Lecture10: Plasma Physics 1. APPH E6101x Columbia University

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1 Lecture10: Plasma Physics 1 APPH E6101x Columbia University

2 Last Lecture - Conservation principles in magnetized plasma frozen-in and conservation of particles/flux tubes) - Alfvén waves without plasma pressure)

3 This Lecture Introduction to plasma waves Basic review of electromagnetic waves in various media conducting, dielectric, gyrotropic, ) Basic waves concepts especially plane waves) Electromagnetic waves in unmagnetized plasma Electrostatic waves in unmagnetized plasma

4 Wave Helmholtz) Equation ' '-I'" n l" II "" V 'J '41 N t' r" with c constant) ", 'P tj lt X '- 4Īi,. "- 'l 'ḷ l- J ") o ln " '; " '- It " ''K "", C\ \ n '- ' U +1 'l J. " \J\ -P.l C" \ rw \ t\, _l.1 u1 1 Q ': t i Q 1 J i 'U - J r" ') '1 t ei ) eta e " \l 3,. "" oq '- 'I'" \t 'l \,.

5 Plane Waves y. '- u. q: ù: Q, \A S- r l I- 't Ul,. u '" "- "" \Q \c ') Q v: t '- :r '- L.ø J -. ).. \ 'l " vi,. ' ;\ tj il \- 1 t 0,. '- T \1 tj i \..t \1 J \- t1 i \& '" \i v 'c ii ct 'X ) +i J! 1 'l 'I.J '", \) 0 /' VI r. v 1 '- \l u "- 1: \l \ '" '" Q Q " '" i4 l.) ') i' t ei é \i cy\ " ", t '4 :i j t \f ϕ = k r ωt "I '" I- J Ii i ta t J Q \I i i '4,- \l t t

6 Wave Eq in Multiple Dimensions The wave equation has solutions that are plane monochromatic waves of the form E = Ê exp[ik r ωt)] B = ˆB exp[ik r ωt)] j = ĵ exp[ik r ωt)]. 6.8) U '\, ':, i ",'-J.; S a '1 i l \Ò i,. ': -l r ñ ;t t L J "' \: lj.. 1. tv '" Vi ϕ = k r ωt \ tv -- y. "- i "- tv l " Here, k is the wave vector, which describes the direction of wave propagation. The magnitude of the wave vector is related to the wavelength by k = 2. The wave J- e: -i " \\ \,. Yo P \ il.a 1- \ t1, '- 3 r+\ )\ i: l c: \ ). r II '". 'l V 0 '0 C' :, r: l I, j '- \ - A. \. l 1 N t: Ql 'I "" '" \o '" \e '),, 'c. \)., 1 /" \ Q. \l,. '- "Q i II i': i. '- ", I "" r- '- "" C1 l '" 'e I 'Q '. it I l " i' 0 ".J J x "- a: 'ë.. ) " r

7 Wave Eq in Cylindrical Coordinates. '- '" í --,. 'i li rc\ "-. "' "' 1 + fi ro t" " l.".t \ "" tl " -i fl Q) "' +. ti 4- \ "" " " "',, N '\ '- t' t '- '- ') y -l, Q I, t"\ t I" L -l "Q,. '\J L u. '\,. "- U Cl., If 0 r: II l '- tv C" "\ ' \N \I - 't Vl ve -i 0.) -; 2: J tj -l r '-, i \I.- Ml - ṙ- t ". 0 i -" - '- ". n " II,,. oj \ " '- \\ i i \ IV.J v '+ O "- T J /' ),.. ii Vl - + N t'\j W V\ l \ 'ò I V\ ci\ v it td \U \t ) -\ L. a J \\ g 'l D tj '1 -

8 Wave Packets and Group Velocity '" l:. V \i \l "" W :; -- \. Cc '1 t tv i'" i J t 'l J" 1-, i - I-.l r. "' J :i,. '- II u J\ i " tj Vt 31n V tj '" t ) "" Jil C "". t /', 0,,:. X 'u ') oj 3," -- 0 rl :i II '" i I V \. t :f l1., t\ 1:. 'l '- -. J. I/ 0 Q W ø:" "' '- ) J l tt t. l) '- '" '" T,., 4- V i 'ii ". f '-.- of,,. /' )t '0 '" l' "" ij "-. " \j" :i il l :k - 'l I ') 0, ) 1 t ' -t 3'" 4 1 II 11 ti i -l.,. lc '" \",. \0- "j '- I- t.- ;i - t - ') ) q \L \! \. \U \L J

9 Wave Dispersion - P Q.. 'Q :i j J t4 \A s q ") :e \l \l t ') ") \;, 1; J.3 " \i ") i- o \ J "; \J. -a i- '" 'e.j \J '" ) "\ Q V' '" r l- ') a) ω \I '0 "" \I \J l: α b) ω \J α a- ;; dispersive medium k k dω/dk β ω non-dispersive medium dω/dk ω ). " \ i \L k \i C: J t o V\ 3i'" k i 1 \ i l. v '0 v 'l i i "" J ) q. \0. J \0 t l.l '\l "" vt, i. '" '. I': l. 0 1 '" "" t- l., '- n 't! i l'" '. '\ i: t -- u "" J VI \l 0 " \. t li i- ' \! ') ), \A,.

10 E ik Ê, E ik Ê, PDE s become algebraic! "- \. \ " \ - 'o C r: 0 \. o o ) '0 "" 'J f' \f t' Phasors, Sine, Cosine 'i "Î c: 'i \ 0-./ Ii 'i 'l J j IJ '" ; "" fv 0 J 0 ) 0 \ L \ "- '- I j I ii I '" i "- v- li c: \) \. ". \. -v \\ /' /' '+ \) -J '\ j J t E iωê J \ \D "- -L. Î \\ \ "- t- \ ;: \: :) " \) ' L Q l \J 'J,,,.. ) U 'V '- ' o. '" 'i J l x +- i' '\ x If,J "- Q l" '- "- \/ 1I. ' \) r- l./ /" -- Y. l./ y. ') N 't lj /' i. -\ ' II l :r 1 ':r Cl - '- J J _- 0 r. Q v' r 't ) -irj 1I t; r- " c- l/ '- c.- - VI ct r" -- 0 Il 2$ II \1 "" I -;rj c: Il) \\ "v \\ ". tj j \. \l V\ c;. CD 1 't F QJ "" cr i \l C C: \)

11 Review of EM Waves E = B t ) E B = µ 0 j + ε 0 t E) = B t = t B) = µ 0 ε 0 2 E t 2 µ 0 j t

12 Review of EM Waves E = B t ) E B = µ 0 j + ε 0 t ε ω = I + i ωε 0 σ ω. ε 0 E t + j = ε 0 εω) E t jω) = σ ω) Eω) εω) = 1 + i ωε 0 σ ω)

13 Review of EM Waves E) + 1 c 2 2 E t 2 = µ 0 ik Ê = iω ˆB j t ik ˆB = iωε 0 µ 0 Ê + µ 0 ĵ n, we discuss the wave equat 0 k k Ê) = kk k ] 2 I)Ê udes 6.7) can be transforme {kk k 2 I + ω2 c 2 I + iωµ 0σ ω) } } {kk k 2 I + ω2 c 2 εω) All of the plasma physics here Ê = 0 Ê = 0

14 k x k x k 2 + ω2 c 2 ε xx k y k x + ω2 c 2 ε yx k z k x + ω2 c 2 ε zx Normal Modes Dispersion Relation) k x k y + ω2 c 2 ε xy k y k y k 2 + ω2 c 2 ε yy k z k y + ω2 c 2 ε zy = 0 = Dω, k) = det k x k z + ω2 c 2 ε xz k y k z + ω2 c 2 ε yz k z k z k 2 + ω2 c 2 ε zz ] [kk k 2 I + ω2 c 2 εω) Ê x Ê y Ê z = ). 6.29)

15 Waves in Unmagnetized Plasma Collisionless reactive) m dv dt = qêeik r ωt) ĵ = nqˆv = i ne2 ωm Ê σ xx = σ yy = σ zz = i ne2 ωm 90 out of phase ˆv = m iω + ν m ) ˆv = q Ê [ ν m ω 2 + ν 2 m In phase low-frequency) Collisional resistive) + iω ω 2 + ν 2 m ] q m Ê ε xx = ε yy = ε zz = 1 + i ωε 0 σ yy = 1 ω2 pe ω 2 ω pe = ne 2 ε 0 m e ) 1/2

16 Waves in Unmagnetized Plasma ] 0 = Dω, k) = det [kk k = = = 2 I + ω2 c 2 εω). ω 2 c 2 1 ω2 pe ω 2 ) 0 k 2 + ω2 c 2 1 ω2 pe ω 2 ) k 2 + ω2 c 2 1 ω2 pe ω 2 ) 0 ) ) Ê x Ê y Ê z = 0. ω 2 = ω 2 pe + k2 c ) v ϕ = ω 2 pe k 2 + c2 ) 1/2 v gr = kc 2 ω 2 pe + k2 c 2 ) 1/2

17 Waves in Unmagnetized Plasma ) ˆv = m iω + ν m ) ˆv = q Ê [ ] ν m iω q ω 2 + νm 2 + ω 2 + νm 2 m Ê ω 2 = ω 2 pe + k2 c 2 k = 1 c ω 2 ωpe 2 ) 1/2 1 + iν m /ω) Collisional resistive)

18 Interferometry a) laser Plasma detector microwave source µw Plasma ϕ = 2π λ horn antenna [N x) 1] dx b) directional coupler phase shifter attenuator detector Fig. 6.5 a) Laser interferometer in Mach-Zehnder arrangement, b) microwave interferometer. The optical arrangement uses partially-reflecting and fully-reflecting mirrors. The analog to a partially reflecting mirror is the directional coupler for microwaves

19 N = Interferometry ϕ = 2π λ 1 ω 2 pe /ω [N x) 1] dx ω 2 pe ω 2 = Table 6.1 Cut-off densities for microwave and laser interferometers Wavelength Frequency Cut-off-density Source λ f n co m 3 ) Microwave 3 cm 10 GHz mm 37 GHz mm 75 GHz HCN-laser 337 µm 890GHz CO 2 laser 10.6 µm 28THz He-Ne laser 3.39 µm 88THz µm 474THz n n co

20 Interferometry N = ϕ = 2π λ [N x) 1] dx 1 ω 2 pe /ω ω 2 pe ω 2 = n n co interferometer signal 0 current pulse cut-off a) 4096 necm 3 ) b) ncut-off t µs) t ms) Fig. 6.6 a) Interferogram in a pulsed gas discharge. b) Reconstruction of the decaying electron density by counting interferometer fringes

21 Electrostatic Waves n t + x nv) = 0 m v = q dφ dx γ n dnk B T ) dx Electron Pressure Force

22 Electrostatic Plasma Waves n t + x nv) = 0 iω ˆn + ikn 0 ˆv = 0 m v = q dφ dx γ n dnk B T ) dx iωm ˆv = ikq ˆφ ikγ k B T ˆn Electron Pressure Force

23 Electrostatic Plasma Waves ω = ω 2 pe k2 v 2 Te) 1/2 = ω pe 1 + 3k 2 λ 2 ) 1/2 De Electron Pressure Force Electron Pressure Force

24 Electrostatic Ion Sound Waves Ion Pressure Force iωm i ˆv i = eê ik n i0 γ i k B T i ) ˆn i 0 = eê ik n e0 k B T e ) ˆn e ˆn i = ˆn e = ek iω 2 m i + ik 2 Ê γ i k B T i e Ê, ikk B T e Look! No electron acceleration Electron Pressure Force ik Ê = ni0 e 2 ) ε 0 m i k iω 2 + ik 2 γ i k B T i /m i Ê + ne0 e 2 ε 0 k B T e ) 1 ik Ê

25 Electrostatic Ion Sound Waves εk, ω) = 1 ω 2 pi ω 2 k 2 γ i k B T i /m i + 1 k 2 λ 2 De ) ω 2 = k 2 γ i k B T i + ω2 pi λ2 De m i 1 + k 2 λ 2 De ω kc s 1 + k 2 λ 2 De T e T i. d C s = ω pi λ De stic wave.

26 Important Wave Concepts Linear vs. nonlinear Dispersion Phase and group velocity W = Polarization and wave structure Energy & intensity Poynting s Theorem) Inhomogeneity W t dv H B + E D) V N ds = dv J E, S V

27 Next Lecture Chapter 6: Plasma Waves Waves in magnetized plasma

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