplst atomic systm, i.., th hydrogn atom. Many approximation mthods wr dvlopd to calculat th atomic nrgy lvls, currntly, th nrgy lvls of many atomic systms ar idntifid and tabulatd in th form of Grotrian diagram. On way to dsignat th various nrgy lvls in th Grotrian diagram is calld LS (also known as Russll-Saundrs) coupling. Howvr, it should b notd that th LS coupling schm dos not ncssarily spcify ach nrgy lvl uniquly, thus on should b carful about using LS coupling schm. According to th LS coupling schm, ach stat is dnotd by its orbital angular momntum and th spin stat along with ach lctron s configuration stat. For xampl, th ground stat nutral hlium dscribd by LS coupling schm is (1s) 2 1 S 0. (1s) 2 dnots that two lctrons occupy th 1s stat, and 1 S 0 is th spctroscopy trm whr th suprscript 1 dnots th nt spin stat is singlt, S dnots that
126 Fig. 1. Atomic nrgy lvls for th cntral fild modl of an atom (.g., Na). Fig. 2. An lctron atom lastic collision. Part B3 th total orbital angular momntum is 0, and th subscript 0 dnots that th total angular momntum is 0 (J=L+S=0). Anothr xampl: th ground stat nutral carbon and oxygn dscribd by LS coupling schm ar (1s) 2 (2s) 2 (2p) 2 3 P 0 and (1s) 2 (2s) 2 (2p) 4 3 P 2. Hr w hav 2 lctrons in th 1s stat, 2 lctrons in th 2s stat, and 2 or 4 lctrons in th 2p stat. In addition, th nt spin stat is triplt (suprscript 3), th orbital angular momntum stat is P (angular momntum quantum numbr is 1) and th total angular momntum stat is 0 or 2 (for lss than half filld orbital, J=L-S=0; for half filld orbital or mor than half-filld orbital, J=L+S= 1 + 1 =2). In summary, th dsignation of atomic nrgy lvls can b don using th spctroscopic dsignation of an atomic stat: 2S + 1 X I n L J whr X is th lmnt symbol, I is th ionization stat (I: not ionizd, II singly ionizd, II: doubly ionizd, tc.), n is th principal quantum numbr, 2S+1 is th multiplicity (S=0: singlt; S=½: doublt; S=1: triplt, tc), L is th total orbital angular momntum (S, P, D, F, G for L = 0, 1,2,3,4), and J = L + S is th total lctronic angular momntum. Th atomic nrgy lvls of Na is shown in Fig. 1 as an xampl. II. ATOMIC COLLISIONS In a homognous plasma, nrgtic lctrons undrgo collision with th nutrals to gnrat xcitd nutrals, atoms, fr radicals, ions, and additional lctrons. Ths lctron collision procsss mak th plasma chmistry complx and intrsting. Du to th larg mass diffrnc, th lctron-particl collision can b viwd as an lastic collision procss, as shown in Fig. 2. Svral othr lctron-atom collision procsss ar listd: 1. Excitation procsss a) Elctron impact ionization (Fig. 3):
Atomic Collisions and Spctra 127 Fig. 3. Elctron impact ionization. Fig. 4. Elctron impact xcitation. Fig. 5. Elctron impact dissociation. * Fig. 6. Elctron mtastabl ionization. + A! + + A + Elctrons with sufficint nrgy can rmov an lctron from an atom and produc on xtra lctron and an ion. This xtra lctron can again b acclratd to gain nough nrgy and ioniz anothr atom. This multiplication procss lads to a continuous gnration of ionizd spcis and th plasma is sustaind. Th ionization procsss gnrally hav th highst nrgy barrirs, on th ordr of 10 V. b) Elctron impact xcitation (Fig. 4): + A! + A* Elctrons with sufficint nrgy can also xcit th lctrons of an atom from th lowr nrgy lvl to a highr nrgy lvl. This procss producs an xcitd nutral spcis whos chmical ractivity towards th surfac could b quit diffrnt from th ground stat atoms. Th thrshold nrgy ndd to produc xcitd spcis can vary gratly, dpnding on th molcul and th typ of xcitation. Som xcitd atoms hav vry long liftims (~ 1-10 msc) bcaus th slction ruls forbid its rlaxation to th ground stat. Ths xcitd atoms ar thus calld mtastabls. All nobl gass hav mtastabl stats. c) Elctron impact dissociation of diatomic molculs (A 2 ) (Fig. 5): + A 2! + A + A Elctrons with sufficint nrgy can also brak th chmical bonds of a molcul and produc atomic spcis. Ths atomic spcis could gain nough nrgy and b at a highr nrgy lvl than th ground stat atoms. Dissociativ procsss usually hav lowr thrshold nrgis than ionization procsss. Dissociativ thrshold nrgis vary from 0 to abov 10 V, dpnding upon th strngth of th bond that is brokn and th mchanism by which th procss occurs. This procss is mostly rsponsibl for th production of chmically activ radicals in most of th plasmas. d) Elctron mtastabl ionization (Fig. 6):
128 * Fig. 7. Pnning Ionization. + A*! + + A + Part B3 Elctrons with sufficint nrgy can also rmov an lctron from a mtastabl atom and produc on xtra lctron and an ion. Sinc th mtastabl atom is alrady xcitd, lss nrgy is rquird hr to ioniz th atom. ) Mtastabl-nutral ionization (Fig. 7): A* + B! A + + B + Mtastabl atom can collid with a nutral and ioniz it if th ionization nrgy of th nutral (B) is lss than th xcitation nrgy of th mtastabl (A*). This is also calld th Pnning Ionization procss. 2. Rlaxation and Rcombination Procsss a) D-xcitation (Fig. 8): A*! A + hν Fig. 8. D-xcitation. Th xcitd stats of atoms ar usually unstabl and th lctron configuration can soon rturn to its original ground sat, accompanid by th mission of a photon with a spcific nrgy that quals th nrgy diffrnc btwn th two quantum lvls. b) Elctron-ion rcombination (Fig. 9): + A + + A! A* + A Fig. 9. Thr-body rcombination. For lctron-ion rcombination, a third-body must b involvd to consrv th nrgy and momntum consrvation. Abundant nutral spcis or ractor walls ar idal third-bodis. This rcombination procss typically rsults in xcitd nutrals. c) Radiativ rcombination (Fig. 10): + A +! A + hν Fig. 10. Radiativ rcombination (3-body procss). Photon can also b gnratd during th coalscnc procss of rcombination. This is also a thr-body rcombination procss, sinc th two-body coalscnc is highly unlikly from th standpoint of nrgy and momntum consrvations. d) Elctron attachmnt (Fig. 11): + A! A
Atomic Collisions and Spctra 129 Fig. 11. Elctron attachmnt. + Elctron can attach to an lctrongativ atom to form a ngativ ion, for xampl, a halogn atom or an oxygn atom. Complx gas molculs such as SF 6 can also undrgo dissociativ attachmnt to form ngativ SF 5 ions. This could also b a thr-body rcombination procss. ) Ion-ion rcombination (Fig. 12): A + + A! A + A _ Fig. 12. Ion-ion rcombination. With ngativ ions gnratd, positiv ions and ngativ ions can collid with finit (usually small) probabilitis. In ion-ion rcombination, on lctron transfrs and two nutrals ar formd. III. ELASTIC COLLISIONS 1. Coulomb collisions In gnral, collisions btwn ion-ion, ionlctron, and lctron-lctron ar all Coulombic collision. Th coulomb potntial is: q1q2 U () r = (1) 4πε 0r Following th drivation abov, th diffrntial collision cross-sction can b dtrmind to b: I ( v, φ ) R 2 2 b 0 = 2 φ 4sin 2 (2) Not: this is th Ruthrford Back Scattring (RBS) cross-sction whr b o is th distanc of th closst approach. 2 qq 1 2 ZZ 1 2 b0 = = 4πε 1 0WR 2 4πε 0 mv R R 2 (3) From this analysis, Coulombic scattring could lad to a singl larg-angl scattring (lss likly) or caus a sris of small-angl scattrings. 2. Polarization scattring With a point charg, q o, approachs an atom whos atomic radius is a with a point positiv charg of q and a uniform ngativ charg cloud -q,
130 q o r a d q Fig. 13. Polarization of an atom (atomic radius a) by a point charg q o. Tabl 1. Rlativ polarizability. Spcis α R H 4.5 C 12 N 7.5 O 5.4 Ar 11 CCl 4 69 CF 4 19 CO 13 CO 2 17 Cl 2 31 H 2 O 9.8 NH 3 14.8 O 2 10.6 SF 6 30 Part B3 th point charg can polariz th atom by displacing th uniform charg cloud through quasistatic intractions. Th inducd lctric fild du to a small displacmnt, d, around th cntr of th atom is: qd E = ind 3 4πε 0a (4) Th inducd dipol is thrfor: 3 qa o Pind = qd = (5) 2 r Th attractiv potntial du to th incoming charg q o is: 2 3 qo a U( r) = (6) 4 8πε0r Th polarizability in this simpl atomic modl 3 is: α = a p, and th rlativ polarizability is: α p α (7) R 3 a 0 Tabl 1 summarizs th rlativ polarizability of svral atomic spcis. Not again that a o is th Bohr radius. If th impact paramtr, h is small nough, i.., smallr than th critical impact paramtr, h L, th particl will b capturd by th atom during this typ of collision. This critical impact paramtr is: 1 2 α 4 pq o hl = 2 (8) vrπε0m R m m m R m + m 1 2 = (9) 1 v R = v 1 v 2 (10) Th Langvin or captur cross-sction can thus b dtrmind as: IV. INELASTIC COLLISIONS L 2 L 2 σ = πh (11) 1. Constraints on lctronic transitions Atoms mit lctromagntic radiation (photons) whn th lctrons undrgo transitions btwn various nrgy lvls. Sinc th typically radiation tim is on th ordr of 1 ns, much shortr than th charactristic tim btwn collisions,
Atomic Collisions and Spctra 131 which ar on th ordr of 100 ns, th xcitd stats will gnrally b d-xcitd by lctric dipol radiation rathr than by collision. Howvr, not vry transition occurs as frquntly as othrs do. Th most frqunt transition btwn various nrgy lvls is th lctric dipol transition and th following conditions should b satisfid for th lctric dipol transition. Th gnral rul of thumb includs: Enrgy consrvation: th nrgy of mittd radiation (photons) should b qual to th nrgy diffrnc btwn th uppr nrgy lvl and th lowr nrgy lvl, hν = E i E j, whr h is th Planck's constant, ν is th frquncy of th mittd photon, E i is th nrgy of th uppr lvl th lctron occupis prior to th transition, and E j is th nrgy of th lowr lvl th lctron occupis aftr th transition. Slction Ruls: during th lctric dipol transition, th following changs for angular momntum stat nd to occur: Chang in th orbital angular momntum stat: L = 0, ±l (0 is not allowd for a transition involving only on lctron) Chang in th spin angular momntum stat: S = 0 Chang in th total angular momntum stat: J = 0, ± 1 (xcpt that J=0 to J=0 transition is strictly forbiddn). From th slction rul, th nrgy lvls of H, can b shown dividd into singlt (para-hlium) and triplt (ortho-hlium) stats, sinc th transitions btwn thm ar forbiddn. Sinc L=0! L=0 is forbiddn, th 2 1 S and 2 3 S stats ar mtastabls (Fig. 14). A mor dtaild Grotrian Fig. 14. Atomic nrgy lvls of H, showing th diagram is includd at th nd of this sction. division into singlt and triplt stats. It is notd that th slction ruls ar not prfct, unlik nrgy consrvation. For xampl, th vry intns mrcury rsonanc lin at 253.7 nm is du to th transition from 3 P 1! 1 S 0. If th abov two conditions ar satisfid, th lctrons can spontanously undrgo transition from th uppr nrgy lvl, i, to th lowr nrgy lvl, j, with a crtain probability pr unit tim. This
132 Intnsity 402.6 nm H I 2 3 P-5 3 D 447.1 nm H I 2 3 P-4 3 D 501.6 nm H I 2 1 P-3 1 P 587.5 nm H I 2 3 P-3 3 D 667.8 nm H I 2 1 P-3 1 D 706.5 nm H I 2 3 P-3 3 S Part B3 probability is calld th transition probability for spontanous mission (also known as Einstin A cofficint) and can b asily found in th litratur for many transitions. For xampl, th transition probability of hydrogn atom btwn 2p stat and 1s stat is 6.28 10 8 sc -1. 2. Idntification of atomic spctra Basd on th abov discussion, w can now undrstand th ssntial faturs of atomic spctrum and obtain som usful information about th plasma systm. As shown in Fig. 15, atomic spctrum usually consists of a numbr of vry sharp lins on th constant background. Whn th spctrum is masurd, th first task is to idntify vry mission lin in th spctrum. This is don by comparing th wavlngth of th mission lins with th nrgy diffrncs btwn two lctronic lvls using th publishd spctral databas (NIST databas). It is notd that in som cass vn this first stp is not vry straightforward and rquirs additional considration. Onc this stp is compltd, w can hav at last two (mayb mor) vry usful information about th plasma. Thy ar: Idntification of xisting atomic spcis in th plasma. Idntification of crtain xcitd atomic stats and thir dnsity in th plasma. 350 400 450 500 550 600 650 700 750 Wavlngth (nm) Fig. 15. Emission from a H Plasma. Latr, w will larn how to us th information to undrstand th plasma stat. Light mission is a major charactristic of plasmas. To mit th light, th atoms in th plasma hav to b in th xcitd stats. Thr ar two diffrnt ways to xcit th atoms in th plasma to th xcitd stats. Th first on is to us th kintic nrgy of th particls in th plasma (in particular lctrons) and to transfr this nrgy to th atoms in th ground stat (or anothr xcitd stat) by collision. This procss is calld collisional xcitation. Th scond procss is to us th nrgy of th photons and to transfr thir nrgy to th atoms by absorption of photons. This procss is calld radiativ xcitation. In most plasma systms, th frquncy of th radiativ xcitation is much smallr than th collisional xcitation, thus can b nglctd. Not again that it is not vry asy to xcit th
Atomic Collisions and Spctra 133 ground stat lctron in th atom to an xcitd stat. Th nrgy rquird for this xcitation is fairly larg. For xampl, in hydrogn atom, a minimum nrgy of 10.2 V is rquird to mov th lctron from th ground stat (1s) to th lowst xcitd stat (2p) from which atom can mit th photons. An ionization procss rquirs mor nrgy than th xcitation procss (for xampl, th ionization potntial of hydrogn is 13.6 V). Th ionization cross-sctions of svral nobl gass and th xcitation corss-sction for H ar shown in Fig. 17 and Fig. 18 as xampls. Onc th atoms in th plasma ar xcitd abov th ground stat, it will vntually b dxcitd to th ground stat. Thr ar thr diffrnt ways to d-xcit th atoms in th plasma. Th first on is th spontanous mission whn th lctron in th xcitd lvl maks a transition to Fig. 16. Ionization cross-sction of nobl gass. th ground lvl or anothr xcitd lvl without any xtrnal influnc. As brifly mntiond arlir, th tim scal for this d-xcitation is vry short if th transition is lctric dipol transition, on th ordr of 10-8 sc to 10-7 sc. In this cas, th nrgy consrvation is satisfid by mitting th photon whos nrgy is qual to th nrgy diffrnc btwn th initial stat and th final stat. In many plasma systms, this is th most important d-xcitation mchanism. On th othr hand, th lctron in th xcitd lvl also maks a transition if thr ar othr photons around th xcitd atoms. This procss is calld stimulatd mission. Though th stimulatd mission is th ky lmnt for th lasr, in most plasma systm, Fig. 17. Excitation cross-sction of lctrons in th stimulatd mission can b nglctd. Th third hydrogn. procss for th d-xcitation is th invrs procss. of th collisional xcitation and is calld collisional d-xcitation. In collisional d-xcitation, th colliding particls will gain nrgy from th xcitd atoms into thir kintic nrgy. Th importanc of collisional d-xcitation is a function of plasma dnsity and lctron tmpratur and it varis for various xcitd stats. 3. A simplifid modl for mission To simplify th discussion, w will mak a numbr of assumptions on our plasma systm. 1. Plasma dnsity (n ) is uniform throughout th volum. 2. Elctron nrgy distribution is Maxwllian and
134 R 1 R 2 A 21 A 2o A 1o E i E 2 E 1 E o Fig. 18. Enrgy diagram of an atom with limitd nrgy lvls. Part B3 its tmpratur is givn as T. 3. Our plasmas ar mad of hypothtical atoms that hav only 4 nrgy lvls, ground stat, first and scond xcitd stat and ionizd stat. 4. Th rat of collisional d-xcitations ar small compard to th rat of spontanous mission, thus will b nglctd. 5. Th systm is in stady stat. As shown in Fig. 18, E o, E 1, E 2, and E i ar th ground stat, xcitd stat 1, xcitd stat 2, and th ionizd stat. R 1, and R 2 ar rat of collisional xcitation from th ground stat, and A 1o, A 2o, and A 21 ar rat of spontanous mission (Not that thy ar also calld Einstin A cofficint). R 1, and R 2 can b calculatd using th collisional crosssctions: R = n n < σ v> 1 0 1 R = n n < σ v> 2 0 2 Rmmbr <σv> is th collision rat avragd ovr th MBD. From ths rats, th quations govrning th dnsity of ach stat can b dtrmind: dn0 = n σ dt dn1 = nn0 < σ 1v > + n2 A21 n1 A dt dn2 = n n0 < σ 2v > n2 A21 + A dt ( < σ 1v > + < 2v > ) + n1 A10 n2 A20 n0 + 10 ( ) In stady stat, th tim drivativs in th LHS ar zro, and w hav two indpndnt quations with fiv unknowns (n, n 0, n 1, n 2, T ). From charg quasi-nutrality and particl consrvation, w hav on mor quation: n = + n0 + n1 + n2 n g 20 whr n g is th gas dnsity without th plasma. During spontanous mission, th xcitd stat 1 and 2 will mit photons at th following frquncis: hν = E hν hν 10 1 E0 20 = E2 E0 21 = E2 E1 If w can masur th numbr of photons mitting
Atomic Collisions and Spctra 135 at ths frquncis, w can thn dtrmin th dnsity of xcitd stats (n 1 and n 2 ) indpndntly: # photons at tim # photons at ν tim ν 10 = na 1 10 20 2 20 volum = na volum Now w hav thr unknowns (n, n 0, T ) for thr quations: n n0 < σ 1v > + n2 A21 n1 A10 = n n0 < σ 2v > n2 A21 + A20 = n + n0 + n1 + n2 = n g 0 ( ) 0 Thrfor n, n 0, and T can b calculatd. Unfortunatly th situation in ral systms is vry diffrnt from this simplifid modl. Thus, th us of plasma spctroscopy alon may not provid th nough information about th plasma systm that w want to know. Howvr, w can still obtain som vry valuabl information on our plasma systm from plasma spctroscopy.
136 Part B3 Fig. 19. H Grotrian Diagram