003...............................3 Debye................. 3.4................ 3 3 3 3. Larmor Cyclotron... 3 3................ 4 3.3.......... 4 3.3............ 4 3.3...... 4 3.3.3............ 5 3.4......... 5 3.4............ 5 3.4.............. 5 3.4.3.......... 5 3.4.4............. 5 3.4.5............ 6 4 6 4.................... 6 4.................... 6 4.3 random walk......... 7 4.4.......... 7 5 8 5................ 8 5. MHD................ 9 5.3................. 9 5.4 MHD MHD........... 9 6 0 6............. 0 6................ 0 6.3.......... 6.4... 7 7.................. 7........... 7..... 7.................. 7..3........ 3 7.3... 3 7.4.............. 4 7.5.......... 4 7.6 R L O X............ 4 8 5 8. Landau................. 5 8. Cyclotron............... 6
. ev=,604k.5 0 5 m 3 Debye ɛ 0 kt e λ D = n e n e Ze 4 ln Λ ν ei = 5.6π / ɛ 0 m/ e kt e /3 ν ei v e H + +e H 0 +3.6 ev Saha x x = 3/ πme kt g i g e n h exp eɛ i /kt 3 g 0 x n i /n = n i /n 0 + n i ɛ i n e 0 7 m 3, T e ev Ar Hg Ar Hg Hg 53nm, 85nm Hg + e Hg +e Hg Hg + hν 0% 5% Si SiO n e =0 7 0 0 m 3, T e = 0eV CVD Si SiC D+T He 4 3.5MeV+n4MeV T 0keV, n 0 0 m 3 E mc e Π e 4 Π e = ne /m e ɛ 0 E 0 V/m. Γ= Ze /a 4πɛ 0 kt 5 Γ > Γ < N D N D 4π 3 nλ3 D 6 Γ= 7 3N /3 D Fermi ɛ F h /m e 3π n /3 8 ɛ F >kt
Fermi Maxwell Fermi 3/4πn /3 4πɛ 0 h /m e e 4σ/3ckT 4 nkt T =.5keV, n =0 3 m 3 T e 0.eV, n =0 0 m 3 HF 3-30MHz HI, HII HII HI HII Saha 50% HI T kev, n 0 3 m 3.3 Debye Ze φ Maxwell eφ n = n 0 exp kt e n 0 + eφ kt e 9 n 0 Poisson φ = ρ = Ze δ r+ n 0e eφ ɛ 0 ɛ 0 ɛ 0 kt = Ze δ r+ φ ɛ 0 λ D φr = Ze exp r/λ D 4πɛ 0 r λ D.4 0 n Poisson E = en ɛ 0, m e v t = e E, n t + n 0 v n exp ikx ωt ω = n 0e Π e 3 m e ɛ 0 3 Larmor Larmor Guidinig Center 3. Larmor Cyclotron m d v dt = q E + v B 4 E =0,B: v x = v sin Ωt + δ v y = v cos Ωt + δ v z = v z0 5 ρ Ω= qb m, ρ = mv 6 qb 3
Ω 3. E B B E v = u E + u, Eq.4 u E E B B 7 m d u dt = q u B 8 E B u E m g u g = m g B qb = g B ΩB 9 R v mv R B u curv = v R n ΩB n 0 B B B Larmor B B 0 + ρ B v ρ B v, ρ B 0 B z 0, ρ = ρ cos Ωt, ρ sin Ωt, 0 v = ρω sin Ωt, ρω cos Ωt, 0 B z x 0 v ρ B = ρ Ω B z x e x B u B = = v Ω B 3 v B B ΩB 4 j 0 B b B = b b B + B b b = B l b B R n 5 b = B/B B B u B = v ΩB / R B v /+v R n ΩB 3.3 n 6 7 3.3. pdq µ µ = IS = qω π πρ = mv / B q = pdq = const. 8 4πm 3.3. 0 v + v = const. 9, B v 0, v 0, v, v, 8,9 0 < v = v 0 + v 0 v = v 0 + v 0 B v 0 v 0 > B 30 v 0 v 0 < B 3 4
B B q v ρ B = µ B l 3 3.3.3 mv dl mv l 33 v 3.4 3 E Penning Trap λ D 3.4. dx, dy, dz dx B x = dy B y = dz B z 34 Ψ r Ψ B =0 Ψ=const. Ψ B = A B r = r A z θ A θ z B θ = A r z A z r B z = r r ra θ r A r θ 35 z A z r, θ =const. θ ra θ r, z =const. 3.4. I r A, B A z = µ 0I ln r, π B θ = µ 0I πr r = const. 3.4.3 36 I B d s = µ0 I B = µ 0 In 37 n R B φ = µ 0I πr 38 I /R u B, u curv E B 3.4.4 I a r, z A A θ = µ 0 a πk I k /Kk Ek r k 4ar a + r + z 39 Kk, Ek ra θ = const. 5
r a + z = const. k Iπa ra θ µ 0Ia r 40 4 r + z 3/ r 3 z 3 u B, u curv 3.4.5 ra θ tokamak θ B Φ, B p B φ,b p B φ /R u B, u curv ω dr dt = ωz dz dt = ωr + v d v d = m q R v + v 4 R, v d = const., ω = const. r + v d /ω + z = const. 4 ω a B/ a/r 3 v v < a R 43 4 4. r 0 m eve Ze 4πɛ 0 r 0 r 0 Ze ɛ 0 m e v e σ = πr 0 Z e 4 ɛ 0 m e v4 e 44 45 τ ei τ ei = ɛ 0 m e v3 e nσv e nz e 4 ɛ 0 me kt e 3/ nz e 4 46 - τ ii = ɛ 0m i v3 i nz 4 e 4 ɛ 0 mi kt i 3/ nz 4 e 4 47 - τ ie τ ei m i m e 48 mi τ ei : τ ii : τ ie =: : m i 49 m e m e 4. τ ei E m e v e = eeτ ei v e = ee m e τ ei 50 6
E = ηj = ηenv e 46 η m e ne τ ei 5 η Z e 3 ɛ 0 m ev 3 e Ze ln Λ η 5.6 πɛ 0 m eve 3 4.3 random walk Γ n D 5 53 Γ= D n 54 nx, tdx t x x + dx nx, t = n t = D n x 55 exp x 4πDt 4Dt 56 item W l, n n l n W l, n = n C n+l 57 n! = 58 n + l/! n l/! πn exp l n l 59 n W l, n x = x = la, t = nτ 60 exp x x 6 4πDt 4Dt D = a τ = [L] [T ] 6 4.4 a τ p Sec.4.3 D x t D = x t Random Walk a a Dτ p τ p a D a t 63 x / Q-machine T i =0.eV, n i = 0 7 m 3 λ ii = 0 4 Ti /n i 4[mm] /ν ii = 0. 0 9 Z 4 / AT /3 i n i 0.5[µs] a =m τ p a t 30[ms] 64 x τ p n i T 5/ i - - - - ρ e D = ρ e n τ ei B 65 kt 7
65 ρ e factor 4 v d ω mv ebr ω 66 a ω a πrq q ω v 67 Rq mv Rq ebr v m vq ρq 68 eb q 5 N 6N Boltzmann f x, v, t f t + v f + q m E + v B δf v f = 69 δt c Boltzmann n x = fd v nm v = m vd v 3 nkt = mv /d v 5. ve n e m e t + v e v e n e t + n ev e =0 n i t + n iv i = 0 70 = p e en e E + v e B+ R 7 vi n i m i t + v i v i = p i + Zen i E + v i B R 7 R = en e η j - ρ m = n e m e + n i m i v = ρ m n e m e v e + n i m i v i 73 ρ = en e + Zen i j = en e v e + Zen i v i 74 70, 7,7 ρ m + ρ m v =0 t ρ t + j =0 v ρ m t + n em e v e v e + n i m i v i v i = p + ρe + j B 75 v v i en e Zen i j en e v e v i en e v e v v e = v j/en e 7 E + v B j B en + p e R = 0 76 e en e en e 8
ρ en e j en e B p e en e = ρ m Dv Dt + p i en e ρ E v B 76 E + v B = η j 77 Ohm v B 5. MHD Magnetohydrodynami Equation E + v B = η j 78 D v ρ m Dt = p + j B 79 B = µ 0 j 80 E = B t 8 B = 0 8 ρ m + ρ m v t 83 MHD 79,80 v A = B µ0 ρ m 84 5.3 78,79 B t = v B+ η µ 0 B 85 η µ 0 a τ η = a µ 0 η 86 Navier-Stockes D v Dt = ρ p + ν v 87 η µ 0 Reynolds R v B η/µ 0 B vb/a η/µ 0 B/a µ 0va R 88 η 85 R = µ 0va = µ 0a v η η a = τ η 89 a/v A ds Φ η 0 dφ B dt = t ds + B v d s 90 78 dφ = E + v B ds dt η = η j ds = B ds 9 µ 0 η 0 Φ=const. 5.4 MHD MHD Ohm MHD 400 C Ohm v B MHD j j B v MHD 79 j B 9
j r B θ [ B z + Bθ ] µ 0 r=a B = p + z 98 µ 0 6 β p [ B z +B θ µ 0 ]r=a, β p p [ B θ µ 0 ]r=a 99 MHD 79 p = v B 9 B p =0, j p =0 Maxwell B = µ 0 j p = B µ B = B 0 µ B B 93 0 p + B µ 0 = B µ B = B B 0 µ 0 l B R n 94 p + B µ 0 const. 95 6. a z θ 94 p + B z + B θ = B θ 96 r µ 0 rµ 0 r /a [ p + B z + ] B θ p + B z + B θ B = θ 97 µ 0 r=a µ 0 µ 0 B z Bz >B z r=a β p < Bz <B z r=a β p > 6. MHD MHD MHD 0
6.3 g g E B g E B g B θ B θ B z B z B z B θ z B θ 6.4 x<0 x>0 g x z z / t =0, v 0, n 0 x v, E e iky ωt 7 0=en 0 v 0 +m i n 0 g 00 y v 0 = m i g B e B = g e y, Ω i Ω i = eb 0 < 0 m i 0 z ω kv 0 v = ie E m + v B 0 0 i E x =0, Ω i ω kv 0 v v ix, v iy v ix = E y v ix = i ω kv 0 Ω i E y 03 iωn + ikn 0 v ix + ikv 0 n + v ix n 0 x = 0 04 Eq.03 ω kv 0 n + i E y n 0 x ikn ω kv 0 E y 0 05 Ω i Ω e Ω i v e0 v 0, v ey v ex v e0 0, v ey 0 05 E y = iωn n0 06 x 05 0 ω kv 0 ω g n 0 n 0 x = 0 07 ω = kv 0 ± k v0 + g n 0 4 n 0 x g n 0 n 0 x > k v0 4 08
g n 0 Imω g n 0 n 0 x D = ɛ 0E + P = ɛ0 + i j ω = ɛ 0 K E 0 K Maxwell 7 7. Coherent Π Ω kt. E = φ = i kφ E k, B =0 B 0. kt 3. R L 4. O X 5. Fast Wave Slow Wave 3, 4, 5 7. 7.. B, E, v k e i k r ωt k E = ω B k H = ωɛ 0 K E k k E+ ω K E =0 c N N E+ K E = 0 N = kc ω E 0 ω, k K K 7.. B, E, v k e i k r ωt k z d v k m k dt = q k E + v k B iωm k v k = q k E + v k B 0 3 v kx = i E x v ky = E x Ω k ω ω Ω k Ω k ω Ω k v kz = i E z Ω k ω E y i E y Ω k = q k m k K ik 0 K E = ik K 0 0 0 K Ω k ω Ω k Ω k ω ω Ω k 4 09, 0 E x E y E z 5 j = k n k q k v k 09 K Π e ω Ω e Π i ω Ω i 6
K Π e Ω e ω Ω e ω Π i ω Ω i K Π e +Π i Π e Π e R L ω n e e ɛ 0 m e, Π i Ω i ω 7 ω 8 n e q ɛ 0 m i 9 Π e ωω Ω e Π i ωω Ω i = K + K 0 Π e ωω +Ω e Π i ωω +Ω i = K K k, N xz z z θ K N cos θ ik N sin θ cos θ ik K N 0 N sin θ cos θ 0 K N sin θ Ex E y E z E 0 0 AN 4 BN + C = 0 3 N = B ± B 4AC 4 A A K sin θ + K cos θ B K K sin θ + K K + cos θ C K K K =K RL 5 7..3 θ =0,π/ 0 <θ=0<π/ θ =0 3 K N 4 K N + K K 6 K = 0 7 N = K + K = R 8 N = K K = L 9 K =0 ω =Π k z, E z y =0 N = R, N = L ie x E y = ± z E x, E y θ = π/ 3 K N 4 K K + K K N + K K K 3 N = K K = RL 3 K K N = K 33 N = K E x = E y =0,E z 0 N = RL K E x = ikx K E y, E z =0 7.3 WKB WKB v E t = v E x 34 ω E = Exe iωt E + ω v E = E + k 0N E = 0 35 N = c v = kc ω Sch oredinger h m Φ +E V Φ = 0 Φ=e is/ h, S = S 0 + h i S +... Ex =e iφx φ iφ N k0 = 0 36 φ = φ 0 + φ ik E x K N E y =0 ie x E y = N K K 30 φ 0 = N k 0, φ 0 = k = ±Nk 0 φ 0 φ = iφ 0, φ = iφ 0 φ 0 37 3
= φ 0 φ 0 d dx = d dx k 38 φ 0 φ 0 = ± Nk 0 dx φ = i log φ 0 = i log Nk 0 39 E = e iφ = Nk0 e ±i Nk 0dx 40 WKB WKB 35 7.4 0 N > 0 N < 0 40 N E e ±i Nk 0dx Nk0 N = sine, cosine Cutoff N < 0 E e ± Nk0 x N 0 WKB N Cutoff Airy N > 0 N > 0 Cutoff 0 0 Landau 7.5 7.6 R L O X R L O X Π e Π i,ω e Ω i, Pi eω i + Pi i Ω e =0 K, R, L K Π e ω R ω ω Rω + ω L ω Ω e ω ω i L ω ω Rω + ω L ω +Ω e ω + ω i ω R = Ω e Ω + e ω L = Ω e + 4 4 43 44 4 +Π e Ω eω i 45 Ω e 4 +Π e Ω eω i 46 kck ω ω k ω k θ =0R L R N = R ω c k = R = ω + Ω i ω Ω e ω ω R ω + ω L 47 4
ω>0 ω = ω R 0 ω =Ω e ω L N = L ω c k = L = ω Ω i ω +Ω e ω ω L ω + ω R 48 ω = ω L 0 ω = Ω i ω R L ω c k < 0 evanescent θ = π/ O X O N = ω c k = K = Π e /ω =+ Π e c k 49 ω =Π e ω X N = ω c k = K RL = R + L RL ω4 Π e +Ω eω +Ω eω i Π eω e Ω i ω ωl ω ωr = ω ωlh ω ωuh ω ωl ω ωr 50 ω UH = Ω e +Π e 5 ωlh = Π e Ω eω i Ω e Ω i Π = e +Ω e Ω 5 i +Π i Ω i Ω e ω = ω R, ω = ω L ω = ω UH ω = ω LH 8 Landau cyclotron 8. Landau z z v 0 ω/k v 0 E = E cos kz v 0 t v = v 0 +v +v + 0 v = v 0, z = z 0 + v 0 t v = qe m cos kv 0t + k z 0 ωt = qe m cos αt + φ 0 53 α kv 0 ωt, φ 0 kz 0 v = qe sin αt + φ 0 sin φ 0 m α z = qe cos φ0 cos αt + φ 0 m α sin φ 0 α t 54 + v + v = qe m cos αt + φ 0 + kz qe m cos αt + φ 0 sin αt + φ 0 kz55 v = qe m sin αt + φ 0kz = q E m k sin αt + φ 0 cos φ0 cos αt + φ 0 α sin φ 0 α t 56 d mv dt = vm v = v 0 mv v mv + v 0 mv 0 = v 0 qe cos αt + φ 0 5
+ q E m cos αt + φ 0 sin αt + φ 0 sin φ 0 α q E kv 0 m sin αt + φ 0 cos φ0 cos αt + φ 0 α sin φ 0 α t φ 0 d mv = q E ω sin αt dt m α + kv 0 cos αt α 57 58 sin x/x cos x sin x dx π, cos xdx 0 x x 0 sin x/x > cos x α 0 fv fv 0 + α k f v d mv = πq E ω f dt m k k v 59 8. Cyclotron B 0 R L E x e ikz ωt, Ey ikz ωt B = k E/ω V m v + mv v z = q E + v B0 V B 60 x y v x + ikv v x = qe x m Vk/ω Ωv y v y + ikv v y = qe y m Vk/ω+Ωv x 6 v ± = v x ± iv y, E ± =E x ± ie y e ikz ωt v ± = iqe± ω kv mω iω kv ±Ωt e ω kv ± Ω 63 v = dv v c ± = dvfv kv/ω eiω kv ±Ω ω kv ± Ω v x v y = v ± = iq m c± E ± e ikz ωt = iq c + E + c E 4m ic + E + ic E v + +v v + v i e ikz ωt q Rev x ReE x e ikz ωt + Rev y ReE y e ikz ωt + = q Imc + E + Imc E 64 4m Imc ± Imc ± = ± πω ω k f kv/ω sin ω kv ± Ω dvfv ω kv ± Ω ω ± Ω k 65 E +, E θ =0 L R v ω±ω k Ω R E + =0,E 0 + q πω ω Ω 4m ω k f E k ω/k Ω e > 0 Ω i < 0 L E + =0,E 0 q πω ω +Ω 4m ω k f E + k R v ± =±iω ikv v ± + qω Vk E ± 6 mω 6