Analytic Models of Near-Field RF Sheaths

Transcription

1 Aalytic Model of Near-Field RF Sheath D. A. D Ippolito ad J. R. Myra Lodetar Reearch Corporatio Preeted at the 18th Topical Coferece o Radio Frequecy Power i Plama, Ghet, Belgium, Jue 4 6, 9

2 Backgroud We eed quatitative calculatio of rf atea heath with full plama effect; heath BC i a promiig approach. Here we decribe a aalytic model of atea heath uig thi ew BC (ueful for code bechmarkig); two cae: teuou plama limit (D Ippolito & Myra, PoP 56, 9) full plama dielectric Quatitative reult of the model: heath capacitace effect, importat whe Λ ε ( / L) >> 1 limit of validity of the vacuum field appro. = creeig of by heath whe ε >> 1 ( (vac) ) vac dz dad APS-DPP 7 Lodetar

3 Approimatio Simple model of ICRF wave laucher: electromagetic wave propagatio i a plama-filled waveguide cotat-deity plama (e.g. aume trog rf covectio) FW i polarized i the y directio ad propagate i the + directio. equilibrium magetic field i B = B z e z + B y e y perturbatio epaio aumig weak field lie tilt (b y B y /B << 1) Aume k y = keep magetic flu from curret trap, but ot effect of feeder or bo curret (model valid ear ceter of atea at y = ). apply the heath BC at z = ±L, where B field lie iterect the heath field lie tilt, heath BC couplig of FW to SW Ue low deity ( teuou plama ) approimatio (ε = 1, ε = but ε >> 1) or full plama dielectric (ε 1,ε, ε >> 1) dad APS-DPP 7 3 Lodetar

4 Liear rf wave propagatio Wave operator: L = ω c ( ) + ε = ( ) + ε = ( I + ε) where = k c / ω ad we epad the plama dielectric i power of b y: ε = ε + ε 1, ε I + ( ε ε ) ezez, = ε ε ε )( b e + e ) 1 = ( y z zby. Homogeeou plama diperio relatio: ( ε f )( ε ) ε = Firt, give the reult for the teuou plama approimatio ad the give geeralizatio to fiite-deity cae. dad APS-DPP 7 4 Lodetar

5 RF field olutio I lowet order, aume a propagatig FW ik f = f = {, Êy co(k z δ), } e BC o y require co(k L δ) = η L = π / + δ k Parameter δ determie the parity of y ( atea phaig) δ = moopole phaig, δ = π/ dipole phaig, I firt order, the FW drive a wave with SW polarizatio but atifyig FW diperio relatio ik f 1p = C {igf i(k z δ),, co(k z δ)} e C = b y Ê y G z /( z 1) dad APS-DPP 7 5 Lodetar

6 Sheath BC The um of thee wave ( f + 1p ) doe NOT atify the full heath BC. Need to add a SW olutio 1 ik 1 = A {ig i(kzz δ),, co(kzz δ)} e SW polarizatio SW waveumber (couplig by heath) The heath BC determie the amplitude A ad require k = k f : + ε z z= L = A = C D D f = b y Ê y G D f D where D i the coupled heath-plama diperio relatio D( η zj ) G j i( η zj δ) + k j ε co( η zj δ) plama iductace heath capacitace heath-plama reoace whe D dad APS-DPP 7 6 Lodetar

7 Computig heath voltage Commoly ued vacuum field heath approimatio give the followig heath voltage (for moopole phaig) vac ( δ = ) = L dz b y Ê y co k z e ik = b π We iclude plama effect by uig the SBC ad itegratig acro the heath (h) dz = ε (L) h h h vac z = dz (vac) Ratio of the two model predictio i give by z y y L e ik (o plama ide of heath-plama boudary) ˆ ( δ) h vac ( δ) () = ( / L)( ( / L) 1) α η η co( η co( η δ) δ) + αi( η δ) η ωl / c α 1+ (δ / π) dad APS-DPP 7 7 Lodetar

8 Screeig of plama by heath Cotiuity of D ormal acro heath-plama iterface creeig of = () vac Λ α L i( η ( / L) η δ) + ( / L)( co(k z 1) η z δ) co( η δ), Λ ε L For moopole phaig with >> 1 ad η = ωl/c<< 1 () = () L vac R Λ co k z z where creeig i trog heath limit ( a Λ ) R ( / L) 1+ ( / L) 1 voltage plit pl h L dz z= L h dz = ta η Λ η 1 Λ a η dad APS-DPP 7 8 Lodetar

9 Self-coitet (Child-Lagmuir) heath olutio Sheath width mut atify the Child-Lagmuir (CL) Law = λ D e T e 3/ 4 where rectified (dc) heath voltage i + = 3Te.6h Ue oliear rootfider to olve the followig cotrait (obtaied by combiig the CL Law with the epreio for h derived above where C ˆ 1/ 4 A B ˆ A = η co( η λ = L δ) D e T vac e 3/ 4 = ( 1) α η co( η δ) C = αi( η δ) B dad APS-DPP 7 9 Lodetar

10 Numerical olutio for elf-coitet h 1. h / vac e vac / T e Plot of the elf-coitet (Child-Lagmuir) heath voltage, h / vac, (obtaied from olutio of eq. o previou page) v the vacuum field heath drive, e vac /T e, for moopole phaig with parameter e = 1 1 cm -3, T e = 5 e, ad = 1. dad APS-DPP 7 1 Lodetar

11 Geeralizatio to full plama dielectric (i moopole phaig) Geeral reult: h Qf = ε Êy i η + A coηz Ff b y ad ε = (recover teuou plama reult) b y ad ε (ew fiite-deity effect) From defiitio of h heath BC + SW polarizatio ˆ = ε (1 Q f R ( / L) η )i η z z coη z ( / L) η z coη z from particular olutio, drive by FW heath-plama-wave phyic dad APS-DPP 7 11 Lodetar

12 Special cae (co t) b y =, ε, ad ε 1 (fiite deity term ehace heath drive; competitive whe ε > b y L / ) h = Ê y Q f i η f 1+ (1 Q f R k z coη )i η k z coη Here h Qf k Êy ε Ê i comparable to vacuum heath voltage whe k y ε b y L Defiitio: F f = ε, F = ε z, R ε = Q f dad APS-DPP 7 1 Lodetar

13 Numerical reult plot approimate reult for h / vac with b y ad ε, keepig oly term b y i umerator CL oliear cotrait NOT eforced here parameter: k L = π/ (lowet mode), b y =.1, B = 3 kg, T e = 3 e, L = 5 cm, f = 45 MHz.» hêvac» FW cut-off Log@ e Hcm -3 LD» hêvac» e = 1 8 cm 3 e = 1 1 cm LHcmLD dad APS-DPP 7 13 Lodetar

14 Summary We have derived a epreio for the atea heath voltage (uig a waveguide model) icludig heath BC [D Ippolito ad Myra, PoP 6, Myra et al., PoP 1994] fiite plama effect (plama dielectric) Child-Lagmuir phyic (elf-coitet heath width) The model how that the magetic field tilt (ot ormal to FW vector) couple the FW & SW heath BC FW-SW couplig i modified by the heath capacitace whe Λ = ε /L >> 1. the heath capacitace alo cree from the plama whe ε >> 1. fiite deity at the atea ha order uity effect o h teuou plama verio i accurate for e below the FW cut-off Applicatio of the model iclude calculatig the elf-coitet heath voltage for bechmarkig heath BC i rf atea code evaluatig validity of the approimate vacuum rf field heath model: valid whe ( /L) >> 1; thi appear to be hard to atify. dad APS-DPP 7 14 Lodetar

15 All Lodetar preprit are available for dowloadig at Recet paper o rf heath modelig uig the heath BC: A Radio-Frequecy Sheath Boudary Coditio ad it ffect o Slow Wave Propagatio, D. A. D Ippolito ad J. R. Myra, Phy. Plama 13, 158 (6). Far Field Sheath Due to Fat Wave Icidet o Material Boudarie, D. A. D Ippolito, J. R. Myra,. F. Jaeger, ad L. A. Berry, Phy. Plama 15, 151 (8). Reoace Coe Iteractio With a Self-Coitet Radio-Frequecy Sheath, J. R. Myra ad D. A. D Ippolito, Phy. Rev. Lett. 11, 1954 (8). "Aalytic model of ear-field radio-frequecy heath: I. Teuou plama limit," D. A. D Ippolito ad J. R. Myra, Phy. Plama 16, 56 (9). "Aalytic model of ear-field radio-frequecy heath: II. Full Plama Dielectric," D. A. D Ippolito ad J. R. Myra, maucript i progre. dad APS-DPP 7 15 Lodetar

16 Appedi Defiitio of dielectric teor elemet (ad approimatio) ε = 1 ω pi ( ω Ω ) i 1 (teuou plama) ωpi ω ε = << 1 Ω i ( ω Ω ) i (teuou plama) ε = >> ωpe 1 (for all e) ω dad APS-DPP 7 16 Lodetar