Introduction to plasma physics

Transcription

1 Introduction to plasma physics 1 Plasma dfinition (S. Ichimaru, Statistical Plasma Physics, Vol I) Plasma is any statistical systm containing mobil chargd particls. Not Statistical mans macroscopic, for scal lngth L and fr lctron dnsity th rlation L >> n -1/3 holds. Not Narly any systm contains som mobil chargd particls, but if thir impact on th systm bhavior is ngligibl, it has no sns to dscrib it using mthods of plasma physics. Not According to this dfinition, systms ar classifid as plasmas vn if thy ar not macroscopically nutral, so calld non-nutral plasmas (.g. chargd particl bams). Ths systms do not mt othr plasma dfinitions. Othr Plasma dfinition (F.F. Chn, Introduction to plasma physics) Plasma is quasi-nutral systm of mobil chargd (and possibly also nutral) particls that xhibits collctiv bhavior. Not. Hr basic plasma proprtis ar mntiond. Not. Collctiv bhavior is thus ssntial, but nd not b dominant. Collctiv bhavior is dominant for idal plasmas.

2 Plasma formation ionization procsss 1. Ionization by cosmic rays E.g. Ionosphr a plasma layr is around th Earth from th hight of about 60 km up to th hight ca 500 km that is formd du to ionization by cosmic rays (both lctromagntic, and corpuscular), th main sourc of which is th Sun. Elctron dnsity maximum of about 10 6 cm -3 is in th uppr F-layr (abov 00 km), whr lctron tmpratur is T 000 K. Not. Crtain amount of mobil charg particls is in any mattr around us du to th action of ionizing radiation. Evn at a small hight abov th sa lvl, 10 fr lctrons and positiv ions ar formd pr scond in air in natur. A fraction of lctrons attach to molculs and ngativ ions ar cratd. Typically, thr ar ~10 3 positiv ions in cm 3 in th outsid air and th ratio of th positiv to th ngativ ion numbr is about Ionization is du to th photon action is calld photoionization. Photon nrgy must b highr than th ionization nrgy U i ( ω > U i > 3.9 V ).

3 . Ionization in lctric fild (collisional) If gas is placd in a strong lctric fild, xisting fr lctrons ar acclratd and whn thy gain a sufficint nrgy to caus ionization during collision by sparation of outr orbital lctrons of nutral atoms or molculs. Elctrons rlasd by impact ionization (collisional ionization) ar again acclratd by th lctric fild and ionization avalanch is formd. Thus, lctric discharg occurs. E.g. in a glow discharg at th prssur of 1 Torr, th lctron dnsity is n = cm -3 and lctron tmpratur is T 10 4 K. 3. Ionization by hating Without using lctric fild, a plasma stat may b attaind by raising th tmpratur of a nutral gas. Binding nrgy of outr lctrons in an atom/molcul is a fw V, and thus th thrmal nrgy of lctrons is sufficint for impact ionization whn lctron tmpratur is K. Morovr, th nrgy mittd photons is thn sufficint for photoionization. In th thrmodynamic quilibrium ionization is givn by ionization quilibrium. 3

4 Ionization quilibrium - Saha quation [SI units] nn n i 1 3/ i.4 10 T xp n U = kt B Boltzmann constant is k B = R/N A = J/K = V/K, and thus k B T = 1 V at th tmpratur T = K, ionization potntial is for instanc for nitrogn atom U i = 14.5 V (15.58 for molcul N ), for Argon it is U i = V. Th atom dnsity in pur argon at th standard atmosphric prssur and tmpratur 0 C (Loschmidt constant) is n 0 = m -3 = n n + n i n n and according to (1) quilibrium ionization is n i /n n = Evn at tmpratur of 1 V, th argon ionization is n i /n n Not. Plasma tmpratur is mostly high, and thus is usually givn in V or in kv. It is also practical for th comparison of tmpratur with ionization nrgis. At highr tmpraturs plasma is oftn multiply ionizd. 4. Prssur ionization At highr dnsitis, th orbital radius of valnc lctrons may b intratomic distancs and thn valnc lctrons may b frd at room tmpratur. (1) 4

5 5 In mtals at room tmpratur, th dnsity of fr lctrons is of ordr 10 3 cm -3. Elctron Frmi nrgy E F is at such dnsity E F /3 π 3n = m π 7.9 V T, () and thus lctron gas in mtals is dgnrat. Th ratio Θ =T/E F is calld th dgnration paramtr. For dgnrat lctron gas (Θ << 1) E F is a good stimat of lctron kintic nrgy. In smiconductors, th dnsity of fr lctrons and hols is much lowr. Typical xampl of plasma originatd by prssur ionization is th intrior of burnt-out star. It is comprssd to such high dnsity that lctron Frmi nrgy is >> binding nrgy of lctrons in atom, and consquntly all atoms ar fully ionizd. On-componnt plasma (OCP) approximation systm of singl spcis of chargd particls mbddd in a uniform background of nutralizing chargs. Not. Proprtis of lctrons and ions may diffr considrably; thrfor focus to on spcis of chargd particls is somtims usful.

6 6 Typical valus of lctron dnsity and tmpratur of som plasmas IG intrstllar gas N gasous nbula I ionosphr GD glow discharg SA sun atmosphr AD arc discharg SC sun corona AGN activ galactic nuclus MF magntic fusion X X-ray star ICF inrtial confinmnt fusion SI sun intrior dgnrat M mtal Fig. 1 J - Jovian intrior WD whit dwarf r s is th ratio of man intr-lctronic distanc to Bohr radius (W-S radius)

7 Coupling paramtr, wakly and strongly coupld plasma 7 Coupling paramtr for OCP is th ratio of Coulomb nrgy at th avrag intrparticl distanc to thir avrag kintic nrgy max(3/ T, E F ). Th avrag distanc R of particls of th dnsity n is 3 R = 4π n 1/3. (3) For ions R i is usually calld ion sphr radius or also Wignr-Sitz radius. Ion sphr contains all bound and fr lctrons blonging to particular ion atomic physics dscription for dns plasmas For dgnrat lctrons coupling paramtr Γ is xprssd, as follows 1/3 7/3 3 m R 4/3 /3 S 0RE F n 0 a, (4) B Γ = = = = r 4π 3 π 4π 4π whr a B is th Bohr radius. Avrag lctron distanc is qual to th Bohr radius for dnsity n = cm -3. Coupling paramtr of dgnrat lctrons dcrass with dnsity!!

8 For classical plasmas (particls with charg Z) th xprssion is 1/3 6 n Z 18 3 ( Z) 10 K Γ= = 4π 0RT 10 cm T 8, (5) and thus coupling paramtr riss with dnsity and dcrass with tmpratur. For lctrons and hydrogn ions Γ = at th blu lin in th prvious figur. Elctrons ar thus strongly coupld only in rd hatchd triangl. For ions, Frmi nrgy is vry small, and this hydrogn ions ar strongly coupld vrywhr blow th blu lin. W will mostly study classical wakly coupld plasmas. Espcially for multiply ionizd plasmas, it is mor probabl that ions ar strongly coupld. Consquntly, ion coupling paramtr Γ i is usd as coupling indicator. In wakly coupld plasmas, mutual potntial nrgy of particls is small compard to thir kintic nrgy, and thus its thrmodynamic proprtis ar clos to a gas and th quation of stat can oftn b approximatd by quation of stat of idal gas.

9 Plasma proprtis quasi-nutrality 9 Systm is quasi-nutral, if th total charg in volums comparabl with th cub of its scal lngth L is much lss than total charg of all positiv chargs (and absolut valu of total ngativ charg). Not. Th scal lngth L of th systm must b much gratr than th distanc, to which ngativ chargs may b sparatd from positiv chargs (usually lctrons from ions). Crtain nrgy is ndd for th sparation of chargs of th opposit signs. Macroscopic charg clouds may sparat only to th distanc, whr thir thrmal nrgy is fully convrtd to th potntial nrgy.

10 10 Simpl physical modl what is th maximum thicknss of an infinit planar lctron layr that can mov against static ions by its full thicknss? (classical statistics is assumd) Fig. Shift of layr Planar capacitor mrgs with surfac charg dnsity σ and lctric fild E is insid σ = n E= σ / Elctron potntial nrgy is qual to its thrmal nrgy n Upot = E = = kbt This is calld lctron Dby lngth λ D λ D kt =D= 0 B n Elctron Dby lngth grows with th squar root of lctron tmpratur T and dcrass with th squar root of lctron numbr dnsity n. Thus, plasma is quasi-nutral at th distancs that ar significantly largr than th Dby lngth; th quasi-nutrality condition is a scal lngth L λ D. 1/ 0 0 (6)

11 Dby scrning Static charg is scrnd in plasmas, bcaus it attracts opposit chargs and kps away chargs of th sam sign. Not. Dby drivd scrning in th thory of lctrolyts. W shall assum that th lctron tmpratur T nd not b in gnral qual to th ion tmpratur T i. This is frqunt in plasmas as (w show it latr) th nrgy transfr btwn lctrons and ions is rathr slow. W shall assum that plasma may b multiply ionizd (in diffrnc th Chn book); w dnot man ion charg by Z. Thus, th lctron charg is q = and th ion charg is q i = Z. Elctrostatic fild around th charg q T placd in th coordinat origin is dscribd by Poisson quation r q ϕ = = n Zn r T ( ) δ( ) i Lt in (whr ϕ = 0) th charg dnsity is r = 0. Thus, n = n 0 = Z n i in. Th thrmal lctron nrgy must gratr than th Frmi nrgy so that th Boltzmann statistics for lctrons would b applicabl. Thus, (7) 11

12 kt π 3n > E = m π B F Not. In mtals, typical lctron dnsity is n = 10 9 m -3, thn E F = 7.9 V, for singly ionizd gas of dnsity n = m -3 is E F = V = 440 K. In Boltzmann statistics, th probability of stat occupation is ~xp(-u/k B T) ϕ n 0 Zϕ n = n0 xp ni = xp kt B Z kt B i Elctron and ion dnsitis may now b substitutd into th Poisson quation. W shall simplify quation by linarization, w shall assum potntial nrgy kintic. For x 1, it holds xp(x) 1 + x and quation (7) is convrtd to Aftr substitution 1 d dϕ n 0 1 Z ϕ = r pro r 0 = + ϕ r dr dr 0 T Ti /3 ϕ = ϕ / r Poisson quation has th form d d ϕ ϕ = r λ D 1 (8) (9)

13 Potntial of a static charg q T in plasma is thus xprssd, as follows q T r ϕ = xp 4 p0r λd 13 (10) At th distanc λ D potntial is scrnd to 1/ of its vacuum valu. Scrning is th sum of lctron scrning with λ D and ion on with λ Di. Dby lngth λ D is kt kt kt λ = λ + λ λ = λ = = (11) B 0 B i 0 B i 0 D D Di D Di n ni Z n Z For T > T i /Z ion scrning of a static charg dominats. Thr is crtain scrning around vry chargd particl in plasma, so calld dynamic scrning. Th static ion scrning occurs only of th particl vlocity is ion thrmal vlocity. If th particl is fastr than th thrmal ions, but much slowr than th thrmal lctron vlocity, lctron static scrning is formd, but ion scrning is than for a static charg.

14 Assumption includd in th drivation W hav usd dnsitis of chargd particls that can b usd with rasonabl accuracy only whn th distancs (λ D in this cas) ar larg compard to avrag intrparticl distanc. It is usually rquird that numbr N D of lctrons in lctron Dby sphr N 4π 4π k T = = 1 3/ 3/ 3/ 3 0 B D λd n 3 1/ 3 3 n 14 (1) Quantity N D or its som small multipl is calld plasma paramtr. For N D 1 plasma is idal and scrning is collctiv procss. Not. Whn N D <1, scrning also xists, but its fluctuations ar > man scrning valu. Whn Poisson quation was linarizd, potntial nrgy of chargd particls ϕ than thir thrmal nrgy k B T was assumd. Surly, this dos not hold nar to origin, but vn th prvious assumption is not valid thr. It is nough to assum that q T is so small that th inquality holds at th avrag distanc among lctrons R 3/ ( 4 ) 1/3 = π n.

15 Collctiv bhavior By th trm collctiv bhavior on dnots mutual particl intractions via macroscopic lctromagntic fild in contrast to microscopic filds that caus particl intraction during binary collisions. Du to scrning, binary intractions ar fficint in plasmas only up to th distanc of Dby lngth, but intractions on longr distancs ar prsnt in plasmas du to macroscopic lctromagntic filds formd by macroscopic collctiv chargs and currnts. Fluctuations with wavlngth > Dby lngth hav mainly collctiv charactr, whil short-wavlngth fluctuations ar mainly controlld by motion of individual particls with th prdominanc of binary intractions (mor in dtail in th book by Ichimaru). Th spd of systm variations du to binary collisions is givn by th collision frquncy n c. Th significanc of binary intractions incrass with th collision frquncy n c. Thr xists a larg varity of collctiv motions in plasmas, but th fastst is th motion of lctron cloud with rspct to ions du to thir mutual attraction. For simplicity w considr ions as static homognous nutralizing background (OCP approximation). 15

16 W us th modl of planar layrs again (Fig. ). Th vlocity of ordrd lctron motion is v d /dt = and th lctron quation of motion is 16 m dv n d n = E= = dt t m 0 d 0 Thus, plasma oscillations occur with th lctron plasma frquncy ω p = n m 0. (13) (14) Elctron plasma frquncy ω p charactrizs th strngth of th collctiv action and for ω p > n c, th collctiv bhavior dominats in th plasma. Th lctron plasma frquncy ω p, th lctron Dby lngth λ D and th lctron thrmal vlocity v T mt th simpl rlation v = kt/ m = ω λ T B p D Not. Whn on includs also ion motion, thn th frquncy of plasma oscillations is ωp = ωp + ωpi, whr ωpi = Zni /( 0 Mi ) = Zωp m / Mi.

17 Fig. 3 Collision schmatics ( ˆr unit vctor in dirction r, b collision paramtr) Collision frquncy of chargd particls W shall assum for simplicity that th componnt of vlocity v 0 in th dirction of motion of flying-in particl bfor collision is constant (valid for larg b whn th altration of th particl motion is small). Th normal componnt of particl momntum is obtaind by tmporal intgration of forc impuls ( ) mv = F t d t. Th normal forc componnt is xprssd via th rlation qq qq F = 4πε = 4 whr th rlation r = b/sinθ was utilizd sinq sin q, 0r πε0b Tim dpndnc of F is givn by th dpndnc on angl θ. A stady motion in th dirction x is assumd, and thus t = x/v 0 = r cosθ/v 0 = b ( ) cosθ/(v 0 sinθ) and dt = bd θ / v0 sin θ. Consquntly 17

18 v v qq sin ( )d qq sin d b = = = 4πε 4 v whr b 0 is th Landau lngth π q t t q q 0mb πε0mb 0 b 0, b = qq / (πε mv ) Th collision paramtr b 0 corrsponds to scattring to 90, thus to th loss of original dirction of vlocity. Th ffctiv cross-sction for 90 is σ = πb 0. Th collision frquncy (for larg-angl scattring) is thus Small-angl scattring n L nqq v0b0 3 4πε0 m v0 = π n = 18 (15) Elctrostatic fild long-rang forc sum of small-angl scattrings oftn dominats ovr larg-angl scattring. Th loss of original dirction of motion happns probably du to many small changs of th vlocity vctor arlir than on larg-angl scattring vnt occurs. Th collision frquncy is thn dfind as 1 ovr th avrag tim in which th particl loss th original dirction of vlocity.

19 19 History of th particl motion may b considrd a random walk in th vlocity spac. If N collisions happn in a crtain tim intrval, th variation of.g. y componnt of vlocity is v = v + v + + v, y y1 y yn and th avrag valu is vy = vyi = 0. As individual collisions may b considrd uncorrlatd, th disprsion of v y can b xprssd D N N = D = D = D = D ( v ) v ( v ) ( v ) y yi yi N y vy 1 i= 1 i= 1 For on collision with collision paramtr b thr holds ( ) v ( ) ( ) v = v + b y v = z vy1 b = Numbr of collisions with paramtr in th intrval db is dn = n0v 0 πbdb and thus th total disprsion of th normal vlocity componnt is xprssd d ( ) 3 db 3 bmax v y = π n0 v0b0 π n0 v0b0 ln dt tot = b b min v b b

20 0 W had to rstrict th divrging intgral. Th lowr boundary is du to th assumption of small-angl scattring, and this is not valid for collision paramtrs b lss than Landau lngth b 0. Th assumption of Coulomb intractions dos not hold for larg collision paramtrs b as th fild is rducd by th Dby scrning, thrfor th uppr boundary is b max = λ D. Lt us dnot Λ this ratio for lctron collision with thrmal vlocity v T λd π0λd m vt 3 3 Λ= = = π n λd = ND b (16) 0 If th plasma paramtr N D is larg, thn Λ is also larg. Th quantity lnλ is calld th Coulomb logarithm. It is th ratio of collision frquncy du to small-angl scattring to collision frquncy of scattring to angls 90. Collision frquncy for collisions of lctrons of vlocity v 0 with lctrons is 4 8 π n 0 n = ln Λ 3 4π m v. (17) ( ) 0 0 Collision frquncy of Coulomb collisions is v 3 and man fr path is v 4, thus rlativly fast lctrons from vlocity distribution tail do not collid frquntly and thy can fly rlativly long distanc without dirction altration.

21 1 Th collision frquncy of lctrons with thrmal vlocity v 0 = v T = (k B T /m ) 1/ is calld th ffctiv collision frquncy 4 8 π n n c = ln Λ 1/ 3/ 4π m k T (18) ( ) ( ) 0 B Th ratio of th ffctiv collision frquncy to th plasma frquncy is n ln ( 3 / c 1 ln Λ ND ) = = 3 ( 01 for ND 1) n 3 N / ω p l p 0 D D (19) For larg N D, th collctiv bhavior charactrizd by ω p dominats ovr th impact of binary intractions charactrizd by n c. Such plasma is calld idal plasma. Som phnomna can b thn dscribd in th approximation of collisionlss plasma. Idal plasma is quasi-nutral and collctiv ffcts causd by macroscopic chargs and currnts dominat in it.

22 Th ratio of th potntial to th kintic nrgy Lt s compar lctron nrgy in fild of th narst lctron, placd in th distanc R = [3/(4π n )] 1/3 with its kintic nrgy (non-dgnrat plasma is assumd) 1/3 n 3 Wp = W 1/3 /3 k kt B 4p R 3 4p W ( ) 0 0 /3 3 1/ p 3 n 3/ 3/ 3/ = /3 k 9 4p0 B 9 D (0) W k T N In idal plasma N D 1 and kintic nrgy of particls is thus thir binding (potntial) nrgy. Idal plasma is wakly coupld. Idal plasma is in this rspct similar to a gas, on oftn spaks about ionizd gas. Th quation of stat of idal gas is thn a good approximation of lctron quation of stat in idal plasma. If vn ions ar wakly coupld (Γ i << 1), thn th quation of stat of idal gas may b usd also for ions.

23 Plasma in natur Idal - Various typs of plasmas dischargs; ionosphr; solar wind; outr layrs of stars; intrstllar gas Idal or non-idal star intriors (cntr of sun is narly idal plasma r = 150 g/cm 3, T = 1.35 kv, Γ = 0.14) Non-idal - lctron gas in mtals (dgnrat plasma), lctrolyts, cntrs of larg plants (Jovian plants) Plasma in laboratory Idal - dischargs of various typs (vacuum tubs, dischargs for gas lasr pumping, pinchs, capillary discharg); MHD gnrators; ion ngins, lasr plasma from gas targts Idal or non-idal - lasr plasma from solid (or liquid) targts Non-idal suprcold plasma (plasma with tmpratur ca 1 K can b obtaind by non-linar photoionization of lasr-coold vapor, lctron dnsitis of cm -3 ) 3

24 4 Numbr of particls (lctrons + ions) in Dby sphr of radius λ D Takn from R.P. Drak, High-Enrgy-Dnsity Physics, Springr 006 (a) Plasma of matrials with high atomic numbr, whr man ion charg Z = 0.63 T is assumd (T is in V). (b) Plasma of matrials with low atomic numbr, whr man ion charg Z=4 is assumd

25 Typical paramtrs of various typs of plasmas hr always n λ D 3 > 1 and ω p > n i. 5

26 6 Typical tmpraturs and dnsitis of various plasma typs