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The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b /    0 l(  lf l 0"`H l 0޽h ? )))̙̙380___PPT10.VPrI  0 t(  tf t 0"`H t 0޽h ? )))̙̙380___PPT10.VPrIM  0 d\ H(  Hr H S =K   r H S |V0   8 d ! Hd H" \Adriftphys !  H2 NA ? ?`     `  H 0A? -  H 0d R  Z*Physics picture behind the drift velocity ++H H 0޽h ? )))̙̙3___PPT10i.=V?+D='  = @B +  0 p(  pX p C    r  p S (Rr  0  r   H p 0޽h ? 3380___PPT10.V@wI  0 x((  x^ x S     x c $H  0    H x 0޽h ? 3380___PPT10.V@wIr$av@s=u8|L~`l1(   Equation Equation.30,Microsoft Equation 3.0/ 0DArialngsRomanTTxE ՜.+,0H    On-screen Show91Ӱƽa ArialTimes New Roman WingdingsSymbolAxisMicrosoft Equation 3.0Physics of fusion powerSlide 2 Gyro motion Typical valuesAdditional Force FSlide 6Drift velocityPhysical picture of the driftElectric field $Electric field that depends on timeMeaning of the drifts "Drift leads to charge separation Electric field evolution "The next d_0phsfarphsfarrift : The ExB velocityThe ExB velocity Meaning of the drifts Slide 17Inhomogeneous magnetic fields Inhomogeneous magnetic field Curvature drift 'Drifts due tWHTCc⏻W}+?<<\S.Qy\@a[q*$ bP姪D}IC J@Ph`hpT"a)b>ղ5RW<%"2~X1Z $j tnQۇ:9)Q蒂Pz PCX4”*$Z7;$;V S[ *ۛ])+ pЬ!рꓗwځ鸋IRWڷԳܷ'3ΩOQ#T?Vz?ް՝TR>ţޓ>5e<yz7rLJrǿQ}G'o?/Upw?eg+US_+j>/č GƸcƿmhl}^v^|*glƖ]>~q!#]*֗XX+G1@ uy5rU14m΍&+5K𻎁mc㾹4&2geQʡD>)vUqC5>9x=:СsZ%qL, uk mjݦα1@NMHjߓ{>WO%slؓu{?rl>?.i%U*) jV`}Ɔ~o}=qv,9`9d>;gFӆeeo㉪dqmE}6>R#yS6h;]Ƶ96}}OcI|=IX0^lc1ƴԸ̤V QGNo.q8^z<wf`9={_mO;[6ix zx=|Otǡ{MO'Ot#9}86qɧ'qXO>{Pm-\}63k&Vi 9}h`k87M;5,<?M~s?m8ڋ[{ǢC_gwÒr>.Jm,eйH丬zKAZ+c`:zM]l>y^gǻk~jS} 6~bic=ޔn:R{SFIxzԿ8شgzCϧ5}غ;?Nղъm9Y;5>j휱:|)mզĎmXy'ڐ=~'~۷5==ѾZҿݷׯk[k봥+u.οO謴܁j➰k[Lf՗7~X=lɰ敽ӫƗ%2ͧ_>!ڰ<3~r޿V|ΌkǏEΞ~O],˹O>Xgސ9g1ͯŹ!Rk~*N.96XShy:^S}Z+0ϱ5?n`ސljsjz:@ɺA_$!K^z_zݮM||wكKP SY\Q:6^92ϥ9ksρϟ[%57k :˱87mHfurߦX]Ȝ5q5s5}niɪV`m>}SoZC{khލ `Z5ŷkׁ}u2-fԱY_Q+sN-Xq-fXn&k6Tpmv @_d O>N}5+9~|\n͛ s]XJ>.guVbVXSM,!Ǧ{W٧kȱkckQ+ZrE>q.c 8XK>N;kyc!8XS͛j`յϗW+Х'ɥP')ط'&!*Y$h]n֔cq(-`-96&J@|!f 4vIu~B,0Tt90Z jV@T t'~~~Cx|M#GON>~Y>X#{/{rЕBhFFB(B(BEaa!l!l!"QQ66FB(B(BEa^|yqqqzzWWW#u{{{~~.vy}}}wwn_vZYUW^^^ڷFj;k ̩gggbv]|,N`5zel!#111Ķa7=@L989i`h{WәH$zh 3 6'l :27c̎ܐMA5|ktf:tH"TR[JvejlU9$AlF!%ΪlN[Hj? 9#\5}zv Ieg 66#PMx2#:w4FvaBO:+#əBHzԦ *?lg6͑9iJ Is@Iflrzd򯜼f8,3ִf*>KG_j!_6mEˑ`_N]?X-t||W*^7N.U E3Pې)?9 泰Q4Yǵ<]WQ0J^JPzC#^SXZ[ 4u-mk!gm0Nbf8I,խcK I5ksg_a/Q]AJ 7h!ih=+¸b*5E,FR5c[ʐu%7oBRlԍ}ׂ]p/.$=J. 5.PGqӆ΅tKÔf/ N.sBM6U4Wvnk^>K0z~3- ߶k4yMjUˉQOܟ*d ^D2B/j Iۦ'i6FX_LOT!lz"$Efv Qlp8Q=W~]UsdV܊)JYp'(6mM `va0O_zj}ǵzI;L f7:n~2GT\ԌM=m:$:s"T:Bg;ѩ-Ltptu]("7#m` Y쮕Q1rBTiw=\ϱ.ʍ":aa!l!l!"QQ66FZnߺEB;X7RXlmUjbl[F\e2qbh/(6f,۳swFaO雼cbEh4}.˃C*]Q4uHKal8$upah!' E I]#jESibhT@ IQ4אֹ.l/$LHC҄v .lMJy.*}+6moa/wbh&mƠ(BCUuUga ~ߧіeQ4هnljEI] 6BRw+Q4\St&V6a gjE$6"Q66FB(B(BEEa!l!l!""QP>~@Ʌ`!}Eմ=XRWc2eJ \H+2`: Kx흻o)7/;,lA`wh oԤp#E:UG T΀Jo@ 䐗|z\<μ5?mG9nȍ6Oч6W <N|?W`O_++_+|`?\]]5Z/^xpww޾H[# ,W3|O μzU;==m.//9rٙ]rNsq:99i...޸CAsTs}raKxޘ/H>T"S׶>=3˗G$*T$®촳Oy;mױ)3)ҩ˿+<W[6.𸻞ׅ y OΗRq,}e?MH!}Wu?IʮKH(eN;#J]Zkkt\_۵]+t\ER\\ 9Xr_=fH= T+pSx|22ݮߕjt0샑ۙ.[.o.[׶p9)ۙpHQpr,L)75vʵM5 RWr-'ps>:bzKwIR%qtzC݇jgtυw5|ޮܹ*3K?OOW}:Ѝ;S}pm7u_ 7F*ӗzǐgqc\-㉹wL;s2A;b_ڹp>_IYڻkdQW)ըg*շ+^8q\'.6)\|1ve6 Wڣ-,G>pv&f7SGIOۙ0OK;%C l^VR:^bUoli[SsԞrwTwy0 ؤ\;'1fcڙg]&{vkؼwnIsC~Kysq5:V'.YN)>EW_wi' _R_V270Wճ23C1ͳ;;ʎɳ¨;{1)?<%eT;O> ~G׻cu>(56yu}FO1z.TyUS:zyi֡.=5`:8}%s4[!|pq#1Icy=t֘vfc y1m=G-}i3q3֥Jyn=vFq\ op8?RC<>~T~8׺tegJ7|fSc~kL> jM6#whqv,ݿ̰}j+5N;t>_`R[}`{k$[/Ol[I:HXHSAllJ*`_*>'){.?ý~`߆KkGRt9!,Wમ,W3|v+)x|++_' l |$+ ~"|`)iSͻ~'cc9"sqw4~44_>L>?ef7WGM{ž! !g!=CaB{B3ž!g!=C!BaB{ž! !g!=CaB{B3ž!КՕ^/^{ڳ[oՌI}Zx3c1 !=Þ!g3,m^ze~>;;?2UsҪŅكmG#\- g;.htL8??7_|iůn'eṲ"&L f+9djjtsȦ>QڥUS}m V&7:>&Y&.gZj6FT?ܞk`2,&}|dGjuʴN6}e -x7m2I[=3jlR5i>@a-VPv,=1GM;x iJ o5>R6tD{`^bQmcFX7g-~HU3㏶>Uj}|kd٢dL!.R|" ?`c(?ڎs {'3_vaT٤qNMSEOسE)ӿ-__2W{fn1v2]]F=97qeϦ+lnW3a Gfm@9 vvs+znmМ՞zӞݫ~S<σ6z{I{jG -)k{6dѣ݅a,+K`[iLW*gg:y3SS9ә4;g vq3/۞ADYςI?[[Meo:=ӓ%'s*ف=[HOfgE2cف=ڱmcV:Q<\g9Mo+YPz`fnmף mCBR8뉴h91VvD[}٢o]SDSLe:'N{ʧ۞k]=SAemHf 3Ɯe[튞EM~8g#e'V#)K"9ԝB[=o9KEOU tCLC*Ⱦ_ gV-8gh}vo2cφU{E/;?01ffWں`7-˶덭~΄ɔbse[}޻aSwv97 vN fO|<{Ԟ|SHg+?u${IdG;7>S^$U%z7{Uh2Զ>*;RpܿC1ݦ _8}7HW5䘉[&hؐ_=o]ǬR!Q{78$#ne;_QWIjX-T{;zLϛc;.ΧMmCV[?ћ-՞7Atvqgu[kb?z/}~\wŞK#)IQ86=n a_ш= }b>E=Dh|hjk{fgrԽ,֞y7L.ҢM_ZޔfwG9.y٬iemPmK,ƞ#] 7]@4hjkUߧg.hIvA;_z3C,B O;Oާ0?]=MwPoXԒڛvOj;=kۏځnD8ٲj; aKK%FkӋ sGtO77mNb&$b\LS!"B{B3ž! !=C!BaB{ž! !g!=C!B{B3ž! !=C!BaB{B3 !g!=C!B{B3ž! !g!BaB{B%Q#?Z"n8 ip}b+jm0PNG  IHDR%} pHYsK`PLTE     , !"#$%&'()*+4-./0123D56789:;<=>?@ABCXEFGHIJKLMNOPQRSTUVWfYZ[\]^_`abcdeghijklmnopqrstuvwyz{|}~ܖ 0ܖDTimes New RomanTTxEܖ 0ܖ DWingdingsRomanTTxEܖ 0ܖ@ .  @n?" dd@  @@`` @'        !% &?D2$f"6 bD@2$Onp_f3IG2$b%T;C0L NbM.2$~0CoD \$ =2$oQQ|q* E2$YQD|0e O2$iȊZVi-]2$exɺYb$j-=vTb$,(-+IE 2$Uefl y}BOb$~hh~v2C b$!u4>m(L2$n'rKy?0b$$YUt+C}|2$q!.2$~YY˱ۙ-̀ }B&\2$"p/ }+_2$Eմ=XRWc2eJ$2 0AA@83ʚ;ʚ;g4YdYd7 0ppp@ <4dddd w 0TTE 80___PPT10 pp?  %0 Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the driftsTotal drift velocity Assume one pushes all the ions in the plasma with the same force F in the x-direction. This leads to an ion drift  Meaning of the drifts Force leads to drift of the ions in the negative y-direction Electrons stationary -> charge separation But the plasma is to remain charge neutral /P     $  0 d$(  dr d S =K   r d S hV0  H d 0޽h ? )))̙̙380___PPT10.CV   0 |(  |r | S +r =K  r  x | c $LSV0 r  R | s *A  H | 0޽h ? )))̙̙380___PPT10.V`ƚR   0 PR(  r  S `=K  r  r  S BV   r  S | 0  R  s *A`tM R  s *AV G H  0޽h ? )))̙̙380___PPT10.V([e   0 (  X  C    r   S  0  r   H  0޽h ? 3380___PPT10.V`7   0 `(  X  C      S  0    H  0޽h ? 3380___PPT10.V([er$@; 1(   Equation Equation.30,Microsoft Equation 3.0/ 0DArialngsRomanTTxEܖ 0ܖDTimes New RomanTTxEܖ 0ܖ DWingdingsRomanTTxEܖ 0ܖ@ .  @n?" dd@  @@`` @'        !% &?D2$f"6 bD@2$Onp_f3IG2$b%T;C0L NbM.2$~0CoD \$ =2$oQQ|q* E2$YQD|0e O2$iȊZVi-]2$exɺYb$j-=vTb$,(-+IE 2$Uefl y}BOb$~hh~v2C b$!u4>m(L2$n'rKy?0b$$YUt+C}|2$q!.2$~YY˱ۙ-̀ }B&\2$"p/ }+_2$Eմ=XRWc2eJ$2 0AA@83ʚ;ʚ;g4YdYd7 0ppp@ <4dddd w 0TTE 80___PPT10 pp?  %0 Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the driftsTotal drift velocity Assume one pushes all the ions in the plasma with the same force F in the x-direction. This leads to an ion drift  Meaning of the drifts Force leads to drift of the ions in the negative y-direction Electrons stationary -> charge separation But the plasma is to remain charge neutral /P     R   0 PR(  r  S `=K  r  r  S BV   r  S | 0  R  s *A2G R  s *AV G H  0޽h ? )))̙̙380___PPT10.V([er Ը .1(   Equation Equation.30,Microsoft Equation 3.0/ 0DArialngsRomanTTxEܖ 0ܖDTimes New RomanTTxEܖ  5=3(.&"""333BM@O\LUUUDDD\kYMfj{ffffwwwwsײǦIDmIDATx݉v:qI.7iY,oyYl hBuJbҘ υ|Y~hJgɡ%(A JP%(A JP%(A JPZRi@JAzÍJ186HCV-+9%SRZ(i6\eK\JkPR*(َy~X{Ҋ\%FT(Ũ]rJ"VJm JP%(A J^ H (؆&ɽ%$dÈzvx&((}rtC_((YJn5JF&>J[MO\^{ƍ@ZQ؜qiJOcf)LU3tLM'7w*,HłS(q*UZJN?.%1V뫫NI+,>~;aQUIǔ5.qJ)i) S%'Tw)uرԥNޛW,x]jJ{Af=SY/hn)qniDJglf=Li)qlJBJͬg438@ά%YIά%RfփͬgZ/_]Nf=z(Y%^rJmf=zIIRQPJCjD^( %zi()0co KɺV"(Ai%A Jӕhkq@_T"icJ)U -4bZh6kCVٕf72#NJbDA(Mh֗yP'j%+#NP8ziqdk i*gWĈދ(9]}JӣHP%/Dr](A JP%(A JPU2/N^I.JW&TKBLꓘRkG]ytGrP{+czC떸8*Sٷ*'N>̹ -}*U;LNm3/ [9qx%>_2pPf?VʯGuͧ5RĹ|Qte+ܮtnNNJA #LjzRu;1qJםNN^J7 ~B9(A<(r S亞&%W:qqBc=6'G'0* XRJu]uŚ]{iҁ>)ElroJbȵŝ8XCJY҈~78J)q+=ɣ_SLhl$P]qw%8J#QzWr)cPXǡ_*֦t5*7r2gZYFf8%(Aɬ乬RiB!P%(A JP%W2:CIŴا+r_Jg͔+*sJdۛ83y2.jgO~(LllJRe/(gzY %YܺT6Jm'}[R= SqʖQ"9Xu2nWK)Iפ=}M~=?S[gWTWG@EE~vrX%e)GRGT{2O[c\* JU(RJfJ&҈4WU-#*fu) EVoW!Ԏ^OilJQCe3q7%rhK)uJRQ'wV\&ҶYvƟ;YST]|V榴8.^4YoK/Δ\տ)M9RR9t63^.Pt;hduJY~.m:sJw71eOQB-f/MOjIENDB`n ,i6ͼ(P_PNG  IHDR'+F pHYs#™PLTE  "5=3(.&"""333).36=DBM@O\LUUUDDDDMUR\f\kY_kwMfj{ffffwwwm{wsz̹ײǦݾqIDATx݋v8IqMƛI/6nK 0ܖ DWingdingsRomanTTxEܖ 0ܖ@ .  @n?" dd@  @@`` P)        !%&()o2$f"6 bD@2$Onp_f3IG2$b%T;C0L NbM.2$~0CoD \$ =2$oQQ|q* E2$YQD|0e O2$iȊZVi-]2$exɺYb$j-=vTb$,(-+IE 2$Uefl y}BOb$~hh~v2C b$!u4>m(L2$n'rKy?0b$$YUt+C}|2$q!.2$~YY˱ۙ-̀ }B&\2$"p/ }+_$$$b$ip}b+jm0@ C 0AA@83ʚ;ʚ;g4YdYd7 0ppp@ <4dddd w 0TTE 80___PPT10 pp?  %H Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the driftsTotal drift velocity Assume one pushes all the ions in the plasma with the same force F in the x-direction. 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The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. 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The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the driftsTotal drift velocity Assume one pushes all the ions in the plasma with the same force F in the x-direction. This leads to an ion drift  Meaning of the drifts UForce leads to drift of the ions in the negative y-direction Electrons stationary  !Drift leads to charge separation The drift of the ions leads to charge separation. 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The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the driftsTotal drift velocity Assume one pushes all the ions in the plasma with the same force F in the x-direction. This leads to an ion drift  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma /      ! #%'   0 |(  |r | S +r =K  r  x | c $LSV0 r  R | s *A &G H | 0޽h ? )))̙̙380___PPT10.V`ƚt   0 Pt(  r  S `=K  r  r  S BV   R  s *AG   jA"drift_meaning_01m 7   0r v   |LDrawing of the slab of plasma with a force F on the ions in the x-direction MMX  0A?t~ H  0޽h ? )))̙̙380___PPT10.V([e  0 (  r  S =K  r  x  c $r V0 r  X  0A?tz X  0A? 9 X  0A?S tcd X  0A?  H  0޽h ? )))̙̙380___PPT10.V0o  0   o(  r  S E=K  r  r  S ;EV  E X  0A?tM X  0A?p}   jA"drift_meaning_04   r   0  qMotion due to the ExB velocity   H  0޽h ? )))̙̙380___PPT10.Vp簾  0 4,P (  r  S d}=K   r  S \V0  X  0A?t=  0% f  U!Free motion along the field line " 2"  0xr ) *j Y%Fast gyration around the field lines & 2&  0`8= S H nExB drift velocity. Provides for a motion of the plasma as a whole (no difference between electrons and ions) o 2ol.  0H= S ZH Polarization drift. Allows for the calculation of the electric field evolution under the quasi-neutrality assumption. Provides for momentum conservation.  2RB  s *D% RB  s *Dz ] S RB  s *Dj) RB  s *DCWS H  0޽h ? )))̙̙380___PPT10.VS  0 (  X  C    r   S  r  0  r   H  0޽h ? 3380___PPT10.V0  0 0(  X  C      S tL 0    H  0޽h ? 3380___PPT10.Vp`  0 `(  X  C    E  S 8E 0  E  H  0޽h ? 3380___PPT10.VSr4w58"`<tL1@NDPR'1(   Equation Equation.30,Microsoft Equation 3.0o the inhomogeneous field All together .  Fonts UsedDesign TemplateEmbedded OLE Servers Slide Titles/ 0DArialngsRomanTTxEܖ 0ܖ"DTimes New RomanTTxEܖ 0ܖ DWingdingsRomanTTxEܖ 0ܖ@ .  @n?" dd@  @@`` x6#       %&( )* +./2 35 6T2$f"6 bD@2$Onp_f3IG2$b%T;C0L NbM.2$~0CoD \$ =2$oQQ|q* E2$YQD|0e O2$iȊZVi-]2$exɺYb$j-=vTb$,(-+IE 2$Uefl y}BOb$~hh~v2C b$!u4>m(L2$n'rKy?0b$$YUt+C}|2$q!.$2$"p/ }+_b$,i6ͼ(P_ Lb$4oû'5V:ݔ \b$ 'zvAu8Pk1Vb$ip}b+jm0@ C2$ .`,k$6R-{h2${UF>j`#2$#FB/@"Xr#xn2$Qt"%WbQ_d꩷d2$ +qQC0cG%b$,-V>,.-wgǹ X2$WzD%\>$ћxM X$b$ͥƚefRR{   0AA@83ʚ;ʚ;g4YdYd7 0ppp@ <4dddd w 0TTE 80___PPT10 pp?  %&Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma /     ! #%'rSRo'111\o+0e^M|g,B;NDr3[p}y`\3g^]]UP~I^9.iwy7;du:O X-חw%LVJAE:{qY|  cvLsTrEJΌ1у4dMɕr Cᙖ\oy&Ǒ/R@/}<k (|ckj65[:>܃>r}RȍiSCNaꮐ~ m5p;=Rt95Յ>-CXwB~܍tۮR`R cszI9?XkmKr+ֿ\}%8 Ju苽gbw})훣uT- a.l95䞔ź^}d c96ru}GŹk˜U@'Z-&^XbZʸ`(G_w/fu Su^]?vJߔ99>̍HwJ.{G37ж`JӼ1{DPQ]O%^Y{ڿX2N]"s)Ƹٵ'?-uXC3lقzmɐ]t9>O̭S(;5 FOmp|j?R;l1|P u5 '-<1AEz֭} kM]BⳫ7':st袹sߕ㳉NWzb/WW@َ!Rp\[WP^ @5tu},칡+YgJahwΙYYOq)7w_\ۥ@ret(Ů m)у)%mpW~5Qy﮶wR98yoSs C]:E--S !؛\^a,.gtg=dM2anL/|_]&aKi0X_=gwgݻdq\O؃uhߴdʔdK˥5(s&ԁscWK<t C%NB{fm{bsPh}OR8[XC"-ЁvhƠJ>Z^g1u 3] 2ЁvZ}U봔s/ܧ#D/ڐ}65:ER }_MY k鱤\ϧ /(?LN9Mmٮ-'c݇`~kh)GW,o>cс/;KсaZ7SiɆnArсRrst^t~@$w9{ٚ/.t`u>aV⇩sz^85tɵF_#9 5͍Ը /HM8e%: Rs){]J 3>?sq.ÿbZսձYmᾍc.Ұ‹!ݓk:Ě>9i-Ĕ檮»t{t/Cj zOɥN`Zõi$\ļ\yM>2=:)VaSٳg-)C8Hśwl~5!Jb.Ғ ڞjybR׼)hCKHo{5 eki@ӌ^ V74]$xMW=/;dě V̯gr}Uٴ/04Hƴd/-p5i ˋH7C\b$%ܞ7,_ADZv"?&ܴj8 lcq='Kt--4%i~!"믓H Ky,&d:uud 3Ok&W~z^i+zyOnN/벆٠z:C1^MbZ, 7˿5c wXŌ/580#2$%>w v尟[=D' 48 smդsXN[M^shݜε)R㢳px{+=jv(.1nRtcD7k-ݍ\b8Eqk}{gnYUp gdZӕ{Af疬&{]Pe|He;%xRR7t\HG=xͻR1žHn)47G`(E}QEC7/^= >[t Oe]┵=%nx giŔ>bokᕴ~IuXk|fɵR\bӭW|뗽Mvk0G]h|ڹ U& ô@6G-I3(n\s3v4mvC`ch {]AsmpxcĿQ}+eSnމO L%CwV$ /߲^м#sa#=8fh!eE@LJ$M)M5-oknxզcx&VvpbY1ߟĚ,uJrpc4YR#%f~5koHJ-dEKlEon$V8U4nL,"qid /ϿV8|Q4nLY;y&qid)Ͽ8I4nբ%NjùnÛp=_Oϻcskܖ)Ipb3@ |ș\XǥO7Y[mJO?Nٺ3Ț%'UK&pjƜe^o~oCYCHKKad֍,3ɏYYPYbk,x{JrDMV`I쒦A[ib ,ⰟKCz Y럲bͪHI{F8_܊1mÃp uu4aHEa8)lzk % ȡ9|:=/Ŷfn aKJۑjnD ;p[/ {i^Cܠ쵅 {p0|:?c|مZu~F;]q 0- ^ٯPL߂qt _Y3s ArPRURt*Np&vA10lgXD#Ǐ>}` v˶ ˞bןi ׼Dܡa>fkj3j>E|&X["XP}e8|PeS D)s4ޫeV̛!2a#% >)k|xLuKyrFs>C ڵ20?8ft#yi2=z<}4_DM\uT=3rnJWlʳF\_7r8>į<H.qMt} 6R0ޗ/pWǥ}zRxtb`84 PjRsJIra878dauP;Oc?0RF}':8#{v}eaĢk6Hy8=a縯*ÊO^>u#z*眷u[Jo_׍`o"+x|i^ apBFX{1j>{51a]TϟE[;Xl`V*r!F[hn"qxTX% JvFH-,?OGc_ _^_:DC46|3 R1-02Wl75ʎKz\ujL__;eܤp>l CC<[^bn F{vCSX׋ǜ YVshU0mXVӲ2DRh0dv9罶*]8ak~D5Fzk#HuNaOw7\(8<Éia:s09lL`1Fb2t> vj'oado͗8 swD=ii] d\Gjh|vvfye#rD? XnFi lBBPNG  IHDRF.%[sRGB pHYs0PLTE999SSSzzzhhhxHIDAThؿo@GSB"AHzAxy@Dxjb1!""ĞN\;ɀ$G{+(RT"?rTgG+K`4;v(ax񴙑 ;އ+j]nwG HE~$jID9!G"Hl6Hk;?RK%q>TpNb ]"8֊Ga=J%-j_~^ Im "Mj_:F *|o*qt`VN (r؍}{ \"|U.y37GA[鲰Fv-Jx$~FLlҲ 8!_wH[t&Ļj4r.GZ)"["ReAdtChʥxTX(Y twF]9Me'C#[$]M3hovG~^N A9>Y4@_|ҩ\Kfʹ1||FW䛹e QR|(RT"m5sIENDB`nx 䭱PNG  IHDRsC,qsRGB pHYsߠ0PLTE999SSSzzzhhh4Kh IDATx]l#GDZ'*wTMRJu-1*C *=_'@UTpR_z.HF|N )oE mSyuA˩ı7g>ľ<<|YQ1^J|#DX {džz}lX!,<=7c8 7E(ClH%?ba75?bCrʗia'7}MqEw4l=$oոpb7 v+b?@_,ܼ7ئTEls[e 5?MX V׸r`U~^.b9GS2GXYW߈AvhF^6- c!o:ץةu>9ѱ5d&:&ɲҔžٳܳ٩ηi*/1ܖ-*ž6R$ձʱ7)"|øc'0_ޱ5"$9,Lᰪϱ杘Ѱѻa†n0,m,ꕵe=(609%(ۛ+̶Ů0ᾱ,<mǮ1E7-f#ՎriB;Gb4<~:ۘӈG?"]x$V.av %Y#"˯;M*0%bz8/kd~?#ؗžer5RZ2/sZ/ڏjӕ*tYoB [=K^?/ˑa_a\% 5 x3s꽡m1t͓X6z3qyx)ءƈxDvzz^!d` [HHמG<ka.RbG#rtz-{km &k{~gLZoa8ۥdCnխt[g@o!u+,BG~@D-ZK/f"qw7L>w]kދ o+xG?TK[_xqHR7^HNěEOӪ]DWwg?vOWCRK&Z]=ڟ=Y>6<,ٹ?^:M%uq۰~ki=Y`7M~gj )xn[km]=#|{ ELɶ, SK.*-ݮPw$Kh/ci-Z[}`ߚGɳDǨv'vjp/9i(\[uїl$EӔE[ee6XaDiAHсM#ƾif`UZ]=os=N3>aA_\͝1K#UvHc^k.Ž= 43?q{;vE($G7j w ogO}y={\}?ƦjE ^\P7PDت0?Utb.dvrrTHQ}}*g& :anlKby1㵭.tBl]HlV]3nn&S"~- {=`\؛n&ivyLj.Yz{dyOjDD"39M$ˈz[2=^ܳ۲*atp) rc#*aoh{&k{iKjchMc%G8= E'TqMVnAJV^KqfꮋrOb=$~+j_ݠ[/)I7]w=ͤz= P]M3y5ԛkeᶡs6۰j2Q©v]AݮtNW? vbǸjtaKI`嶫h$RIͤ2#Tm3?bOj(ƾ `CP/#R=˵fHyAlH`^)ˍFɫTs#~~˯L>EsJEF5՛~ 5?_nJڳw7rQq> B!B!B!£fV |B!B!B!BZS+!B!5xzzh3۷o;{snSWWW{nێm]__^=/..M     @ !"#$%&'()*+,-./0123456789:;<=>?zABCDEFGHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghijklmnopqrstuvwx{|}~(   Equation Equation.30,Microsoft Equation 3.0/ 0DArialngsRomanTTxEܖ 0ܖ"DTimes New RomanTTxEܖ 0ܖ DWingdingsRomanTTxEܖ 0ܖ@ .  @n?" dd@  @@`` 6#        %&( )* +./2 35 6/2$f"6 bD@2$Onp_f3IG2$b%T;C0L NbM.2$~0CoD \$ =2$oQQ|q* E2$YQD|0e O2$iȊZVi-]2$exɺYb$j-=vTb$,(-+IE 2$Uefl y}BOb$~hh~v2C b$!u4>m(L2$n'rKy?0b$$YUt+C}|2$q!.2$;+׵z" Йm dV).2$"p/ }+_b$,i6ͼ(P_ Lb$4oû'5V:ݔ \b$ 'zvAu8Pk1Vb$ip}b+jm0@ C2$ .`,k$6R-{h2${UF>j`#2$#FB/@"Xr#xn2$Qt"%WbQ_d꩷d2$ +qQC0cG%b$,-V>,.-wgǹ X2$WzD%\>$ћxM X$b$ͥƚefRR{  b$i lBBNF$b$䭱 I 0AA@83ʚ;ʚ;g4YdYd7 0ppp@ <4dddd w 0TTE 80___PPT10 pp?  %&Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma /     ! #%'  0 -%d(  dr d S =K   r d S hV0  X d 0A?tz X d 0A ? 7 X d 0A"? t  d 0=r  3 , Polarization drift. Note this drift is proportional to the mass and therefore much larger for the ions compared with the electrons  2RB d s *Df # 3 H d 0޽h ? )))̙̙380___PPT10.CVro2o'1(   Equation Equation.30,Microsoft Equation 3.0/ 0DArialngsRomanTTxEܖ 0ܖ"DTimes New RomanTTxEܖ 0ܖ DWingdingsRoman @!={S+`je#kװ):9)󶩹<qcMG=_^D?Oh2g I_]Yzo52\oKk/R׍xA|9TrhOֶ,9inD8~5Qb5Y)q{Zn%lA+b>Ƚ6֖8>)MX%6FyM[kfrr--ڇ>HQf}Ɛ{CNՔ_5e-Ho6sH{g=5ƅy^VŤClZloИeP{{E--]VLc㭵3ERo ̡5v~ս_j(7j̍Nb)R}koiY}8)sFY:s{F8?ChN Ё؉r5^rOYj^/6֠)56akvovBUNY^svP |xrCkSYw֡ @}}>@-ԏ72M>Vnwvea@1h[>@}~@1tasZ}vFXۭj0\Bcׇ?uvvvC0 B[iRZwww:ЍV?1:}v-g-ĄXjvʙLG"֦[t.kDav?-KD\{q}Z)hNqѦ38FۻCdZ%w;fA_PĢ${SXyf  !IDۼ9őF9%>%B78K8'?ȅ}bm+zE)24gЇv 0, nvDfȧMGtM_A. =85+3̐!lDy9]\[1ga=-6sRIILgK̋a"O+DDG/ı}6e{C`k[y$Rđ_ʿhNJEG}R\Q!?tU|jΊïy+և-pka9k]ՇoRנ tB퍺mFCW\uX} &ïv-ZI !}E*+-g#MTa4} x˵6? 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XH@۬!sV;gIENDB``!W--]5*/+*epUS x?uGp$v i@8n $P.lnM Tm *-)P jܐ9;×3{gW}߹39g?XG_=Z>Pӟ._'˟Y_|,>b?/7?e~>ǿ`YUz?[h_o.?S'󽯖wgCIC!B!TTxEܖ 0ܖ@ .  @n?" dd@  @@`` H@9%        %&( )* +./2 35 67 8?2$f"6 bD@2$Onp_f3IG2$b%T;C0L NbM.2$~0CoD \$ =2$oQQ|q* E2$YQD|0e O2$iȊZVi-]2$exɺYb$j-=vTb$,(-+IE 2$Uefl y}BOb$~hh~v2C b$!u4>m(L2$n'rKy?0b$$YUt+C}|2$q!.2$;+׵z" Йm dV).2$"p/ }+_b$,i6ͼ(P_ Lb$4oû'5V:ݔ \b$ 'zvAu8Pk1Vb$ip}b+jm0@ C2$ .`,k$6R-{h2${UF>j`#2$#FB/@"Xr#xn2$Qt"%WbQ_d꩷d2$ +qQC0cG%b$,-V>,.-wgǹ X2$WzD%\>$ћxM X$b$ͥƚefRR{  b$i lBBNF2$of#ltN}l MTb$䭱 Ib${B:m.r ` 0AA@83ʚ;ʚ;g4YdYd7 0ppp@ <4dddd w 0TTE 80___PPT10 pp?  %8Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma (In homogenous magnetic fields |When the magnetic field strength is a function of position the Lorentz force varies over the orbit Taking two points A and B,?1/     ! #%')8  0 p 8(  r  S  = =K  =  r  S V  =   8  Y    ,`  0A!? ]  04 Y  LDrawing of the Grad-B force   " XA#grad-B1`   r H  0޽h ? )))̙̙380___PPT10.W`-d  0 (  X  C    =   S =  0  =   H  0޽h ? 3380___PPT10.W`-dr3( ް2)1(   Equation Equation.30,Microsoft Equation 3.0/ 0DArialngsRomanTTxEܖ 0ܖ"DTimes New RomanTTxEܖ 0ܖ DWingdingsRomanTTxEܖ 0ܖ@ .  @n?" dd@  @@`` H@9%        %&( )* +./2 35 67 8?2$f"6 bD@2$Onp_f3IG2$b%T;C0L NbM.2$~0CoD \$ =2$oQQ|q* E2$YQD|0e O2$iȊZVi-]2$exɺYb$j-=vTb$,(-+IE 2$Uefl y}BOb$~hh~v2C b$!u4>m(L2$n'rKy?0b$$YUt+C}|2$q!.2$;+׵z" Йm dV).2$"p/ }+_b$,i6ͼ(P_ Lb$4oû'5V:ݔ \b$ 'zvAu8Pk1Vb$ip}b+jm0@ C2$ .`,k$6R-{h2${UF>j`#2$#FB/@"Xr#xn2$Qt"%WbQ_d꩷d2$ +qQC0cG%b$,-V>,.-wgǹ X2$WzD%\>$ћxM X$b$ͥƚefRR{  b$i lBBNF2$of#ltN}l MTb$䭱 Ib${B:m.r ` 0AA@83ʚ;ʚ;g4YdYd7 0ppp@ <4dddd w 0TTE 80___PPT10 pp?  %&Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma (Inhomogeneous magnetic fields |When the magnetic field strength is a function of position the Lorentz force varies over the orbit Taking two points A and B?7/     ! #%')8  0 p 8(  r  S  = =K  =  r  S V  =   8  Y    ,`  0A!? ]  04 Y  LDrawing of the Grad-B force   " XA#grad-B1`   r H  0޽h ? )))̙̙380___PPT10.W`-drr( NK)1 (   Equation Equation.30,Microsoft Equation 3.0/ 0DArialngsRomanTTؠKܖ 0ܖDTimes New RB!B!CsV <6B!B!B!B8I#j B!B!B!™jV!!!!V B!B!B!BTP+!B!B!B!NU+5B!B!B!BTϩB!B!B!Bxz^wgϞ ?L-K ۷o?oλ;']bw rsOW^|p7_~Mo|K ij^h]3ƏS烥|q=k ~8;_||>žqZw 0]{gqv_QGff?}9B>< vjџǹ#gډ=a[tK kX Svcze稤Ou1}>w|[Z;i׶QWS\Y:aۻU+ZVe]>3z윁OlԱkB}]uרei7v=AB7%{r#Hŧdɑz7ڧ6-a>no4r[:N3:>{~Tڟg4ݻ l(KKgG>{OWg̱yky=x>XƮ[-TXF}7?{Em wͽ7_['?SjLm)5i\ KSSP<Կ^,5uX+r(mϏcsƹE5Aro,=mh/{ⷽY7VZ<:vX?omlmVMc_䄎_[ymL~6%H*.s픻썛_?#w}a1y? 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The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma (Inhomogeneous magnetic fields |When the magnetic field strength is a function of position the Lorentz force varies over the orbit Taking two points A and B?7*Inhomogeneous magnetic field Force due to magnetic field gradient is directed such that the particle tries to escape the magnetic field Leads to the grad-B drift /     ! #%')+  0 zp (  r  S n=K   r  S n    8  Y   ` `  0A!? ]  0tp Y  LDrawing of the Grad-B force   " XA#grad-B1`   X  0A?%  X  0A&?  X  0A?  RB @ s *D  X  0A*? 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The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. 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0AA@83ʚ;ʚ;g4YdYdK 0ppp@ <4dddd w 0TK 80___PPT10 pp?  %Physics of fusion powerLecture 5: particle motion  Gyro motion sThe Lorentz force leads to a gyration of the particles around the magnetic field We will write the motion as i Typical valuesFor 10 keV and B = 5T. The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma ,(Inhomogeneous magnetic fields |When the magnetic field strength is a function of position the Lorentz force varies over the orbit Taking two points A and B?7*Inhomogeneous magnetic field Force due to magnetic field gradient is directed such that the particle tries to escape the magnetic field Leads to the grad-B drift -Curvature drift fA particle moving along a curved field line experiences a centrifugal force For a low beta plasma /0     ! #%')+./ 0 (  f  0"`H  0޽h ? )))̙̙380___PPT10.W.ah  0  h(  r  S t2=K  2 x  c $ 2V  2   \A%curvature` I  2  0<9  rBCentrifugal force due to the motion along a curved magnetic field CCX  0A)?CtL g X  0A+? ^ H  0޽h ? )))̙̙380___PPT10.W   0 (  X  C    2  S L  0  2  H  0޽h ? 3380___PPT10.W.a  0 (  X  C    2  S ,2 0  2  H  0޽h ? 3380___PPT10.W r-,@OwQUW_-Z/1.$(   Equation Equation.30,Microsoft Equation 3.0/ 00DArialgsRomanTTؠKܖ 0ܖ"DTimes New RomanTTؠKܖ 0ܖ -p?ҳ1{*ǹ?  { So(qgn_׾|^y`k? -׮s*ˈ`ӇT^z}=m߮s ֈ'sj羂Uz]s Bj(=VTMy\\\p_*GHЃq_[p s\|R#?Wt;Ia|$G}$5?V? y)a |$> ZlX;^f{ ߂m?B{#0:j8j/cJC9\ŸO׿]si s{?YCmIm#I#k_FUo[|XyCuǯS۾[ zomM!cm,oٹiǨ>j z]~}> Onj8=ϾJ*'iǥ+5ҫm"䰨ܱӰsܟ°㠴܁kݟuo>%=V(ǩCn8k?<^ݜgJԒ9Ďp]s\U_YorB?gs]v^vT+:xȚ֏d'ׂ,Yx:(ClZ߿xeiN\srٴA'zgsXvuΧCDK8Fo뺑#Jl|^vkmyQpjΚ1uZީ~IRboSYkcSسC9K=P v"b%NͽOح9r?>-bl!謁mS6T zx>`M Zk-!~s sn.3ҷX=9r+7An;Fkĩc]cCk>xjnuqj:z؜uе+wjũ9zjP[Bk!UCdu.S}qܮj!ߵ?"8u uum?kWSC7+|TJg6o}-麺5pjµ;z\ 6 5Dp PC"V}^oO'+0 v ;-ǖGg;y|$4L@^Im>#糄ΝPshZsjzPߧOl}\`۟Raִ':vXu呧`\]=$HC{Ot >{~Gw%&v+lZ1T3 Lơl6 hv|3c_揰HICE(/{)s? %̥GJbm^5<}t΄M`2|? h }ҹtT_hwM7?-8={Gs3o+{o];w}~<*5ǿdrtwC;!C;!C!)%'sx"/؇l.ruuy5CFI!޵24Ey\!cξ~y .wXkCn^j]}b{wrqq2f_(BwQLվͨSoE7OɱDŽw->В9ߡ#,ګÔᛕɵZ!,׻Y6.ИRhby,7/+2<_Y I"gnʷ[E$_LGl7ML ?gWj~~HxwKz]p{Z7r y-37e,4إeNطPzļRtJ.6# eDn?T{hs[pPٕ6ͻWw1׀7jt/7 ]+;k#mZW\̗m3N͸@Uy$b+dy$:n w(xWtJ.l4urlq e|܈ku#b^qk<\×wہtqepK͏vtxɎ?|xY~(2!"Ov+cwn|ڠYJ9b}dԫaF·9^ɖ6'pƞI..6`;ˎDd___m!}\J.]cᛜ_YF iy猊( ]S"\O Yvx;wCx;ʻ,Fn Ҷ<lTPNG  IHDRP5HsRGB pHYsdc0PLTE999SSShhhzzz*4 IDATx]lYį4m,[#U+~ZEl`ҷ/⡰EZH*BeHĭ8ˏOqA8xrծV${ܙ;{x̑'3w9390PT* $ @Pgѱ:lqUگAYko">}@]Gtvյ/N;Џ (9is]u%'m~&̻2=̠y >ۼ#PإJ=L2(#e| N~ ]UP5MiG&Wo*;n Ӟ9lɓ;]vܤ2f֚"&͠j$TIP1T^fydxlnH6r ol#h2Ǧ"(e:[2iTWvL"\1(E~Qpa F2ư @>2 vpZޙ0I޸{zL8x@5B޵3&N2(*GΔp|fWۤʇE.u%4!lx_+u<0k]r@M_iL̑[*,^-ߝ!R.8t%6Q¨ɢMd¡Sv+]^9&g|Pae\y DJf$Kpϰw%49ԧ z1#=5dN GYo@]BC6x>_}>܉lu%6ɜQs &aJ\3·$'S6Keي,tlhյIQH4P-I Z>),:T=w E:[؂Er_҉^Aumh(kjzS^BAάPb{]J6 FE?28ǚ@\Wdi7PwNo^Auk=Go5P*LE:8:ݠA7,=ĦYo12KۋD"cQM}  z&wI.Q8LxO;d`k:wHO;cħyM^з\Rc)']Sf5?nA6<m#:5 )v0v۳&'\q\ TNnIlH'w&D.~ԙ} ZzhFP%Y-H.Je2A@UPOU`&Uz_{%4!b|ںQ)= ;8tHR&Bw\gvHC=UMƥH_ȣp&2 IKlBHTʬPuEh5vށ*"C><.9%4AUɵܑMQpt_9 ɶ mELi0&璫DU·ޥ*wCi,5F3'[׏|y*5{:ꂲcP *vIrv;)EV&\E0OUEJ]MϦ7opQFPQnO[<‘v0 uz(e7Piԍ3_kaԁyjaPՓ=©YT +s˻<̠ =Yܬz[<&j@^m4LY)eBo*Qa_QJTJ Vʆd_:gΜ><~z 碷"{pA6DP`UES/ts,jP:uS9"PL q%lC+pcPNZ|oz~T* @)8#9/0wb뺁4>+k. 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The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma ,(Inhomogeneous magnetic fields |When the magnetic field strength is a function of position the Lorentz force varies over the orbit Taking two points A and B?7*Inhomogeneous magnetic field Force due to magnetic field gradient is directed such that the particle tries to escape the magnetic field Leads to the grad-B drift -Curvature drift fA particle moving along a curved field line experiences a centrifugal force For a low beta plasma 0&Drifts due to the inhomogeneous field The drifts due to the inhomogeneous field (curvature and grad-B) The drift due to the magnetic field in homogeneity is in general much smaller than the thermal velocity /L     ! #%')+./1  0  (  r  S )2=K  2 r  S t'2V0 2  8 o    gk HN v   =S t   6V"`'-n   0"`v`  0A-?o    0@     t@Scales as 1/L where L is the scale length of the magnetic field A 2A:@ v  F t  6V"`'-n  0"`v   0 ~  } Scales as rv 2 2  X  0A,?9 EH  0޽h ? )))̙̙380___PPT10.Wpґ  0 0(  X  C    2  S ,2 0  2  H  0޽h ? 3380___PPT10.WpCrWZ0 ~,3Z@11\%(   Equation Equation.30,Microsoft Equation 3.0/ 00DArialgsRomanTTؠKܖ 0ܖ"DTimes New Romar~c\)WG\*W\*W{\*W;\*W\*Wk\*W+\*W?+5/^x(C[ T (Wݻwo޼̫WQgo߾^`~X&W׍M sm+ W+\/h0(k^GN[bbmW8+n/_j0j~S+ŋWľ~ؿuQ5g;57}ky3QJ+`\ 8'?Üii揄O>{ψ>Xu1/b_V?rl*9եUg_=oe 0_v p=_LνGgsq.b<CE1AM-q_yXY9t+۹|}qŝ/ޕGK ͯU_|yb١~qy6i{`Ľm=r1m՜1RbS~H_ͯs@rmS?#|QYkrYcm||3c5Ti x\S_iJUW?yY0_v >uyԮ3ݳٳ}6־2?4<Ґ-Vxf_ͯ\+`goH|.*hk>4ķnjj~՞33SG5te{J5cؘUԕ^]6 }_9ǧ]ae-~ߪ_ͯs@\BUw)W`3ѮTӟmeqںZ 3mŵ[Qmj۟9,A_vuQWtmWym6NC-9OKǷڿsZ`W{LFǽb9G~HgֈY%;V |Cm1)0c~>gljUSn{ދs9{VF{}UWuC jvo_6fV9Ժzn,url=}8j Semx^om[ᱹ\\+^9cS}ɿ_+?r鿓rke|@qfiZ__x\7;6.e]_?e365pps C}hckظe~խ}cCm"qϹY}}mG7%e*wy0^c~5Q$Hs޷4=ϩql#RcjC=9ğ1_M|e uW]sƦĬp!ϝ辛}QDM9?垲}G;&^` jsvyXeۿu\ \`/-ܟrX͕ckXm͙5z:X6h²ߊ608Aja~3e36 'kM93Ŭʽ%ke]$3qS=ޱoks٣cq񪲶:+W+d[iS|ɺZ\)msjsߎ}}??&{[_̯l ?sglJ91s|rL {:u53cq6ǰz,;B엜cny(.ﱋ1\yoşs?o]Z-y~FʸUh=>ݭ5䰭xsиՒ1*EΥ⳾)VuZ۳۵*ı19gO0έ찭͗Ǯ銳q>93މؼ=#;%?DZƕ9#<φ^|f6>OCjͭߪ9 {GzFbL+;|[+pa1Q[iUmQx5y~Fj|bpwr<VFǷr\ck16^*!?n3&.y&nj#]kۼk?#y(,K K0-phm9e]ܓ-%(?je1̭8Бm޽=#| 8<t6*c e.uy.+/94$Ɠk+1u=i_yw BS)UIΐs614R|seGYmi^KO 0_/t嚴ϻuJWsP|3r@h9aqL?y>=Dt0X^Iƨ߄5nŘ|{9\GY[+QZqLbVQU?sgٳLD$E[ e:3m>uٴɵHq[l1wRf[y읖.{լ{$|Y;m|D羇LD+T>bڹ=Ƭn]8wݎT Iu:=<`/6CMIߏi1+ja_̪]w/NVY/T=ټW|F|Ap'Wߺ`'܊A]rxmm5Oa[g;xJr㿏>S[]wq/ޟ~`Opsqc 9&>&gAߝW ?K{Ƭ^J w_0ϧ\34&$QK\1Yy}[1|Ȓ~<`M♲Gt伙H_mX(D1Y YIzTx"WrݐZ}krxbO!kד"UrUƻw]:V|gO-5G#bjW8<rv@|}]|B| @ 1.+\2ķ/z|YY]7w@| 1QKokJ|+ޡg0Әpں?F]oŷ  fv+1rķ0?o 9Y0[gdzVKkȭ=V^ǗUƜ)mlsGj3s[sH(Ik~az\rkosķpV'\Y췣c܆ٮXظT{O /Xk.튵=1rԺW5>U6k}ѡoo1%'1+Y1r*}dU%cSꆵϙ.u? |+Ʒʵi\u}|n[({CkdUvhlSY'-!Ʒ_>gɎb j_K}v?Nybs?>|rbOZ5zZKygv?TM=2Sښ v1'1+!fuwݘU>]&չGnsےsպuP ؏st>SV3R]n30Խl]kߚs~߂*a>Ɏi yYK7U+?UOs95yOk~)jɘ)g= nZ{~y@V C~'y|z1ܕ؜q#ܞW[pfmv\RMskc휛sf~<{fgʼ)#8b|kI|vܷ]߉M?<>-s##ssչf/sNCK3gj <5m%?s}Ǭ-EмU rT_jn[AY\֕Y;{Ogמ<%r檕՞׎xf*W;`O.cUjjg&v|@W;GkJi!؋͹3KCahxt~){/>ߵֹRf䎶ٵ%ľ6ّdZ:/ZۥFı𳦳ձn׬[*ѤӴ8?ɵZڰԮ(5>_3Ĭ;+8m411Z%մ;=cᚣV۞ֳի=ŷ{ݫG.ʶ.9a6]>߹߷᳢Ҝ#<_SjN{|ubCg׼!uԸS*Su- b.}h3k鼬B}eXpU޿k>朊kg{U[nYSk,Ur_WHRj9SO׵NDo](熚񒟅{y[;K63(lkגgߎb7ϙt^֔{ yXfstKb.F{d>ֹGڲySdGC7:\iktV 8S|k k~ߠ~@95ZuYjXɱ1D~Ι=r%Ie]#Ձ&;-Or'yYSje[[{ոxUp;津k)sm2ovskrϼ/5~Zlu ˚Z+kھ #_n.ںJn]ߙ_932fcgX2]ojl}Z*/}p3c퐪\ oc-.ssϱ-]^^ pi9b:D5m^jwz~,(gn1n~-i?qΗT^V;.5\-C-C-g:!fe~ڒ쿷矶roiV;ֺ~Msșݘ+fU+\#Gkn TM*9sgjkk[Kڶ<[=췷s(e!Yy5wJ;:'O{ eNƎqynytɼ69yrseaǎ3ZǬ-WmSlŬs۾&@Uw \9e.[wϽoҮacڿܳί;^O#oTѧ=ŭ[G;W7{~;V>c$#&;\๭新=幸k;mʰ/L>ڷ槲<8Ź,o}^rݓ>t=6GĿ>߸}Y|8K1+sq?G^ l_XkkmdH^wʼ% |#W-)z){mOhGb,{e'=%ڹ499?r|(c&_okeq19♪Mr~+`-K>Xs0=y\bs1oKl }٪۸ɵkVY9cW2X{W |/W) 7f/& g/cZ1*XК_vM4W;NYYRθ9=cb]{gy*mgq[*W\/FϘ5qsqL+\,KO\{s-"`521p+^[ ƍq+\96<x>|xuT_yD~駇 vm2%WS* Z>5{= Е?zzL_hn?&.l慶J6z26 -xDb@"F$}l\ġleZN6T˗F"}'}]~Ѽ&]s[͹d 2D/Xkn"fG5Çv.=A L`5MF~ KͬZq~^BJA)/Gs ;9oe_Teil$2 h6*4M Ok=L_j]ʸݾx؆4F~ EeZѭxsC))Y_Kή&ϰǶkњҔTnu,cZ<@d4hl?j4lfQTs{XBK ͦ ;"k$<@d4hGTf׳:6hX 5vMqlJWyhіCSCWX޼~"OMs2@DwN҄Uyf]hzh /F~ H‹<Ӷ" ~`-D<@d4h__a6ג39o_hz&OA2@DtK/ 5PXkVtʛ*<@d4hKKo/^z`ci:i{#'2@"X#;LCu &zĘ}Atyh'E~ MUq`47qw'2#%nXZeNj]ˣL@о :<@ "?xz|||2,6q@ k}D j0u kҍҘ$D;@`~g].6QM~vÏ:L߿'I]ـ Z#QBS4)9"XbDvb9f3k>|P[̩虋\ CLJz[' v6]<K jo<@ ρH45\Oty /L\X"m= hn}伕FC;o8~w%69s^]a{U_V֬^Mr"̬H /S2T_wN#h6m6]2 v2j~WSeºRh覣7]9 5iyhρxuC|&Z58"秼n\Ĕ>4F1<@x>>|QY&'5w Z>+6Of~t׶5*S,LTw+M\A}}2?|L2m4`OIbͧ5&h2:Xd~`sDn 6d~U0MjAps] UCoZ]@G<@x)UJaNyY–j뛫Z 1̿ }}>?1*ǰ#<@xTya2VM5/uy}8鷲U?}K5]g h` *CJsm ̢Aw $F޻q/B_W*g%{6MƋ@~Ji;`,Gwf7Y隍&ʸ]h=r osGz#9xٌ6v2/b~_:2YN|:H)wYÛƿk0zW*#@2WD;1?Ç)Paod9 Rs~+r󋌊u:)MX?Ȓti?04Ylf&L.ep3LF34 vc܋ SVuFtMNpQqY6~RQ='RaWp?yt;BFU ; P{g ũv .6*@""KETUO=axՔQL^삹o&T0*""Eҵ_009ɳܴڳnTC+:9ȸ%"ʲ⳦㌮ő^ոڼʸ=rr"[""|Ǧq0f rw+m=lM,ݰ իW '$YNdTqKDD[JcƙzR*w\ ڕݰb9QAdk:fNku)es5pr"[""?il6Kѵ[)S -dU֎M?Pۓ<КhϨĸ%ϸơ4B߾}{kǦ\Kޛw5/lxݭ=ý] K.NV-` Zɨĸ%"~ lv酵ӈ?m CRYp5YZkdhm= M["^;6G2'ʰO55K),+u)k3fEth9QCFDD:ѳcSɚ5ۥmNCoFCDFDDtr? 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The Larmor radius of the Deuterium ions is around 4 mm for the electrons around 0.07 mm Note that the alpha particles have an energy of 3.5 MeV and consequently a Larmor radius of 5.4 cm Typical values of the cyclotron frequency are 80 MHz for Hydrogen and 130 GHz for the electrons Often the frequency is much larger than that of the physics processes of interest. One can average over time One can not however neglect the finite Larmor radius since it lead to specific effects (although it is small)b A Additional Force FConsider now a finite additional force F For the parallel motion this leads to a trivial acceleration Perpendicular motion: The equation above is a linear ordinary differential equation for the velocity. The gyro-motion is the homogeneous solution. The inhomogeneous solution   Drift velocity6Inhomogeneous solution Solution of the equation Physical picture of the driftThe force accelerates the particle leading to a higher velocity The higher velocity however means a larger Larmor radius The circular orbit no longer closes on itself A drift results. lJElectric field qUsing the formula And the force due to the electric field One directly obtains the so-called ExB velocity b #Electric field that depends on timeCIf the electric field depends on time, an additional drift appears  Meaning of the drifts NAssume a Force F on each ion in the x-direction Electrons are stationary   !Drift leads to charge separation The drift of the ions leads to charge separation. A small charge separation will lead to a large electric field, i.e. a build up of an electric field can be expected This would lead to a polarization drift Quasi-neutrality  Electric field evolution The polarization drift balances the drift due to the force The plasma remains quasi-neutral, and the electric field can be calculated from the polarization drift"!The next drift : The ExB velocity \The electric field evolution leads to an ExB velocity Substituting the electric field ,.$The ExB velocity  The ExB velocity Satisfies the equation Chain. Force leads to drift. Polarization drift balances the drift and leads to electric field, ExB velocity is in the direction of the force ,-&Meaning of the drifts In a homogeneous plasma ,(Inhomogeneous magnetic fields |When the magnetic field strength is a function of position the Lorentz force varies over the orbit Taking two points A and B?7*Inhomogeneous magnetic field Force due to magnetic field gradient is directed such that the particle tries to escape the magnetic field Leads to the grad-B drift -Curvature drift fA particle moving along a curved field line experiences a centrifugal force For a low beta plasma 0&Drifts due to the inhomogeneous field The drifts due to the inhomogeneous field (curvature and grad-B) The drift due to the magnetic field in homogeneity is in general much smaller than the thermal velocity 2 All together & .  /h     ! #%')+./13  0 e]@  (   r   S $c=K  c    TA. ?] 2   <tp ]W  BParallel motion    <ڰ`nG ; Gyration     < g  V ExB drift     <8"w  `Pololarization drift   <c t LGrad-B and curvature driftdB   <DjJ?PF)XB   0DjJ"XB   0DjJZ XB   0DjJ CXB   0DjJ}tH   0޽h ? )))̙̙380___PPT10.W(  0 P(  X  C    c  S (c 0  c  H  0޽h ? 3380___PPT10.Wpr2 \31