PLASMA PHYSICS II. MOTION OF IONS AND ELECTRONS IN E AND B FIELDS

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1 PLASMA PHYSICS II. MOTION OF IONS AND ELECTRONS IN E AND FIELDS We onsider he pahs of ions and elerons in E and fields for some simple ases. ( ) The Loren fore on a poin harge is F E+. E is measured in V m. in T (ofen in gauss. gauss T). E onsan, uniform Suppose E E$. The Loren fore euaion beomes m E. This desribes a onsan aeleraion along.. onsan, uniform Suppose $. Here is how we migh produe a uniform magnei field. m m. (3) () ()

2 Take d of (), subsiue using () m m m. Wrie m he (angular) loron freuen or grofreuen. (Noe he smbol Ω is ofen used.) Similarl, The soluions an be wrien as sin m os (The signs and phase angles hae been hosen o mah he skehes below. The upper sign is for a posiie harge, he lower for a negaie.) Inegrae again r L os rlos m sin mrl sin is alled he Larmor radius, radius of graion, or groradius. So a harge in a onsan, uniform moes in a irle wih onsan speed. Noe ha he loron freuen does no depend on how fas he harge is moing. You do. Chek ha he direions of moion are orre and ha he euaions aboe mah he skehes. You do. Calulae he loron freuen (in H) for (a) hdrogen ions and (b) elerons in a magnei field of T.

3 If he harge has a, his -omponen of he moion is unhanged. The harge moes in a helial pah. 3. E onsan, uniform. onsan, uniform. Suppose $. ( ) m E + ( ) m E m E. (3) () () (3) gies onsan aeleraion along. You do. Suppose E has a -omponen onl. Desribe he moion and skeh he pah for his ase. () and () are manipulaed as before, he gie E + and E +, wih soluions E sin + E m os. The pah of an eleron is a ombinaion of uniform irular moion plus a drif, alled an E drif. E E ( ) E E $ $. Noe ha he drif erm is independen of he harge and is sign, so all he harges will drif ogeher. The pahs are loids. 3

4 If E hen E E. Here is an eample where he E drif an ause a plasma o roae. 4. onsan, non-uniform. We will onsider wo disin ases. Case (a): 4

5 ( ) α + α α where α is small. You do. Show Mawell s euaions. and are saisfied as long as we inlude hese small and erms. The Loren fore euaion beomes ( ) m α + + α m ( ) m α α m ( ) m α α + m where indiaes he uniform onsan or ero-order par and a We will wrie + small firs-order orreion, of he same order as α, and subsiue in he euaions. This is a sandard approah and we will use i freuenl. The ero-order euaions. If we wrie down he ero-order erms, i.e., he erms in and hose ha do no onain α, he euaions ha remain desribe moion in a onsan, uniform. This was disussed earlier. You do. Show his. The firs-order euaions. We firs sole he ero-order euaions o obain ; hen we wrie down he firs order erms, i.e., he erms in and hose,,,,, onaining α; hen subsiue for,,,,,. This gies α + + rl os α sin m α rl os sin α m m r α α r L + sin sin os L os whih an be wrien as 5

6 Adiabai inarian α ± α os sin α m os m ± α sin α upper sign ions, lower sign elerons, wih soluions α α os sin α α m os ± sin α. m is a onsan of he moion or adiabai inarian. Adiabai arries he idea of slowl- hanging. ( ) ( ) You do. Show his is rue o firs order in α. Sar wih and subsiue using he soluions aboe. (Noe. Chen p 3 gies an alernaie deriaion.) You do. Use he definiion of r L and his resul o show ha he magnei flu enirled b orbi A is onsan. Φ M I follows ha he magnei momen of he graing harge is onsan. The magnei momen is defined as µ ia where i, where is he harge and is he ime for one graion and A is he area enirled b he orbi. m e µ πrl. π So µ is onsan. Magnei mirror As an eleron spirals ino a higher region, inreases and rl dereases. Sine he oal energ m is a onsan, mus derease. Eenuall and he eleron reerses direion. I has been refleed b a magnei mirror. 6

7 e.g., magnei mirror used o rap plasma in an eperimenal deie. To do a magnei mirror alulaion use m + Case (b): m onsan. m onsan and onseraion of energ ( α ) + α You do. Show Mawell s euaions are saisfied. αdesribes he gradien. αdesribes he uraure. In he deriaions below, he uraure erms are underlined. ( + α) m ( α α) m ( α) m 7

8 Proeed as before. ± α os ± α m ± α α os upper sign ions, lower sign elerons, wih soluions α sin + α α α α m os ± sin os ( os sin ) α ± Consider. There are onsan drif erms. α ± due o he gradien The ombine o gie ± α due o he uraure. d ± α +. This is he epression we will use. I is perhaps unforunae ha hese drifs are in he same direion. We anno deise a suh ha he anel. We an epress α in erms of he gradien of he magnei field or he radius of uraure R of he field lines. (i) From he euaions for aboe, α. (ii) From he skeh aboe, α R You do. Show his. 8

9 gradien drif I is eas o see wh a gradien gies rise o a drif. Consider he pah of a harge where he field is large aboe he line and small below i. Aboe, he Larmor radius is small and below, i is large. We an skeh he drif. The drif, for posiie ions, is in he direion of or R. e.g. in a oroidal magnei field µ Ni πr so and r. i.e., inreases as ou go radiall in owards he ais. r In his aseα. r 9

10 e.g., radiaion bels in he earh s magnei field. This illusraes he magnei mirror as well. 5. Magnei field wih ime ariaion Drif and mirroring euaions do no allow he long range prediion of rajeories, pariularl if here is no smmer. I is nie o hae onsans of he moion or inarians. Again i an be shown ha een when he magnei field aries in ime, he magnei momen µ is onsan. This is he firs adiabai inarian. e.g. Adiabai ompression as a mehod of heaing a plasma Suppose a plasma is rapped b a magnei field. If he magnei field is inreased hen inreases. Collisions will disribue his era energ. The plasma is heaed. There are wo oher inarians. The are illusraed b he following eample. () µ onsan () Longiudinal (or seond) adiabai inarian J oer a pah dl onsan bak and forward beween mirrors So if he loaion of he mirrors hanges slowl wih ime, due o he solar wind, his remains onsan. (3) Third adiabai inarian The guiding enre ma preess going from one field line o anoher. u he field lines all lie on a flu surfae - a barrel-shaped surfae suh ha he enlosed flu is onsan.

11 Eerises for Chaper onsan, uniform. Calulae loron freuenies and Larmor radii for (i) 8 kev deueron in a fusion reaor. 5.7 T (ii) 5 ev eleron in a plasma CVD soure. gauss. (iii) kev eleron in he earh s magnei field..5 gauss. E onsan, uniform. onsan, uniform.. In a low emperaure plasma deie alled a magneron, is piall 3 gauss, he poenial differene V is 5 V oer mm in he region of ineres and he E is perpendiular o he. Esimae he drif eloi of he elerons. onsan, non-uniform. Case (a): 3. (a) In a magnei mirror where he magnei field is, he rajeor makes an angle θ wih he magnei field line. Show ha refleion ours where he magnei field is sin θ. (b) Suppose now ma is he maimum alue of he magnei field. Show ha if θ < sin ma hen here is no refleion. In a magnei mirror deie, his would mean he parile was no rapped - i would be los. This angle defines a one in eloi spae - he loss one 4. In our groron millimere-wae soure mos of he elerons rael from he eleron gun hrough he resonan ai. u some elerons are refleed. A he eleron gun (.3 m) he magnei field is.5 T, a he resonan ai ( m) i is. T. Consider a pariular eleron ha is refleed a he ai. I leaes he gun wih an energ of kev. Is guiding enre is 3. mm from he ais. Calulae (i) a he ai, (ii) and hene a he gun, (iii) he disane o he guiding enre from he ais a he ai, (i) he Larmor radius a he ai, () he Larmor radius a he gun. onsan, non-uniform. Case (b):

12 5. In a small eperimenal plasma deie a oroidal is produed b uniforml winding urns around a oroidal auum essel, and passing a urren of 5 A hrough i. The major radius is.6 m. A plasma is produed in hdrogen b a radiofreuen field. The eleron emperaure is 8 ev and he ion emperaure is ev. The wo emperaure disribuions are Mawellian. The plasma densi a he enre of he essel is 6 m 3. (i) Calulae he field a he enre of he essel (ii) Calulae he oal drif for boh ions and elerons a he enre of he essel. On a skeh, show he direions of hese drifs. Re 6. The earh s magnei field, in he euaorial plane, is 3 5 T. r A 5 R e in one of he Van Allen radiaion bels, he elerons hae an energ of 3 kev and he proons an energ of ev. (i) Calulae he oal drif for boh proons and elerons. On a skeh show he direions of hese drifs. (ii) If he plasma densi is 5 m 3, alulae he ring urren densi. R e m. 7. For he Van Allen bel in Eerise 6, esimae he following imes. (i) loron period, (ii) ime beween mirror refleions, (iii) ime o drif one around he earh. 3