PLASMA PHYSICS VII. PLASMA DIAGNOSTICS

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1 PLASMA PHYSICS VII. PLASMA DIAGNOSTICS Elctrostatic or Langmuir Probs Probs ar insrtd into th plasma to masur T and n. Th probs prturb th plasma lctrically and may b, in th cas of tokamak plasmas, an intolrabl sourc of impuritis. Singl prob Typically a tungstn prob in a glass tub. Prob prturbs th plasma but th ffcts ar small outsid a thin shath surrounding th ε 0kT prob. Th shath thicknss is of th ordr of a Dby lngth λ D = (s Chaptr I) n and is << R, th prob radius.. I-V charactristic 0 1

2 (i) V S is th spac or plasma potntial (th potntial of th plasma in th absnc of a prob). Thr is no E. Th currnt is du mainly to th random motion of lctrons (th random motion of th ions is much slowr). (ii) If th prob is mor positiv than th plasma, lctrons ar attractd towards th prob and all th ions ar rplld. An lctron shath is formd and saturation lctron currnt is rachd. (iii) If th prob is mor ngativ than th plasma, lctrons ar rplld (but th fastr ons still rach th prob) and ions ar attractd. Th shap of this part of th curv dpnds on th lctron vlocity distribution. For a Maxwllian distribution with T > T i, th slop of ln I plottd against V is kt (iv) V F is th floating potntial (an insulatd lctrod would assum this potntial). Th ion flux = th lctron flux so I = 0. (v) All th lctrons ar rplld. An ion shath is formd and saturation ion currnt is rachd. Collisions If thr ar collisions, particls may hav to rly on diffusion to trach th prob. D From bfor λ m. Collisions ar important if λ m << R. v Magntic fild rms Th ffct of a magntic fild is xtrmly complicatd. Th ions and lctrons gyrat and this affcts thir random motions and collisions. Th bhaviour of th prob will dpnd on its orintation in th magntic fild. Th magntic fild can b ignord if r L >> R. Points to not Swpt Langmuir probs may giv tim bhaviour of plasma paramtrs, but thr ar difficultis. Any scondary mission, photomission will lad to rrors Car is rquird if distribution is non-maxwllian. For instanc if thr is a drift. Doubl prob

3 I-V charactristic Assum symmtry. Th systm floats and follows any changs in V S. If V is slightly positiv, thr ar mor lctrons raching 1 and fwr raching. If V is vry positiv, saturation ion currnt into. Th currnt into 1 cannot xcd this valu. It can b shown that and th slop of th charactristic is Points to not I V = Isat itanh kt di dv Isat i =. kt In a magntic fild, shadowing of th probs must b avoidd. Magntic probs Magntic coil (i) A small coil is insrtd into th plasma and orintd to masur a particular componnt of B or to pick up MHD wavs. Typically th small coil is in a glass tub it has no lctrical contact with th plasma. You do. Driv this. Us intgrator. V = NA db. dt 3

4 NAB Vout =. RC (ii) Flux loop or diamagntic loop. A larg coil surrounding th plasma vssl is usd to masur total magntic flux. Rcall that if I z = 0, Bz ( r) pr ( ) + = constant. µ 0 Th prsnc of th plasma acts to dcras th magntic fild insid it (hnc diamagntic). If w masur B ( a) z using a small coil and <B z > using a diamagntic loop thn B ( a) < B > z z < p > = µ 0 givs an stimat of th total kintic nrgy in th plasma (using p = nkt). You do. Show this. (iii) Monitor position of plasma in a tokamak If th plasma movs up, th flux in th uppr loop incrass whil th flux in th lowr loop dcrass. Tak diffrnc and intgrat. If th plasma movs out, th flux in th outr solnoid incrass whil th flux in th innr solnoid dcrass. Tak diffrnc and intgrat. (iv) Loop voltag masurmnts 4

5 Th mf inducd in th loop quals th mf in th plasma loop. Masur plasma currnt I l indpndntly and calculat th plasma rsistivity using R = η. η dpnds on lctron A tmpratur so T can b stimatd. Rogowski coil To masur I th total currnt. Th larg loop compltly surrounds I. Th masurmnt is indpndnt of how th currnt is distributd and it has th advantag of making no lctrical contact with th currnt bing masurd. Not th rturn wir to minimiz th ffct of any flux thrading th larg loop. di V = naµ 0. dt whr n is th numbr of turns pr unit lngth, A is th ara of th small loop. You do. Driv this. You hav to assum B is uniform across A. Us intgrator. Th coil should b shildd to avoid lctrostatic pickup from th plasma. (Rogowski coil can b usd as a currnt prob in non-plasma applications.) Microwav intrfromtry This is a non-prturbing way of masuring n. Th intrfromtr shown is th microwav vrsion of th Mach-Zhndr intrfromtr. Th optical path lngth in th prob arm changs as th plasma dnsity varis. 5

6 Th wavs in th plasma ar usually th O wavs whos rfractiv indx n N = 1 dpnds only on th dnsity and not on th magntic fild. Fild at dtctor du to signal passing through prob arm E1 = A1cos ωt + φ1 + φ plasma l whr φ plasma = ( k k ) dx 0, k = ω 0, k =µ ω. 0 c c n c ( ) Fild at dtctor du to signal passing through rfrnc arm E = A cos ωt+ φ. Th output of th squar-law dtctor is V = E + E ( ) 1 ( ) ( ω φ1 φ plasma ) 1 ( ω φ1 φ plasma ) ( ω φ) = A cos t A A cos t+ + cos t+ 1 ( ωt φ ) + A cos + ( ω φ1 φ plasma ) A1 + A1 cos t+ + = ( ω φ φ plasma φ ) cos( φ φ plasma φ ) + AA cos t AA ( ω φ ) A + A cos t+ + Th capacitanc of th dtctor shorts th microwav frquncy signals to ground. Th rmaining slowly-varying part is A1 A V = + AA 1 cos( φ1 + φ φ) + Points to not plasma. As th drivation shows, th intrfromtr masurs avrag dnsity along th chord. To obtain a dnsity profil would rquir masurmnts along many chords. Th longr th wavlngth th bttr is th snsitivity to small dnsitis but th poorr is th spatial rsolution. Bam bnding may occur. 6

7 Microwav rflctomtry Masurs n. Diffrnt frquncis will b rflctd from diffrnt layrs in th plasma. Point to not Thr is a problm if dnsity is not monotonically incrasing. Lasr intrfromtry Masurs <n>. Two forms of intrfromtrs ar th Mach-Zhndr intrfromtr and th Michlson intrfromtr. Som lasrs that hav bn usd in th School of Physics ar H-N lasr (visibl 633 nm, infrard 3.39 µm), CO lasr (infra-rd 10.6 µm), formic acid molcular vapour lasr (farinfrard 433 µm), HCN lasr (far-infrard 337 µm). 7

8 Scattring of lctromagntic radiation from a plasma (a) Thomson scattring Scattring of lasr light from lctrons in th plasma to mak a non-prturbing masurmnt of T. Th scattrd wav satisfis Sinc k s = k i, k = k + k and ω = ω + ω s i k = k i s sin θ. i Thomson scattring can b dscribd classically. Th principl is that th incidnt wav acclrats th lctrons; In th non-rlativistic cas, &v = Ei cos( ki r ω t), m and bcaus th lctrons ar acclrating, thy radiat; At larg distancs, in th far fild, r ( r v& ) Es = 3 c r whr th quantitis on th rhs ar valuatd at rtardd tim. (At high nrgis a quantum mchanical tratmnt is mor appropriat. Compton scattring.) Th ions ar too massiv to radiat - thy still hav an ffct howvr. Th contributions from all th lctrons must b combind. 1 α = << 1. Thr ar two ways of looking at this cas k λ D (a) k is larg so th rsolution is high, individual lctrons can b sn. (b) λ D is larg so plasma ffcts ar unimportant. In othr words, th ffcts of individual lctrons ar important. Th phass of th wav arriving at th diffrnt lctrons is random as will b th phass of thir scattrd wavs. So th scattring is incohrnt. Th total scattrd intnsity is givn by adding th scattrd intnsitis from ach lctron. 8

9 (If w think of 1 k as a scal lngth, thn α = scal lngth Dby lngth.) 1 α = >> 1 k λ D (a) k is larg so th rsolution is low, only a cloud can b sn, (b) th scattring is from fluctuations in charg dnsity (ion acoustic wavs) and is collctiv or cohrnt. Th total scattrd intnsity is found by calculating th total far fild and squaring it. Th intnsity in th incohrnt cas is n and in th cohrnt cas is much largr, n. Th intnsity of th scattrd radiation is givn by I ω = I r 1 cos φ ns k, ω ( ) ( ) ( ) 0 whr I 0 is th incidnt intnsity, r = is th classical radius of th lctron (which 4πε 0mc is vry small), ( 1 cos φ) is a gomtrical factor in which φ is th angl btwn E i and k s, n S k, ω is th dynamic form factor. is th plasma dnsity and ( ) From this xprssion w s (i) that th scattrd intnsity is gratst for φ = 90 So choos φ = 90. (ii) th scattrd intnsity is xtrmly small So us an nrgtic pulsd lasr (ruby lasr). (iii) th spctrum gos as S( ) S( ) k, ω which dpnds on α. k, ω has bn plottd blow for two valus of α. 9

10 ( ) S k, ω has an lctron trm and an ion trm. If α << 1, th ion trm is ngligibl and th shap is du to Dopplr broadning. Elctron thrmal motion givs a dopplr shift and th width of th lin givs T. n could b obtaind from th absolut valu of th scattrd intnsity. If α 1 th lctron pak can giv n but at still highr α dpnds on n and T. At α 1 th ion trm bcoms important and, sinc th pak is intrprtd as th xistanc of an ion acoustic wav, dpnds on T. Undr diffrnt conditions th width can dpnd on T i or T. Th condition α 1 can b obtaind by using a long wavlngth and/or a small scattring angl. Points to not Avoid parasitic light. But not th light of intrst is at a slightly diffrnt wavlngth from th incidnt light. Us a bam dump (a nar prfct absorbr). A full tratmnt must includ magntic filds and rlativistic ffcts. (b) Scattring from macroscopic dnsity fluctuations Scattring from wavs, instabilitis, turbulnc. Choic of frquncy/wavlngth. Assum you know what rang of frquncis/wavlngths you want to study. Givn λ i, th minimum λ that can b invstigatd is λ i / (or maximum k is k i ). If λ is too larg (or if k is too small), th scattring angl θ is small and spatial rsolution is poor. 10

11 If ω is too low (or th wavlngth of th wavs you ar studying is too larg), bam bnding may occur (s abov), th width of th bam which is st by diffraction may b too larg and spatial rsolution is poor. You do. Find a rfrnc to gaussian bams. What dtrmins how th bam sprads with distanc Elctron cyclotron mission Elctron cyclotron mission from a singl gyrating lctron is at th lctron cyclotron frquncy and its harmonics. Elctron cyclotron mission from th plasma. (i) Th spctrum is broadnd principally bcaus th magntic fild is not uniform..g., A tokamak whos major radius R = 0.54 m, minor radius a = 0.10 m and in which th magntic fild B = 6.1 T. At th insid wall of torus B = 7. T, at th outsid wall B = 5.0 T. You do. Show this. (S Chaptr II) What rang of frquncis might w xpct for ach harmonic? (ii) Intnsity of th radiation. At low frquncis, th plasma is optically thick. Any radiation is rabsorbd. Th plasma is lik a black-body ω kt I( ω) = Ibb ( ω) = 3 8π c so T can b stimatd from th intnsity. 11

12 At high frquncis, th plasma is optically thin. Rflctions from th walls bcom important. τ( ω) 1 I( ω) = Ibb( ω) τ( ω) 1 r whr τ is th optical thicknss and r is th rflction cofficint. Not that optical thicknss cannot b dducd from th arlir cold plasma disprsion rlations. (iii) Usually obsrv at 90 to th magntic fild. In this dirction, X wavs dominat. Th spctrum of lctron cyclotron mission is obtaind using Fourir transform spctroscopy. Suppos plasma mits a singl frquncy Fild at dtctor du to bam rflctd off stationary mirror E = A cos 1 ( kd ω t). Fild at dtctor du to bam rflctd off moving mirror E = A cos ( k( d + x) ω t). Th output of th squar-law dtctor is ( E E ) A k x x 1 + = 4 k d t + cos cos ω A k x 1 x = 4 k d t 1+ + cos cos ω Th low-frquncy part is V A k x = cos = A ( 1 + cos kx). Exprss this as 1

13 intnsity I = S( k)( + kx) 1 cos. Th plasma mits a rang of frquncis. Using a similar approach intnsity Ix ( ) = Sk ( )( 1+ cos kxdk 0 ) S(k) is th spctrum. In a masurmnt, I(x) is rcordd for a rang of x, th avrag valu is subtractd out laving what is calld th intrfrogram Int( x) = S( k) cos kx dk. 0 This is a Fourir transform. Carry out th invrs transform to obtain th spctrum S(k) Sk ( ) = Intx ( ) cos kxdx. π 0 W do not hav radings for x = 0 to, only to x m. This mans that th rsolution of th instrumnt is limitd to k = π. x m Plasma spctroscopy Important procsss that dtrmin populations includ 1 Radiativ, involving photons bound-bound transitions - absorption and (its invrs procss) mission of photons A+ hν A fr-bound transitions - photoionization and rcombination with th mission of a photon A+ hν A + + Collisional, involving lctrons. Sinc th procsss involv lctrons T is important. bound-bound transitions - lctron impact causing xcitation or dxcitation A+ A + fr-bound transitions - lctron impact causing ionization or thr-body rcombination + A+ A + + Two modls ar of particular intrst. 1. Local thrmal quilibrium (LTE) High dnsity, low tmpratur plasmas. Collisions xcit and dxcit. Th populations of th nrgy lvls is givn by Boltzmann distribution for tmpratut T. For two lvls n and m En Em nn gn kt = nm gm whr th ns ar th numbr dnsitis, th gs ar th statistical wights and th Es ar th nrgis. This can b gnralizd to giv ratios of numbr dnsitis of nrgy lvls for atoms in diffrnt stats of ionization and as a spcial cas th Saha quation (s Chaptr I).. Coronal quilibrium. Lik th sun s corona. Low dnsity, high tmpratur plasmas. Collisions xcit and photons dxcit. 13

14 Sinc collision rat dpnds on dnsity, which is low, th only way for a downward transition is by spontanous mission A low dnsity plasma is optically thin so any photons scap bfor thy can xcit anothr atom and th only way for an upward transition is by collisions. Most plasmas li outsid LTE. Th coronal modl is appropriat for tokamak plasmas as long as thr has bn sufficint tim for quilibrium to hav bn rachd. Coronal quilibrium may apply to lowr nrgy lvls but LTE may still apply to th highr. Lin spctra Nutral atoms and ions that still hav bound lctrons mit radiation whnvr thy mak a transition from a highr nrgy stat to a lowr on. This provids a way of studying th working gas and any impuritis in th plasma. Howvr intrprtation (bsids simply rvaling which spcis ar prsnt) is gnrally vry difficult. W nd to know th populations in th various possibl stats (complicatd functions of n and T and th composition of th plasma) and undrstand th procsss that maintain thm. A ** A * + hν Diagnostics frquntly us a monochromator to masur th absolut intnsity of lin ratio of intnsitis of two lins lin shap and/or width Lin broadning Th width of spctral lins is principally du to Dopplr broadning - particl thrmal motion givs a dopplr shift. This would giv T a or T i. Prssur broadning which includs collisional broadning and Stark broadning. dpnds on th influnc of narby particls on th mitting atom. Collisional broadning. Most of th tim th atom radiats undisturbd, occasionaly thr is a collision and an abrupt phas chang. f FWHM 1 whr τ is th man tim πτ btwn collisions. atoms. Stark broadning. Th most important prturbing ffct is th E fild of narby 14

15 Instrumntal broadning. Th masuring instrumnt has a finit rsolution. Us convolution to combin th ffcts of th diffrnt kinds of broadning Continuum spctra Diagnostics may b basd on absolut intnsity ratio of intnsitis at two wavlngths In visibl, uv and x-ray. brmsstrahlung fr-fr transitions - intraction of lctrons with ions of ffctiv nuclar charg Z. rcombination radiation missivity (radiatd powr /unit volum /unit angular frquncy /unit solid angl) hω kt j = n n Z 1. T i No brmsstrahlung from non-rlativistic lik particl intractions. Radiation losss by a fusion plasma ar wors if th impuritis mak th Z ffctiv highr. A + + A j is similar but th xponnt is rplacd by a sris of trms with Z hω E n kt 1 G Z whr E 1 is th ionization potntial of hydrogn and G = 1 if h ω >> E 1, n othrwis 0. Rcombination radiation is lss important than brmsstrahlung if kt > 3 E1Z Th combind spctrum is shown blow.. 15

16 Not at low frquncis j T 1, th rcombination dgs. Th highst frquncy dg is whn th atom nds up in th ground stat aftr th lctron is capturd. Points to not Nd to calibrat stup to find its snsitivity at th wavlngths bing usd. Nd to b awar of: any lins in th wavlngth rang, radiation from vssl walls. 16